# Exposition of Symbolic Logic with Kalish-Montague derivations

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2 Preface The system of logic used here is essentially that of Kalish & Montague 1964 and Kalish, Montague and Mar, Harcourt Brace Jovanovich, The principle difference is that written justifications are required for boxing and canceling: 'dd' for a direct derivation, 'id' for an indirect derivation, etc. This text is written to be used along with the UCLA Logic 2010 software program, but that program is not mentioned, and the text can be used independently (although you would want to supplement the exercises). The system of notation is almost the same as KK&M; major differences are that the signs ' ' and ' ' are used for the quantifiers, name and operation symbols are the small letters between a and h, and variables are the small letters between i and z. The exercises are new. Chapters 1-3 cover pretty much the same material as KM&M except that the rule allowing for the use of previously proved theorems is now in chapter 2, immediately following the section on theorems. (Previous versions of this text used the terminology tautological implication in section This has been changed to tautological validity to agree with the logic program.) Chapters 4-6 include invalidity problems with infinite universes, where one specifies the interpretation of notation "by description"; e.g. "R( ): ". These are discussed in the final section of each chapter, so they may easily be avoided. (They are not currently implemented in the logic program.) Chapter 4 covers material from KK&M chapter IV, but without operation symbols. Chapter 4 also includes material from KK&M chapter VII, namely interchange of equivalents, biconditional derivations, monadic sentences without quantifier overlay, and prenex form. Chapter 5 covers identity and operation symbols. Chapter 6 covers Fregean definite descriptions, as in KK&M chapter VI. Version Aug 2013 of An Exposition of Symbolic Logic is a lightly revised version of the August 2012 version of An Introduction to Symbolic Logic (also known as Terry-Text). Copyrighted material Introduction -- 2 Version of Aug 2013

3 CONTENTS Chapter One Sentential Logic with 'if' and 'not' 1 SYMBOLIC NOTATION 2 MEANINGS OF THE SYMBOLIC NOTATION 3 SYMBOLIZATION: TRANSLATING COMPLEX SENTENCES INTO SYMBOLIC NOTATION 4 RULES 5 DIRECT DERIVATIONS 6 CONDITIONAL DERIVATIONS 7 INDIRECT DERIVATIONS 8 SUBDERIVATIONS 9 SHORTCUTS 10 STRATEGY HINTS FOR DERIVATIONS 11 THEOREMS 12 USING PREVIOUSLY PROVED THEOREMS IN DERIVATIONS Chapter Two Sentential Logic with 'and', 'or', if-and-only-if' 1 SYMBOLIC NOTATION 2 ENGLISH EQUIVALENTS OF THE CONNECTIVES 3 COMPLEX SENTENCES 4 RULES 5 SOME DERIVATIONS USING RULES S, ADJ, CB 6 ABBREVIATING DERIVATIONS 7 USING THEOREMS AS RULES 8 DERIVED RULES 9 OFFICIAL CONDITIONS FOR DERIVATIONS 10 TRUTH TABLES AND TAUTOLOGIES 11 TAUTOLOGICAL VALIDITY Chapter Three Individual constants, Predicates, Variables and Quantifiers 1 INDIVIDUAL CONSTANTS AND PREDICATES 2 QUANTIFIERS, VARIABLES, AND FORMULAS 3 SCOPE AND BINDING 4 MEANINGS OF THE QUANTIFIERS 5 SYMBOLIZING SENTENCES WITH QUANTIFIERS 6 DERIVATIONS WITH QUANTIFIERS 7 UNIVERSAL DERIVATIONS 8 SOME DERIVATIONS 9 DERIVED RULES 10 INVALIDITIES 11 EXPANSIONS Copyrighted material Introduction -- 3 Version of Aug 2013

