If...then An Introduction to Logic Kent Slinker February 18, 2014
Part I. Informal Logic 2
Contents I. Informal Logic 2 1. Introduction to Part One 4 2. Relations between statements 6 2.1. Entailment................................... 7 2.2. Relevance.................................... 8 2.3. Independence.................................. 9 3. Arguments 13 3.1. Combining propositions............................ 13 3.2. The Argument................................. 14 3.3. Validity, Inference, and Invalidity....................... 18 4. More properties of arguments 25 4.1. Truth versus Inference, Soundness, Strength and Weakness........ 25 4.2. Valid Argument Forms............................. 27 4.3. Pseudo-valid argument forms.......................... 32 4.4. The Counter Argument............................ 34 5. Induction 40 5.1. Evaluating Inductive Arguments....................... 42 5.1.1. Exercises Chapter 5.......................... 45 Answers to Selected Problems 47 3
1. Introduction to Part One Logic, like Mathematics, Psychology, History, Physics, and many other academic fields is a broad subject, with many specialized areas and sub-disciplines. The aim of the first part of this text is to introduce you to the main ideas behind what is sometimes called Philosophical Logic in an informal way. In general, Philosophical logic, at the basic level we will deal with in Part One, is a study of the principles of correct reasoning. Of course, to judge something as being correct is to assume that there is an incorrect form as well, and so it is with logic - so to state matters at once as clearly as possible: In Part 1 we will be primarily concerned with an examination of the process of reasoning, and developing tools and procedures, concepts and definitions, indispensable for the process of distinguishing correct from incorrect reasoning. To start us down that road, consider the following question: Suppose I roll two dice and adding together the numbers of each face I get a sum of 12. What number must be on the face of each die? 1 Before you answer, think carefully about what the question is asking and what information is given. What are the relevant parts of the question? What words cause you to be careful with your response, and given a possible answer, what examples convince you that your answer is correct? What examples or considerations may cause you to retract your answer? Consider your answer carefully before looking at the answer given in the footnote below. Was your answer correct? If not, do you understand why? The process of looking at information, and drawing conclusions from that information invites us to examine relations between statements, and our primary question for the next few chapters will be to learn how to determine whether the truth of one statement affects the truth of another statement, and when it does, what accounts for that relationship. In going down this road, we will build foundational conceptual tools, which we will master by doing 1 It is impossible to know for certain which number is on the face of each dice. Since there is nothing about the word dice that insists on it being a standard dice - like the ones used in a game of craps or monopoly - it leaves open the possibility that one dice might have all zeros and the other dice all 12s, or one dice might have only 6 or 5 on each face, while the other dice might have 6 or 7 (giving two possibilities 6,6 or 5,7) - this is just a few of the many many possibilities that make certainty in this case impossible. However, if it were stipulated that the dice were standard dice, then the answer would be 6 on each face. This tells us to be careful about assuming too much from a given statement, or at least to state any assumptions in our conclusion, such as, If the dice are standard, then the answer is 12, otherwise no certain answer can be given..." 4
1. Introduction to Part One exercises, and building on these foundational concepts we will eventually reach a point where a more formal treatment of our subject seems both desirable and necessary. This formal treatment will be given in Part Two of this text. 5
2. Relations between statements Consider the following two statements: 1. I have a dime, a quarter, and 11 pennies in my pocket. 2. I have some change in my pocket. For simplicity s sake, let s call the first statement p and the second statement q. Now we pose our first question related to logical analysis, Does the truth of p have anything to do with the truth of q? - By this, we simply mean, in the broadest of terms, whether the truth of the statement, I have a dime, a quarter and 11 pennies in my pocket lends credence to the statement, I have some change in my pocket. Clearly the answer is yes, for if the first is true, then certainly the second must be true. Why is this so? The answer to this question clearly has to do with the connection between the terms, dime, quarter and pennies and the word change when used in this context. Now lets turn the question around and ask if the truth of q has anything to do with the truth of p, or more plainly, does the truth of, I have some change in my pocket affect in any way the truth of, I have a dime, a quarter, and 11 pennies in my pocket. To make this as clear as possible, what we are asking is whether the truth of q gives us any reason whatsoever to suspect that p might be true. Again, the answer is yes, as one possible way to have change in one s pocket is indeed to have a dime, a quarter and 11 pennies, but as this is just one of many combinations of coins which makes the statement true, the truth of q does not guarantee the truth of p. Let s consider another example: 1. Today is October 1. 2. The word, ostentatious has 12 letters. As above we will refer to the first sentence as p and the second as q, as ask the same questions: Does the truth of p have anything to do with the truth of q, and similarly does the truth of q have anything to do with the truth of p? In both cases, we are inclined to say no. The fact that today is the first of October is unrelated to the fact that the word ostentatious has 12 letters, and the fact that the word ostentatious has 12 letters as nothing to do with today s date. As above, the reason is related to the connection between the meanings of the terms, but for now we will put aside that connection and simply record our observations. Given two statements, p and q, then we have at least the following three cases: 1. The truth of one statement guarantees the truth of another statement. 6
