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1 2006

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Referential semantics: duality and applications

Jansana R., Palmigiano A.

Reports on Mathematical Logic

2006

Nr 41,spec. iss.

63-93

In this paper, Wojcicki's characterization of selfex- tensional logics as those logics that are endowed with a complete local referential semantics is extended to a fully edged duality between atlas-models (i.e. generalized matrix models) and refer- ential models of an arbitrary selfextensional logic S. This duality serves as a general template where a wide range of Stone- and Priestley-style dualities related with concrete logics can t. The rst application of this duality is a characterization of the fully selfextensional logics among the selfextensional ones. Fully selfex- tensional logics form a subclass of particularly well-behaved selfex- tensional logics, and only recently [1] this inclusion was shown to be proper. In this paper, fully selfextensional logics are character- ized as those selfextensional logics S whose algebraic counterpart Alg(S) { seen as a category { is dually equivalent to the reduced referential models of S. This implies that if S is fully selfexten- sional, then every algebra in Alg(S) is isomorphic to an algebra of sets.