Chapter 5 Methods of Proof for Boolean Logic ?· our proof depends on more than just the Boolean operators.…

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<ul><li><p>Chapter 5</p><p>Methods of Proof for Boolean</p><p>Logic</p><p>Truth tables give us powerful techniques for investigating the logic of the</p><p>Boolean operators. But they are by no means the end of the story. Truth</p><p>tables are fine for showing the validity of simple arguments that depend only</p><p>on truth-functional connectives, but the method has two very significant lim-</p><p>itations.</p><p>First, truth tables get extremely large as the number of atomic sentenceslimitations of truthtable methods goes up. An argument involving seven atomic sentences is hardly unusual, but</p><p>testing it for validity would call for a truth table with 27 = 128 rows. Testing</p><p>an argument with 14 atomic sentences, just twice as many, would take a table</p><p>containing over 16 thousand rows. You could probably get a Ph.D. in logic for</p><p>building a truth table that size. This exponential growth severely limits the</p><p>practical value of the truth table method.</p><p>The second limitation is, surprisingly enough, even more significant. Truth</p><p>table methods cant be easily extended to reasoning whose validity depends</p><p>on more than just truth-functional connectives. As you might guess from the</p><p>artificiality of the arguments looked at in the previous chapter, this rules out</p><p>most kinds of reasoning youll encounter in everyday life. Ordinary reasoning</p><p>relies heavily on the logic of the Boolean connectives, make no mistake about</p><p>that. But it also relies on the logic of other kinds of expressions. Since the</p><p>truth table method detects only tautological consequence, we need a method</p><p>of applying Boolean logic that can work along with other valid principles of</p><p>reasoning.</p><p>Methods of proof, both formal and informal, give us the required exten-</p><p>sibility. In this chapter we will discuss legitimate patterns of inference that</p><p>arise when we introduce the Boolean connectives into a language, and show</p><p>how to apply the patterns in informal proofs. In Chapter 6, well extend our</p><p>formal system with corresponding rules. The key advantage of proof methods</p><p>over truth tables is that well be able to use them even when the validity of</p><p>our proof depends on more than just the Boolean operators.</p><p>The Boolean connectives give rise to many valid patterns of inference.</p><p>Some of these are extremely simple, like the entailment from the sentence</p><p>P Q to P. These we will refer to as valid inference steps, and will discuss</p><p>128</p></li><li><p>Valid inference steps / 129</p><p>them briefly in the first section. Much more interesting are two new methods</p><p>of proof that are allowed by the new expressions: proof by cases and proof by</p><p>contradiction. We will discuss these later, one at a time.</p><p>Section 5.1</p><p>Valid inference steps</p><p>Heres an important rule of thumb: In an informal proof, it is always legiti-</p><p>mate to move from a sentence P to another sentence Q if both you and your</p><p>audience (the person or people youre trying to convince) already know important rule of thumb</p><p>that Q is a logical consequence of P. The main exception to this rule is when</p><p>you give informal proofs to your logic instructor: presumably, your instructor</p><p>knows the assigned argument is valid, so in these circumstances, you have to</p><p>pretend youre addressing the proof to someone who doesnt already know</p><p>that. What youre really doing is convincing your instructor that you see that</p><p>the argument is valid and that you could prove it to someone who did not.</p><p>The reason we start with this rule of thumb is that youve already learned</p><p>several well-known logical equivalences that you should feel free to use when</p><p>giving informal proofs. For example, you can freely use double negation or</p><p>idempotence if the need arises in a proof. Thus a chain of equivalences of the</p><p>sort we gave on page 120 is a legitimate component of an informal proof. Of</p><p>course, if you are asked to prove one of the named equivalences, say one of</p><p>the distribution or DeMorgan laws, then you shouldnt presuppose it in your</p><p>proof. Youll have to figure out a way to prove it to someone who doesnt</p><p>already know that it is valid.</p><p>A special case of this rule of thumb is the following: If you already know</p><p>that a sentence Q is a logical truth, then you may assert Q at any point in</p><p>your proof. We already saw this principle at work in Chapter 2, when we</p><p>discussed the reflexivity of identity, the principle that allowed us to assert a</p><p>sentence of the form a = a at any point in a proof. It also allows us to assert</p><p>other simple logical truths, like excluded middle (P P), at any point in aproof. Of course, the logical truths have to be simple enough that you can be</p><p>sure your audience will recognize them.