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**1 - 6**of**6**### Răzvan Diaconescu An Institution-independent Proof of Craig Interpolation Theorem

"... Abstract. We formulate a general institution-independent (i.e. independent of the details of the actual logic formalised as institution) version of the Craig Interpolation Theorem and prove it in dependence of Birkhoff-style axiomatizability properties of the actual logic. We formalise Birkhoff-styl ..."

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Abstract. We formulate a general institution-independent (i.e. independent of the details of the actual logic formalised as institution) version of the Craig Interpolation Theorem and prove it in dependence of Birkhoff-style axiomatizability properties of the actual logic. We formalise Birkhoff-style axiomatizability within the general abstract model theoretic framework of institution theory by the novel concept of Birkhoff institution. Our proof destills a set of conditions behind the Craig Interpolation Property, which are easy to establish in the applications. Together with the generality of our approach, this leads to a wide range of applications for our result, including conventional and nonconventional logics (many of them from algebraic specification theory), such as general algebra, classical model theory, partial algebra, rewriting logic, membership algebra, etc. all of them in various versions and with various types of sentences (including infinitary ones). In dependence of axiomatizability properties many other applications are expected for various institutions or logics.

### IOS Press Institution-independent Ultraproducts

"... Abstract. We generalise the ultraproducts method from conventional model theory to an institutionindependent (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, mode ..."

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Abstract. We generalise the ultraproducts method from conventional model theory to an institutionindependent (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, 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.

### Ultraproducts and possible worlds semantics in institutions

"... We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called ‘institutions’. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We al ..."

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We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called ‘institutions’. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We also extend the institution-independent method of ultraproducts of [R. Diaconescu, Institution-independent ultraproducts, Fundamenta Informaticæ55 (3–4) (2003) 321–348] to possible worlds semantics and prove a fundamental preservation result for abstract modal satisfaction. As a consequence we develop a generic compactness result for possible worlds semantics. c ○ 2007 Elsevier B.V. All rights reserved.

### From Universal Logic to Computer Science, and back

"... Abstract. Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic. 1 ..."

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Abstract. Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic. 1 From Universal Logic... Although universal logic has been clearly recognised as a trend in mathematical logic since about one decade only, mainly due to the efforts of Jean-Yves Béziau and his colleagues, it had a presence here and there since much longer. For example the anthology [9] traces universal logic ideas back to the work of Paul Herz in 1922. In fact there is a whole string of famous names in logic that have been involved with universal logic in the last century, including Paul Bernays,