Equivalence

Equivalence is a 1. (material equivalence) the combination of two formulas with the connective &lt;-&gt; (if and only if, iff), which is only true if both formulas have the same truth value. phi &lt;-&gt; psi can also be defined as the conjunction of two implications: phi -&gt; psi and psi -&gt; phi. For this reason, the connective of material equivalence is sometimes called the biconditional. The truth table for material equivalence is as follows:

(i)	phi		psi	 phi &lt;-&gt; psi 1		 1	     1 	 1		 0	      0 	 0		 1	      0 	 0	 	 0	      1

See Connective. 2. (logical equivalence) a relation obtaining between two formulas phi and psi if their material equivalence phi &lt;-&gt; Psi is a tautology. In other words, two formulas which are logically equivalent have the same truth value for every possible model.

Example
phi -&gt; psi is logically equivalent with Neg [ phi &amp; Neg psi ] in propositional logic and ThereIs(x) [ P(x) ] is equivalent with Neg All(x) [ Neg P(x) ] in predicate logic. When two expressions are logically equivalent, it is possible to substitute them for each other, without changing the truth values of the proposition they are contained in.

Link
Utrecht Lexicon of Linguistics