A Modal Logic of Supervenience

TL;DR

This paper introduces a new modal logic of supervenience with a dyadic operator, providing semantics, axiomatizations, and comparisons to related logics, advancing the formal understanding of supervenience in philosophy.

Contribution

It presents a novel modal logic of supervenience, with a natural semantics, a sound proof system, and extends to more general logics, addressing open research questions.

Findings

The new logic is more expressive than the modal logic of agreement.

A bisimulation notion for the logic of agreement is proposed.

Axiomatizations of propositional logic of determinacy are provided for various frames.

Abstract

Supervenience is an important philosophical concept. In this paper, inspired by the supervenience-determined consequence relation and the semantics of agreement operator, we introduce a modal logic of supervenience, which has a dyadic operator of supervenience as a sole modality. The semantics of supervenience modality is very natural to correspond to the supervenience-determined consequence relation, in a quite similar way that the strict implication corresponds to the inference-determined consequence relation. We show that this new logic is more expressive than the modal logic of agreement, by proposing a notion of bisimulation for the latter logic. We provide a sound proof system for our new logic. We also lift on to more general logics of supervenience. Related to this, we compare propositional logic of determinacy and non-contingency in expressive powers, and give axiomatizations of propositional logic of determinacy over various classes of frames, thereby resolving an open research direction listed in~\cite[Sec.~8.2]{Gorankoetal:2016}. As a corollary, we also present an alternative axiomatization for propositional logic of determinacy over universal models. We conclude with a lot of future work.