Studia Logica

, Volume 101, Issue 4, pp 849–899 | Cite as

Categorical Abstract Algebraic Logic: Referential Algebraic Semantics

  • George VoutsadakisEmail author


Wójcicki has provided a characterization of selfextensional logics as those that can be endowed with a complete local referential semantics. His result was extended by Jansana and Palmigiano, who developed a duality between the category of reduced congruential atlases and that of reduced referential algebras over a fixed similarity type. This duality restricts to one between reduced atlas models and reduced referential algebra models of selfextensional logics. In this paper referential algebraic systems and congruential atlas systems are introduced, which abstract referential algebras and congruential atlases, respectively. This enables the formulation of an analog of Wójcicki’s Theorem for logics formalized as π-institutions. Moreover, the results of Jansana and Palmigiano are generalized to obtain a duality between congruential atlas systems and referential algebraic systems over a fixed categorical algebraic signature. In future work, the duality obtained in this paper will be used to obtain one between atlas system models and referential algebraic system models of an arbitrary selfextensional π-institution. Using this latter duality, the characterization of fully selfextensional deductive systems among the selfextensional ones, that was obtained by Jansana and Palmigiano, can be extended to a similar characterization of fully selfextensional π-institutions among appropriately chosen classes of selfextensional ones.


Abstract Algebraic Logic Referential Algebraic Semantics Wójcicki’s Theorem Generalized Matrix Model Atlas Algebraic Semantics Duality π-Institution Selfextensional Logic Fully Selfextensional Logic 


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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceLake Superior State UniversitySault Sainte MarieU.S.A

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