An Alternative Semantics for Relevant Logic with Identity

Shawn Standefer (University of Melbourne)

Shawn Standefer (Melbourne) will present "An Alternative Semantics for Relevant Logic with Identity" at 11 on 10 August in Old Arts 209.

Fine provided models for quantified relevant logics that admitted completeness results in his “Semantics for Quantified Relevance Logic”, although his models require expanding domains. In “An Alternative Semantics for Quantified Relevant Logic”, Mares and Goldblatt provided a new frame semantics for the quantified relevant logics QR and RQ. Mares-Goldblatt admissible proposition frames are constant domain Routley-Meyer frames for R that come equipped with a set of propositions and propositional functions. The truth conditions for the universal quantifier depart from the classical, Tarskian conditions, according to which satisfaction of a universally quantified formula is being satisfied by every instance. Instead, satisfaction of a universally quantified formula requires the existence of a certain sort of proposition that entails every instance. The logics QR and RQ are sound and complete for their respective classes of frames. Goldblatt and Kane adapted these frames to provide frames for a range of propositional relevant logics with propositional quantifiers and proved  soundness and completeness results.   

In “Semantics for Relevant Logic with Identity”, Mares extended Fine’s models for quantified relevant logics to deal with identity with substitution axioms. In this talk, I will provide a brief overview of models for quantified relevant logics with, and without, identity. I will then define Mares-Goldblatt admissible proposition models for QR with identity and substitution axioms and indicate points of interest in the soundness and completeness proofs. I will close with directions for future work.