# Methods of Proof for Boolean Logic

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1 Chapter 5 Methods of Proof for Boolean Logic limitations of truth table methods Truth tables give us powerful techniques for investigating the logic of the Boolean operators. But they are by no means the end of the story. Truth tables are fine for showing the validity of simple arguments that depend only on truth-functional connectives, but the method has two very significant limitations. First, truth tables get extremely large as the number of atomic sentences goes up. An argument involving seven atomic sentences is hardly unusual, but testing it for validity would call for a truth table with 2 7 = 128 rows. Testing an argument with 14 atomic sentences, just twice as many, would take a table containing over 16 thousand rows. You could probably get a Ph.D. in logic for building a truth table that size. This exponential growth severely limits the practical value of the truth table method. The second limitation is, surprisingly enough, even more significant. Truth table methods can t be easily extended to reasoning whose validity depends on more than just truth-functional connectives. As you might guess from the artificiality of the arguments looked at in the previous chapter, this rules out most kinds of reasoning you ll encounter in everyday life. Ordinary reasoning relies heavily on the logic of the Boolean connectives, make no mistake about that. But it also relies on the logic of other kinds of expressions. Since the truth table method detects only tautological consequence, we need a method of applying Boolean logic that can work along with other valid principles of reasoning. Methods of proof, both formal and informal, give us the required extensibility. In this chapter we will discuss legitimate patterns of inference that arise when we introduce the Boolean connectives into a language, and show how to apply the patterns in informal proofs. In Chapter 6, we ll extend our formal system with corresponding rules. The key advantage of proof methods over truth tables is that we ll be able to use them even when the validity of our proof depends on more than just the Boolean operators. The Boolean connectives give rise to many valid patterns of inference. Some of these are extremely simple, like the entailment from the sentence P Q to P. These we will refer to as valid inference steps, and will discuss 128

3 130 / Methods of Proof for Boolean Logic conjunction elimination (simplification) conjunction introduction disjunction introduction are justified in asserting either. More generally, we are justified in inferring, from a conjunction of any number of sentences, any one of its conjuncts. This inference pattern is sometimes called conjunction elimination or simplification, when it is presented in the context of a formal system of deduction. When it is used in informal proofs, however, it usually goes by without comment, since it is so obvious. Only slightly more interesting is the converse. Given the meaning of, it is clear that P Q is a logical consequence of the pair of sentences P and Q: there is no way the latter could be true without former also being true. Thus if we have managed to prove P and to prove Q from the same premises, then we are entitled to infer the conjunction P Q. More generally, if we want to prove a conjunction of a bunch of sentences, we may do so by proving each conjunct separately. In a formal system of deduction, steps of this sort are sometimes called conjunction introduction or just conjunction. Once again, in real life reasoning, these steps are too simple to warrant mention. In our informal proofs, we will seldom point them out explicitly. Finally, let us look at one valid inference pattern involving. It is a simple step, but one that strikes students as peculiar. Suppose that you have proven Cube(b). Then you can conclude Cube(a) Cube(b) Cube(c), if you should want to for some reason, since the latter is a consequence of the former. More generally, if you have proven some sentence P then you can infer any disjunction that has P as one of its disjuncts. After all, if P is true, so is any such disjunction. What strikes newcomers to logic as peculiar about such a step is that using it amounts to throwing away information. Why in the world would you want to conclude P Q when you already know the more informative claim P? But as we will see, this step is actually quite useful when combined with some of the methods of proof to be discussed later. Still, in mathematical proofs, it generally goes by unnoticed. In formal systems, it is dubbed disjunction introduction, or (rather unfortunately) addition. Matters of style Informal proofs serve two purposes. On the one hand, they are a method of discovery; they allow us to extract new information from information already obtained. On the other hand, they are a method of communication; they allow us to convey our discoveries to others. As with all forms of communication, this can be done well or done poorly. When we learn to write, we learn certain basic rules of punctuation, capitalization, paragraph structure and so forth. But beyond the basic rules, there are also matters of style. Different writers have different styles. And it is a Chapter 5

