Two propositions are logically equivalent if they share the same truth value across all possible scenarios.

Not to be confused with equivalence relation and logical biconditional

Common Logical Equivalences

EquivalenceName

Identity Law

Domination laws

Idempotent or tautology laws
Double Negation

Commutative laws

Associative laws
Distributive laws
De Morgan’s laws

Absorption laws

Negation laws
1

Relationship with Biconditional

See: logical biconditional

and are logical equivalent if and only if when the biconditional is always true (a tautology).

Substitutability

In propositional logic, if a proposition is logically equivalent to a propositional , then we can substitute for and the result will be logically equivalent to the original sentence. However, this is no longer true in predicate logic.

Footnotes

  1. Logical equivalence - Wikipedia