Introduction

Abstract

This book is an example of fruitful interaction between (non-classical) propositional logics and (classical) model theory which was made possible due to categorical logic. Its main aim consists in investigating the existence of model-completions for equational theories arising from propositional logics (such as the theory of Heyting algebras and various kinds of theories related to propositional modal logic). The existence of model-completions turns out to be related to proof-theoretic facts concerning interpretability of second order propositional logic into ordinary propositional logic through the so-called ‘Pitts’ quantifiers’ or ‘bisimulation quantifiers’. On the other hand, the book develops a large number of topics concerning the categorical structure of finitely presented algebras, with related applications to propositional logics, both standard (like Beth’s theorems) and new (like effectiveness of internal equivalence relations, projectivity and definability of dual connectives such as difference). A special emphasis is put on sheaf representation, showing that much of the nice categorical structure of finitely presented algebras is in fact only a restriction of natural structure in sheaves. Applications to the theory of classifying toposes are also covered, yielding new examples.KeywordsPropositional LogicWinning StrategyHeyting AlgebraModal AlgebraAmalgamation PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.