An Introduction to TRADITIONAL LOGIC. Scott M. Sullivan

Transcription

1 An Introduction to TRADITIONAL LOGIC Scott M. Sullivan

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3 AN INTRODUCTION TO TRADITIONAL LOGIC: Classical Reasoning for Contemporary Minds

4 An Introduction to Traditional Logic: Classical Reasoning for Contemporary Minds 3 rd Edition Copyright 2013 by Scott M. Sullivan All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission of the author, except in the case of brief quotations embodied in articles or reviews. Published April 2013 by Classical Theist Publishing, Houston, TX., in the United States of America ISBN

5 To my wife, whose sacrificial love and care of the family provided the time for me to write.

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7 I want to acknowledge all those to whom I owe my interest and knowledge in logic, especially the in-depth insights of Jacques Maritain and Henry B. Veatch, whose work has undoubtedly been a heavy influence on me. While I have tried to avoid a mere transfer of their ideas, I am so imbued with their thought that any reader familiar with their writings will see those ideas reflected here as well.

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9 Table of Contents Introduction xiii Chapter 1: Fundamental Notions in Logic 1 Logic as the Science of Reason 1 The Three Acts of the Human Intellect 2 Two Main Branches of Logic 8 Chapter 1 Summary 9 Exercises 10 Part I: Definition 11 Chapter 2: Signification, Concepts & Terms 13 Intentionality and Signs 13 Logic as a Science of Second Intentions 16 Concepts 18 Terms 24 Chapter 2 Summary 28 Exercises 29 Chapter 3: The Categories & the Predicables 31 The Categories 32 The Predicables 35 Chapter 3 Summary 41 Exercises 41 Chapter 4: Forming Definitions 45 Division and Classification 45 Forming Definitions 49 Chapter 4 Summary 57 Exercises 58 Table of Contents ix

10 Part II: Propositions 63 Chapter 5: Propositions 65 What is a Proposition? 65 Truth and Falsity 66 The Structure of Categorical Propositions 70 The Four Types of Categorical Propositions 73 Distribution 74 Supposition 76 Putting Everyday Language Into Proper Logical Form 80 Chapter 5 Summary 87 Exercises 88 Chapter 6: Propositional Properties & Compound Propositions 91 Overview of Propositional Properties 91 Opposition 92 Compound Propositions 102 Chapter 6 Summary 105 Exercises 106 Part III: Argument 109 Chapter 7: Argumentation & the Syllogism 111 Argumentation Introduced 111 The Syllogism 114 Testing Syllogisms for Validity 1: Rules of the Syllogism 120 Testing Syllogisms for Validity 2: Venn Diagrams 126 Three Conditions for Successful Argumentation 131 Chapter 7 Summary 132 Exercises 133 Chapter 8: Valid Syllogistic Forms & Reduction to the First Figure 139 Reducing Syllogistic Forms to the First Figure 139 Chapter 8 Summary 146 Exercises 147 x An Introduction to Traditional Logic

11 Chapter 9: Other Types of Syllogisms 149 Compound Syllogisms 149 The Dilemma 156 The Enthymeme 159 Sorites 160 The Epicheirema 162 Expository Syllogisms 163 Chapter 9 Summary 164 Exercises 164 Chapter 10: Some Final Aspects on Argumentation 169 Different Types of Syllogisms Based on the Strength of the Premises 169 Demonstrations through Effects and Causes 170 Induction 172 Inference to the Best Explanation 181 Chapter 10 Summary 184 Exercises 184 Chapter 11: Fallacies 187 Fallacies of Language 188 Formal Fallacies 189 Material Fallacies 192 Exercises 203 Appendix: In Defense of the Square of Opposition 207 Index 223 Table of Contents xi

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13 Introduction This is a logic book for beginners. It is written both for the college student and the do-it yourselfer. You may be asking yourself, why do we need another book on traditional logic? It is true that there are many traditional logic books out there, but often these works of wisdom are out of print and collecting dust in libraries since the trend today is contemporary symbolic logic, and an additional problem is that these older texts are often inaccessible to today s beginner because the language is too technical for one not well versed in Aristotelian philosophy. This coupled with a need for more diagrams in such a text that might prove helpful to the visual learner amounts to what seems to me to be a need for yet another text on traditional logic. I hope to remedy these defects here. Now of course, a logic text by its very nature is always going to be a tough read for the beginner, but I have tried as much as possible not to make it any tougher than it needs to be. Logical terms are, I hope, patiently explained in easy to understand everyday language, numerous diagrams are provided to help the beginner to visually process the information, and finally a plethora of exercises will give the student plenty of opportunity for practice. Now a note about the term, traditional logic. This is a relatively loose term intended to cover the logical systems from ancient Greece to the heyday of medieval scholasticism. The logic we cover here is the logic that enjoyed an organic development from its founding by Aristotle and input from the Stoics and Megarian logicians to a later flourishing the Middle Ages; a logic that, although having some minor differences, nevertheless retained a common core and can generally be considered as the logic used and studied by the Neoplatonists, Boethius, St. Albert the Great, St. Thomas Aquinas, Duns Scotus, John of St. Thomas, etc. This system of logic gradually decreased in popularity as it went through a transitional period around the time of Leibniz, and eventually suffered a substantial change into the more mathematical and symbolic varieties of the logic that we see in Frege, Russell, and Whitehead. Introduction xiii

14 Traditional logic is a relevant logic, it is a logic based on philosophical realism, which is simply a fancy way of saying it is a logic based on common experience. Logic is a part of philosophy and cannot get by without presupposing philosophical views about reality and how we come to know it. Space does not permit a full explanation of the differences between traditional logic and the modern symbolic versions, but some brief remarks are in order. In the traditional view, logic is inherently about things, that is, it is not an abstract game of symbolic manipulation about well-formed formulas (WFF s), but it is about real things and real consequences. In the traditional view, the principles of logic are not laws of thought per se, but reproduce principles of existence that must be observed if our thought is to apply to reality. Traditional logic deals with the instruments of thought (concepts, judgments and arguments) as intrinsically intentional and referential to reality and so is very concerned with how they are to be correctly applied to reality. Thus, this logic is not concerned with logical forms without any real reference. In other words, the traditional logic is a relevant logic, it is a logic based on philosophical realism, which is simply a fancy way of saying it is a logic based on common experience. What common experience? The common experience that there is an external world out there full of different kinds of things and that we can know this world and the kinds of things it contains. We all know, for example, what apples and oranges are, and we know they are both not the same kind of thing in one way (apples are not oranges) but in another way they are the same kind of thing (insofar as they are both fruits) and hence they are different from other kinds of things such as dirt, mosquitoes, and anteaters. Sure this is all common sense, so why mention it? The point is that logic is a part of philosophy and cannot get by without presupposing philosophical views about reality and how we come to know it. We cannot help but to address some concerns about the matter or content of logic, and this side of things is what makes the philosophy behind the logic so important. So let this serve as an unapologetic admission; while the underlying philosophical issues cannot be thoroughly discussed in the space we have here, at least let us come forth and say that the traditional logic presented here presupposes what can be called a classical realist philosophy, which, in my view, is the same as saying it takes for granted the common sense mentioned above. This presupposition should serve the average reader well, since it is the same presupposition by which they already conduct their daily lives. Still the focus here is on what is called formal logic, the structure of thought, and even those with philosophies sharply opposed to a classical realist philosophy can agree with the realists in this area. Studying the formal arrangement of reasoning is especially important nowadays, since merely studying disconnected informal fallacies and trying to apply them to arguments in prose without a coherent overarching system (like in a critical thinking class) does not give the student enough tools for argument formulation and analysis. In my view, although humans have always needed good thinking and logic skills in order to grow in learning and to be able to examine positions critically, nevertheless the need is more desperate today. As a professor I am often unhappily surprised at xiv An Introduction to Traditional Logic

15 how little most college students know about argumentation, fallacies, proof, etc., and this greatly handicaps their ability to analyze important positions and formulate arguments of their own. It seems to me that the lack of requiring logic in high schools and universities is partially responsible for a widespread relativism and the distrust of reason so rampant in contemporary society. It is my hope that this book will play some role in curing this ailment. Introduction xv

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17 Chapter 1: Fundamental Notions in Logic àà Logic as the Science of Reason àà Three Acts of the Intellect àà The Two Main Branches of Logic: Material and Formal Logic as the Science of Reason The philosopher Aristotle once wrote that all human beings by nature desire to know. We are knowing creatures and when confronted with questions and problems, we seek to find solutions. But although we are all knowers and thinkers by nature, it is of course true that we can always improve our reasoning skills. This is where logic comes in. Logic studies reason itself as a tool of knowledge. Expertise with this tool allows one to proceed with ease and correctness in acquiring truth. The study of logic is unique in that by studying the very process of reasoning itself, the fruits of this work will pay off in all areas that use human reason; i.e., philosophy, the physical sciences, theology, history, etc. This is so because all valid human reasoning follows common forms and structures. By studying logic, your ability to think, make proper distinctions, organize your thoughts, and deduce conclusions will be greatly enhanced. Logic studies reason itself as a tool of knowledge. Expertise with this tool allows one to proceed with ease and correctness in acquiring truth. Fundamental Notions in Logic 1

18 All of us already use logic all the time. Now you may be surprised to know that all of us already use logic all the time. Natural human reasoning instinctively recognizes things like inferences and contradictions, and as a thinker you have been doing this from the time you became able to think. So in this sense, logic should not be that intimidating because you have already been using it! What the formal study of logic really does is help you get better at what you are already doing. Of course, reasoning well is not easy and logic can be a tough subject. It will cause you at times to strain your mind, and that is actually a good way to think about it, for studying logic builds the mind much in the same way that weightlifting builds the muscles. Hence, the pain involved in working through difficult logic problems will pay big gains in the end. By sharpening your reasoning skills, logic provides an enormous benefit to the mind that will carry over into many areas of life. We now take a look at the three ways the human mind operates in order to understand the basis for the three parts of logic. The Three Acts of the Human Intellect Philosophers in the classical tradition recognized that the human intellect has three fundamental operations or acts. These three acts are called simple apprehension, judgment, and reasoning. Since logic is based upon these three acts, it is important to get a brief understanding of them here in the beginning. Simple Apprehension Human knowledge begins with your sense experiences. There is nothing in your mind that was not at first, in some way, from your senses. Now this doesn t mean that all knowledge ends with sensation, only that it begins there. From this sense experience one derives concepts that reflect the kind of thing one is sensing. Humans naturally desire to know kinds or whats and we even see this in small children, for as soon as they are old enough they begin incessantly asking, what s that?, in an attempt to get to the kind or whatness of something. Of course adults do this too. It is just human nature to want to know what things are. But how do we do this? When you sense a poodle, a greyhound, and a rottweiler, your mind is able to abstract from these individual experiences the concept of dog. It is not that you actually sense this dogness with your five senses, you see or hear only the individual 2 An Introduction to Traditional Logic

19 animal, yet nevertheless it is clear that the human intellect can know kinds and this is a good thing because the world around us is full of different kinds of things. But what is a what and how do we know it? In philosophical terminology, we call this what an essence or nature. An essence or nature simply refers to what something is. It might be helpful to add the suffix -ness or -ity to a word to sort of bring out what we mean by the nature of something. A white fence, white snow, and a white piece of paper all share the same color whiteness. A triangle here and a triangle there both have the nature of triangularity. The same is true for other natures as well. Now when you know the nature of something you have a concept of that nature. These concepts are universal in that by referring to one nature, they can be said of many individuals. Universal concepts of natures are quite interesting in that they can be realized or instantiated in several places at the same time. Because different individuals of the same kind share the same nature, multiple instances of that kind can occur simultaneously at different places. An apple nature can be manifested on a tree, in a basket, in your refrigerator, and at the store all at the same time. Because of this universal aspect, this nature of an apple does not change regardless of what happens to individual apples. Even if their numbers doubled or if all apples became extinct, the concept of apple remains the same as it always was. Individual apples come and go but the nature of an apple (to grow on trees, to be sweet, etc.) stays the same. Thus, the unchanging concept of apple is universal in the sense that it refers to all apples whether past, present, or future. Only the individuals change, not the nature or the universal concept derived from it. In philosophical terminology, we call this what an essence or nature. An essence or nature simply refers to what something is. Now when you know the nature of something you have a concept of that nature. From this we can see that concepts have necessary features about them. A square necessarily has four sides, after all, being four-sided is what its nature requires! There cannot be a three-sided square - that would violate the very essence of what a square is. Hence, it is the natures of things that dictate certain necessary characteristics about them and this is important because this is what grounds the stability and predictability of our knowledge. Think about it. Because you can know the nature of a thing, you can know in advance what a rose will smell like next spring, not because you have smelled that particular rose (it doesn t even exist yet), but because you have smelled other roses that are of the same nature as those that will bloom next spring. Similarly, a doctor knows how to treat a new patient he has never even seen before in his life because he has studied the nature of the human body, the effects of medicine upon it, etc. We know that dogs are mortal kinds of things, they don t live forever, and once you know a dog s nature you know this, again, in advance and don t have to wait around and see if all individual dogs will die or not. To know that a dog is mortal comes from knowing what kind of thing it is. Now this necessity is not just made up and foisted onto things, it is discovered and derived from the Fundamental Notions in Logic 3

20 things themselves. To summarize this, we need to know these three aspects about our concepts: Concepts of natures are universal, unchanging, and have necessary characteristics. Now when we create a word for these concepts in our minds, in logic we call that word a term. The term is the actual word used to signify the concept and it is concepts that gives terms their meanings. For example, the English term apple means the same thing as the German term apfel. While terms may change from language to language, the universal concept stays the same. The Germans have the same concept of an apple as do English speaking people, even though they use a different term to signify it. It is this very fact that makes translating languages possible. Simple apprehension is the initial knowing or taking in of an object. The goal of the first act of the intellect, to define a term. This whole process of acquiring concepts and defining the meaning of terms is part of the first act of the intellect known as simple apprehension. Simple apprehension is the initial knowing or taking in of an object. In an act of simple apprehension, our mind merely grasps a thing without affirming or denying anything about it. 1 When you see an apple for example, your mind is able to derive from your sense impressions the concepts of apple, red round, etc. This apprehending of concepts is what is meant by simple apprehension. Simple apprehension is obtaining the essence or whatness of something which can be expressed in a term, and this is the goal of the first act of the intellect, to define a term. As we will see, terms can be clear or unclear, ambiguous or unambiguous, and other properties of this sort, but it is important to know now that terms cannot be true or false. This is because simple apprehension is not saying how something really is or really is not; rather this act simply grasps the nature or essence of something and then deals with defining terms. To summarize: The first part of logic is based on the intellectual act of simple apprehension and aims at the defining of terms. 1 Jacques Maritain, Formal Logic (New York: Sheed and Ward, 1946) p. 4 4 An Introduction to Traditional Logic

21 Judgment Once you have concepts in your mind you can begin to make judgments. We do this all the time, but to break it down logically, a judgment occurs from joining or dividing two or more concepts and is a statement about the way things really are. For example, by eating an apple one can arrive at the judgment, This apple is sweet. Notice how in this judgment, the concept of apple is united with the concept sweet. If we said, The apple is not red, you can see how apple would be divided from the concept red. So the first thing to know about judgments is that they consist of a uniting or dividing. The second thing to know is that judgments refer our concepts back to the source from which they came. In the act of judging, we either unite or divide two concepts and say this is the way they are. For example we may say, This apple is red or we may say, This apple is not red, but in either case we are talking about the way something really is, and this makes the judgment susceptible to truth or falsity. So it is important to know that the product of a judgment is a proposition and it is only in a proposition that truth and falsity occurs. Logic is not concerned with statements such as commands ( Do this! ) or questions ( Do you like apple pie? ) but only with declarative statements that claim to be true. The second part of logic is based on the intellectual act of judgment and aims at forming propositions. Reasoning Intellectual activity does not stop with simple apprehension and judgment. We can progress or infer from what is known to something that is not known through logical reasoning or argumentation. This is a step-by-step advance in knowledge and the laws of logic govern the proper procedures for these types of advances. So all reasoning is a sort of advance in knowledge but there are two basic kinds of reasoning known as inductive and deductive. Inductive argumentation often moves from particular premises to a general conclusion; i.e., This swan is white, that swan is white, and every swan I have seen is white, therefore, all swans are white. Deductive argumentation on the other hand moves from more universal premises to a particular conclusion. Take the classic example: Every man is mortal Socrates is a man Therefore Socrates is mortal Fundamental Notions in Logic 5

22 The first two propositions are the premises, and the last proposition the conclusion that is inferred from them. In an argument, the conclusion is what one is trying to prove to be true, while the premises are the reasons or support for the truth of that conclusion. Now, any good deductive argument must meet three qualifications; the terms must be clear, the premises must be true, and the reasoning logically valid. A little reflection on the nature of trying to argue a point will show why. The terms must be clear, because if not, ambiguity, confusion, and an outright fallacy may result. The premises must be true in order to prove anything, since obviously an argument that contains a false premise or premises cannot prove anything. Finally, the reasoning must be valid which simply means that the conclusion must follow from the premises. If any one of these elements fails, the whole argument fails as well. In other words, if there is a problem with the clarity of a term, the truth of a proposition, or the validity of the reasoning, the conclusion is not established and thus remains uncertain (it could still be true, but it is not shown to be true by a bad argument and so the conclusion remains uncertain or unsubstantiated). In a deductive argument, if all three conditions are met, the argument is said to be sound. We will look at these three elements of a good argument in more detail later, for now you should know the nature of the third part of logic: The third part of logic is based upon the intellectual act of reasoning and aims at forming arguments. The three acts of the intellect are activities that produce something. Each act of the intellect has its respective product. The products of each act are the instruments of human knowledge and serve as the main tools of logic. After all, in order to know anything we must first have a concept of what something is. We must also be able to combine and divide different concepts as to make true statements about things, and so we need propositions. Finally, for human knowledge we need evidence or support for a conclusion, and so the tool of argument is a third necessary factor for human knowledge. To put all of this in a nutshell, the following listing should be useful: Simple apprehension produces terms from concepts: Concepts are internal signs that are universal and signify a nature or what something is. A term is a name we give to that concept, i.e. apple for the nature of an apple. Terms can be either clear or unclear (unambiguous or ambiguous). Judgments produce propositions: Propositions intend an act of existence in that they say the way something is or is not. Propositions can be either true or false. Reasoning produces arguments: Arguments intend causes or reasons that demonstrate why a conclusion is true. Arguments can be either valid or invalid. 6 An Introduction to Traditional Logic

23 Another way of showing this is that these three tools of logic correspond to three basic questions everyone asks when they want to know something; what something is (meaning one is seeking the nature of something), whether something is (indicating that one is wanting to know if something is or is not so), and why something is so (indicating that one is looking for explanations, evidence, or reasons for something). And so these three questions of what, whether, and why correspond to the logical tools of concepts, propositions, and arguments. 2 2 I owe this way of explaining the logical tools to Francis Parker and Henry Babcock Veatch, Logic as a Human Instrument (New York: Harper and Brothers Publishers, 1959) pp Fundamental Notions in Logic 7

24 Two Main Branches of Logic Material logic covers the process of determining whether or not a certain proposition is true or false. Formal logic on the other hand studies the form of reasoning, that is, rules governing the structure and validity of argumentation. Logic is a broad science actually containing two main divisions that work hand in hand with each other. First, material logic covers the process of determining whether or not a certain proposition is true or false. This area of logic is concerned with the content and process of acquiring knowledge. In other words, material logic deals with the actual facts about the things we know. These facts are the stuff or matter with which the reasoning process works (hence the term material logic ). The scholastic logician John of St. Thomas used the analogy of a builder to explain. A builder does not make the bricks and wood for a house, those are the building materials, but the builder does two things, he first inspects the materials and then introduces the form of a house into this matter by structuring it in a certain way. Now logic is the art of building good arguments. To build a good argument, just as in building a house, you have material considerations and formal considerations. The things we are reasoning about make up the matter of arguments and the builder of the argument (the logician) has to determine what matter is suitable and then give that matter its form by arranging it in a certain way. So just as a good house depends both on its matter and form, so too does a good argument. The logician s task in the realm of material logic is like the builder who inspects his materials to make sure they are stable (in other words, true) enough to support the house, and the roof is like the conclusion of an argument. If the materials are faulty, the conclusion will cave in! Formal logic on the other hand studies the form of reasoning, that is, rules governing the structure and validity of argumentation. The form of an argument is the way the content is arranged. Formal logic is more abstract than material logic, but both are needed. Material logic is concerned with truth and falsity of the content since an argument may be true or false because of its matter, yet the same argument considered formally deals with validity and it is valid or invalid because of its form: Material logic is concerned with the content of arguments, while formal logic is concerned with the way in which the contents are arranged. This again brings us to the important distinction between truth and falsity on one hand, and valid and invalid on the other. It is vital to know that true in logic does not mean valid, and false does not mean invalid. Rather validity/invalidity is something that applies only to whole arguments while truth/falsity applies only to premises and conclusions. How so? An argument is valid if certain things being 8 An Introduction to Traditional Logic

25 stated, something other than what is stated necessarily follows. 3 A valid argument means that the conclusion follows logically from the premises, regardless whether or not those premises happen to be true or not. Truth and falsity are properties that only apply to premises (the propositions used in an argument) not arguments. In this way, an argument can be valid and still have a false conclusion, or just the opposite; it can be invalid yet still have a true conclusion. For example: Every man is immortal Socrates is a man Therefore Socrates is immortal This is a valid argument. The conclusion follows from the premises because the form (the arrangement of the terms and the types of premises) is a valid form. But notice even though this argument is valid, the conclusion is not true. The conclusion is false because materially speaking the first premise is false. It is false that men are immortal. So we can see that logical validity has nothing to do with the truth of the premises. With this understood, next take a look at an example of the opposite case where we have an invalid argument with a true conclusion: Snow is in Alaska Alaska is part of the United States Therefore, snow is white It is easy to see here that even though the premises and conclusion are true, still the conclusion does not follow from those premises. The key point here to remember is that validity and truth do not mean the same thing. Chapter 1 Summary Logic is the science of correct reasoning The three acts of the human mind are simple apprehension, judgment, and reasoning. Simple apprehension is simply taking in knowledge and producing concepts. The first part of logic is based on this act and aims at defining terms. Judgment is the uniting or dividing of two concepts. The act of judgment produces a proposition, which when used in an argument is called a premise. Reasoning is the process of moving from two known propositions to a conclusion that was not previously known, and thus the product of reasoning is an argument. Terms can be unambiguous or ambiguous Propositions can be true or false 3 Aristotle, Prior Analytics 24b15-20 Fundamental Notions in Logic 9

26 Arguments can be valid or invalid An argument is sound when the terms are unambiguous, the propositions are true, and the reasoning valid. If the argument fails in any of these areas, the conclusion is uncertain. Inductive arguments normally move from particular premises to a general conclusion. Deductive arguments move from general premises to a more particular conclusion. There are two main branches of logic: Formal logic deals with the arrangement and structure of arguments while material logic deals with knowledge and the content of propositions Exercises 1. Name the three acts of the intellect; what they do and what they produce. 2. Explain how it is that terms, propositions, and arguments correspond to the three questions of what, whether, and why. 3. What is a universal concept? 4. What is the difference between truth and validity? 5. Describe the difference between formal and material logic. 10 An Introduction to Traditional Logic

27 PART I: DEFINITION

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29 Chapter 2: Signification, Concepts & Terms àà Intentionality and Signs àà Concepts àà Terms Intentionality and Signs The first part of logic is aimed at defining terms, but in order to understand precisely what concepts and terms do, it is important to first understand intentionality and signification. Knowledge is always about, of, or for something other than itself. We know about dolphins, have reservations about an event, acquire facts for a certain action, know of a war, etc. This having an aboutness or being for something else is called intentionality. When you know what a square is, your knowledge is about squares, but the square isn t about anything, it s just a square and that s all. All knowledge is intentional in that it involves ideas in the mind that are for or about something other than themselves. Knowledge is always about, of, or for something other than itself. This having an aboutness or being for something else is called intentionality. Now since knowledge is inherently intentional, and logic is a tool of knowledge, it is clear that the logical tools must be intentional tools. Take the three main tools of logic; concepts, propositions, and arguments. Concepts are of something, propositions are about things, arguments in proof of something. Logic, as an instrument Signification, Concepts & Terms 13

30 of knowledge, is inherently wrapped up in intentionality and so if we are going to understand logic we have to understand intentionality. Knowledge and logic involve signification. A sign is that which represents something other than itself. In other words, knowledge and logic involve signification. To be intentional is to signify. Since our knowledge is always about things, whatever it is that we know should be significant of reality, which means that our instruments of knowing are signs. A sign is that which represents something other than itself. Smoke is a sign of fire, sirens are signs of an emergency, a red traffic light is a sign indicating a law to stop at an intersection, a morning frost is a sign that the evening was cool, and words are signs that refer to the concepts in our minds. Thus, we can see that signs are inherently intentional since the sign points away from itself to something else. A sign leads us to whatever the sign signifies, which we call the signatum. All signs represent their signatum. Artificial and Natural Signs Signs can be natural or artificial. But there are different kinds of signs! First, we can make a distinction between different kinds of signs based on the relation between the sign and the signatum. Signs can be natural or artificial. An example of a natural sign is the smoke that naturally indicates a fire. This sign is true for all people at all times throughout history. This is because real causes have real effects and these effects naturally point to their causes and are signs of them. However, when the foundational relation between sign and signatum is not natural, but comes from the work of human beings, the sign is artificial. The significance of artificial signs comes not from their own nature but by way of human custom. For example, artificial signs, such as the traffic light, acquire their meaning by convention. So with this distinction we can say that the foundational relation between a sign and its signatum can be either natural (in the sense of a cause-effect relationship like smoke is to fire or morning frost is to a cool night), or artificial, where the signification relation comes from human custom (like in the case of the red traffic light). Natural signs signify by a natural relationship Human language is based upon the artificial sign. The connection between the English utterance dog and the actual canine in reality is a relation based upon convention and agreement. That s why language differs from culture to the next. The German word hund or the Latin canis are different conventional signs used to refer to the same reality that the English term calls dog. We will discuss more on this relationship later. while artificial signs signify by convention. 14 An Introduction to Traditional Logic: Definition

31 Material and Formal Signs Another distinction between types of signs can be made regarding the nature of the sign itself. In this way, signs may be either material or formal. Material signs are signs that are more than mere signs. For example, the smoke is first an existing reality in itself and any signification or signing that it may do afterwards is secondary. Smoke is a sign of fire to be sure, but smoke is not merely a sign, it is also a gaseous entity. The same could be said of frost; it is an actual thing first and a sign of a cold night second. A stop sign must first be apprehended itself, and then the notion it represents (to stop) is understood. 4 This is the essence of material signs, that they first be known themselves before whatever else they signify can be known. In other words, one first apprehends the thing that is a sign and then only after that does the knower apprehend the signatum. Material signs are signs that are more than mere signs. Material signs are first known as things in themselves and then subsequently signify a signatum. Formal signs on the other hand are just signs and nothing more. An example of a formal sign is an idea. Ideas represent other things; you can have an idea of a plant, car, or a hunting trip. Ideas are always about something else and only about something else. To know an idea is to automatically know what it is about. It is not as if ideas have certain traits that are known before what they represent is known. They themselves are the meanings and that is all. In other words, ideas are signs and only signs and this is precisely what a formal sign is. The formal sign is a thing whose very nature is to point to something else. They reveal something other than themselves while not revealing themselves, like a clean window shows the outdoors. Formal signs on the other hand are just signs and nothing more. An example of a formal sign is an idea. The cornea of your eye can serve as an analogy. You see things through the cornea of your eye without that cornea getting in the way and obstructing your vision with its own qualities. The cornea does not present itself in your vision, it s job is entirely to act as a lens to present other things. Ideas as formal signs work this way. A formal sign is transparent and has no other nature except to signify something else. They are completely self-effacing and inherently directed towards others. This is not to say that a formal sign cannot be made into an object of study after all we are doing that right now. When we do that however, it is no longer a sign in itself but it becomes the signatum of another formal sign, that is, another idea. 4 What we are calling material signs is what John of St. Thomas called an instrumental sign. Following Veatch however, material sign better conveys the difference between this kind of sign and its formal sign counterpart. (See Parker and Veatch, op. cit., p. 17). Signification, Concepts & Terms 15

32 We cannot do without the formal sign in the process of knowledge. Just try to think of something without using a formal sign. Material signs must first be known in themselves before what they signify can be known. If everything was a material sign, the sign by which you know the material sign would have to be a material sign, and then another material sign would be needed to know that, etc., etc., and this would mean that we would never know anything at all because we would be trapped in an infinite regress! Knowing a material sign presupposes one can know its nature without having to use another material sign as an intermediary. To know anything at all, there must be an idea at some point which reveals the nature of the thing exactly as it is without having to first be known itself. So remember: Formal signs are nothing but meanings, all they do is Given these different types of signs; between natural and artificial on one hand and formal and material on the other, there are four possible combinations which we illustrate by the following chart: reveal. Their only nature is to be about something else. As you can see, this chart shows that there is no such thing as an artificial formal sign. Only something like a concept can be a formal sign, while the two examples of material signs should be pretty straightforward. Logic as a Science of Second Intentions Now logic is primarily concerned with entities like the concept. These logical entities are beings of reason meaning that they exist only in the mind and so are distinct from real beings. This distinction is simply a matter of common sense. Real being is the kind of being that is not dependent upon a mind for its existence while a being 16 An Introduction to Traditional Logic: Definition

33 of reason is mind dependent. A dog is a real being while the concept of a dog is a being of reason. Recognizing the difference between real beings and beings of reason brings us to the scholastic distinction between a first and second intention. We have said that all knowledge is intentional because it is always about some object. Since we now say that there are two kinds of beings; real and mental, this opens up the possibility that our knowledge can be about two different kinds of objects. An act of knowing a real being refers to an object of first intention. Knowledge referring to real beings is first in that the objects of that knowledge exist on their own regardless if anyone knows about them or not. An act of knowing that refers to something as it exists in the mind refers to an object of second intention. These objects as known are second in the sense that existence in the mind is a secondary way of being when compared with real existence. All knowledge begins in sensation, so if there were no objects of first intention (real things) there wouldn t be any objects of second intention (mental beings) either. For example, an act of knowing that refers to a real dog is an act that refers to an object of first intention, while an act of knowing that refers to the concept of a dog refers to an object of second intention. Why Intentionality is Important to Logic Why does all this matter? To understand precisely what logic is we need to keep in mind two things; first, that logic is a science of second intentions. If you ll remember, logic is the science of thought. When we are talking about formal logic, the objects of logic are beings of a second intention. Concepts, propositions and arguments exist only in the mind and so the subject matter of logic is of things as known. Secondly, we must remember that entities like concepts are formal signs. Formal signs are not things in themselves but are all about something else. If logic is to be a tool to help us know, then its value ought to be appraised by how well it does just that. Our logic must be fit to help us grasp reality. In this way, logic as a second intention science is not self-sufficient anymore than second intentions are without the first. In other words, logic finds its value not in itself but only insofar as it represents the real. So even though the objects of logic are second intentional beings, we must not forget their connection to the real world. Logic is concerned with mental beings like concepts, propositions and arguments that in turn signify reality. Although formal logic doesn t deal directly with extra-mental existence, its objects nevertheless do in turn refer to real things (first intentions). 5 5 We don t want to have a logical approach so formalized that it neglects material logic. Logic is concerned with mental beings like concepts, propositions and arguments that in turn signify reality. Signification, Concepts & Terms 17

34 The Difference Between the Material and Formal Objects of Knowledge This brings us to a distinction about the object of our knowledge. The actual thing being perceived is called the material object of our perception. The material object would be for example, a particular apple sitting on the table. Whatever aspect we are considering about this object is called the formal object of our knowledge. The formal object is the thing considered as known and under a certain aspect. For example, if we are talking about an apple s color of red, the aspect of its color as red is the formal object of our apprehension. We are considering just the redness of an apple when we derive the concept red. So with this distinction, the same material object such as an apple can have different formal objects insofar as one considers the color, taste, size, etc., of that apple. We turn now to that first tool of logic, concepts and their relationship to terms and real beings. Concepts When these natures are known they are called concepts. Try to imagine having an understanding of something with no idea or concept of the thing whatsoever. It is impossible. We cannot know anything unless we get an inkling of what it is and this means that we must have a concept, but that concept must also be of a thing s nature. In order to show this, try to imagine anything real that isn t a what of some kind or another. Can you think of anything? Red, heavy, fast, angry, dinosaur, gasoline, old, etc., all of these terms signify a nature or a whatness. Anything, in order to be real, must be something with a certain nature. There are different kinds of things which means nothing other than things have different natures, and the human intellect is able to know these natures. When these natures are known they are called concepts. Concepts can have different properties, and we now look at two; namely, the property of universality and the property of extension and comprehension. Concepts are Universal Universal concepts, or just universals for short, are one notion that can be said of many. One of the properties of a concept is universality. When we have a concept in our mind, that concept is applicable to many individuals, or in other words, that concept is universal. Universal concepts, or just universals for short, are one notion that can be said of many. Universals can justifiably do this because they always pertain to a certain nature. When you learn about triangles in geometry, you are learning about the nature of triangles and as a result when you know something about the 18 An Introduction to Traditional Logic: Definition

