Combinations of tense and deontic modality: On the \(R_{t}\) approach to temporal logic with historical necessity and conditional obligation (Q2568340)

This paper presents three hierarchies of `two-dimensional temporal logics with explicit realization operators'. The first provides the basic temporal framework, in which the two dimensions are one for moments in time and the other for histories or worlds over times. The realization operator, \(R_{th}A\), expresses that \(A\) is realized at time \(t\) in history \(h\). This allows reference to times without indexing other modal operators. The second level adds a monadic operator \(N\) for historical necessity or inevitability (truth in all histories), and the third adds dyadic deontic operators for conditional obligation and permission as well as a set of `frame constants' that represent levels of value or perfection. For all three levels, logics are defined both axiomatically and semantically, and strong and weak soundness and completeness theorems are proved.