Historical/temporal necessities/possibilities, and a logical theory of them in branching time
Fengkui Ju, Woxuan Zhou
TL;DR
The paper develops a rigorous framework distinguishing six time-related modalities: four notions of necessity (strong/weak historical/temporal) and two notions of possibility (historical/temporal). It grounds these notions in an ontic-rule, branching-time semantics, introducing a last-moment and next-moment operator along with four necessity operators, and formalizes them in a contextualized model structure. An axiomatic system for the logic HTNP is provided and proven sound and complete, with detailed treatment of dualities, monotonicity, and coexistence properties, as well as a discussion of will-sentences and declaration dynamics. The work advances a linguistically informed, formally precise account of how agents' ontic rules shape what must, may, or should hold across alternative timelines, offering a foundation for future conditional and cross-linguistic investigations with potential applications in linguistics, philosophy of time, and formal semantics.
Abstract
In this paper, we do three kinds of work. First, we recognize four notions of necessity and two notions of possibility related to time flow, namely strong/weak historical/temporal necessities, as well as historical/temporal possibilities, which are motivated more from a linguistic perspective than from a philosophical one. Strong/weak historical necessities and historical possibility typically concern the possible futures of the present world, and strong/weak temporal necessities and temporal possibility concern possible timelines of alternatives of the present world. Second, we provide our approach to the six notions and present a logical theory of them in branching time. Our approach to the six notions is as follows. The agent has a system of ontic rules that determine expected timelines. She treats some ontic rules as undefeatable, determining accepted timelines. The domains of strong/weak historical necessities, respectively, consist of accepted and expected timelines passing through the present moment, and historical possibility is the dual of strong historical necessity. The domains of strong/weak temporal necessities, respectively, consist of accepted and expected timelines, and temporal possibility is the dual of strong temporal necessity. The logical theory has six operators: a last-moment operator, a next-moment operator, and four operators for the four notions of necessity. Formulas' evaluation contexts consist of a tree-like model representing a time flow, a context representing the agent's system of ontic rules, a timeline, and an instant. Third, we offer an axiomatic system for the logical theory and show its soundness and completeness.