2. Relations between statements 2. The truth of one statement gives some support to the truth of another statement, but does not guarantee it. 3. The truth of one statement has nothing to do with the truth of another statement. Let us now give informal definitions to these types of relationships between statements, as they will be used again and again in our informal analysis of Logic. 2.1. Entailment Definition 1. If the truth of a statement p guarantees that another statement q must be true, then we say that p entails q, or that q is entailed by p. Examples help us understand definitions, which will play a major part in our study of logic. Before we give some examples, it might be helpful to point out that our use of the letters p and q is arbitrary. We could just as well use k, m, #, or any other mark, as long as it is understood what these symbols stand for. In our case, our symbols are just going to stand for sentences which have the property of being either true or false, where false is understood to mean the same thing as not true". Sentences with the property of being either true or false are called propositions in Logic. Prudence suggests that the symbols we choose cause as little confusion as possible, and in general it is a good idea to realize that p and q are dummy variables, choices which are arbitrary and hence definitions which use them do not depend on our choice of symbol. For the most part, we will stick with tradition in this text, and use letters like p, q, r, s, and so forth to stand for propositions. However, for reasons which will become clear when we turn to our formal analysis of Logic, using the letters t, f, or v to stand for propositions might cause confusion, so we will not use them in this textbook. Notice also the key term, must be true in our above definition of entailment. Must be is another way of stating that something is necessary, something for which no other option or possibility exists. Must be encodes one of the most fundamental concepts of Logic, the concept of logical necessity. Here are some more examples of statements which entail other statements: Example. Let p be, Mary knows all the capitals of the United States, and let q be, Mary knows the capital of Kentucky, then p entails q. Example. Let q be, Every one in the race ran the mile in under 5 minutes, and let p be, John, a runner in the race, ran the mile in less than 5 minutes, then q entails p. Example. Let k be, The questionnaire had a total of 20 questions, and Mary answered only 13, and let n be the statement, The questionnaire answered by Mary had 7 unanswered questions, then n is entailed by k. 7
2. Relations between statements Example. Let j be, The winning ticket starts with 3 7 9, and let c be the statement, Mary s ticket starts with 3 7 9 so Mary s ticket is the winning ticket. In this case c is not entailed by j. If the relationship of entailment seems conceptually familiar, there is probably a good reason for this. We say that true conditional statements, or true implications are examples of entailment. Hence any conditional statement, (e.g. a statement in the form if.... then ) which is true, is an example of entailment. In Mathematics, the more familiar term is implication. The exercises at the end of this chapter will give you more practice with the concept of entailment, which is central to our study of Logic. Let us now turn to the concept of relevance by providing an informal definition. 2.2. Relevance Definition 2. Let p and q be two propositions, then p is relevant to q if the truth of p counts for the truth of q but does not guarantee it. This is an informal definition indeed, as seemingly all we have done is replace the word relevance with the phrase, counts for the truth of. However providing a more detailed definition would take away from the basic concepts we seek to cultivate at this juncture. One important point to note about this definition is that entailment is not a specific form of relevance. Other texts may define this term differently, so if you consult another text, make sure you check how they define relevance. This underscores the need to stipulate precise definitions and allow others to know the intended use of specific terms. Example. Let p be, Jane knows the capital of Arizona and let q be, Jane knows all the capitals of the United States, then the truth of p is relevant to the truth of q. Here q could be made true by the conjunction of several cases; Jane knows the capital of California, and Jane knows the capital of Nevada, and Jane knows the capital of..., and clearly Jane knowing the capital of Arizona is one such case. In this case the relevance results in the fact that p is one of the many ways which together make q true. Note that on one hand p is relevant to q, but on the other hand q entails p. The order of terms (p first, then q, or q first then p) makes a difference. We will visit this important characteristic of order of terms in more detail in Part 2 of the text. Example. Let p be, John ate uncooked meat and later came down with a case of e-coli infection and let q be, Eating uncooked meat is a cause of e-coli infection, then p is relevant to q. Clearly p alone does not establish q, but the fact that John ate uncooked meat and later came down with a case of e-coli is evidence which supports the truth of q. 8
