Equivalence rules

Recall that two propositions are logically equivalent if and only if they entail each other. In other words, equivalent propositions have the same truth value in all possible circumstances: whenever one is true, so is the other; and whenever one is false, so is the other. The proposition P is equivalent to the proposition ~~P, for example. In fact, it is somewhat misleading to say that P and ~~P are two different propositions. They mean exactly the same thing; they are just different ways of representing the same proposition. If any two well-formed formulas (WFFs) are logically equivalent, they represent the same proposition.

An equivalence rule is a pair of equivalent proposition forms, with lowercase letters used as variables for which we can substitute any WFF (just as we did previously with inference rules). By memorizing a few simple equivalence rules, we can more easily recognize when two sentences mean the same thing—a useful skill in philosophy. Familiarity with equivalence rules is also necessary for constructing logical proofs, as we’ll see on the next page.

Here are six inference rules worth memorizing:

Since logically equivalent WFFs represent the same proposition, they can be substituted for one another in any context, even when they appear as components of a larger WFF. Since P is equivalent to ~~P by the “double negation” rule, for example, (Q • P) is likewise equivalent to (Q • ~~P), by that same rule. Moreover, the substitution can go in either direction. For example, by De Morgan’s law, we can replace ~(A • B) with (~A ∨ ~B) and vice versa: we can replace (~A ∨ ~B) with ~(A • B). Here are a few more examples:

((P • Q) ≡ ~~R) is equivalent to ((P • Q) ≡ R) by double-negation. (A pair of tildes is removed from the right side of the biconditional by DN.)

(A ⊃ (B ∨ C)) is equivalent to (A ⊃ (C ∨ B)) by commutation. (The consequent of the conditional is replaced with an equivalent formula by Com.)

(~A ∨ (C ∨ ~D)) is equivalent to ((~A ∨ C) ∨ ~D) by association.

(~A • ~(B ⊃ C)) is equivalent to ~(A ∨ (B ⊃ C)) by De Morgan’s law. (A tilde is “factored out” from the two conjuncts and the ‘•’ is replaced with a ‘∨’.)

((~A • ~B) ⊃ C) is equivalent to (~(A ∨ B) ⊃ C) by De Morgan’s law. (The antecedent of the conditional is replaced with an equivalent formula by DM.)

(~(Q • R) ⊃ ~P) is equivalent to (P ⊃ (Q • R)) by contraposition.

(P • (Q ⊃ R)) is equivalent to (P • (~R ⊃ ~Q)) by contraposition. (The second conjunct is replaced with an equivalent formula by Contra.)

(P ⊃ (Q • R)) is equivalent to (~P ∨ (Q • R)) by implication.

(~(A ∨ B) ∨ C) is equivalent to ((A ∨ B) ⊃ C) by implication.