SECTION 2 BASIC CONCEPTS 2.1 Getting Started...9 2.2 Object Language and Metalanguage...10 2.3 Propositions...12 2.4 Arguments...20 2.5 Arguments and Corresponding Conditionals...29 2.6 Valid and Invalid, A Closer Look...32 Basic Definitions...32 Valid, Invalid, True, and False...34 Two Foundational Issues...38 2.7 Non-technical Proofs of Invalidity and Validity...41 Establishing Invalidity...42 Establishing Validity...45 2.8 Wrap-up...48 2.9 Exercises...51 3.0 Quick Reference: Summary of Basic Concepts...55 2.1 GETTING STARTED The first question you might ask in an introductory logic course is: What is logic? At first, the answer is likely to seem perplexing because answering the question requires some knowledge of logic. The key is that logic, like other areas of knowledge, has its own specialized vocabulary. Understanding some of logic s basic vocabulary is necessary to answer your question. Logic is the same as any other subject matter in this respect penetrating the essential vocabulary is the first, and possibly the most important, step. The purpose of Section 2 is to introduce and explain some of the central concepts necessary for understanding logic. In order to get a foot in the door, let us begin by saying that one of the main tasks of logic is the analysis of arguments and the development of techniques for distinguishing between good and bad arguments. Focusing on techniques suggests a view of logic as a toolbox containing methods for assessing - 9 -
and constructing arguments. Certainly, this is a central part of logic. However, as you progress in your study of the subject you will recognize that logic also presents issues that go substantially beyond the notion of logic as simply a toolbox. An important word of caution is in order at this point. There is an enormous difference between simply understanding a new concept and being able to use it fluently. The basic vocabulary of logic introduced in Section 2 is not difficult to understand. However, it is vital that you be able to think using these terms without having to stop and ponder their meaning. In short, they must become part of your working vocabulary, and your use of them must be effortless and comfortable. Carefully reading and studying Section 2 is necessary to achieve this goal, but competence with the new vocabulary will emerge with continued use and practice. 2.2 OBJECT LANGUAGE AND METALANGUAGE Logic investigates arguments, and arguments occur in language. Because arguments occur in language, part of our task involves the study of structures in a language. In talking about structures in language we also make use of language. For example, books on English grammar contain statements about English words and sentences, and those statements may be written in English. Occasionally this may result in confusion because statements about English are written in English. A simple distinction is useful here and is best introduced by example. (1) Bison are four-legged animals. (2) Bison is a five-letter word. Sentence (1) is true, but sentence (2) is false; (2) is false because bison are animals not words and do not have any number of letters in them. However, the intent of sentence (2) is clear. If sentence (2) is to be true, then it must be understood to be making a statement about the name of a kind of animal, not about an animal itself. And, that name is indeed a fiveletter word. In order for a sentence to make a claim about an object, the object itself cannot occur in the sentence. What does occur in the sentence is the name of the object. If we now reconsider sentence (2) the problem is - 10 -
obvious. The first word in that sentence is the name of an animal. To make an assertion about a word we need a name for the word. A common convention for creating the name of a word is to enclose the word in single quotation marks. 1 Thus, the name of the first word in sentence (2) is bison. Using this convention, sentence (2) may be rewritten with the desired results. (3) Bison is a five-letter word. To construct the name of a word or phrase, enclose it in single quotation marks. All of the following sentences are true. (4) Rose is a noun. (5) The first long word in many dictionaries is aardvark. (6) Texas is a very big state even though Texas is short. The same method is used to talk about a collection of words, a part of a sentence, a complete sentence, or a collection of sentences enclose the expression to be named in single quotation marks. (7) War and Peace contains many thousands of words even though War and Peace contains only three words. (8) My face in thine eye, thine in mine appears, And true plain hearts do in the faces rest are words penned by John Donne. The distinction that we are considering is frequently referred as the difference between using and mentioning a word or phrase. A word or phrase is used when it functions to make a claim about whatever it names or describes. In sentence (1) the word bison is used. A word or phrase is mentioned when an assertion is being made regarding it. In sentence (4) Rose is mentioned. The distinction between using and mentioning a word or phrase is usually clear from context, and the lack of single quotation marks, or a similar device, is ordinarily not a problem. No one would misunderstand the intent of sentence (2). However, when we introduce variables to represent different objects and entertain some very abstract expressions 1 Other conventions are frequently encountered, e.g., italics, a distinctive font, underline, bold print, and placing an expression on its own print-line to distinguish it from other expressions. - 11 -
