6. RESULTADOS
6.1 Revisión de los parámetros requisitos o estándares
One of our goals in this chapter is to deduce, by means of our semantic analysis of standard-sensitive predicates, that all standard-sensitive context dependent predicates are susceptible to sorites paradoxes. Therefore, we will need to refor- mulate the notion of “sorites susceptibility,” using the concepts made available within our formal semantics, so that such a deduction within our formal theory might be possible. For this task, we will first propose a definition which comes rather close to our more informal and concrete notion of “sorites susceptibility” presented in Section 1.1. This conservative reformulation, we will then extend into greater abstraction.
Concept 3.1 (Sorites Series I) LetP be a standard-sensitive predicate,OP
be its densely ordered set of degrees, andhK,hw, δii, K = (W0, R0, V0), W0 ⊆
W×OP be a context. Asorites seriesforP athK,hw, δiiis a subsetSof the
domain of objects athK,hw, δii5, well-ordered under a relationQsuch thatQ has a first elementα, a last elementβ,hK,hw, δii°DP α,hK,hw, δii°D¬P β and for allγ, η∈S, ifQγη, then itappears 6to the speakers in the context that
hK,hw, δii°D(P γ →P η).
Definition 3.2 (Sorites Paradox) LetP be a standard-sensitive predicate, OPbe its densely ordered set of degrees, andhK,hw, δii,K= (W0, R0, V0), W0⊆
W ×OP be a context. There exists asorites paradox forP at hK,hw, δiiif
there exists a sorites series forP athK,hw, δii.
Definition 3.3 (Sorites Susceptible) LetP be a standard-sensitive pred- icate, and OP be its densely ordered set of degrees. P issorites susceptible
if there is some contexthK,hw, δii, K= (W0, R0, V0), W0 ⊆W ×OP at which
there is a sorites paradox forP.
We hope that the reader finds these definitions of “sorites series,” “sorites paradox” and “sorites susceptible” intuitively acceptable, and a natural for- malization of the notions appearing in Section 1.1. Definition 3.1 states, for
5By “the domain of objects athK,hw, δii”, we mean the domain of objects atw, which
may include more than simply those entities which exist inw.
6The notion of a context’s “appearance” to its speakers will be left unformulated here,
this constituting the grossest of the lacunae mentioned in the Chapter Overview and the sole reason why Concept 3.1 isn’t named “Definition 3.1” (since it isn’t, strictly speaking, a correct definition). Ideally, I would like it that itappearsto the speakers in contexthK,hw, δii
that hK,hw, δii Φ iff hK,hw, δii DΦ. As will become clear later, I would therefore like my logic forD to be such thathK,hw, δiiD∀x, y((DP x∧Qxy)→ ¬D¬P y) implies
hK,hw, δii DD∀x, y((P x∧Qxy)→P y). The logic I have now obviously doesn’t permit me this, and I can think of no easy augmentations of it which do. On the other hand, this concept of “appearance” would also seem to require that for any such contexthK,hw, δii, if
hK,hw, δiiDφ, then it appears to the speakers inhK,hw, δiithathK,hw, δiiDφ. Thus, this concept of appearance is not presently in a very happy state.
example, that a sorites series for a predicate P at a context is simply a well- ordered series of objects taken from the domain available at the context, having a first and last member such that the speakers in the context may assert that the first member is P and that the last member is ¬P, and it appears to the speakers that, for any two objects directly adjacent in the series, they may assert that if the one isP then the other is P as well. This certainly feels to capture the essential features of a sorites series.
The reader, however, should not become attached to these definitions; they are merely a stepping stool. For the purposes of achieving a greater degree of abstraction, of simplifying our discussion, and of easing the length of our proofs, we will discard Concept 3.1 and use instead a more abstract formal representa- tion of sorites series. The intuition behind this more abstract definition is that, in all actual sorites series for a vague predicate P, the ordering of the objects making up the series is isomorphic to the ordering of their degrees of P. In our newer definition, then, we “strip away” the concrete objects making up the sorites series, and leave only the ordering of their degrees in place.
First, however, we must introduce some new notation. If S ⊆ OP, then
let ≺s
p ⊂ S×S be the relation {hα, βi: α <P β and there is no γ ∈ S such
that α <P γ <P β}. Furthermore, we add “≺sp” as a symbol within our object
language, giving it the obvious semantic definition that hK,hw, δii°α≺s pβ iff
α≺s pβ.
