Modal and Temporal Logics for Abstract Space-time Structures

In the 4th century BC, the Greek philosopher Diodoros Chronos gave a temporal definition of necessity. Because it connects modality and temporality, this definition is of interest to philosophers working within branching time or branching spacetime models. This definition of necessity can be formalized and treated within a logical framework. We give a survey of the several known modal and temporal logics of abstract space-time structures based on the real numbers and the integers, considering three different accessibility relations between spatio-temporal points. 1. A Temporal Notion of Necessity Of the many different interpretations which can be given to the concept of necessity (logical necessity, physical necessity, deontic necessity, etc.), one of the most interesting from the formal point of view is the definition of necessity put forward by the Stoic logician Diodoros Chronos. Diodoros, a member of the Dialectical School who taught in Athens and Alexandria around 315–284 BC, is said to have defined the possible as that which either is or will be true, and the necessary as that which is true and will never be false.1 This definition of necessity was first analyzed formally by Prior, in Prior (1955) and Prior (1967), who provided an axiomatization of Diodorean necessity within the context of linear time structures. But it is within the context of branching time (or branching-space time) that Diodorean necessity has generated the most interest. This interest comes from two angles: The logicians who study abstract structures find the problem of axiomatizing the logics of time and necessity for branching time models an interesting theoretical question, and the philosophers who use branching time structures to model theories of agency and knowledge find the logical systems of these structures useful for their applicability to certain problems.