4 Chapter Four Many-Place Predicates 1 MANY-PLACE PREDICATES 2 SYMBOLIZING SENTENCES USING MANY-PLACE PREDICATES 3 DERIVATIONS 4 THE RULE "INTERCHANGE OF EQUIVALENTS" 5 BICONDITIONAL DERIVATIONS 6 SENTENCES WITHOUT OVERLAY OF QUANTIFIERS 7 PRENEX NORMAL FORMS 8 SOME THEOREMS 9 SHOWING INVALIDITY 10 COUNTER-EXAMPLES WITH INFINITE UNIVERSES Chapter Five Identity and Operation Symbols 1 IDENTITY 2 AT LEAST AND AT MOST, EXACTLY, AND ONLY 3 DERIVATIONAL RULES FOR IDENTITY 4 INVALIDITIES WITH IDENTITY 5 OPERATION SYMBOLS 6 DERIVATIONS WITH COMPLEX TERMS 7 INVALID ARGUMENTS WITH OPERATION SYMBOLS 8 COUNTER-EXAMPLES WITH INFINITE UNIVERSES Chapter Six Definite Descriptions 1 DEFINITE DESCRIPTIONS 2 SYMBOLIZING SENTENCES WITH DEFINITE DESCRIPTIONS 3 DERIVATIONAL RULES FOR DEFINITE DESCRIPTIONS: PROPER DESCRIPTIONS 4 SYMBOLIZING ORDINARY LANGUAGE 5 DERIVATIONAL RULES FOR DEFINITE DESCRIPTIONS: IMPROPER DESCRIPTIONS 6 INVALIDITIES WITH DEFINITE DESCRIPTIONS 7 UNIVERSAL DERIVATIONS 8 COUNTER-EXAMPLES WITH INFINITE UNIVERSES Copyrighted material Introduction -- 4 Version of Aug 2013

7 and Whitney was a president by `W'. We will abbreviate Whitney was not a president by `not W'. So the argument can be shortened to: P or W not W P A major point of this book is to explore the notion of deductive validity. Since the deductive kind is the only one considered here, we simply refer to it as "validity". In this section we will go over certain consequences of the following definition of validity: An argument is valid if, and only if, there is no logically possible situation in which all of its premises are true and its conclusion false. When we talk about "truth" here we do not have anything deep or mysterious in mind. For example, we say that the sentence 'There is beer in the refrigerator' is true if there is beer in the refrigerator, and false if there isn't beer in the refrigerator. That's all there is to it. We have already seen one case of a valid argument which has all of its premises true and its conclusion true as well: P or W not W P True True True What other possibilities are there? Well, as we noted above, it is possible to have some of the premises false and the conclusion false too. (This is sometimes referred to as a case of the "garbage in, garbage out" principle.) Suppose we use `R' to abbreviate Robert E. Lee was a president. Then this argument does not have all of its premises true, nor is its conclusion true: R or W not W R False True False Yet this argument is just as good, as far as its validity is concerned, as the first one. If its premises were true, then that would guarantee that its conclusion would be true too. There is no logically possible situation in which the premises are all true and the conclusion false. This argument, though it starts with a false premise and ends up with a false conclusion, has exactly the same logical form as the first one. This sameness of logical form lies at the foundation of the theory in this book; it is discussed in the following section. Although false inputs can lead to false outputs, there is no guarantee that this will happen, for you can reason validly from false information and accidentally end up with a conclusion that is true. Here is an example of that: P or not W W P True False True In this example, one of the premises is false, but the conclusion happens to be true anyway. Mistaken assumptions can sometimes lead to a true conclusion by chance. The one combination that we cannot have is a valid argument which has all true premises and a false conclusion. This is in keeping with the definition given above: a deductively valid argument is one for which it is logically impossible for its conclusion to be false if its premises are all true. We have seen that there are valid arguments of each of these sorts: PREMISES all true not all true not all true CONCLUSION true false true What about invalid arguments? (That is, what about arguments that are not deductively valid?) What combination of truth-values can the parts of invalid arguments have? The answer is that they can have any combination of truth-values whatsoever. Here are some examples: Copyrighted material Introduction -- 7 Version of Aug 2013

