Some results in non-monotonic proof-theoretic semantics

TL;DR

This paper investigates the relationships between different proof-theoretic semantics for non-monotonic logic, establishing equivalences under certain conditions and demonstrating the limitations of intuitionistic logic's completeness within these frameworks.

Contribution

It clarifies the connections between reducibility and base semantics in non-monotonic proof-theoretic validity, and discusses the notions of point-wise soundness and completeness.

Findings

Reducibility semantics and standard base semantics are equivalent under certain conditions.

Sandqvist's base semantics completeness implies the inverse equivalence.

Intuitionistic logic is not point-wise complete in non-monotonic proof-theoretic semantics.

Abstract

I explore the relationships between Prawitz's approach to non-monotonic proof-theoretic validity, which I call reducibility semantics, and some later proof-theoretic approaches, which I call standard base semantics and Sandqvist's base semantics respectively. I show that, if suitable conditions are met, reducibility semantics and standard base semantics are equivalent, and that, if Sandqvist's variant is complete over reducibility semantics, then also the inverse holds. Finally, notions of "point-wise" soundness and completeness (called base-soundness and base-completeness) are discussed against some known principles from the proof-theoretic literature, as well as against monotonic proof-theoretic semantics. Intuitionistic logic is proved not to be "point-wise" complete on any kind of non-monotonic proof-theoretic semantics. The way in which this result is proved, as well as the overall behaviour of "point-wise" soundness and completeness, is significantly different in the non-monotonic framework as compared to what happens in the monotonic one - where the notions at issue can be used too to prove the "point-wise" incompleteness of intuitionistic logic.