where precision is important, it is possible to hopelessly confuse issues if the difference between use and mention is not clearly maintained. In using language to talk about language, the terms object language and metalanguage are frequently used to mark the distinction between the language being discussed and the language used to conduct the discussion. The object language is the language we are talking about and the metalanguage is the language we use to talk about the object language. Object language and metalanguage may be the same natural language. In an English grammar book written in English, the object language is English and the metalanguage is English. Although they are the same natural language, their functions are logically distinct. Object language and metalanguage may be different natural languages. For example, in a textbook written to teach German to non-german speakers, the object language is German, but the metalanguage may be English, Spanish, or Chinese. When we begin to study the particulars of elementary symbolic logic, the logic itself will be presented as an artificial language. At that point, the importance of the distinction between object language and metalanguage will be clearer. In studying logic as a language, sometimes expressions written in the language of logic will be used to solve problems, and at other times the language of logic itself will be the subject of study. In that situation, clarity regarding object language and metalanguage will be essential. 2.3 PROPOSITIONS Arguments are composed of certain types of sentences called propositions. Because arguments are intended to provide evidence for the truth of their conclusions, not all types of sentences can function in arguments. Open the door is not a sentence one would expect to find playing a role in an argument because sentences such as commands, requests, and questions are neither true nor false. The sentences of importance in arguments are those that are either true or false. A proposition is a sentence (or a fragment of a sentence able to function as a sentence), which is understood to have a fixed meaning and is either true or false; in addition, the truth or falsity of a proposition holds at all times and is independent of the proposition s author. Admittedly, the - 12 -
last characteristic is not abundantly clear; however, examples will clarify the definition. Suppose that George W. Bush in a news conference on July 4, 2008 stated I am the President of the United Sates of America. Given the context that is provided, this statement is clearly true. But, the sentence as quoted does not meet the conditions for being a proposition. If a proposition is true we want it to be true no matter who states it. The example is not true if anyone other than George W. Bush makes the statement. Although the context makes it unnecessary to rewrite the statement in question, it can easily be restated so that its truth does not depend on specific person stating it, viz., George W. Bush is the President of the United Sates of America. But this new version still fails as a proposition because its truth is dependent on the time at which it is said. When stated on July 4, 2008, it is true; however, if stated on March 4, 2009 it will be false. Earlier we stated that a true proposition should be true no matter who states it; now we want to add that it should be true at any time whatsoever. True propositions are, in this sense, timeless. This problem can also be addressed by a simple restatement: George W. Bush is the President of the United Sates of America on July 4, 2008. Sentences in arguments routinely fail to meet the stringent definition of proposition. However, it is rarely necessary to rewrite sentences to make their truth conditions explicit because sentences usually occur in a context we recognize and automatically incorporate into our understanding of the sentence. On rare occasions, explicit re-writing may be necessary. For example, suppose a complex discussion involving a number of people; someone says I am a Chief Financial Officer, later someone else says I am not a Chief Financial Officer. If there is enough noise in the discussion and you lose track of who said what, then there is a chance of misinterpreting the statements and concluding that they are inconsistent. They obviously are not inconsistent unless said by the same person. Propositions can be divided into three mutually exclusive and exhaustive groups. Analytic propositions (or logically true propositions) Inconsistent propositions (or logically false propositions) Synthetic propositions (or contingent prepositions) - 13 -
An analytic proposition is a proposition that is necessarily true, and its truth depends only on the meanings of the words in the proposition. The truth of an analytic proposition is not established by gathering data, making observations, or conducting scientific experiments. All of the following propositions are analytic. (1) All roses are roses. (2) The litmus paper is red or the litmus paper is not red. (3) All zithers are musical instruments. (4) England is north of Malta or England is not north of Malta. An inconsistent proposition is a proposition that is necessarily false, and its falsity depends only on the meaning of the words in the proposition. An inconsistent proposition can also be defined as the negation of an analytic proposition. All of the following are inconsistent propositions (5) No mammals are mammals. (6) The Mississippi river is the longest river in the United States and the Mississippi river is not the longest river in the United States. (7) All yawls are sloops. A synthetic proposition is a proposition that is neither necessarily true nor necessarily false. i.e., it is neither analytic nor inconsistent. Synthetic propositions are sometimes called contingent because their truth depends on empirical circumstances. Making observations and collecting data are methods relevant to establishing the truth or falsity of synthetic propositions. The following are all examples of synthetic propositions. (8) All dogs are friendly. (9) There are more dairy cows in Vermont than people. (10) Flowers are found in many homes. (11) Turnips are nourishing. One additional frequently used term requires definition. Some propositions are possible propositions or logically possible propositions. All true propositions are, of course, possible propositions. But some, false propositions are also possible. For example, Hudson is the capital of New York is false, but it might have been true and in that sense the proposition - 14 -
is correctly labeled as possible. A possible proposition (or logically possible proposition) is a proposition that is either analytic or synthetic. This is of course equivalent to defining a logically possible proposition as any proposition that is not inconsistent. CHALLENGE Analytic propositions are of central interest in logic. However, logic is not simply a listing of specific analytic propositions. Logic, like physics, is a general science. You will not find the statement A six ounce apple dropped from the 25th floor of the Empire State Building on April 23rd, 1998 accelerates at 32 feet per second per second listed in a physics book as a law of physics. It is too specific. Physics is interested in a general rule covering all free-falling bodies. In a similar manner, the logician is not particularly concerned with individual analytic propositions, but with investigating general laws regarding analytic propositions. Before reading beyond this point, consider the problem of how logic can attain the desired generality. In approaching this question, it will be instructive to review some of your own answers to the questions posed in 1 Beginning. Properly done your answer will apply to both propositions and arguments. Generality is achieved in logic by introducing the concept of form or structure. Propositions (2) and (4) are clearly different, yet they have a good deal in common. What they have in common is their form or structure; they differ in content. One of the central tasks of logic is to devise methods to reveal the form of propositions. No simple definition of form will make the concept precise. In part, this is because there are different ways of analyzing the form of a proposition. 2 Replacing a proposition s content words with variables and retaining the logical form words will frequently reveal the logical framework of a proposition. Logical form words shape the skeleton of propositions. Expressions such as all, 2 Notice that this uses the expression the form of a proposition. It is virtually impossible to consistently avoid using the phrase the form of... ; however, this expression is misleading in its singularity. The notion that all propositions have a single, unique form is not defensible. - 15 -
none, some, most, if...then, and, but, only if, if and only if, or, unless, provided that, neither nor, and not are a few examples of logical form language found in propositions. Replacing the content words and retaining the logical form words in propositions (1) through (4) yields the following structures. 3 (1.1) All X are X (1.2) p or not p (1.3) All X are Z (1.4) q or not q The letters X, Z, p, and q in the above examples are variables. Variables are of fundamental importance in achieving generality, and they are best thought of as place-holders. Variables indicate a position within a structure where appropriate expressions may be substituted. In our examples, X, and Z may be replaced by class names such as apples, men over 6 feet tall, and positive integers. While X and Z in (1.1) and (1.3) cannot represent complete sentences, the occurrences of p and q in (1.2) and (1.4) can be replaced by complete sentences, but not by class names. It is important that you are not misled by the apparently literal nature of a variable. The form of All logicians are fascinating people, may be represented by All L are F, All Z are X, All D are B, All F are L, and All are. All of these represent the same logical form. (1.1) through (4.1) are the forms of propositions (1) through (4), and the forms succeed in showing that (2) and (4) have the same logical form. Although (1.1) through (4.1) are not propositions, propositions may be obtained from them by uniformly substituting appropriate linguistic 3 (1.1) and (1.3) are easily created by replacing words in (1) and (3) with variables, the situation is more complex with regard to the relation of (2) and (4) to (2.1) and (4.1). In these cases, we did more than replace a string of words with a variable. The problem revolves around the word not. In symbolizing the litmus paper is not red the not was removed and placed in front of the variable, i.e., not p. We have taken the litmus paper is not red to be equivalent to it is false that the litmus paper is red, and replaced it is false that with not. English does not have simple and invariable rules for the placement of not. For example, consider how to express the negation of all that glitters is gold. You may recall that we said logic would be presented as an artificial language. It is precisely to avoid the vagaries and ambiguities of English (or any other natural language) that logic is developed as a very simple and very precise artificial language. - 16 -
expressions for the variables. Expressions such as (1.1) through (4.1) are propositional forms; a propositional form is the form of a proposition. When is it correct to say that a proposition has a specific form? A proposition exhibits a given form if that proposition can be obtained from the form by uniformly substituting appropriate linguistic expressions in place of the form s variables; the resulting proposition is a substitution instance of the propositional form. All propositions may be viewed as a substitution instance of at least one propositional form. A proposition is a substitution instance of a given propositional form if and only if it can be obtained by uniform substitution from that form. All cows are ducks and All aardvarks are animals are both substitution instances of propositional form (1.3). They are not substitution instances of propositional form (1.1) because they cannot be obtained from it by uniform substitution. Uniform substitution requires that when an expression is substituted for a specific variable in a form, it must be substituted for all occurrences of that variable in that form. We cannot substitute different linguistic expressions for different occurrences of the same variable. However, we can substitute the same linguistic expression for different variables. For example, All roses are roses can be obtained by uniformly substituting into All X are Y or All X are X. Propositions (1) and (3) above are both analytic, although with an important difference. It is not necessary to know what rose means in order to know that proposition (1) is true. It is only necessary to understand All...are... and to note that the subject and predicate terms in the proposition are identical. For example, it is not necessary to know what yataghan means in order to recognize that All yataghans are yataghans is true. 4 The truth of proposition (1) is a function of its logical form words alone. The same is also true of propositions (2) and (4). It is not necessary to know what litmus paper is, or where England and Malta are located; it is only necessary to know how or and not function. Propositions of this type are formally analytic propositions. A formally analytic proposition is a proposition that is necessarily true simply by virtue of its logical form words. Unlike proposition (1), recognizing the analyticity of proposition (3) requires knowing what zither and musical instruments mean in addition to understanding the logical form words. Propositions of this type are semantically analytic propositions. A semantically analytic proposition is a 4 Provided, of course, that yataghan is a meaningful English word. - 17 -
