In the following chapters, we shall attend to a number of logical issues that arise concerning the treatment of vague propositions.
The first order of business will be to outline a theory of vague propositions.
We will begin by observing that sets of possible worlds are not suitable for the modelling of vague propositions: the view that necessarily equivalent propositions are identical, along with the assumption that the vague supervenes on the precise, entails that all propositions are precise.
Drawing from the supervaluationist literature, an extremely natural alternative to sets of possible worlds is sets of ordered pairs consisting of worlds and precisifications. I shall argue in chapters 12, 14, and 15 that there are problems with these sorts of theories as well. The purpose of this chapter, then, is to develop a theory of vague propositions that does not rest on either of these identifications. The approach adopted here can be traced back to the likes of Frank Ramsey and Arthur Prior, and might naturally be called the propositions- first approach. Instead of attempting to analyse propositions in other terms—as sets of worlds or world-precisification pairs—we simply take propositions as primitive. Theoretical entities, such as possible worlds or precisifications, are in good standing only if they can be understood as certain sorts of constructions out of propositions, and not the other way around.In section 11.1, I begin by introducing a framework for discussing propositional fineness of grain: we introduce a binary individuation connective, A ≡ B, that can be read as saying that A is the same proposition as B. From this connective we can define a necessity operator that is extremely broad: one can show that it is broader than any other normal operator definable in the language, and that propositions that are necessarily equivalent according to that operator must be identical.
In section 11.2, I show that determinacy and metaphysical necessity are both independent sources of fineness of grain: propositions are not individuated by necessary equivalence or determinate equivalence. Moreover, due to higher-order vagueness, they cannot be individuated by any finite combinations of these operators either. I note that, as a result, the individuation connective and the broadest necessity cannot be straightforwardly defined from these operators.
In section 11.3, I outline my own theory of vague propositions, in which they are individuated by their role in thought. Finally, in section 11.4, we attempt to get clearer on the sorts of problems this theory is supposed to address.
class=32 style='margin-left:0cm;page-break-after:avoid'>11.1