2. Relations between statements Example. Let p be, Mr. Smith is a millionaire and let q be, The Smiths live in a mansion, then p is relevant to q. In all of the above cases, we have two statements, p and q, where the truth of p is relevant to the truth of q. In other words, knowing that p is true lends some credence to the truth of q, but unlike entailment, the truth of one statement does not guarantee the truth of the other. We might think of relevance in terms of degrees of evidence. When we want to establish that a certain statement is true, many times we offer evidence from various sources. Each single piece of evidence is related to the truth of the statement we want to establish (is relevant to it), but in many cases no single piece of evidence by itself firmly establishes it. 2.3. Independence After becoming acquainted with the concept of relevance, it is natural to ask about the concept of irrelevance. If q is irrelevant to p, then the truth of q does not support the truth or falsity of p. Instead of using the term irrelevance for this case, it is customary to use the term independent instead. Our last relation between statements is that of truth independence, which we now define. Definition 3. Let p and q be two statements. We say p and q are logically independent if the truth of one does not effect the the truth of the other. The notion of truth independence will play a central role in formal logic, but for now we will just examine some examples to get a feel for what it means for two propositions to be be independent. Example. Let p be, Tom s first child was born October 9, 2010, and let q be, Pianos have 88 keys, then the truth of q is independent of the truth p. Example. Let p be, At normal pressure water freezes at 0 degrees C, and let q be, February has 28 days except for leap years, then q is logically independent of p. Example. Let p be, Columbus sailed for the Americas in 1492, and let q be, The concert ended at 10:15pm, then q is logically independent of p. 9
2. Relations between statements Sometimes it is difficult to say whether two statements are logically independent or relevant, as relevance allows for the truth of two propositions to be connected even in remote ways. As we will soon see, this poses no problem for the Logician, who is primarily interested in hypothetical cases where the question, If...then is of more interest many times than the question of is actually. As a result, when questions concerning relevance versus independence arise, the Logician can just consider what follows if p is relevant to q, and similarly what follows if q is independent of p. A similar problem arises between the relations of entailment versus relevance. Exploring this connection has been one of the most fruitful chapters of Logic, and in a sense has defined the direction of logical investigation for years. In particular the question of entailment versus very strong relevance characterizes what is known as the problem of induction, which we will encounter in some detail in Chapter 5. Exercises Section 2 Preliminaries: a) Without looking (but with having actually read the text) write out the definition of Entailment, Relevance and Independence. Compare your answers to the actual definitions given in the text. Are your definitions equivalent? If not, highlight areas in your definition which are different than the original definition. b) Write out the actual definitions several times, until you get them correct. State that you have done so. c) Find five examples of statements p and q, such that p entails q, five examples where p is relevant to q, and 5 examples where p and q are independent. d) Give reasons why these distinctions between relations between statements are important to the process of reasoning, Main Exercises: State the relation between p and q. Write E if p entails q, R if p is relevant to q, and I if q is logically independent of p. Recall the order of p and q in the question matters in the cases of Relevance and Entailment. 1.* Let p be, The capital of Arizona is Phoenix and let q be January has 31 days. 2. Let p be, The Mississippi river is free from ice and let q be Today is May 8. 3.* Let p be, Mary is a grandmother of 10 and let q be At least one of Mary s children has children. 4. Let p be, Tom s favorite color is green and let q be John owns a green truck. 10
2. Relations between statements 5.* Let p be, The shortest driving distance from Tucson to Phoenix is 100 miles, and John drove from Tucson to Phoenix and let q be John drove at least 100 miles on his trip from Tucson to Phoenix. 6. Let p be, The book has a total of 237 pages and let q be The book weighs more than 6 ounces. 7.* Let p be, Mary knows the capital of French Guyana and let q be Mary knows every world Capital. 8. Let p be, 27 out of 30 students graduated in May and let q be The tally of monthly traffic accidents increased in October by 3%. 9.* Let p be, If John gets a raise, then he will buy a new car, and he did get a raise and let q be John bought a new car. 10. Let p be, John bought a new car and let q be If John gets a raise, then he will buy a new car, and he did get a raise. Answer True or False. Justify your answer by appealing to definitions or examples or other reasons. Example. If p entails q, then q can be false even if p is true. FALSE: The definition of entailment states that if p entails q, then if p is true, q must be true. Example. If p entails q, then q can be false. TRUE: Let p be, "Yesterday was Wednesday", and let q be "Today is Thursday", so p entails q (by definition of weekdays), but if today is some other day rather than Thursday, we have that p entails q, where q is false. This example highlights the If part in the definition of entailment, since If p is true, then q must be true, is not the same as, p is actually true, so q is also true. 11.* If p entails q, then q entails p. 12. If p is relevant to q, then q is relevant to p. 13. If q is logically independent of p, then p is logically independent of q. 14. p entails q if it is impossible for q to be false and p to be true. 15.* Since our choice of letters to represent propositions is arbitrary, we can let p stand for one proposition, and a different proposition at the same time. (hint: what would happen to our definitions if this were allowed?) 16. Not all sentences in English are propositions. 11
2. Relations between statements 17.* Propositions are sentences which have the property of being true or false. 18. Given three propositions p, q, and r, it may be the case that neither p or q alone entail r, but taken together they do. In other words, it may be the case that p does not entail r, and q does not entail r, but p and q together entail r. 19.* If two statements are logically independent, then both p and q can be false. 20. If p entails q, then both p and q can both be false. 12
