Relation To Material Equivalence
Logical equivalence is different from material equivalence. The material equivalence of p and q (often written p↔q) is itself another statement, call it r, in same object language as p and q. r expresses the idea "p if and only if q". In particular, the truth value of p↔q can change from one model to another.
The claim that two formulas are logically equivalent is a statement in the metalanguage, expressing a relationship between two statements p and q. The claim that p and q are semantically equivalent does not depend on any particular model; it says that in every possible model, p will have the same truth value as q. The claim that p and q are syntactically equivalent does not depend on models at all; it states that there is a deduction of q from p and a deduction of p from q.
There is a close relationship between material equivalence and logical equivalence. Formulas p and q are syntactically equivalent if and only if p↔q is a theorem, while p and q are semantically equivalent if and only if p↔q is true in every model (that is, p↔q is logically valid).
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