Title of the course: Logical Consequence
Instructor’s Name: Prof. Alexei Muravitsky
Institution: Northwestern State University
Dates: 20-31 July 2020
Prerequisites: An introductory course in mathematical logic is a plus; otherwise, some familiarity with the understanding of mathematical proofs and their implementation is expected.
Level: Graduate, advanced undergraduate
Abstract: The concept of logical consequence in formal languages will be discussed. Most languages in focus will be propositional, but we will also consider first-order language as an auxiliary tool. The following topics will be covered: formal languages and their semantics; logical matrices (the following examples of logical matrices will be presented: the two-valued matrix, the Lukasiewicz three-valued matrix, the Lukasiewicz three-valued matrix with modality, the Gödel n-valued matrices, the Dummett denumerable matrix, the Lukasiewicz denumerable matrix. An implementation of the logical consequence will be presented as a consequence relation and as a consequence operator, and various types of realization of the consequence relation will be discussed, including definitions by closure systems, by logical matrices, and by inference rules. The following theorems will be proved: Lindenbaum’s theorem, Los-Suszko-Wojcicki’s theorem and Wojcicki’s theorem. If time permits, the concept of Lindenbaum-Tarski algebra will be discussed.
Language: English