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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).

Any information about this would be appreciated.

Thank you for your time, Levon

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    $\begingroup$ 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. $\endgroup$
    – Emil Jeřábek
    Mar 18, 2011 at 15:19
  • $\begingroup$ Thank you for the answer. Initially I was concerned with the decidability of whether two axioms imply the same theorems of a specific form. Then I came to the above question and its triviality slipped my attention. Sorry for a stupid question. $\endgroup$
    – Levon
    Mar 18, 2011 at 17:08

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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 $\top$.

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