Substitution

Substituting logical equivalents The law of double negation tells us that A and ~~A are equivalent. So, it seems pretty clear that A ∧ B and ~~A ∧ B are also equivalent. Why? Because the only way that the components of compound sentences effect those compound sentences is through their truth- values. So if you switch one component with another that has exactly the same truth-values in all cases, you won’t effect the compound sentence’s truth valuations at all.

The law of substitution of logical equivalents This important insight gives us the law of substitution of logical equivalents (or SLE) for short. SLE: Suppose that X and Y are logically equivalent, and suppose that X occurs as a sub-sentence or component of some larger sentence Z. Let Z* be the new sentence obtained by substituting Y for X in Z. Then Z is logically equivalent to Z*.

Proving equivalence using the laws Now that we have SLE, and the laws of logical equivalence that we discussed in the previous session, we have enough for a new method of showing logical equivalence. (This is pretty useful, because though truth tables and Venn diagrams are illustrative for simple examples, once we start dealing with longer sentences of sentence logic they become wildly impractical.)

Proving equivalence using the laws The idea is fairly simple at heart, but it’s important to get it right. Those laws we reviewed last session give us some general rules for what sorts of sentences are logically equivalent to each other. (E.g. the law of double negation tells us that anything of the form ~~X, where X is just any sentence of sentence logic, is logically equivalent to X.)

Proving equivalence using the laws And then the law of substitution of logical equivalents tells us that we can substitute any component of a compound sentence with a logical equivalent to that component and end up with something that is logically equivalent to the original compound sentence. So we can apply these laws together to show that two different sentences of sentence logic are logically equivalent.

An example Let’s look at an example. Let’s prove that ~(~P ∧ Q) is logically equivalent to P ∨ ~Q We start with ~(~P ∧ Q). We’re going to have to get from this disjunction to a conjunction. And that’s just what those De Morgan Laws we proved last session can help us to do. Let’s look back at those laws.

An example One of those laws says that ~(X ∧ Y) is logically equivalent to ~X ∨ ~Y. The sentence we are starting with is ~(~P ∧ Q). But this is just an instance of ~(X ∧ Y), with ~P as X and Q as Y. So De Morgan’s Law tells us that ~(~P ∧ Q) is logically equivalent to ~~P ∨ ~Q. We’re close to our final destination of P ∨ ~Q now.

An example We just need to turn that ~~P into a P. But that’s easy! Because the law of double negation tells us that ~~P is logically equivalent to P. And the law of substitution of logical equivalents tells us that we can therefore substitute our ~~P for a P and end up with a logically equivalent sentence. So ~~P ∨ ~Q is logically equivalent to P ∨ ~Q.

An example So now we have shown that ~(~P ∧ Q) is logically equivalent to ~~P ∨ ~Q and that ~~P ∨ Q is logically equivalent to P ∨ ~Q. So we can (because of a law we will introduce shortly) conclude that ~(~P ∧ Q) is logically equivalent to P ∨ ~Q .

An example The standard way of writing such a proof is like this: ~(~P ∧ Q) ~~P ∨ ~Q DM P ∨ ~Q DN, SLE

Some more laws A couple of slides back I said that because we had proved that ~(~P ∧ Q) is logically equivalent to ~~P ∨ ~Q and that ~~P ∨ Q is logically equivalent to P ∨ ~Q, we could conclude that ~(~P ∧ Q) is logically equivalent to P ∨ ~Q. This is because of the law of transitivity of logical equivalents, which just says that for any three sentences X, Y and Z, if X is logically equivalent to Y and Y is logically equivalent to Z, then X is logically equivalent to Z.

More laws Like the law of transitivity of logical equivalents, these laws are pretty intuitive – but they’re important! Don’t worry if they seem perplexing to you now, once you get the hang of proofs you’ll start to see the rationale for them. The commutative law: for any sentences X and Y, X ∧ Y is logically equivalent to Y ∧ X. And X ∨ Y is logically equivalent to Y ∨ X.

More laws The associative law: For any sentences X, Y, and Z, X ∧ (Y ∧ Z), (X ∧ Y) ∧ Z and X ∧ Y ∧ Z are logically equivalent to each other. And X ∨ (Y ∨ Z), (X ∨ Y) ∨ Z and X ∨ Y ∨ Z are logically equivalent to each other. (Similarly, conjunctions with four or more components may be arbitrarily grouped and similarly for disjunctions with four or more disjuncts.)

More laws The law of redundancy: for any sentence X, X ∧ X is logically equivalent to X. Likewise, X ∨ X is logically equivalent to X.

Getting the hang of it This all might seem a little confusing. Don’t panic – seeing all these laws at once is a confusing experience. The best way to understand them is just to jump in and start doing examples. Go through Teller’s exercises in the text if you need to – this will help a lot. The more you practice, the easier it will get.

Examples Let’s do some examples together on the board now. ‘~~A ∨ B’ is logically equivalent to ‘B ∨ A’ ‘X ∧ (~~Y ∨ Z)’ is logically equivalent to ‘(X ∧ Z) ∨ (X ∧ Y)’