# Equivalence of monadic axioms

Call two axioms equivalent if they imply the same set of theorems. I am interested in decidability of so defined equivalence. In this generality the problem is obviously undecidable since it can be used to decide Entscheidungsproblem. So I am interested in cases where Entscheidungsproblem is decidable, particularly in case of monadic axioms (i.e. axioms containing only monadic functional and predicate symbols).

Two sentences $\phi$ and $\psi$ imply the same sets of theorems if and only if $\phi\leftrightarrow\psi$ is a tautology. Thus when the Entscheidungsproblem is decidable for a Boolean-closed class (such as the monadic case you mention), this equivalence is also decidable. – Emil Jeřábek Mar 18 '11 at 15:19
The comment showed that decidability of axiom equivalence is implied by decidability of pure logic. (I.e. to decide if $\Theta$ and $\Theta'$ are equivalent, it is equivalent to decide if $\vdash \Theta \leftrightarrow \Theta'$. Conversely, if one can decide axiom equivalence, then one can decide pure logic. Namely, $\vdash \Theta$ iff the axiom $\Theta$ is equivalent to $\top$.