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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,