3. Arguments If Philosophical Logic has a single most important object of study, that object is certainly the argument. In this chapter we will learn what an argument is in the Logical sense, and use the relations of entailment, relevance and independence to understand the logical concept of inference. This will allow us to classify all arguments into two basic types, valid and invalid. Take care, many of the terms that are central to Chapter 3 will have a very different meaning in the context of Logic than elsewhere. It will be important to keep this in mind to avoid errors and to master the definitions and concepts in this chapter. 3.1. Combining propositions In our analysis in the previous section, we allowed ourselves to refer to specific propositions with single letters, such as p, q, and r. We will continue to use this convention throughout this book, indeed, when we start our formal analysis of arguments and logical concepts, such single letters will dominate our discussion, together with other symbols which do not stand for propositions but instead words like, and, or, if...then and others. To anticipate such usage, let us informally stipulate that two propositions p and q are equivalent only if p and q are true (or false) under the exact same conditions or circumstances. For example, if p stands for, Exactly 24 hours have passed since we put the petri dish in the incubator, and if q stands for, Exactly one day has passed since we put the petri dish in this incubator, then p is equivalent to q. Keeping this in mind, suppose we have two propositions, p and q, which are not equivalent, can we form another proposition whose truth is connected to both p and q but not equivalent to either? The answer is yes, as a matter of fact there are many ways this can be done, but for the moment we will just focus on one way (and look at others when we turn to our formal analysis of Logic). To illustrate how this is done, consider two specific propositions p and q and let p be, John is a student at Pima Community College and let q be, Mary is a student at the University of Arizona. Supposing that John and Mary are not connected in any way, we can stipulate that p and q are independent propositions. We want to create another proposition, which we will call r, whose truth is connected to both p and q. The natural way to do this is to join p and q together with the conjunction, and. In other words, let r stand for, John is a student at Pima Community College and Mary is a student at the University of Arizona. Symbolically we might say that r = p and q. Clearly r is not equivalent to p or to q alone, but the truth of r entails the truth of both p and q individually! The process of joining two propositions with the word and is called conjunction. In the above example, we restricted the use of conjunction 13
3. Arguments to two propositions which are not logically equivalent. This restriction assured us that the resulting proposition was not equivalent to either of the original two. In general, we can ignore this restriction and can use conjunction freely between any number of propositions to create another proposition whose truth entails the truth of each of the individual propositions conjoined with the word, and. As it turns out, we can repeat the above process and replace the word and with the word or, hence r would stand for, John is a student at Pima Community College or Mary is a student at the University of Arizona. When we use the word or we say the resulting proposition r is the disjunction of p and q, but in this case the truth of r only entails the truth of at least one of individual disjunctions p or q. To summarize, we can take any number of propositions, join them all together with multiple uses of the word and or or to form another proposition which is the conjunction or disjucntion of all of the individual ones. We will learn more about this fundamental process when we study formally the truth table definitions of both and and or later on. 3.2. The Argument Suppose someone wants to assert a claim that something is true. Following convention we will call that claim q (recall that our letters stand for propositions, sentences which are either true or false). Now if q is not obviously true, it seems appropriate for the person asserting the claim to support it with some type of evidence, or at the very least provide reasons for claiming q is true. Since evidence comes in the form of propositions which are either true or false, then we can also denote each piece of evidence as p, r, s, etc. When we do this, we are saying that because of p, r, s, etc. it is reasonable to accept the claim q as true. One of the major goals of Philosophical Logic is classifying arguments and discovering whether an argument really gives good reasons for accepting its main claim. This process of reflection and analysis is called reasoning, and as a matter of fact, it is of such importance that many textbooks define Philosophical Logic as the process of distinguishing between good and bad reasoning! Let us now informally define the argument, our central object of study in Part 1. Definition 4. An argument consists of a claim, which is called a conclusion, together with at least one proposition, called a premise, that is given to support the truth of the conclusion. It is important to note that an argument consists of two parts, a conclusion and at least one premise given to support the conclusion. Informally, this is saying something similar to: because p is true, then it is reasonable to conclude q is true. In this case, p is the premise, and q is the conclusion. In real life, arguments are far more complex than this simple but abstract example, and we will examine many in this chapter. In particular, most arguments have more than one premise, and many times more than one conclusion. When this happens, we can try to identify the main conclusion, and consider the other conclusions as acting as premises (in context of the entire argument) 14
