Can We Get By without Semantic Indecision?

Let us begin by asking what would happen if we attempted to assimilate these cases to our theory of vagueness. The relevant move, in each case, would be to insist that the linguistic items in question are not semantically undecided about which of a number of determinate propositions to express.

How might we apply our framework to each of the above examples? The first example looks like it lends itself naturally to an analysis in terms of symmetries. The fact that there is an automorphism of the complex numbers that maps i to — i is effectively a symmetry of the complex numbers—one that preserves all mathematical relations between complex numbers.

If we think of propositions as being determined by sets of indices, and that the indeterminacy described above is a source of fine-grainedness, then the picture will be as follows. All the indices will agree that there are two square roots of — 1, although according to some i and / will be identified with the same root (and thus — i and \ will be identified with the other root), and according to others i and \ will be identified with the same root (and — i and / to the other).

Given this picture, an automorphism of propositions is induced by the permutation of indices that maps an index in which i and \ are identified to the index where i and / are identified but otherwise agrees about the other facts.

In our second example we stipulated that a number is nice* if it is less than 15 and not nice* if it is greater. To see how this can modelled using symmetries, let us suppose that these stipulations allowed us to refer to an indeterminate property—the property of being nice*—which is neither identical to the property of being less than 15 nor the property of being greater than 15. If I can do this, then it seems that I could now introduce, via exactly the same method, a name for another vague property, being nice*, as follows:

A number is nice* iff it is either less than 15, or equal to 15 and not nice*.

Notice that a number is nice* if it is less than 15, it is not nice* if it is greater than 15, and 15 is nice* if and only if 15 is not nice*.

A number is nice* iff it is either less than 15, or equal to 15 and not nice*.

Now we can easily see that the proposition that 15 is nice* is distinct from the proposition that 15 is nice*: the truth of one proposition implies the falsity of the other and at least one of them must be true, so they cannot be identical. Similarly, neither proposition is identical to the proposition that 15 is less than or equal to 15, or the proposition that 15 is less than or equal to 14, since we presumably don't know whether 15 is nice* or nice*, but we surely know that 15 is less than or equal to 15 but not less than or equal to 14. That each proposition implies the negation of the other seems to be the only important difference between them. Anything else that we might think or say to distinguish them would involve the new propositions in some way or another. The proposition that 15 is nice* and the proposition that 15 is nice* can't be distinguished in terms of the role they play in our mental lives. They bear exactly the same evidential relations to other propositions and cannot be treated asymmetrically by a rational agent within her system of beliefs and preferences.

The process by which we expand the space of propositions to include the new proposition that 15 is nice* can be modelled using a well-known mathematical method. Let us suppose that we begin with a complete Boolean algebra of propos­itions, B, containing no indeterminate or vague propositions (let's suppose they are isomorphic to sets of possible worlds), and we want to see what happens when we ‘introduce' a new indeterminate proposition p (that 15 is nice*). The mathematical way to do this is to consider the Boolean algebra, B[p], freely generated by adding p to B.

Freely generated finite extensions of a complete Boolean algebra, like B[p], have a particularly simple characterization in terms of automorphisms. Let G be the set of automorphisms of B [p] that fix B. Then, the set of elements of B [p] that are fixed by every element of G is just B itself. (More generally, if A is a subalgebra of B, then B is a Galois extension of A iff A is the set of elements of B fixed by all automorphisms that fix A.^[191])

The set of automorphisms of B [p] that fix B are plausibly symmetries in the sense of chapter 13 (they preserve rational probabilities and values). Indeed, this group is particularly simple—it only contains two automorphisms: the identity automorphism, and an automorphism that switches the proposition that 15 is nice* with the propos­ition that 15 is nice*, but leaves logically independent propositions alone.

Applying our definitions of precision and vagueness, we get that neither the proposition that 15 is nice* nor the proposition that / is positive are precise. Both of these can be mapped, via a symmetry, to a distinct proposition and so are not fixed by all symmetries. Moreover, both propositions are indeterminate. The claim that it's determinate that p is just the conjunction of all propositions p is mapped to under symmetries. Thus, the claim that it's determinate that 15 is nice* is just the conjunctive proposition that 15 is both nice* and nice*, which is always false. Analogously, the claim that it’s determinate that i = / ends up entailing both i = / and i = \ and so must be false, since / = \.

At any rate, insofar as our formal framework goes, the first two examples of semantic indecision lend themselves quite naturally to our model theory that is spelt out in terms of symmetries.

