# If $\alpha\implies\beta$ and $\beta\implies\alpha$ are satisfiable, then $\alpha\iff \beta$ must be satisfiable?

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?

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

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.)

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.