Recipe

Check whether two sentences are equivalent

Use when: you are given two sentences and asked whether they are logically equivalent, and they use at most four or five letters

Goal: a verdict: equivalent, because every row matches, or not equivalent, with one row where they disagree

Taught in Checking equivalence with truth tables

1. 1

   Collect the sentence letters of both sentences into one list, in alphabetical order.

   Example

   For P \to \neg Q and \neg(P \wedge Q) the list is P, Q. For P \vee Q and P \vee Q \vee R it is P, Q, R, even though R appears in only one sentence.

   Why: equivalence is agreement in every situation, and a situation fixes every letter either sentence depends on.

2. 2

   With n letters, write 2^n rows in the standard order: all T first, the rightmost letter alternating fastest.

   Why: the standard order guarantees no valuation is skipped or repeated.

3. 3

   For each sentence, compute its value on each row, from the innermost brackets out to the main connective.

   ⚡ Shortcut

   If each sentence is false on only a few rows, find those rows directly. A conditional is false only when its antecedent is true and its consequent false. A disjunction is false only when every disjunct is false. Equal sets of false rows mean equivalence.

   Why: a compound's value depends only on its parts' values, so working inside out gives the right value on every row.

4. 4

   Go down the rows comparing the two values, or scan the biconditional column for an F.

   • every row matches — the sentences are logically equivalent.
   • some row differs — they are not equivalent; stop and report that row.

   Why: equivalence needs agreement on every row, and one disagreement is a situation where the sentences come apart.