Definition:Semantic Equivalence/Boolean Interpretations
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Definition
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.
Definition 1
Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:
- $\mathbf A \models_{\mathrm{BI}} \mathbf B$ and $\mathbf B \models_{\mathrm{BI}} \mathbf A$
that is, each is a semantic consequence of the other.
That is to say, $\mathbf A$ and $\mathbf B$ are equivalent if and only if:
- $\map v {\mathbf A} = T$ if and only if $\map v {\mathbf B} = T$
for all boolean interpretations $v$.
Definition 2
Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:
- $\map v {\mathbf A} = \map v {\mathbf B}$
for all boolean interpretations $v$.
Definition 3
Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:
- $\mathbf A \iff \mathbf B$ is a tautology
where $\iff$ is the biconditional connective.
Equivalence of Definitions
The definitions above are all equivalent, as shown on Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations.