Nonclassical logics have been introduced for philosophical reasons, for reasons arising in the foundations of mathematics, and for certain computer science/informatics applications. They are applicable in further disciplines as well, for example, linguistics.
In this course we study a wide variety of nonclassical logics by systematizing them from the point of view of their set theoretical semantics. Normal modal logics (e.g., K, T, B, S4, S5) can be axiomatized as extensions of classical logic and interpreted by Kripke-style (or ``possible worlds'') semantics. Relevance logics (e.g., T, E and R) may be interpreted by Routley-Meyer-style semantics. Generalized Galois logics, introduced by Dunn in the early 90s, treat these (and other) nonclassical logics in a unified framework. Some other well-known logics that we will look at that have specific computer science applications are linear logic, action logic, combinatory logic and dynamic logic. We will also look at the Lambek calculi, which arise in the categorial approach to grammar in linguistics. Beyond concentrating on modeling various proof-theoretic formulations of logics in relational semantics (and so proving soundness and completeness theorems), we will address problems of canonicity and duality as well.
A familiarity with some nonclassical logics (or with elements of universal algebra) might be helpful for understanding the course, whereas solid knowledge of classical first-order logic is highly recommended, as is basic knowledge of set theory.. The primary text is the book "Generalized Galois Logics" (in preparation by the instructors), that will be supplemented with other recommended readings, if necessary. Draft copies of the book will be made available to the students.