A real-world interpretation for $\{\neg(\phi\leftrightarrow\psi)\}\vdash((\neg\phi)\leftrightarrow\psi)$ One of the exercises in Chiswell's _Mathematical Logic_ is to prove the following sequent > $\\{\neg(\phi\leftrightarrow\psi)\\}\vdash((\neg\phi)\leftrightarrow\psi).$ I'm not interested in deriving this statement (which Chiswell says is "hard"), but giving it a natural interpretation. It's the type of statement which seems easy to apply, and yet, with a fallacious result. The following, for example, is manifestly false: > If it's not the case that rain is the same as precipitation, then, non-rain is the same as precipitation. Here "is the same as" is a gloss for "iff-then". What's my mistake here? And, what would be a correct, natural interpretation of this sequent?

Rain and precipitation may be thought of as defined by unary predicates $R, P$ respectively true of precisely that which is rain and precipitation. To say that rain is not the same as precipitation essentially means $\
eg \forall x (R(x) \leftrightarrow P(x))$. This is equivalent to $\exists x \
eg (R(x) \leftrightarrow P(x))$, which by the result of the exercise implies $\exists x( \
eg R(x) \leftrightarrow P(x))$. Glossing, if rain is not the same as precipitation, there's something which is not rain if and only if it is precipitation. (Snow, for instance, is such a thing.) That there are statements like $\exists x( \
eg R(x) \leftrightarrow P(x))$ which don't gloss into very simple statements of ordinary language suggests the language of logical symbolism is sometimes more expressive than ordinary language.

To say that non-rain is the same as precipitation would be to say $\forall x( \
eg R(x) \leftrightarrow P(x))$.