Dynamic reasoning without variables

A variable free notation for dynamic logic is proposed which takes its cue from De Bruijn's variable free notation for lambda calculus. De Bruijn indexing replaces variables by indices which indicate the distance to their binders. We propose to use reverse De Bruijn indexing, which works almost the same, only now the indices refer to the depth of the binding operator in the formula. The resulting system is analysed at length and applied to a new rational reconstruction of discourse representation theory. It is argued that the present system of dynamic logic without variables provides an explicit account of anaphoric context and yields new insight into the dynamics of anaphoric linking in reasoning. A calculus for dynamic reasoning with anaphora is presented and its soundness and completeness are established

Plural pronominal anaphora in context : dynamic aspects of quantification

This dissertation presents a formal model of plural pronominal discourse anaphora. It focuses on the question of how to model the contextual interpretation of plural pronouns. Specifically, it develops a non-representational dynamic semantics of quantification. A quantificational sentence has several sets associated with it, each of which is a potential antecedent for a subsequent plural pronoun. Three such sets play an important role in the thesis. If `D(A)(B)' is a quantificational sentence, where `D' is the determiner, `A' the restrictor and `B' the nuclear scope, then we say that `A' is the maximal set, A^B is the reference set and A-B is the complement set. It is argued that one of these sets, the complement set, is only accessible through inference. This is in constrast to the other sets, which are contextually introduced by quantificational structures as salient antecedents. The formal model of context and context change presented in this work deals with these sets made salient by quantification and, as such, excludes the possibility of reference to the complement set. A complicating factor in constructing the model is that quantificational structures do not only introduce (singular or plural) individuals in the discourse; they also introduce dependencies. That is, pronouns have access to correspondences between salient sets. Accordingly, the notion of context has to involve structure. Furthermore, it turns out that a more complex account of the accessibility of (dependent) antecedents is also called for. Consideration of these and other issues ultimately leads to the development of an incremental dynamic semantics of quantification. Specifically, a variable-free formalism is proposed where the introduction of an individual in discourse boils down to incrementing a stack with this individual, following Van Eijck's 2001 framework of incremental dynamics. It is shown that adjusting the framework to enable it to deal with plurals succesfully solves some serious problems of more standard approaches. Moreover, as far as empirical coverage is concerned, the formalism is comparable to the popular discourse representation theory of plural reference in Kamp and Reyle 1993. In contrast to this theory, however, the formalism proposed here is in no need of operations on representations in order to obtain certain kinds of anaphora. In fact, it is in no need of a level of representation at all. This study is of interest to researchers concerned with the formal semantics of quantification, plurality and anaphora, and to scholars interested in more general issues concerning the semantics and pragmatics of discourse anaphora

Contexts in Dynamic Predicate Logic

In this paper we introduce a notion of context for Groenendijk & Stokhof's Dynamic Predicate Logic DPL. We use these contexts to give a characterization of the relations on assignments that can be generated by composition from tests/conditions and random resettings in the case that we are working over an infnite domain. These relations are precisely the ones definable in DPL if we allow ourselves arbitrary tests as a starting point. We discuss some possible extensions of DPL and the way these extensions interact with our notion of context