10 7. (a) Give an example of a "reversing" argument, that is, one which is guaranteed to have a false conclusion if its premises are true, and is guaranteed to have a true conclusion if any of its premises are false. (b) Give an example of an argument that must have a true conclusion no matter what the truth-values of its premises. Is this argument valid? 3 LOGICAL FORM If you want to show that an argument is invalid, you can describe a possible situation in which the premises are all true and the conclusion false. We illustrated this above with the argument: Either Roosevelt or Truman (or perhaps both) was a president. Truman was a president. Roosevelt was a president. But this direct appeal to possible situations is sometimes difficult to articulate, and judgments of possibility can differ. Fortunately, there is another technique that is often more useful. You could challenge the above reasoning by saying: "That reasoning is no good. If that reasoning were good, we could prove that McGovern was a president! For we know that: and we know: Either McGovern or Truman was a president, Truman was a president. So, by your reasoning we should be able to conclude that McGovern was a president too!" This challenge, like the first one, also shows that the argument given above is invalid. But whereas the first type of challenge focuses on how the ORIGINAL argument works in some POSSIBLE situation, this second challenge is based on how some OTHER argument works in the ACTUAL situation. What we do in this second technique is to give an argument that is different than the first, but closely related to it. In the case in question, the new argument is: Either McGovern or Truman was a president. Truman was a president. McGovern was a president. We know the new argument is invalid because it actually has all true premises and a false conclusion (we chose it on purpose to be this way). Since the new argument is invalid, so is the original one. But why should the original argument be invalid just because this second argument is invalid? The answer is that, intuitively speaking, they both employ the same reasoning, and it is the reasoning that is being assessed when we make a judgment about validity. But how can we tell that they employ the same reasoning? The answer is that they both have the same form. Each argument is one in which one of the premises is an "or" statement, with the other premise being one of the parts of the "or" statement and the conclusion being the other part. This sameness of structure or form indicates a sameness of the reasoning involved. A key assumption on which all of modern logical theory is based is that goodness of deductive reasoning is a matter of form. Any argument which has just the same form as the argument we were just discussing is invalid, no matter whether its subject matter is religion, politics, mathematics, or baseball. Likewise, any argument which has this form: P or W not W P Copyrighted material Introduction Version of Aug 2013

11 is valid, regardless of its subject matter. With this in mind, we can give a modern account of validity due to form: An argument is formally valid if and only if every argument with exactly the same form is valid. It follows from this definition that if an argument is formally valid, so is any argument with exactly that form, if an argument is not formally valid, neither is any argument with exactly that form. A central preoccupation of modern logic, then, is the investigation and classification of logical forms. (That is why this logic is called "formal logic".) This will be our business throughout the chapters that follow. EXERCISES 1. Decide whether each of the following arguments is valid or invalid. If the argument is invalid then give an argument which has the same form, and which actually has all true premises and a false conclusion. a. Either Polk or Lee was a president. Either Whitney or Lee was a president. Either Polk or Whitney was a president. b. Lee wasn't a president, and Polk was. Either Polk or Whitney was a president. Whitney was a president. c. Polk was a president and so was Lee. Whitney was a president. Polk was a president and so was Whitney. d. Either Polk or Whitney was a president. Lee was not a president. Lee wasn't a president and Polk was. 2. Which of these are true, and which are false: a. Some invalid arguments have the same forms as valid ones. b. You can show an argument valid by producing another argument which has the same form and which has true premises and a true conclusion. c. If you are wondering whether an argument is valid or not, and you fail to find another argument which has the same form and all true premises and a false conclusion, that shows the original argument to be valid. 3. Here are some argument forms. For each, say whether every argument with that form is valid. If it is not valid, give an example of an argument with the given form that has true premises and a false conclusion. a. If A then B A B b. If A then B B A c. not (A and B) not-b not A d. A or B B A Copyrighted material Introduction Version of Aug 2013

