We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract (...) properties and are preserved and reflected by translations under suitable conditions. (shrink)

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.

This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the classical Beth theorem by relying on the interpolation properties of the institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many new actual definability results, (...) including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for 'borrowing' definability properties from another institution via an 'adequate' encoding between institutions. The power of our general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-)varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalised as institutions from computing science and logic. (shrink)

Institution Theory Institution theory is a very general mathematical study of formal logical systems—with emphasis on semantics—that is not committed to any particular concrete logical system. This is based upon a mathematical definition for the informal notion of logical system, called institution, which includes both syntax and semantics as well as the relationship between … Continue reading Institution Theory →.

This is a short survey on the development of the formal specification and verification language H with emphasis on the scientific part. H is a modern highly expressive language solidly based upon advanced mathematical theories such as the internalisation of Kripke semantics within institution theory.

We extend the concept of quasi-variety of first-order models from classical logic to multiple valued logic and study the relationship between quasi-varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi-varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Saturated models constitute one of the powerful methods of conventional model theory, with many applications. Here we develop a categorical abstract model theoretic approach to saturated models within the theory of institutions. The most important consequence is that the method of saturated models becomes thus available to a multitude of logical systems from logic or from computing science. In this paper we define the concept of saturated model at an abstract institution-independent level and develop the fundamental existence and uniqueness theorems. (...) As an application we prove a general institution-independent version of the Keisler–Shelah isomorphism theorem “any two elementarily equivalent models have isomorphic ultrapowers” (assuming Generalized Continuum Hypothesis). (shrink)

In this essay we analyse and elucidate the method to establish and clarify the scope of logic theorems offered within the theory of institutions. The method presented pervades a lot of abstract model theoretic developments carried out within institution theory. The power of the proposed general method is illustrated with the examples of interpolation and definability, as they appear in the literature of institutional model theory. Both case studies illustrate a considerable extension of the original scopes of the two classical (...) theorems. Our presentation is rather narrative with the relevant logic and institution theory concepts introduced and explained gradually to the non-expert reader. (shrink)