# Introduction Symbolic Logic

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2 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 IMPLICATION 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 ITH QUANTIFIERS 6 DERIVATIONS ITH QUANTIFIERS 7 UNIVERSAL DERIVATIONS 8 SOME DERIVATIONS 9 DERIVED RULES 10 INVALIDITIES 11 EXPANSIONS Copyrighted material Introduction -- 2

5 The triangle made of three dots is an abbreviation of the word `therefore', and is a way of identifying the conclusion of an argument. In order to save on writing, and also to begin displaying the form of the arguments under discussion, we will start abbreviating simple sentences by capital letters. For the time being we will abbreviate Polk was a president by `P', and hitney was a president by `'. e will abbreviate hitney was not a president by `not '. So the argument can be shortened to: P or not 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. hen 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. e have already seen one case of a valid argument which has all of its premises true and its conclusion true as well: P or not P hat other possibilities are there? ell, 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 not R False 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 P False 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. e 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 Copyrighted material Introduction -- 5

7 1. Decide whether each of the following arguments is valid or invalid. If the argument is invalid then describe a possible situation in which its premises are all true and its conclusion false. a. Either Polk or Lee was a president. Either Lee or hitney was a president. Either Polk or hitney was a president. b. Lee wasn't a president, and Polk was. Either Polk or hitney was a president. hitney was a president. c. Polk was a president and so was Lee. hitney was a president. Polk was a president and so was hitney. d. Either Polk or hitney was a president. Lee was not a president. Lee wasn't a president and Polk was. 2. hich of these are true, and which are false: a. Some valid arguments have false conclusions. b. No invalid argument has all true premises and a false conclusion. c. If an argument is valid, and you produce a new argument from it by adding one or more premises to it, the resulting argument will still be valid. d. If an argument is invalid, and you produce a new argument from it by adding one or more premises to it, the resulting argument must still be invalid. e. If an argument has an impossible premise, it is valid. (An example of an impossible sentence is `Some giraffes aren't giraffes'.) f. If an argument has a necessarily true conclusion, it is valid. (An example of a necessarily true sentence is `Every giraffe is a giraffe'.) g. If an argument has a false premise, it is valid. 3. An argument which is valid and which also has all of its premises true is called sound. Based on this definition, which of the following are true, and which false: a. All valid arguments are sound. b. All invalid arguments are unsound. c. All sound arguments have true conclusions. d. If an argument is sound, and you produce a new argument from it by adding one or more premises to it, the resulting argument will still be sound. e. All unsound arguments are invalid. f. If an argument has a necessarily true conclusion, it is sound. 4. Suppose that we have two arguments which are both valid: Q Q S Copyrighted material Introduction -- 7

8 what do we know about this argument? S 5. Suppose that the following two simple arguments are both sound: Q Q S what do we know about this argument? S 6. Suppose that the following argument is invalid: S what do we know about these arguments? R R S 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. 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 have challenged 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. hat 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. e 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. Copyrighted material Introduction -- 8

9 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, not a matter of content. 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 not P is valid, regardless of its subject matter. ith 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 hitney or Lee was a president. Either Polk or hitney was a president. b. Lee wasn't a president, and Polk was. Either Polk or hitney was a president. hitney was a president. c. Polk was a president and so was Lee. hitney was a president. Polk was a president and so was hitney. d. Either Polk or hitney was a president. Lee was not a president. Lee wasn't a president and Polk was. 2. hich 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. Copyrighted material Introduction -- 9

10 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 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? hy? 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? hy? 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? hy? 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 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: Copyrighted material Introduction -- 10

12 6 THE PLAN OF THE TEXT Introduction 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. Copyrighted material Introduction -- 12

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