INTRODUCTION TO PART II

In this second part, one of the central issues will be the question of whether or not the intuitive notion of Specified Quantity of A ([+sqa]) can be captured by the Theory of Generalized Quantification. As it is possible to circumscribe this notion quite precisely at the informal level, it should be possible to give it a formal interpretation in terms of existing or new notions of this theory, where it may serve its purpose as explaining part of the compositional machinery involved in the interpretation of sentences expressing terminative aspect. The same applies mutatis mutandis to the notion of Unspecified Quantity ([-sqa]).

It may be helpful to begin the search for a formal articulation of the notion of (Un-) Specified Quantity of A by having a closer look at its informal characterization in Verkuyl (1972):

Introduction

Aspectual classes appear to play an important role in the linguistic and philosophical literature these days, judging from the many references to Vendler's ‘Verbs and Times’. In the literature on aspect, it is hard to find a contribution that does not refer to this philosophical essay.

What did Vendler actually propose? The question is of some importance because it is a rather striking feature of his analysis that he kept the quadri partition at the lexical level. He used the term ‘term’ to denote verbs, even though he seemed to be aware of the fact that his categories are complex in the sense that, for example, the direct object appears to co-determine whether or not a transitive verb belongs to one of the four categories. His inclination to stay at the lexical level enforces the idea that what he really did was to propose ontological categories: if knowledge of the world and knowledge of a language tie up intimately at some place, they do that at the lexical level and not so much at the structural level (Putnam 1978; Dowty 1979; Partee 1980).

Vendler's classification, however, turned out to relate to linguistic work on aspect. Some of his criteria were well known in the literature on the imperfective and perfective aspect in Slavic languages. As said earlier, the implication that each of his lexical classes can be used in the theory of aspectuality actually constitutes a setback.

The idea that aspectuality, roughly, the property which makes it possible for a sentence to signal whether or not it pertains to something bounded, can be characterized compositionally by a system of rules of interpretation has been around for some time now. However, its articulation appears to be difficult. Not only is it an area of investigation in which several disciplines are deeply involved – linguistics, logic, cognitive science, computer science and philosophy – the phenomena are also very hard to capture because the systematic formal study of temporality as expressed by sentences of natural languages is relatively new. Of course, there is a tradition in which at least a lot of valuable data and generalizations have been made available. However, when it comes to making a theory, soon one is bound to recognize that a lot of preparatory work is to be done before one can even think about theory formation. The picture is also complicated by the fact that the formal semantic study of discourse structure has developed very rapidly in the past ten years, so that all of a sudden all sorts of relatively new factors are drawn into an area of investigation whose scope was mostly restricted to sentences only. A lot of sorting out has to be done against the currents of shifting fashions.

Introduction

There is a line of analysis of NP-quantification in GQT in which NPs are taken to be of type <<<,e,t>,t>,t>. Scha (1981) and Verkuyl (1981) are among the earliest publications in this tradition, both building further on Bartsch (1973) and Bennett (1975). The main goal of Verkuyl (1981) was to build a sort of bridge between the framework of Montague Grammar and the Chomskian pre- Government-Binding X-bar Syntax. Its descriptive purpose was not to find out what [+sqa] stands for, rather it was focussed on the treatment of numerals and quantifiers. Yet, somewhat to my own surprise, some of its results turn out to bear directly on the issue of quantification in aspect construal, especially as the sort of NP-representations proposed (recall: at the <<<e,t>,t>,t>-level) offer a quite workable point of departure for more sophisticated extensions in the area of temporality. Therefore, a short summary of my 1981 grammar is in order. The present chapter is divided into two sections: section 5.1 gives a description of an X-bar grammar à la Montague, in which syntactically and semantically two structural positions are available for the Standard GQT-determiners, whereas section 5.2 shows that this grammar has certain concrete things to say and solutions to offer about the Russell-Strawson controversy. Apart from the contribution it may have to the solution of the philosophical controversy, it is also relevant for the question of whether or not it is necessary to exclude terminative aspect from occurring in tautological sentences.

