On a Decidable Generalized Quantifier Logic Corresponding to a Decidable Fragment of First-Order Logic

Van Lambalgen (1990) proposed a translation from a language containing a generalized quantifier Q into a first-order language enriched with a family of predicates Rs, for every arity i (or an infinitary predicate R) which takes Qz~(z, yl, . . . , y,~) to Vz(R(z, yl,..., g,~) --" ~b(x, Yl,..., Y,~)) (gl . . . . . g~ are precisely the free variables of Qzq~). The logic of Q (without ordinary quantifiers) corresponds therefore to the fragment of first-order logic which contains only specially restricted quantification. We prove that it is decidable using the method of analytic tableaux. Related results were obtained by Andr6ka and N6meti (1994) using the methods of algebraic logic.