Commentary Combination of Tense and Modality

Richmond Thomason’s article Combinations of tense and modality provides the first clear exposition of T × W semantics. A T × W frame is defined in terms of a set T of times, a set W of worlds, a linear order < on T , and a set {≈t: t ∈ T} of equivalence relations on W. A T × W structure is defined by specifying a valuation on the frame that assigns truth-values to formulas relative to time-world pairs. Thus, a formula α turns out true or false at any pair (t, w), where t ∈ T and w ∈ W. The temporal operators behave as in linear tense logic. For example, Fα is true at (t, w) if and only if α is true at (tʹ, w) for some tʹ such that t < tʹ. In addition, a modal operator □ is defined in such a way that □α holds at (t, w) when α holds at (t, wʹ) for all wʹ such that w ≈t wʹ. 1 Thomasonʹs exposition is accompanied by some comments that are far from enthusiastic. About the systems based on T × W semantics, Thomason says that they “do not seem particularly interesting from a philosophical point of view”. According to him, the only interesting case is that in which the set {≈t: t ∈ T} is so defined that for any t, tʹ, if w ≈t wʹ and tʹ < t then w ≈tʹ wʹ. For in that case the relation ≈t obtains between w and wʹ when w and wʹ “share the same past up to and including tʹʹ. This way □ expresses historical necessity, the property that is usually indicated by words such as ‘settled’ or ‘inevitable’ . Moreover, not even in that case Thomason is satisfied with T × W semantics. He finds more congenial tree-like semantics, where a frame is defined in terms of a set T of times and a nonlinear order < on T that branches forward, that is, such that it may happen for distinct t, tʹ, tʹʹ ∈ T that t < tʹ and t < tʹʹ but neither tʹ < tʹʹ nor tʹʹ < tʹ. Thomason says “I like to think of possible worlds as overlapping, so that the same moment may have alternative futures”. And again: “Intuitions may differ, but to me the natural notion is that of a possible future -not that of a possible course of events” . Tomasonʹs attitude towards T × W semantics is largely shared. Although some technical work has been done to investigate the properties of systems based on T × W structures, in the current debate on time T × W semantics is either ignored or treated with nonchalanche. Tree-