Conjunctive, Disjunctive, Negative Objects and Generalized Quantification

Abstract

This paper presents shadow theory, according to which, for every object of some type, σ – object in the broadest sense of the term, including truth values and functions – there is another object of the same type, the negative shadow of the object, and for every (finite or infinite) set of objects of a single type, σ, there are two other objects of the same type, the conjunctive shadow and the disjunctive shadow of the set. For instance, Adam has his negative shadow, not-Adam, and Adam and Betty have their conjunctive and disjunctive shadows, Adam-and-Betty and Adam-or-Betty. These are negative, conjunctive, and disjunctive objects in the sense in which they distribute over the objects of one type higher, $\sigma \rightarrow t$, where t is the type of truth values; thus, for instance, Adam-or-Betty is a professor if and only if Adam is a professor or Betty is a professor. The shadows of the same type are divided into infinitely many ranks. The usual infinite hierarchy of types, σ, $\sigma \rightarrow t$, $(\sigma \rightarrow t) \rightarrow t$, $((\sigma \rightarrow t) \rightarrow t) \rightarrow t$, $(((\sigma \rightarrow t) \rightarrow t) \rightarrow t) \rightarrow t$, …, are reducible to the bottom two types, σ and $\sigma \rightarrow t$, with the rank distinction among shadows; thus, shadows help simplify type theory. This paper also presents a deductive system based on shadow theory that can be used for the formalization of natural language inferences that involve compound noun phrases and quantifiers. Unlike in Montague’s theory or generalized quantifier theory, in this theory such expressions denote objects (shadows) of type e, the type of individuals. There is no essential difference between quantification over individuals (or objects in general) and denotations to them.