proposition that is necessarily true by virtue of the meanings of its content and logical form words. We have already observed that logic is a general discipline, and generality is achieved by concentrating on propositional forms rather than individual propositions. Therefore, we will extend our notion of formal analyticity to propositional forms. Propositional forms (1.1), (1.2) and (1.4) are analytic propositional forms. A propositional form is an analytic propositional form if and only if it is the form of a formally analytic proposition. There is another way to distinguish formally and semantically analytic propositions, which will further clarify the relationship between these types of proposition and their corresponding forms. Each of the following propositions has the same form as proposition (2) and all of them are analytic. (12) Everest is taller than K-2 or Everest is not taller than K-2. (13) Spenser wrote The Faerie Queene or Spenser did not write The Faerie Queene. (14) Montpelier is the capital of Vermont or Montpelier is not the capital of Vermont. Contrast the above three propositions with the following four propositions. Each of the four has the same form as proposition (3), and proposition (3) is semantically analytic. (15) All triangles are triangles. (16) All red apples are colored objects. (17) All umpires are men. (18) All peonies are fragrant. Observe that (15) and (16) are analytic, but (17) and (18) are not. If a proposition is formally analytic then all propositions of that form are analytic, but this is not true of semantically analytic propositions. A proposition is formally analytic if and only if every proposition of the same form is analytic. A proposition is semantically analytic if and only if it is analytic and not all propositions of that form are analytic. - 18 -
If a proposition is formally analytic, then all propositions of that form are analytic, and if a group of formally analytic propositions all have the same form, then they are substitution instances of that form. We now have another, and perhaps clearer, way to define an analytic propositional form. A propositional form is analytic if and only if all substitution instances of that form are analytic. 5 Following is a summary of some, but not all, of the central points regarding propositions. A proposition is sentence (or a fragment of a sentence able to function as a sentence), which is understood to have a fixed meaning and is either true or false. The truth or falsity of a proposition holds at all times and is independent of the proposition s author. A propositional form is the form of a proposition. A proposition is a substitution instance of a propositional form if and only if it can be obtained by uniform substitution from that form. An analytic proposition is a proposition that is necessarily true. An inconsistent proposition is a proposition that is necessarily false. A synthetic proposition is a proposition that is neither necessarily true nor necessarily false. A proposition is formally analytic if and only if every proposition of the same form is analytic. A proposition is semantically analytic if and only if it is analytic and not all propositions of that form are analytic. 5 The concept of analyticity is considerably more intricate than has been suggested here. Many questions have not been addressed. For example: Are all semantically analytic propositions reducible to formally analytic propositions? What makes an analytic proposition analytic? What is the fundamental nature of analyticity? Is analyticity a viable concept? Philosophers have considered a number of theories to explain the nature of analyticity and at present there is no generally agreed upon answer to these fundamental questions. - 19 -
A propositional form is analytic if and only if all substitution instances of that form are analytic. If you look carefully at the summary of major points and the more detailed information preceding that summary, you may notice that there are some interesting concepts and relationships that were not explicitly defined. Before moving on to the section on arguments, try to answer the questions in the challenge box. CHALLENGE 1. Is there an inconsistent propositional form? If so, provide a definition. 2. Is there a synthetic propositional form? If so, provide a definition. 3. If you defined inconsistent propositional form, what is the nature of the relationship of the form and its substitution instances? 4. If you defined synthetic propositional form, what is the nature of the relationship of the form and its substitution instances? 5. A distinction was made between formally analytic propositions and semantically analytic propositions. Can a similar distinction be applied to inconsistent propositions? In this section we have developed one major dimension of the subject matter of logic logic is the study of analytic propositions and analytic propositional forms. This undertaking has two important aspects. First, logic formulates general procedures for identifying analytic propositions and analytic propositional forms. Second, logic devises techniques for the construction of analytic propositions and analytic propositional forms. 2.4 ARGUMENTS An argument is an organization of propositions making a distinctive claim. Specifically, in an argument the claim is made that some of the - 20 -
