The course deals with systems for the axiomatic rejection of propositions. Rejection systems are also referred to as complementary calculi as they axiomatise the complement of the valid formulas of a given logic. Thus, in a formal derivation in a complementary calculus, invalid statements are deduced from other invalid statements.
The first mention of the term rejection in modern logic was done by Jan Lukasiewicz in 1920 and he subsequently developed the first axiomatic rejection system in the context of his study on Aristotelian syllogistic.
In this course, we will discuss rejection calculi for different logics, like classical logic, many-valued logics, modal logics, etc. Moreover, we also discuss the concept of an anti-consequence operator, as introduced by Lukasiewicz's student Jerzy Slupecki and subsequently studied, e.g., also together with Grzegorz Bryll and Urszula Wybraniec-Skardowska, all well-known researchers from the famous Lvov-Warsaw school of logic. Finally, we also discuss applications of rejection systems for nonmonotonic logics.
