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Consider this sentence:

If $\alpha\implies\beta$ and $\beta\implies\alpha$ are satisfiable, then $\alpha\iff \beta$ is satisfiable.

I think the above sentence is correct because

$$\alpha\iff \beta\equiv (\alpha\implies\beta)\wedge (\beta\implies\alpha).$$

Is my argument valid?

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  • $\begingroup$ Why do you think that $P \land Q$ needs to be satisfiable if $P$ and $Q$ are satisfiable? $\endgroup$ Commented Apr 9, 2022 at 13:34

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If $\alpha\implies\beta$ and $\beta\implies\alpha$ are satisfiable, then $\alpha\iff \beta$ is satisfiable.

This is false: for a counterexample, just let $\beta=\lnot\alpha.$

(On the other hand, the disjunction of two satisfiable sentences is indeed satisfiable, and the conjunction or disjunction of two validities is indeed a validity.)

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The sentence isn't necessarily correct.

Let $\beta \equiv \neg \alpha$. For interpretation $I$ for which $I(\alpha) = 1$, you'd get $I(\beta) = 0$, and therefore $\beta \rightarrow \alpha$ is satisfied by interpretation $I$.

Analogous to that, interpretation $J$ for which $J(\alpha) = 0$ satisfies $\alpha \rightarrow \beta$.

However, for every interpretation where $\alpha$ is true, $\beta$ is false, so $\alpha \leftrightarrow \beta$ is never satisfied.

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