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Equivalence
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If A,B are formulas are such that
v ( A ) = v ( B )
for all interpretations v , A is (logically) equivalent to B:
A ≡ B
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Example: ¬p V q ≡ p → q since both formulas are true in all interpretations except when v ( p ) = T, v ( q ) = F and are false for that particular interpretation
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Caution : ≡ does not mean the same thing as ↔ :
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A ↔ B is a formula (syntax)
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A ≡ B is a relation between two formula (semantics)
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Theorem : A ≡ B if and only if A ↔ B is true in every interpretation; i.e. A ↔ B is a tautology .