35 nature of triangles, say a certain formula, you know something about them all. It is unnecessary to have to go out and verify in each and every individual instance of a triangle to see if a certain geometrical formula will apply to all of them. Likewise, being an expert in roses does not mean that you have studied every single rose in existence, it simply means that you are well acquainted with the nature of a rose, viz., you have studied enough roses to have a clear grasp of what they are and hence you even know a lot about the particular roses you have yet to come across or will never directly experience - all because you know their nature beforehand. It is only because of the human ability to know by way of universal concepts that makes this possible; in fact knowing universals is indispensable for knowledge. Without them, it would be impossible to say what anything is (try it and see). Visualize how impossible knowledge would be if you could not know universals. Without universal concepts, you wouldn t be able to speak collectively about anything. You would not be able to meaningfully use any common nouns like dog, apple, human, rose etc. Imagine if a botanist could not think with the concept plant. Instead of using one concept to apply to many individuals, the botanist would have to think about each individual one by one without having any concept of a commonality between them. Each individual instance would have to have its own proper name and that s all, and so knowledge of plants would be impossible. 6 Moreover, universal concepts are only meaningful and essential to knowledge be- Without universal concepts, you wouldn t be able to speak collectively about anything. 6 This point stands in stark opposition to the philosophy of nominalism. The nominalist thinks there is nothing common in reality or no common natures or properties at all. But nominalism is untenable and although this is actually a metaphysical issue and not one of formal logic, nevertheless a few arguments against nominalism can be made here. First, nominalism is counterintuitive and simply denies common experience because it holds that two things cannot be the same kind. Common sense tells most of us that two apples for instance, although distinct in number, are nevertheless the same kind of thing. Moreover, nominalism cannot account for the stability and predictability of knowledge for since there are allegedly no two things of the same kind, anything from past experience will be a different kind of thing from what one will experience in the future. Thirdly, nominalism suffers from a drastically bloated ontology. If nominalism were true, it would mean every molecule of H2O in every drop of rain or fragment of water in every bathtub, faucet, pipe, dishwasher, swimming pool, public fountain, car wash, municipal water facility, lake, river, and ocean on the planet is a different kind of water. Surely this scenario is overwhelmingly counterintuitive and strikes one as prima facie bizarre to a degree in which a man of common sense can be well within his epistemic rights to confidently reject. Unquestionably, the proponent of such a view shoulders a massive burden of proof for such a claim. Finally, the nominalist ontology cannot account for even the simplest recognition of color. For if no two things can be the same kind, then no two things can be the same color. But take a piece of blue construction paper. It seems the nominalist would say this whole is the same color, and that means one part of it is the same color as another part. As long as there is spatial continuity, the nominalist holds that the paper is the same identical color. Yet, suppose a small child cuts a one-inch snip in the paper, leaving the majority intact. It certainly seems that no color change has occurred at all. As the child cuts further into the paper, there is still no apparent color change occurring, but of course quantitative continuity is still maintained down to the very last thread. But if this is true what happens then? Does the nominalist really want us to believe that this paper remains the same color all along until the child snips that very last millimeter of thread? Such a claim would seem as preposterous as if the paper were not blue at all but rather bright cherry red! But perhaps the nominalist may want to avoid this absurdity by saying the different portions of the blue paper were not the same color to begin with. But then we ask just where then is the color? All extension is divisible ex hypothesi and all surfaces are extensions. Blue must have surface and thus parts. So either the paper changes colors when cut, which is absurd, or the spatially separated parts of the paper are different colors to begin with, which means there is no one color anywhere, which is also absurd. Signification, Concepts & Terms 19

36 cause they reflect reality. It is not enough just to know universals; those universals must in some way reflect reality. If things did not really have a common essence, if there were no such thing as real kinds like plant, the universal concept would not refer to anything in reality. This point is worth repeating. In order for these common concepts to be meaningful, the things to which they refer must really have a common nature. So in order for a universal like plant to have any meaning, there must really be things that are plants. In fact, the reality is prior to our knowing. It is first because there really are things like plants that we can then truly have a meaningful concept such as plant. 7 Since all knowledge begins with sensation, universal concepts are known by being abstracted from the individual things themselves. A universal concept is simply a common concept which stands for many individuals and it is able to do this because it signifies the kind or nature of those individuals. So to summarize, a universal concept is simply a common concept which stands for many individuals and it is able to do this because it signifies the kind or nature of those individuals. What we have been describing is the act of simple apprehension. Simple apprehension is the first act of human knowledge which consists of a mental grasping of an essence without affirming or denying anything about it. In other words, this operation of the mind is a simple reception of some intelligible data, like thinking about swan or water or white. In this act of simple apprehension, the knower is merely being passive in the sense that he or she is just grasping an essence of something and that s all. Extension and Comprehension of Concepts The comprehension of a concept is simply a qualitative consideration of it, that is, what the concept means. Extension is simply the sum of all the things to which the concept applies. Another logical property that concepts have is their extension and comprehension. The comprehension of a concept is simply a qualitative consideration of it, that is, what the concept means. Given any concept, there are a certain number of characteristics that make it up. In other words, every concept contains certain elements or notes. Take for example the concept of a swan. The concept of a swan contains the notes of being a living thing, an animal, able to fly, etc. The collection of all these notes is the comprehension of this concept. To put it another way, the comprehension of a concept is its intelligible content that serves as its meaning. The extension of a concept, however, refers to a quantitative consideration. Extension is simply the sum of all the things to which the concept applies. In other words, a concept can refer not only to its meaning, but also to all of the individuals that have that nature. So to use the above example, the concept swan can refer not only to its comprehension, but also to all of the beautiful white animals floating on the lake. The extension of a concept extends to all of the individuals and this is why it is called 7 This point is in opposition to the philosophy of conceptualism, like that held by William of Ockham, which holds that universals in the mind refer only to singulars and that in reality there is no commonality. Conceptualism falls victim to many of the same problems as nominalism. 20 An Introduction to Traditional Logic: Definition

37 extension. The extension also includes all the beings which could possibly be characterized by the comprehension. So the extension of dog applies to both all the actual dogs and to any possible dogs that may or may not exist in the future. This is because concepts signify the natures of things and not their actual existence. Restricted to a mere whatness, the extension of a concept includes all possible individuals that have the traits included in the comprehension, whether past, present, or future as well as those they may never really exist at all. 8 To encapsulate these two properties: The comprehension of a concept is what the concept means (its characteristics), while the extension of a concept refers to all the individuals to which the concept applies. There is a certain priority here in that extension presupposes comprehension. The comprehension of a concept determines which things the concept can apply to. Without comprehension there wouldn t be any extension. It is because concepts first have a certain comprehension that reflects the nature of things that they can in turn be applicable to only some things and not others. So only things that are dogs satisfy the comprehension of dog and thus only these things can belong to the extension of the concept dog. The concept then is primarily characterized by its comprehension because it presents a real nature to the mind, and so any extension follows in accordance with the constitutive notes of that nature. 9 This will become more important later for definitions, because we will see that an essential definition primarily expresses the comprehension of a concept. The Inverse Relation Between Extension and Comprehension The more we start to qualify an idea, the less the number of things it will apply to becomes. In other words, the more notes we pack into a concept the range of things to which it applies diminishes. For example, take liquid. This concept has a certain 8 So the extension of the concept animal applies to all animals past, present, and future as well as the purely possible. Thus the extension would include a wide range of things like dinosaurs, cows, humans, and unicorns. 9 A point for the more advanced; again we must rule out any hint of nominalism in logic. Nominalism conflates comprehension with extension, in other words, nominalism makes the mistake of saying the meaning of a term is simply all the individual things to which it refers, but this would reduce the syllogism to a mere tautology or non sequitur, as pointed out by the skeptic Sextus Empiricus. If by saying, man is mortal, Socrates is a man and therefore Socrates is mortal you mean by man only every individual man, then Socrates either was or was not already actually included in the first premise. If he was so explicitly included, the conclusion is a needless repetition, and if he was not included, then the conclusion does not follow. The only way out of this dilemma is to reject nominalism and say man in the first premise refers to the comprehension of man and that comprehension is not merely the actual sum total of its extension but refers primarily to a nature and possible instances of it. In this way, Socrates would only be virtually included in the term man as used in the first premise. The more notes we pack into a concept the range of things to which it applies diminishes. Signification, Concepts & Terms 21

38 comprehension and its extension is quite broad, applying to everything from water to gasoline. If we add the note of being a beverage, this certainly falls under the concept of liquid but with the additional note of being a beverage we notice that the extension has drastically decreased. Liquids such as gasoline are now ruled out. So we have increased our comprehension but reduced our extension. Imagine again that we add to this comprehension the notion of alcoholic. Again the comprehension (what it is we are talking about) is fuller, more complete, yet the extension reduces even more. Now we are no longer talking about things like water or soda, but with the increased comprehension we have a reduced extension to only alcoholic beverages. So again, the more comprehension we pack into a concept the less its extension becomes. Thus, there is a logical principle at work here called the inverse relation between extension and comprehension which can be stated: The greater the comprehension of a concept the less its extension becomes. The greater the extension of a concept the less the comprehension becomes. The extension of a concept can vary in another way as well. So far we have been assuming the extension of a term as universal meaning referring to all. The extension of the simple concept bird in a universal sense is all birds whether past, present, or future. But quantifying concepts can change the extension, i.e., some birds is a particular extension since it is not referring to all, while this bird is a singular extension because it only refers to one. We discuss this aspect in more detail later when we look at the amplification and restriction of terms below in this chapter as well as the quantity of propositions on page 76. Concepts can also be classified in different ways depending on different points of view. Here we consider concepts as simple vs. complex, concrete vs. abstract, and collective vs. divisive. Simple and Complex Concepts Concepts can be classified as simple or complex. Concepts can be classified as simple or complex. For instance, some concepts are simple, such as red or apple, other concepts are complex like bright pretty red or sour apple on the tree. Simple concepts contain only one essence, while complex concepts refer to a group of essences. Accordingly, the act of simple apprehension may produce concepts that are either simple or complex and yet they are both still the result of simple apprehension, since the word simple here in simple apprehension means without any additional work or constructive activity of the mind. Remember, simple apprehension means only a passive taking in without any affirmations or denials. So do not confuse simple and complex concepts with simple apprehension. Simple apprehension produces both simple and complex concepts. 22 An Introduction to Traditional Logic: Definition

39 Concrete and Abstract Concepts Concepts can also be either concrete or abstract while keeping the same content. We could think of say animal in both a concrete sense and an abstract sense. We can think of this concept in a concrete sense when it pertains to animals at the zoo. On the other hand, we could also think of it in the more abstract sense of animality. Animality has the same intelligible content as animal yet it does not allow itself to be thought of as being present at the zoo. Animality as such is not at the zoo even though animals are. This is because animality is an abstract concept; it cuts away from or prescinds from all individual instances. Abstract concepts are completely disengaged from concrete instances. It is important to understand precisely what is being said here. All concepts are universals and in that sense they are all abstract. Animal is abstract in the sense that it is a universal and can apply to anything from a rabbit to a human. But this concept is not abstract in the same way that animality is abstract. Animality is more abstract than animal because this concept necessitates a breaking away or separation from all individual instances that animal does not. The lion and the tiger at the zoo are animals but they are not animalities! So do not confuse a general notion of all concepts being abstract with a stricter notion that some concepts are more abstract than others. 10 It is this strict sense of abstraction that we are talking about here when we say a concept can be concrete or abstract. In the same way, do not confuse a concrete concept with an individual. The concept animal is not an individual animal; rather the concept just conveys the nature of an animal that is capable of being in or constituting an individual animal. So to sum up this point, although all universal concepts are abstract, some senses of a concept (concrete concepts) are not so abstract that they cannot be said of individuals while other senses (abstract concepts) are too abstract to be said of individuals. Collective and Divisive Concepts Concepts can be used in a collective or a divisive sense. The collective sense refers to a collection taken as a whole. To use the term in this way would be to say something like, The apples weigh five hundred pounds. Here the term apples does not mean that each and every apple weighs 500 lbs but only the apples taken collectively as a group. On the other hand, one can use the same term to refer to each and every individual of a group such as, The apples were picked from a tree, meaning that each Concepts can be used in a collective or a divisive sense. The collective sense refers to a collection taken as a whole. 10 St. Thomas Aquinas makes this distinction in his De Ente et Essentia chapter. 3. By the same token, scholastic philosophers make a distinction and say concrete concepts are abstractions without precision (they are abstract but are still able to apply to individuals) while abstract concepts are abstractions with precision (and cannot apply to individuals). Signification, Concepts & Terms 23

40 We use a concept in a divisive sense when we mean it to apply to each and every individual. and every individual apple was picked from a tree, not as a group, but one by one. This is to use a concept in a divisive sense. We use a concept in a divisive sense when we mean it to apply to each and every individual. This is important because without realizing this distinction, one can make errors of equivocation like this example: The grains of sand in that box weigh 300 lbs This thing is a grain of sand in that box Therefore this thing weighs 300 lbs Obviously grain of sand is being used in two different senses here; first collectively and second individually, so this argument would not be valid. The main point of going into some of the diverse ways in which concepts are used is important to avoid ambiguities. Human thought is a vastly complex operation capable of subtle distinctions and numerous senses all while using the same concept. Consistency in thought means consistency in the way concepts are used. The logician must be careful not to lapse into a mechanical rigidity when analyzing concepts and must always get at the intended sense. Studying the different senses of concepts will help achieve this. Terms Terms are verbal or written expressions that conventionally signify concepts. Terms are verbal or written expressions that conventionally signify concepts. 11 As expressions of concepts, terms can have the same divisions and properties that concepts do. For example, terms can be simple or complex, collective or divisive, concrete or abstract, have the properties of extension and comprehension, etc. Yet there are still important differences between terms and concepts. In order to see this, we must understand the relationship terms have to concepts and things. The Relationship Between Terms, Concepts and Things Since we human beings are social animals, we naturally interact with others and in order to communicate, our concepts must be expressed by way of terms. Now all the terms we use in writing and speaking are signs, but more precisely, terms, either written or verbal, are first something in themselves (either marks on a paper or a sound) and so are material signs. They are also artificial signs because terms attain their meaning through convention and agreement. For example, when we see a dog 11 With the qualification that some terms do not really signify concepts as such, but signify qualifications of concepts like most, few, slow, hastily, etc. Such terms are called syncategorematic terms. Our main concern here is the categorematic term that signifies something as a thing in its own right and not merely as a qualifier. 24 An Introduction to Traditional Logic: Definition

41 we come to know the dog s nature and derive a concept. English-speaking people have agreed by convention to call this known nature a dog. So while a term like dog is an artificial/material sign, the concept of dog is not. 12 Concepts are natural/ formal signs. This concept signifies the nature of the animal itself, and that concept is the same that is in the mind of the speaker of any language. So while a German or Latin speaker uses different conventional signs for it, i.e., hund and canis, nevertheless the concept of a dog in their mind is the same as the concept in the mind of the English speaker, and it is precisely this that makes translation possible. If the concept of a dog were not the same from say the English speaker to the German speaker, there would be no common ground for translating hund as dog. So basically this means there are three different things to consider; the terms we use, our concepts about things, and the things themselves. This threefold structure can be looked at from two different ways, namely, the order that these things have in the process of knowing and the order they take in the process of signifying. Taken from the side of knowing, first there is the thing in reality, say an actual dog. Secondly, there is the concept of dog that results when a dog nature becomes known. While the actual dog is not a sign, the concept of a dog is a natural sign (and hence not conventional) that signifies the nature of a dog. Third in the order of knowing is the conventional term, written or spoken, that we use to stand for that concept. This term too is a sign but it is an artificial or man made sign and the term is the only conventional sign of the three. So you can see that when we come to know about something, the actual thing is first, then when known the concept is in our mind, and then we can come up with a written or spoken term for it. Going the other way however in the order of signifying, like when we speak about what we already know, the priority is reversed. In the order of signifying, terms signify concepts and concepts signify things. In other words, terms are first and they signify real things by way of signifying concepts. So while a spoken word signifies both the concept and the actual thing, there is still a type of priority here in that the spoken word immediately signifies the concept in the mind and then through that, or mediately, the term signifies the actual thing. The following chart is designed to show this relationship: In the order of signifying, terms signify concepts and concepts signify things. 12 The reader may want to refer to the chart again on p. 16 Signification, Concepts & Terms 25

42 With this relationship understood, we now look at the major divisions of terms for the logician, the division between subject, copula, and predicate on one hand, and the division between univocal, equivocal, and analogical terms on the other. Subject, Copula, and Predicate When terms are considered from the role that they play in a sentence, the primary division of terms for the logician divides into subject and predicate. If you ll remember from your English grammar, the subject of a sentence is that of which something is said and the predicate is what is said of something. If we say The dog is black, the term dog is the subject and black is the predicate. As we will see, in logic, the subject will always be united with the predicate by a form of the verb to be. Often, this is done by using the verb is and since this verb is what joins the subject to the predicate, this term is called a copula and it always signifies some mode of existence. The subject and predicate terms united by a copula is what makes for a proposition in logic, and this will be the topic of a later chapter. Univocal, Equivocal, and Analogical Terms Univocal terms are terms used in exactly the same sense in each instance. An equivocal use of a term is when the same term is used but with a different meaning. When the meaning of a term is similar in some respects but dissimilar in others, it is called an analogical term. Because we simply do not have enough terms to cover all the things we want to talk about, the number of concepts outnumbers the words we have. Thus, sometimes the same word can have more than one meaning. Univocal terms are terms used in exactly the same sense in each instance. For example, when we say the poodle barks, the German Shepherd barks, the Great Dane barks, etc., the term bark is being used in the same way each time. But we can also talk about the bark of a dog and the bark of a tree. In this case, the same term bark is being used in a completely different sense in both cases. When this happens, logicians call the term equivocal. An equivocal use of a term is when the same term is used but with a different meaning. Next, there is a third middle type of usage where the term does not completely have the same meaning, but the meaning is not completely different either. If we say running is healthy, a person is healthy, and a complexion is healthy, the word healthy is similar in each case but not completely the same. Running is a cause of being healthy, a person is the thing that is healthy, and a complexion is an effect of being healthy. When the meaning of a term is similar in some respects but dissimilar in others, it is called an analogical term. The good thing about analogical terms is they can help us better understand the less familiar. If we were to say God is love, love here is used analogously in reference to the way human beings love. Traditionally understood, God s love is not exactly like human love, but it is not completely different either. Likewise if we say the dog knows, the man knows and God knows, the term know is used analogously. Humans do not know in the same way as dogs do (although both know in the sense that they can internally possess 26 An Introduction to Traditional Logic: Definition

43 something signifying the external world) and God does not know in the way human beings know, yet there is a similarity between the two. Finally, we must point out that strictly speaking it is not that terms in themselves are univocal, equivocal, or analogous but rather the usage of these terms can be this way. Summarizing these usages then, we say that univocal terms have the same meaning, equivocal terms have entirely different meanings, and analogical terms have partly the same and partly different meanings. The use of equivocation is fallacious and to be avoided in logic. We must be aware of these different uses of terms in our reasoning to avoid ambiguity and to be able to both convey our thoughts and understand the thoughts of others more clearly. We now consider different properties of terms, namely that the properties of amplification and restriction, reimposition and alienation. 13 Amplification and Restriction of Terms Terms can be qualified in such a way that either broadens or restricts what they would be representing on their own. For example, if one says, Every dog (in the world) is hungry vs. Every dog (in this yard) is hungry we can see that the words in parentheses modify what is meant by every dog. Amplification (ampliatio) is simply a broadening of a range or domain of a term, while restriction (restrictio) is the opposite of this. This modification may be either implicit or stated explicitly. 14 Reimposition and Alienation of Terms A reimposition of a term (called appellatio by the scholastic logicians) covers or names a term with a different determination than the term would have by itself. To put it simply, this is just naming a term under a certain aspect. For example, by calling someone a heavy sleeper we are not just calling them heavy in an unqualified sense. After all people can be quite thin and underweight and still be heavy as a sleeper! So you can see how sleeper here sort of reimposes a meaning on heavy that heavy would not have by itself. The same is true if we said Peter is a great logician we don t mean Peter is completely great, only that as a logician he is great. 13 There is though another property of terms that arises when terms are put into a proposition, called supposition, that will be discussed in a later section (see page 80). 14 The modification of terms both here in amplification/restriction and below in reimposition and alienation, are done through terms that are called syncategorematic. In other words, syncategorematic terms like all, some, only, insofar as, do not signify anything on their own but serve only as a modifier of things. In this way, a proposition may contain both categorematic (signifying things) and syncategorematic terms. Signification, Concepts & Terms 27

44 So a reimposition of terms is a property by which their meanings are qualified in some way. 15 Some logicians have noted that it was by a misuse of reimposition that the sophists would try to trap one in a contradiction. 16 Called the sophism of the veiled man they would ask, Can you see the veiled man? (assume there is a man hiding under a blanket right in front of you). If you say yes, then these sophists would say that you contradict yourself, since it is a contradiction to say you see what you do not see, a veiled man! But of course, to see a veiled man is to see a man only in a qualified way, like the man shaped object under a blanket and not to see the man in an unqualified sense. The alienation of a term (alienatio) is when the proper meaning of a term is altered or transferred to an improper or metaphorical usage, like when we refer to the man in the painting, or when we talk about a boxer having hands of stone. Man and stone here are used in an improper sense. In the first we mean the picture of a man and in the second we mean that his hands are just very hard. Chapter 2 Summary Intentionality is a term that means something is about or for something else. Knowledge, for example, is an intentional activity. The instruments by which we know things are signs. A sign is that which represents something other than itself. The signatum is what the sign signifies. Signs are either artificial or natural, and they can be either formal or material. Natural signs are not man made, while artificial signs are. Material signs are more than just signs. They are first something in themselves before whatever it is they signify. Formal signs are signs and only signs they only signify another. A first intention refers to a real being existing independently of thought while a second intention refers to a mental being. Concepts are what the intellect abstracts from the essences of things. Concepts signify the natures of things and are universal. The ability to grasp universal concepts is indispensable for knowledge. The material object of knowledge is a particular being, while the formal object is the aspect under which that being is considered (like considering only the color of an apple). 15 The scholastic understanding of appellatio seems to have varied from one logician to the next (see Bochenski pp ). Here we follow John of St. Thomas. 16 See for example, Maritain, op. cit., p An Introduction to Traditional Logic: Definition

45 Comprehension is what a concept means; extension is the things to which the concept applies. Extension presupposes comprehension, and can be singular, particular, or universal. Extension and comprehension are inversely related. Concepts can be simple ( apple ) or complex ( sour apple ). Concepts can be either concrete ( human ) or abstract ( humanity ). Concepts can be used collectively (applying to a group) or divisively (applying to each individual member of a group). Terms are the written or verbal expressions of concepts. Terms are conventional signs that signify things through concepts. Concepts are natural and not conventional, and they signify things directly. The subject term is who or what the sentence is about, the predicate is what is being said about the subject, and the copula is the verb is (to be ) that joins the two. Univocal terms have the same meaning, equivocal terms have entirely different meanings, and analogical terms have partly the same and partly different meanings. Terms can be amplified or restricted, explicitly or implicitly. A reimposition of a term is a property by which a term is taken under a certain aspect or qualification. Alienation gives an improper or metaphorical meaning to a term. Exercises A. Terminology: 1. What is a sign? Describe the different types of signs. 2. What is the difference between a material object of knowledge and a formal object of knowledge? 3. What is a concept? 4. How do second intentions differ from first intentions? 5. What is a universal? 6. Describe the relationship between terms, concepts, and things. 7. What is the extension and comprehension of a term? 8. Describe the difference between the univocal, equivocal, and analogical uses of a term. B. Determine whether the following term in bold is being used collectively or divisively: 1. Most of the class did a great job on the exam. 2. The feathers in that barrel weigh 500 pounds. 3. The people of this country have voted in a new president. Signification, Concepts & Terms 29

46 4. Every individual man has a mother therefore mankind has a mother. 5. Americans are fed up with this president. 6. Men have been around for thousands of years. 7. Everyone knows that babies tend to walk around 10 months. 8. The students, without exception, loved logic class. 9. Those weightlifters can lift a bus! 10. None of these balls are inflated to their proper level. C. Determine which of the following signs are natural formal, natural material, or artificial material: 1. A smile 2. A stop sign 3. The word cat 4. The concept of a cat 5. The rattling of a rattlesnake 6. A grimace 7. A hug 8. Thunder 9. An ancient form of writing 10. The idea of white 11. Crying D. Determine if the terms in bold are used in a univocal, equivocal, or analogical sense: 1. Cinderella went to the ball and danced on the balls of her feet and had a ball. 2. I love my daughter and I love Italian food. 3. I went to the bank to get some money and then sat on the bank of the river. 4. I took two aspirin and then called two doctors. 5. This car is good, and I have a good wife. 6. My thoughts exist and this rock exists too. 7. Sam pushed the legs of the table with his legs. 8. Logic is hard, and this cherry wood tabletop is hard too. 9. She is a fast learner and a fast runner. 10. He is a fast runner but during Lent he likes to fast. 30 An Introduction to Traditional Logic: Definition

47 Chapter 3: The Categories & the Predicables àà The Categories àà The Predicables The goal of the first part of logic is to get a clear understanding of the meaning of terms and to develop proper definitions. To do this, we must organize and order our thoughts about the things that we know. Categorizing things according to what they are is the primary way we achieve this. We said in the last chapter that it is quite obvious that there are different kinds of things in the world. Here we will see that not only are there different kinds of things, but also there are different ways in which something can exist, and this understanding of the different modes of existence leads us to what Aristotle called the Categories. In addition to the different categories, something also needs to be said about how these categorical concepts can be said of a subject. When we say, The horse is brown, just how does this predicate brown apply to the horse? A horse is one thing and brown is another, and so what does it mean to say that a horse is brown? This question led the Greek cynic Antisthenes to say that simple statements like A horse is brown does not even make sense. He thought one can say, Brown is brown or A horse is a horse, but one cannot say, The horse is brown because that would ac- The goal of the first part of logic is to get a clear understanding of the meaning of terms and to develop proper definitions. To do this, we must organize and order our thoughts about the things that we know. Categorizing things according to what they are is the primary way we achieve this. The Categories & the Predicables 31

48 tually be saying The horse is not a horse! 17 However, with an understanding of the Predicables, we can avoid Antisthenes skepticism about such common sense statements and so we will see that understanding the predicables will pay big dividends in both definition and predication. The Categories Have you ever wondered about the different ways something can exist? In other words, what are the basic categories of being and how might we determine them? Classifications in the physical sciences might provide a starting point by putting things into categories of living and non-living, vertebrate and invertebrates, vegetable, animal, or mineral, etc. But this schema is not broad enough to cover all reality. Where would the property ten feet high or to the left of go in the schema of the physical sciences? Is the color red a vegetable animal or mineral? Obviously, these sorts of properties are real; things really are ten feet high, to the left of other things, and sometimes red. Thus, the common classificatory scheme used in the physical sciences is not broad enough to cover all the ways of being. We need a broader philosophical cataloging if we want to cover all of reality. Substance and Accidents Substances are things that exist in themselves and not in another. Aristotle proposed an ultimate categorization of all the different ways something can exist. The first distinction he made, coming just by reflection on ordinary experience, is that there are some entities that are able to exist in themselves and other entities that can only exist in others. Some things like horses and trees are individual existing things that exist in themselves and not in another. Yet, there are other modes of existence like shape, color size that do not exist on their own but only in something else (imagine coming across a weight that was not the weight of anything!). In other words, common experience tells us there are both things and modifications of things. In philosophical terminology, these are called substances and accidents respectively. Substances are things that exist in themselves and not in another. The most principal characteristic of substances is that they have the mark of unity and individuality - their own existence per se. Yet we must make a distinction of two senses of the term substance. A primary substance is the existing individual thing. Socrates the individual person would be an example of a primary substance. A secondary substance is simply the nature of a primary substance, such as a man. Secondary substances are always universals while primary substances are always 17 This either tautology or contradiction argument of Antisthenes still menaces any philosophy or logical system that does not incorporate the doctrine of the predicables. Often the argument is ignored, but of course simply ignoring the problem is not a refutation of it. 32 An Introduction to Traditional Logic: Definition

49 particular individuals. So the most basic reality is a primary substance, the actual existing individual (e.g., Socrates) and following that is the secondary substance, the nature of that primary substance (e.g., man). Now primary substances are individual things and never predicates. To be a predicate means to be said of another, and an individual thing can never be said of another. It doesn t make any sense to say, This man is a Socrates. However the secondary substance man is a universal and makes a fine predicate. It makes perfect sense to say, Socrates is a man. So while secondary substances can be predicates, primary substances cannot. An accident on the other hand is that which exists only in a substance and not in itself. The term substance literally means to stand under or to ground the various types of accidents, hence, substances are necessary for accidents to exist; e.g., there cannot be a color red, without there being a red something. If the substance does not exist, it would be impossible for any of the accidents to exist. If we notice Socrates with a tan, Socrates is a substance, and the color tan is an accident. Socrates can be without the accident tan but tan cannot be without a substance like Socrates. Now upon reflection there are different types of accidents because there are different ways something can be modified. A color is not the same kind of accident as a place or size. Aristotle, basing his thought on common experience, said there were ultimately nine different ways something could be modified or in other words, nine irreducibly different kinds of accidents. An accident on the other hand is that which exists only in a substance and not in itself. The Categories Taking substance as the first category and then adding the nine accidental Categories, Aristotle s Categories totaled to ten. He proposed these as the ten irreducibly different ways in which something can exist. The Categories then are the ten ultimate kinds or whats. While new discoveries in the future might suggest a change in this enumeration, his list seems comprehensive enough for everyday experience and our use in logic: The Categories then are the ten ultimate kinds or whats. 1. Substance stating a whatness or essence e.g., dog, man, rose 2. Quantity stating how much or how many e.g., numbers, surfaces, a kilometer, etc. 3. Quality stating a qualification e.g., shape, color, happy, hot 4. Relation stating connections between things e.g., left, on top of, before 5. Action stating what a subject is doing e.g., pushing, heating 6. Passion stating what s being done to a subject e.g., being pushed, being heated The Categories & the Predicables 33

50 7. Location stating the place of a subject e.g., here, United States, at home 8. Position stating the arrangement of a subject e.g., sitting, crouching, prone 9. Time stating when a subject is operating e.g., now, last year, tomorrow 10. Possession stating how the subject is garbed e.g., armed, dressed Another point to keep in mind is that the intent here is for the Categories to contain only predicates. So granting this, we can say what we have here are the ten ultimate kinds of predicates Category Details For the sake of clarity, we expand upon only on the first three Categories. We do not need to go into detail on the other seven because they should be straightforward enough to not require elaboration in an introductory logic text. Substance is the main category because all the others are accidents that can only exist in a substance. Substance: Substance is the main category because all the others are accidents that can only exist in a substance. Again, this category should not be confused with primary substances (individual concrete things). As we said, primary substances can never be a predicate. Since only universals can be predicates, only universals representing the nature of an individual substance, (secondary substances) go into the category of substance. Secondary substances can always be predicates of some subject because they are the natures of individuals and thus are always universal. So if one says, Socrates is a man, the subject Socrates is a primary substance and the predicate man is a secondary substance belonging to the category of substance. Quantity: What belongs properly to quantity is some sort of measure. Quantity is always some sort of multitude that is able to be numbered or that is already numbered. This category refers to the material parts of a thing regarding its extension in space and measurable by mathematics. This extension in space that can be measured is either continuous (no breaks, like a line, surface, or a body) or discrete (with breaks, like numbers or the syllables of speech). The parts of continuous bodies have common boundaries while the parts of discrete bodies do not. Quality: A quality is the disposition or mode of a substance, in other words it refers to how something is determined. Quality is subdivided into four kinds of accidents. Habits or dispositions are the accidents for inclinations towards a certain trait; i.e. handsome, strong, aggressive, courageous, loving, etc. Abilities or inabilities qualify the subject s source of operation; i.e., agreeable, sensitive, to be imaginative, a runner, a basketball player, a bagpipe player, a logician, etc. Sense qualities qualify the subject s color, temperature, odor, etc. Finally form or figure refers to a thing s shape; round, square, spherical, etc. 34 An Introduction to Traditional Logic: Definition

51 Three Rules for Placement in the Categories There are three rules or conditions to be met before placing something in a category: 1. Only simple realities belong in the Categories; i.e., not sour apple but apple 2. Only universals (not individuals) belong in the Categories 3. Only finite natures belong in the Categories (not an unlimited essence such as God or inherently unlimited concepts such as being, good, true, and unity. 18 ). Again we have to remember that the Categories contain predicates and only a predicate is a thing said of something other than itself. These rules reflect this. 19 Obviously, a complex reality like sour apple doesn t fit into one category because it is a complex of two: sour is a quality and apple is a substance. Particular objects cannot be in a category because as we saw, a particular cannot be said of something other than itself. Finally, an unlimited nature such as the nature of God exhausts the span of all existence and cannot be limited to any one category. Logical Relevance of the Categories The Categories are an attempt at an exhaustive classification of the ultimate modes of existence and so defining any term will ultimately reduce back to one of these categories. Given any finite thing whatsoever, if we want to know what it is, it will have a home somewhere under one of the ten Categories. Thus, insofar as the Categories provide us with a classification of all existents, they can also be said to provide us a schema for classifying anything that we know and that is crucial for defining terms. The Predicables Concepts can be related to one another in different ways, and these ways are the topic of the Predicables. The Predicables tell us how the various kinds of predicates are said of a subject, that is, they give the way in which the predicate is related to the 18 In scholastic philosophy, these concepts are called transcendentals because they transcend the categories. 19 The listing here differs from what may be found in other logic texts and this is intentional. Additional rules that are sometimes given such as univocity or not allowing artifacts are superfluous. The rule of univocity is not needed because only natures are allowed as a category and natures cannot be equivocal, only words can. Artifacts can be allowed in a sense as long as the first rule is followed; i.e., the shape of a chair is a quality, while the material of a chair (say wood) is a substance. The Categories & the Predicables 35

