Definition
Two statements are logically equivalent when they have the same truth value under every possible assignment of truth values to their component propositions. The relation is written with a double-arrow connective and tested by comparing the columns of their truth tables: if the columns match row for row, the statements are equivalent. Logically equivalent statements are interchangeable in any proof — substituting one for the other never changes whether an argument is valid.
Equivalence is sharper than mere material biconditional truth in a single case. It is a structural relation: not "they happen to match here" but "they must match in every conceivable scenario the propositional vocabulary can describe."
Why it matters
How it works
The canonical test is the truth table. Build a table covering every combination of truth values for the atomic propositions in both statements, compute each statement's value row by row, and compare the resulting columns. Identical columns confirm equivalence; any divergence refutes it. Several standard equivalences recur often enough to be treated as named transformation rules: De Morgan's laws relate negated conjunctions to disjunctions of negations; double negation removes a stacked pair of negations; contraposition flips a conditional by negating both sides and swapping their order; and the implication-as-disjunction equivalence rewrites a conditional in purely disjunctive form.
The practical payoff is proof economy. A proof step that seems to require a long detour often collapses to a single substitution once an equivalence is recognised. Equivalence also underwrites the project of reducing arbitrary statements to canonical normal forms — conjunctive or disjunctive normal form — which makes them mechanically checkable. In propositional calculus, recognising equivalence is the first skill that separates rote symbol-pushing from genuine reasoning about logical structure.