Concept 3.4 (Sorites Series II) LetP be a standard-sensitive predicate, OPbe its densely ordered set of degrees, andhK,hw, δii,K= (W0, R0, V0), W0⊆
W ×OP be a context. A sorites series for P at hK,hw, δiiis a subset S of
OP, such that ≺s
p is a well-ordering with a first elementα and a last element
β7, hK,hw, δii°DP α, hK,hw, δii°D¬P β and it appears to the speakers in
hK,hw, δiithathK,hw, δii°D∀x, y((P x∧y≺s
px)→P y).
Throughout the rest of this chapter, we will use the term “sorites series” with the meaning supplied by Concept 3.4. Moreover, Definitions 3.2 and 3.3 are reinterpreted in light of this new definition.
It should be clear to the reader that at any context at which there is for a predicate P a sorites series by Concept 3.4, and at which there are objects in the domain witnessing each of the degrees in that sorites series, there will be a sorites series for P by Concept 3.1. Therefore, we maintain that Concept 3.4 handily formalizes the notion of there being at a context animaginable sorites for P. The reader may observe the resemblance here to our formulation in Section 3.7.3 of the notion of there being an imaginable borderline case forP.
Finally, to further reduce the size of our arguments, we will introduce here the concept of a “constraining selection.”
7We violate the standard conventions here, and use the term “first element” to mean an α∈Ssuch that for allγ∈S, whereγ6=αand≺sT
p is the transitive closure of≺sp,γ≺sTp α.
Definition 3.5 LethK,hw, δii,K= (W0, R0, V0), W0⊆W×OP be a context,
andhK0, wi ∈vagsel,K0= (W, R, V). hK0, wiis aconstraining selection for
hK,hw, δiiif for allhw, δi ∈W0,{w0:∃δ0∈OP R0hw, δihw0, δ0i}={w0:Rww0} and{δ0:∃w0∈W R0hw, δihw0, δ0i}={δ0:hK0, wi°sel(δ0)}.
By the second Vagueness Principle (Principle 3.7.12), there exists a constraining selection for every context.
Now, it is easily shown that, by our dynamic semantics, the four premises of the sorites argument are jointly inconsistent.
Proposition 3.6 There is no contexthK,hw, δiisuch that there exists a well- ordering Q with a first element α and a last element β, hK,hw, δii ° DP α,
hK,hw, δii°D¬P β andhK,hw, δii°D∀x, y((P x∧Qxy)→P y).
proof: Suppose there were such a contexthK,hw, δii. The second Vagueness Principle implies the existence of a constraining selection forhK,hw, δii. Now, by the semantics ofDand the existence of a constraining selection, there is some
hw0, δ0i such that R0hw, δihw0, δ0i and hK,hw0, δ0ii ° P α, hK,hw0, δ0ii ° ¬P β andhK,hw0, δ0ii°∀x, y((P x∧Qxy)→P y). Induction on the elements of the ordering Q is now sufficient to prove that hK,hw0, δ0ii° P β, contrary to hy- pothesis. Note that this argument is indifferent as to whether the elements of Qare degrees inOP or objects in the domain ofhK,hw, δii.
Just as we had adumbrated, then, our dynamic semantics implies that one of the premises of the sorites argument must not be true. Recalling Graff 2000’s psychological question, the task now before our semantic theory is to explain why all four premises of the argument should appear to be simultaneously true. That is, we must explain why it seems to us as speakers that there are contexts which Proposition 3.6 demonstrates not to exist.
Now, by either Concept 3.1 or 3.4, a context in which there is a sorites series for a standard-sensitive predicateP will be one that appears to its speakers to support all the premises of the sorites argument. Thus, if we could somehow prove that for any standard-sensitive predicateP and contexthK,hw, δii, there exists a sorites series for P at hK,hw, δii, we could accomplish two goals of this chapter at one stroke. First, by Definition 3.3, we will have shown that all standard-sensitive predicates are sorites susceptible8. Secondly, our hypothet- ical proof would need to show that, for any context hK,hw, δii, it appears to speakers athK,hw, δiithathK,hw, δii°D∀x, y((P x∧y≺s
px)→P y). Such a
proof would, in effect, have to explain why speakers find the inductive premise of the sorites argument so darn plausible, and so would answer Graff 2000’s psychological question.
Our goal, then, is some argument to the effect that for any standard-sensitive
8Although this point will not be pursued here, we will have also shown, by our comments
above regarding Concept 3.4, that for any standard-sensitive predicateP, there exists in any context animaginablesorites series forP.
predicatePand any contexthK,hw, δii, a sorites serieshS,≺s
piforPathK,hw, δii
may be constructed. The crafting of this argument, however, will require the composition of some rather disparate facts; Sections 3.4 and 3.5 bring those facts to light.