5 132 / Methods of Proof for Boolean Logic Exercises In the following exercises we list a number of patterns of inference, only some of which are valid. For each pattern, determine whether it is valid. If it is, explain why it is valid, appealing to the truth tables for the connectives involved. If it is not, give a specific example of how the step could be used to get from true premises to a false conclusion From P Q and P, infer Q. 5.2 From (P Q), infer P. 5.4 From P Q and Q, infer P. From (P Q) and P, infer Q. 5.5 From (P Q), infer P. 5.6 From P Q and P, infer R. Section 5.2 Proof by cases The simple forms of inference discussed in the last section are all instances of the principle that you can use already established cases of logical consequence in informal proofs. But the Boolean connectives also give rise to two entirely new methods of proof, methods that are explicitly applied in all types of rigorous reasoning. The first of these is the method of proof by cases. In our formal system F, this method will be called disjunction elimination, but don t be misled by the ordinary sounding name: it is far more significant than, say, disjunction introduction or conjunction elimination. We begin by illustrating proof by cases with a well-known piece of mathematical reasoning. The reasoning proves that there are irrational numbers b and c such that b c is rational. First, let s review what this means. A number is said to be rational if it can be expressed as a fraction n/m, for integers n and m. If it can t be so expressed, then it is irrational. Thus 2 is rational (2 = 2/1), but 2 is irrational. (We will prove this latter fact in the next section, to illustrate proof by contradiction; for now, just take it as a well-known truth.) Here now is our proof: Proof: To show that there are irrational numbers b and c such that b c is rational, we will consider the number 2 2. We note that this number is either rational or irrational. Chapter 5

6 Proof by cases / 133 If 2 2 is rational, then we have found our b and c; namely, we take b = c = 2. Suppose, on the other hand, that 2 2 is irrational. Then we take b = 2 2 and c = 2 and compute b c : b c = ( 2 2) 2 = 2 ( 2 2) = 2 2 = 2 Thus, we see that in this case, too, b c is rational. Consequently, whether 2 2 is rational or irrational, we know that there are irrational numbers b and c such that b c is rational. What interests us here is not the result itself but the general structure of the argument. We begin with a desired goal that we want to prove, say S, and a disjunction we already know, say P Q. We then show two things: that S follows if we assume that P is the case, and that S follows if we assume that Q is the case. Since we know that one of these must hold, we then conclude that S must be the case. This is the pattern of reasoning known as proof by cases. In proof by cases, we aren t limited to breaking into just two cases, as we did in the example. If at any stage in a proof we have a disjunction containing n disjuncts, say P 1... P n, then we can break into n cases. In the first we assume P 1, in the second P 2, and so forth for each disjunct. If we are able to prove our desired result S in each of these cases, we are justified in concluding that S holds. Let s look at an even simpler example of proof by cases. Suppose we want to prove that Small(c) is a logical consequence of proof by cases (Cube(c) Small(c)) (Tet(c) Small(c)) This is pretty obvious, but the proof involves breaking into cases, as you will notice if you think carefully about how you recognize this. For the record, here is how we would write out the proof. Proof: We are given (Cube(c) Small(c)) (Tet(c) Small(c)) as a premise. We will break into two cases, corresponding to the two disjuncts. First, assume that Cube(c) Small(c) holds. But then (by Section 5.2

7 134 / Methods of Proof for Boolean Logic conjunction elimination, which we really shouldn t even mention) we have Small(c). But likewise, if we assume Tet(c) Small(c), then it follows that Small(c). So, in either case, we have Small(c), as desired. Our next example shows how the odd step of disjunction introduction (from P infer P Q) can be used fruitfully with proof by cases. Suppose we know that either Max is home and Carl is happy, or Claire is home and Scruffy is happy, i.e., (Home(max) Happy(carl)) (Home(claire) Happy(scruffy)) We want to prove that either Carl or Scruffy is happy, that is, Happy(carl) Happy(scruffy) A rather pedantic, step-by-step proof would look like this: Proof: Assume the disjunction: (Home(max) Happy(carl)) (Home(claire) Happy(scruffy)) Then either: or: Home(max) Happy(carl) Home(claire) Happy(scruffy). If the first alternative holds, then Happy(carl), and so we have Happy(carl) Happy(scruffy) by disjunction introduction. Similarly, if the second alternative holds, we have Happy(scruffy), and so Happy(carl) Happy(scruffy) So, in either case, we have our desired conclusion. Thus our conclusion follows by proof by cases. Arguing by cases is extremely useful in everyday reasoning. For example, one of the authors (call him J) and his wife recently realized that their parking meter had expired several hours earlier. J argued in the following way that there was no point in rushing back to the car (logicians argue this way; don t marry one): Chapter 5

