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14
General logics
 In Logic Colloquium 87
, 1989
"... theory, categorical logic. 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 repre ..."
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Cited by 11 (5 self)
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theory, categorical logic. 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
"... ..."
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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Cited by 1 (1 self)
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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.
Borrowing Interpolation
, 2011
"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a ..."
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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.
BOOTSTRAPPING TYPES AND COTYPES IN HASCASL
"... We discuss the treatment of initial datatypes and final process types in the widespectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL’s type class mechanism, and we describe constructions o ..."
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We discuss the treatment of initial datatypes and final process types in the widespectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL’s type class mechanism, and we describe constructions of types of finite and infinite trees that establish the conservativity of datatype and process type declarations adhering to certain reasonable formats. The latter amounts to modifying known constructions from HOL to avoid unique choice; in categorical terminology, this means that we establish that quasitoposes with an internal natural numbers object support initial algebras and final coalgebras for a range of polynomial functors, thereby partially generalizing corresponding results from topos theory. Moreover, we present similar constructions in categories of internal complete partial orders.
Coalgebraic Modal Logic in COCASL
"... We extend the algebraiccoalgebraic specification language CoCasl by full coalgebraic modal logic based on predicate liftings for functors. This logic is more general than the modal logic previously used in CoCasl and supports the specification of a variety of modal logics, such as graded modal log ..."
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We extend the algebraiccoalgebraic specification language CoCasl by full coalgebraic modal logic based on predicate liftings for functors. This logic is more general than the modal logic previously used in CoCasl and supports the specification of a variety of modal logics, such as graded modal logic, majority logic, and probabilistic modal logic. CoCasl thus becomes a modern modal language that covers a wide range of Kripke and nonKripke semantics of modal logics via the coalgebraic interpretation.
Ultraproducts and possible worlds semantics in institutions
"... We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the socalled ‘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 socalled ‘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 institutionindependent method of ultraproducts of [R. Diaconescu, Institutionindependent 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.
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. We dedicate this work to the memory of our dear friend and colleague Joseph Goguen who passed away during its preparation. 1