52 subject. So here with the Predicables, we are focusing not so much on what is said of the subject but how it is said. Accordingly, there is a clear difference between the Categories and the Predicables: The Categories what is being predicated The Predicables how it is being predicated Despite their differences, the Predicables work hand in hand with the Categories and we use both in logic in order to define more clearly. When we have both the ultimate classifications of predicates and the way in which they are linked we have an invaluable aid for definition. We will first go over a brief description of these Predicables and then cover them in more detail. The Predicables are genus, specific difference, species, property, and accident: A genus is that which is predicated of many which differ only according to species. A specific difference is the identifying characteristic that makes something a special kind within its genus. A species is that which is predicated of many which differ only according to number. Genus: A genus is that which is predicated of many which differ only according to species. In other words, a genus is a general class of different kinds of things, e.g., animal is the genus of humans, lions, fish, bears, etc. Specific Difference: A specific difference is the identifying characteristic that makes something a special kind within its genus. The specific difference is what sets one species of thing apart from another within a general class, e.g., rationality sets human beings apart from brute animals within the genus of animal, or three sided is the specifying difference of triangles within the genus of polygon. Species: A species is that which is predicated of many which differ only according to number. In other words, the species is the specific kind of thing being defined and is what is predicated of many individuals, e.g., human is the species of Joe, Bill, and Terri. The species of a thing is its nature, and as we will see, it is the result of combining the genus with the specific difference. Property: A quality naturally arising from and unique to a certain species. The property is naturally present simply from the existence of the nature of the thing in question, e.g., only human beings have the ability to laugh. In other words, simply because a thing is of a certain kind, it will have certain properties and only things of this kind have that property. So animals and only animals are able to have sense experiences, persons and only persons are capable of doing philosophy, etc. Note it is sometimes difficult to tell a property from a specific difference, but the distinction is that the specific difference is more essential and the cause of the property. So the specific difference of being a man is rational which sets humans apart from all other animals, and because man is rational he has the property of being able to laugh. 36 An Introduction to Traditional Logic: Definition

53 Accident: A quality that is non-essential and not unique to a species, e.g., to be tan. When we say, The horse is brown, the predicate brown is predicated as an accident. Accidents here in the predicables differ from properties in that properties are particular to only one species while any species can have an accident (many different kinds of things can be brown). We now look at the first three predicables in more detail. Genus, Species, and Specific Difference The genus states the essence of a thing in a way common to different species by saying something essential about a thing in general. When we categorize the kind of something in a relatively broad way, we are stating its genus. To put it differently, the genus is the whatness of differing species. Some examples of a genus (indicated by italics) are: A triangle is a polygon A mammal is a vertebrate Purple is a color A plant is a living thing As the genus is the kind of something in a general way, the species is the kind of something in a relatively more specific way. Thus, genus and species are the technical terms for general and specific kinds. The species falls within the genus and that genus is part of what makes up the species, thus, a species is a particular kind of thing within a genus. Take the genus polygon. A triangle is a species of polygon, and so polygon will always be included as part of what a triangle is. But of course, a triangle is not the only kind of polygon. There are many other species of polygons as well. So what makes a triangle different from other species within a genus is the specific difference. You can see the role of a specific difference in the very term; a specific difference is simply that which makes a species different within a genus. The specific difference of a triangle is three-sided. Now to derive the species of something, all that needs to be done is combine the genus with the specific difference, e.g., three-sided polygon is the species of a triangle. The genus states the essence of a thing in a way common to different species by saying something essential about a thing in general. As the genus is the kind of something in a general way, the species is the kind of something in a relatively more specific way. Thus, genus and species are the technical terms for general and specific kinds. Genus and species refer to the general and specific kind of something respectively. To define the species, combine the genus with the specific difference. The Categories & the Predicables 37

54 We stress these first three predicables because they play an essential role in definition. The Predicables can be tough to understand for beginners in logic, but it may help to schematize them with questions and examples. So below is a diagram of a series of questions to which the correct answer is a certain type of predicable. When put this way, the difference between the types of predicables should become much more clear to the beginning logician: All of this should make even more sense when we see how the Predicables work together in a classic diagram called the Tree of Porphyry (which gets its name from the 3rd Century A.D. Neoplatonic logician, Porphyry). This diagram appeared in almost every medieval logic text and is an arrangement of genera and species in a way that looks something like an upside down tree: From this diagram you can see the relationship between genus and species. Living is a species of the genus corporeal (bodily) substance, but it is also a genus relative 38 An Introduction to Traditional Logic: Definition

55 to animals and plants. Likewise animal is a species of living thing while it is also the genus relative to rational (human beings) and irrational animals (brutes). In fact with this last, rational is the specific difference that sets humans apart from the other species of animals. You may see how this sort of arrangement looks like other similar groupings from say biology. Now although scientific groupings have more terms like kingdom, phylum, class, order, etc., the predicables of genus and species, etc., are sufficient for doing logic. Comprehension and Extension Regarding the Predicables Given the relationship between genus, species, and specific difference is hierarchical, with some being more general and others more specific, this means that these terms are also inversely related to comprehension and extension. Genera have more extension and less comprehension than species, and vice versa. The following diagram illustrates this: 20 To be the genus of another concept is to have a greater extension and less comprehension. Likewise, to be a species of another concept is to have a smaller extension and greater comprehension. So it should be clear that genus and species are relative concepts, yet this must be taken within certain limits. The broadest or most general kinds of things are the Categories. Above these, no distinctions of essence and nature can be made. The Categories are the highest kinds of things. They are 20 This diagram is derived from Veatch, op. cit., p. 62 The Categories & the Predicables 39

56 called the summa genera, each of them being the highest genus for all the species that fall under it. At the other extreme we have a case of maximum comprehension and minimum extension, the lowest or infimae species are the most specific types. The only thing lower than the lowest species are just individuals, not kinds. In other words, according to this layout, there is no kind of thing broader than one of the ten Categories and there is no lower species than the infimae species. Now just what species are actually the most specific is sometimes a difficult question and the details of some of the trickier cases really do not concern us here. Sticking with the clear examples we can say that man is an infima species of the category substance and there are not any subspecies underneath it. It is the specific difference that adds to the comprehension and decreases the extension. So if we think back to our earlier example on the tree of Porphyry, the specific difference of having sense perception (not listed in diagram) would be what sets animals off apart from plants within the genus living. And the same can be said for the next division, adding a specific difference to a genus is what generates a species of that genus. For example adding rational to the genus animal generates the species rational animal ( human ) as set apart from brute animals, and this adds to the comprehension of a concept and thereby decreases its extension. Now these three predicables we have been stressing; genus, species, and specific difference all refer to the essential nature of something. But of course, we can also talk about the non-essential characteristics of things and this involves the other two predicables of property and accident. Property and Accident A property is a characteristic that always accompanies a particular nature. An accident however, is anything else that is non-essential and can be found in numerous things. The predicables of property and accident do not refer to a thing s essential nature. Rather, they either naturally derive from the nature (as in a property) or simply belong to the thing in a non-essential way (as in an accident). A property is a characteristic that always accompanies a particular nature. Given a certain nature, a particular property will always subsequently be there too. Given the nature of a triangle is to be a three sided shape, it will always be the case that the property of the sum of the triangle s three interior angles equals two right angles. This is a trait unique to triangles and always accompanies them, so it is a property in the strict sense. An accident however, is anything else that is non-essential and can be found in numerous things. 21 To say, Socrates is tan, is to predicate by way of accident because many things can be tan other than Socrates. We will see in the next chapter that accidents are important for definition however, because even though they are not essential to 21 Again note the different senses of the term accident. In the Categories, an accident is marked off from a substance in that an accident means that which exists only in another. In the Predicables, an accident means a non-essential quality that is capable of being had by many different kinds of things. 40 An Introduction to Traditional Logic: Definition

57 a thing, many times we do not precisely know what the specific difference is between two species. For example, what exactly is the specific difference between a cat and a mouse? Our intellects are capable of knowing what both a cat and a mouse are, but our mind is not capable of knowing their essence so well that we can say precisely what the one specific difference between them is, even though we know that a cat and a mouse are not the same kind of thing. When this happens, it is sufficient to use an aggregate of various accidents (i.e., range of size, habits, sounds, etc.) to substitute for the specific difference. Chapter 3 Summary The Categories are the ultimate ways something can be and provide a schema to help with definition. There are 10, the first being substance, and the other 9 are accidents. A substance is that which exists in itself and not in another. There are two types. A primary substance is an individual existing thing while a secondary substance is the nature of a primary substance. An accident is a modification and exists only in a substance. The Predicables genus, species, specific difference, property, and accident give the modes or ways a predicate relates to a subject. The genus is that which is predicated of many which differ only according to species. The species is that which is predicated of many which differ only according to number. The specific difference is what sets one species of thing apart from another within a genus. In practice, the species is derived from the combination of the genus plus the specific difference. The predicable of property is a quality unique to a species while the predicable of accident is a quality that is common to different species. The Tree of Porphyry is a diagrammatic way of laying out the relationships of genus and species. A genus has more extension and less comprehension than a species. The highest genera, the summa genera, have no genera above them while the lowest species, the infimae species, have no lower species, beneath them, only individuals of that species fall below the infimae species. Exercises A. Identify the Category in which the following concepts belong. If it cannot be placed into a Category, indicate that as well: 1. man 2. hot The Categories & the Predicables 41

58 3. hitting 4. nighttime 5. at home 6. tomorrow 7. immediately 8. heavy 9. metal 10. on top of 11. friendship 12. being pushed 13. chopping (like in chopping wood) 14. in class 15. between 16. wrestling 17. able to knit 18. Friday 19. horse 20. tan feet long 22. Socrates 23. angry 24. wearing a sword kilograms B. Indicate the type of predicable: 1. John is tan. 2. Sam is rational. 3. Sally is a human. 4. Steve is a living thing. 5. Red is a color. 6. Philosophy is an area of study. 7. Ben is a jerk. 8. Steve is fast. 9. George is a substance. 10. Astronomy is a science. 11. A square is a shape. 12. A table is furniture. 13. A triangle is three-sided. 14. A tiger is a predator. 15. A tiger is a carnivore. 16. A cougar walks on all fours. 17. Species is a predicable. 18. A dolphin is a mammal. 42 An Introduction to Traditional Logic: Definition

59 19. A grasshopper is an insect. 20. Michelangelo is an artist. 21. A rifle is a firearm. 22. A rifle is a relatively long firearm with a rifled barrel, stock, and is fired from the shoulder. 23. A rattlesnake has a poisonous bite. 24. An obtuse angle is greater than 90 degrees and less than 180 degrees. 25. An acute angle is an angle measuring less than 90 degrees. 26. A right angle is 90 degrees. 27. Humans have a written language. 28. Human beings have the ability to build machines. 29. Animals are sentient. 30. Humans are capable of doing logic. The Categories & the Predicables 43

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61 Chapter 4: Forming Definitions àà Division and Classification àà Forming Definitions This chapter explains the logical techniques of division and classification and then brings these techniques together with what was contained in the previous chapters for the purpose of forming proper definitions. Division and Classification Dividing a general concept up into its component parts plays an important role in forming definitions. In order to define a thing properly, we seek an elaboration for it in terms of genus and species. We saw in the last chapter that the Categories were the highest one can go in terms of a genus, so any essential definition will fall into one of these Categories. But the Categories are quite broad. Quality for example, includes many types of things and so if we wanted to define a certain quality like hot, we would have to divide up quality somehow to show where hot would fit in. This is where division and classification come in. We were actually doing this already in the last chapter when we looked at the tree of Porphyry. Here we look at in more detail. Forming Definitions 45

62 Opposition is simply setting concepts apart from one another. We divide concepts through opposition. Opposition is simply setting concepts apart from one another. While there are actually four types of opposition in logic, we discuss only two here, contradictory and contrary opposition. Contradictory Opposition The strongest and most complete type of opposition is contradictory opposition. The strongest and most complete type of opposition is contradictory opposition. Contradictory opposition is the strongest type because it allows for no intermediate states; it is a simple affirmation or denial of some property. For example, one can be alive or not-alive, present or not-present, red or not-red, round or not-round, existing or not existing, etc. It should be easy to see that this type of division is exclusive and exhaustive there is no other alternative with contradictories and all the bases are covered with the two stated pairs. Something can be either red or not-red and those are the only two options, there is not even a middle ground between them. Contradictory opposition brings about a strict either-or set of options. Because of this, one of the elements of contradictory opposition must be true about something and the other false, a feature that will be put to powerful use later when we look at disjunctive arguments and dilemmas. Contrary Opposition A weaker form of opposition is contrary opposition. This opposition is an opposite of extremes within the same genus. A weaker form of opposition is contrary opposition. This opposition is an opposite of extremes within the same genus. For example, hot and cold are contrary pairs within the genus temperature. Sick and well, virtue and vice, fast and slow are also examples of contrary opposition. Even and odd are contrary pairs within the genus of whole numbers. Contraries are mutually exclusive in that a thing cannot have both contraries at the same time. For example, whole numbers can be either even or odd but not both. Now although contrary opposition is exclusive, it is not always exhaustive. This is because many times (but not always) contraries allow for intermediate states. With temperature for example, something can be either hot or cold, these are exclusive in that something can t be both (given whatever temperature we assign to hot and cold) but they are not exhaustive because there are numerous degrees between hot and cold. So unlike contradictory opposition, contrary opposition usually does not result in an exhaustive set of alternatives. Rather it is often the case that contraries allow for shades of gray or intermediate alternatives between them. Let s look at some sets of examples to make sure we understand the difference between these two types of opposition: Even and not-even (contradictory opposition) Even and odd (contrary opposition) 46 An Introduction to Traditional Logic: Definition

63 Black and not-black (contradictory opposition) Black and white (contrary opposition) Healthy and not-healthy (contradictory opposition) Healthy and sick (contrary opposition) Division We have just seen two ways to bring about a logical division of a concept. What we were doing was dividing a genus into parts through two different types of opposition. Now logical division is a good way of knowing what things are and how they differ from one another. We use these types of opposition as a way of clarifying a concept that we already know by seeing where it fits within the whole scheme of things. Looking at the tree of Porphyry in the last chapter, you can see how by starting with a broad category of substance, concepts are divided by opposition and this upside down tree works its way down to the species of a human being ( rational animal ). Both types of opposition are used in this tree. Dividing corporeal into living and non-living is contradictory opposition, while dividing living into animal and plant is an example of contrary opposition. To make a good logical division of a concept, certain rules must be followed: Rules for Making a Good Division 1. The basis of the division must remain constant. It is ok to use a different basis for different divisions, but any one division must be made on the same basis. If we wanted to divide religion on the basis of geography, you may divide it into East and West, but then you would not want to go on and add another consideration within this same division, say one that is based on the number of adherents like over one million and under one million. Shifting the basis of division like this might go on forever and will likely result in an inefficient overlap. Hence, only one basis of division should be used for any one division. 2. The division must exhaust the genus. It almost goes without saying that if we want to understand a concept by dividing it up, that understanding would be incomplete if we leave anything out. With any good opposition, all the divided members taken together must fill up the whole. In other words, when dividing a genus, all the species should be covered. Contradictory opposition is the best way to exhaust a genus (dividing a bodily substance into living and non-living leaves nothing Forming Definitions 47

64 out). Dividing temperature into hot and cold does not exhaust the genus because it leaves out all the degrees in between. So we must be careful that our use of contrary opposition does not violate this rule. 3. The divided parts must be less universal than the thing divided. A part cannot be greater than a whole, so if we are dividing something into parts, the divided part cannot be greater (more universal) than the whole. For example, it would be an error to divide animal into corporeal and incorporeal substance, because the parts would be more general than their genus. Following this rule often means we stick to dividing a genus into its species. 4. The parts must exclude one another. Members of a division are opposed (by a type of opposition) and so they must be distinct. In other words, the parts cannot overlap. For example, it would be wrong to divide living things into animals, plants, and mammals. Mammals is already covered under the part animals and so this division is no good since its elements do not exclude one another. To put it simply, a good division does not include any element more than once. Again, contradictory opposition is the best way to avoid overlapping. It is not necessary to start with one of the main Categories in order to use opposition in making a division (although such a start may be useful if you do not know where else to begin), but you can make a logical division with any general concept. The division process can begin anywhere from the highest Categories down to the lowest or infimae species which divides into individuals. Classification We can achieve the same result that division gives us by moving from the bottomup. In other words, instead of going from top (most universal) to bottom (least universal) like we do in division, we can work from the specific to the general. When approached this way, the process is called classification. You can classify Socrates as a man, man as a rational type of animal, animal as a sentient type of living body, etc. In classification all the same rules apply as they do in division. One can begin classification starting with individuals and proceed up to any level lower than the main Categories. A more detailed tree of Porphyry will help bring these principles together: From Veatch, p An Introduction to Traditional Logic: Definition

65 Finally, we need to clarify some more terms. The genus closest to a species is called the proximate genus while a genus further away is the remote genus. In this case, mammals is the proximate genus of man and vertebrate is the remote genus of man. Now in the same way, any species of a species within a given genus is a subspecies of that genus. So concepts like vertebrate and invertebrate, mammal and reptile, etc., are subspecies of the genus organic, and of course here it makes fine sense to say some subspecies are more proximate and others are more remote. So one can have a proximate species and a remote species of a certain genus. Additionally, when dividing something that is already the result of a prior division, this product is called a subdivision, e.g., animal is a subdivision of organic (a subdivision is really the same as the subspecies). If classifying something that is the result of a prior classification, the resulting category is called a superclassification, which is really just another term for remote genus, e.g., vertebrate is a superclassification of man. 23 Forming Definitions A definition is a phrase that makes explicit what a single concept means or what a term signifies. To define a term is to set limits to it (In Latin, de-fine is about a limit ). If we do not define a term, we run the risk of attributing too much or too little to what 23 The reason there are two terms, remote genus and superclassification, is because one may make use of a remote genus without using classification at all, as in the process of division. The same could be said about the difference between a subdivision and a subspecies. Even though they are both the same, one can have a subspecies through classification and so strictly speaking it would not be a subdivision because classification is not using division. A definition is a phrase that makes explicit what a single concept means or what a term signifies. Forming Definitions 49

66 we are supposed to be talking about. Now we must remember that the definition of terms belongs to the first part of logic. Hence, definition is not a proposition; rather it is merely an elaboration on the meaning of a term. For example, the definition of a triangle is three-sided polygon. The definition itself is not a proposition, but of course, it can be put into a proposition. If we say, A triangle is a three-sided polygon, even though the proposition contains a definition and even if the two seem to occur simultaneously, nevertheless a definition and a proposition are not the same thing. With regard to definition, there are two terms that every student of logic should know. The definiendum is the term or concept being defined and the definiens is the definition given to it. If we say, A triangle is a three sided polygon, triangle as used here is the definiendum and three sided polygon is the definiens. It should come as no surprise that the human intellect is capable of making numerous types of definitions and again not surprisingly, it belongs to the study of logic to enumerate and understand these various types. The major distinction is the difference between nominal definitions and real definitions. We judge between the nominal and real definition by what comes about or unfolds directly from the definition, in other words, whatever the definition immediately refers to. If the definition immediately defines a word, the definition is nominal and if the definition immediately refers to a thing the definition is real. Nominal definitions are primarily definitions of words in that they directly refer to how a word is used rather than to an essence. Nominal Definitions Nominal definitions are primarily definitions of words in that they directly refer to how a word is used rather than to an essence (It may be that a nominal definition can be traced back to an essence, but not directly). There are three main types of nominal definition. First, nominal definitions can refer to convention, namely, how people have meant a certain term in the past. Conventional definitions are the kind we often find in dictionaries, which often use synonyms. Dictionaries tell us how a word is used in the relatively recent past and because of this, dictionaries have to be occasionally revised. Another type of nominal definition is looking at the source of the term or etymology. For example, defining a word based on its Latin or Greek roots is an etymological definition. Other times it is the case that we have a hard time saying what a word means or perhaps we simply wish to coin a new term. When this happens we have to come up with a meaning for the sake of the discussion at hand, like when one says something like Let us define moral as such and such. These suggested meanings are called stipulative definitions. Nominal definitions primarily refer to word usage and there are three types; conventional, etymological, and stipulative. 50 An Introduction to Traditional Logic: Definition

67 Real Definitions Real definitions directly relay information about the actual thing signified. There are four main types. The best kind of definition is one that gets at the essence of something. An essential definition then is the goal towards which the logician usually aims because to define an essence is precisely to define what something is. What a term means is important, and essential definitions give precisely that, the whatness or comprehension of a concept or term. A famous dialogue in Plato s Meno makes this point. Here Socrates is asking Meno for the definition of virtue : Real definitions directly relay information about the actual thing signified. [Socrates] By the gods, Meno, be generous, and tell me what you say that virtue is; for I shall be truly delighted to find that I have been mistaken, and that you and Gorgias do really have this knowledge; although I have been just saying that I have never found anybody who had. [Meno] There will be no difficulty, Socrates, in answering your question. Let us take first the virtue of a man-he should know how to administer the state, and in the administration of it to benefit his friends and harm his enemies; and he must also be careful not to suffer harm himself. A woman s virtue, if you wish to know about that, may also be easily described: her duty is to order her house, and keep what is indoors, and obey her husband. Every age, every condition of life, young or old, male or female, bond or free, has a different virtue: there are virtues numberless, and no lack of definitions of them; for virtue is relative to the actions and ages of each of us in all that we do. And the same may be said of vice, Socrates. [Socrates] How fortunate I am, Meno! When I ask you for one virtue, you present me with a swarm of them, which are in your keeping. Suppose that I carry on the figure of the swarm, and ask of you, What is the nature of the bee? and you answer that there are many kinds of bees, and I reply: But do bees differ as bees, because there are many and different kinds of them; or are they not rather to be distinguished by some other quality, as for example beauty, size, or shape? How would you answer me? [Meno] I should answer that bees do not differ from one another, as bees. [Socrates] And if I went on to say: That is what I desire to know, Meno; tell me what is the quality in which they do not differ, but are all alike - would you be able to answer? [Meno] I should. Socrates wants to know what virtue is. In other words, he is looking for an essential definition of virtue, viz., what is the nature or essence of virtue that all virtues have in common and justifies us calling them virtues? Meno s response gives only some Forming Definitions 51

68 particular examples of virtue but does not define its essence. As a result, Meno fails to say what virtue is, and so Socrates chides him. The reason essential definitions are so important and useful is that they tell us what is most fundamental and basic about a term or concept, that is, they get down to the core of what definitions are supposed to do. Take this example: 24 (1) Man means rational animal. (2) Man means animal with a sense of humor. (3) Man means naturally featherless biped. These definitions, as accurate as they are, are not equally as useful. As a matter of fact, they are ordered from the most essential to the least. Rational animal is the classic definition of a man. It tells us precisely what a man is. The second definition above is a definition by property. The ability to laugh is unique to human beings, but it is not as essential as rationality is to human nature because rationality is the cause of the ability to laugh while the ability to laugh is not the cause of rationality. The third definition above is even less essential than the others, for being featherless or bipedal are accidental characteristics that can be individually had by many things other than men. For these reasons, the first definition is considered better than the second and third. It is not that the last two are incorrect; they are just not as essential or basic. To say we want an essential definition means nothing else than we want the species of the definiendum. As indicated previously, the species is the result of the genus plus the specific difference, and that is what we want with an essential definition. The essential definition is made up of two parts, a common part (the genus) and a particular part (the specific difference). An essential definition consists of a common part and a particular part, the genus + the specific difference. The genus should be the proximate genus, and when we put this with a specific difference, we get a species. In fact, we instinctively and naturally think of defining things in terms of genus and species when asked for a definition of something. If asked what a triangle is, spontaneously one will often answer back, A type of shape. Notice that this response appeals to the genus shape and vaguely points to the species/specific difference by saying type of. It is a very natural response to define rose as a type of flower, a dog as a type of animal, physics as a type of science, etc. The study of logic simply teaches us to hone and perfect this natural and intuitive way of defining things through a better understanding of the predicables. 24 From Veatch, op. cit., An Introduction to Traditional Logic: Definition

69 In practice, finding the genus of something is rather easy, while finding what makes it different from others within the genus is more difficult. This is because it is easier to see the bigger picture of a thing s essence than it is to see the smaller details about it. So usually the sticking point in defining is coming up with a specific difference that sets the thing apart from other species within the same genus. Ideally we should use the specific difference whenever possible, however as mentioned in the last chapter, the human intellect is not always able to do this. We know that dogs are not the same thing as mice, but the exact specific difference between them is obscure. For practical purposes then, often we do the best we can by using properties or an aggregate of accidents that suffice for the difference. All that is needed here is some characteristic or set of characteristics that distinguishes the kind of thing being defined apart from the others within the same genus, and properties or a group of accidents can work quite well for this. But there is more to real definitions than just the essential. It may be useful to define something in a way other than its essence. Accordingly, another type of real definition is a descriptive definition. The descriptive definition is simply a description of a thing by listing its accidents or properties, and perhaps along with its genus, like we saw in saying man means animal with a sense of humor or naturally featherless biped. above. A descriptive definition doesn t say essentially what something is because it is a description in terms of non-essential characteristics. So although the descriptive definition is a type of real definition because it refers primarily to a thing, it cannot be called an essential definition because it includes these non-essential accidents in a way that does not suffice as a specific difference. 25 A third type of real definition is a causal definition in the sense of a definition by extrinsic cause. 26 One can define something by what made it or what its purpose is. Such a definition by extrinsic cause does not constitute a thing s nature, but refers to something extrinsic to the thing itself. For example, the scholastic logicians defined the human soul as a form created by God for beatitude, and in the same way, we could define a simple hammer as, that which is made by a toolmaker for pounding nails into wood. These are definitions based on who or what made the thing and the purpose or end for which it was made. A fourth type is definition by material cause. This is simply defining something by the type of material that constitutes it; a chair can be a wood thing and a sword a steel instrument. Finally, these types Another type of real definition is a descriptive definition. The descriptive definition is simply a description of a thing by listing its accidents or properties. A third type of real definition is a causal definition in the sense of a definition by extrinsic cause. A fourth type is definition by material cause. 25 It is important not to confuse a descriptive definition that uses accidents, with the aggregate of accidents used as a specific difference when making an essential definition. In other words, just because accidents appear in a definition, doesn t automatically mean the definition is only descriptive. As was said above, it may be that these accidents are serving the role of specific difference and marking a species off from everything else within the genus. So how do you tell the difference? Follow this rule of thumb: if the listing of accidents appears with a genus, and if that listing of accidents suffices in setting that kind of thing off from everything else under the genus, then we have an essential definition. Anything less, the definition is only descriptive. 26 In Aristotelian terminology, this is definition by either efficient or final causality. The essential definition is a definition by formal causality. Forming Definitions 53

70 of definitions may be combined in various ways, thus a hammer can be defined as a wood and metal thing made by a toolmaker for pounding nails into wood (Combining both definition by material cause and extrinsic cause together.) So to summarize real definitions: Real definitions refer primarily to a thing and there are four types; essential, descriptive, extrinsic cause, and material cause. The logician stresses real definitions above nominal definitions and within the types of real definition, the essential is most important. The main goal of the first part of logic is to say what something is, to explain the essences of things, since this type is the most informative. In order to correctly derive such an essential definition, one must follow certain rules. Rules for an Essential Definition: 1. The definition should be through genus and specific difference. The ideal definition seeks the species through a combination of something common (genus) and something particular (specific difference). For example, saying triangle means three-sided polygon gives both that which is common (polygon) and that which is particular (three-sided). 2. The definition should be clearer than the thing defined. Definitions are supposed to clarify. It would defeat the whole point of defining if definitions were more obscure than what was being defined. This rules out overly technical terms, unnecessary jargon, and other types of obscure language when making definitions. A definition should make a concept easier to understand, not harder. This also means that definitions should avoid poetical metaphors. The use of poetry and metaphor is fine in certain contexts, but not for logic. To say, love means the scent of a rose on a warm summer breeze does not clarify the essence of what love means. Also, this rule also means that the definition should not be circular or synonymous. It would not be too helpful if someone defined joyful as feeling joy. The definition is supposed to clarify, not simply repeat (either with the same word or synonym) the term being defined. 3. The definition should be neither too broad nor too narrow. Essential definitions should apply to all possible instances of what is being defined 54 An Introduction to Traditional Logic: Definition

71 and only those instances. When successful, an essential definition is interchangeable with the term being defined. To put it another way, when the definition is essential, the definiens will be identical with the definiendum. So if a definition is too broad, it will include additional things that fall outside the scope of what it is supposed to define. To define a bear as four-legged mammal is too broad because that definition also includes rabbits and wolves. By the same token, a definition should not be too narrow by excluding things that should be included in it. To define chair as four legged piece of furniture, used for sitting, and made of wood is too narrow because that definition rules out chairs made of other materials. 4. The thing to be defined must be definable. There are certain conditions that need to be met in order for something to be definable by an essential definition: 1. The thing must be essentially one, that is one essence. If a definition refers to many essences, it is not one definition but many. In order to have a proper definition, the confusion of plurality must be removed and one thing defined Every defined thing must be a species under a genus, that is always a universal. Since every essential definition consists of a genus plus specific difference, this point is clear. If a concept is not under a genus (like the Categories), there can be no essential definition of it. This rule also means there cannot be essential definitions of individuals. Strictly speaking, we do not ask for a definition of a Socrates. Definitions are meanings, and it makes no sense to say, What is the meaning of Socrates! In short, we define universal terms but we describe individuals (We will see more about the individual as not-definable below). If a definition is too broad, it will include additional things that fall outside the scope of what it is supposed to define. By the same token, a definition should not be too narrow by excluding things that should be included in it. Terms Which Do Not Permit of an Essential Definition Given these rules, we can see that there are certain concepts that cannot be strictly defined. Again, the highest genera, the Categories, are not susceptible to a strict definition simply because there is no genus above the highest genus. Thus, the Categories are indefinable in the strict sense, and this is to be expected. Some concepts must be given and fundamentally basic. If it were otherwise, we would always have to define a term by something else, and that by something else, and so on to infinity. To avoid this infinite regress, the regress comes to a stop with the Categories. The Categories, are not susceptible to a strict definition. 27 Several things can come together of course to form one substantial unit and thus have one essence; i.e., a horse is one substance even though it consists of parts. Forming Definitions 55

72 Individual things do not permit a proper definition. Also individual things do not permit a proper definition. It is usually the case that we describe individuals (a certain height, a certain weight, hair color, etc.) but we do not define individuals. Individuals are not essences. Only terms that refer to essences or whats are able to be divided and explained by means of concepts. It does not make sense to ask, What is a Napoleon? in the same way we might ask, What is an emperor?. The first cannot be defined because it is an individual, while the latter can because it is a kind or type of thing. The best we can say about individuals is that they have an essence (i.e., Napoleon is a human ), but singular terms like Napoleon refer primarily to individuals, not their essence. It is quite easy in fact to keep this point in mind by remembering that essential definitions are always definitions of an essence. Finally, concepts or essences that are inherently unlimited like being, good, true, unity and God cannot be strictly defined. To define is to set limits on a concept, and the limitless cannot have limits. God for example, cannot be defined as to what he is, only negatively by saying what he is not, i.e., immaterial (not-material), infinite (not-limited), etc., or by saying what he is like analogically or metaphorically, i.e., all-knowing, all-loving, a consuming fire, etc. Summing Up the Tools of Definition and their Interaction The logical tools we have mentioned provide us with a means of ordering our knowledge and arriving at a better understanding of things. These tools have their limitations to the extent that their power lies within the limitations of human knowledge itself. Although defining in logic is not done with mathematical certitude, nevertheless these tools can go a long way in aiding our comprehension of concepts. Our knowledge and the ability to express it are much improved when we can say what a thing is and how it differs from something else. The following is a summary of these tools: Concepts - Concepts are both the objects and the building blocks of a definition. We define concepts in terms of other clarifying concepts since the goal of definition is to make explicit, through several concepts, what a single concept means or what a term signifies. Categories - The Categories help us in defining because defining aims at the essence or whatness of things and the Categories represent the ultimate types of whats. Any definable concept will belong to a category, and since it is easier sometimes to think in generalities, the Categories often give us a good starting point in definition. 56 An Introduction to Traditional Logic: Definition

73 Division and Classification - Concepts within the Categories may be divided or classified as a way of furthering our knowledge. When we do this, we order and clarify our knowledge of terms and make our own tree of Porphyry by organizing concepts into genus and species. Predicables - The Predicables help us in defining because each predicable is a way of saying something about the thing. As mentioned above, this is a natural and instinctive approach to definitions which logic simply refines. Rules of Definition - It is rather obvious how understanding the rules governing the defining process can aid us in forming proper definitions. Chapter 4 Summary The task of definition is to clarify a concept s component parts. This is done through division and classification. Contradictory opposition is the strongest in that it is exhaustive and allows for no intermediate states. Contrary opposition is an opposition between two extremes within the same genus. Many contraries allow for middle positions. Logical division is a way of knowing what things are and how they differ from one another, and the same is true for classification. The proximate genus is the genus closest the species while the remote genus is further away in universality. A definition is a phrase that makes explicit what a single concept means or what a term signifies. The definiendum is the concept being defined and the definiens is the definition given to it. Nominal definitions are definitions of words that primarily refer to how a word is used rather than to an essence. Three types of nominal definitions are conventional, etymological, and stipulative. Real definitions relay information about the actual thing signified. An essential definition says what the essence of something is, a descriptive definition defines a thing in terms of its accidental characteristics or properties, a definition by extrinsic cause defines a thing in terms of who or what made it and what it was made for, and a definition by material cause defines a thing in terms of the matter that constitutes it. Forming Definitions 57