8 Proof by cases / 135 Proof: At this point, either we ve already gotten a ticket or we haven t. If we ve gotten a ticket, we won t get another one in the time it takes us to get to the car, so rushing would serve no purpose. If we haven t gotten a ticket in the past several hours, it is extremely unlikely that we will get one in the next few minutes, so again, rushing would be pointless. In either event, there s no need to rush. J s wife responded with the following counterargument (showing that many years of marriage to a logician has an impact): Proof: Either we are going to get a ticket in the next few minutes or we aren t. If we are, then rushing might prevent it, which would be a good thing. If we aren t, then it will still be good exercise and will also show our respect for the law, both of which are good things. So in either event, rushing back to the car is a good thing to do. J s wife won the argument. The validity of proof by cases cannot be demonstrated by the simple truth table method introduced in Chapter 4. The reason is that we infer the conclusion S from the fact that S is provable from each of the disjuncts P and Q. It relies on the principle that if S is a logical consequence of P, and also a logical consequence of Q, then it is a logical consequence of P Q. This holds because any circumstance that makes P Q true must make at least one of P or Q true, and hence S as well, by the fact that S is a consequence of both. Remember Proof by cases: To prove S from P 1... P n using this method, prove S from each of P 1,..., P n. Exercises The next two exercises present valid arguments. Turn in informal proofs of the arguments validity. Your proofs should be phrased in complete, well-formed English sentences, making use of first-order sentences as convenient, much in the style we have used above. Whenever you use proof by cases, say so. You don t have to be explicit about the use of simple proof steps like conjunction elimination. By the way, there is typically more than one way to prove a given result. Section 5.2

9 136 / Methods of Proof for Boolean Logic 5.7 Home(max) Home(claire) Home(max) Happy(carl) Home(claire) Happy(carl) Happy(carl) 5.8 LeftOf(a, b) RightOf(a, b) BackOf(a, b) LeftOf(a, b) FrontOf(b, a) RightOf(a, b) SameCol(c, a) SameRow(c, b) BackOf(a, b) Assume the same four premises as in Exercise 5.8. Is LeftOf(b, c) a logical consequence of these premises? If so, turn in an informal proof of the argument s validity. If not, submit a counterexample world. Suppose Max s favorite basketball team is the Chicago Bulls and favorite football team is the Denver Broncos. Max s father John is returning from Indianapolis to San Francisco on United Airlines, and promises that he will buy Max a souvenir from one of his favorite teams on the way. Explain John s reasoning, appealing to the annoying fact that all United flights between Indianapolis and San Francisco stop in either Denver or Chicago. Make explicit the role proof by cases plays in this reasoning. Suppose the police are investigating a burglary and discover the following facts. All the doors to the house were bolted from the inside and show no sign of forced entry. In fact, the only possible ways in and out of the house were a small bathroom window on the first floor that was left open and an unlocked bedroom window on the second floor. On the basis of this, the detectives rule out a well-known burglar, Julius, who weighs two hundred and fifty pounds and is arthritic. Explain their reasoning. In our proof that there are irrational numbers b and c where b c is rational, one of our steps was to assert that 2 2 is either rational or irrational. What justifies the introduction of this claim into our proof? Describe an everyday example of reasoning by cases that you have performed in the last few days. Give an informal proof that if S is a tautological consequence of P and a tautological consequence of Q, then S is a tautological consequence of P Q. Remember that the joint truth table for P Q and S may have more rows than either the joint truth table for P and S, or the joint truth table for Q and S. [Hint: Assume you are looking at a single row of the joint truth table for P Q and S in which P Q is true. Break into cases based on whether P is true or Q is true and prove that S must be true in either case.] Chapter 5

