Eliminating the Universe

The following sections are included:

• Dedication
• Reader Alert
• Reading this Book
• Contents

We begin with a brief discussion of the insightful and prescient vision of the relation between language and mind that George Boole (1815–1864) presents in his 1854 magnum opus:

In this chapter, I provide the motivation for taking standard logics — Sentential Logic, First Order Logic, Higher Order Logic — as models of linguistic analysis. Our presentation is informal, designed to be intelligible and hopefully interesting to scholars who are not familiar with the mathematical perspectives that logic brings to linguistic analysis. The later chapters presuppose some basic mathematical proficiency, manipulating functions and relations.

We are concerned first in this chapter to present a descriptively oriented overview of expressible types of nominal quantifier phrases in English. We supply some modest information concerning the distribution of these types in other languages. Our English inventory is largely drawn from Keenan (1996), Keenan and Stavi (1986), Keenan and Moss (1985), and Keenan and Westerståhl (1997). Our cross language claims are based largely on KP (Keenan and Paperno 2012) and PK (Paperno and Keenan 2017), which profited from Matthewson (2008), Suihkonen (2007), Suihkonen and Solovyev (2013), and Bach et al. (1995).

This chapter focuses on generalized quantifiers of type 1. They are expressed by DPs which combine with P_n+1s to form P_ns.

Here we discuss briefly, two entailment paradigms naturally involving proportionality quantifiers (Keenan 2005, 2008). I had originally wondered if these paradigms might not be characteristic of proportionality quantifiers, but Peters and Westerståhl (2006) generalize insightfully, the crucial paradigm showing that it applies to a more general class of quantifiers, though initially it is awkward to find properly non-proportional examples that are naturally expressible. We provide information concerning the boolean structure of midpoint quantifiers which enable us to find a few others.

We ended Chapter 4 with a generalization concerning the relation between classical syntactic structure, possessive DPs, and their boolean semantic properties. Here, we refine and extend that generalization and then present two further generalizations. In all cases, the generalizations illustrate the utility of boolean concepts (but we are not remotely suggesting that there are no other useful and enlightening semantic concepts).

Boolean preliminaries Here we formulate some mathematical generalizations about the classes of quantifiers presented earlier. For this we need some boolean background. For linguistic naturalness we present boolean structure as relational lattices, interdefinable with boolean algebras.

Can we characterize syntactic categories of English expressions in terms of the boolean structure of the sets in which they denote? Surely not, at least not completely. There is much in natural language semantics that is not specifically boolean. But we can I think make some inroads to see how some categories are distinguished in boolean terms. This may help us get a grip on the possible semantic architecture of grammatical categories in natural language. The relatively few grammatical categories we have investigated do present diverse boolean structures. And possibly the many categories we do not study here can be at least partially understood as functions or refinements of those we can say something substantive about.

In this chapter we study some as yet unexpected properties of the Noun arguments of Dets, e.g. the logical role of poet in some/all/most poets or the pair (student, teacher) in more students than teachers. To be sure the Noun argument(s) specify the conservativity domain — the set(s) of things we can limit ourselves to for purposes of the DP(s) at hand. But just how essential is this restriction? When the Det is existential (intersective) or universal (co-intersective) the Noun argument can be eliminated in favor of a neutral expression denoting the entire universe and we quantify over all the entities in the model.

We are initially concerned here with how to characterize the “logical” character of quantifiers like some, all, most, most but not all, etc. compared with “non-logical” Dets such as John’s two and more male than female. The issue may seem “philosophical” but we see that theoretical linguistics builds in a kind of logical/non-logical distinction into syntactic theories. We conclude by extending our notion of semantic invariance to syntax and speculate concerning the relation between syntactic and semantic invariants.

Chapter 4 concerned, in effect, type (2) functions built by composing two type (1) functions. So type (2) functions express properties of binary relations (and more generally map k+2-ary relations to k-ary ones). We see here that very few type (2) functions are compositions of type (1) functions (Keenan 1987c). We explore empirically other means English provides of expressing type (2) functions. Often in English, they are superficially similar to pairs of type (1) DPs, but on close inspection, identity fails.

This chapter is adumbrated in Keenan (1982) and builds on Keenan (2015), from which the semantic problem we address is taken.

The following sections are included:

• Bibliography
• Author Index
• Subject Index