Inconsistency, paraconsistency and ω-inconsistency

Abstract

In this paper I'll explore the relation between ω-inconsistency and plain inconsistency, in the context of theories that intend to capture semantic concepts. In particular, I'll focus on two very well known inconsistent but non-trivial theories of truth: LP and STTT. Both have the interesting feature of being able to handle semantic and arithmetic concepts, maintaining the standard model. However, it can be easily shown that both theories are ω-inconsistent. Although usually a theory of truth is generally expected to be ω-consistent, all conceptual concerns don't apply to inconsistent theories. Finally, I'll explore if it's possible to have an inconsistent, but ω-consistent theory of truth, restricting my analysis to substructural theories.