13 140 / Methods of Proof for Boolean Logic Exercises In the following exercises, decide whether the displayed argument is valid. If it is, turn in an informal proof, phrased in complete, well-formed English sentences, making use of first-order sentences as convenient. Whenever you use proof by cases or proof by contradiction, say so. You don t have to be explicit about the use of simple proof steps like conjunction elimination. If the argument is invalid, construct a counterexample world in Tarski s World. (Argument 5.16 is valid, and so will not require a counterexample.) 5.15 b is a tetrahedron. c is a cube. Either c is larger than b or else they are identical. b is smaller than c Max or Claire is at home but either Scruffy or Carl is unhappy. Either Max is not home or Carl is happy. Either Claire is not home or Scruffy is unhappy. Scruffy is unhappy Cube(a) Tet(a) Large(a) Cube(a) a = b Large(a) Large(a) a = c (c = c Tet(a)) 5.18 Cube(a) Tet(a) Large(a) Cube(a) a = b Large(a) Large(a) a = c (c = c Tet(a)) a = b a = c (Large(a) Tet(a)) 5.19 Consider the following sentences. 1. Folly was Claire s pet at 2 pm or at 2:05 pm. 2. Folly was Max s pet at 2 pm. 3. Folly was Claire s pet at 2:05 pm. Does (3) follow from (1) and (2)? Does (2) follow from (1) and (3)? Does (1) follow from (2) and (3)? In each case, give either a proof of consequence, or describe a situation that makes the premises true and the conclusion false. You may assume that Folly can only be one person s pet at any given time Suppose it is Friday night and you are going out with your boyfriend. He wants to see a romantic comedy, while you want to see the latest Wes Craven slasher movie. He points out that if he watches the Wes Craven movie, he will not be able to sleep because he can t stand the sight of blood, and he has to take the MCAT test tomorrow. If he does not do well on the MCAT, he won t get into medical school. Analyze your boyfriend s argument, pointing out where indirect proof is being used. How would you rebut his argument? Chapter 5

14 Arguments with inconsistent premises / Describe an everyday example of an indirect proof that you have used in the last few days. Prove that indirect proof is a tautologically valid method of proof. That is, show that if P 1,..., P n, S is tt-contradictory, then S is a tautological consequence of P 1,..., P n. In the next three exercises we ask you to prove simple facts about the natural numbers. We do not expect you to phrase the proofs in fol. You will have to appeal to basic facts of arithmetic plus the definitions of even and odd number. This is OK, but make these appeals explicit. Also make explicit any use of proof by contradiction Assume that n 2 is odd. Prove that n is odd Assume that n + m is odd. Prove that n m is even Assume that n 2 is divisible by 3. Prove that n 2 is divisible by A good way to make sure you understand a proof is to try to generalize it. Prove that 3 is irrational. [Hint: You will need to figure out some facts about divisibility by 3 that parallel the facts we used about even and odd, for example, the fact expressed in Exercise 5.25.] Can you generalize these two results? Section 5.4 Arguments with inconsistent premises What follows from an inconsistent set of premises? If you look back at our definition of logical consequence, you will see that every sentence is a consequence of such a set. After all, if the premises are contradictory, then there are no circumstances in which they are all true. Thus, there are no circumstances in which the premises are true and the conclusion is false. Which is to say, in any situation in which the premises are all true (there aren t any of these!), the conclusion will be true as well. Hence any argument with an inconsistent set of premises is trivially valid. In particular, if one can establish a contradiction on the basis of the premises, then one is entitled to assert any sentence at all. This often strikes students as a very odd method of reasoning, and for very good reason. For recall the distinction between a valid argument and a sound one. A sound argument is a valid argument with true premises. Even though any argument with an inconsistent set of premises is valid, no such argument is sound, since there is no way the premises of the argument can all be true. For this reason, an argument with an inconsistent set of premises is not worth always valid Section 5.4

15 142 / Methods of Proof for Boolean Logic never sound much on its own. After all, the reason we are interested in logical consequence is because of its relation to truth. If the premises can t possibly be true, then even knowing that the argument is valid gives us no clue as to the truth or falsity of the conclusion. An unsound argument gives no more support for its conclusion than an invalid one. In general, methods of proof don t allow us to show that an argument is unsound. After all, the truth or falsity of the premises is not a matter of logic, but of how the world happens to be. But in the case of arguments with inconsistent premises, our methods of proof do give us a way to show that at least one of the premises is false (though we might not know which one), and hence that the argument is unsound. To do this, we prove that the premises are inconsistent by deriving a contradiction. Suppose, for example, you are given a proof that the following argument is valid: Home(max) Home(claire) Home(max) Home(claire) Home(max) Happy(carl) While it is true that this conclusion is a consequence of the premises, your reaction should not be to believe the conclusion. Indeed, using proof by cases we can show that the premises are inconsistent, and hence that the argument is unsound. There is no reason to be convinced of the conclusion of an unsound argument. Remember A proof of a contradiction from premises P 1,..., P n (without additional assumptions) shows that the premises are inconsistent. An argument with inconsistent premises is always valid, but more importantly, always unsound. Exercises 5.27 Give two different proofs that the premises of the above argument are inconsistent. Your first should use proof by cases but not DeMorgan s law, while your second can use DeMorgan but not proof by cases. Chapter 5

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