propositions in the argument provide evidence for the truth of another proposition in the argument. The propositions in an argument fall into two functionally different categories. The proposition(s) providing the evidence are the premise(s) of the argument; the proposition supported by the premises is the conclusion of the argument. No proposition by itself is either a premise or a conclusion. Within an argument, a proposition may function as a premise (or conclusion) only in relation to another that functions as conclusion (or premise). And, of course, the same proposition may be a premise in one argument and a conclusion in another argument. There are two important characteristics of an argument. First, as mentioned above, in an argument the claim is made that the truth of the premises provides evidence for the truth of the conclusion. Second, the premises in an argument are asserted as true. An example will help explain the last point. (1) If the fuel injectors in John s car are blocked, then his car will not run. (2) The fuel injectors in John s car are blocked. Therefore, his car will not run. Number (2) is an argument. In number (2) the proposition The fuel injectors in John s car are blocked is asserted. No conditions are attached; it is simply stated as true. This is not the case in number (1) where no premise is asserted and no conclusion is drawn. Number (1) is a single conditional sentence, and although it may be true, it does not follow that the individual sentences that make up the if then conditional are themselves true. Number (1) does not assert the proposition The fuel injectors in John s car are blocked. What is put forth as true in number (1) is the single conditional sentence: If the fuel injectors in John s car are blocked, then his car will not run. In short, arguments are to be distinguished from conditional propositions. 6 To say that an argument asserts its premises are true does not mean that they are true, or are necessarily believed to be true by whoever puts forth the argument. It does mean that in an argument the premises are simply declared as true, and their truth is presented as evidence for the truth of the conclusion. This does not prohibit a conditional proposition from being a premise or a conclusion in an 6 Although arguments and conditional propositions are not the same, it is clear that there is an important relationship between them. This relationship will be explained in section 2.5-21 -
argument. For example, following is an argument whose first premise happens to be a conditional proposition. (3) If the minimum airspeed at which the wing of the Skycar generates lift is 85 miles per hour, then at airspeeds below 85 miles per hour the wing will stall. On its test flight, the Skycar flew successfully at 75 miles per hour. Therefore, the stall speed of the Skycar s wing is not 85 miles per hour. Logic develops techniques for evaluating arguments. This presupposes the ability to correctly identify an argument and differentiate premises and conclusion. Although this sounds relatively simple, it is not. The technical aspects of logic are, in a sense, easy to understand because they are always precise and frequently mechanical. The identification of arguments, premises, and conclusions is not a mechanical process. The successful identification and analysis of arguments requires interpreting language with a reasonable amount of sensitivity. There are no hard and fast rules here, no methods to guarantee success, and no substitute for practice. Some logical form words are helpful in disentangling the premises and conclusions in arguments. For example, therefore, hence, consequently, so, it follows that, it must be true that and this implies that are some common words and phrases that may introduce the conclusion in an argument. Since, for, because, and for these reasons may introduce premises. However, the presence of these logical form words does not necessarily mean that there is an argument present. For example the following is not an argument. (4) Steve gave his wife flowers because she was sick. It is not an argument because the truth of she was sick is not offered as evidence for the truth of Steve gave his wife flowers. In this case, the word because does not introduce a premise; it introduces an explanation for why Steve gave his wife flowers. In the arguments we have used so far, the premises were stated first and the conclusion last. But, premises and conclusions may occur in any order; the conclusion may be first, last, or in the middle of an argument. - 22 -
Careful reading and attention to context is the only way to identify the premises and conclusion in an argument. Arguments are generally divided into two classes: deductive and inductive. 7 This distinction reflects the relative strength of the evidence an argument claims to exist between its premises and conclusion. A deductive argument claims that its premises provide conclusive evidence for its conclusion, conclusive in the sense that it is not logically possible for the premises to be true and the conclusion false. On the other hand, an inductive argument puts forth a weaker claim. An inductive argument alleges that its premises provide sufficient evidence for rendering the conclusion probable for the purposes at hand. An inductive argument claims that if the premises are true, the conclusion is probably true. Because deductive and inductive arguments make such different claims, the criteria used to evaluate them are not the same. An argument interpreted deductively might be a bad deductive argument, while the same argument interpreted inductively might be a good inductive argument. In the following examples, (5) is deductive and (6) is inductive. (5) Since there are more than 365 people in the room and there are only 365 days in a year, it must be the case that there are two people in the room with the same birthday. (6) If Einstein is correct, the velocity of light in any given medium is a constant. We know that the velocity of light in air is a constant. And this evidence supports Einstein s claim. It is possible to evaluate any argument using criteria appropriate to deductive arguments. One could evaluate the presentation of evidence in a courtroom using deductive criteria; however, doing so is insensitive to the function of this type of argument. Most courtroom arguments are probably best evaluated from an inductive standpoint. We will focus on that part of logic concerned with the evaluation of deductive arguments. From now on, when reference is made to arguments it will be understood to mean deductive arguments, unless explicitly stated to the contrary. 7 Some logicians have argued that not all arguments are deductive or inductive, i.e., deduction and induction are mutually exclusive, but not exhaustive categories of arguments. See Stephen Barker, The Elements of Logic. - 23 -