The first problem is that it seems that examples analogous to the above examples suggest that there are actually a large number of indeterminate propositions mapped by symmetries to the proposition that 15 is nice* or that / is positive. For instance, suppose that instead of two splintered mathematical communities there were 20 who each introduce their own names for the roots of —1. Are we to think that once the 20 communities reintegrate, there will be 2²⁰ propositions expressed by the various combinations of identity statements between the newly introduced names? If so, how is it that these propositions came about?

Note that this situation is unlike the evidential roles that we took to be sufficient for the postulation of a vague proposition in chapter 6—these were determined by the kinds of effects inexact evidence can have on our credences in the precise, and so there are no analogous arguments for multiplying vague propositions in the same way.

Anyway, the crucial question for this picture is to explain how propositions can be multiplied so easily. One could say that the propositions only existed because of the linguistic practices of the 20 different communities, and had there only been two communities there would have only been 2 relevant propositions. On this story, the existence of propositions is highly dependent on the contingent practices of linguistic communities. An account of propositions along these lines, however, would not be friendly to the non-linguistic theory of propositions I have been endorsing here— propositions would be language-dependent in a way that strips the theory of its primary advantages.

A better thing to say, then, is that the propositions would have existed whether or not the communities had introduced names for the roots of — 1 in the way they did.

While a non-linguistic account of these examples is not hopeless, both the examples discussed above will find a natural analysis in terms of genuine semantic indecision which I shall sketch later. What of examples involving scientific terms? In broad outline a non-linguistic account of this example would posit, in addition to the properties of proper mass and relativistic mass, a third vague property, Newtonian mass. The value of an object's Newtonian mass is always either its proper mass or its relativistic mass, but when these differ it is always indeterminate which.

A couple of other features of Newtonian mass seem reasonable. Firstly, it is natural to think that there is a penumbral connection between the Newtonian mass of any two objects: determinately, if the Newtonian mass of the first object is its proper mass then the Newtonian mass of the second is its proper mass, and similarly for relativistic masses. Secondly, it is also natural to think, given that Newtonian mass turned out not to be a fundamental quantity, that the values of the Newtonian masses of objects supervene on the values of the more fundamental quantities of proper and relativistic mass: either it's necessary that the Newtonian mass of every object is identical to its proper mass or it's necessary that the Newtonian mass of every object is its relativistic mass. Although Newtonian mass is necessarily coextensive with another more fundamental quantity, it is indeterminate which of these quantities it is coextensive with, and, therefore, there is ignorance about which of these quantities Newtonian mass coincides with. Thus, Newtonian mass might have a distinctive conceptual role which neither proper mass nor relativistic mass has: perhaps someone who was fairly, if not fully, confident that Newtonian physics is the correct description of reality, and therefore also pretty confident that Newtonian, proper, and relativistic mass are coextensive, might nonetheless treat the first differently from either of the other two, conditional on the supposition that the theory of relativity is true.

At any rate, on this way of understanding the example, it was determinate that Newton was neither giving a theory of proper mass nor of relativistic mass—he was giving a theory of Newtonian mass. As we later discovered, some of the consequences of his theory are determinately false: they neither hold of proper nor relativistic mass and thus do not hold of Newtonian mass. Other consequences are indeterminate: they would have been true, or approximately true, if he had been talking about proper mass and not relativistic mass, or vice versa. When we eventually did discover the nature of relativity, our use of the word ‘mass' changed. It no longer referred to Newtonian mass, and we introduced disambiguations of the word ‘mass' to refer to the two properties which we believe do exist.

The picture described seems at least coherent, but is it plausible? One source of implausibility is the idea that physicists frequently find themselves theorizing about vague properties, instead of the perfectly natural and fundamental properties we typically take them to be talking about. Presumably, this issue isn't confined just to previous physical theories we know to be false—it also applies to our present physical theories which, for all we know, are false. Presumably, the most we know about our present theories is that they are approximately true, but this allows for the possibility that our present physical concepts refer to indeterminate concepts much like the notion of ‘mass' did amongst physicists before the theory of relativity. Note, however, that if physicists working with false theories are theorizing about properties at all, it is independently natural to think that they are not theorizing about fundamental properties. Note also that we have already cast doubt on the hypothesis that physicists usually find themselves theorizing about precise properties. In section 12.2, for example, I argued that even physical properties like being an electron are vague.

The story here is also subject to the multiplicity worry raised in connection with the first two examples. To prevent the number of vague properties depending on the number of incorrect physical theories proposed throughout history, we would have to posit an abundance of vague properties to account for all of the possible incorrect physical concepts that could be introduced by false physical theories. Perhaps in this case thereismorehopeofdelineating afixedclass ofconceptual roles, andarguing that there is a property for each of those conceptual roles, thus answering the arbitrariness worry. However, I do not know whether the mechanism by which we posit such properties seems different enough from the case of vagueness to warrant treating this example in a different way.

17.3