In this chapter, we shall be investigating the interaction of metaphysical modalities with vagueness.
As with the surrounding chapters, our main point of contrast will be the supervaluationist semantics outlined in section 12.1.2.
To address these issues, I shall adopt the framework of a modal propositional calculus with two primitive operators, □ and Δ, representing metaphysical necessity and determinacy respectively.
The standard way to model either of these operators in isolation is to specify a set of indices of some sort and an accessibility relation over those indices (see Kripke [84]). Each formula of the language is assigned a truth value relative to each index. A formula of the form □A or ΔA is true at an index iff A is true at every index accessible to it. To generalize this sort of semantics to deal with both operators at once, one equips a single set of indices with two accessibility relations, one for each operator.It would be nice to say something a bit more informative about what these indices are, and this is where matters become more complicated. In practice, we choose a set of indices that can model variation in the kinds of truths that the operator in question cares about. If we are studying the logic of a single modal necessity operator, the indices are usually worlds. If we are studying determinacy on its own, the indices are precisifications. If we are doing temporal logic, the indices are times. And so on. Propositions get identified with sets of worlds, precisifications, and times in each respective framework. Yet, I think it should be clear that none of these identifications are satisfactory. If propositions are sets of times, we run into trouble modelling contingency or borderlineness. If they’re sets of worlds, there are difficulties modelling temporary or borderline propositions.
And if they’re sets of precisifications, we run into trouble with contingent or temporary propositions.When we are considering logics that combine two or more of these operators, the indices cannot be either times, worlds, or precisifications, but something that is more fine-grained than any of these taken alone. A standard approach—the one adopted, for example, in the supervaluationist semantics we outlined in section 12.1.2—is to treat the indices for a multi-modal system as ordered tuples of the indices of each mono-modal system (see MarxandVenema [101], for example). Inmodaltense logic, for example, when we evaluate a tense operator at a world-time pair, we operate on the time coordinate and leave the world alone, and when we apply a modal operator, we operate on the world coordinate and leave the time alone. However, it is worth emphasizing that this approach encodes some fairly strong assumptions about the logical interaction of the two kinds of operators. In this framework, indices have two degrees of freedom, only one of which each operator can affect. This means that the two operators exhibit a kind of independence from one another that gives rise to a distinctive and controversial combined logic of vagueness and modality.
The strategy I have been pursuing in the last few chapters has been to take propositions as primitive, and treat the indices as maximally strong consistent propositions.1 In this alternative setting, there is no guarantee that our indices can be decomposed into ordered pairs of entities in such a way that necessity corresponds to variance in one of the coordinates and determinacy to variance in the other. In this chapter, we shall investigate some of the logical freedom that this approach affords us.
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