Logical equivalence implication between Kleene and Classical logic

Question

For any propositional assertions, $\phi$ and $\psi$, expressed using only the standard propositional logical connectives $\{\lnot,\land,\lor,\rightarrow,\iff\}$, if $\phi$ and $\psi$ are logically equivalent in Classical logic, are they necessarily logically equivalent in Kleene logic, and vice versa? That is, is it the case that $$\phi \equiv_C \psi \iff \phi \equiv_K \psi,$$ (where C=Classical and K=Kleene)? Logical equivalence is taken to mean sameness of truth table. I've worked out one direction of the biconditional, which is that $$\phi \equiv_K \psi \rightarrow \phi \equiv_C \psi.$$ This is because the specified set of logical connectives are classical-truth preserving in Kleene logic. So, for any two propositional assertions, $\phi$ and $\psi$, if $\phi \equiv_K \psi$, then $\phi$ and $\psi$ have the same truth table, by definition of logical equivalence. They therefore agree on all classical rows of the truth table, by the classical-truth preserving property of the set of connectives. But this set of classical rows just is the truth table for $\phi$ and $\psi$ in Classical logic, and since these rows are the same, $\phi \equiv_C \psi$. Therefore, $\phi \equiv_K \psi \rightarrow \phi \equiv_C \psi.$

But I can't figure out how to prove the other direction. I can argue that if $\phi \equiv_C \psi$, then $\phi$ and $\psi$ will agree on all classical rows of their truth tables in Kleene logic. But, I don't know what to say about the remaining rows?

Answer 1

Counterexample: $\phi=P$, $\psi=P\land(Q\lor\lnot Q)$.

Answer 2

I assume you are talking about Kleene's three-valued logic of indeterminacy. If so , the equivalence you are trying to prove does not hold, because classical logic proves the law of excluded middle $\phi \lor \lnot \phi$, but Kleene's logic does not: if $U$ is the undefined truth value, $U \lor \lnot U$ is $U$ not $\top$.

[This needs to be combined with the trick in David C. Ullrich's answer as discussed below.]