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Here’s 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 you’re 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 you’re addressing the proof to someone who doesn’t already know that. What you’re 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 you’ve 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 119 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 shouldn’t presuppose it in your proof. You’ll have to figure out a way to prove it to someone who doesn’t 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 a proof. 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 don’t involve logical equivalences or logical truths, but that are clearly supported by the meanings of ∧ and ∨. First, suppose we have managed to prove a conjunction, say P ∧ Q, in the course of our proof. The individual conjuncts P 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

Valid inference steps / 129

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 conjunction elimination (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 ∧, 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 conjunction introduction 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 disjunction introduction 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, 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

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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 reader’s 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, if Q is already known to be a logical consequence of sentences P1, . . . , Pnand each of P1, . . . , Pn has been proven from the premises, then you may assert Q in your proof.

2. Each step in an informal proof should be significant but easily under-stood.

3. Whether a step is significant or easily understood depends on the audience to whom it is addressed.

4. The following are valid patterns of inference that generally go unmen-tioned in informal proofs:

◦ From P ∧ Q, infer P.

◦ From P and Q, infer P ∧ Q.

◦ From P, infer P ∨ Q.