arXiv:2208.11922v2 Announce Type: replace 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.