Say a set of sentences in first-order logic has the *finite countermodel property* if any sentence in the set that is falsifiable is falsifiable on a finite domain. (Two examples: the set of sentences with only unary predicates; the *dual* class to the Bernays–Schönfinkel class, i.e., those sentences in prenex form whose universal quantifiers, if any, precede existential quantifiers.) It's obvious that logical validity for any such set is decidable.

But is the converse true? Or are there decidable fragments of first-order logic that lack the finite countermodel property? (I think there ought to be: I'm imagining sentences that may be false only on infinite domains -- i.e. which admit only infinite countermodels -- but which nevertheless permit the computation of an upper bound on the time it takes some semidecision procedure to finish. Though I suppose my question amounts to asking whether *that's* possible.)

satisfiability. Validity of $\exists^\*\forall^\*$-sentences is in fact undecidable. If you want FMP for validity (or falsifiability), you need to define the class dually: $\forall^\*\exists^\*$. $\endgroup$ – Emil Jeřábek supports Monica Dec 14 '11 at 12:51