3.7 The Reduction, Formal
3.7.3 Downsizing Our Goals
As both the Law of Higher Order Vagueness and the Ineluctability of Vague- ness are presently formulated, we are just asking too much from our dynamic semantics in demanding that it derive these generalizations. Simply put, given the way we’ve phrased them, both of our targeted generalizations are false. The Ineluctability of Vagueness, for example, literally states that in every con- text, every standard-sensitive predicate has borderline cases. But, that’s just not true. As I look around me, there are currently no borderline cases for the standard-sensitive predicate “blue”; everything I can see is either clearly blue or clearly not blue. What we’ve fudged in our formulation of this generalization is that, intuitively, we want it to state that in every context there areimaginable
borderline cases for any standard-sensitive predicate. The reason for our initial, sloppy formulation was that we hadn’t yet offered any formal definition of what an “imaginable borderline case” is. Similarly, the Law of Higher Order Vague- ness is currently stated too strongly. Take any standard-sensitive predicate you like. There certainly aren’t, as I look around my office, objects to witnessevery
vagueness order for that predicate. Again, what we would like our semantics to capture is the weaker claim that in every context, for every standard-sensitive predicateP and natural number n≥1, there is someimaginable objectαsuch that InP(α).
How, though, can we represent there being in a context an imaginable bor- derline case forPor an imaginable witness forInPwithout terribly complicating
our theory of contexts? Let us start by extending our semantics of standard- sensitive predicates in such a way that they may also range over their own degrees.
Definition 7.13 Let P be a standard-sensitive predicate, hOP, <
Pi be its
dense ordering of degrees, α, δ ∈ OP, and hK,hw, δii be a context such that
K = (W0, R0, V0) where W0 ⊆ W ×OP. hK,hw, δii ° P(α) if and only if
α >P δ.
Definition 7.13 could be viewed as an analysis of locutions such as “Six feet is tall” or “A BMI of 30 would definitely be fat.” We would state, for example, that at a particular context “Six feet is tall” is true if and only if six feet happens to be taller than every degree available at the context. Of course, one may prefer a more sophisticated analysis of such locutions, perhaps one in which they are taken as “shorthand” for more complex propositions quantifying over the domain of individuals, but for our purposes here we choose to keep the analysis simpler.
Now, with Definition 7.13 in place, let us introduce the notion of a “border- line degree.”
Definition 7.14 (Borderline Degree) LetP be a standard-sensitive pred- icate,OP be its densely ordered set of degrees,α, δ∈OP, andhK,hw, δiibe a
context such that K = (W0, R0, V0) whereW0 ⊆W ×OP. αis a borderline
degreeforP athK,hw, δiiif and only ifhK,hw, δii°IP(α).
We claim that the notion of a “borderline degree” captures effectively what is meant by an “imaginable borderline case” for a predicate. This point is really driven home by the following proposition.
Proposition 7.15 Let P be a standard-sensitive predicate, fP be its asso-
ciated function, OP be its densely ordered set of degrees, α, δ ∈ OP, β be
some object, and hK,hw, δii be a context such that K = (W0, R0, V0) where
W0 ⊆ W ×OP. If α is a borderline degree for P at hK,hw, δii, and in all
case forP athK,hw, δii.
proof: First, we will show thatP(β) is I athK,hw, δii. Sinceαis a borderline degree forP athK,hw, δii, by our semantics forI, there existhu, γi,hv, ²i ∈W0 such that R0hw, δihu, γi, R0hw, δihv, ²i, hK,hu, γii ° P(α) and hK,hv, ²ii 6°
P(α). Thus, by Definition 7.13,α >P γandα6>P ². Therefore, by assumption,
fP(u)(β)>
P γ andfP(v)(β)6>P ², and sohK,hu, γii°P(β) andhK,hv, ²ii 6°
P(β). By our semantics for I, then, hK,hw, δii° IP(β), and by Proposition 7.9,P(β) is I athK,hw, δii.
Now, lethK0,hw0, δ0ii,K0= (W00, R00, V00),W00⊆W×OP be an information
state such that the worlds available at hK0,hw0, δ0iiare a subset of the worlds available at hK,hw, δii, the degrees available at hK0,hw0, δ0ii and hK,hw, δii are the same, and for any uin the set of worlds available at hK0,hw0, δ0ii, if
R0hw, δihu, γi, thenR00hw0, δ0ihu, γi. We will show thatP(β) is I athK0,hw0, δ0ii. Now, since γ and ² are degrees available at hK,hw, δii, they are also degrees available athK0,hw0, δ0ii. Thus, there arehu0, γi,hv0, ²i ∈W00such that
R00hw0, δ0ihu0, γi,R00hw0, δ0ihv0, ²i. Moreover, since the worlds available at
hK0,hw0, δ0iiare a subset of the worlds available at hK,hw, δii, then we know thatfP(u0)(β) =αandfP(v0)(β) =α. Thus,fP(u0)(β)>
P γandfP(v0)(β)6>P
², and sohK,hu0, γii°P(β) whilehK,hv0, ²ii 6°P(β). Therefore,hK0,hw0, δ0ii°
IP(β), and so again by Proposition 7.9,P(β) is I athK0,hw0, δ0ii.
Proposition 7.15 establishes that if α is a borderline degree forP in some context, then any object for which it’s known that its degree of P is α will be a borderline case for P at that context. Now, what precisely does one do when one imagines a possible borderline case for a predicate P? Intuitively, one pictures in their mind some object whose degree of P is precisely known to them, but yet there is an indeterminacy in whether P applies to it. One way of characterizing this act is that for some borderline degree α forP, one pictures an object for which it’s known that its degree ofP isα. That is, in my imagining a borderline case forP, I find the “gap” forP and imagine to myself an object whose degree ofPfalls precisely in that gap. Clearly, if such an object were to exist, then by Proposition 7.15 there would be a borderline case forP. Thus, it seems that for there to be an imaginable borderline case forP at some context it is sufficient that there be at that context a borderline degree for P. Furthermore, it’s clear that the existence of an imaginable borderline case forP implies the presence of a borderline degree forP; if in some context there were no borderline degrees forP, thenP must in that context be used with a precise sense, and so intuitively there could be no imaginable borderline cases forP.
Thus, for the notion of an “imaginable borderline case for P,” we shall throughout this thesis substitute that of a “borderline degree forP”. Because of our equation of imaginable borderline cases and borderline degrees, we can now formulate the following as our desired results.
Law of Higher Order Vagueness LetP be a vague predicate andOP be
its densely ordered set of degrees. For any context hK,hw, δii in which P is being used, and any natural number n≥1 there is some degree α∈OP such
that hK,hw, δii°InP(α).
Ineluctability of Vagueness Let P be a vague predicate and OP be its
densely ordered set of degrees. For any contexthK,hw, δiiin which P is being used, there is some degreeα∈OP such that hK,hw, δii°IP(α).
Clearly, when formulated in this way, the Law of Higher Order Vagueness implies the Ineluctability of Vagueness. Therefore, in order to overcome the challenges raised in Sections 3.2.1 and 3.2.2, it is sufficient that we prove the Law of Higher Order Vagueness. In the next section, we do just that.