This is exercise 1.3.22 from Hinman's Fundamentals of Mathematical Logic. Let $\mathrm{Sent}_{\neg, \vee, \wedge}$ be the set of all sentences from propositional logic closed under negation, disjunction, and conjunction and consider a mapping $\circ: \mathrm{Sent}_{\neg, \vee, \wedge} \to \mathrm{Sent}_{\neg, \vee, \wedge}$ defined as: if $\phi$ is an atomic sentence, then $\phi^\circ = \phi$; if $\phi$ is $\neg \psi$, then $\phi^\circ = \neg \psi^\circ$; if $\phi$ is $\psi \wedge \theta$, then $\phi^\circ = \psi^\circ \vee \theta^\circ$; finally, if $\phi$ is $\psi \vee \theta$, then $\phi^\circ = \psi^\circ \wedge \theta^\circ$. Basically, $\circ$ interchanges $\vee$ and $\wedge$ while leaving everything else the same (that's why I called it a "semi-dual" -- I don't know if there's a more appropriate name for it).
Given this setting, Hinman asks us to prove that, if $\models \neg \phi$, then $\models \phi^\circ$. This seems simple enough, but I'm having troubles actually carrying out the proof. It seems that the obvious way is by proceeding by induction, yet I don't see a way of framing the induction hypothesis strongly enough to get what I want.
Specifically, in the disjunction case, the result is straightforward:
\begin{align*} \models \neg \phi &\implies \models \neg(\psi \vee \theta)\\ &\implies \models \neg \psi \wedge \neg \theta\\ &\implies \models \neg \psi \text{ and } \models \neg \theta\\ &\implies \models \psi^\circ \text{ and } \models \theta^\circ\\ &\implies \models \psi^\circ \wedge \theta^\circ \end{align*}
But what about the other cases? For negation, for instance, I just have that $\models \neg \phi \implies \models \neg \neg \psi \implies \models \psi$, but then I can't apply the induction hypothesis. Similarly, for the conjunction case, I just get $\models \neg \phi \implies \models \neg(\psi \wedge \theta) \implies \models \neg \psi \vee \neg \theta$, so I can't apply the induction hypothesis either.
Therefore, it seems that, for these other cases, I need a lemma or something to be able to get the induction off the ground. In the conjunction case, I tried to show that if a conjunction is contradictory, either one of its conjuncts is contradictory or else each conjunct is equivalent to the negation of the other, but that didn't help much. Any hints or ideas?
