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

Abstract:
On one decidable generalized quantifier logic corresponding to a decidable 
  fragment of first­order logic
Natasha Alechina

Van Lambalgen (1990) proposed a translation from a language containing a 
generalized quantifier Q into a first­order language enriched with a family 
of predicates R_i , for every arity i (or an infinitary predicate R) which 
takes $Qx\phi(x, y_1, ..., y_n)$ to $\forall x (R(x, y_1, ..., y_n) \implies 
\phi(x, y_1, ..., y_n) )$ ($y_1, ..., y_n$ are precisely the free variables of
$Qx\phi$). The logic of Q (without ordinary quantifiers) corresponds therefore 
to the fragment of first­order logic which contains only specially restricted 
quantification. We prove that it is decidable using the method of semantic 
tableaux. Similar results were obtained by Andreka and Nemeti (1994) using 
the methods of algebraic logic.