Logical Equivalence
Two wffs are said to be logically equivalent just in case they have the same truth
value on every row of their truth tables. So if two wffs are logically equivalent, then
whenever one is true, the other will be true, and whenever one is false the other will be
false.
To determine whether two wffs are logically equivalent, we construct truth tables for each and compare the two to make sure that the truth values "match" on each row. To save space, we can have the two wffs share the base columns and present both wffs in the same table.
Let's see whether (~A v B) is logically equivalent to (A ⊃ B).
| A | B | ~A | (~A v B) | (A ⊃ B) |
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |
The truth values match at each row, so the two wffs are logically equivalent. Let's see whether (A ⊃ B) is logically equivalent to (B ⊃ A).
| A | B | (A ⊃ B) | (B ⊃ A) |
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
The result? The truth values of the two wffs differ on the second and third rows. Hence they are not logically equivalent. That should be no surprise, considering that the two wffs have reversed antecedents and consequents!
Let's put together some of the semantical concepts we've been using in this chapter. What happens when we construct a biconditional of two logically equivalent wffs? We'll have a biconditional whose parts always have the same truth value. But a biconditional is true on every row where its parts have the same truth value. Since that happens on every row, the biconditional will be a tautology. What about tautologies? Are any two tautologies logically equivalent? Yes! What about any two contradictions? Suppose we form a biconditional using a tautology and a contradiction. What will be the logical status of the biconditional?
Here's an exercise to test wffs for logical equivalence.
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