A First Introduction to the Algebra of Sentences

These notes cover the content of a basic course in propositional algebraic logic given by the author at the italian School of Logic held in Cesena, September 18-23, 2000. They are addressed to people having few background in Symbolic Logic and they are mostly intended to develop algebraic methods for establishing basic metamathematical results (like representation theory and completeness, finite model property, disjunction property, Beth definability and Craig interpolation). For the sake of simplicity, only the case of propositional intuitionistic logic is covered, although the methods we are explaining apply to other logics (e.g. modal) with minor modifications. The purity and the conceptual clarity of such methods is our main concern; for these reasons we shall feel free to apply basic mathematical tools from category theory. The choice of the material and the presentation style for a basic course are always motivated by the occasion and by the lecturer’s taste (that’s the only reason why they might be stimulating) and these notes follow such a rule. They are expecially intended to provide an introduction to category theoretic methods, by applying them to topics suggested by propositional logic. Unfortunately, time was not sufficient to the author to include all material he planned to include; for this reason, important and interesting developements are mentioned in the final Section of these notes, where the reader can however only find suggestions for further readings. Lack of time is also responsible of the fact that sometimes only key points of proofs are provided,