In (polyadic) first-order logic, for any sentence $\psi(x,y)$ with variables $x$ and $y$ free, we have $(\exists y)(\forall x)\psi(x,y)\vDash (\forall x)(\exists y)\psi(x,y)$ but not the reverse entailment. Is the reverse entailment true in *monadic* predicate logic, however? And if so, how would a proof go?


It strikes me as a little bizarre that I can't find a reference on this point, but perhaps the answer is obvious and I'm just not seeing it. (Heuristically, though, it would seem to be worth making explicit.)



quantifier order in monadic first-order logic
With a monadic predicate $P$, the sentence $(\forall x)(\exists y)(P(x)\iff P(y))$ does not entail $(\exists y)(\forall x)(P(x)\iff P(y))$. In fact, the former is logically valid but the latter fals in any structure where the interpretation of $P$ is neither the empty set nor the whole universe.


