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. (shrink)

The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful (...) marriage of possible world semantics and algebra and its evolution leading to very general construction of Wójcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions. (shrink)

A logic is selfextensional if its interderivability (or mutual consequence) relation is a congruence relation on the algebra of formulas. In the paper we characterize the selfextensional logics with a conjunction as the logics that can be defined using the semilattice order induced by the interpretation of the conjunction in the algebras of their algebraic counterpart. Using the charactrization we provide simpler proofs of several results on selfextensional logics with a conjunction obtained in [13] using Gentzen systems. We also obtain (...) some results on Fregean logics with conjunction. (shrink)

A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart of any finitary and congruential logic . This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical (...) in this new sense. (shrink)

This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows (...) us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowa's approach. I hope in this way to give the reader a general view of Rasiowa's key position in the evolution of Algebraic Logic during the twentieth century. (shrink)