Non-Reflexive Logics

Abstract

This chapter is about non-reflexive logics in the “tolerant-strict” family. We present them both from a proof-theoretic and a model theoretic point of view. For the first, we define new sequent calculi in which identity is not admissible nor derivable, that are obtained by removing the identity axiom from the calculus LK and possibly adding or removing other rules. For the second, we build on some notions already introduced in the previous chapter and present different semantics, which are then related to the calculi through soundness and completeness results both at the inferential and metainferential level. The level of metainferences is further discussed showing that the non-reflexive calculi are structurally incomplete and then proving some results regarding their global, local and absolute global ranges. Next, the non-reflexive logic TS is related to other logics introduced in this and the previous chapters, namely CL, ST, LP and K3. In particular, several notions of translation and duality which can be found in the literature are presented and compared. Some space is also devoted to the introduction of metainferential mixed logics. Finally, we build on the notion of anti-inference central to the discussion on dualities and show that, given a Tarskian logic, one can construct a non-reflexive logic with the same anti-inferences. We conclude by briefly presenting some three-sided sequent calculi used in the literature to characterize some of the logics discussed here and in the previous chapters.