Properties, Propositions and Conditionals

Section 1 discusses properties and propositions, and some of the motivation for an account in which property instantiation and propositional truth behave naively . Section 2 generalizes a standard Kripke construction for naive properties and propositions, in a language with modal operators but no conditionals. Whereas Kripke uses a 3-valued value space, the generalized account allows for a broad array of value spaces, including the unit interval [0,1]. This is put to use in Section 3, where I add to the language a conditional suitable for restricting quanti cation. The shift from a value space based on the mini-space {0, 1 2 , 1} to one based on the mini-space [0,1] leads to more satisfactory results than I was able to achieve in previous work: a vast variety of paradoxical sentences can now be treated very simply. In Section 4 I make a further addition to the language, a conditional modeled on the ordinary English conditional, paying particular attention to how it interacts with the restricted quanti er conditional. This is all done in the [0,1] framework, and two alternatives are considered for how the ordinary conditional is to be handled; one of them results from adding a tweak to a construction by Ross Brady. Section 5 discusses a further alternative, a standard relevance conditional (for the ordinary conditional, perhaps for use with a di erent quanti er-restricting conditional), but argues that it is not promising. Section 6 discusses the identity conditions of properties and propositions (again in the setting of a value space based on [0,1]); the issue of achieving naivety for coarse-grained properties is seen to be more complicated than some brief remarks in Field 2010 suggested, but a way to get a fair degree of coarse-grainedness is shown. 1. Naivety in a Theory of Properties and Propositions 1.1. Properties and Propositions. I take it that the main point of talking about propositions and properties is to provide a natural framework for talking about language and the mind. In the language case, this goes as follows: (EXP): A sentence (as used on a given occasion) is true (at possible or impossible world w) if and only if it expresses (on that occasion) a proposition that is true (at w). A formula (as used on a given occasion) is true of something o (at world w) if and only if it expresses (on that occasion) a property that is instantiated (at w) by o. Linguistic truth and satisfaction (the converse of truth-of) are reduced to propositional truth and property-instantiation, via the expression relation. In the mental case, e.g. of beliefs, it's similar. In the abundant sense in which there are highly unnatural properties like being either hairy or purple or a small ocelet or an African country. A sparse conception of properties, e.g. one con ned to basic physical properties, might well have another point, but such a conception will not be under discussion.