Abstract

This chapter is concerned with representing the semantics of natural language plurals and mass terms in property theory; a weak first-order theory of Truth, Propositions and Properties with fine-grained intensionality (Turner 1990, Turner 1992, Aczel 1980). The theory allows apparently coreferring items to corefer without inconsistency. This is achieved by using property modifiers which keep track of the property used to refer to a term, much like Landman’s roles (Landman 1989). We can thus predicate apparently contradictory properties of “the judge” and “the cleaner,” for example, even if they turn out to be the same individual. The same device can also be used to control distribution into mereological terms: when we say “the dirty water is liquid,” we can infer that those parts that are dirty water are liquid without inferring that the dirt is liquid. The theory shows how we can formalise some aspects of natural language semantics without being forced to make certain ontological commitments. This is achieved in part by adopting an axiomatic methodology. Axioms are proposed that are just strong enough to support intuitively acceptable inferences, whilst being weak enough for some ontological choices to be avoided (such as whether or not the extensions of mass terms should be homogeneous or atomic). The axioms are deliberately incomplete, just as in basic PT, where incomplete axioms are used to avoid the logical paradoxes. The axioms presented are deliberately too weak to say much about ”non-denoting’ definite descriptors. For example, we cannot erroneously prove that they are all equal. Neither can we prove that predication of such definites results in a proposition. This means that we cannot question the truth of sentences such as “the present king of France is bald.”