80-601 Formal Methods: Algebraic Logic
6 units, Carnegie Mellon University
The relationship between logic and algebra has a long history: algebraic methods have been employed in logic since at least Leibniz. Various well-known logical calculi admit algebraic semantics, and the fruitful interplay between logic and algebra has been recognized in a number of distinct areas, from philosophy to mathematics to computer science. Algebraic logic studies formal systems using algebraic tools and techniques, revealing some striking connections between algebra, logic, and topology. In this class we will introduce some of the main results and methods of algebraic logic, focusing primarily on propositional languages. We will first cover the basic notions of universal algebra and elementary topology. We will then discuss algebraic completeness results for classical, intuitionistic, and many-valued logics, highlighting the role played by duality theory — the study of dualities between classes of algebras and topological spaces (and especially Stone duality for Boolean algebras). We will then discuss the fundamental notions of algebraic modal logic: we will introduce algebraic semantics for modal logic and prove the Jónsson-Tarski representation theorem, which extends Stone duality to Boolean algebras with operators. Finally, we will ask some more general questions: when does a logic admit an algebraic semantics? How do we associate a class of algebras to a logic in general? Can we provide a general account of the "algebraizability"" of a logic? These are some of the fundamental questions studied in abstract algebraic logic. In the last part of the class, we will discuss some of the fundamental results in that area. "
Prerequisites: none listed
Offered: Fall 2025
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