Equivalence Glossary
Definition
Two formulae are logically equivalent iff each can be derived from the other, which is the case iff they are true for exaxtly the same interpretations.
Material equivalence is a much weaker notion: formulae are materially equivalent on an interpretation iff they have the same truth value on that interpretation.
The equivalence connective or biconditional expresses material equivalence: A IFF B is true if A and B are both true or both false; it is false if A and B have opposite truth values.
Comments
We write
A
⊣⊢ B
to mean
that A and B are logically equivalent.
Two formulae which are logically equivalent can replace each other in any context (as subformulae of longer things) without changing the logical consequences of the formula in which they occur. This property of replacement of provable equivalents is a fundamental principle of logic.
Examples
- p IFF q is equivalent to (p IMP q) AND (q IMP p). It is also equivalent to (p AND q) OR NOT(p OR q)
- ALLx(Fx IMP Gx) is equivalent to NOTSOMEx(Fx AND NOTGx).
- On any interpretation for which p is true, p AND q is materially equivalent to q.