Asymmetric consequence relations and many-valued semantics

Abstract

This work is concerned with the definition and motivation of asymmetric logics. These are many-valued logics with different sets of designated values for the premises and the conclusions. Their existence will be motivated both philosophically and formally. Not only do they constitute examples of novel logics that can model some aspects of our reasoning, but they are formally and historically important as counterexamples to the famous thesis of Roman Suszko. Suszko’s Thesis states that every logic is (logically) two-valued. Importantly, asymmetric logics can be given a many-valued (Suszko-style) semantics (which was first suggested by Grzegorz Malinowski in his notion of q-consequence). However, these logics are non-Tarskian, which forces the need to reevaluate the standard definition of validity. The asymmetric logics ST and wST are discussed and motivated in further detail.