12 e. A and not-a B f. A B or not-b g. A or B not-b or C A or C 4. Recall that an argument which is valid and which also has all of its premises true is called sound. a. If you are wondering whether an argument is sound, and you manage to find another one with the same form and having all true premises and a false conclusion, does that show the original argument to be unsound? Why? b. If you are wondering whether an argument is sound, and you manage to find another one with the same form and having all true premises and a true conclusion, does that show the original argument to be sound? Why? c. If you are wondering whether an argument is sound, and you manage to find another one with the same form and having all false premises and a false conclusion, does that show the original argument to be sound? To be unsound? Why? 5. For each of the examples in 3, say whether or not every argument with that form is sound, and also say whether some argument with that form is sound. 6. {This question is speculative, and does not necessarily have a straightforward answer} Could there be an argument that is valid but not formally valid? Could there be an argument that is formally valid but not valid? 4 SYMBOLIC NOTATION Our investigation of logical forms will take an indirect route, but one that has proved to be worthwhile. Instead of attempting a direct classification of the logical forms of sentences of English, we will develop an artificial language that is considerably simpler than English. It will in some ways be like English without some of the logically irrelevant aspects of English. And it will lack some of the characteristics that make the use of English confusing when used in argumentation. For example, the artificial language will lack some of the structural ambiguity of English. Consider this English sentence: Mary teaches little girls and boys. Does this tell us that Mary teaches little girls and little boys, or that she teaches little girls and regular-size boys? If this sentence occurred in an argument, the validity of the argument might turn on how the sentence was read. In the artificial language to be developed, structural ambiguities of this sort will be absent. The artificial language will be especially designed to make logical form perspicuous. You are already familiar with this from arithmetic. Consider the partly symbolic sentence: For any two numbers x and y, x+y = y+x. It is clear what this says. The same thing can also be said without any symbols: Given any two numbers, the result of adding them together in one order is the same as the result of adding them together in the reverse order. It is apparent that the use of symbols makes the claim clearly and vividly. Our logical symbolism will be like this. Copyrighted material Introduction Version of Aug 2013

13 In fact, it will be possible in the symbolic language to tell the logical forms of its sentences just by examining the shapes and arrangements of its symbols. And it will be possible to evaluate formal validity and invalidity of arguments expressed in it by a variety of techniques that appeal directly to the visible arrangements of the symbols in the sentences used. This does not mean that we will lose sight of reasoning that is expressed in our native tongue. One of the tasks of learning the artificial language will be to learn how to take sentences of English and re-express them in the artificial language we are learning. One of our goals, after all, is to learn about reasoning that we encounter every day, in its natural habitat. 5 IDEALIZATIONS The data that we have to deal with are incredibly complex, and this is only an introductory text. So we will idealize from time to time. This is no different from any other art or science. In physics you usually begin by studying the behavior of bodies falling in a perfectly uniform gravitational field, or sliding down frictionless planes. There are no perfectly uniform gravitational fields, and no frictionless planes, yet studying these things gives us clear and simple models that can be applied to real phenomena as approximations. And then in the advanced courses you can learn how friction affects the sliding, and how non-uniform fields affect the movement of things in them. Here are some of the idealizations that we will make in this book: We will look only at arguments with indicative sentences, not with imperative or interrogative. We will ignore any problems due to vagueness. For example, given a perfect understanding of the situation, you may still be unsure whether to say that Mary loves John, because of the vagueness of distinguishing between loving and liking. We will also totally ignore the fact that sentences may change truth-value over time and with differing situations. If I say today: I'm feeling great! this may be true, but the very same sentence may be false tomorrow. And it may be true when I say it, yet false when someone else utters it. This "context dependence" of truth has aroused a great deal of interest, and there are many theories about how it works. They all presuppose that their readers have already learned the material in this book. We will pretend in our investigations that sentences come with unique truth-values that do not change with context. The effects of context constitute an advanced study. We will also assume that each sentence is either true or false. Again, the question of whether, and which, sentences lack truth-value is interesting, but is not to be pursued at the beginning. Many other idealizations will become apparent as we proceed. 6 THE PLAN OF THE TEXT In this text we will develop a symbolic notation in a step-by-step process. In chapter 1 we consider simple sentences that are combined together with the "connectives" 'or' and 'if...then...'. Even with this very austere notation we can formulate and study a number of formal validities. In chapter 2 we expand this notation by introducing further connectives: 'and', 'or', and 'if and only if'. In chapter 3 we vastly expand our symbolic language with the introduction of variables and quantifiers. Each expansion of the notation builds on what has gone before, so we are continually increasing our ability to validate formally valid arguments and invalidate arguments that are not formally valid. Chapters 4-6 contain additional expansions. Copyrighted material Introduction Version of Aug 2013

