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α|==|β iff |= (α↔β)

Does anyone know how to go about showing that this is true?

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Truth table?${}$ – peterwhy Sep 14 '13 at 18:17
up vote 2 down vote accepted


First, assume that $\alpha\leftrightarrow\beta$ is a tautology. Suppose that $g$ is some assignment of truth values to the relevant propositional variables. Since $\alpha\leftrightarrow\beta$ is true in this assignment, conclude that if $\alpha$ was true then $\beta$ is true, and vice versa.

In the other direction, assume that $\alpha$ is logically equivalent to $\beta$. Take some assignment $g$ and show that if $\alpha$ is false then $\beta$ is false, and so $\alpha\leftrightarrow\beta$ is true; and if $\alpha$ is true then $\beta$ is true, and so $\alpha\leftrightarrow\beta$ is true. These are the only two options, so $\alpha\leftrightarrow\beta$ is a tautology, as wanted.

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