A Logical Theory for Conditional Weak Ontic Necessity in Branching Time

Weak ontic necessity is the ontic necessity expressed by should or ought to. An example of it is I should be dead by now. A feature of this necessity is that whether it holds is irrelevant to whether its underlying proposition holds. This necessity essentially involves time. This paper presents a logic for conditional weak ontic necessity in branching time. The logic's language includes the next instant operator, the last instant operator, and the operator for conditional weak ontic necessity. Formulas are evaluated at tuples consisting of a tree-based model, a context, a timeline, and an instant. A context is a set of ordered ontic laws determining expected timelines. When evaluating conditional weak ontic necessity, we first update the context with the antecedent, then check whether the consequent holds with respect to the updated context. We discuss some consequences of the formalization and compare it to some related work. We study the logic's expressivity and axiomatize a special validity of it.

Automated summary

This paper presents a logic for conditional weak ontic necessity in branching time and discusses its consequences and comparisons with related works.