74 The ideal definition is an essential definition and these should be clearer than the thing defined and neither too broad nor too narrow. The Categories are the highest genera and do not admit of an essential definition. Also, individuals and inherently unlimited essences cannot be strictly defined either. Exercises A. Make good divisions of the following that extend to two subdivisions: 1. Temperature 2. Shape 3. Religion 4. Sports 5. Color 6. Plants 7. Languages 8. Money 9. Beverages 10. Government B. Construct a Tree of Porphyry: In the dialogue The Sophist, Plato has a well-known exchange between the Stranger and Theaetetus on the definition of angling. 28 Construct a tree of Porphyry concerning the term angling based on this text: Stranger: What is there which is well known and not great, and is yet as susceptible of definition as any larger thing? Shall I say an angler? He is familiar to all of us, and not a very interesting or important person. Theaetetus: He is not. Stranger: Yet I suspect that he will furnish us with the sort of definition and line of enquiry which we want. Theaetetus: Very good. Stranger: Let us begin by asking whether he is a man having art or not having art, but some other power. Theaetetus: He is clearly a man of art. Stranger: And of arts there are two kinds? 28 I am grateful to John Oesterle s Logic: The Art of Defining and Reasoning (New Jersey: Prentice hall, 1963) p. 61 for this example, which is too good to pass over. The text from Plato is edited and abbreviated. 58 An Introduction to Traditional Logic: Definition

75 Theaetetus: What are they? Stranger: Seeing, then, that all arts are either acquisitive or creative, in which class shall we place the art of the angler? Theaetetus: Clearly in the acquisitive class. Stranger: And the acquisitive may be subdivided into two parts: there is exchange, which is voluntary and is effected by gifts, hire, purchase; and the other part of acquisitive, which takes by force of word or deed, may be termed conquest? Theaetetus: That is implied in what has been said. Stranger: And may not conquest be again subdivided? Open force may be called fighting, and secret force may have the general name of hunting? Theaetetus: Yes. Stranger: And there is no reason why the art of hunting should not be further divided into the hunting of living and of lifeless prey. Theaetetus: Yes, if both kinds exist. Stranger: Of course they exist; but the hunting after lifeless things having no special name, except some sorts of diving, and other small matters, may be omitted; the hunting after living things may be called animal hunting. And animal hunting may be truly said to have two divisions, land-animal hunting, which has many kinds and names, and water-animals hunting, or the hunting after animals who swim? Theaetetus: True. Stranger: And of swimming animals, one class lives on the wing and the other in the water? Theaetetus: Certainly. Stranger: Fowling is the general term under which the hunting of all birds is included. The hunting of animals who live in the water has the general name of fishing. And this sort of hunting may be further divided also into two principal kinds? Theaetetus: What are they? Stranger: There is one kind which takes them in nets, another which takes them by a blow. Theaetetus: What do you mean, and how do you distinguish them? Stranger: As to the first kind-all that surrounds and encloses anything to prevent egress, may be rightly called an enclosure, for which reason twig Forming Definitions 59

76 baskets, casting nets, nooses, creels, and the like may all be termed enclosures? And therefore this first kind of capture may be called by us capture with enclosures, or something of that sort? Theaetetus: Yes. Stranger: The other kind, which is practiced by a blow with hooks and three pronged spears, when summed up under one name, may be called striking, which is done at night, and by the light of a fire, and is by the hunters themselves called firing, or spearing by firelight. And the fishing by day is called by the general name of barbing because the spears, too, are barbed at the point. Theaetetus: Yes, that is the term. Stranger: Of this barb-fishing, that which strikes the fish who is below from above is called spearing, because this is the way in which the threepronged spears are mostly used. Theaetetus: Yes, it is often called so. Stranger: Then now there is only one kind remaining. When a hook is used, and the fish is not struck in any chance part of his body-he as be is with the spear, but only about the head and mouth, and is then drawn out from below upwards with reeds and rods:-what is the right name of that mode of fish, Theaetetus? Theaetetus: I suspect that we have now discovered the object of our search. C. Construct a Tree of Porphyry: Take the following text on sounds from the medieval logician Peter of Spain and construct a tree of Porphyry from it: A sound is whatever is properly perceived by hearing; for though a man or a bell may be heard, this is only by means of a sound. Of sounds, one is voice another not voice. Sound-voice is the same as voice; so voice is sound produced from the mouth of an animal, formed by the natural organs Of voices, one is literate, another not literate. Literate voice is that which can be written, e.g., man ; not literate is that which cannot be written. Of literate voices one is significant, another not significant. Significant voice is that which represents something to the hearing, e.g., man or the groans of the sick which signify pain. Not significant voice is that which represents nothing to the hearing, e.g., bu, ba. Of significant voices one signifies naturally, another conventionally. Conventionally significant voice is that which represents something at the will of one who originates it, e.g., man. Naturally significant voice is that which represents the same thing to all, e.g., the groans of the sick, the bark of dogs. Of conventionally significant voices 60 An Introduction to Traditional Logic: Definition

77 one is simple or not complex, e.g., a noun or a verb, another composite or complex, e.g., a speech. 29 D. State the kind of definition 1. Philosophy: The Greek roots philo + sophia means love of wisdom 2. Square: A four-sided polygon with all equal lengths 3. Screwdriver: An instrument for driving screws into a hard surface 4. Horse: a relatively large, hoofed, herbivorous mammal 5. Football: An American sport where two teams of 11 players, using padded equipment, maneuver across a rectangular, 100-yard-long field with goal lines and goal posts at either end, the object being to gain possession of an oblong shaped ball and advance it across the opponent s goal line by running or passing plays or by kicking it through the air between the opponent s goal posts, all within a time period of 4 quarters, each consisting of 15 minutes of playing time. 6. Red: A hue within the visible color spectrum with a wavelength of approximately 630 to 750 nanometers 7. Throne: A piece of furniture designed to be a surface upon which a royal occupant may sit. 8. Shotgun: a wood and metal thing 9. Automobile: an enclosed thing made primarily of metal on rubber wheels designed for carrying passengers from one point to another 10. Spider: a type of arthropod with eight legs, two venomous fangs and able to spin webs that ensnare food. E. Define the following and label the kind of definition you are giving (although you should try to focus on the essential definition) 1. Door 2. Grandfather 3. Illness 4. Apple 5. Beetle 6. Rose 7. Computer 8. Abortion 9. Insult 10. Book 11. Yardstick 12. Planet 13. Grass 29 Peter of Spain, Summulae Logicales tr. from I.M. Bochenski, A History of Formal Logic (Notre Dame: University of Notre Dame Press, 1961) p. 153 Forming Definitions 61

78 14. Shovel 15. Alligator 16. Government 17. Whale 18. Igloo 19. Map 20. Rattlesnake 21. Camouflage 22. Chokehold 23. Violin 24. University 25. Mercury 62 An Introduction to Traditional Logic: Definition

79 PART II: PROPOSITIONS

80

81 Chapter 5: Propositions àà What is a Proposition? àà Truth and Falsity àà The Structure of Categorical Propositions àà Distribution àà Supposition àà Putting Everyday Language into Proper Logical Form What is a Proposition? Human knowledge is not complete with just terms, concepts and definitions. These are just the beginning. We need something more than mere concepts. We need a tool that not only discloses a nature but one also that affirms or denies a nature to a subject, in other words, a tool that says something about existence and the way things are. The mental act that does this is the judgment, and a judgment is expressed in a proposition. A proposition is a sentence that signifies something as true or false by telling it like it is (or is not) and because of this, such propositions are susceptible to truth and falsity. Thus the technical definition of a judgment is that act of the mind that unites by affirming or divides by denying, and a proposition is simply an oral or written expression of that judgment. 30 There can be no judgment (and hence no proposition) unless the intellect affirms or denies something. You may affirm that the apple is red, or you may deny the apple is red, but in both cases you are judging about reality, that is, the actual color of A proposition is a sentence that signifies something as true or false by telling it like it is (or is not) and because of this, such propositions are susceptible to truth and falsity. 30 So in the same way that terms express concepts, propositions express judgments. Propositions 65

82 the apple. So unlike concepts and terms, propositions are inherently susceptible to truth and falsity. This means that not every sentence is a proposition. Imperatives, requests, interrogatives, and exclamatory utterances do not have a true/false value and so are not propositions. Take these for example: Charge the enemy on the right flank! Can you help me? What is your name? Ouch! None of these utterances are propositions because there is no affirmation or denial here. If someone says to you Would you mind passing the salt?, to reply with, Yes because I think what you just said is false, would not make any sense. Requests and other utterances like this do not make any claims about reality in a way that allows them to be susceptible to truth and falsity. Only a proposition makes a claim There are different types of propositions. Most simply, there is the categorical proposition which is a simple assertion or denial where one thing is said of another. Next, there is the compound proposition which is one sentence that contains two propositions, usually joined by either or, and, if then, etc. We will discuss the compound proposition in the next chapter and focus on the categorical proposition here. First we will discuss in more detail what we mean about the truth and falsity of propositions, then we will elaborate on the structure of propositions, their properties of distribution and supposition, and end with a look at how to translate ordinary language into proper logical form. that is susceptible to truth and falsity. Truth and Falsity We have said that only judgments and the propositions that they express are the types of sentences that are capable of being true or false. In other words, all propositions have a truth-value. The truth-value of a proposition simply refers to whether a proposition is true, false, or undetermined (meaning that we simply don t know). Now at this point it will be very beneficial to discuss what truth and falsity really mean, and we could hardly do better than to begin with Aristotle, who gave this famous answer to the question of what is truth and falsity: 31 To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true Aristotle, Metaphysics Book IV 66 An Introduction to Traditional Logic: Propositions

83 This is the classic formulation of the correspondence theory of truth. Truth rests on the conformity or correspondence of our mind with reality; in other words, truth is telling it like it is. When what is in one s mind agrees, harmonizes, or corresponds with reality, the judgment is true, and if not, then it is false. To signify something as true is to signify how it is in reality, while to signify something as false is to signify a thing otherwise than it is in reality. Truth rests on the conformity or correspondence of our mind with reality; in other words, truth is telling it like it is. How does this work? All judgments affirm or deny a predicate with a subject. We do this by uniting or dividing two concepts. If John is wearing a red hat, the simple concept red can truly be affirmed of the subject John s hat. In this case, saying, John s hat is red is true. If you see a red apple, you first apprehend it and then you can make a judgment about it by either affirming that the concepts apple and red are united by a copula is, which is expressed by the proposition, The apple is red ; or you can deny that the concepts red and apple go together in reality by saying The apple is not red. In either case, two concepts are either united or divided by using the verb for to be and it is this element that makes the statement a judgment about how things really are. Take another example: George Bush was elected President in 2004 Now the predicate of being elected President in 2004 is being affirmed of the subject George Bush. Because this affirmation agrees with reality (The real George Bush was really elected President in 2004) the proposition is true. Now an error is the lack of conformity between what is in the mind and what is in the world. Consider for instance: The United States is not a country Here the predicate country is being denied of the subject United States. But this denial of the predicate is not true. The United States really is a country, thus this proposition is false. So we can see how judgments affirm or deny predicates of their subjects and when this agrees with reality the judgment is true and when not, it is false. Now we can checkout many fact-claims by simple observation. In order to apply this correspondence test, we require only two things: the judgment we make about something and the real objects or events to which the judgment corresponds. If reality corresponds, the judgment is true; if not, the judgment is false. Remember, however, there are some judgments about which we are uncertain. When this occurs, the judgment is an opinion. Opinions occur because we are not certain whether or not the judgment is true or false. Propositions 67

84 But How Can There Be Truth When There is Disagreement? Reality doesn t change just because two people do not agree. Reality doesn t change just because two people do not agree. In other words, real things are what they are regardless of what anyone happens to think about it. The Earth is round, in spite of one s inability to convince another that it is not flat. The Earth doesn t become flat just because someone thinks it is. So even if the flat-earther doesn t know the Earth is round, or won t admit the Earth is round, or even if a poll taken indicates that most people believe the Earth is flat still the idea that the Earth is round is the true one and anyone who thinks the Earth is flat is flat wrong. Truth is grounded in reality, and what people happen to think about it doesn t change what is real. Truth is true because reality is what it is. So truth does not depend upon anyone else agreeing with it, knowing it, or admitting it. In sum, it is the actual existing thing is the ground for the verification of judgments. Things do not change just because opinions do. The Principle of Non-Contradiction The principle of non-contradiction; which states that a thing cannot both be and not be at the same time and in the same sense. From the correspondence theory of truth follows a fundamental principle. It is a commonly recognized fact that things exist in a certain way, and cannot exist in a way that is contradictory to that mode. For example, a window can be completely closed, or it may be completely open, or somewhere in between, but the window cannot be both closed and open at the same time and in the same way. The window simply cannot exist in both states at the same time. Likewise, a square cannot exist as a circle, nor can a dog be not a dog at the same time and in the same sense. From this fundamental and necessary truth about any existence, we get what is known in logic as the principle of non-contradiction; which states that a thing cannot both be and not be at the same time and in the same sense. It s not a law that logicians just made up out of the blue, and imposed on reality, rather it s a law derived from the way things really are. Of course this all sounds like common sense to most of us, and indeed it is. It is quite obvious, so obvious in fact that Aristotle called this principle self evident. In other words, it is a principle about which we cannot even be mistaken. Why? One cannot even deny the principle without assuming it. Think about it; to deny this is to say that a thing can both be and not be at the same time. So when they might say, I think the law of non-contradiction is false they really want that statement, that very denial of the principle, to be accepted as true. But if things can both be and not be, that very denial also can both be true and not true! It would be like saying square circle or my brother is an only child, the mere utterance of these phrases entails their falsity. Hence, that is why Aristotle rightly says one couldn t even deny this principle without assuming it, and when they do that, they speak nonsense. 68 An Introduction to Traditional Logic: Propositions

85 When it comes to truth, contradictory views are mutually exclusive. Only one of them can be correct, and we have a name for this rule in logic, it is called the principle of tertium exclusum, or better known as the principle of excluded middle. Given any pair of contradictories, one of the two must be true and the other false. Take the example of an atheist and a theist. The atheist says, God does not exist, and the theist says just the opposite, God does exist. Only one of these two can be right, but not both. God can t both exist and not exist at the same time. Between the atheist and theist, somebody must be wrong. When it comes to truth, contradictory views are mutually exclusive. Now in some cases it may be that we do not know which side is correct, but that does not mean we do not know that there is a truth to the matter. Knowing the actual truth of the matter and knowing that there is a truth to the matter are two different things. Take the following: 4356 x 345 x 8 x 345,686 = even or odd number? Without a calculator, most of us cannot say if the answer is even or odd. We may not know which one it is, but that the answer is either even or odd we do know. So even if we don t know whether the number is even or odd, we do know that it must be one or the other and not both. This will always be the case when the alternatives are mutually exclusive. If Bill says it s odd and Sally says it s even, we at least know somebody is wrong and somebody is right. So even when we don t know something, we must not confuse an inability to know the truth of the matter with there not being a truth to the matter. Self-Refuting Propositions If the content of a proposition violates the principle of non-contradiction, it is selfrefuting or self-contradictory. Propositions that refute themselves negate their own truth-value and so commit intellectual suicide and pull the rug out from underneath themselves. As soon as these types of statements are uttered they are false. If the content of a proposition violates the principle of non-contradiction, it is selfrefuting or selfcontradictory. Here are some examples of self-refuting propositions: All English sentences are only three words long. My brother is an only child. Every statement is false. (This one too?) There is no such thing as truth. (Really? Is that true?) There is no knowledge! (Do you know this?) Propositions 69

86 No one can know anything. (Then how do you know that no one can know?) The nerve of you! What gives you the right to judge other people? (If judging others is wrong, then why are you judging me for judging?) Don t impose your morality on others - that s wrong! (Is not this statement itself an imposition of morality?) Having seen now what is meant by truth and falsity and the fundamental principle of non-contradiction which follows, we now turn to analyzing the categorical proposition. The Structure of Categorical Propositions Categorical propositions are simple, that is, they consist of three parts; the subject, the predicate, and the copula. These three parts in addition to the things to which the subject and predicate signify make up the matter of the proposition. Propositions also have qualitative and quantitative considerations. We now look at each of these in detail. The Matter of Propositions The matter of a proposition can be of three kinds; necessary, contingent, or impossible. The matter of propositions consists of its terms and their relation to the copula, since the copula is the link that joins the subject and the predicate together. The matter of a proposition can be of three kinds; necessary, contingent, or impossible. Necessary or natural : This type of matter is when one term is included in the essence of another such as red is a color, man is an animal, square is a polygon with four equal sides. This type of matter is necessary in that such a proposition is necessarily true or true by definition. For example, it is true, by definition, that a square be a polygon with four equal sides. 32 Contingent: Contingent matter is when the predicate attaches accidentally to the subject. In other words, the predicate is not necessarily attached to the subject. In this case, the predicate is contingent in that it can be either 32 For a more advanced point, with necessary matter, either the predicate can appear essentially in the subject or the subject can appear essentially in the predicate. 70 An Introduction to Traditional Logic: Propositions

87 present or absent without the subject ceasing to be; e.g., Peter is just, the square is red, The apostle is old, etc. Impossible or incompatible : Impossible matter is when the predicate cannot be attached to the subject because it is naturally repugnant to it; e.g., man is a plant, the square has three sides, this color has no surface, etc. The matter of propositions is primarily a concern for material logic. So quite often in formal logic, one is concerned with only formal validity and so variables are used to stand for the subject and predicate. This is done so the logician can focus more easily on the form of the proposition. Thus, every dog is mortal becomes simplified to Every S is P which is just shorthand for Every subject is predicate. 33 The Quality of Propositions: To Affirm or Deny Understood on the basis of quality, a proposition is said to affirm or deny. The subject and predicate can be united or divided by using a form of is or is-not. An affirmative proposition is one that unites the predicate with the subject and a negative proposition denies the predicate of the subject; e.g., The apple is red is a proposition with an affirmative quality and The apple is-not red is a proposition of a negative quality. A negative proposition may also be expressed by saying No apples are red. Even though the negative element no does not appear next to the copula, still the proposition expresses the idea that the predicate red is separated from the subject apples. An affirmative proposition is one that unites the predicate with the subject and a negative proposition denies the predicate of the subject. One must remember that propositions derive their affirmative or negative quality from the copula verb and not some other verb in the proposition. A proposition may contain more than one verb but only the copula unites or divides the predicate from the subject. Take this proposition with complex subjects and predicates, He who is not with me is against me. The subject in this proposition is the phrase, He who is not with me. 34 Notice there is a verb in the subject, and it is even the verb is not, but this first verb is not the copula. It is the second is in this example that unites the predicate (one who) is against me. So in spite of the appearance of the is not in the subject, this proposition is nonetheless affirmative. Note also that the presence of negative terms, that is, negated subjects or predicates, do not make the proposition have a negative quality. If one were to say, All non- Christians are people needing baptism, we should recognize that this proposition is affirmative. It unites the predicate people needing baptism with the subject non- Christians. 33 Although we do discuss material logic in this text to some extent, nevertheless the focus is on formal logic. 34 Remember, concepts may be simple or complex (see ch. 2). Both the subject and predicate in this example signify complex concepts. Propositions 71

88 The Quantity of Propositions: How Much? So the quantity of a proposition simply refers to how much (the extension) of the subject a given proposition is talking about. The quantity of a proposition is universal if it is refers to all of the subject. Particular propositions, however, limit their scope by talking about only a portion of the subject. Propositions with singular subjects refer to only one individual. Think of the different ways in which one can talk about triangles in terms of how many. One could discuss a geometrical formula that would apply to all triangles, past, present, and future, one could talk about just isosceles triangles, or one could talk about just this particular triangle in front of them which is painted red. So the quantity of a proposition simply refers to how much (the extension) of the subject a given proposition is talking about. Strictly speaking, the quantity of a proposition can be universal, particular, or singular. The quantity of a proposition is universal if it is refers to all of the subject. Universal propositions are exhaustive; they refer to the entire populations which they represent. For example, Every man is mortal is a universal proposition because it refers to every man, even possible men that may appear in the future. Likewise, No cats are things that can fly is also universal as it refers to each and every cat. 35 Particular propositions, however, limit their scope by talking about only a portion of the subject, as in saying Some dogs are animals that can swim. Propositions with singular subjects refer to only one individual, e.g., Fido can swim. In logic, singular subject propositions are usually treated as universal; Socrates is mortal means the whole of Socrates and so the subject is not restricted like in particular propositions. The subject is a singular concept, and a singular concept taken in all of its extension (all of only one!). 36 So while strictly speaking there are three types of quantity, nevertheless since singular subject propositions function for the most part the same way as universal propositions do. Later however we will see that there will be some special treatment given to singular subject propositions. Of course a quantity could also be vague or indefinite. Indefinite quantities could go either way. With these propositions, it is not explicit how much of the subject is meant. Books are useful, usually means only some books while Dogs pant usually refers to all dogs and hence is universal. There is no mechanical means of figuring out indefinite quantities in advance, so context and intent of the speaker is important here, but in the end they are either universal or particular and so again do not constitute a different type of quantity. 35 Remember the amplification or restriction properties of terms can modify a term so that a universal term may still be restricted in a sense. For example, All the animals on Noah s Ark were hungry is still a universal proposition, even though technically it is called incompletely universal since it is limited to the animals on the ark. 36 This is not to say that singular propositions are always treated the same way as universal propositions. Exceptions will be discussed in the chapters to follow. 72 An Introduction to Traditional Logic: Propositions

89 The Four Types of Categorical Propositions To simplify things we now use the variables mentioned above; S for the subject and P for the predicate. Now when we combine the two types of quality (affirmative and negative) with the two types of quantity (universal and particular 37 ) we get four basic types of categorical propositions: Universal Affirmative Every S is P Universal Negative No S is P Particular Affirmative Some S is P Particular Negative Some S is not P A note must be added regarding the proper form of the universal negative No S is P. We do not say, Every S is not P because that is ambiguous. For example, it could take the form Every woman is not a bad driver meaning only that there are some exceptions. Here one would simply mean that it is not the case that all women are bad drivers, because Shirley, Jane and others happen to be good drivers. Our common manner of speaking usually intends this statement to mean only that there are exceptions. So as a simple rule of thumb, all factors being equal, Every S is not P translates into the particular negative, Some S is not P. However, one could possibly mean by this statement a stronger form like All women are not bad drivers. With the emphasis on the all the speaker may intend to mean each and every woman driver. To avoid this ambiguity, logicians prefer the formulation, No S is P to stand for the universal negative. No women are bad drivers gets the universal point across much more clearly. The scholastic logicians used vowels from the Latin affirmo (I affirm) and nego (I deny) to stand as an abbreviation for the four types of categorical propositions. The first two vowels in AffIrmo ( A and I ) stand for universal and particular affirmations while the first two vowels in nego ( E and O ) stand for universal and particular denials: A stands for a Universal Affirmative proposition E stands for a Universal Negative proposition I stands for a Particular Affirmative proposition O stands for a Particular Negative proposition 37 As mentioned, for now we are treating singular quantity propositions as universal. Propositions 73

90 These letters standing for the different types of propositions are a shorthand method of referring to them. We will often refer to this shorthand, especially in the next chapter when the square of opposition is covered. Reducing Propositions to the Four Forms The reason why logic uses these four types of proposition is that a) most statements will fit into one of these four and b) when placed in this logical form, relationships and inferences can be made with relative ease. When we reduce utterances to one of the four types, the meaning of statements becomes more explicit and the rules governing inferences about them is much more simplified than if not reduced to these forms. Thus, there is a great advantage in reducing utterances to one of these four types of propositions. This is not to say that all statements can be reduced to these four. Although most statements susceptible to truth and falsity are capable of being put in this way, human language is capable of producing other utterances that will not fit into A, E, I, and O statements. Modal propositions for example (statements that deal with things that are necessarily, possibly, impossibly, or contingently true), will not reduce to these four. However the study of modal propositions belongs to modal logic, which is more advanced and beyond the scope of this book. Distribution Distribution refers to a term taken in all of its extension. When a term applies to each and every member, the term is said to be distributed. Distribution refers to a term taken in all of its extension. When a term applies to each and every member, the term is said to be distributed. A term is distributed if it claims to know something about all of the members of a certain category. If the term only refers to a part of its extension, that is if the term applies to only some of the members, the term is said to be undistributed. Both the subject and the predicate can be either distributed or undistributed. Four Rules of Distribution The role of distribution is of great importance when it comes to determining the validity of arguments. Because of this, the following rules should be memorized: 1. A universal proposition always distributes its subject 2. A particular proposition never distributes its subject 3. An affirmative proposition never distributes its predicate 4. A negative proposition always distributes its predicate 74 An Introduction to Traditional Logic: Propositions

91 Distribution Rule #1: A Universal Proposition Always Distributes its Subject It is clear that universal propositions always refer to all of the subject. To say, Every student is a good logician is clearly talking about all of the students. The distribution here covers each and every student. Likewise if one says, No students are good logicians again we are talking about the entire group. Distribution Rule # 2: A Particular Proposition Never Distributes its Subject In a similar way, it is also quite clear that particular propositions are only talking about a part of the subject and not the whole. Some snakes are dangerous is not speaking of all snakes, but only a subsection of things known as snakes. Determining the distribution of predicates is not as clear as determining the distribution of subjects. This is because there is not a quantifying term like all sitting there right in front of the predicate to tell us about its distribution. So keep in mind that these next two rules apply to the distribution of predicates: Distribution Rule #3: An Affirmative Proposition Never Distributes its Predicate In an affirmative proposition, whether universal or particular, we are only speaking of part of the predicate. For example in, All birds are things that fly, or Some birds are things that fly, we are not speaking exhaustively of things that fly. The extension of the predicate is only partial. There could be other things that fly that are not birds (say bats for example). So in any affirmative proposition, the predicate is only covered partially and thus is undistributed. Distribution Rule #4: A Negative Proposition Always Distributes its Predicate In negative propositions, the predicate is universally divided from the subject. This means that any time you have a negative proposition, you will always be talking about all of the predicate. All negative propositions take either the form, None of these things are any of those things, or if particular, Some of these things are not any of those things. Thus, the predicate of negative propositions is always distributed because whenever this occurs you will always be talking about each and every one. When someone says, No rats are things that fly, the predicate is distributed in its entirety. In other words, out of the whole category of things that fly, you will not find any rats. Take another example, Some animals are not mammals, says there are some animals that are not any of those mammal things. In this case, the predicate mammals is distributed because part of the subject is excluded from everything that falls under the category of mammals. To abridge the distribution of predicates we can say; an affirmative proposition never distributes its predicate, while a negative proposition always distributes its predicate. Propositions 75

92 Now when we apply these rules to the four types of propositions, we get the following distribution schema: Supposition Supposition is the substitution of a term for a thing. Since propositions always contain the verb for to be, they will always refer to some form of existence. When terms are used in a proposition, they refer to something as existing, and so it can be said that the term acquires a new property when placed in a proposition. The property by which we refer to the existing thing or things we are talking about in a given proposition is called supposition. Simply speaking, supposition is the substitution of a term for a thing; and this substitution must be done in conformity with the requirements set forth by the copula. Supposition is a property belonging to terms only as they occur in a proposition. Since propositions always contain the verb for to be, they will always refer to some form of existence. Now existence can be used in many ways, we can talk about mental existence (beings of reason existing only in the mind like Santa Claus ), real existence, or past and future existence. So the copula verb to be has a variety of meanings depending on the intention of the speaker. So technically speaking supposition is a property of terms, but we discuss it here because it is a property that only arises when a term is joined with a copula. 38 This means we must not confuse signification with supposition. Terms can signify and have meaning on their own without being joined with a copula (that is the whole point of definition!). Take the term man for example. This term alone signifies and has meaning, it means rational animal, but when we add the copula is to it, the new property of supposition comes into play because now we are not just saying man but man is, in other words, an existing man of some sort. Hence, the supposition of a term (referring to the existing object) is not to be confused with the signification 38 Propositions have properties, some of which are of its parts (supposition, ampliation, restriction) while other properties, as we will see in the next chapter, apply to the whole proposition. 76 An Introduction to Traditional Logic: Propositions

93 of a term (referring to meaning), which the term has outside of a proposition. To state the difference succinctly: Signification that from which the meaning of the term springs (comes from the first act of the intellect; an essence or meaning ) Supposition that to which the term is given (comes from a judgment; the things to which the term applies in a proposition) In fact, supposition presupposes signification. The term man signifies an intelligible content first, and then next when one says man is we already have an idea of what is because the term man had meaning to begin with. So in short, concepts first signify essences, and then secondarily, when placed into a proposition, they supposit certain things as existing. 39 There are three types of supposition: Supposition presupposes signification. Material supposition is when a term supposits for itself. For example, in the statement, Man is a three letter word, the term man supposits simply for itself. Simple Supposition is when a term supposits for what it primarily and immediately signifies, not mediately. This type of supposition is simple in that it stops with the immediate essence and doesn t pass on to individual things. For example in, Man is a species, the term man supposits by simple supposition. Man in this proposition stands only or simply for human nature without passing on to the individuals that have that nature. Personal Supposition is when a term supposits for actual individuals, i.e., those things that are signified mediately. It is called personal because it carries through or extends to the individual instances of some nature. To say, Every man is an animal, every man supposits first for the essence of man but then goes on mediately by extending to any individual that has this nature The implication of the copula as to the mode of existence we are talking about is very relevant to logic. Since supposition is the use of a term in such a way as to make it stand for something existing, it can be the case that sometimes the thing we are 39 The astute reader may wonder how supposition differs from extension. Extension refers to what is potentially supposited, while supposition refers to what is actually supposited. This distinction is traceable to the difference between signification and supposition. A term has extension prior to being used in a proposition, while supposition refers to something as existing. No mere term in isolation can do this. Outside a proposition, it is impossible to tell what a term supposits for, e.g., human has extension but no supposition until put into a proposition. Propositions 77

94 talking about does not exist in the way intended by the copula. Take this proposition for example: Aristotle will be a great philosopher. Now this proposition is false, but what makes it false? It is not because the predicate does not apply to the subject, after all great philosopher is a correct characteristic of Aristotle. What makes this proposition false is that it fails to supposit for anything. Aristotle will not be anything because he is no longer alive. 40 The same is true here: World War IX was a terrible episode in human history. Of course, there hasn t been a World War IX but because this proposition does not supposit for anything, the statement is false. This brings us to our first rule of supposition: Supposition Rule 1: Given any affirmative proposition, if the subject of the proposition does not refer to anything, the proposition is false This is a static rule of supposition which simply says no supposition means no proposition. For a proposition to be legitimate, the substitution of a term for thing must fit what is intended by the verb to be. A proposition about nothing is not really a proposition. It s like saying, is worthless or is bald. These are not propositions because they do not really have a subject. With this in mind, we can say that a proposition can be false in two ways; 1) the normal way, namely because the predicate does not apply to the subject in the way stated or 2) because the subject does not exist in the way demanded by the copula. This rule does not apply to negative propositions, for these propositions may be true even if the subject does not exist (because they do not exist), e.g., Socrates is not a President of the United States is true because Socrates does not exist and cannot be a President unless he does. Also, this does not mean that one cannot intend to talk about beings of reason. One can legitimately say, Santa Claus is a big man that wears a red suit, because the is here simply refers to a mental existence. 40 Of course I am leaving aside theological and philosophical questions about the immortality of the soul and bodily resurrection. 78 An Introduction to Traditional Logic: Propositions

95 It should go without saying that simply abiding by this first rule of supposition does not then mean the proposition is true, only that it will really stand for something. Yet, there is a second aspect of supposition we have to consider. Suppose now we have an argument with two propositions that check out all right with regard to our first rule of supposition, as in this classic: Man is a species Socrates is a man Therefore Socrates is a species Something is clearly wrong here, yet it is not the signification of the term man, nor is it the supposition of the premises in themselves. The problem here is that the first premise does not use the same type of supposition as the second. In Man is a species, man supposits with simple supposition, but in Socrates is a man, the term man supposits with personal supposition, and to make a shift in supposition like that invalidates the conclusion. Take a similar example from Peter of Spain: Therefore this does not follow: every animal other than man is irrational, therefore every animal other than this man is irrational but in fact this is the fallacy of the figure of speech, by proceeding from simple to personal (supposition). 41 You can see the same fallacious inference here again. The inference shifts from talking about man as a nature to an individual man, once more shifting from simple to personal supposition. To make this point more clearly, take one more example: Animals exist A unicorn is an animal Therefore a unicorn exists The supposition in the first premise indicates a real existence (personal supposition), but the supposition in the second refers to only a mental existence by talking about the nature of a unicorn (simple supposition), yet the conclusion infers real existence! This shift in supposition is invalid because the conclusion of real existence does not follow from the premises. To jump from one type of supposition to another in the same argument is fallacious. Thus to guard against this error, we have a second rule: Supposition Rule 2: An argument is invalid if the type of supposition varies from premise to conclusion 41 Peter of Spain, Summulae Logicales, VI.6 Propositions 79

