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  • Chapter 5

    Methods of Proof for Boolean

    Logic

    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 lim-

    itations.

    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

    testing it for validity would call for a truth table with 27 = 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 cant 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 youll 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 exten-

    sibility. 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, well extend our

    formal system with corresponding rules. The key advantage of proof methods

    over truth tables is that well 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

  • Valid inference steps / 129

    them briefly in the first section. Much more interesting are two new methods

    of proof that are allowed by the new expressions: proof by cases and proof by

    contradiction. We will discuss these later, one at a time.

    Section 5.1

    Valid inference steps

    Heres an important rule of thumb: In an informal proof, it is always legiti-

    mate to move from a sentence P to another sentence Q if both you and your

    audience (the person or people youre trying to convince) already know important rule of thumb

    that Q is a logical consequence of P. The main exception to this rule is when

    you give informal proofs to your logic instructor: presumably, your instructor

    knows the assigned argument is valid, so in these circumstances, you have to

    pretend youre addressing the proof to someone who doesnt already know

    that. What youre really doing is convincing your instructor that you see that

    the argument is valid and that you could prove it to someone who did not.

    The reason we start with this rule of thumb is that youve already learned

    several well-known logical equivalences that you should feel free to use when

    giving informal proofs. For example, you can freely use double negation or

    idempotence if the need arises in a proof. Thus a chain of equivalences of the

    sort we gave on page 120 is a legitimate component of an informal proof. Of

    course, if you are asked to prove one of the named equivalences, say one of

    the distribution or DeMorgan laws, then you shouldnt presuppose it in your

    proof. Youll have to figure out a way to prove it to someone who doesnt

    already know that it is valid.

    A special case of this rule of thumb is the following: If you already know

    that a sentence Q is a logical truth, then you may assert Q at any point in

    your proof. We already saw this principle at work in Chapter 2, when we

    discussed the reflexivity of identity, the principle that allowed us to assert a

    sentence of the form a = a at any point in a proof. It also allows us to assert

    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

    sure your audience will recognize them.

    There are three simple inference steps that we will mention here that dont

    involve logical equivalences or logical truths, but that are clearly supported

    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

    for the conjunction to be true without each conjunct being true. Thus, we

    Section 5.1

  • 130 / Methods of Proof for Boolean Logic

    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. Thisconjunctionelimination

    (simplification)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 , 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

    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 toconjunctionintroduction 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 simplestep, 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 shoulddisjunctionintroduction 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? Butas 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, capi-

    talization, 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

  • Valid inference steps / 131

    good thing, since we would get pretty tired of reading if everyone wrote with

    the very same style. So too in giving proofs. If you go on to study mathemat-

    ics, you will read lots of proofs, and you will find that every writer has his or

    her own style. You will even develop a style of your own.

    Every step in a good proof, besides being correct, should have two prop-

    erties. It should be easily understood and significant. By easily understood

    we mean that other people should be able to follow the step without undue

    difficulty: they should be able to see that the step is valid without having to

    engage in a piece of complex reasoning of their own. By significant we mean

    that the step should be informative, not a waste of the readers time.

    These two criteria pull in opposite directions. Typically, the more signif-

    icant the step, the harder it is to follow. Good style requires a reasonable

    balance between the two. And that in turn requires some sense of who your knowing your audience

    audience is. For example, if you and your audience have been working with

    logic for a while, you will recognize a number of equivalences that you will

    want to use without further proof. But if you or your audience are beginners,

    the same inference may require several steps.

    Remember

    1. In giving an informal proof from some premises,