3. Arguments which are supposed to support the main conclusion. Let s illustrate this process with some examples. Example. Because of heavy rush-hour traffic, Tom missed his flight. In this case, our conclusion (claim) is, Tom missed his flight, and the reason given for this claim is the single premise, Because of rush hour traffic. In this simple case, it is easy to see which part of the argument is the conclusion and which is the premise. Example. If Henry is the person who damaged the rental car, then he must have been in San Diego during Spring Break. If Henry were in San Diego during Spring Break, then he could not have been in Tucson at the same time. But we know Henry was in Tucson during Spring Break, so Henry is not the person who damaged the rental car. In this example, the conclusion is that Henry is not the person who damaged the rental car. The premise that Henry was in Tucson during Spring Break supports the claim that Henry was not in San Diego, which in turn leads directly to the conclusion that Henry was not responsible for the damaged rental car, since after all, the very first premise asserts that, If Henry is the person who damaged the rental car, then he must have been in San Diego during Spring Break. This is an example of a chain of reasoning, where one premise leads to another conclusion which together with another premise leads to the final conclusion. Chains of reasoning of a special type will be studied in detail in Chapter X. Example. It is a good idea to make sure you have working fire alarms in your house. Just look at the family whose house burned down on Christmas. They lost everything in their house, and almost lost their daughter who nearly died of smoke inhalation. Their house did not have working fire alarms, and for that reason, the fire itself went unnoticed while the family slept. They were saved only by the chance occurrence of a neighbor s teenage son arriving home late from a Christmas-eve date who noticed the fire and woke the family up. This argument is a bit more complicated than the previous examples, but it is more like arguments we encounter in our daily lives. In order to start our analysis of the argument, we have to break it down into its parts, which means finding its premises and the principle conclusion. To find the main conclusion, consider each sentence in turn, and ask, Is this the main claim all the other sentences taken together wish to establish? Asking ourselves this question with respect to the above argument, we see that the main conclusion is stated at the very start of the argument, namely, Its a good idea to make sure you have working fire alarms in your house. The remainder of the sentences basically give reasons that support this conclusion. Lets turn to those reasons now and examine how they support this claim. In doing so, we may have to identify some unwritten assumptions that the author has made, and re-write some sentences so that they take the form of propositions (sentences which are 15
3. Arguments either true or false). To help with this process, lets enumerate each sentence of the argument and consider each in turn: 1. It is a good idea to make sure your have working fire alarms in your house. 2. Just look at the family whose house burned down on Christmas. 3. They lost everything in their house, and almost lost their daughter who nearly died of smoke inhalation. 4. Their house did not have working fire alarms, and for that reason, the fire itself went unnoticed while the family slept. 5. They were saved only by the chance occurrence of a neighbor s teenage soon arriving home late from a Christmas-eve date who noticed the fire and woke the family up. Considering each sentence in turn, we discover that 2-5 support 1, which is just saying that 1 is the main conclusion and 2-5 are premises that support that conclusion. However, the really attentive student might point out that, according to our definition of an argument, sentence number 2 does not seem to be part of the argument! Why? According to the definition of an argument, premises are propositions, and propositions are statements which are either true or false. In this case, sentence number 2 is a command, not a proposition. Commands are neither true or false (whether or not you obey the command is a proposition, but the command itself is not). According to our definition, 2 can not form part of our argument. But surely its truth is connected in some way to the to the argument as a whole and should not be ignored. How do we resolve the problem? Of course, we could just simply re-define the term argument to include cases like these, or just throw out sentence 2, but there is a less drastic option which Logicians employ constantly, we simply re-state the essence of the sentence in such a form as to keep its connection to the main conclusion, while turning it into a proposition. Such a restatement of 2 may be: A family s house burned down on Christmas. Now clearly, this statement is either true or false, and is important to the conclusion and the remaining sentences, and this re-writing of 2 fully preserves the original s connection to the whole, hence Logicians consider this an acceptable change. After we have made this change, we see the remaining sentences are indeed propositions and relevant to the main conclusion, hence they are indeed premises. But what makes them relevant to the main conclusion? To answer this question, the concept of unstated assumptions, or unstated premises, made by the argument is helpful. 1 What are some of these assumptions which make the premises relevant? Let s start with our modified premise 2, A family s house burned down on Christmas. What is it about this premise that makes it relevant to the claim that, It is a good idea to make sure your have working fire alarms in your house? There are several possibilities, but 1 Unstated premises are technically called enthymemes, but we will just call them unstated premises or assumptions in this text. 16