96 This is the dynamic rule of supposition, and it is needed because the first rule could be met (both proportions refer to something via a certain mode) yet moving from these premises to a conclusion is invalid when the type of supposition shifts somewhere in mid-argument. We turn now to see how to translate ordinary language into proper logical form. Putting Everyday Language Into Proper Logical Form 42 There is a great advantage to reducing human statements susceptible to truth and falsity to one of the four types of propositions. Proper logical form is the result of rewording a statement in such a way that highlights its cognitive aspects, making it suitable for logical inference. We said above that there is a great advantage to reducing human statements susceptible to truth and falsity to one of the four types of propositions. When we reduce propositions in this way it means we are putting them in proper logical form. Proper logical form is the result of rewording a statement in such a way that highlights its cognitive aspects, making it suitable for logical inference. When we put statements in proper logical form, it clarifies what is being said and it allows us to make inferences with more accuracy and ease. When placed in proper form, we can then use the rules of validity that are based on that form and so will apply to any instance of that form. Thus it is quite advantageous to be able to have rules regarding certain molds or forms of statements because these rules abstract from and are not bogged down in the actual content or matter of the propositions. So when we have logical rules regarding say the A proposition, Every S is P, those rules will apply to any content that is placed in this abstracted form. In other words, rules about the A proposition will apply to anything we put in place of the variables S and P. It does not matter if we mean, Every man is funny, Every alligator is vicious, Every logic class is boring or what have you. The rules apply to the form, Every S is P and so will apply to any utterance that can be stated in that form. So we cannot stress enough the importance of both understanding the rules regarding the forms of propositions and then being able to reduce everyday language to one of these forms. Take for example the simple statement, Birds fly. How could we state this in one of the four types of propositions? After all, as it stands now there is no copula and the predicate is incomplete since it consists only of the verb fly. Now if we grant the speaker the benefit of the doubt and allow that he knows about penguins and ostriches, we can say that the statement is implicitly referring to most birds, which means it is particular and affirmative. Thus, we can use the quantifier some and by turning the verb into a noun phrase and making the tacit copula explicit we get, 42 Much of the material in this section is indebted to Veatch, op. cit., ch An Introduction to Traditional Logic: Propositions

97 Some birds are things that fly. In this case, the logical equivalence of Birds fly is Some birds are things that fly. The second is in proper logical form because all the implicit elements that are relevant to logic are now explicitly stated. Now not every case is that easy. Human language is a highly versatile tool capable of a wide variety of distinctions and nuances, and in fact can be slippery at times. Remember, language consists of artificial signs whose connections can be as changeable as human wills. So given that there is a notable advantage to putting ordinary language in proper logical form, we must be aware that this task can be tough. Yet do not be daunted, for the study of logic provides us with some guidelines on this issue. We begin this task with the primary rule of all logical reformulation: we can only change the wording of a statement, not its meaning. When putting everyday language in proper logical form, it will not do to change the meaning of a statement. Statements reduced to proper logical form reword something only in a way that preserves its original meaning. Otherwise, one ends up with a different statement than the one they started with. This point is so important it is worth repeating: The primary rule of reducing statements to proper logical form is: change only the wording, not the meaning. With this rule always in mind, we now will look at some details on how to express common language into what is logically proper. So the emotional or noncognitive aspects of a statement are not relevant here. It is true that some sentences may be regarded as poetic, ceremonial, or propagandistic, but for logic we abstract only the aspects having to do with knowledge. The Procedure of Propositional Formulation The procedure seems to work well for most people is to isolate and make the elements of a proposition explicit, and this is done in three steps: 1. Isolate the S and P 2. Identify the copula and its quality 3. Identify the quantity Propositions 81

98 Of course, many propositions can be formulated without following these steps, but since the more complicated ones are not so easily formulated without them, you should make a habit of always following these steps in their particular order. We now explain each step in more detail. Step 1: Isolate the S and P Given any proposition, isolate the subject and predicate and put it in brackets or parentheses. This will help you immensely in determining what type of proposition you are dealing with. So a simple example like, These apples are bright red things, comes out as, (These apples) are (bright red things), by clearly bracketing off the subject and predicate. Now often it is the case that the predicate is not stated in a way suitable for logic. Proper logical form requires subject and predicate be a noun, noun-phrase, or pronoun. Look at this example for instance: The Vikings were big and nasty. As stated, the predicate big and nasty is not explicitly a noun or equivalent, but can easily be made into such by adding those who were big and nasty. This addition only makes explicit what was already implicit and so the reformulated proposition, looks like: (The Vikings) were (those who were big and nasty). Predicates then are often restated and made into nouns or noun phrases by adding words like that which, that which is, one who, one who is, those who, those who are, etc. Since this does not alter the meaning but only clarifies what is already intended in the predicate, this addition may be necessary for proper logical form. Step 2: Identify the Copula and its Quality This step is rather straightforward. To identify the copula and its quality is simply to find the verb is or any of its variations and then see if its quality is affirmative (is) or negative (is not). Sometimes the copula is already explicit but quite often it is only implicit and needs to be made explicit. So if we may pick on our Nordic friends again: Big and nasty Vikings came from the sea. Given what we said about the predicate needing to be a noun, pronoun, or noun phrase, we can correct the predicate to those who came from the sea and that would show the need for the copula: (Big and nasty Vikings) (those who came from the sea.) 82 An Introduction to Traditional Logic: Propositions

99 Now all we have is a subject and predicate with no copula, but that is quite all right because the copula was implied all along. All we need to do is add it and we have: (Big and nasty Vikings) are (those who came from the sea.) Of course we could use were and have the same effect (since this is just the past tense for the verb to be ). The point here is that if the copula is not explicitly there, it needs to be so in order to have proper logical form. Since repetition is the mother of learning, we can best show how to derive proper logical form with more examples: 43 I helped Joe I didn t hurt Joe They will harm Joe You won t tell Joe He works for Joe She doesn t like Joe (I) am (one who helped Joe) (I) am-not (one who hurt Joe) (They) are (ones who will harm Joe) (You) are-not (one who will tell Joe) (He) is (one who works for Joe) (She) is-not (one who likes Joe) Notice that in these examples the not is connected with the copula by a hyphen, and this is a good way to show that the copula and therefore the whole proposition is negative and not the predicate. Notice however that some of these propositions could be put another way without changing their meaning. For example, (She) is-not (one who likes Joe) is equivalent to (She) is (not one who likes Joe). It really does not affect the meaning if the not is put with the copula or with the predicate. Nevertheless it is probably easier to put it with the copula when other parts of the predicate are negative, since too many nots in one predicate can be confusing. Step 3: Identify the Quantity As we have said, the quantity of a proposition refers to how much of the subject we are talking about. Proper logical form seeks to discover the quantity in an expression and indicate it at the beginning of the sentence. For example the statement, Virtuous acts are good things says that all virtuous acts are good things (how else could they be virtuous?) and thus this is a universal affirmative proposition and reformulated as All (virtuous acts) are (good things). As in this example, we can see that sometimes the quantifier is disguised and so must be made clear and explicit. This should not be too difficult since there are only three choices of quantifying expressions; Every, no, and some. Taking another example: He who lives by the sword, dies by the sword. 43 See Veatch, op. cit., p. 156 Propositions 83

100 Here the he seems to suggest anyone and everyone who happens to be living by the sword, and thus this would be a universal affirmative A proposition (Every S is P) which when reformulated looks like: Everyone (who lives by the sword) is (one who dies by the sword) Now there are some words that serve as tip-offs as to the quantity of a proposition. We use many qualifying expressions in English indicating all or some and so there are sometimes clues as to what the quantity might be in a proposition. We call these universal and particular quantifiers, and here are some examples of each: Universal Quantifiers: All - All Greeks are wise. Everyone Everyone who is Greek is wise. Any Any Greek is wise. He who He who is a Greek is wise. One who One who is a Greek is wise. Whoever or Whosoever Whoever is a Greek is wise. Whatever or Whatsoever Whatever is a Greek is wise. Never Greeks are never wise. No No Greek is wise. None None of the Greeks are wise. Nobody Nobody who is Greek is wise. Particular Quantifiers: Some Some Greeks are wise. Several Several Greeks are wise. Often Often Greeks are wise. Frequently Frequently Greeks are wise. Usually Usually Greeks are wise. Sometimes Sometimes Greeks are wise. Most Most Greeks are wise. A few A few Greeks are wise. The point here is that when you see universal quantifiers like any, whosoever, or nobody, etc., it is highly probable that you are dealing with a universal proposition of some sort, and the same is true for the particular quantifiers. Remembering and recognizing these quantifiers will be of great help in determining the quantity of a proposition. Yet we must always be careful and avoid using a mechanically rigid way of interpreting language. The intention of the speaker is always the key here. Even with what is usually a particular quantifier like few, still it might indicate a universal proposition. Take the proposition, Few men are perfect. Now when someone says this, 84 An Introduction to Traditional Logic: Propositions

101 they probably do not mean to assert that some men are perfect, in fact the speaker most likely means to say that most if not all are not perfect. So notice the simple difference from saying, Few men are perfect (possibly implying a universal negative E proposition No men are perfect ) and A few men are perfect (a particular affirmative I Some men are perfect ). A few always indicates an I proposition while few indicates an O or even an E proposition. The simple addition of a one-letter word can make a huge difference in the meaning. As we said, it may be the case that the quantity cannot be determined. When the quantity is unclear, we have an indefinite proposition. Birds fly, Dogs bark, French people are bad drivers etc. do not have an explicitly stated quantity. In these cases, the logician makes use of context, common sense, and perhaps further interaction with the speaker in order to determine the quantity. The point here is that we must never forget the primary rule in translating everyday language into proper logical form, the wording can be changed but not the meaning. To do this accurately, one must ask and re-ask; what precisely is it that the speaker intends to say in this sentence? There is no predetermined logical rule or mechanical replacement for this reflection. Difficult Expressions There are common ways of linguistic expression that are sometimes difficult to capture in logical form. One type takes the form of saying Not every. If we say, Not every student likes logic, what are we really saying in terms of the four main propositions? Not every student likes logic does not mean that no student likes logic. The claim is not that strong, rather it seems to say that only some students do not like logic and does not make any claim about all students. This means that this sentence is an O proposition and when put logically would read something like, Some students are-not ones who like logic. The same is true for all are not. All students are not ones who like logic, really only means that some students do not like logic. So Not every and All are not are roughly equivalent, they both indicate O proposition. 44 Exclusive and Exceptive Propositions Scholastic logicians have noted other phrases that fit into the discussion here. An exclusive proposition is a term that explicitly excludes other things, such as when terms contain words like alone, only, etc. An exceptive proposition is simply a 44 The technical reason for this is because in these cases the not functions to contradict an A proposition, and the contradictory of an A is an O. Propositions 85

102 proposition with terms stating an exception, as when terms contain words like, except, besides, etc. In these cases, there is a word that excepts or excludes a thing or group of things from a subject, which means we are really dealing with two groups, the fundamental or unexcluded group and the group that is excepted or excluded from it. When we formulate a proposition for each group, we get two; an excluded proposition (the proposition containing the excluded group) and the unexcluded proposition (the proposition containing the unexcluded group). 45 Take these two examples: Only the Spartans are tough fighters. Everyone except the Athenians were ignorant of philosophy. Because exclusive and exceptive propositions are really dealing with two subjects, they are in reality two propositions in one and so break down into two categorical statements united by the word and. Doing this involves a three-step process: Step 1: Formulate the excluded proposition (the one containing the excepted or excluded subject) by using a negation like non- or not. The first example excludes non-spartans and the second excludes non-athenians. Using this rule then we derive, No non-spartan is a tough fighter, for the first example and, Every non- Athenian was ignorant of philosophy, for the second example. Step 2: Formulate the unexcluded proposition (the one containing the unexcluded subject). The unexcluded subject in the first example is Spartans and in the second is Athenians. So from this rule we get, Spartans are tough fighters and The Athenians were not ignorant of philosophy. Step 3: Unite the propositions from the first and second steps with the conjunction and and we end up with the following: Only the Spartans are tough fighters, reformulates to, Spartans are tough fighters and no non-spartan is a tough fighter. Everyone except the Athenians were ignorant of philosophy, reformulates to, Every non-athenian was ignorant of philosophy and the Athenians were not ignorant of philosophy. Again, these exceptive and exclusive propositions are really two propositions in one and so when reduced to proper logical form actually form a compound proposition, the rules of which will be a topic for the next chapter. For now all the student should know is that these expressions call for two conjoined propositions. 45 What we are calling the unexcluded proposition; John of St. Thomas calls the fundamental proposition. 86 An Introduction to Traditional Logic: Propositions

103 Now of course it may be the case that the exception or exclusion occurs in the predicate, but the same rule applies. Two conjoined propositions are still called for; one that attributes the predicate to the subject and one that handles whatever was intended by the excepted predicate, e.g., Every Athenian was ignorant except for philosophy reduces to Every Athenian was ignorant of non-philosophy and no Athenian was ignorant of philosophy. If you find this last section difficult, do not be troubled. Figuring out the semantics of these sorts of phrases and putting them in proper logical form can be very difficult task and in fact is perhaps one of the trickiest aspects of logic. Different scholastic logicians had different distinctions and solutions to the matter which go beyond this text. So while we have not by any means covered all the possibilities and solutions, we have provided some of the basics sufficient for an introduction to logic. Chapter 5 Summary Judgments unite or divide concepts by means of a copula. Expressed judgments are propositions. Only judgments and the propositions expressed by them can be true or false. Truth is an agreement between the mind and reality, and thus a discord between mind and reality results in falsity. This is the correspondence theory of truth. The truth-value of a proposition can be true, false, or undetermined. A disagreement over an issue does not mean there is no truth to the matter. The principle of non-contradiction holds that a thing cannot both be and notbe at the same time and in the same way. Between positions that are mutually exclusive, the principle of excluded middle says one must be false and the other true. The matter of propositions can be necessary, contingent, or impossible. The quality of propositions is affirmative or negative, and the quantity can be either particular or universal. Propositions with a singular subject are very often treated as universal statements. The quantity of propositions with indeterminate or indefinite subjects can be ascertained by context, common sense, or clarification by the one making the proposition. The four main types of categorical propositions are universal affirmative (A), universal negative (E), particular affirmative (I), and particular negative (O). When a term claims to know something about all its members, the term is distributed, and if not, the term is undistributed. Propositions 87

104 The property by which terms refer to an existing thing or things in a given proposition is called supposition. Supposition can be simple, material, or personal. Putting everyday language into proper logical form is very helpful because logical rules of inference apply to any instance of those forms. The primary rule for this procedure is that one can change the wording but not the meaning. Exercises A. Recognizing Self-Contradictions: signify which of the following are selfrefuting: 1. All truth is relative to the individual 2. Nobody knows anything 3. Some knowledge comes from sensation 4. You shouldn t judge others! 5. I like chocolate ice cream 6. Abortion is morally wrong 7. All English sentences are only three words long 8. Don t impose your morality on others! 9. The war in Iraq was not a just war 10. Sometimes truth is relative B. Determine whether the following propositions are of the type A, E, I, or O. Put in proper logical form where necessary: 1. Every dog has ears. 2. Some theologians are heretics. 3. Peter is very intelligent. 4. No person should get sick from this food. 5. Caesar was an emperor. 6. Not all dogs bite. 7. Few snakes climb trees. 8. Every good boy does fine. 9. No bad guy gets away with his evil deeds. 10. Virtuous acts make a man good. 11. The will is a human faculty. 12. Socrates taught the people at the marketplace. 13. Not all doctors are incompetent. 14. Triangles have three sides. 15. Americans are tired of this policy. 16. There are those who would disagree with that statement. 88 An Introduction to Traditional Logic: Propositions

105 17. The family that prays together, stays together. 18. Some of the players were worn out after that game. 19. Frequently it happens that one is punished when rules are broken. 20. Anyone who thinks that is crazy. 21. Whoever calls me better have a good reason. 22. Fallacies are never good arguments. 23. Usually a woman takes a security officer with her. 24. All are not happy with the new building. 25. Some of those fellows like to go to the beach. C. Give the supposition type of the term in italics: 1. The genus is a part of the concept. 2. Angel is a five-letter word. 3. Every man is a type of animal. 4. Some dogs are half crazy. 5. Human is a universal concept. 6. Rottweiler is a German term. D. Reformulate the following into a compound proposition with proper logical form: 1. Everyone except the Jones had a good time at the party. 2. Everyone save for Bill was laid off today. 3. Only women may enter this restroom. 4. Priests alone were allowed to enter the temple. 5. Nobody besides those two got away. Propositions 89

106

107 Chapter 6: Propositional Properties & Compound Propositions àà Overview of Propositional Properties àà Opposition àà Conversion àà Obversion àà Compound Propositions Overview of Propositional Properties Categorical propositions have properties that allow them to be immediately related to other propositions. This means that the truth or falsity of one proposition can imply the truth or falsity of another. It is simply a matter of common sense that if, All birds are things that lay eggs, is true, another proposition that says, Some birds are things that lay eggs, must also be true, and the proposition, No birds are things that lay eggs, must be false. Most people who have never even studied logic intuit these sorts of implications and in fact make them all the time. This is a very useful The truth or falsity of one proposition can imply the truth or falsity of another. Propositional Properties & Compound Propositions 91

108 technique. One can make themselves more clearly understood by saying the same thing in a different way and logic gives us some rules for doing this through the various properties of propositions. Just as terms have properties like comprehension and extension, so too do whole propositions have their own properties and this chapter covers three of them; opposition, conversion, and obversion. Opposition Opposition is a property of propositions that comes from the different ways of affirming and denying the same predicate to the same subject. Simply speaking, opposition is a property of propositions that comes from the different ways of affirming and denying the same predicate to the same subject. In looking at the four types of propositions (A, E, I, O), we can see they have the same terms, yet when they differ in quality or quantity, they may differ in truth-value or they may not. As a result, there are relative truth-values between the four types of propositions. For example, if it is true that, All dogs like to run, then the corresponding E using the same terms which says, No dogs like to run, must be false. The relations of opposition are best seen on a diagram known as the Square of Opposition. The square of opposition covers all four logical propositions and shows their various relationships in diagrammatic form. Since there are four types of propositions, each corner of the square indicates each type as you see here: 92 An Introduction to Traditional Logic: Propositions

109 Note that universal propositions occupy the top part of the square and particular propositions the bottom. Affirmative propositions occupy the left half of the square, while negative propositions occupy the right half. The various opposing relationships between the four propositions; contradictory, contrary, subcontrary, and subalternation are also illustrated. We will now discuss the details of these different types of opposition, moving from the strongest to weakest. Contradictory Opposition We saw in the last chapter with the principle of non-contradiction, that it is impossible for two contradictory statements to both be true and it is also impossible for them to both be false. This type of opposition is represented on the diagonal lines of the square (from Every S is P to Some S is not P, or from, Some S is P to No S is P ). In contradictory opposition, one proposition must be true and the other must be false. Contradictory opposition is an absolute denial of what another proposition asserts, which means that this denial of one proposition to another occurs both in the quality and quantity. The opposition here is the most drastic and the result is an exclusive either/or situation. In contradictory opposition, one proposition must be true and the other must be false. Let s use an example to clarify. If it is true that, Every chair is red (A), then it must be false that, Some chair is not red (O). In other words, if A is true, O must be false. The reverse is also the case, that if, Some chair is not red, is true, then of course it cannot be the case that, Every chair is red. This is just common sense. We all know that if there is some chair (say in your dining room) that is not red, it cannot be the case then that every chair in there is red. So if O is true then A must be false. You can also make this inference when you know that a proposition is false. So if it is false that, Some chair is not red, then it must be the case that, Every chair is red. So from the falsity of O you can infer the truth of A, and vice versa. These same conditions also apply to the E and I propositions since they too are contradictories. The good thing about contradictory opposition is that it never leaves us undetermined regarding the truth-value of the opposed proposition. It is the strongest type of opposition because exactly one and only one of two contradictory propositions must be true and the other false. Contrary Opposition This type of opposition occurs between the A and the E (between the two top corners of the square). In contrary opposition, both contraries cannot be true, but both In contrary opposition, both contraries cannot be true, but both can be false. Propositional Properties & Compound Propositions 93

110 can be false. Do not confuse contrary opposition with the contradictory opposition just discussed. Contrary opposition is not purely an either/or situation. Contrary opposition is an opposition within the same universality (both the A and the E are universal propositions), and so there is a middle ground between these extremes. Now if one extreme is true, the contrary must be false. If, Every chair is red, is true then the contrary, No chair is red, must be false. Yet both can be false at the same time. For example, if it is false that, Every chair is red, this falsity does not necessarily entail the truth that, No chair is red. Just because it is not true that every chair is red does not mean that no chairs are. After all, maybe just a couple chairs are red and the rest not! So from the falsity of the first contrary, the truth of the other could be either true or false, and we cannot infer which it is just from the falsity of the first. Hence we say in logic, that from the falsity of one contrary, the other is undetermined. It is important to keep this point in mind. In contrary opposition, we can only make a truth determination based upon the truth of one contrary. If one is true, the other must be false but both can be false. Subcontrary Opposition In subcontrary opposition, both cannot be false, but both can be true. This type of opposition occurs between the two bottom corners of the square (I and O). In subcontrary opposition, both cannot be false, but both can be true. This rule for subcontrary opposition is just the opposite of the rule for contrary opposition. This is because subcontrary opposition is an opposition between particular propositions, and so it is always possible that the particular subjects may be referring to different parts of the same whole. Some chairs are not red and Some chairs are red, can both be true if each refers to a different part of the whole group of chairs. There is nothing wrong with saying some chairs are red and some are not as long as we are not talking about the same some each time! So if one subcontrary is true, this only leaves the other undetermined. Yet these subcontraries cannot both be false because the falsity of one implies the truth of the other. If it is false that, Some chairs are red, then it must be true that, Some chairs are not red. So with subcontraries, if one is false the other must be true, but both can be true A technical point for the more interested: subcontrary opposition is not opposition in the strict sense, because two particulars may be referring to different subjects or different subgroups of an overall category (and as we said the strict definition of opposition occurs between propositions having the same subject). But nevertheless, the rule for subcontraries can be shown to hold based upon contradictory and contrary opposition. This comes from a valid yet roundabout way of reasoning. If it is false that Some dogs are cats then the contradictory of this proposition No dog is a cat must be true, and thus the contrary of this second proposition, Every dog is a cat must be false, which in turn means that the contradictory of this third proposition, Some dogs are not cats must be true! So in short, if Some dogs are cats is false, it must be the case that Some dogs are not cats is true. 94 An Introduction to Traditional Logic: Propositions

111 Subalternation This type of opposition occurs when two propositions agree in quality but not in quantity, and moves between the universal and the specific (looking at the diagram, this moves between the vertical sides of the square). 47 You can think of this as the vertical up and down relationship on the square because it refers to the relationship between the A and the I on one side, and between the E and the O on the other. With subalternation, if Every chair is red is true, then a smaller set of that same subject, viz., Some chairs are red, must also be true. In other words, if the universal proposition is true, then so is the particular version of that proposition. The reverse however is not the case, just because the particular, Some chairs are red, is true, that does not entail that the universal, Every chair is red is also true. Yet, if the particular is false, then it must also be the case that the universal is also false. If Some chairs are red is false, then it must also be the case that the universal, Every chair is red is also false. So to pull this together by looking at our diagram, remember this: if the universal top proposition is true, you infer the truth of the bottom one directly underneath it as also true. If the bottom proposition is false, you infer the falsity of the proposition directly above it. Therefore, the rule for subalternation is that one may descend with truth but rise with falsity. 48 Summary of Opposition The rule for subalternation is that one may descend with truth but rise with falsity. Thus we can see now that the truth-values of the four types of propositions are so interrelated that if we knew only the truth-value of one of them, we might determine the truth-value of the others. Knowing only that A is false, we can know that O is true (through contradictory opposition) and that E and I are undetermined. Take another example, if E is true, then O is true (through subalternation) and A is false (either through contrary opposition with the E or contradictory opposition from O) and consequently I is false (contradictory opposition from the E). We could go on and on with this and it is good practice to do so. Proficiency in being able to move from the truth-value of one proposition to the truth-values of the other types is very helpful both in logic as well in everyday thinking. So it is important to know the rules of opposition well and to be able to move with accuracy 47 Another technical point, although strictly speaking subalternation is not a form of opposition (for the two are not opposed in either truth or quantity), it is nevertheless considered under opposition because the relation is one that unavoidably arises. 48 Or as one of my students put it, Troublemakers go down to hell and fathers go up to heaven! The t in troublemakers of course standing for truth and the f in fathers standing for falsity. Propositional Properties & Compound Propositions 95

112 and ease between the different points of the square. To summarize the rules of opposition then: Contradiction: Two contradictory propositions cannot both be true or both be false at the same time. If one is true the other must be false, and if one is false the other must be true. Contrary: Two contraries cannot both be true, but they can both be false. If one is true the other must be false, but if one is false, the other is undetermined. Subcontrary: Two subcontrary propositions cannot both be false, yet they both can be true. If one is false the other must be true, but if one is true the other is undetermined. Subalternation: Descend with truth and rise with falsity. If the universal A proposition is true, so too the particular I under it is also true; but if the universal A is false, the particular I is undetermined. If the particular I is true, the universal A is undetermined, but if the particular I is false, the universal A must also be false. The same follows for the E and O relationship. Two Issues with the Square: Singular Subjects and Existence Propositions So far we have been talking about universal and particular statements on the square, but what about propositions with singular subjects? Take the example: Lincoln was a president. Singular subject propositions such as these do not need a square. A simple contradictory relation will do. To oppose this we would say, Lincoln was not a president, but it would be odd to say Some of Lincoln was a president, Some of Lincoln was not a president or No Lincoln was a president. The other three options provided by the square are either inaccurate or awkward. In this case, we simply say singular subject propositions such as these do not need a square. A simple contradictory relation will do. Lincoln either was or was not a president, and that is all that needs to be considered in these cases where the subject is considered without any qualifications. So with singular subject propositions, we oppose only with a contradiction, so this example would be opposed like this: Lincoln was a President. or Lincoln was not a president. Now if some qualifier comes into use, say a temporal qualifier such as sometimes never etc., then we can use the square. In this case, one could say something like Lincoln was always a president (A) Lincoln was never a president (E) Lincoln 96 An Introduction to Traditional Logic: Propositions

113 was sometimes a president (I) or Lincoln was sometimes not a president. When a qualifier such as this is entered, these propositions will relate to each other normally on the square. The same procedure is used for existence propositions. One would not want to relate the proposition God exists to Some of God exists or Some of God does not exist. Like in the example above, unless there are some temporal qualifications in the proposition, in which case one may use the square, existence propositions also only oppose each other by a simple contradiction, God exists vs. God does not exist. So although we have said that singular propositions should be treated as universal statements, this rule of thumb has its limits. They are nevertheless a distinct category of proposition. In conversion (which we will see next) and the syllogism, singular propositions do function like universal propositions. Singular propositions do not however work like universal propositions in opposition. Barring anything like temporal qualifiers, they admit of only one kind of opposition contradiction. There is a more advanced objection to the square of opposition, which we treat in an appendix, 49 however that objection goes beyond our purposes here. Conversion Conversion is the process where the subject and predicate trade places while keeping the same quality and truth-value. Unlike opposition that used the same subject and predicate, conversion changes them around. For example, No dog is a cat converts to No cat is a dog. The conversion is valid and really means the same thing, because if no dog is a cat it is also clear that no cat is a dog. The result of a conversion is called the converse of the original, so No cat is a dog is the converse of No dog is a cat. Conversion is the process where the subject and predicate trade places while keeping the same quality and truth-value. However, not every proposition converts in the same way. Logic sets up rules for valid conversions. One cannot validly convert, All dogs are animals into All animals are dogs. The reason why this does not validly convert has to do with the distribution of the terms. If a term is not distributed before the conversion, then it cannot be distributed after the conversion either. Valid conversion is like getting water from a well. You can t get something out of nothing. You can t get more water from the well than what was already in there. The water here in logic is the distribution. You can only draw as much as the original distribution allows. One infers or takes out too much when they say that because, all dogs are animals, that therefore, all animals are dogs. The original predicate animals was not distributed in the original proposition, and so cannot exceed that amount and be distributed in the converse. So no valid conversion can pass from some to all. 49 See the appendix, In Defense of the Square of Opposition. Propositional Properties & Compound Propositions 97

114 Conversion is the process where the subject is switched with the predicate which makes the converse, but this must be done in a way that doesn t violate the distribution of the original. In a nutshell then, conversion is the process where the subject is switched with the predicate which makes the converse, but this must be done in a way that doesn t violate the distribution of the original. There are two kinds of conversion, simple and conversion by limitation. Simple Conversion Simple conversion is a valid process in which a subject is simply interchanged with the predicate without making any changes in quantity or quality. Only E and I propositions validly convert in this way. No Catholic is a Protestant, simply converts to, No Protestant is a Catholic. Likewise, Some reptiles are fast runners, converts simply to, Some fast runners are reptiles. The reason why E and I propositions convert by simple conversion is because only in these propositions do the subject and predicate have the same distribution value. So it is easy to see that since these propositions have the same distribution in both the subject and predicate, switching them will not alter the distribution. Conversion by Limitation Universal affirmative A propositions do not validly convert by simple conversion. With an A proposition, one must do something extra to make the conversion valid. The reason why is because the distribution of the subject and predicate in A propositions differ and the conversion process must account for this difference. 50 Whatever distribution value a term has in the previous proposition, that term must retain that value in the converted proposition. For example, Every bird is an animal, does not validly convert simply to, Every animal is a bird. In the first, the predicate animal was not distributed but in the converse it is, which means the conversion went beyond the original distribution. What A propositions really say when distribution is taken into account is that Every bird is (some) animal, and it is invalid to go from just some animals to all of them. 51 To convert a proposition by limitation, one switches the subject and predicate and then changes the quantity of the subject. To convert a proposition by limitation, one switches the subject and predicate and then changes the quantity of the subject. Every bird is an animal, now becomes through conversion by limitation, Some animals are birds. So the steps for conver- 50 It may be helpful to review the distribution section on page There are some instances where a simply converted A proposition would be true, but this would not be because of the logical form, hence the simple conversion of such propositions would remain formally invalid. For example, Every triangle is a three sided polygon, could convert simply to, Every three sided polygon is a triangle. The converted proposition is true, but not because of the formal relationship between these propositions, but because of the subject matter. That is, good definitions are interchangeable with the term they are defining. As propositions, the universal affirmative converts only accidentally. The same would be true for Every man is a rational animal, for this proposition is convertible with, Every rational animal is a man. But again, these conversions are possible not because of the logical form they have but because of their matter. 98 An Introduction to Traditional Logic: Propositions

115 sion by limitation are; 1) switch the subject and predicate 2) change the quantity of the subject. 52 While E and I propositions convert simply, and A propositions convert by limitation, the O proposition does not convert at all. Any switching of the subject and predicate of O propositions will always result in an invalid shift in distribution. This is because the predicate is always distributed in a negative proposition, and the subject is always undistributed in particular propositions. There is no way to convert, Some dancers are not ballerinas, either simply or by limitation because changing the distribution in the process cannot be avoided. Ballerinas would go from distributed to undistributed, while dancers would go from undistributed to distributed. Finally one last point; it is true that E propositions are not limited to simple conversion, but also can convert by limitation. No reptile is a bird, converts to, No bird is a reptile could also convert to, Some birds are not reptiles. Although this is technically valid, E propositions are almost always converted simply because simple conversion accomplishes more than conversion by limitation does. It doesn t make much sense to infer less when one could validly infer more. We can succinctly summarize conversion by looking at the three rules involved: 1. Universal Negative (E) and Particular Affirmative (I) propositions convert simply. This is because the distribution is the same in the subject and predicate in these propositions. 2. Universal Affirmative (A) propositions only convert by limitation. Universal affirmative propositions do not have the same distribution in regards to both subject and predicate and so cannot be simply exchanged. The converted or new subject must be limited in quality to talk only about some of that subject. 3. Particular Negative (O) propositions never validly convert. Converting this proposition either simply or accidentally would violate the principle that the distribution must remain the same in converse propositions. Converting False and Singular Propositions Converting false propositions might be confusing because sometimes the conversion of a false proposition will wind up true and sometimes it will remain false. Every musician is a woman, is false but it validly converts to, Some musicians are 52 Logicians call this conversion per accidens because the process limits the predicate, but that limitation is nothing essential to the predicate, but merely a modification of it. Propositional Properties & Compound Propositions 99

116 women, which is true. What this means is that the rule that a converted proposition must maintain the truth-value of the original only applies to true statements; false ones are allowed to go either way. Singular subjects are treated as universal subjects in conversion. Thus, affirmative singulars are treated as A propositions while negative singulars are treated as E propositions. For example, Peter is a saint, converts to, Some saint is Peter, and, Peter is not a saint (which is just a clearer way of saying None of this man Peter is a saint ), converts to, No saint is Peter. Obversion Obversion is based on the axiom that if a proposition is true, then so is the denial of its contradictory. Obversion is based on the axiom that if a proposition is true, then so is the denial of its contradictory. Propositions are related through obversion by changing the quality of the proposition (i.e., from affirmative to negative) and negating the predicate. This is different from what we did in opposition, above. In opposition the same subject and predicate are kept and the truth is opposed, but in obversion the predicate is changed by way of negation and the truth of the proposition is maintained. So obverted propositions maintain the same truth value. If the original proposition is true, so must the obverted proposition also be true; and if the original proposition is false, so too must the obverted proposition be false. To do obversion properly, we must follow two steps. 1. First change the quality of the proposition. The proposition must go from affirmative to negative or vice versa. 2. Second, negate the predicate. One must use the prefix such as non- to negate the predicate For example, let s begin with this proposition: Every Scottish soldier is courageous. Step 1: We apply the first rule of obversion by changing the quality: 53 No Scottish soldier is courageous. Step 2: and next we apply the second rule that negates the predicate: No Scottish soldier is non-courageous. By following these two steps we have completed the obversion. The statement, Every Scottish soldier is courageous, is equivalent to the truth of the obverted proposition; No Scottish soldier is non-courageous. 53 Notice this example is correctly stated in the proper form of the universal negative, No S is P. 100 An Introduction to Traditional Logic: Propositions

