A Modal Logic of Supervenience

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 [11, Sec. 8.2]. 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.

Logics for Propositional Determinacy and Independence

“…We do not want to claim that the natural language word or has always a unique interpretation. 9 However, L D works quite nicely in the above examples, as we will next demonstrate.…”

“…Studying D over different classes of Kripke models provides an interesting research direction. In fact, the recent study [Fan16] has already taken up that direction.…”

“…The two papers have at least partially been written as a response to [GK16]. The first one of them is [Fan16] which develops a formal modal logic of supervenience and also addresses some research questions raised in [GK16] and reiterated here in Section 8. The other one is [Hum17] which explores, inter alia, connections between supervenience and dependence and discusses in detail some aspects of [GK16].…”

“…We thank them for their comments and discussion. In particular, Humberstone subsequently commented in depth some aspects of our work in [Hum17] and Jie Fan produced the manuscript [Fan16].…”