3. Arguments let s explore the obvious by first asking, Would it matter if the premise stated that the house burned down on another day rather than Christmas? If not, then the fact that it burned down on Christmas is not as important as another fact, that being that it burned down (to see this, just change 2 to state, A family s house did not burn down..., and ask if that changes its connection to the conclusion). So, the fact that a house has burned down is at least relevant to having working fire alarms, but we need to pursue the issue more. Suppose the family did have working fire alarms, would that fact result in the house not burning down? This one is harder to answer, as it requires more information than we are given. Whether or not having a working fire alarm would prevent the house from burning down, the argument gives us another line of reasoning - premise 3 suggests that losing everything, including the life of a loved one is undesirable, and premise 4 argues that without working fire alarms, one might sleep through a fire, and as a result lose one s life. Finally premise 5 suggests that unless one wants to leave one s alert system to chance, then one should have a working fire alarm. What has been assumed in all of this? At least the following (and probably more): It is possible that houses burn down. It is better that family members live than die in house fires. One can die from house fires which go unnoticed. Many things go unnoticed while sleeping. A working fire detector can sound the alarm, which, at the very least will alert even sleeping people to a fire which might otherwise go unnoticed. A fire detector can monitor one s house at all times, especially late at night when neighbors sleep. It is better not to leave the fire alarm alert to chance. With these assumptions in mind, we can re-read Example 2 and see that each sentence in our argument supports each of the above assumptions, and the natural conclusion is that it is better to have a working fire alarm than none at all. It is, of course, a legitimate question as to whether the person making the above argument should just do so by using the above assumptions for premises, but the point of example 2 was to examine an argument that is more like one encountered in our daily lives. Notice that we have not stated anything about the strength of the above argument, or just how relevant the premises are to the conclusion, whether any single premise or conjunction of premises entails the conclusion nor have we said anything about the likelihood of the conclusion being true, or in general, whether a given argument presents a good case for its specific conclusion(s). These concepts will be the detailed subject of later investigations, but to get us started down that road, we now turn to the very important concept of inference. 17
3. Arguments 3.3. Validity, Inference, and Invalidity Let us begin our investigation of inference by considering three examples: Example 1 If John makes the free throw, then the U of A will win the game. John made the free throw, so the U of A won the game. Example 2 If John makes the free throw, then the U of A will win the game. John did not make the free throw, so the U of A lost the game. Example 3 If John makes the free throw, then the U of A will win the game. John did not make the free throw, so yesterday my neighbor ate oatmeal for breakfast. Let s sort out the premises and conclusion to each of the above examples and use our method of connecting premises together with the word and as we studied in Section 3.1 with the goal of determining whether the conjunction of all of the premises is relevant, entails, or is independent of the conclusion. To simplify our analysis, let p be, If John makes the free throw, then the U of A will win the game, and let q be, John made the free throw, and let r be, The U of A won the game. This allows us to re-write Example 1 as, p and q, therefore r. 2 Now we pose the following question, When is p and q true? If you are tempted to consider specifics about the rules of basketball, game conditions, and players, then STOP. These indeed are important questions, but surprisingly they are of minor importance to our immediate goal. Let s assume we know all of these answers (even though we do not) and ask the question again, when is p and q true? Hopefully we can all agree that there is no hope for the conjoined statement p and q being true unless both p and q are individually true. This important point applies in general, not just when p and q stand for propositions related to free throws and basketball games, but in every case! As a matter of fact, the conjunction of any number of propositions is true only when every single individual proposition in the conjunction is true, as we discussed at the end of Section 3.1. With this in mind, let us now consider the argument given in Example 1, and ask, using the vocabulary we have learned, whether the conjunction p and q entail r, is relevant to r or is independent to r? In order to answer this question, it is a good idea to just assume (pretend) every single premise is true, which means in this particular case, we assume that the statement, If John makes the free throw, then the U of A will win the game is indeed true, and the statement, John made the free throw is also true. Given these assumptions it is natural to infer that the U of A will win the 2 Notice that we have replaced the word, so with the word therefore to highlight the conclusion to the argument. It turns out that there are many words which help us decide which sentences in an argument are conclusions. If one sees (or can add unwritten) words and phrases such as; so, then, therefore, hence, as a result, it follows that, then what follows is usually a conclusion (major or minor) of the argument. On the other hand words and phrases like, because, since, for the reason that, mark premises in an argument. 18
3. Arguments game. Is it possible for the U of A to lose, if our assumptions are true? Clearly not, for it that were possible, then at least one of our premises must be false. Make sure you clearly understand this point before you go on. To repeat; if we assume the following two premises are true, If John makes the free throw, then the U of A will win the game and John makes the free throw, then it must be the case that the U of A wins the game. Of course to assume something is true is not the same as to assert that it really is true. However, these types of assumptions are key to understanding one of the major tools in the Logician s tool box. Clearly if we know that if all of the premises to an argument happen to be true, then the conclusion to that argument must be true, such knowledge is important if one wants to analyze arguments! This brings us to a very important definition, which is of such importance that the student should spend as much time as needed to fully understand the definition, which requires not only knowing what the definition says, but what it means. Definition 5. An argument which has the property that the conjunction of all of its premises entail its conclusion is said to be a valid argument. Equivalently an argument is valid if it has the property that it is impossible for the conclusion to be false and all of the