INTRODUCTION TO PART III

In this part, the interaction between temporal structure and atemporal structure in sentences expressing terminative and durative aspect will be under investigation. In particular, our compositional aspectual theory will be set out to give an explanation for why it is that the unbounded progress expressed by [+add to]-verbs, for example walk, eat and play, can be brought to an end by the [+sqa]-nature of their arguments, and it will try to give a formal account of how complex sentential meanings are formed out of the temporal and atemporal meanings of their constituents.

Some would even like to say that an atemporal grammar like plug in chapter 7.3 might be sufficient if one could agree upon the simple observation that the number of elements of a partition never exceeds the cardinality of the set underlying it. However, this would amount to refusing to take into account the role of temporal structure. It might take Part III to convince those who want to stick to atemporal representations. It is only after going through four separate issues crucially involved in theory formation about temporal structure of sentences that in chapter 13 a formal account can be given because its ingredients and the choices behind them are built up stepwise in the preceding sections. In view of this, the organization of Part III is as follows.

Introduction

An important issue emerging from the discussion so far is the status of events. In the present aspectual theory, events – here taken as eventualities in Bach's sense – are not primitive: they are construed. In the above extrapolation of Jackendoff's theory, events result from a mapping of Things and Paths into the set of events. There are also theories in which they are taken as primitives.

Let me underscore here the pointlessness of a debate about whether or not events are primitive individuals or complex dividuals. That matter can be settled by belief, or maybe by purely philosophical arguments. However, a debate about events as possible denotations of expressions in a natural language is meaningful if the debate is focussed on the question of whether or not they explain things that cannot be explained otherwise. I would like to operate on this assumption, which is another way of saying that it makes no sense to explain the notion of terminativity in sentences like (47), (48), (50) and (52) in terms of expressions like ∃(…e…), where e is a variable over eventualities or events. Of course, it is very handy to speak about events or eventualities but this does not mean that one is committed to their existence. It is just handy to have terms for things whose nature we do not understand well and whose existence we cannot be sure of.

Introduction

The visual connotation of the English term aspect is surpassed in Slavistic circles by that of the term vid, which means ‘appearance’, or ‘view’. The need to formalize exactly the representations of our basic stock of examples, such as the sentences in (47)–(52), cannot but lead into a position in which a visual connotation is appropriate, in particular by an appeal to the notion of perspective. This is partly due to the technical decision to make the extensional logic underlying the plug-representations partly intensional. In particular, the notion world in Intensional Logic (IL) makes it possible to admit intensional objects into the domain: they are defined as functions from worlds to entities of some sort, whose extension can be obtained by applying this function at a given world-index w. It is not difficult and even illuminating to make this characterization of these abstract semantic objects concrete by imagining oneself to have a look from w into the domain where this extension is present.

In my view, aspect construal should indeed be analysed in terms of perspective, in the sense that indices are playing a crucial role in determining sentential aspect, where index is the neutral term for semantic entities on which other semantic entities are made dependent. To the set of indices belong numbers, worlds, models, intervals, situations, points, etc. Perspective comes in at the moment at which it depends on the choice of an index whether or not a sentence is used to express terminative aspect or durative aspect.

Introduction

Chapter 13 has three goals. The first one is comparable with the goal of chapter 3.1: to introduce machinery and explain some basic technical type-logical notions for those readers who may find it convenient to go through it in order to refresh their logical memory. This will be done in section 13.1. This goal has been combined with the need to justify each step formally. This is necessary because the EL-logic which has been employed up to now will be extended with some intensional rules. In many formal semantic analyses, intensionality is given by making use of the rules of IL, the intensional logic used in Montague's PTQ. However, the plug-grammar of chapter 7.3 will only be partially intensionalized and not by IL-rules but rather by rules of the system called Ty2. As IL and Ty2 can be translated into each other, I will first introduce the relevant EL-rules, and then make a transition to the more generally formulated two-sorted logic Ty2, which has become increasingly standard because it turns out to be an easier language to work with, especially in the GQT direct representation of plug. In section 13.2 the second goal becomes visible: making a weak intensional version of plug on the basis of Ty2, that is, a version which is intensionally weaker than Ty2. This is to say that we do not need the full intensional capacity of IL and Ty2 to deal with the varied forms of index-dependency discussed in chapter 12.