</p><p>There are three simple inference steps that we will mention here that dont</p><p>involve logical equivalences or logical truths, but that are clearly supported</p><p>by the meanings of and . First, suppose we have managed to prove aconjunction, say P Q, in the course of our proof. The individual conjunctsP and Q are clearly consequences of this conjunction, because there is no way</p><p>for the conjunction to be true without each conjunct being true. Thus, we</p><p>Section 5.1</p></li><li><p>130 / Methods of Proof for Boolean Logic</p><p>are justified in asserting either. More generally, we are justified in inferring,</p><p>from a conjunction of any number of sentences, any one of its conjuncts. Thisconjunctionelimination</p><p>(simplification)inference pattern is sometimes called conjunction elimination or simplification,</p><p>when it is presented in the context of a formal system of deduction. When it</p><p>is used in informal proofs, however, it usually goes by without comment, since</p><p>it is so obvious.</p><p>Only slightly more interesting is the converse. Given the meaning of , itis 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</p><p>if we have managed to prove P and to prove Q from the same premises, then</p><p>we are entitled to infer the conjunction P Q. More generally, if we want toconjunctionintroduction prove a conjunction of a bunch of sentences, we may do so by proving each</p><p>conjunct separately. In a formal system of deduction, steps of this sort are</p><p>sometimes called conjunction introduction or just conjunction. Once again,</p><p>in real life reasoning, these steps are too simple to warrant mention. In our</p><p>informal proofs, we will seldom point them out explicitly.</p><p>Finally, let us look at one valid inference pattern involving . It is a simplestep, but one that strikes students as peculiar. Suppose that you have proven</p><p>Cube(b). Then you can conclude Cube(a) Cube(b) Cube(c), if you shoulddisjunctionintroduction want to for some reason, since the latter is a consequence of the former.</p><p>More generally, if you have proven some sentence P then you can infer any</p><p>disjunction that has P as one of its disjuncts. After all, if P is true, so is any</p><p>such disjunction.</p><p>What strikes newcomers to logic as peculiar about such a step is that using</p><p>it amounts to throwing away information. Why in the world would you want</p><p>to conclude P Q when you already know the more informative claim P? Butas we will see, this step is actually quite useful when combined with some</p><p>of the methods of proof to be discussed later. Still, in mathematical proofs,</p><p>it generally goes by unnoticed. In formal systems, it is dubbed disjunction</p><p>introduction, or (rather unfortunately) addition.</p><p>Matters of style</p><p>Informal proofs serve two purposes. On the one hand, they are a method of</p><p>discovery; they allow us to extract new information from information already</p><p>obtained. On the other hand, they are a method of communication; they allow</p><p>us to convey our discoveries to others. As with all forms of communication,</p><p>this can be done well or done poorly.</p><p>When we learn to write, we learn certain basic rules of punctuation, capi-</p><p>talization, paragraph structure and so forth. But beyond the basic rules, there</p><p>are also matters of style. Different writers have different styles. And it is a</p><p>Chapter 5</p></li><li><p>Valid inference steps / 131</p><p>good thing, since we would get pretty tired of reading if everyone wrote with</p><p>the very same style. So too in giving proofs. If you go on to study mathemat-</p><p>ics, you will read lots of proofs, and you will find that every writer has his or</p><p>her own style. You will even develop a style of your own.</p><p>Every step in a good proof, besides being correct, should have two prop-</p><p>erties. It should be easily understood and significant. By easily understood</p><p>we mean that other people should be able to follow the step without undue</p><p>difficulty: they should be able to see that the step is valid without having to</p><p>engage in a piece of complex reasoning of their own. By significant we mean</p><p>that the step should be informative, not a waste of the readers time.</p><p>These two criteria pull in opposite directions. Typically, the more signif-</p><p>icant the step, the harder it is to follow. Good style requires a reasonable</p><p>balance between the two. And that in turn requires some sense of who your knowing your audience</p><p>audience is. For example, if you and your audience have been working with</p><p>logic for a while, you will recognize a number of equivalences that you will</p><p>want to use without further proof. But if you or your audience are beginners,</p><p>the same inference may require several steps.</p><p>Remember</p><p>1. In giving an informal proof from some premises, if Q is already</p><p>known to be a logical consequence of sentences P1, . . . ,Pn and each of</p><p>P1, . . . ,Pn has been proven from the premises, then you may assert Q</p><p>in your proof.</p><p>2. Each step in an informal proof should be significant but easily under-</p><p>stood.</p><p>3. Whether a step is significant or easily understood depends on the</p><p>audience to whom it is addressed.</p><p>4. The following are valid patterns of inference that generally go unmen-</p><p>tioned in informal proofs:</p><p> From P Q, infer P. From P and Q, infer P Q. From P, infer P Q.</p><p>Section 5.1</p></li><li><p>132 / Methods of Proof for Boolean Logic</p><p>Exercises</p><p>In the following exercises we list a number of patterns of inference, only some of which are valid. For</p><p>each pattern, determine whether it is valid. If it is, explain why it is valid, appealing to the truth tables</p><p>for the connectives involved. If it is not, give a specific example of how the step could be used to get from</p><p>true premises to a false conclusion.