Definition des fragments modaux de la logique des predicats a partir de formules du premier ordre qui sont des traductions des proprietes poly-modales elementaires. Distinguant les fragments variables et finis des fragments lies a un quantificateur, l'A. developpe une version semantique des fragments gardes en remplacant les liens syntaxiques par des restrictions sur les types d'attribution dans les modeles generalises. Se referant a l'algebre cylindrique, l'A. indique les nouvelles directions que peuvent prendre les theoremes de Tarski dans un environnement mathematique: celle des contraintes structurelles speciales, des extensions infinies, de la logique modale etendue et d'une semantique dynamique

Modal languages and bounded fragments of predicate logic

In the rst part of this paper we introduced products of modal logics and proved basic results on their axiomatisability and the f.m.p. In this continuation paper we prove a stronger result - the product f.m.p. holds for products of modal logics in which some of the modalities are reflexive or serial. This theorem is applied in classical rst-order logic; we identify a new Square Fragment (SF) of the classical logic, where the basic predicates are binary and all quantiers are relativised, and for which we show the f.m.p. in the classical sense. Also we prove that SF not included in Guarded Fragment (and in Packed Fragment) and that it can be embedded into the equational theory of relation algebras. 1

Products of modal logics. Part 2: relativised quantifiers in classical logic

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

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Generalized quantification as substructural logic

The semantics for quantifiers described in this paper can be viewed both as a new semantics for generalized quantifiers and as a new look at standard first-order quantification, bringing the latter closer to modal logic.

Modal quantification over structured domains

Van Lambalgen (1991) proposed a translation from a language containing a generalized quantifier Q into a first-order language enriched with a family of predicates Ri, for every arity i (or an infinitary predicate R) which takes Qxφ(x,y1,...,yn) to ∀x(R(x, y1,...,yn) → φ(x,y1,..., yn)) (y1,..., yn are precisely the free variables of Qxφ). 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 this logic (and the corresponding fragment) has the interpolation property.

Interpolation for a Sequent Calculus of Generalized Quantifiers

Van Lambalgen (1990) proposed a translation from a language containing a general- ized quantifier Q into a first-order language enriched with a family of predicates ifc, for every arity t (or an infinitary predicate R) which takes Qx^(x,yi, . . . ,yn) to Vx(#(x,yi, . . . ,yn) - ♦ (x,y\, . . . ,yn)) (yi, - • • ,Vn are precisely the free variables of Qx ). 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.

First-Order Logic

Papers

On a Decidable Generalized Quantifier Logic Corresponding to a Decidable Fragment of