117 Particulars are obverted in the same way. Some modern logicians are silly, becomes, Some modern logicians are not non-silly. Likewise, singular propositions can also be obverted. St. Augustine was a theologian obverts to St. Augustine was not a non-theologian. Some more examples for each type of proposition are: (A) All men are mortal. obverts to (E) No men are not- mortal. (E) No men are perfect. obverts to (A) All men are not-perfect. (I) Some philosophers are logical. obverts to (O) Some philosophers are not not-logical. (O) Some castles are not inhabited obverts to (I) Some castles are uninhabited. This does not mean it is not advisable to express oneself in negative terms. Talking that way is often difficult to understand. One of the main reasons why obversion is useful is to change a negatively expressed predicate into a positive one that has a clearer meaning. So if one has a proposition that said, No politicians are non-liars, this would obvert to Every politician is a liar (non-non-liar), and the obverted proposition is more understandable. Summary of Propositional Properties One of the main reasons why obversion is useful is to change a negatively expressed predicate into a positive one that has a clearer meaning. We have covered three different properties of propositions that allow us to express the same truth in a different way and/or infer the truth-value of another proposition. 54 To pull all of these together and to avoid confusing them, the following diagram summarizing these properties might be helpful: 54 There is what is sometimes called a fourth, namely contraposition, but this is only the sequential combination of obversion, conversion, and then a second obversion. Thus, contraposition is not really a distinct propositional property. Propositional Properties & Compound Propositions 101

118 Compound Propositions Some propositions have as their parts not just two concepts but two whole propositions that are united by something other than a copula. These propositions are called compound propositions. So far we have been focusing on the categorical proposition, which is a statement that is simple and unqualified. Yet, there are other types of propositions that are more complex. Some propositions have as their parts not just two concepts but two whole propositions that are united by something other than a copula. These propositions are called compound propositions because they contain more than one statement. When someone says, The cat was fast and it ran up into a tree, they are actually making two propositions, one about the speed of a cat, and another about what the cat did. Likewise, when one says, If that noise is thunder, then it is going to rain, they are not stating something categorically or unconditionally, but rather hypothetically by joining two simple propositions. In logic we divide these compound expressions into what is explicitly compound; and these are conjunctive, disjunctive, and conditional propositions; or by what is implicitly compound and these are exclusive, exceptive, and reduplicative propositions. Laid out in a diagram, the different types of propositions look like this: 55 We now discuss these compound propositions in more detail. Conjunctive Propositions A conjunctive proposition is a compound of two or more propositions joined by the word and. A conjunctive proposition is a compound of two or more propositions joined by the word and. Roses are red and violets are blue is a conjunctive proposition. There is no causal connection inherent between the two; they are simply two factual assertions coexisting side by side. Since the two statements in this type are accidentally united and not causally connected, any two statements, however unrelated, can be joined in this way, e.g., I was born in Alabama and I am having fried chicken for dinner. Now in a conjunctive proposition, each proposition that makes up a part is called a conjunct and for a conjunctive proposition to be true, each of its conjuncts must be true. For the above to be true, it must be true that both roses are red and that violets are blue. If any one of these is false, the whole proposition is false as well. 55 Diagram derived from Maritain, op. cit., p An Introduction to Traditional Logic: Propositions

119 Take another example, John went to the store and he had a wreck on the way home. For this statement to be true, it must be true that both John actually went to the store and that he indeed had a wreck on the way home. Just one conjunct being false is enough to falsify the whole statement. For that reason, the rule for the truth-value of conjunctive propositions is: Conjunctive Rule: Both conjuncts of a conjunctive proposition must be true in order for the whole proposition to be true Disjunctive Propositions Disjunctive propositions are the either-or types of statements, e.g., Either Sam went to the store at that time or he did not. In disjunctive propositions we have two propositions united by the word or and each part is called a disjunct. Disjunctive propositions can be either strong or weak. The weak disjunction allows for the possibility of both disjuncts being true at the same time. When someone says, Sam would like something to eat or drink, they intend that Sam might desire both, something to eat and drink. We call the or here weak because it permits the possibility that both disjuncts might be true. On the other hand, strong disjunctions assert that both disjuncts cannot be true at the same time nor can they both be false at the same time. With strong disjunctions, one disjunct must be true and the other false. For instance, He will either recover from cancer or it will be a fatal illness. Here the or is exclusive and will not permit the possibility of both disjuncts being true nor will it permit both to be false. It must be one or the other and not both. Naturally, it cannot be the case that he both recovers from cancer and that it also is fatal to him. Some more examples of a strong disjunction are, This whole number is either even or it is odd, and Either God exists or he does not exist, since a whole number cannot both be even or odd but must be one or the other, and likewise God either exists or he does not. Disjunctive propositions are the either-or types of statements. The weak disjunction allows for the possibility of both disjuncts being true at the same time. With strong disjunctions, one disjunct must be true and the other false. The truth-value rules for disjunctions are: Strong Disjunctive Rule: In a strong disjunction, one of the disjuncts must be true and the other false in order for the whole proposition to be true. In other words, in a strong disjunction both disjuncts cannot be true and both cannot be false. Weak Disjunctive Rule: In a weak disjunction, the whole proposition is true when just one disjunct is true. Propositional Properties & Compound Propositions 103

120 Conditional Propositions The conditional proposition is the typical If-then statement. A conditional proposition asserts that the antecedent cannot be true without the consequent also being true. The conditional proposition is the typical If-then statement, for instance, If Peter goes to Rome, then he will be martyred, or take another example, If you do not study hard in logic, then you will get a bad grade. The first part of the conditional proposition (the if part) is called the antecedent, and the latter part (the then part), is called the consequent. So in the first example above, If Peter goes to Rome is the antecedent and then he will be martyred is the consequent. A conditional proposition asserts that the antecedent cannot be true without the consequent also being true. So for a conditional proposition to be true, the connection between the antecedent and consequent must be necessary. The consequent must really follow from the antecedent. 56 If the antecedent could occur without the consequent happening, then this shows that the connection is not necessary and the whole proposition is false. To put it another way, for a conditional proposition to be true, the sequence needs only to be valid. If the sequence is invalid, the proposition is false. For example, If it rains today, my mail will be late. In order for that proposition to be true, we ask if the antecedent (raining today) can occur without the consequent (my mail being late) occurring. If the antecedent can be true with the consequent being false, then the conditional proposition is invalid. Consequently, the truth-value rule for conditional propositions is: Conditional Rule: For a conditional proposition to be true, the connection between antecedent and consequent must be necessary. The conditional proposition is false if the antecedent can be true and the consequent false. 56 There is a large debate about implication that goes as far back as the ancient Stoic and Megarian logicians. Historically, there are two basic kinds of implication. The first is Philonian implication (named after Philo of Megara) that says conditional propositions are true except when the antecedent is true and the consequent false. This is a very loose interpretation of implication and allows for true propositions to be made even when there is no connection between the antecedent and the consequent. So it would be valid in this interpretation to say, If = 5, then I m having chicken for dinner (!). This is clearly counterintuitive because of course there is no connection between mathematics and what I m having for dinner; yet, most contemporary symbolic logicians follow this Philonian understanding (also called material implication ). The second major kind of implication is a stricter Diodorean implication (named after Diodorus Cronus of Iasus) which says a conditional proposition is true when there is a necessary connection between the antecedent and the consequent, so here the antecedent and consequent have to be causally related (like in saying, If the temperature is below 0 degrees centigrade, then water will freeze. ). With Diodorean implication, one cannot make just factual or happenstance relations between antecedent and consequent. While it is still the case with Diodorean implication that the antecedent cannot be true and the consequent false, there is the additional criterion that there needs to be some causal connection between the antecedent and consequent as well. We follow this more common sensical Diodorean notion of implication (also called strict implication ) here. 104 An Introduction to Traditional Logic: Propositions

121 Implicitly Compound Propositions We have already seen these sorts of propositions in the last chapter. Statements that include words like except or only break down into two propositions united by the word and, which means these statements are implicitly compound. Like any other compound proposition, these statements must follow the same rule; both conjuncts must be true in order for the whole compound proposition to be true. However there is one type that we have not looked at yet, and that is the reduplicative proposition. The reduplicative proposition attributes a predicate to a subject under a certain aspect. For example, take the proposition: Pedro as a boxer is great. The reduplicative proposition attributes a predicate to a subject under a certain aspect. This proposition actually is saying two things, and when we break it down we unite them with the conjunction and : Pedro is a boxer and that boxer aspect is great. Both propositions have to be true for the original proposition to be true. Pedro must both be a boxer and then he must be great when considered under this aspect. The predicate is attributed to the subject under the aspect of the redoubling or reduplicative particle. From the above, we cannot derive Pedro is great absolutely, but only as he is a boxer. So in reduplicative propositions, it is important to remember that the predicate only applies to the subject under the condition of the reduplicative notion. Chapter 6 Summary The properties of propositions; opposition, conversion, and obversion allow one to restate the truth of one proposition in a different way and/or implicate the truth-values of other propositions Opposition is a property of propositions that arises from the different ways of affirming and denying the same predicate to the same subject. The relationships on the square of opposition show the four types of opposition; contradictory, contrary, subcontrary, and subalternation. Propositions with singular subjects or existence propositions, unless they contain temporal qualifiers, are not opposed with all four relationships on the square, but rather a simple contradictory opposition. Conversion is a process of switching the subject with the predicate, with the result being the converse of the original. E and I propositions convert Propositional Properties & Compound Propositions 105

122 simply, A propositions convert by limitation, and O propositions do not convert at all. Obversion is the process by which one states an equivalent proposition by denying the contradictory of the original proposition. Propositions are related through obversion by changing the quality and negating the predicate. Compound propositions contain two or more propositions. The propositions that are explicitly compound can be conjunctive, disjunctive, and conditional; and each have their respective rules for their truth-value. The implicitly compound propositions are exceptive, exclusive, and reduplicative, and since they break down into conjunctive propositions, they follow the conjunctive rule for their truth-value. Exercises A. Indicate the type of opposition 1. Every dog is scared. No dog is scared. 2. Every horse is a fast runner. Some horses are not fast runners. 3. Every philosopher is goofy. Some philosophers are goofy. 4. Some Africans live in tribes. No Africans live in tribes. 5. Some mammals can fly. Some mammals cannot fly. B. Opposition - Assume all the following propositions are true. Indicate the relevant contrary, subcontrary, contradictory and subaltern propositions by letter (A, E, I, O) and list their truth-values: 1. Every S is P 2. No S is P 3. Some S is P 4. Some S is not P 5. Every man is a liar. 6. No dogs go to heaven. 7. Some Romans ate meat. 8. Some Scottish clans did not fight bravely. 9. Sometimes baseball is an interesting game. 10. Every concert is crowded. 11. Some countries require visas. 12. Every man is a social animal. 13. Whatever is moved is moved by another. 14. Few of the books from that publisher are good. 15. Virtuous acts are what make you a good person. 16. Some things are not right. 106 An Introduction to Traditional Logic: Propositions

123 C. Opposition - Assume all the following propositions are false. Indicate the relevant contrary, subcontrary, contradictory and subaltern propositions by letter (A, E, I, O) and list their truth-values: 1. Every S is P 2. No S is P 3. Some S is P 4. Some S is not P 5. Every man is a liar. 6. No dogs go to heaven. 7. Some Romans ate meat. 8. Some Scottish clans did not fight bravely. 9. Sometimes baseball is an interesting game. 10. Every concert is crowded. 11. Some countries require visas. 12. Every man is a social animal. 13. Whatever is moved is moved by another. 14. Few of the books from that publisher are good. 15. Virtuous acts are what make you a good person. 16. Some things are not right. D. Convert the following. If conversion is not possible indicate with not convertible : 1. Some S is not P. 2. Every S is P. 3. No S is P. 4. Some S is P. 5. Every man is a winner. 6. No person is able to know if God exists. 7. Some animals are dangerous. 8. Some hunters are-not fishermen. 9. All cattle like to graze. 10. Most lawyers are not as bad as people say they are. 11. Some policemen are college graduates. 12. The Smith and Wesson.44 magnum is the world s most powerful handgun. 13. Boxers who win are boxers who train. 14. None of the athletes are gold medallists. E. Obvert the following propositions: 1. Some S is not P. 2. Every S is P. 3. No S is P. Propositional Properties & Compound Propositions 107

124 4. Some S is P. 5. Some politicians are good men. 6. Some politicians are not good men. 7. Every cat is one that catches mice. 8. No jewels are inexpensive. 9. Every pigeon can fly. 10. Some books are harmful. 11. All logic is fun. F. Indicate the type of proposition (categorical, conjunctive, disjunctive, conditional, exclusive, exceptive, reduplicative) the following: 1. Socrates as philosopher was quite brilliant. 2. Peter and Paul were Apostles. 3. Humans alone are rational animals. 4. If you want to excel in logic, then you must study. 5. Either aliens from other planets exist or they do not. 6. All of the people except for a few believed him. 7. He told me that and I believe him. 8. Either the gladiator will win or he will be killed. 9. Every categorical proposition is unqualified. 10. All those guys, save for one, won the award. 11. If you don t believe you will perish. 12. Only women are allowed free of charge into the club. 13. As an artist, Michelangelo was fantastic. 14. Christ, insofar as he was a man, suffered and died. 15. The Germans alone conquered Poland. 108 An Introduction to Traditional Logic: Propositions

125 PART III: ARGUMENT

126

127 Chapter 7: Argumentation & the Syllogism àà Argumentation Introduced àà The Syllogism: Rules and Arrangement àà Testing Syllogisms for Validity àà Three Conditions for Successful Argumentation Argumentation Introduced The third part of logic is about argumentation, and argumentation gives reasons in support of a conclusion. By argumentation, supporting propositions called premises serve as evidence or support for a conclusion that was not explicitly or directly known before. In a valid argument, granting certain things as true means other things must also be true. In this sense, argumentation is a sort of mental movement from the known to the unknown, and this is why the conclusion is said to follow from the premises. Argumentation gives reasons in support of a conclusion. How does this work? Suppose someone wondered what good reasons there are for thinking God exists. One might respond by saying something like; Well, we see things in the universe that depend upon causes for their existence, and this can t go on to infinity, so it must stop somewhere at a first cause, a cause for everything, which would be God. Such a demonstration put logically, might take this form: This example is known as the Kalam cosmological argument. Argumentation & the Syllogism 111

128 Whatever begins to exist has a cause. The universe began to exist. Therefore, the universe has a cause. Naturally, one would have to give further argumentation to show that the nature of this first cause of the universe is changeless, timeless, powerful, personal, etc., but the point here is to show that the conclusion is reached and validly supported by reasons or premises. In a properly structured argument, the premises are better known than the conclusion and that is how they are used to show the conclusion that is not known (if the premises are not known any better than the conclusion, what good are they in proving the conclusion?). Let s try another more obvious example. Suppose one didn t know Socrates had the ability to laugh, but they did know that all men have this ability to laugh and they also knew that Socrates was a man. This is all one would need to set up an argument that demonstrates Socrates has the ability to laugh: Every man is one who has the ability to laugh Socrates is a man Therefore Socrates is one who has the ability to laugh. So you see, if at first someone knew only the premises and not the conclusion, those premises they did know can be arranged to demonstrate something that they did not. We ll see how this works in detail a little later. Having support for a conclusion is what makes an argument different from an assertion. An assertion is just a statement asserted as true by itself with no support. If you think of a conclusion as being like the roof of a house, a good argument has premises that support a conclusion like the walls of a house support a roof, while an assertion is just a statement thrown out as true with no support at all, like a roof just sitting on the ground. A good argument should support a conclusion similar to the way in which walls support a house, and naturally the premises or walls have to be both sturdy (true) and arranged in a certain form or pattern in order to do that. Correct reasoning follows certain patterns and the third part of logic deals with determining the validity of those patterns. Deductive Argumentation Given that we want to know the truth of some proposition, and if the truth of that proposition is not immediately apparent, then its truth obviously needs to be established by something else. This means that these other supporting truths mediately establish the truth of that proposition; in other words, the truth of the conclusion is established by means of the premises. But there are two different ways in which 112 An Introduction to Traditional Logic: Argument

129 premises can do this. The first way is to enumerate several instances of something and use that enumeration to support a general contention. This process is called induction and looks something like this: Tom is a man and he is mortal Dick is a man and he is mortal Harry is a man and he is mortal Etc., Therefore, all men are mortal Very often in our experience we see the truth of a repeated number of instances and so infer a general rule. So this method of induction usually involves a movement from an enumeration of particular instances of some truth to a universal conclusion about all instances. The defining element of induction is that a grouping of particular experiences occurs within the context of the argument itself. The use of induction is called inductive argumentation or inductive reasoning and will be discussed later. The second way premises can be used to support the conclusion is the method we have been using all along in our examples. One could also establish a conclusion by linking two concepts together by means of a third. Take this old standby for instance: Induction usually involves a movement from an enumeration of particular instances of some truth to a universal conclusion about all instances. Every man is mortal Socrates is a man Therefore Socrates is mortal The type of support or mediation here is different than in induction. Within the context of a deductive argument, there is no enumeration of individual instances, but rather the conclusion links the subject Socrates to the predicate mortal by way of a middle linking concept of man. This process of mediation through concepts is called deduction, and the process is called deductive argumentation or deductive reasoning. Now although deductive arguments do not use the mediation of individual experiences within the context of the argument itself (like inductive arguments do) this is not to say the premises of deduction do not need individual experiences at all (which would be false since all knowledge originally comes from individual experiences). It is just that the deductive argument itself does not appeal to a set of individual experiences like inductive arguments do. Within the context of a deductive argument, there is no enumeration of individual instances. Deductive arguments are our main focus here. The key point to deduction is that the conclusion is established from what is given in the argument itself and this conclusion follows purely intellectually. The tool used in deductive argument is called the syllogism and is the principal topic of this chapter. Argumentation & the Syllogism 113

130 The Syllogism A syllogism is a complex expression in which given certain things being stated, something else necessarily follows. The major term is the predicate of the conclusion. The minor term is the subject of the conclusion. The middle term is the link between the major and minor terms. The syllogism is the foundation of logic, but what exactly is it? Stated formally, a syllogism is a complex expression in which given certain things being stated, something else necessarily follows. Let s look at the parts of a syllogism. A basic syllogism contains three propositions, two of which are premises and one that is a conclusion. The conclusion is the proposition that is being established. The other two propositions are the reasons that support the conclusion and are called premises. The syllogism consists of three and only three terms; the major, middle, and minor. The major term is the predicate of the conclusion and ideally is the one with the most extension or universality. The minor term is the subject of the conclusion and ideally is the term with the least extension or universality. The middle term is the link between the major and minor terms and is the middle in regards to universality. The middle term occurs twice in the premises but not in the conclusion. The major premise is the premise containing the major term while the minor premise is the premise that contains the minor term. In each of the premises, the major and minor terms are compared separately with the middle term; and then in the conclusion the major and minor terms are compared with one another. So every syllogism consists of three and only three terms, the middle term occurs twice in the premises and does not appear at all in the conclusion, while the major and minor terms appear only once in the premises and once in the conclusion. Since the syllogism is demonstrative reasoning, the reasoning ideally moves from the more universal to the less. The major premise, since it contains the term with the most universality, should be placed first and the minor premise comes second. The classic example above does just this. But now let s look at this argument again with annotations to highlight the various parts: Major premise: Every man [middle term] is mortal [major term] Minor premise: Socrates [minor term] is a man [middle term] Conclusion: Therefore Socrates [minor term] is mortal [major term] Summarizing these essential elements of the syllogism: A syllogism consists of three propositions: two of which are premises and one which is the conclusion A syllogism consists of three terms: major, minor, and middle. Each term is used twice. The subject of the conclusion is the minor term. The minor term appears once in the conclusion and once in the premises. The premise that contains this minor term is the minor premise. 114 An Introduction to Traditional Logic: Argument

131 The predicate of the conclusion is the major term. The major term appears once in the conclusion and once in the premises. The premise that contains this major term is the major premise. The middle term appears in both premises but not in the conclusion. The Middle Term: a Key Concern for Validity Logical validity depends upon a connection between the premises and conclusion. Since the validity of an argument relies so much on this connection, we should know something about how this works. The conclusion of a deductive argument is made from a mediating universal concept. If we want to join S and P in a conclusion, then we need to connect them by way of some third concept. For example, suppose we wanted to link the subject this shape to the predicate has four sides. The mediating concept for this example is the nature square. The syllogism goes about this by using the middle term (abbreviated as M ) as the link to join S and P: Logical validity depends upon a connection between the premises and conclusion. Every square (M) has four sides (P) This shape (S) is a square (M) Therefore, this shape (S) has four sides (P) In this example, S is joined to P by way of the middle term that links the two together. We can illustrate this with a technique known as Euler s circles: Euler s circles is a technique from the 18 th century German mathematician Leonhard Euler designed to graphically illustrate propositions and arguments. The diagram is essentially an arrangement of three circles that represent the three terms of a syllogism, and by putting them together the group symbolizes the three propositions of an argument. Argumentation & the Syllogism 115

132 Illustrating this relationship with circles, we can see how this shape (S) is connected to has four sides (P) by way of the middle term square (M) which links the subject and predicate together. The two outermost circles represent the major premise, Every square has four sides, and the two innermost circles represent the minor premise, This shape is a square. Notice how the middle term square is precisely in the middle regarding universality (extension) and it is from this that the middle term derives its name. Square has more extension than this shape, and has four sides has more extension than square. This mediation of the middle term is the key element in validity. 59 This type of syllogism, when the middle is really middle in universality, is considered by logicians like Aristotle to be the most perfect type of syllogism because each term, according to being a major, minor, or middle, is doing precisely what it is supposed to do. The middle term has this same importance in a denial. Supposing we wanted to show this shape does not have three sides. We can use the same major premise this time to deny the union between the middle and minor terms: No square (M) has three sides (P) This shape (S) is a square (M) Therefore, this shape (S) is not one that has three sides (P) Here we have a separation of the major term has three sides from the linking middle term square and thus the argument effects a separation of subject and predicate in the conclusion, This shape is not one that has three sides. With Euler s circles the result looks like this: 59 The precise way of stating this connection is by a relation of identity. This shape is identified with square which in turn is identified with having four sides. The relation of triple identity that results (as will be mentioned in more detail below, that two things identified with some third are identical to each other) grounds the inference. 116 An Introduction to Traditional Logic: Argument

133 Notice again that the middle term is the linking term, and if that link is broken in the premises there will be a break in the conclusion. Also since each term here again is playing its proper role in universality, this is another example of what Aristotle called a most perfect syllogism. This notion of a perfect syllogism is important in that perfect syllogisms are more easily recognized as valid, and the ability to turn valid yet imperfect syllogisms into perfect ones will be discussed later in the next chapter. The main point here is that you see the special linking role the middle term serves. The Form of the Syllogism The above discussion about the major, minor, and middle terms as well as the major and minor premises all concern the matter or building blocks of the syllogism. We now move to discuss the form of a syllogism or the way in which these elements are arranged and the types of propositions used. The form of a syllogism consists of what is called mood and figure and is another consideration for a syllogism s validity. As we ve said, the basic syllogism only allows for two premises and one conclusion, giving us three propositions in all. The mood of the syllogism is its arrangement of the types of propositions that make it up. Taking the propositions in order, we indicate the mood by the letter for each type of proposition. So for example, a syllogism that uses three universal A propositions in a row is the AAA mood: The form of a syllogism consists of what is called mood and figure. The mood of the syllogism is its arrangement of the types of propositions that make it up. A Every mortal thing is that which dies A Every man is a mortal thing A Every man is that which dies This syllogism has the AAA mood because those are the types of propositions used, and of course, using other types of propositions means a different mood. Let s try another example, this time we use the mood EIO: E No wrestlers are weak I Some high school kids are wrestlers O Some high school kids are not weak So you can see the mood of the syllogism is simply the three letters for the propositional types that correspond to the sequence of major premise, minor premise, and conclusion. This does not by any means mean that all moods are valid. There are actually a total of 64 possible combinations that could be the moods of the syllogism, Argumentation & the Syllogism 117

134 but we don t need to go into all the invalid moods, rather we are concerned only with the valid. The other factor to consider when discussing the form of the syllogism is the position of the middle term. The positioning of the middle term is known as the figure of a syllogism. Since the middle term belongs in the two premises, it could be in either the subject or the predicate position. So two possible places within two premises gives us four possible arrangements or figures of the syllogism, (although we will cover just three of these as explained below). Again using the variables S for the minor, P for the major, and M for the middle term, we can look at the possible figures: The reader may be wondering if we missed one. Although there is a possible fourth figure (with the middle term in the predicate position in the major premise and in the subject position in the minor premise), we do not need to consider it for two reasons. Speaking in terms of validity, just inverting the major premise with the minor gives us a first figure syllogism (also called an indirect first figures). Secondly, the fourth figure, although valid, violates the nature of a syllogism in that the middle term is not middle at all. It is less in universality than the minor term and more in universality than the major. Hence, we do not need to discuss the fourth figure and can get by just fine without it. 118 An Introduction to Traditional Logic: Argument

135 The form of the syllogism is the combination of the mood with the figure of the syllogism. These syllogistic forms will be discussed with much more detail in the next chapter. Formulating Syllogisms The point of studying logic should always be in light of everyday life, and we all know that not everyone goes around speaking in terms of perfect syllogisms. We saw an inkling of this in the last chapter with the section on putting everyday language into proper logical form, and a similar point can be made about formulating arguments. To put your logical skills to use in both your own reasoning and in understanding the arguments of others, one must be able to formulate thoughts into a proper syllogism, and there are a few tips that can help one do this. In most cases we do not know which is the major and minor premise unless we first know which is the major and minor term, and we cannot know that unless we know the conclusion. Hence, a good rule of thumb for formulating or examining an argument that is not put into a syllogism is to first determine the conclusion. From that, one can derive the major and minor terms, and from that, one can derive the major and minor premises. So the point is that once one first nails down the conclusion the argument is trying to establish, it becomes easier to determine what the premises are. Try this on the following statement: To put your logical skills to use in both your own reasoning and in understanding the arguments of others, one must be able to formulate thoughts into a proper syllogism. All logic professors make me sick, and no wonder, since all logic professors are grouchy and all grouchy people make me sick. Can you pull out the syllogism here? First identify what the speaker wants to prove. This is the conclusion, All logic professors make me sick. Next, find the major premise (remember this is the premise containing the predicate of the conclusion, which in this case is make me sick ), and we see that the major premise is, All grouchy people make me sick. Thus, the minor premise is all that is left, All logic professors are grouchy and when we arrange them in a proper order we get: Major premise: All grouchy people make me sick Minor premise: All logic professors are grouchy people Conclusion: Therefore all logic professors make me sick It sometimes helps when reading arguments to be on the lookout for indicators for the premises and conclusion. A conclusion indicator is a word or phrase that indicates what is intended to be the conclusion to an argument. For example, when you see words like thus, therefore, so, then, etc., this should tip you off that a Argumentation & the Syllogism 119

136 conclusion is soon to follow. By the same token, premise indicators indicate which propositions are being used to support a conclusion. For example, words like since, if, on account of, because etc., indicate that what follows is supposed to support a conclusion. Frequent practice with looking for key words will help one formulate common speech into syllogisms. By keeping these words in mind that tip us off to premises and conclusions, we can put this process of formulating syllogisms into three steps: Three step process for forming syllogisms: 1. Find the conclusion 2. Determine what the premises are that are supposed to support this conclusion 3. Arrange the premises in order of major and minor. Testing Syllogisms for Validity 1: Rules of the Syllogism A valid deductive argument is one where if the premises are true, the conclusion necessarily follows. Now that we know how to formulate syllogisms, we are ready to test them for validity. To review the earlier material; terms can be clear or unclear, propositions can be true or false, and arguments can be either valid or invalid. If the inference is valid, then the conclusion really follows from the premises. In other words, with a valid argument there is a connection between what is given in the premises and what is inferred in the conclusion. A valid deductive argument is one where if the premises are true, the conclusion necessarily follows and we can be certain of the conclusion. If all mammals are animals and all dogs are mammals, then we know for certain that all dogs are animals. Strong deductive arguments then have this power to produce certitude. Invalid inferences, on the other hand, occur when this connection between premises and conclusion is only apparent. If the premises can be true and the conclusion false, the argument is invalid, and that means it contains a fallacy somewhere. The first and most important way to test a syllogism for validity is to examine it in light of the rules of the syllogism. In order to have a valid argument, any syllogism must adhere to a set of logical rules. These rules are the best technique for testing syllogisms for validity, but they require some work in the beginning in that they should be memorized. We have eight rules for the syllogism: 120 An Introduction to Traditional Logic: Argument

137 The Eight Rules of the Syllogism Only three terms: major, middle, and minor 2. The terms can never be broader in the conclusion than they are in the premises 3. The middle term must never be in the conclusion 4. The middle term must be distributed at least once 5. From two negative premises, nothing follows 6. When both premises are affirmative, the conclusion cannot be negative 7. The conclusion always follows the inferior premise (particular and negative) 8. From two particular premises, nothing follows We now explain each rule in more detail. Rule 1: Three and only three terms The syllogism aims to link a subject to a predicate by means of a third middle term. A fourth term is unneeded and would undermine everything the syllogism is trying to do. This rule also disallows terms used equivocally, that is, with a different meaning in each of the premises. If the term is used equivocally, the syllogism will be fallacious. For example: Violation of Rule 1: Every dog barks A constellation is a dog Therefore a constellation barks A fourth term is unneeded and would undermine everything the syllogism is trying to do. Violating rule 1 is called the four term fallacy. Just because the same word is used, doesn t mean each one has the same meaning. The term dog in this example has two different meanings, the first being an animal and the second the shape of a set of stars, so the conclusion is fallacious. Rule 2: The terms can never be broader in the conclusion than they are in the premises You cannot get something from nothing and this is as true in logic as it is everywhere else. The syllogism seeks to show what follows from the premises. If there is more of something in the conclusion than what was given in the premises, You cannot get something from nothing and this is as true in logic as it is everywhere else. 60 The enumeration of these rules varies with different logicians. Although having more members, this listing here is best for beginners due to its explicitness. We include as rules what are often called corollaries. This listing then simplifies matters at the cost of having some overlap. Argumentation & the Syllogism 121

138 obviously this extra material did not come from the premises and so the inference would be invalid: Violation of Rule 2: No sea-sponge walks Every sea-sponge is an animal Therefore, no animals walk You cannot get all from merely some. Notice that the term animal in the minor premise is not distributed since it is only talking about a part of the genus animal, yet that same term is distributed in the conclusion and there suddenly it jumped to talking about all animals. In other words, the conclusion is talking about more animals than the minor premise was, and so this term is broader in the conclusion than it was in the premises. Since the violation here occurred with the minor term, the error is called the fallacy of the illicit minor. If it had been the major term that contained the error, it is called the fallacy of the illicit major. The middle term must never be in the conclusion. Rule 3: The middle term must never be in the conclusion. This problem should be easy to spot and avoid. The whole point of a syllogistic argument is to link a subject with a predicate by means of a third term. It defeats the purpose if that link turns up again a third time in the conclusion: Violation of Rule 3: Every plant is a living thing Every animal is a living thing Therefore, every living thing is a plant or animal The middle term must be distributed at least once. Rule 4: The middle term must be distributed at least once The middle term is the link between the major and minor term. If this middle term is not distributed, we cannot know if we are talking about all the instances of that middle term and so we cannot have a valid link. In other words, unless the content of the middle term is exhausted, we could be talking about two separate parts of that group and thus we would not have a true link. For example: Violation of Rule 4: All robins are birds All ducks are birds Therefore, all ducks are robins The problem here is that the middle term birds is never distributed. 61 Although both premises are true, they are only talking about a portion of the genus birds, 61 Review the rules for the distribution of terms on page An Introduction to Traditional Logic: Argument

139 and these two portions of course are not connected. This error is known as the fallacy of the undistributed middle. Rule 5: From two negative premises, nothing follows This rule is straightforward and easily memorized. If the two premises are negative in quality, there cannot be a valid inference as the following example shows: From two negative premises, nothing follows. Violation of Rule 5: No rich people are nice No southerners are rich people Therefore all southerners are nice Rule 6: When both premises are affirmative, the conclusion cannot be negative Again, you can t get something out of nothing, and if there is no negation in the premise, the argument is invalid if a negation pops up out of nowhere in the conclusion: When both premises are affirmative, the conclusion cannot be negative. Violation of Rule 6: Every man is mortal Socrates is a man Therefore Socrates is not mortal Rule 7: The conclusion always follows the inferior premise This is actually a corollary of earlier rules, but we make it a full fledged rule here because it is so helpful in recognizing invalid arguments. By inferior we mean particular and negative. If at least one premise is particular, then the conclusion cannot be universal (you cannot get all from only some ). By the same token, if at least one premise is negative, the conclusion cannot be affirmative. The violation of this rule results in the fallacy of an affirmative conclusion with a negative premise or the fallacy of a universal conclusion with a particular premise The conclusion always follows the inferior. Violation of Rule 7: All weird people need help Some people from Texas are weird Therefore, all people from Texas need help As you can see, by drawing a universal conclusion from a particular premise we have a minor that is distributed in the conclusion that was not so in the premises (hence in this case by violating rule 7 we also violate rule 2). Rule 8: From two particular premises, nothing follows This rule is also a corollary of earlier rules, but again it is helpful and less complicated to consider it as a rule by From two particular premises, nothing follows. Argumentation & the Syllogism 123