</p><p>5.1.</p><p>From P Q and P, infer Q. 5.2.</p><p>From P Q and Q, infer P.</p><p>5.3.</p><p>From (P Q), infer P. 5.4.</p><p>From (P Q) and P, infer Q.</p><p>5.5.</p><p>From (P Q), infer P. 5.6.</p><p>From P Q and P, infer R.</p><p>Section 5.2</p><p>Proof by cases</p><p>The simple forms of inference discussed in the last section are all instances of</p><p>the principle that you can use already established cases of logical consequence</p><p>in informal proofs. But the Boolean connectives also give rise to two entirely</p><p>new methods of proof, methods that are explicitly applied in all types of</p><p>rigorous reasoning. The first of these is the method of proof by cases. In our</p><p>formal system F , this method will be called disjunction elimination, but dontbe misled by the ordinary sounding name: it is far more significant than, say,</p><p>disjunction introduction or conjunction elimination.</p><p>We begin by illustrating proof by cases with a well-known piece of math-</p><p>ematical reasoning. The reasoning proves that there are irrational numbers b</p><p>and c such that bc is rational. First, lets review what this means. A number</p><p>is said to be rational if it can be expressed as a fraction n/m, for integers</p><p>n and m. If it cant be so expressed, then it is irrational. Thus 2 is rational</p><p>(2 = 2/1), but</p><p>2 is irrational. (We will prove this latter fact in the next sec-</p><p>tion, to illustrate proof by contradiction; for now, just take it as a well-known</p><p>truth.) Here now is our proof:</p><p>Proof: To show that there are irrational numbers b and c such that</p><p>bc is rational, we will consider the number</p><p>22. We note that this</p><p>number is either rational or irrational.</p><p>Chapter 5</p></li><li><p>Proof by cases / 133</p><p>If</p><p>22</p><p>is rational, then we have found our b and c; namely, we take</p><p>b = c =</p><p>2.</p><p>Suppose, on the other hand, that</p><p>22</p><p>is irrational. Then we take</p><p>b =</p><p>22</p><p>and c =</p><p>2 and compute bc:</p><p>bc = (</p><p>22)2</p><p>=</p><p>2(22)</p><p>=</p><p>22</p><p>= 2</p><p>Thus, we see that in this case, too, bc is rational.</p><p>Consequently, whether</p><p>22</p><p>is rational or irrational, we know that</p><p>there are irrational numbers b and c such that bc is rational.</p><p>What interests us here is not the result itself but the general structure of</p><p>the argument. We begin with a desired goal that we want to prove, say S, and</p><p>a disjunction we already know, say P Q. We then show two things: that S proof by casesfollows if we assume that P is the case, and that S follows if we assume that</p><p>Q is the case. Since we know that one of these must hold, we then conclude</p><p>that S must be the case. This is the pattern of reasoning known as proof by</p><p>cases.</p><p>In proof by cases, we arent limited to breaking into just two cases, as we</p><p>did in the example. If at any stage in a proof we have a disjunction containing</p><p>n disjuncts, say P1 . . . Pn, then we can break into n cases. In the first weassume P1, in the second P2, and so forth for each disjunct. If we are able to</p><p>prove our desired result S in each of these cases, we are justified in concluding</p><p>that S holds.</p><p>Lets look at an even simpler example of proof by cases. Suppose we want</p><p>to prove that Small(c) is a logical consequence of</p><p>(Cube(c) Small(c)) (Tet(c) Small(c))</p><p>This is pretty obvious, but the proof involves breaking into cases, as you will</p><p>notice if you think carefully about how you recognize this. For the record,</p><p>here is how we would write out the proof.</p><p>Proof: We are given</p><p>(Cube(c) Small(c)) (Tet(c) Small(c))</p><p>as a premise. We will break into two cases, corresponding to the two</p><p>disjuncts. First, assume that Cube(c) Small(c) holds. But then (by</p><p>Section 5.2</p></li><li><p>134 / Methods of Proof for Boolean Logic</p><p>conjunction elimination, which we really shouldnt even mention) we</p><p>have Small(c). But likewise, if we assume Tet(c) Small(c), then itfollows that Small(c). So, in either case, we have Small(c), as desired.</p><p>Our next example shows how the odd step of disjunction introduction</p><p>(from P infer P Q) can be used fruitfully with proof by cases. Suppose weknow that either Max is home and Carl is happy, or Claire is home and Scruffy</p><p>is happy, i.e.,</p><p>(Home(max) Happy(carl)) (Home(claire) Happy(scruffy))</p><p>We want to prove that either Carl or Scruffy is happy, that is,</p><p>Happy(carl) Happy(scruffy)</p><p>A rather pedantic, step-by-step proof would look like this:</p><p>Proof: Assume the disjunction:</p><p>(Home(max) Happy(carl)) (Home(claire) Happy(scruffy))</p><p>Then either:</p><p>Home(max) Happy(carl)</p><p>or:</p><p>Home(claire) Happy(scruffy).</p><p>If the first alternative holds, then Happy(carl), and so we have</p><p>Happy(carl) Happy(scruffy)</p><p>by disjunction introduction. Similarly, if the second alternative holds,</p><p>we have Happy(scruffy), and so</p><p>Happy(carl) Happy(scruffy)</p><p>So, in either case, we have our desired conclusion. Thus our conclu-</p><p>sion follows by proof by cases.</p><p>Arguing by cases is extremely useful in everyday reasoning. For example,</p><p>one of the authors (call him J) and his wife recently realized that their parking</p><p>meter had expired several hours earlier. J argued in the following way...</p></li></ul>


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