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:
- Double negation (DN) says that a pair of tildes can be added or removed from any WFF:
x is equivalent to ~~x
- Commutation (Com) says that the two component propositions of a conjunction, disjunction, or biconditional can switch places:
(x • y) is equivalent to (y • x)
(x ∨ y) is equivalent to (y ∨ x)
(x ≡ y) is equivalent to (y ≡ x)
This rule is similar to the commutative property of addition and multiplication in mathematics: 1+2 = 2+1 and 2×3 = 3×2.
- Association (Assoc) allows us to rearrange the parentheses that associate the components of two conjunctions, two disjunctions, or two biconditionals:
(x • (y • z)) is equivalent to ((x • y) • z)
(x ∨ (y ∨ z)) is equivalent to ((x ∨ y) ∨ z)
(x ≡ (y ≡ z)) is equivalent to ((x ≡ y) ≡ z)
This rule is similar to the associative property of addition and multiplication: (1+2)+3 = 1+(2+3) and (1×2)×3 = 1×(2×3).
- De Morgan’s law (DM) says that a negated conjunction is equivalent to a disjunction with both components negated, and vice versa:
~(x • y) is equivalent to (~x ∨ ~y)
~(x ∨ y) is equivalent to (~x • ~y)
This rule is analogous in some ways to the distributive property of addition over multiplication. We can think of the ‘~’ (tilde) outside the parentheses as being “distributed” to each of the components inside the parentheses, in the same way we distribute a multiple: 2×(3+4) = 2×3 + 2×4. Alternatively, we can imagine “factoring out” the tilde from each component inside the parentheses, just as we factor out a multiple: 2×3 + 2×4 = 2×(3+4). The analogy isn’t perfect, though. When “distributing” or “factoring out” the tilde, we also have to change the ‘•’ to a ‘∨’ or vice versa, whereas we don’t change the ‘+’ when distributing or factoring a multiple in mathematics.
- Contraposition (Contra) says that the antecedent and the consequent of a conditional can switch places if we negate both of them:
(x ⊃ y) is equivalent to (~y ⊃ ~x)
- Implication (Imp) says that a conditional is equivalent to a disjunction in which the first disjunct is the negation of the conditional’s antecedent:
(x ⊃ y) is equivalent to (~x ∨ y)
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.