Dynamic squares

This paper examines various propositional logics in which the dynamic implication connective ⇒ (discussed in Groenendijk and Stokhof’s (1992) Dynamic Predicate Logic and Kamp’s (1981) Discourse Representation Theory) plays the central role. Our approach is modal: we view ⇒ as a binary modal operator in a two dimensional modal logic. Two dimensional modal logics are systems in which formulas are evaluated at pairs of points. Such logics provide a natural setting for studying ⇒, for we can think of the first point in the pair as an input state, and the second point as an output state. Although the basic idea is very simple, as soon as one begins to explore it one faces a number of choices. First, what type of two dimensional system should be employed? For example, should we look at ⇒ in the setting of ‘full square’ (or ‘classical’) two dimensional modal logics such as those studied by Segerberg (1980) and Venema (1991), or is a more general setting that uses ‘relativised squares’ (see van Benthem (1991) and Marx et al. (1992)) appropriate? In addition, are there connections with Boolean Modal Logics such as those studied by Gargov et al. (1987) and de Rijke (1992a)? These distinctions are all explained in the course of the paper; for now it suffices to note that the choices involved are rather interesting. Somewhat loosely, opting for a ‘square’ two dimensional treatment of ⇒ reflects most accurately Groenendijk and Stokhof’s original system; viewing ⇒ in a ‘relativised’ setting suggests modifications of the original system of both technical and philosophical interest; whereas making the simple notational change that leads to Boolean Modal Logic yields a natural system that reflects the computational origins of dynamic semantics. A second choice that confronts us concerns the Boolean connectives (that is, ∧, ∨, ¬, → and ↔). Should they be present or not? They are not present in Groenendijk and Stokhof’s system, and indeed if one adopts a ‘strongly dynamic’ perspective on natural language interpretation (such as that underlying the work of Vermeulen (1991, 1992) and Visser (1992a, 1992b)) this is a natural path to explore. On the other hand, in a number of publications van Benthem has emphasized the need to consider the interplay of static and dynamic notions (see van Benthem (1991), for example). Viewed from this perspective, to discard the Boolean connectives is to discard static logic, which results in a lop-sided system which doesn’t reflect the interesting interplay. This paper is more concerned with exploring options than making choices. By limiting our discussion to one relatively simple operator (namely ⇒) we will be able to discuss the various forms of two dimensionality, the differences between systems with and without the Boolean operators, and, perhaps most importantly, to make clear why we think the two