14 - Answers to the Exercises Answers to the Exercises -- Introduction SECTION 2 1. a. INVALID Any possible situation in which Lee was a president but neither of the others was. b. INVALID Any possible situation in which Polk was a president but neither of the others was. c. VALID d. INVALID Any possible situation in which Whitney was a president and neither of the others was. 2. a. True. (Such an argument will always have at least one false premise.) b. False. Some do; some don't. c. True. d. False. Sometimes adding a premise converts an invalid argument into a valid one, and sometimes it does not. It depends on what you add. e. True. There can't be a possible situation in which it has all true premises and a false conclusion because there can't be a possible situation in which it has all true premises. f. True. There can't be a possible situation in which it has all true premises and a false conclusion because there can't be a possible situation in which it has a false conclusion. g. False. It might be valid, or it might be invalid. 3. a. False. Valid arguments with false premises aren't sound. b. True. An invalid argument isn't sound because it isn't even valid. c. True. The premises are all true, and it's valid, so its conclusion must be true too. d. False. If you add a true premise it will remain sound, but if you add a false premise it will become unsound. e. False. A valid argument is unsound if it has a false premise. f. False. If the conclusion is necessarily true the argument will be valid, but it still might have a false premise, and thus be unsound. 4. It has to be valid. For suppose it were not. Then there would be a possible situation in which A is true and C is false. Since the first argument is valid, B is true in this situation; but then since the second argument is valid, C is also true in that situation, contradicting our supposition that there is a situation in which A is true and C is false. 5. It has to be sound. It has to be valid for the same reason as in the previous example. And since the first argument is sound, A is true. So its premise is true. 6. We know that at least one of them is invalid, but we don't know which. If they were both valid, the first argument would have to be valid, as in exercise 4. So they aren't both valid. But there are cases in which the first is valid and the second invalid, and cases in which the first is invalid and the second valid, and cases in which they are both invalid. First valid and second invalid: A B C First invalid and second valid: A B C Both invalid: Polk was a president Polk or Lee was a president Lee was a president Polk was a president Polk and Lee were presidents Lee was a president Copyrighted material Answers to the Exercises Version of Aug 2013

15 - Answers to the Exercises A B C Polk was a president Nixon was a president Lee was a president 7. a. Polk was a president. Polk wasn't a president. [Naturally, the argument is invalid.] b. Polk was a president. Either Whitney was a president or he wasn't. This argument is valid; it cannot have all true premises and a false conclusion because it cannot have a false conclusion. SECTION 3 1. a. Either McGovern or Nixon was president. Either Nixon or Goldwater was president Either McGovern or Goldwater was president. b. The original argument will do; it already has all true premises and a false conclusion. c. VALID. d. Either Whitney or Polk was a president. Lee was not a president. Lee wasn't a president and Whitney was. 2. a. False. b. False. This does not show that no argument with that form has true premises and a false conclusion. c. False. You might not have looked hard enough. 3. a. VALID b. INVALID If Polk and Lee were both presidents, Polk was a president. Polk was a president. Polk and Lee were both presidents. c. INVALID not (Polk was a president and Lee was a president) not Lee was a president not Polk was a president d. INVALID Lee or Polk was a president. Polk was a president. Lee was a president. e. VALID f. VALID g. VALID (This depends interpreting `or' inclusively; this is discussed in chapter 2 below.) 4. a. Yes. It shows the original argument invalid, and an invalid argument is not sound. b. No. The original argument could still be invalid, or have a false premise, or both. Example: Original argument: "Found" argument: Lee was a president Nixon was a president. Whitney was a president Kennedy was a president. Copyrighted material Answers to the Exercises Version of Aug 2013

16 - Answers to the Exercises c. It shows neither. Examples: Original unsound argument: Lee wasn't a president Whitney wasn't a president Original sound argument: Lee wasn't a president Lee wasn't a president "Found" argument: Nixon wasn't a president. Kennedy wasn't a president. "Found" argument: Nixon wasn't a president. Nixon wasn't a president. 5. a. Some arguments with this form are sound: the ones with true premises. But not all; some of them have false premises. b. None are sound, since none are valid. c. None are sound, since none are valid. But not all; some of them have false premises. d. None are sound, since none are valid. e. None are sound, since none has a true premise. f. Some arguments with this form are sound: the ones with true premises. But not all; some of them have false premises. g. Some arguments with this form are sound: the ones with true premises. But not all; some of them have false premises. 6. Many logicians think that there are arguments that are valid, but not formally valid. An example is: Herman is a bachelor Herman is unmarried The validity of this argument comes from the meaning of the word 'bachelor', and not from the form of the sentences in the argument. As we have defined 'formally valid', any argument that is formally valid is automatically valid. Copyrighted material Answers to the Exercises Version of Aug 2013

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