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Generalized Interpolation in First Order Logic

Borzyszkowski T.

Fundamenta Informaticae

2005

Vol. 66, nr 3

199--219

We consider the Craig Interpolation Property for many sorted first-order logic. The Craig Interpolation Property explored in this paper is inspired by the institution independent generalization of this property presented in [21]. In [3] the author presents the interpolation result for the institution of many sorted first-order logic, with both morphisms in the pushout square being injective on sort names. The author also shows that the Craig Interpolation Property does not hold when both morphisms are certain morphisms which are noninjective on sort names. An open question in that paper was whether the interpolation property holds with only one morphism being injective on sort names. In this paper we give answer to this question. Following the overall structure of the classical proof presented in [7] for single sorted first-order logic, but with new technicalities concerning the many sorted case, we show that many sorted first-order logic has the interpolation property when just one (left or right) morphism is injective on sort names.

Institution-independent Ultraproducts

Diaconescu R.

Fundamenta Informaticae

2003

Vol. 55, nr 3,4

321-348

We generalise the ultraproducts method from conventional model theory to an institution-independent (i.e. independent of the details of the actual logic formalised as an institution) framework based on a novel very general treatment of the semantics of some important concepts in logic, such as quantification, logical connectives, and ground atomic sentences. Unlike previous abstract model theoretic approaches to ultraproducts based on category theory, our work makes essential use of concepts central to institution theory, such as signature morphisms and model reducts. The institution-independent fundamental theorem on ultraproducts is presented in a modular manner, different combinations of its various parts giving different results in different logics or institutions. We present applications to institution-independent compactness, axiomatizability, and higher order sentences, and illustrate our concepts and results with examples from four different algebraic specification logics. In the introduction we also discuss the relevance of our institution-independent approach to the model theory of algebraic specification and computing science, but also to classical and abstract model theory.