A deductive argument claims that its premises furnish conclusive evidence for its conclusion. The evaluation of deductive arguments consists in ascertaining if that claim is correct. The words valid and invalid are used in evaluating the claim of deductive arguments. 8 The fundamental objective of the definition of valid is to absolutely guarantee that arguments constructed in accord with the rules of the logic will never lead from true to false statements. In other words, validity is defined in a as a truth-preserving relationship,.i.e., if you start with true propositions and argue validly, the result must be a true proposition. If a deductive argument succeeds in furnishing conclusive evidence for its conclusion, then it is valid. A valid deductive argument is an argument in which it is not logically possible for its premises to be true and its conclusion false. A deductive argument is invalid if and only if it is not valid. In other words, if the claim of conclusive evidence is not correct then it is possible for the premises to be true and the conclusion false, and the argument is an invalid deductive argument. A deductive argument is invalid if and only if it is logically possible for its premises to be true and its conclusion false. 9 In section 2.3 a distinction was made between propositions and propositional forms. This distinction introduced generality into the discussion of propositions and logic. It also led to the observation of an important relationship between formally analytic propositions and their propositional forms. For similar reasons, we now distinguish between arguments and argument forms. As you follow the discussion, you should observe a very close parallel between the concepts used in the discussion of propositions and those used for arguments. An argument is composed of propositions, and the form of an argument consists of the form of those propositions together with the logical connections between those propositions. An argument form is the form of an argument. 8 The terms valid and invalid have a broader use in ordinary English than in logic. In our use, these terms are restricted to describing deductive arguments, and they are both exhaustive and exclusive. 9 There is much more by way of both explanation and qualification that needs to be said regarding the concepts of validity and invalidity and their relation to truth and falsity; section 2.6 will address these issues in more detail. - 24 -
Following are examples of arguments (not necessarily valid) and their corresponding argument forms. We adopt the convention of stating the premises of the argument above the line and the conclusion below it. 10 (7) If the sky is blue, it will be a pleasant day. The sky is blue. It will be a pleasant day. (7.1) If p, then q p q (8) No emus are wallabies. No wallabies are emus. (8.1) No X are Y No Y are X (9) All tomatoes are red. All tomatoes are colored. (9.1) All X are Y All X are Z (10) The picnic was a success or the guests were unhappy. The picnic was a success. The guests were not unhappy. (10.1) p or q p not q (11) No chemists are fools. Some illiterates are fools. No chemists are illiterates. (11.1) No Z are Y Some X are Y No Z are X (12) Some cats are pets. Most cats are pets. (12.1) Some X are Y Most X are Y (7.1) through (12.1) are not arguments because they are not made up of propositions. They are argument forms. Specifically, they are the forms of arguments (7) through (12). We previously observed that specific propositions are substitution instances of propositional forms. The same relationship holds between arguments and argument forms. Arguments are substitution instances of argument forms. Arguments (7) through (12) are substitution instances of argument forms (7.1) through (12.1) 10 The symbol is shorthand for therefore. - 25 -
respectively. An argument is a substitution instance of a given argument form if and only if it can be obtained by uniform substitution from that form. For example, by substituting the appropriate linguistic expressions for the variables in (11.1) we may generate an infinite number of arguments all having the same form as argument (11). In the above examples, arguments (7) through (9) are valid, and (10) through (12) are invalid. A careful reading of the arguments should make this intuitively clear. The terms valid and invalid, as they have been defined, apply only to arguments. Since we wish to talk about argument forms as well as arguments, these terms will be extended to cover argument forms. An argument form is valid if and only if all substitution instances of that form are valid arguments. An argument form is invalid if and only if not all substitution instances of that form are valid, i.e., there is at least one substitution instance of the argument form with true premises and a false conclusion. Although arguments (7), (8) and (9) are all valid, only argument forms (7.1) and (8.1) are valid. At first glance, it may be surprising that argument (9) is valid but its form is invalid. But not all substitution instances of (9.1) are valid, and therefore, it is an invalid argument form. For example, substitute cats for X, animals for Y and boats for Z. The result is all cats are animals; therefore, all cats are boats, which is obviously invalid. On the other hand, all substitution instances of (7.1) and (8.1) are valid arguments, so these are valid argument forms. While arguments (8) and (9) are both valid there is an important difference between them. The validity of (8) does not depend on the meaning of emus and wallabies ; it is only necessary to understand the logical form words used in the argument. In argument (9), knowledge of the logical form words alone is not sufficient to ascertain its validity; it is also necessary to understand the meaning and relationship of red and colored. Argument (8) is a formally valid argument. Argument (9) is a semantically valid argument. A formally valid argument is an argument whose validity is only a function of its logical form words. A semantically valid argument is an argument whose validity is not only a function of its logical form words. The distinction between formally and semantically valid arguments may also be defined by the relation of the arguments to their argument forms. An argument is formally valid if and only if all arguments of that - 26 -
