Referential semantics: duality and applications
Ramón Jansana, Alessandra Palmigiano · 2006
A b s t r a c t. In this paper, Wójcicki’s characterization of selfextensional logics as those logics that are endowed with a complete local referential semantics is extended to a fully fledged duality between atlas-models (i.e. generalized matrix models) and referential 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 fit. The first application of this duality is a characterization of the fully selfextensional logics among the selfextensional ones. Fully selfextensional logics form a subclass of particularly well-behaved selfextensional logics, and only recently [1] this inclusion was shown to be proper. In this paper, fully selfextensional logics are characterized 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 selfextensional, then every algebra in Alg(S) is isomorphic to an algebra of sets.