Quantiiers and Partiality

1 Varieties of Partiality Quantiication can involve partiality in several ways. Quantiiers loaded with presuppositions give rise to partiality by introducing truth value gaps (if there are no A, and the quantiier all + carries an existence presupposi-tion, then all + A are B will lack a truth value). The study of quantiiers in contexts of incomplete information involves a quite diierent kind of partial-ity. This paper investigates the behaviour of binary quantiiers in settings of incomplete information, i.e., in partial models. The framework to handle informational partiality that is presented in this paper is relevant for the behaviour of quantiiers in the natural language semantics of propositional attitudes and perception reports, for theories of vagueness in natural language and for semantic accounts of natural language fragments containing a truth predicate. First, the constraints on binary relations on a universe that make these relations qualify for the honoriic title quantiicational are generalized to the partial case. Special consideration is given to quantiiers deened via supervaluation from quantiiers on total models. Next, properties of partial quantiiers are studied, with particular emphasis on behaviour under growth of information and growth of domain. Quantiier interpretations for natural language determiners like all, some, most, exactly two, at least three and at most ve, pick out binary relations on sets of individuals, on arbitrary universes E. Notation: Q E AB. We call A the restriction of the quantiier and B its body. Many quantiier relations satisfy the familiar constraints EXT, CONS and ISOM (cf. the introduction to this volume). We will put generalized quantiiers in a three valued setting. In such a setting, quantiiers themselves can introduce truth value gaps, but they can also havèthree valued sets' as arguments. Our framework deals with both these kinds of partiality. Below, in Section 2, new versions of EXT, CONS and ISOM will be proposed that also cover the cases where only partial information concerning the extensions of the body and the restriction of a quantiier is available. The deenite description the n A are B can be viewed as a partial quan-tiier that is undeened if the number of As is not equal to n, and that behaves like the universal quantiier otherwise (see ?). Consider the partial quantiier in (1). (1) The four A are B.