form are valid. An argument is semantically valid if and only if it is valid, and not all arguments of the same form are valid. The following bullets are a summary of the major points developed in this section regarding arguments. The parallelism of language between propositions and arguments should be obvious. An argument is an organization of propositions claiming that the truth of some of its propositions (the premise or premises) constitutes evidence for the truth of another proposition in the argument (the conclusion). An argument form is the form or structure of an argument. An argument is a substitution instance of a given form if and only if it can be obtained by uniform substitution from that form. A deductive argument claims that it is not logically possible for its premises to be true and its conclusion false. An inductive argument claims that if its premises are true, then acceptable evidence has been provided establishing the probable truth of the conclusion. A valid argument is one in which it is not logically possible for the premises to be true and the conclusion false. An argument is formally valid if and only if all arguments of that form are valid. An argument form is valid if and only if all substitution instances of that form are valid arguments. An argument is semantically valid if and only if it is valid, and not all arguments of the same form are valid. An invalid deductive argument is one in which it is logically possible for the premises to be true and the conclusion false. An argument form is invalid if and only if not all substitution instances of that form are valid, i.e., there is at least one substitution instance of the argument form with true premises and a false conclusion. - 27 -
At the end of the previous section, logic was characterized as the study of certain types of propositions and propositional forms. We may now give a description of logic in terms of the study of arguments. Logic is, in part, concerned with the analysis and evaluation of arguments. Formal deductive logic is the examination of the principles of formal validity. As was the case with propositions, this undertaking also has two aspects. First, logic develops general procedures for identifying valid and invalid arguments and argument forms. Second, logic develops techniques for the construction of valid arguments and valid argument forms. In section 2.4, a footnote stated Although arguments and conditional propositions are not the same, it is clear that there is an important relationship between them. The nature of that relationship is addressed in section 2.5. However, before you begin reading section 2.5, try to answer the question posed in the following Challenge Box. CHALLENGE In 2.3 logic was characterized in terms of propositions, and in 2.4 logic was characterized in terms of arguments. These two approaches can be linked. The question is what is the connection between propositions and arguments? See if you can establish a relationship between an argument and a specific proposition that would allow you to connect the validity of the argument to a logical characteristic of the proposition. If you succeed in connecting arguments and propositions, try to do the same for argument forms and propositional forms. In other words, is there a relationship between an argument form and a propositional form that would allow you to connect the validity of the argument form with a logical characteristic of the propositional form? - 28 -
2.5 ARGUMENTS AND CORRESPONDING CONDITIONAL PROPOSITIONS We have developed two ostensibly different conceptions of logic. From one perspective logic is concerned with propositions, and from another it is concerned with arguments. Are these two viewpoints simply different, or is there some link between the two? It should be apparent from the structure of the preceding sections on propositions and arguments that there is a very important connection between arguments and propositions. For every argument there is a specific proposition that bears an important and distinctive relationship to the argument. This proposition is the corresponding conditional proposition for the argument. A conditional proposition is any proposition of the form if then. 11 The corresponding conditional for any deductive argument is the conditional proposition whose antecedent is the conjunction of the premises in the argument and whose consequent is the conclusion of the argument. 12 Following are examples of arguments paired with their corresponding conditionals. (1) Either Jones or Smith will be the regular quarterback. Jones will not be the regular Quarterback. Smith will be the regular quarterback. (2) The ship on the horizon is a Xebec. The ship on the horizon has three masts. (1.1) If either Jones or Smith will be the regular quarterback and Jones will not be the regular quarterback, then Smith will be the regular quarterback. (2.1) If the ship on the horizon is a Xebec, then the ship on the horizon has three masts. 11 The proposition occurring between if and then is the antecedent of the conditional; the proposition following then is the consequent of the conditional. 12 The conjunction of the premises is formed by connecting all of the premises together with and ; that conjunction is true if and only if each conjoined premise is true. If there is only one premise, then that single proposition is the antecedent in the corresponding conditional. - 29 -
(3) Some bloodhounds are excellent pets. Some bloodhounds are not excellent pets. (3.1) If some bloodhounds are excellent pets, then some bloodhounds are not excellent pets. The concept of the corresponding conditional is readily extended from arguments to argument forms. In the case of an argument form, the corresponding conditional is constructed in the same way except that the antecedent and consequent will be propositional forms rather than propositions. Following are two examples of argument forms and their corresponding conditional. (4) All X are T No Y are T No Y are X (5) Some X are Y Some X are not Y (4.1) If all X are T and no Y are T, then no Y are X (5.1) If some X are Y, then some X are not Y In general, given any argument, or argument form, with P n premises and C as conclusion, P 1 P 2. P n-1 P n C the corresponding conditional would be: If P 1 and P 2... and P n-1 and P n, then C. What exactly is the connection between an argument and its corresponding conditional? An argument is valid if and only if the statement the premises are true and the conclusion false, when applied to the argument, is necessarily false. And it follows from this that an argument is valid if and only if the statement if the premises are true, then the conclusion is true, is necessarily true. But notice that the corresponding conditional is essentially the statement if the premises are true, then the conclusion is true. Consequently, if an argument is valid its corresponding conditional is true. However, as we just observed, the corresponding conditional for a valid argument is not merely true, but necessarily true, and a necessarily true statement is analytic. - 30 -