140 itself so one will more quickly recognize an invalid argument in this form. Violation of this rule is known as the fallacy of two particular premises: Violation of Rule 8: Some policemen are good shots Some men are not policemen Therefore some men are not good shots This is invalid because it doesn t follow that just because some policemen are good shots and some men are not policemen that there are some men who are not good shots. The conclusion may in fact be true, but this argument doesn t prove it to be so. In this case, the predicate good shots is distributed in the conclusion but not in the premises (see rule 2). Since this rule is really a corollary of earlier rules, violating this rule will always be a violation of one of the earlier rules. Why the Rules for Validity Work The reason why the above eight rules are good for determining validity is because of a few self-evident principles that form the basis of all logical argumentation. In other words, there are three obviously true principles that are fundamental for demonstration. These are here for the sake of understanding how the syllogism works, and so the memorization of these is not as important as knowing the eight rules: First Principle: Principle of Triple Identity: The syllogism s whole power of inferring consists in the union of two things with a third. Whenever things are identical with a third thing they are identical with each other. All discursive reasoning is grounded in this principle and the strength of the middle term as a link depends upon it. Second Principle Dictum de omni: Said of all : Whatever is said of all of a subject is said of everything contained under that subject. To put it another way, whatever may be affirmed of a whole class may be affirmed of everything contained in that class. If all the chairs are black, it must also be true that some of the chairs are black. Again, if all men have the ability to laugh, then it must be true that any individual man has the ability to laugh. Third Principle Dictum de nullo: Said of none This is the opposite of the second principle. Whatever is denied of all of a subject is also denied of whatever is contained under a subject. To put it another way, whatever may be denied of a whole class may be denied of everything contained in that class. If no men can live forever, then it must also be true that no individual man can live forever. 124 An Introduction to Traditional Logic: Argument

141 Validity and Truth in Syllogisms Now the difference between truth and validity comes up again in talking about what is established or not established in a conclusion to an argument. Just because an argument is invalid, does not mean the conclusion is false. Arguments can be invalid and still have true conclusions. Take this example: Every animal is four legged All four legged things are purple Therefore, all animals are mortal Now this example is quite radical because not only are both premises false, but the reasoning is also invalid and yet the conclusion is still true. Bad arguments do not make their conclusions false; they only fail to establish their conclusions. What should be said of a bad argument is that the truth of its conclusion is undetermined. The conclusion may turn out to be false, but it is not false just because that argument intending to support it fails. It should also be clear that simply having a valid argument does not mean the conclusion is true either. Take this example: Every leprechaun is a god Bill is a leprechaun Therefore, Bill is a god The reasoning here is quite valid, but of course our friend Bill is not a leprechaun nor is he the creator of the universe. So even though this argument is valid, the question of the truth of the conclusion is a different story entirely. Take another example of an invalid argument: Some men are rational Socrates is a man Therefore Socrates is rational This argument is fallacious because it falls victim to the fallacy of the undistributed middle. The conclusion happens to be still true; it is just that it is not proven to be true based upon what is given in the premises. The following guidelines should be kept in mind: Just because an argument is invalid, that alone does not mean its conclusion is false, but only undetermined. An invalid argument can still have a true conclusion. Just because an argument is valid, that alone does not mean its conclusion is true. A valid argument can still have a false conclusion. We now look at an additional technique for determining the validity of an argument. Argumentation & the Syllogism 125

142 Testing Syllogisms for Validity 2: Venn Diagrams Logicians have often employed different diagramming techniques as a way of checking the validity of arguments, and here we look at a popular method called the Venn diagram. Venn diagrams are relatively easy to use and do a fine job in showing validity. The only drawback to them is that they do not tell us what fallacy is involved when an argument turns out to be invalid. For this reason, they should be learned in conjunction with the rules of the syllogism. In a Venn diagram, we represent the two premises of an argument by drawing three overlapping circles. Each circle represents one of the terms, so there is a circle for the major, middle, and minor terms. We can label the circles S, M, and P respectively (although you will probably want to give the three circles more relevant names if you are not using variables). The basic diagram, without taking any arguments into account, looks like this: This is your starting shell for making Venn diagrams. We draw this first and then move on to consider the premises of a syllogism. By either shading out or inserting an x in the various areas of the diagram, we can represent any categorical proposition, and the method is actually quite simple. When a premise is universal, we shade out the areas that the premise does not include. If a premise is particular, we use an x to mark the area of the diagram that the particular premise claims as a part. With these two ways of marking the circles we can diagram both universal and particular premises. 126 An Introduction to Traditional Logic: Argument

143 Let s take a simple example: Every M is P Every S is M Therefore, every S is P The following Venn diagram represents this syllogism: You can see how in the above diagram there are three circles for each term and they are labeled accordingly. Next, we diagram the major premise, Every M is P, by shading out everything of the circle M that is not a part of P, and by doing this we show with the unshaded area (which is the area that has not been ruled out by the premise) that every M is P. Next, we take the minor premise, Every S is M, and do the same thing. We shade out everything of the S circle that is not also a part of M, and again, by doing that we show that everything that remains of S is a part of M. So the point here is that we indicate that we are discarding areas of a circle by shading them out because they are not covered in the premise. What remains after diagramming the two premises is examining the diagram for the conclusion, Every S is P, and we can see that the diagram does indeed warrant this conclusion. Everything that remains of S is also a part of P, Every S is P. So the argument is valid. Taking another example: No M is P All S is M Therefore, no S is P Argumentation & the Syllogism 127

144 We diagram this argument as: The above diagram shows the major premise, No M is P, by shading out, all of the circle M that is also a part of P. The minor, All S is M, is diagrammed by shading out all of the circle S that is not a part of M, and the conclusion, No S is P, is shown to be valid because nothing of what remains of S is a part of P. Let s take another example, this time with a particular premise: Every M is P Some S is M Therefore, some S is P This argument, when diagrammed, looks like this: Here we have shaded out all areas of M that are not part of P to account for the major, Every M is P. Next, we have a particular minor premise and these are represented 128 An Introduction to Traditional Logic: Argument

145 in Venn diagrams by using an x to positively mark the area the particular proposition affirms. Placing an x in the area where S overlaps with M then signifies, Some S is M. In doing that we can see the remaining conclusion clearly, that some S is P (because the x is also in the P circle as well). The argument is valid. Now one important point needs to be made when using Venn diagrams to test syllogisms that contain one universal and one particular premise. It is important that you always diagram the universal premise first, before diagramming the particular premise. Otherwise you might not know where to put the x (try this argument above and see!). Now let s look at a couple of invalid examples to see how the diagram would indicate a fallacious inference. Take this argument for instance: Every P is M Some S is M Therefore some S is P This argument diagrams as: We diagram the major premise by shading out everything of P that is not M. But then when we get to the minor premise we have to place an x for Some S is M, but where? Does it go in the section of M that is a part of P, or in the section of M that is not a part of P? The problem is that the premise doesn t tell us precisely where, and so logically we are not permitted to commit to any one side over the other. Hence, we have to put it on the line between the two. But that shows that the conclusion, Some S is P is invalid. The premises didn t warrant placing the x in the P portion of circle M, and so this conclusion does not follow. So the Venn diagram technique shows that a conclusion is invalid when what the argument states is not borne out by the diagram. Argumentation & the Syllogism 129

146 Looking at one more fallacious example, take this argument: Every P is M Every S is M Therefore every S is P This argument diagrams as: By diagramming the two premises, you can see that the conclusion does not follow because there is a possible part of S that still has not been ruled out by this argument, namely, the possibility of the part of S that is only a part of M but not a part of P. So this argument fails to prove that Every S is P. To summarize the technique of making Venn diagrams, follow these steps: 1. Make three overlapping circles, one for each term in the argument, and label them. 2. Diagram both premises. If there is one universal and one particular premise, diagram the universal first. When diagramming a particular premise, always remember to put the x on the dividing line if the premise does not specifically indicate as to which of the two sections it belongs. 3. Inspect the diagram to see if it verifies the conclusion, if it does the argument is valid. Venn diagrams are a very good way of determining the validity of an argument. As said above though, the only drawback is that they are a little too easy in that they can show you that an argument is valid or invalid but they cannot show you why an argument is invalid. To determine why an argument is invalid, you need the rules of the syllogism, and so no logician can get by with only Venn diagrams. So this technique of Venn diagrams should be thought of as a supplement to analyzing argument and not as a stand-alone solution. 130 An Introduction to Traditional Logic: Argument

147 Having covered the validity of arguments in detail, it is fitting now to cover the difference between a good argument that does establish the truth of its conclusion and a bad one that does not. As it turns out, any successful argument must meet three conditions and now we look into exactly what those conditions are. Three Conditions for Successful Argumentation Since arguments themselves can be valid or invalid, and they consist of terms which can be clear or unclear and premises that can be true or false; these are the three crucial areas for the success of any argument. This point is very important and worth repeating, these are the three places where any argument can succeed or fail. In order to have a successful argument, the terms need to be clear and consistent, the premises true, and the reasoning valid. All three of these conditions need to be met in order to have a successful argument. If any condition fails, the whole argument fails. Taking each criteria individually we can see how this is so. Condition #1: The terms of the argument must be clear and consistent Unclear terms or terms that shift in meaning cannot serve as elements of a successful argument (look again at the four term fallacy above). Condition #2: The premises of the argument must be true This criterion is straightforward. Obviously an argument with false premises cannot prove anything. Condition #3: The reasoning of the argument must be valid Naturally all valid reasoning must avoid false inferences and conclusions that do not follow from the premises. Thus, if all three conditions are met, i.e., the terms are clear, the premises true, and the structure of the argument valid, then the conclusion of a deductive argument follows necessarily and there is no way the conclusion could be falsified by experience. When this happens, the argument is said to be good, conclusive, or sound. A sound argument is one with clear terms, true propositions, and valid reasoning. Bad arguments always contain at least an ambiguous term, a false premise, or fallacious and invalid logic. Memorizing these conditions will be very useful in both formulating arguments of your own and countering those which you oppose. When you know what goes into a successful argument, you know both how to make good ones and how to recognize the bad ones. Argumentation & the Syllogism 131

148 The Sophistic Objection to the Syllogism The sophists of ancient Greece were skeptics and some claimed it was impossible to demonstrate anything with a syllogism. This objection attacks condition #2, because in order for a syllogism to succeed, one must show the premises of that syllogism to be true. But of course that would involve two more syllogisms, one for the major premise and one for the minor, and then we would have to have a proof for the premises of these two new syllogisms, which calls for four more syllogisms, and these in turn need proofs for their (eight) premises, etc., etc., ad infinitum. Since an infinite number of steps cannot be completed, one could never get around to demonstrating anything with a syllogism. Aristotle, however, refutes this objection by denying that the regress is infinite. All knowledge and demonstration comes to a stop and rests on immediate and evident principles or primitives that are not in need of proof. These primitive truths are so obvious that they do not need proof and in fact, any proof would have to assume them anyway. The principle of non-contradiction is a good example. This principle is evidently and necessarily true, it does not need any proof. To try and prove this principle would be folly, since any proof would have to assume this principle and so one would be arguing in a circle trying to prove it. Additionally, clear facts from sense experience like fire is hot and snow is cold are better known than any syllogistic support that could be given to them. In order to have any knowledge, it is necessary that the basis of that knowledge come to a stop at some time. Thus Aristotle s answer amounts to saying that it is false that all propositions need to be proven with a syllogism. Some are so basic and obvious, like the principle of non-contradiction, that they are not provable in this way. All knowledge and demonstrations must eventually reduce to the primitive givens of both self-evident propositions and sense experience. Chapter 7 Summary In argumentation one acquires knowledge of a new truth by means of truths already known. Arguments provide reasons for their conclusions while assertions merely state them without support. The syllogism is the foundation of logic and is a complex expression in which given certain things being stated something else necessarily follows. The essential elements of a syllogism are three propositions (two premises and one conclusion) with three terms (major, minor, and middle). Each term appears only twice. The middle term appears only once in both premises. The major and minor terms appear once in the premises and once in the conclusion. 132 An Introduction to Traditional Logic: Argument

149 The middle term is the link between subject and predicate. The link is affirmed in an affirmative conclusion and denied in a negative conclusion. The mood of the syllogism is its arrangement of the types of propositions that make it up, e.g., EIO The figure of a syllogism is determined by the position of the middle term. Although there are four possibilities, traditionally most logicians use only the first three. The first figure is considered to be the most clear and most perfect because there the middle term is truly middle in universality. To formulate a syllogism, find the conclusion, determine what the premises are that are supposed to support this conclusion, and arrange the premises in order of major and minor. In a valid deductive argument, if the premises are true, the conclusion necessarily follows. If the premises can be true and the conclusion still false, the argument is invalid and commits a fallacy. The rules of the syllogism are the best way to determine if a syllogism is valid. A valid argument does not necessarily mean an argument s conclusion is true. A valid argument can have a false conclusion. Likewise, just because an argument is invalid, that alone does not mean its conclusion is false, but only undetermined. An invalid argument can have a true conclusion. The process of making Venn diagrams to check the validity of syllogisms is first; make a set of three overlapping circles, each one labeled for each term, next; diagram each premise by either shading out an area that the term does not cover or mark an area with an x if the term is particular, then finally; check to see if the diagram confirms the conclusion. The three conditions for any successful argument are that 1) the terms must be clear and consistent 2) the premises must be true 3) the reasoning must be valid. A failure in any of these three conditions will cause the whole argument to fail. Successfully meeting all three conditions means the argument succeeds and is sound. All knowledge and all demonstrations must eventually reduce to the primitive givens of both self-evident propositions and sense experience. Exercises A. True or False? 1. The middle term may appear in the conclusion 2. The major premise contains the minor term. 3. The minor term is the predicate of the conclusion. 4. The major term is the predicate of the conclusion. 5. The major term appears both in the major premise and is the subject of the conclusion. Argumentation & the Syllogism 133

150 6. The minor term appears both in the minor premise and as the predicate of the conclusion. 7. The major term appears only in the premises. 8. The minor term appears in the minor premise and as the subject of the conclusion. 9. In order to have a valid syllogism, the minor term needs to be distributed at least once. 10. In order to have a valid syllogism, the middle term must be in the conclusion. 11. The major premise is the conclusion, that s why it is major. 12. The major term is the link between the predicate and the subject of an affirmative conclusion. B. State the Mood 1. All men are great, No great things are ugly, therefore no men are ugly 2. All birds can sing, All birds can eat, therefore some things that can eat are things that can sing 3. No good boys are thieves, All good boys love their mothers, therefore some who love their mothers are not thieves 4. All saints are in heaven, all martyrs are saints, therefore all martyrs are in heaven 5. No Greeks like ignorance, All Athenians are Greeks, therefore no Athenians like ignorance 6. Every scientist is a smart fellow, some former students of Harvard are scientists, therefore some former students of Harvard are smart fellows 7. No villain can beat Turboman, John is Turboman, therefore no villain can beat John 8. Every intentional killing of an innocent human being is wrong, Elective abortion is the intentional killing of an innocent human being, therefore elective abortion is wrong. 9. All Scottish soldiers carried a pike, some English soldiers are not ones who carried a pike, therefore some English soldiers are not Scottish soldiers. 10. None of these cars are any good, Some of these cars are red, therefore some red cars are not any good C. State the conclusion, figure, and mood 1. No sad man is happy, some old men are happy, therefore 2. Every Mexican dish is hot, some tacos are Mexican dishes, therefore 3. No mammal is a snake, but some mammals live in the ground, therefore 4. Everything beautiful is valuable, virtue is beautiful, therefore 5. No sin is praiseworthy, but some praise is a sin, therefore 6. All dogs go to heaven, some animals are dogs, therefore 7. Every thief is a sinner, but no saint is a sinner, therefore 134 An Introduction to Traditional Logic: Argument

151 8. No soul is corruptible, every soul faces judgment, therefore 9. Some birds have feathers, every bird has legs, therefore 10. All who strive for money are unhappy, no philosopher is unhappy, therefore 11. Some logic classes are not boring affairs, all times spent with Bill are boring affairs, therefore 12. Some doctors are not good, All doctors are those who completed med school, therefore 13. Some propositions are compound, all propositions are made up of words, therefore 14. All surgeons make a lot of money, no philosophers make a lot of money, therefore 15. All pool sharks will rip you off, Sam is a pool shark, therefore 16. Every powerlifter can squat over 500 lbs, some guys at the gym are powerlifters, therefore 17. No law-abiding citizens are those who carry guns in this state, some of those guys down the street carry guns in this state, therefore 18. All country music stars like trucks, some four-wheel drive vehicles are trucks, therefore 19. Every bagpiper wears a kilt, no German wears a kilt, therefore 20. Every donut will make you gain weight, every donut is high in calories, therefore D. Finding middle terms: Construct premises that result in a valid syllogism for the following conclusions: 1. No dog is a cat 2. All plants are alive 3. All logicians are smart fellows 4. Some bad guys are in jail 5. Some professional wrestlers are great body slammers 6. No triangle has four sides 7. Some Americans are good people 8. Some politicians are not arrogant 9. All tigers eat meat 10. Some judges are fair E. Assuming the terms are clear, state which of the following are true or false about syllogisms 1. A false conclusion means the argument is invalid. 2. A true conclusion means the argument is valid. 3. An invalid argument can have a false conclusion. 4. A valid argument can have a true conclusion. 5. An invalid argument can have a true conclusion. Argumentation & the Syllogism 135

152 6. A valid argument can have a false conclusion. 7. When the premises are true and the argument valid, the conclusion is true. 8. When the premises are false and the argument valid, the conclusion is false. 9. When the premises are true and the argument invalid, the conclusion is undetermined. 10. When the premises are false and the argument invalid, the conclusion is false. 11. If the conclusion is false, the premises must be true. 12. If the conclusion is true, the premises must be valid. 13. Valid conclusion means true premises. 14. When one has true premises but a false conclusion, the problem is in the validity of the reasoning. 15. When one has a true conclusion but false premises, the problem is in the validity of the reasoning. F. Determine the validity of the following using only the rules of validity: 1. All rich people are hard working, no philosophers are rich, therefore, no philosophers are hard working. 2. Whatever printed in a book is only man made, historical events are printed in books, therefore historical events are only man made. 3. Most psychologists are atheists, some atheists are morally upright, therefore some psychologists are morally upright. 4. Every cat likes to eat mice, every hawk likes to eat mice, therefore every hawk is a cat. 5. All communists are followers of Marx, some communists like Lenin, therefore some who like Lenin are followers of Marx. 6. Anyone who knows the most is the best, those who win the race are the best, therefore those who win the race are those who know the most. 7. All water conducts electricity, no wood conducts electricity, therefore no wood is metal. 8. Nothing with bad members is good, some political groups have bad members, therefore no political groups are good. 9. Most who work here are not liars, but Bill works here, therefore Bill is not a liar. 10. No one who is promiscuous is virtuous, but every girl suitable for marriage is virtuous, therefore no girl suitable for marriage is promiscuous. 11. No person can get out of paying taxes, but no one who pays taxes is one who is rich, therefore no one can be rich. 12. Whatever begins to exist has a cause, the universe began to exist, therefore the universe has a cause. 13. Every nice guy is a good friend, but Bill is a nice guy, therefore Bill is a good friend. 14. Every nice guy is a good friend, but Bill is not a nice guy, therefore Bill is not a good friend. 136 An Introduction to Traditional Logic: Argument

153 15. All drunk drivers are a menace to society, and anyone who is a menace to a society should be locked up, therefore those who should be locked up are a menace to society. 16. All bark is that which grows on trees, all dogs can release a bark, therefore all dogs can release that which grows on trees. 17. Some Muslims are terrorists, Tim is a terrorist, therefore Tim is a Muslim. 18. Some of the original followers of Jesus were Apostles, no original follower of Jesus was from North America, therefore some Apostles were not from North America. 19. Anyone who is scared is probably guilty, some of those guys from that gang are scared, therefore some of those guys from that gang are probably guilty. 20. Nothing that causes pain can cause gain, but every hitting of your thumb with a hammer causes pain, therefore every hitting of your thumb with a hammer can cause gain. 21. Some animals are snakes, nothing warm blooded is a snake, therefore something warm blooded is not an animal. 22. Whatever contains a high amount of fat is bad for your heart, but some meats contain a high amount of fat, therefore some meats are bad for you heart. 23. Some things bad for your heart contain high amounts of fat, some meats contain high amounts of fat, therefore some things bad for your heart are meats. 24. Some ironworkers are troublemakers, no knitters are troublemakers, therefore no ironworkers are knitters. 25. Some ironworkers are troublemakers, no knitters are troublemakers, therefore some ironworkers are not knitters. 26. All bodies are extended in space, No soul is extended in space, therefore no soul is a body. 27. Violence is wrong, war is violence, therefore war is wrong. 28. Some violence is wrong, some wars are violence, therefore some wars are wrong. 29. Whatever cannot be conceived as not existing must exist, God cannot be conceived as not existing, therefore God must exist. 30. Real knowledge must be about things, philosophy is real knowledge, therefore philosophy must be about things. 31. Man is a word, Socrates is a man, therefore Socrates is a word. 32. Nothing that occurs by chance and accident happens often, all laws of nature occur often, therefore no laws of nature are things that happen by chance and accident. 33. All cats are four-legged, some cats are Siamese, therefore some Siamese are four-legged. 34. All conservatives are ignorant, the majority of this population is ignorant, therefore the majority of this population is conservative. Argumentation & the Syllogism 137

154 35. People who do not philosophize do not know the meaning of life, some who do not know the meaning of life are hopelessly lost, therefore some who are hopelessly lost are those who do not philosophize. 36. No one who is really good at logic is one who doesn t study, some students are really good at logic, therefore some students are not those who don t study. 37. Leaders are not followers, some leaders are mental midgets, therefore some mental midgets are not followers. 38. No chimpanzee can communicate with universal concepts, all beings with an intellect can communicate with universal concepts, therefore no being with an intellect is a chimpanzee. 39. Whatever has different properties cannot be identical, brain events and mind events have different properties, therefore brain events and mind events cannot be identical. 40. No one owes anything to unintelligent objects, John is unintelligent, therefore no one owes anything to John. 41. Everyone must have a license to drive, no one under age sixteen can have a license, no one under sixteen can drive. 42. All power corrupts, knowledge is power, therefore knowledge corrupts. 43. Some people who complain about this country should move away, Some Californians are people who complain about this country, some Californians should move away. 44. Some brides do not like shotgun weddings, Sally is a bride, therefore Sally does not like shotgun weddings. 45. All humans can learn, no dog is human, therefore no dog can learn 46. All heretics should repent, some Frenchmen are heretics, therefore all Frenchmen should repent. 47. Some of the best bikes are made in Canada, this bike here is the best bike, therefore this bike here is made in Canada. 48. No one who goes to Heaven is lost, everyone without a city map is lost, therefore no one without a city map goes to Heaven. 49. Some beautiful music is from Chopin, All music by Mozart is beautiful, therefore some music by Mozart is from Chopin. 50. Some of my dishes are dirty, no unused dishes are dirty, therefore some unused dishes are not some of my dishes. G. Using Venn diagrams, check the arguments in exercise F for validity. 138 An Introduction to Traditional Logic: Argument

155 Chapter 8: Valid Syllogistic Forms & Reduction to the First Figure àà Reducing Syllogistic Forms to the First Figure We have discussed two ways of determining the validity of the syllogism, first through the use of the rules of the syllogism and secondly through the use of Venn diagrams. This chapter covers yet a third technique that was a product of the scholastic logicians for recognizing validity, namely, by memorizing the valid forms of a syllogism. Reducing Syllogistic Forms to the First Figure The last chapter mentioned that the form of a syllogism is the mood plus the figure. If you were to take the possible moods and multiply them by the possible figures you would get a number of 256 possible syllogistic forms. However, we do not bother with all of these because most of them are invalid. We focus on only the valid Valid Syllogistic Forms & Reduction to the First Figure 139

156 ones. The medieval logicians developed an ingenious mnemonic device for not only memorizing the valid forms of a syllogism, but also for determining how to reduce the various forms to the first figure. They did this by giving each of the valid forms a name. These names for each form will sound a bit bizarre, but the reason they have the names they do is because the name contains an impressive amount of information. The vowels in the name give us the mood, while the consonants immediately following the vowels tell us how the syllogism reduces to the first figure, which as we said, is considered the ideal form. The scholastics memorized the different valid forms of syllogisms by learning this phrase: 62 Barbara, Celarent, Darii, Ferioque prioris; Cesare, Camestres, Festino, Baroco secundae; Tertia Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison habet. Below is a chart listing these valid syllogistic forms: 62 We leave out here what some logicians would include, namely the fourth figure names of Bramantip, Camenes, Dimaris, Fesapo, and Fresison. We also do not include the subaltern moods (which go by the name of Barbari and Celaront in Figure 1, Cesaro and Camestrop in Figure 2, and Camenop in Figure 4) that draw only a particular conclusion when a universal could be drawn. 140 An Introduction to Traditional Logic: Argument

157 The first figure is the most perfect, but the second figure, since it always has a negative conclusion, is good for refuting an adversary, and the third figure forms, since they always end with a particular conclusion, are good for showing that something is not universally true because a counterexample can be shown. The reason why it is a good practice to be able to reduce the valid forms to the first figure is because, if you ll remember, the first is the most perfect figure. It is only in the arrangement of the first figure alone that the middle term is truly middle in universality. Ideally then it is good practice to be able to reduce second and third figures back to the first figure. There is a set of rules for validly doing this and these names of the various forms help us here. The good thing about these rather odd Latin names for the different syllogisms is that they indicate not only what mood they represent, but they also pack a tremendous amount of information on how to reduce them to the first figure. The following list of pointers will help you unpack the meaning in these names: Each name consists of three syllables and those syllables correspond to the three propositions in a syllogism. The vowels in each syllable indicate the mood of the syllogism. Take the first one, the syllogism Barbara tells us that the mood of this syllogism is AAA, that is, three universal affirmative propositions are indicated by the vowels ( Bar-bar-a ). The syllogism Celarent (Ce-lar-ent) stands for the mood EAE, and so on. The first letter of the name in the second and third figures indicates the first figure syllogism to which it reduces. So the names under the first figure are the syllogisms to which the second and third figures reduce. For example, Baroco reduces to Barbara, Cesare reduces to Celarent, Festino to Ferio, etc. The consonants after each vowel tells the means by which that proposition, represented by the prior vowel, reduces to the first figure. These can be one of four possibilities; simple conversion, conversion by limitation (per accidens), changing the premises, reduction by contradiction. The first two points above are simple and straightforward and so the remainder of this chapter focuses on the last point. There are four ways in which the proposition reduces to a proposition in the first figure and these ways are indicated by the letters S, P, M, and C. These letters refer to simple conversion, conversion by limitation, transposition, and reduction by contradiction. If there is any other letter, then nothing is done to the proposition at all and it is kept as is. We already saw how to do the first two (simple conversion and conversion by limitation) in Valid Syllogistic Forms & Reduction to the First Figure 141

158 chapter 6, while the last two are new. Nevertheless it is still useful now to summarize all four: S stands for simple conversion. When S follows a vowel, that means the proposition represented by the previous vowel is reduced by simple conversion (simply switching the subject and predicate). This process applies only to E and I propositions; e.g., No snake is a mammal = No mammal is a snake, Some snake is a reptile = Some reptile is a snake. P stands for conversion by limitation (the P comes from per accidens). This applies only to A propositions. The subject and predicate are switched with the new proposition limited to a particular, i.e., Every mammal is an animal = Some animals are mammals. M stands for transposition (from the Latin mutatio). The major and minor premise simply transpose by trading places. The major becomes the minor and the minor becomes the major. C stands for reduction by contradiction. This applies only to Bocardo and Baroco. Reduction by contradiction amounts to a Barbara syllogism that shows if one accepts the premises in either Bocardo or Baroco but not their conclusion, then a contradiction results. This technique is both indirect and a bit complex and so will be explained in detail below. Remember the goal here is to take the valid syllogisms that are not in the first figure and validly reduce them to the first figure. Arguments in the first figure are clearer and their reasoning more evident, and this is because the middle term is occupying the middle position in first figure syllogisms. We will walk through two examples. First, we take the second figure syllogism Cesare: No mammal is a reptile Every snake is a reptile No snake is a mammal You can see the middle term reptile is in the figure II position and the mood is of course EAE represented by the name Cesare. Now from this name we know that Cesare, because it begins with C, must reduce to the first figure syllogism Celarent. To do this, all we need to do is look at the vowels of Cesare, and we see that the first letter following the first vowel is s (Ces) which tells us that the first E, the major premise, reduces by simple conversion. When this is done, the original major, No mammal is a reptile becomes No reptile is a mammal. The letter following the 142 An Introduction to Traditional Logic: Argument

159 next vowel in Cesare is not relevant so we do not have to do anything else. The result of this reduction then is: No reptile is a mammal [simple conversion] Every snake is a reptile No snake is a mammal By simple conversion of the major premise, Cesare becomes Celarent and so the middle term is now in the first figure position. Taking another example, look at an instance of the third figure syllogism Darapti: All logicians are uptight All logicians are very educated people Therefore, some very educated people are uptight Here we have the mood AAI and the first letter of Darapti tells us that it reduces to the first figure syllogism Darii. The letter following the first vowel in Darapti is not one of the four important ones and so we do nothing with the major. However the second vowel in Darapti represents the minor premise and it is followed by an p which stands for conversion by limitation. So we convert the minor of Darapti this way, and so All logicians are very educated people becomes Some very educated people are logicians. Nothing else needs to be done and we now have the first figure syllogism Darii: All logicians are uptight Some very educated people are logicians [conversion by limitation] Therefore, some very educated people are uptight Let us now look at an example that will require us to transpose the premises. Take the second figure syllogism Camestres: Every wrestler is a tough guy No golfer is a tough guy Therefore, no golfer is a wrestler We know we have to reduce this to the first figure Celarent, and the first thing we see after the first vowel in Camestres is m (mutatio) meaning transposition. So this major premise must become the minor premise to make the reduction. Additionally, the letter s follows both the second premise and the original conclusion so these both need to reduce by simple conversion. So here we have to do something to all three propositions to end up with Celarent: Valid Syllogistic Forms & Reduction to the First Figure 143

160 No tough guy is a golfer [simple conversion] Every wrestler is a tough guy [transposed from original major] Therefore, no wrestler is a golfer [simple conversion] Now Baroco from figure II and Bocardo from figure III merit special attention. Technically speaking, one of the reasons is because they both contain A and O premises, and to convert the A means you would have two particular premises (a violation of one of the rules of validity) since the O does not convert at all. So we have to do something different here with these two, and this is called reduction by contradiction. The rule for reduction by contradiction is this: take the contradictory of the conclusion and exchange it with the O premise. When this is done, there will be a new Barbara syllogism with a conclusion that contradicts one of the original premises, which shows one cannot accept the premises of the original syllogism without also accepting its conclusion. Walking through an example of each, first take this example of Baroco: Every animal is sensible Some stone is not sensible Therefore, some stone is not an animal If one denies the conclusion but accepts both premises, one can reduce to Barbara by contradiction. To deny the conclusion is to accept its contradictory, so we state this as Every stone is an animal. This is the contradictory of the conclusion Some stone is not an animal. Next take the conceded major from the former syllogism and use it as a major in the new syllogism with the new premise above as a minor: Every animal is sensible [from the original] Every stone is an animal [contradictory of the original conclusion] Therefore every stone is sensible [contradicts the accepted minor premise of the original] Now we have a new Barbara syllogism with a conclusion that contradicts the minor premise of the previous syllogism, and as we said, our opponent accepted that premise. Therefore to deny the prior conclusion and yet accept the minor premise in the prior syllogism is to affirm a contradiction. It cannot both be true that some stone is not sensible and every stone is sensible! Doing this again, we now look at an example of Bocardo: Some bird is not a flyer Every bird is a thing with a beak Therefore some things with beaks are not flyers 144 An Introduction to Traditional Logic: Argument

161 Again, suppose an opponent accepts the premises but denies the conclusion. So we take the contradictory of the conclusion (because this is what it means to deny the conclusion), Every thing with a beak is a flyer. Next we form a Barbara syllogism by taking this new proposition as a major premise and combining it with the original minor premise: Every thing with a beak is a flyer [contradictory of the original conclusion] Every bird is a thing with a beak [from the original] Therefore, every bird is a flyer [contradicts the accepted major premise of the original] Again this shows that if one accepts the premises in the prior argument but not its conclusion, they contradict themselves. The following charts show how the II and III figures reduce to the first: Valid Syllogistic Forms & Reduction to the First Figure 145

162 Chapter 8 Summary The first figure is more perfect because the arrangement of the middle term makes these syllogisms more clear The Latin names of the valid syllogism provide information about the mood of the syllogism they represent and if they represent second or third figure syllogisms, they also provide the methods of reduction The valid second and third figures reduce to the first through various uses of simple conversion, conversion by limitation, transposition, or reduction by contradiction 146 An Introduction to Traditional Logic: Argument

163 Exercises A. First determine the name of the following forms of the syllogism and then reduce them to the first figure: 1. Every P is M No S is M Therefore, no S is P 2. Some flowers are not pretty All flowers are plants Therefore, some plants are not pretty 3. No good person is a backstabber Every gossiper is a backstabber Therefore, no gossiper is a good person 4. Every sailor is on a ship Every sailor likes to eat Therefore, some on a ship like to eat 5. Some soldiers carry a rifle All soldiers are in the Army Therefore, some in the Army carry a rifle 6. All soldiers are in the Army Some soldiers carry a rifle Therefore, some who carry a rifle are in the Army 7. Some Germans were not part of the Nazi party All Germans lived in Germany Therefore, some who lived in Germany were not part of the Nazi party 8. No musician is without an instrument Some musicians are happy Therefore, some who are happy are not without an instrument 9. Every homemaker cleans the home No businessman cleans the home Therefore, no businessman is a homemaker 10. No adults here like desert Some kids like desert Therefore, some kids are not adults 11. Every baby cries Some adults do not cry Therefore, some adults are not babies Valid Syllogistic Forms & Reduction to the First Figure 147

