Ontic necessities are those modalities universally quantifying over domains of ontic possibilities, whose ``existence'' is independent of our knowledge. An ontic necessity, called the weak ontic necessity, causes interesting questions. An example for it is ``I should be dead by now''. A feature of this necessity is whether it holds at a state has nothing to do with whether its prejacent holds at the state. Is there a weak epistemic necessity expressed by ``should''? Is there a strong ontic necessity expressed by ``must''? How do we make sense of the strong and weak ontic necessities formally? In this paper, we do the following work. Firstly, we recognize strong/weak ontic/epistemic necessities and give our general ideas about them. Secondly, we present a complete logical theory for the strong and weak ontic necessities in branching time. This theory is based on the following approach. The weak ontic necessity quantifies over a domain of expected timelines, determined by the agent's system of ontic rules. The strong ontic necessity quantifies over a domain of accepted timelines, determined by undefeatable ontic rules.

Weak ontic necessity is the ontic necessity expressed by ``should'' or ``ought to'' in English. An example of it is ``I should be dead by now''. A feature of this necessity is whether it holds does not have anything to do with whether its prejacent holds. In this paper, we present a logical theory for conditional weak ontic necessity based on context update. A context is a set of ordered defaults, determining expected possible states of the present world. Sentences are evaluated with respect to contexts. When evaluating the conditional weak ontic necessity with respect to a context, we first update the context with the antecedent, then check whether the consequent holds with respect to the updated context. The logic is complete. Our theory combines premise semantics and update semantics for conditionals.