We now have a fundamental connection between arguments and their corresponding conditionals. An argument is valid if and only if the corresponding conditional is analytic. An argument is invalid if and only if the corresponding conditional is not analytic. Conditionals (1.1) and (2.1) are both analytic, but notice that the first is formally analytic, and the second is semantically analytic. The arguments that correspond to these are both valid, but as you would expect (1) is formally valid and (2) is semantically valid. This relationship holds in general. An argument is formally valid if and only if the corresponding conditional is formally analytic. An argument is semantically valid if and only if the corresponding conditional is semantically analytic. We may also link argument forms and their corresponding conditional forms. Argument form (4) is valid and the corresponding conditional (4.1) is an analytic propositional form. Argument form (5) is not valid and the corresponding conditional (5.1) is not an analytic propositional form. An argument form is valid if and only if the corresponding conditional is an analytic propositional form. Summarized below are some of the major points regarding arguments and their corresponding conditional propositions. The corresponding conditional for any deductive argument is the conditional proposition whose antecedent is the conjunction of all of the premises in the argument and whose consequent is the conclusion of the argument. An argument is valid if and only if its corresponding conditional is analytic. An argument is formally valid if and only if its corresponding conditional is formally analytic. An argument form is valid if and only if its corresponding conditional is an analytic propositional form. An argument is semantically valid if and only if the corresponding conditional is semantically analytic. - 31 -
2.6 VALID AND INVALID, A CLOSER LOOK In section 2.4, the concepts of validity and invalidity as applied to deductive arguments were introduced. This section focuses in considerable detail on those basic concepts, and it is divided into three segments. First, the basic definitions; second, a more detailed look at how valid and invalid as properties of deductive arguments relate to the truth or falsity of the propositions in those arguments; and lastly, comments on what may be termed foundational issues. 13 Basic Definitions A deductive argument is valid if and only if it is not logically possible for the premises to be true and the conclusion false. A deductive argument is invalid if and only if it is logically possible for the premises to be true and the conclusion false. (Or, more concisely: A deductive argument is invalid if and only if it is not valid.) As a direct consequence of the definition of valid, it is necessarily true that a valid argument with true premises will have a true conclusion. A frequently encountered alternative formulation of this statement is: if a valid deductive argument has true premises, then the conclusion is necessarily true. This formulation is not correct due to the placement of the key word necessarily. In a valid argument, what is necessary is the relationship between the premises and the conclusion; it is not the case that the conclusion must itself be necessarily true. For example, the following argument is clearly valid: There are ten apples on the desk; 13 2.4 also placed boundaries on the type of arguments examined in formal deductive logic. They are worth restating with some expansion. Unless otherwise specified, our interest in arguments is now limited to deductive arguments and matters of formal validity and invalidity. When the term argument is used without qualification it will mean deductive argument, valid will mean formally valid, and invalid will mean formally invalid. Parallel considerations hold for propositions; we are interested in formally analytic propositions and analytic propositional forms. - 32 -
therefore, there are at least five apples on the desk. What is necessarily true is If there are ten apples on the desk, then there are at least five apples on the desk. But, there are at least five apples on the desk is not necessarily true. The fundamental objective behind the definition of valid is that valid inferences should guarantee that if an argument has true premises and is valid the conclusion will be true. In short, the relationship of validity is truth-preserving. Another way of expressing this basic point is to say that truth is a hereditary property with respect to validity, i.e., if you start with true propositions and validly deduce other propositions all of the derived proposition will inherit the property of being true. A number of essential points should be emphasized regarding the definition of valid. The definition of valid states that it is not logically possible for a valid argument to have true premises and a false conclusion. The definition of valid implies that if a valid argument has true premises, it must have a true conclusion. The definition of valid implies that if a valid argument has a false conclusion, not all of its premises can be true. The definition of valid neither states nor implies that a valid argument has true premises and a true conclusion. The definition of valid neither states nor implies that if an argument has true premises and a true conclusion, it is valid. Following are critical points that should be underscored concerning the definition of invalid. The definition of invalid asserts that it is logically possible for an invalid argument to have true premises and a false conclusion. The definition of invalid does imply that if the premises of an argument are true and the conclusion is false, then the argument is invalid. - 33 -