Let’s Talk About Canonical Extensions

Abstract

The canonical extension is a fundamental construction in the algebraic theory of modal logics. It plays a key role in the proof of completeness via canonicity for a large class of axiomatic extensions of basic normal modal logic, and it has allowed such results to be generalized from classical modal logic to a vast family of non-classical logics. In this talk, we will introduce the main aspects and results concerning canonical extensions, moving from the completeness of modal logic to a purely algebraic representation of it. We will then discuss and elaborate on a novel presentation of canonical extensions as belonging to a family of algebraic extensions parametrized by the sets of meets and joins of the original poset that one wishes to preserve. We will discuss how, by varying these sets of meets and joins, different types of maps can be extended in ways that guarantee desirable properties, such as having adjoints or preserving meets and joins. We will also examine how other completions, such as the Dedekind–MacNeille completion, fit into this framework and discuss various case studies. Finally, we will show how this construction can be applied to logics such as Propositional Dynamic Logic (PDL), where certain infinite meets are preserved, and how the parametric notion of canonical extension can be used to generalize completeness results for presentations of PDL with rules involving infinitely many assumptions.

About the Speaker

Apostolos Tzimoulis is a Research Scientist in the Department of Computer Science at the University of Luxembourg. His research focuses on algebraic logic, proof theory, and applications of non-classical logics to agent reasoning and agent interaction.