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$G$ and $F$ are formulas, If every assignment that models $G$ also models $F$, does it mean that $F=G$?

The question may be silly, but I'm not sure if there's some obscure scenario where $F\neq G$.

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It depends on what you mean by $=$, really... –  Zhen Lin Dec 25 '12 at 4:09

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up vote 5 down vote accepted

You can always extend a formula without changing its truth value by "anding" it with a tautology. So for instance $G=a \wedge b$ and $F=(a \wedge b) \wedge (a \vee \neg a)$ are satisfied by the same models but are not equal.

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Consequently, the answer to the question is "no", and this implies that there exist an infinity of "obscure scenarios" where F≠G. –  Doug Spoonwood Dec 25 '12 at 5:44

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