164

165 Chapter 9: Other Types of Syllogisms àà Compound Syllogisms àà The Dilemma àà The Enthymeme àà Sorites àà The Epicheirema àà Expository Syllogisms Compound Syllogisms We have been looking at the categorical syllogism where the major premise is a categorical proposition and the key elements are terms. Now in this chapter we consider other kinds of syllogisms that can be collectively called compound syllogisms. A compound syllogism occurs when the major premise is a compound proposition and the minor affirms or destroys one of the parts of the major. Unlike a categorical syllogism that depends upon a middle term, the compound syllogism depends upon a connection of propositions that are related conditionally, disjunctively, or conjunctively, and for this reason there are three kinds of compound syllogisms; the conditional, the disjunctive, and the conjunctive. A compound syllogism occurs when the major premise is a compound proposition and the minor affirms or destroys one of the parts of the major. Other Types of Syllogisms 149

166 The Conditional Syllogism In the conditional syllogism, the major premise is a conditional (If then) proposition while the minor either affirms the antecedent or destroys the consequent. In the conditional syllogism, the major premise is a conditional (If then) proposition while the minor either affirms the antecedent or destroys the consequent (remember in a conditional proposition, the antecedent is the if part and the consequent is the then part). Take for example: If Socrates is thinking, then he exists Socrates is thinking Therefore, Socrates exists This argument reaches a conclusion by first positing a conditional premise and then affirming the antecedent. Since the antecedent is affirmed, this way of using the conditional syllogism is appropriately called the mode of affirming or by its Latin name, modus ponens. The modus ponens argument is the first valid form of the conditional syllogism. The second valid form comes about when one reaches a conclusion by denying the consequent: If Socrates is thinking, then he exists Socrates does not exist Therefore, Socrates is not thinking Since this argument reaches its conclusion by denying the consequent it is called the mode of denying or in Latin, modus tollens. Both the mode of affirming and the mode of denying are valid because the supreme principle to a valid conditional syllogism is the same as a conditional proposition. 63 With the major premise, there must be a necessary connection between the antecedent and the consequent. If the antecedent can be true without the consequent also being true, the major does not contain a valid sequence and the syllogism fails. In order to have a valid sequence, it is impossible that the antecedent be true and the consequent be false. So the idea here is that the mode of affirming is based on this principle in that we know if the antecedent is true, the consequent must also be true. The mode of denying is also based on this principle in that we know if the consequent is false, the antecedent must also be false. But this necessary connection does not work in reverse. It can be the case that the antecedent is false and yet the consequent still be true. This is because there could be some other cause for the truth of the consequent. So we must be aware of two fallacies here. The first is called the fallacy of affirming the consequent and runs like this: Fallacy of Affirming the Consequent: If Socrates is thinking, then he exists Socrates exists Therefore, Socrates is thinking 63 For a review of conditional propositions, see p An Introduction to Traditional Logic: Argument

167 Now this is clearly false and the problem lies in assuming that the connection between antecedent and consequent is symmetrical or bidirectional (the same both ways). Clearly Socrates can exist and not be thinking, and so the consequent can be true and the antecedent still be false. For this reason, you cannot affirm the consequent as true and still expect the antecedent to also be true. The same fundamental error is involved in the second fallacy, the fallacy of denying the antecedent: Fallacy of Denying the Antecedent: If Socrates is thinking, then he exists Socrates is not thinking Therefore, Socrates does not exist Both of these fallacies with the conditional syllogism result from misunderstanding the direction of the conditional relationship. Consequently it is very important not to confuse the valid modus ponens, with the fallacy of affirming the consequent, nor to confuse the valid modus tollens with the fallacy of denying the antecedent. In order to remember the difference between the valid forms and the fallacies, the following rules will help. Rules for the Conditional Syllogism: 1. To affirm the antecedent is to affirm the consequent, but to affirm the consequent is not to affirm the antecedent; e.g., if it rains then the ground will be wet, but the ground may be wet without it having rained. 2. To deny the consequent is to deny the antecedent, but to deny the antecedent is not to deny the consequent; e.g., if the ground is not wet then it has not rained, but just because it has not rained does not mean the ground is not wet. The modus tollens merits special attention and this is because at times it can be beneficial to argue indirectly, that is, one can indirectly establish their own position by disproving the position that is opposed to it. This type of argumentation attempts to show that if an opponent s position is true, absurd consequences follow and therefore the opponent should abandon their original position. This is quite an effective means of argument since it starts with the opponent s own position, while yet showing the absurd conclusions that follow. This form of argument is called a reductio ad absurdum (reduction to absurdity), since basically what it says is, If what you say is true, then ridiculous conclusion X follows, but ridiculous conclusion X is false, therefore what you say must also be false. For example, an argument from moral obligation (that there are some acts in which we are truly and morally obligated to do and others in which we are truly and morally obligated to avoid, in spite of what Other Types of Syllogisms 151

168 other humans may think about it) is sometimes used to disprove atheistic evolution (the position that God does not exist and all humans are a product of unintelligent chance) by a reductio ad absurdum: If atheistic evolution is true, then there are no moral obligations But there are moral obligations (it is absurd to think we are not morally obligated to avoid say, not harming an innocent child for fun) Therefore atheistic evolution is false. Thus, this argument hinges on the recognition that we are more certain of moral obligations than we are of being the products of blind purposeless chance. Take another example. Here is a reductio ad absurdum argument supporting the notion that mathematical entities like numbers are real and exist independently of the mind: If numbers are just ideas in our minds, then we can truly make = 187 (or anything else for that matter) just by thinking about it But we cannot truly make = 187 just by thinking about it, that is absurd Therefore, numbers are not just ideas in our minds. Here the reductio ad absurdum hinges on the recognition that the truths of mathematics are both objective and necessary, that is, we discover mathematical facts and do not make them up, and these facts are always true, so it would be impossible and absurd to say that numbers are things existing only in your mind or grounded in contingent things. In any case, you can see in these examples that modus tollens has a special role of serving as a means of establishing one s position indirectly by means of eliminating the opposition. Boethius Enumeration of Conditional Syllogisms The philosopher Boethius (c ) developed a relatively advanced enumeration of the different types of conditional syllogisms. The full list goes beyond the introductory stage of logic, but we include here his initial enumeration of six valid types: If A is, B is; but A is; therefore B is 2. If A is, B is, but B is not; therefore A is not 3. If A is, B is, and if B is, C must be; but then: if A is, C must be 4. If A is, B is, and if B is, C too must be; but C is not, therefore A is not 5. If A is, B is, but if A is not, C is; I say therefore that if B is not, C is 6. If A is, B is not; if A is not, C is not; I say therefore if B is, C is not 64 Boethius, De Hypotheticis Syllogismis, enumeration and translation taken from Bochenski, p An Introduction to Traditional Logic: Argument

169 Just from this initial listing by Boethius, we can appreciate how complex this topic might become. The Disjunctive Syllogism In a disjunctive syllogism, the major premise is a disjunctive proposition (either or); while the minor either posits or destroys one of the major s disjuncts. Now the section on disjunctive propositions made a distinction between strong disjunctions and weak disjunctions. As you will remember, strong disjunctions exclude the possibility of both disjuncts being true and the possibility of both being false. Strong disjunctions are strong either/or statements, one must be true and one must be false, e.g., either Lincoln was a US president or he was not. Weak disjunctions, however, require only one disjunct to be true but allow for the possibility of both, e.g., either this man is a fool or a criminal. Accordingly, we call the type of syllogism that makes use of a strong disjunction a strong disjunctive syllogism and those that use the weak disjunction a weak disjunctive syllogism. In a disjunctive syllogism, the major premise is a disjunctive proposition (either or); while the minor either posits or destroys one of the major s disjuncts. For instance, here is an example of a strong disjunctive syllogism: Either the product of 33 x 122 is even or odd The product of 33 x 122 is even Therefore the product of 33 x 122 is not odd The nature of the major premise is a strong disjunction, a number cannot be both even and odd, so the truth of one disjunct implies the falsity of the other and vice versa. This is the strength of strong disjunctive syllogisms; the options are exclusive and because of this the minor can be either an affirmation or denial of a disjunct and the conclusion still be valid. This is not true however with weak disjunctive syllogisms. In order for a weak disjunctive syllogism to be valid, the minor can only be a denial of one of the disjuncts. An affirmation of one disjunct in the minor premise of a weak disjunctive syllogism is not enough to rule out the other, since weak disjunctions, by their very nature, allow for both to be true. For instance: Either he is going to work or he is singing a song He is singing a song Therefore, he is not going to work This is invalid since the major premise is not a strong disjunction and the minor is only an affirmation. After all, the man may be singing on the way to work and thus both disjuncts may be true. The point again here is that in a weak disjunctive Other Types of Syllogisms 153

170 syllogism, the affirming of one disjunct does not lead to a denial of the other. Denying a disjunct however, works just fine: Either he is going to work or he is singing a song He is not singing a song Therefore, he is going to work This is valid because given the truth of the major premise, the conclusion necessarily follows, since denying one of two alternatives necessarily leaves the other (remember the truth condition for weak disjunctions is that only one disjunct need be true). By elimination, the minor simply destroys one possible option and the conclusion affirms what remains. To summarize, one ought to know these rules about the disjunctive syllogism: Rules for the Disjunctive Syllogism: A strong disjunctive syllogism is valid when the minor is an affirmation of a disjunct or when it is a denial of one of the disjuncts. Either way, the strong disjunctive syllogism is valid in these cases. A weak disjunctive syllogism is valid only when the minor is a denial of one of the disjuncts. It is not valid when the minor is an affirmation. If this rule is violated, it commits the fallacy of affirming a weak disjunct. Now given that there are at least two disjuncts in a disjunctive proposition, and that one can affirm or deny either disjunct in the minor, there are four possible moods of the disjunctive syllogism. Using P and Q as variables for the disjuncts, the moods are: 1. Either P or Q P Therefore, not Q 2. Either P or Q Not P Therefore Q 3. Either P or Q Q Therefore not P 4. Either P or Q Not Q Therefore P 154 An Introduction to Traditional Logic: Argument

171 As should be clear when we apply the two rules above to these possible moods, when the argument is a strong disjunctive syllogism, all four moods are valid. However, if the argument is a weak disjunctive syllogism, only moods 2 and 4 (where the minor is a denial) are valid. Finally it should be mentioned that weak disjunctive syllogisms with more than two disjuncts are treated like those with only two, with the understanding that we must deny all alternatives but one, as in this example: Sam either went to the store, ate lunch, or took a nap. He did not go to the store or take a nap Therefore Sam ate lunch Strong disjunctive syllogisms can also have multiple disjuncts, that can be valid when affirmed or denied in the minor. Yet since this is a strong disjunction we are talking about, we need only affirm one disjunct in the minor, but if we are going to deny, again we must deny all but one remaining alternative if the argument is going to be valid. The argument from design for God s existence (a designer of the universe) does this: The complexities involved in a life-permitting universe are ultimately due to either necessity, chance, or design. The complexities involved in a life-permitting universe are not ultimately due to necessity or chance Therefore, the complexities involved in a life-permitting universe are ultimately due to design. Here is another example of a strong disjunctive syllogism with a denial in the minor: Sam either received an A, B, C, D, or F on the exam. Sam did not receive an A, B, C, or D Therefore Sam received an F on the exam Conjunctive Syllogism A conjunctive syllogism is simply a syllogism with a conjunctive proposition as a major premise. The major is a denial that both conjuncts can be joined, and the minor affirms or denies one of the conjuncts. For example: No man can serve both God and mammon This man is serving mammon Therefore this man is not serving God. A conjunctive syllogism is simply a syllogism with a conjunctive proposition as a major premise. Other Types of Syllogisms 155

172 But we must be careful. The rule for validity here depends on the nature of the conjuncts. If the conjuncts are exhaustive of the possibilities, a denial in the minor is valid. For example: No whole number can be both odd and even This whole number is not even Therefore this whole number is odd This example is perfectly valid because there are only two possibilities, odd and even, and so here the minor can be a denial. But take this for instance: No man can be in school and at home at the same time This man is not in school Therefore this man is at home This is invalid because the conjuncts are not exhaustive. A man could be in numerous other places than at home when he is not in school. When this type of conjunctive proposition occurs that allows for multiple possibilities not listed in the proposition itself, only an affirmation of one of the conjuncts in the minor is valid: No man can be in school and at home at the same time This man is in school Therefore this man is not at home Thus the rule of validity for conjunctive syllogisms is this: if the conjunctive major exhausts the possibilities, the minor can either affirm or deny one of the conjuncts. If the conjunctive major does not exhaust the possibilities, the minor can only affirm one of the conjuncts. 65 The Dilemma The dilemma states a disjunction by which two alternatives or horns emerge, and then conditional propositions carry these horns to their pointed conclusion. The dilemma is a frequent weapon of attack in philosophy. Fundamentally, the dilemma states a disjunction by which two alternatives or horns emerge, and then conditional propositions carry these horns to their pointed conclusion. The simple formula runs like this: Either A or B If A then C If B then C Therefore C 65 The rules for conjunctive syllogisms with more than one conjunct are complex and go beyond our purposes here. 156 An Introduction to Traditional Logic: Argument

173 The conclusion of a dilemma might also be complex in that it has more than one alternative, as in this formulation: Either A or B If A then C If B then D Therefore either C or D Looking at some examples in the history of philosophy, first, Socrates makes use of a dilemma in a passage from the Meno on the paradox of acquiring knowledge. How can we learn what we do not know? Put into a formal dilemma, Socrates reasons: Either you already know what you need to learn or you do not. If you already know what you need to learn, then you do not need to learn it. So learning is unnecessary. If you do not already know what you need to learn, then you do not know what you need to learn and so learning is impossible. Therefore, learning is either unnecessary or impossible. Perhaps one of the most well known examples of a dilemma, which also originates from Socrates, is the famous Euthyphro dilemma. This dilemma argues that God cannot have anything to do with what counts as morally good or evil because that would mean moral norms are either the result of God s arbitrary whim or that there is a moral law independent of God s will: Either an action is good because God prefers it or God prefers an action because it is good. But if an action is good just because God prefers it, then this is morally disastrous for God could just arbitrarily decide one day that something like genocide is a good act. If God prefers an action because it is good, then this is theologically disastrous since that means there is goodness independent of God. Therefore good actions do not depend on God. Another well-distinguished example of a dilemma is the argument called Pascal s Wager. This dilemma aims to establish the foolishness of atheism: Either God exists or he does not. If God exists and you do not believe in him, your soul will be damned. If God does not exist and you do not believe in him, you gain nothing for this unbelief. Therefore, those who do not believe in God either gain nothing for their unbelief or they will be damned because of it. Other Types of Syllogisms 157

174 This is a dilemma taken from St Thomas Aquinas on why it is impossible for man s happiness to consist in wealth: Wealth is either natural (i.e., food and water) or artificial (i.e., money). If natural, then it is impossible for man s happiness to consist in this kind of wealth, because these are only means to the end of overall health and happiness. Thus, this type of wealth cannot be the ultimate end of happiness. If artificial, then it is also impossible for man s happiness to consist in this type of wealth, since the good of artificial wealth is only good as a means for acquiring natural wealth, and natural wealth, as we have already said, is only a means and not the ultimate end of happiness. Therefore man s happiness cannot consist in wealth. As you can see, fundamental to any dilemma is a major disjunction that sets up two alternatives, followed by conditional propositions that carry out those alternatives to some conclusion (usually a bad one). Note that these conditional minors are what makes a dilemma different from a disjunctive syllogism, since the minor in a disjunctive syllogism only affirms or denies one of the disjuncts. Not surprisingly, there are different ways of stating dilemmas, one may use them to reach a negative conclusion (a destructive dilemma) or one may use them to reach a positive conclusion (a constructive dilemma), and as we said, sometimes the conclusion is simple (only one option) and sometimes it is complex (more than one option). There are three principal ways of opposing a dilemma: 1. Escaping Between the Horns: This method finds an alternative not given in the dilemma itself. In this way, one escapes from being impaled on the horns by choosing a third option not given in the disjunction. It is important then when formulating dilemmas that the alternatives be exhaustive. The fallacy of the false dilemma arises when there are more options than what the dilemma provides for in the major. In this way, Either Socrates is here or he is dead is a false dilemma, since Socrates could be somewhere else and still be alive but Either Socrates is alive or he is not does not leave any third alternative. 2. Taking the Dilemma by the Horns: This method accepts the truth of the major disjunction by admitting there is not another alternative, but still denies one or both of the conditional minors. In other words, the disjunction is accepted but at least one of the conditional minors is disputed. 3. Rebutting a Dilemma: This method constructs a counter dilemma based on the original disjunction, with a conclusion that denies the original conclusion. A classic example from many logic texts is an example from 158 An Introduction to Traditional Logic: Argument

175 ancient Greece. A mother tries to talk her son out of going into politics by arguing, If you tell the truth men will hate you, and if you do not tell the truth the gods will hate you, therefore either way you will be hated. The son replies by posing a counter dilemma, If I tell the truth the gods will love me and if I do not tell the truth men will love me, therefore either way I will be loved. So you can see that this method of responding to a dilemma does not affect the original argument; the original dilemma is accepted but the focus of the alternatives changes. The Enthymeme The enthymeme is simply a syllogism with a missing but implicit proposition. 66 Sometimes the enthymeme is called an imperfect syllogism because one of its propositions is tacit. Common everyday discussion makes use of enthymemes all the time. Take this exchange for example: The enthymeme is simply a syllogism with a missing but implicit proposition. Bill: John will never get a good job Sam: Why? Bill: Because he did not complete college. Now this argument, as it sits, is an enthymeme. There is an assumed but missing major premise. Can you figure out what it is? Put logically in a complete syllogism the reasoning would look like this: Whoever does not complete college will never get a good job John did not complete college John will never get a good job. The major premise assumed in the dialogue is made explicit here. When faced with enthymemes, it is actually not too difficult to discover the missing proposition when we consider that every syllogism has three terms each appearing twice, and since the enthymeme provides two propositions (in this case the conclusion and the minor premise), it also provides all three terms. To formulate the missing proposition, one simply takes whichever term of the conclusion that has appeared only once in the premises and unites it with the middle term. In the above example, the enthymeme provided the minor, John did not complete college and the conclusion, John will never get a good job. The minor term here is John, and since it appears twice, we know we do not need to supply the minor premise. The middle term is the one that 66 The abridged syllogism is called enthymeme from the Greek words en meaning in and thymos meaning mind, thus, enthymeme stands for a syllogism having one proposition that is in the mind. Other Types of Syllogisms 159

176 does not appear in the conclusion and so here is, (one who) did not complete college. The major term (one who) will never get a good job however appears only once, which tells us we need to supply the missing major premise. To formulate this missing major premise, we put the middle term with the major to get, Whoever does not complete college (is one who) will never get a good job. The same procedure would be true if it were the minor that was missing. Now the tacit proposition in a valid enthymeme can be either of the two premises or the conclusion. In fact, depending upon which one it is, there are three different types of enthymeme: First order enthymeme: An enthymeme with an unstated major premise Second order enthymeme: An enthymeme with an unstated minor premise Third order enthymeme: An enthymeme with an unstated conclusion. 67 Since an enthymeme is just a syllogism with an implicit proposition, it does not constitute a different type of argument, but awareness of it is very important since so many arguments have suppressed premises. Sorites Syllogisms can be strung together where the conclusion of one syllogism can become a premise in the next to make a long chain of an argument. The sorites is an extended syllogism that consists of several individual syllogisms strung together. These individual syllogisms that make up the parts of the sorites are usually expressed as third order enthymemes. There are two types of sorites depending on whether the first premise contains the subject or the predicate. Progressive Sorites The progressive or Aristotelian sorites is when the first premise contains the subject and the syllogism progresses to the final predicate. This type follows the pattern S is A, A is B, B is C, C is P, therefore S is P. Spelling this pattern out: Every man is an animal Every animal is a living thing 67 Aristotle recommends using this enthymeme as a tactic of rhetorical persuasion because the listeners are left to draw out the conclusion for themselves. 160 An Introduction to Traditional Logic: Argument

177 Every living thing is a substance Every substance is an existing thing Therefore, every man is an existing thing The subject man in the original premise is brought back in at the end after a series of expanding on the predicate animal culminating in existing thing. The conclusion utilizes the subject of the first proposition and validly unites it with the predicate of the last. This series is actually a string of third order enthymemes since the conclusion is not explicitly stated at each step. To spell out the above sorites, it would look like this: Every man is an animal Every animal is a living thing Therefore, every man is a living thing Every man is a living thing Every living thing is a substance Therefore, every man is a substance Every man is a substance Every substance is an existing thing Therefore, every man is an existing thing The validity of progressive sorites is governed by two special rules: 1. Only the first premise may be particular This is to ensure that the middle terms are always distributed. 2. Only the last premise may be negative This is to ensure that one avoids the fallacy of illicit major. Regressive Sorites The regressive or Goclenian sorites 68 is when the first premise provides the predicate and the syllogism regresses or works its way back to the final subject. The regressive sorites follows the pattern A is P, B is A, C is B, S is C, therefore S is P. For instance: Every substance is an existing thing Every living thing is a substance Every animal is a living thing 68 Named after the Aristotelian commentator Goclenius ( ). Other Types of Syllogisms 161

178 Every man is an animal Therefore, every man is an existing thing Again, the predicate of the first premise is brought back in at the end to validly unite with the final subject man. The regressive sorites, like the progressive, is also actually a string of third order enthymemes and when spelled out looks like this: Every substance is an existing thing Every living thing is a substance Therefore every living thing is an existing thing Every living thing is an existing thing Every animal is a living thing Therefore every animal is an existing thing Every animal is an existing thing Every man is an animal Therefore, every man is an existing thing The special rules of validity for regressive sorites are: 1. Only the first premise may be negative 2. Only the last premise may be particular The Epicheirema The epicheirema is simply a syllogism with a causal premise. The term was originally meant by Aristotle to cover probable argumentation, but today means a syllogism that contains its own support for one or both of its premises. To take an example: Every PhD in botany is one who knows a lot about it, because they have studied that subject for years John is a PhD in botany Therefore, John is one who knows a lot about it The major premise presents a reason for its own truth, and so this is an example of a causal premise. This major premise is actually an enthymematic syllogism and is resolved into: Everyone who has studied botany for years is one who knows a lot about it (implied major) 162 An Introduction to Traditional Logic: Argument

179 Every PhD in botany is one who has studied botany for years Every PhD in botany is one who knows a lot about it The point here is the same as with enthymemes, the causal proposition of an epicheirema is itself resolvable into another syllogism. This is not to say that all epicheiremas contain enthymemes, since they could contain the whole syllogism explicitly stated. Consider another example, this is a contemporary argument that argues for the existence of the human soul by showing that the activities of the mind are not reducible to that of the physical brain: Things that have different properties cannot be identical Mind and brain events have different properties because they differ as regards to exclusive first person access, spatial location, and color Therefore, mind events and brain events cannot be identical This time it is the minor premise that is the causal proposition which resolves into this: Whatever differs in regards to exclusive first person access, spatial location, and color has different properties (implied major) Mind events and brain events differ in regards to exclusive first person access, spatial location, and color. Therefore, mind events and brain events have different properties. Expository Syllogisms In an ordinary syllogism, the middle term is a universal term, however, in the expository syllogism, the middle term is singular. Consider: Socrates was a philosopher Socrates was a Greek Therefore a Greek was a philosopher Notice that the middle term here is Socrates a singular term. This is called an expository syllogism because the middle term in the major premise exposes the union between the major and minor terms. Technically speaking, this is not a syllogism in the proper sense since in every syllogism the middle term is universal and uses the said of all or said of none principles. Nor is it a true syllogism in another sense that we move from the known to the unknown, since there is nothing really Other Types of Syllogisms 163

180 implicit or virtual in the middle term. Yet this syllogism is useful in clarifying something you already know. Chapter 9 Summary The compound syllogism is a syllogism that has a compound proposition as a major premise The conditional syllogism is a type of compound syllogism that has a conditional major premise. Valid moods are modus ponens (affirming the antecedent) and modus tollens (denying the consequent). The disjunctive syllogism is a compound syllogism with a disjunctive proposition as a major premise and an affirmation or denial in the minor. Disjunctive syllogisms can be strong or weak. If strong, then either affirmation or denial in the minor is valid. If weak, then only a denial in the minor is valid. The conjunctive syllogism has a negative conjunctive proposition as a major premise. The minor can be an affirmation or denial if the major is exhaustive, or just an affirmation if the major is not exhaustive. The dilemma is an argument with a disjunctive premise combined with at least two conditional premises. Dealing with a dilemma involves; escaping between the horns, taking the dilemma by the horns, or rebuttal. The enthymeme is a syllogism with an implicit proposition. The suppressed proposition can be the major, minor, or conclusion, and so is called a first order enthymeme, second order enthymeme, or third order enthymeme respectively. The sorites is a chain syllogism where the conclusion of one syllogism is a premise for the next. A sorites can be progressive (where the subject is in the first premise and the predicate in the last) or regressive (where the predicate is in the first premise and the subject in the last). The epicheirema is a syllogism with a causal premise. The causal premise, if not already an explicit syllogism, is resolvable into a syllogism. The expository syllogism is an argument with a singular term serving as the middle. Exercises A. Determine the validity of the following: 1. If P then Q, P, therefore Q 2. If P then Q, Q, therefore P 3. If P then Q, not Q, therefore not P 4. If P then Q, not P, therefore not Q 164 An Introduction to Traditional Logic: Argument

181 5. Strong disjunction A or B, A, therefore not B 6. Weak disjunction A or B, A, therefore not B 7. Strong disjunction A or B, not A, therefore B 8. Weak disjunction A or B, not A, therefore B 9. You cannot be a winner and a loser, you are a winner, therefore you are not a loser 10. No can be running and walking at the same time, this man is not walking, therefore this man is running 11. If the country is at war, they will want a new president, the country is not at war, therefore they do not want a new president. 12. If the bear is hungry, he will be dangerous, the bear is hungry, therefore the bear is dangerous 13. If the musicians are any good, the tickets will be expensive, the tickets are expensive, therefore the musicians must be pretty good. 14. If evolution is true, there will be a gradually progressive fossil record, there is a gradually progressive fossil record, therefore evolution is true. 15. If he went to the shop he bought a candle, he bought a candle, therefore he went to the shop. 16. If there is thunder, then it is raining, there is not thunder, therefore it is not raining. 17. If there is change, then being must come from what it was not, being cannot come from what it is not, therefore there is no change. 18. If colors are only in the mind then we do not see external things, but we do see external things, therefore colors are not only in the mind. 19. If God does not exist, then all things are morally permissible, but all things are not morally permissible, therefore God exists. 20. Either the number is even or it is odd, it is even, therefore it is not odd. 21. Either the number is even or it is odd, it is not even, therefore it is odd. 22. Either the animal is a cat or a dog, it is not a cat, therefore it is a dog. 23. The complexity in the universe is due to either chance or design, it is not due to chance, therefore it is due to design. 24. Either he is alive or he is dead, he is dead, therefore he is not alive. 25. Either she is thirsty or hungry, she is thirsty, therefore she is not hungry. 26. Either she is thirsty or hungry, she is not thirsty, therefore she is hungry. 27. Either the Spartans will win the war or they will lose, they will not lose, therefore they will win. B. Enthymemes - Write a complete syllogism by supplying the missing premise or conclusion: 1. Every M is P, Every S is M, therefore 2. No P is M, Every S is M, therefore 3. Every P is M, Some S is not M, therefore 4. Some S is not P because all M is S Other Types of Syllogisms 165

182 5. Some S is P because all M is S 6. No S is P, since all S is M 7. No food should be wasted because all food is needed 8. Some politicians are trustworthy because some politicians take an oath 9. Some politicians are trustworthy because everyone who takes an oath is trustworthy 10. All cheetahs are hard to outrun for all cheetahs are very fast 11. Some college kids are tired because whoever studies all night is tired 12. Roses do not bloom in winter because no flowers bloom in winter 13. Every rottweiler is a good guard dog for rottweilers come from Germany 14. No people should be unjustly harmed because all people are made in the image of God. 15. Some wars are bad because some things that hurt people are wars C. Epicheiremas Expand the following causal premises into a full syllogism: 1. Some thieves are in jail because thieves do bad things, no good guys are in jail, therefore some good guys are not thieves. 2. Every chair in this room is black because every chair in this room was painted by Johnny, my seat is a chair in this room, therefore my seat is black. 3. All philosophy is necessary because it is good for you, logic is philosophy, therefore logic is necessary. D. Sorites Rewrite the following sorites to explicitly state the implicit conclusions: 1. All rocks are bodies All bodies are substances All substances are real beings All rocks are real beings 2. All roses are flowers All flowers are plants All plants are living All living things have a soul All roses have a soul 3. All living things have a soul All plants are living things All flowers are plants All roses are flowers All roses have a soul 4. No liquids are solids All beverages are liquids 166 An Introduction to Traditional Logic: Argument

183 All beers are beverages All Budweisers are beers No Budweisers are solids 5. All Budweisers are beers All beers are beverages All beverages are liquids No liquid is a solid No Budweisers are solids E. Dilemmas - Construct three of your own dilemmas. Other Types of Syllogisms 167

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185 Chapter 10: Some Final Aspects on Argumentation àà Different Types of Syllogisms Based on the Strength of the Premises àà Demonstrations through Effects and Causes àà Induction Different Types of Syllogisms Based on the Strength of the Premises We can make a distinction between different kinds of syllogisms based upon the strength of the premises they utilize. Propositions constitute the proximate matter of syllogisms, and since propositions can vary in terms of the strength of their truth value, the syllogism can be said to vary in this way as well. In this way, the syllogism can be of four types: Demonstrative Syllogism In a demonstrative syllogism, the premises are true, primary, immediate, prior to the conclusion, better known than the conclusion, Some Final Aspects on Argumentation 169

186 and the cause of the conclusion. 69 When valid, the conclusion of a demonstrative syllogism produces certain knowledge. Probable Syllogism When the premises are probably true, if the reasoning is valid the syllogism is probable. In other words, the premises might be false, but are likely to be or not be true in various degrees. These types of premises produce a conclusion that is more or less probable. Erroneous Syllogism When it is impossible for at least one of the premises be true, the resulting syllogism is erroneous. In an erroneous syllogism, the premises consist of incompatible terms and so it is impossible for them to be true (i.e., like Every square has three sides ). Sophistical Syllogism Sophistic syllogisms often have the appearance of validity yet they do not adhere to the formal rules of inference. Sophisms are often disguised with ambiguous language or other defects. Because of this, the validity of its conclusion is only illusory. The errors of sophistic syllogisms are exposed in the chapter on fallacies. Demonstrations through Effects and Causes The syllogism can be considered in two different ways depending upon the information they provide. A sound syllogism can provide an argument showing merely that something is the case, or a sound syllogism can provide an argument showing both that something is the case and the reason why it is the case. In other words, one can make an argument to show that something is so without at the same time showing why it is so. So there can be arguments simply demonstrating a fact and that s all or there can be arguments showing the fact and why it is a fact. (To show why something is a fact is to show its cause). Granted when you demonstrate both the fact and the cause you have a better argument than if you demonstrate only the fact alone. Suppose we wanted to show that Joe was asleep. We could come up with a good argument showing that Joe was asleep by saying that we see him with eyes closed, 69 Aristotle, Post. An b17. John of St Thomas, following Aquinas, says it suffices for the propositions to be virtually primary and immediate. In other words, it is enough for a demonstration to have premises deduced or resolved from propositions which are by themselves primary and immediately true (See Yves Simon s, The Material Logic of John of St Thomas (Chicago: University of Chicago Press 1955) p. 476). 170 An Introduction to Traditional Logic: Argument

187 snoring on the couch, and that he is not responsive to sounds of a normal volume, etc. We can then formulate an argument: Whoever has their eyes closed, is snoring, and unresponsive to sounds of a certain volume is asleep. Joe is one who has his eyes closed, is snoring, and is unresponsive to sounds of a certain volume. Therefore, Joe is asleep What this argument simply shows is the fact that Joe is asleep. It is an argument through effects. Because of the effects of eyes closed and snoring, we know the conclusion that he is sleeping. This argument does not show the reason why Joe is sleeping; it just shows that Joe is sleeping because of the presence of certain effects. This then is a demonstration through effect, or in Latin, a quia demonstration. But suppose we argue that Joe is sleeping in another way, this time through causes. This time we start with a general statement and then reach the same conclusion: Anyone totally exhausted from studying all night, is asleep. Joe is totally exhausted from studying all night Therefore, Joe is asleep See the difference in approach? This time we not only show that Joe is asleep, but the argument contains the cause or reason why Joe is asleep, because he is totally exhausted from studying all night. So unlike the first argument above that demonstrated only the fact that Joe is asleep, this one demonstrates the fact and why Joe is asleep. This is a demonstration through cause or in Latin, a demonstration propter quid (which means because of this ) The difference between the two arguments lies in the different middle terms. Both arguments have the same conclusion above, that Joe is asleep, and so have the same minor and major terms. The difference is the middle term. The middle term in the demonstration through effect is Whoever has their eyes closed, is snoring, and unresponsive to sounds of a certain volume while the middle term in the demonstration through cause is Anyone is totally exhausted from studying all night. The first middle term is an effect while the second middle term is a cause. So the difference in middle term is the real reason for the two types of demonstration. Some Final Aspects on Argumentation 171