premises to be true. If an argument is valid, we also say that the inference from the premises to the conclusion is valid. Note the introduction of the term inference above. In general, an inference is the act of drawing a conclusion from a set of premises. We will say that an argument is valid or that the inference of an argument is valid interchangeably. Before we examine Examples 2 and 3, let s discuss just what this important definition means, by considering the most common mistakes students make with respect to this new concept. All of the statements below are incorrect or incomplete: i) A valid argument must have a true conclusion. ii) iii) If all premises of a valid argument are false, then the conclusion must be false. A valid argument is an argument with all true premises and a true conclusion. iv) The conjunction of all the premises in a valid argument must be true. The first statement omits a very key part of that definition, that part that says, If all premises are true. Missing or forgetting key parts of definitions is perhaps the number one reason for student errors. The second statement is not the definition of a valid argument or a statement entailed by that definition. The definition of a valid argument is silent about what, if anything, must be the case if all the premises of a valid argument are false, hence no assumption should be made on the part of the student concerning that possibility. In Part 2 of this textbook, we will prove this statement to be incorrect formally. For now, don t make the error and consider this statement as a proper characterization of valid arguments. 19
3. Arguments The third statement omits a very important part of our definition for a valid argument. Why this is an error is left as an exercise for the student. The fourth statement is a bit more difficult. So we will demonstrate is falsity by example. Suppose I state the following argument: The Cullinan diamond weighs more than 100 grams, and was worn by Queen Victoria when she was crowned in 1964, therefore the Cullinan diamond weighs more than a 2 gram feather. Clearly, this argument is valid (it is impossible to have a false conclusion with all true premises), but since Queen Victoria was not crowned in 1964, the conjunction of the premises can t be true (recall this is another way of saying at least one of the premises is not true). The argument is valid because the truth of premise one alone entails the truth of the conclusion, which means it is impossible to have a false conclusion and all true premises. We are now ready to examine Examples 2 and 3. Repeating the procedure as above, we assume that the conjunction of the premises are true (recall, this is just another way of saying that all of the premises are individually true). Assuming that both, If John makes the free throw, then the U of A will win the game. and John did not make the free throw are true, is it possible for the U of A lost the game to be false? Careful thinking shows that indeed this is a possibility. To see this, simply suppose that Mike makes the winning free throw rather than John. Then both premises are true, but since Mike made the winning free throw, the U of A won the game (so the conclusion stating that the U of A lost the game is false). A similar analysis will show that it is also possible for all of the premises to be true but the conclusion to be false in example 3. The possibility that all of the premises to an argument can be true, but the conclusion can still be false leads us to our final definition. Definition 6. If it is possible for an argument to have all true premises but still have a false conclusion, then the argument is said to be invalid. Equivalently, if the premises of an argument do not entail the conclusion, the inference from the premises to the conclusion is called invalid. Again, be careful to not add anything to the definition of an invalid argument that is not stated in the definition. As a rule of thumb, any argument where the premises are relevant (but do not entail) the conclusion is an invalid argument, as well as any argument whose conclusion is logically independent of the truth of the premises - there is one important case which prevents us from saying this is must always be the case, which we will encounter when we examine arguments formally in Part 2. Invalid does not mean false or even bad reasoning - it means only that the possibility is open (no matter how remote) for the conclusion to be false, even if all of the premises are true. What is the case is that all arguments are either valid or invalid, simply because invalid means not valid. Hence, any argument can be classified as either valid or invalid, and determining which is the case will be an important first step in our analysis of arguments. However, like any subject of scope and depth, the first step is not the last - and the same is true in our analysis of arguments. Knowing whether an argument 20
3. Arguments is valid or invalid is important, but we will need to examine and answer many more questions before we can decide whether an argument gives good reasons for supporting its conclusion! An argument is valid iff: Valid argument/inference, three equivalent definitions: 1. The truth of the premises entail the truth of the conclusion. 2. If all the premises are true, the conclusion must be true. 3. It is impossible for the conclusion to be false and all the premises to be true. An argument is invalid iff: Invalid argument/inference, two equivalent definitions. 1. The truth of the premises do not entail the truth of the conclusion. 2. The mere possibility for the conclusion to be false with all true premises makes an argument invalid (no matter how remote that possibility). As always, examples help us to grasp new concepts. Example. If the washes are full of water in Tucson, then it rained. This argument as simple as it gets, with one premise (the washes are full of water) and one conclusion (it rained). To determine whether this argument is valid or invalid, we ask, Is it possible that it rained and the washes are not full of water? It does not take much imagination to conceive of this possibility, (think about it raining for just a few seconds) - hence this argument is invalid. Example. The lights are not working because the circuit breaker has tripped or the electrical storm knocked out the power. The circuit breaker is tripped therefore the electrical storm did not knock out the power. This argument may not be as obviously invalid as the previous example, in order to analyze it, we must know what the conclusion is and what the premises are. The conclusion (the claim the argument wishes to establish) is that the electrical storm did not knock out the power (use the clue that conclusions follow words such as then and therefore ). The premises are: the lights are not working because the circuit breaker has tripped or because the electrical storm knocked out the power and the circuit 21
