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What is a Logic?
 J.Y. BEZIAU (ED.), LOGICA UNIVERSALIS, 113–135
, 2006
"... ... model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual ..."
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... model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum
Heterogeneous theories and the heterogeneous tool set
 Semantic Interoperability and Integration. IBFI, Dagstuhl
, 2005
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Abstract Beth definability in institutions
 Journal of Symbolic Logic
, 2006
"... Abstract. 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 ..."
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Abstract. 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 (manysorted) 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. §1. Introduction. 1.1. Institutionindependent model theory. The theory of “institutions ” [26] is a categorical abstract model theory which formalises the intuitive notion of logical
An institutional view on categorical logic and the CurryHowardTaitisomorphism
"... We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number ..."
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We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.
Information Processing Letters 103 (2007) 5–13 Stratified institutions and elementary homomorphisms
, 2006
"... Communicated by J.L. Fiadeiro www.elsevier.com/locate/ipl For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parameterized. For instance, in the firstorder logic, the satisfaction is parameterized by the valuation of unbound var ..."
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Communicated by J.L. Fiadeiro www.elsevier.com/locate/ipl For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parameterized. For instance, in the firstorder logic, the satisfaction is parameterized by the valuation of unbound variables, while in modal logics it is further by possible worlds. This paper proposes a uniform treatment of such parameterization of the satisfaction relation within the abstract setting of logics as institutions, by defining the new notion of stratified institutions. In this new framework, the notion of elementary model homomorphisms is defined independently of an internal stratification or elementary diagrams. At this level of abstraction, a general Tarski style study of connectives is developed. This is an abstract unified approach to the usual Boolean connectives, to quantifiers, and to modal connectives. A general theorem subsuming Tarski’s elementary chain theorem is then proved for stratified institutions with this new notion of connectives.
An Institutional View on Categorical Logic
"... We introduce a generic notion of categorical propositional logic and provide a construction of a preorderenriched institution out of such a logic, following the CurryHowardTait paradigm. The logics are specified as theories of a metalogic within the logical framework LF such that institution com ..."
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We introduce a generic notion of categorical propositional logic and provide a construction of a preorderenriched institution out of such a logic, following the CurryHowardTait paradigm. The logics are specified as theories of a metalogic within the logical framework LF such that institution comorphisms are obtained from theory morphisms of the metalogic. We prove several logicindependent results including soundness and completeness theorems and instantiate our framework with a number of examples: classical, intuitionistic, linear and modal propositional logic.