Categorical Abstract Algebraic Logic: Selfextensional …-Institutions with Implication
George Voutsadakis · 2007
The work of Jansana on selfextensional deductive systems with an implication satisfying the deduction-detachment property, that was partially based on the well-known work of Font and Jansana on providing a general algebraic semantics for sentential logics, is abstracted to cover selfextensional logics with implication that are formalized as …-institutions. Analogs are provided in this more general context of the main results of Jansana. In the flrst, it was shown that the class of algebras canonically associated with a deductive system with an implication having the deduction-detachment property is a variety. In the second, selfextensionality of a deductive system possessing an implication with the deduction-detachment property was seen to imply full selfextensionality. Finally, the existence of a dual isomorphism between selfextensional deductive systems having an implication with the deduction-detachment property, ordered by extension, and subvarieties of the variety, over the same similarity type, axiomatized by the Hilbert equations is demonstrated. In order to prove analogs of these results at the categorical level, the powerful machinery developed in the last few years in this area is brought to bear. In particular, speciflc use is made for the flrst time, of the theory of varieties and quasi-varieties of algebraic systems, as previously developed by the author.