Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 3.4.7 Using Inference Rules Correctly

We’ll soon see how to approach writing an entire proof that exploits the identities and rules that we’ve just described. But before we do that, let’s just make sure that we understand how individual ones can be applied correctly.

We already know that identities work in both directions, while rules apply only in one.

Here’s another important distinction: We can use the identities to make substitutions, both for entire lines in a proof as well as for subexpressions.

Suppose that we are given:

[1] \((\neg (P \wedge Q)) \rightarrow R \)

We can apply De Morgan to the subexpression (P  Q) and thus rewrite [1] as the equivalent sentence:

[2] \((\neg P \wedge \neg Q) \rightarrow R \)

We can also apply Conditional Disjunction to [1] and thus rewrite it as the equivalent sentence:

\([3] \neg (\neg (P \wedge Q)) \vee R \)

The reason that we can apply the identities anywhere is that they are identities . They allow us to rewrite one expression as another one with equivalent truth value. So applying them never changes truth values.

On the other hand, the inference rules work only in one direction. They do not rewrite expressions as equivalent ones. We know only that the new one is entailed by the original one. Thus we must be careful to use them exactly as they are written.

Big Idea

Inference rules may be applied only to entire logical expressions.

Cake and Ice Cream

Give names to the following statements:

C: I get to have cake.

IC: I get to have ice cream.

H: I’ll be extra happy.

Suppose that we are given:

[1] \((C \wedge IC) \rightarrow H \)

We may not apply Simplification to (C  IC) to derive C and thus to transform [1] into:

[2] \(C \rightarrow H \)

Clearly this is a stronger and thus not equivalent claim. We have not been given any reason to believe that I’ll be extra happy with only cake.

Now that we’ve seen both correct and incorrect ways to apply logical rules, let’s take another look at some of the examples that we considered at the beginning of this course.