Relational semantics, ordered algebras, and quantifiers for deductive systems

Abstract

In this talk we sketch an abstract approach to two-sorted relational semantics based on polarities, whose main advantage is that it applies to arbitrary logics and classes of ordered algebras. Relational semantics has proved to be a fundamental tool in the investigations of non-classical logics and ordered algebras. Very roughly speaking, logicians may find relational semantics appealing because it allows a freedom of construction (i.e. the possibility of deforming models, adding points to them etc.), which is not immediately available in semantics with a more algebraic flavour. On the other hand, algebraists may encounter relational semantics on their way in the form of the skeleton underlying the theory of completions of ordered algebras (including Dedekind-MacNeille completions and canonical extensions) and of Priestley-style dualities. Finally, philosophers may find relational semantics valuable because of its natural interpretation in terms of possible worlds (or states of knowledge, bodies of information etc.), and its relevance in the foundation of constructive mathematics.