Introduction

This chapter contains a brief survey of the formal machinery employed in GQT, a theory which is generally considered as a further development of the treatment of quantification in Montague's PTQ. There is, however, a technically important difference between Montague's treatment of NPs and the GQT-treatment: the former uses the indirect method of translating expressions of natural language into type-logical expressions, which are then semantically interpreted as set-theoretical objects, whereas the latter uses the direct method. On the former, an English sentence like All children came in is translated into a logical formula, say f(All children came in), which is then interpreted. The essence of the direct approach is that all, children and came_in are directly taken as denoting semantic objects. This means that the syntactic structure of All children came in is mapped into constituents which are directly related to semantic objects of the domain. These objects belong to distinguished types (individuals, sets, functions, relations, etc.).

As its machinery will play an important role throughout the present study, I shall first introduce a type-logical language EL which itself is an extension of the standard predicate first-order logic including prepositional logic. EL will be used as a formal logical language representing (the denotations of) expressions of natural language. It will play a role throughout the following chapters as a sort of checkpoint with respect to which elaborations of the system can be measured.

Introduction

In this chapter, three issues are central. The first one follows naturally from the line of argument in chapter 9. It will be argued in section 10.1 that the notion of homogeneity may apply to verbs but that this notion then is no longer of interest for aspectual theory. In terms of Taylor's sultana metaphor: the VP walk to Rome should not bear comparison with a fruit-cake, nor the VP walk in John walked with gold, or cake for that matter. By dealing with the sultana-issue at the level of lexical knowledge, one clears the way for a different approach in which the verb and its complement have different contributions to make.

This opens up the second issue: the verb induces temporal structure to which the information associated with its complement systematically relates. As a result, the notion of Path emerges: the fusion of temporal and atemporal structure which can be bounded (terminative) or unbounded (durative). The contribution of verbs expressing change to the construction of temporal structure will be discussed in section 10.2 as a first leg up to formalization in chapter 13.

The third issue is built on top of the others: the question is what the terminative (47), (48), (50) and (51) have in common, and what the durative sentences (49) and (52). For (48) and (51) this question can be made more concrete: if there is a mapping between the temporal structure induced by eat and the set of three sandwiches, what is the mapping relation between the walk-structure and the denotation of the PP-complement (including the preposition)?

Introduction

Can the notion of [±sqa] be captured in terms of generalized quantification? This question was raised in Verkuyl (1987a) concerning what can be called Standard GQT. This subsumes all work on generalized quantification based on the assumption that NPs are of type <<e,t>,t>, that is, that they denote collections of sets. The answer given was roughly: a determiner Det of an NP yields a [±sq]-NP if and only if the intersection of the two sets standing in the Detrelation is bounded, that is, finite and non-empty. This answer is not satisfactory: (a) it does not cover the question of whether or not the [+sqa]-property is related to any of the well known constraints on quantifiers, such as monotonicity, strength, properness, and so on; (b) there are apparent counterexamples which have not yet been dealt with; and (c) the conditions under which an NP may express [-sqa]-information were not sufficiently described.

The characterization of [+sqa] at the <<e,t>,t>-level will be summarized in section 4.1. It will be made clear that there is an interesting difference between the so-called mathematical meanings of at least n and at most n on the one hand and their meaning in sentences expressing temporal structure. In section 4.2, attention will be paid to some of the major constraints on Generalized Quantifiers having emerged in the eighties, in order to see in section 4.3 whether or not [+sqa] may be associated with one of them.