Monotone Decreasing Quantifiers in a Scope-Free Logical Form

In Hobbs (1983) (henceforth, ITQ) and Hobbs (1985) (OP) I developed the outlines of an approach to semantic representation in which the logical form of an English sentence is a flat (i.e., scope-free) conjunction of existentially quantified, positive literals, with roughly one literal per morpheme. In this representation scheme the logical form of a sentence is vague with respect to quantifier-scoping decisions, and further information about scoping relations is encoded in the form of further existentially quantified positive literals. In the DIALOGIC system for syntactic analysis, developed in the early and middle 1980s, translations into such a logical form were implemented for a great majority of English syntactic constructions, and this system was used successfully in a number of applications. In Hobbs et al. (1993) (IA) my colleagues and I developed an approach to the interpretation of discourse in which to interpret a text is to find the least-cost abductive proof of the logical forms of the sentences of the text, essentially by back-chaining on mostly Horn-clause axioms in the knowledge base and making assumptions when necessary. One shortcoming of the proposal advanced in ITQ was in the treatment of monotone decreasing quantifiers, such as “few” and “no” (cf. Barwise and Cooper, 1981). A monotone increasing quantifier, like “most”, is “monotone increasing” beacuse when the predicate in the body of the quantified expression is made less restrictive, the truth value is preserved. Thus,