We explore the possibility and some potential payoffs of using the theory of accessible categories in the study of categories of logics. We illustrate this by two case studies focusing on the category of finitary structural logics and its subcategory of algebraizable logics.

We give a version of L´os’ ultraproduct result for forcing in Kripke structures in a first-order language with equality and discuss ultrafilters in a topology naturally associated to a partial order. The presentation also includes background material so as to make the exposition accessible to those whose main interest is Computer Science, Artificial Intelligence and/or Philosophy.

. We explore the possibility and some potential payoffs of using the theory of accessible categories in the study of categories of logics. We illustrate this by two case studies focusing on the category of finitary structural logics and its subcategory of algebraizable logics.

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $$\kappa $$, where $$\kappa $$ is a regular cardinal. The corresponding new notion is called $$\kappa $$ -filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $$\kappa $$ -filter pairs give rise to a fixed logic of cardinality $$\kappa $$. To make the (...) question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $$\kappa $$. Along the way we use $$\kappa $$ -filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair. (shrink)

We introduce the Profinite Hull functor of special groups, showing that it gives rise to a new local - global principle, the subform reflection property. We also indicate applications of this principle to the abstract algebraic theory of quadratic forms.