3. Arguments breaker is tripped. Given this information, we pose the question, Is it possible for the conclusion to this argument to be false, even if all the premises are true? If the answer is yes, then according to our definition the argument is invalid. In this case, we can consider the possibility that both the circuit breaker is tripped and the electrical storm knocked out the power. While perhaps a rare occurrence, this is certainly a possibility, and a possibility which clearly makes both premises true but the conclusion false. Hence, the argument is invalid. Exercises Section 3 Preliminaries: a) Without looking (but with having actually read the text) write out the definitions of: Argument, Valid argument, and Invalid argument. Compare your answers to the actual definitions given in the text. Are your definitions equivalent? If not, highlight areas in your definition which are different than the original definition. b) Write out the actual definitions several times, until you get them correct. State that you have done so. c) Make up (create your own) 3 valid arguments and 3 invalid arguments. Clearly state whether each argument is valid or invalid and why (hint, for the why part, appeal to the definitions in each case). d) How are the terms valid and invalid used in Logic differently than elsewhere? Provide an example of the term valid and invalid which clearly do not mean the same as our definitions in class (you may use the internet to aid in your search). Main Exercises: Identify the premises and main conclusion to the following arguments. Use this information to state whether the argument is valid or invalid. If the argument is invalid, modify the premises to make it valid. Remember to know and use the definitions of valid and invalid in your reasoning process (memorize them or write them down so you have them handy when doing the exercises) Example. Malaria is caused by a parasite that lives in mosquitoes that inhabit tropical regions throughout the world. John has malaria, so he must have recently been in a tropical country. Answer: The premises are: 1) Malaria is caused by a parasite that lives in mosquitoes that inhabit tropical regions throughout the world. 2) John has malaria. The conclusion is that John must have recently been in a tropical country. The argument is invalid, as John may have only been in a tropical country 3 years ago, this fact alone would make the conclusion false even if all the premises were true. Another possibility (among others) is that the parasite that causes malaria is also found in other non-tropical regions of the world. This possibility is not excluded by that fact that the parasite lives also in tropical areas, but this fact allows the conclusion to be false (as John may have gotten malaria by a mosquito in a non-tropical region), even if all the premises are true. One way the argument could be made valid would be the modification of premise one 22
3. Arguments to state: Malaria is caused by a parasite that only lives in mosquitoes that inhabit tropical regions throughout the world and is only caught by individuals that reside in tropical regions Example. If John gets a new job, then we will take a vacation in the Summer to Italy. If we take a vacation to Italy, we will have to obtain US passports, which require a birth certificate. In order to get my birth certificate, I will need to look through Mom s boxes, which will require that I visit my parents. John got a new job, so I will have to visit my parents. Answer: The premises are: 1) If John gets a new job, then we will take a vacation in the Summer to Italy, 2). If we take a vacation to Italy, we will have to obtain US passports, which require a birth certificate, 3) n order to get my birth certificate, I will need to look through Mom s boxes, which will require that I visit my parents, 4) John got a new job. The conclusion is that I will have to visit my parents. The argument is valid. In order to see this, assume the conclusion is false. This means that I did not visit my parents, which is turn means I did not get my birth certificate, which means I did not get a new passport, which means I did not get a Summer vacation it Italy, which means John did not get a new job, which means premise 4 would be false, thus making it impossible to have a false conclusion and all true premises. 1.* If it rains in Tucson, then the washes will be full of water. The washes are not full of water, therefore it did not rain in Tucson. 2. Every time I have taken an algebra class in the past, I fail. So if I take an Algebra class this semester, I will just fail again. 3. The suspect in the homicide investigation wore a large hat. The 3rd man to the left in the police line-up has a large head, therefore that man is probably the murderer. 4. The battery is bad, or the alternator is not working. The battery is not bad, so the alternator is not working. 5.* The Power ball has reached a near-record jackpot of $210 million dollars. Almost anyone would like that kind of money, and one thing is for sure, if you don t play, you can t win, so everyone should buy a power ball ticket. 6. Many people who have had a bad case of the flu and took zinc supplements report their flu did not last as long as it would have otherwise. Therefore, zinc supplements help alleviate the flu. 7. It is a good idea to make sure you have working fire alarms in your house. Just look at the family whose house burned down on Christmas. They lost everything in their house, and almost lost their daughter who nearly died of smoke inhalation. Their house did not have working fire alarms, and for that reason, the fire itself went unnoticed while the family slept. They were saved 23