The course deals with systems for the axiomatic rejection of propositions, i.e., with proof systems which characterise the invalid formulas of a logic. Rejection systems, also called refutation systems, therefore axiomatise the complement of the valid formulas of a given logic and in this sense they are also often referred to as complementary calculi. 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.
