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.)



decidable fragments of first-order logic without the finite countermodel property
If you restrict attention to the traditional prefix–vocabulary classes, validity in the following fragments is decidable without having the finite model property (note that it is customary in the literature to discuss satisfiability rather than validity, hence you will find there the dual classes):


* Full FO in a language with equality, unary predicates, and a single unary function.
* The prefix class $\forall^{\*}\exists\forall^{\*}$ (i.e., sentences in prenex normal form with only one existential quantifier) in a language with equality, arbitrary predicates, and a single unary function.
* Any prefix class with a finite prefix in a fixed language with finitely many relations and no functions. (This is a trivial case: if you further normalize the matrix to CNF, there are only finitely many formulas in the class.)


A nice survey is in [this lecture](http://www-mgi.informatik.rwth-aachen.de/~graedel/kalmar.pdf) by Erich Grädel. A comprehensive reference is: E. Börger, E. Grädel, Y. Gurevich, *The Classical Decision Problem*, Springer, 1997 (reprinted 2001), [MR1482227](http://www.ams.org/mathscinet-getitem?mr=1482227).


