6. RESULTADOS
6.3 Experiencias exitosas en Instituciones de Salud a nivel nacional
Our goal is to understand why it appears to us as speakers that there are con- texts of the kind demonstrated by Proposition 3.6 to be impossible. That is, we want to understand why the inductive premise of the sorites argument always appears to us true, even though it never is. It was pointed out that this goal could be accomplished by demonstrating that for any standard-sensitive predi- cateP and any contexthK,hw, δii, a sorites series hS,≺s
piforP at hK,hw, δii
may be constructed. In this section, we will attempt that demonstration, and we begin with the following definition.
Definition 6.1 (Canonical Sorites Series) LetP be a standard-sensitive predicate,OP be its densely ordered set of degrees,hK,hw, δii,K= (W0, R0, V0),
W0 ⊆W ×OP be a context,hK0, wi ∈vagsel be a constraining selection for
hK,hw, δii, µand f be the fixed functions from Definition 5.4, and hS,≺s pibe
a well-ordering of degrees from OP with a first element αand a last element
β. hS,≺s
piis acanonical sorites series for P at hK,hw, δii, ifαis a proper
upper bound andβ is a lower bound for the set{δ0:hK0, wi°sel(δ0)}, and for any twoγ, η∈S, ifγ≺s
oη, thenµ([γ, η])< f(0).
Now we want to prove some facts about canonical sorites series, in particular that they are sorites series by Concept 3.4. First of all, it should be clear that for any contexthK,hw, δiiand standard-sensitive predicateP there is at least one canonical sorites series forP at hK,hw, δii.
Proposition 6.2 Let Let P be a standard-sensitive predicate, OP be its
densely ordered set of degrees, hK,hw, δii, K = (W0, R0, V0), W0 ⊆ W ×OP
hS,≺s
pi be a canonical sorites series for P at hK,hw, δiiwith a first elementα
and a last elementβ. hK,hw, δii°DP α.
proof: Let hw0, δ0i be such that R0hw, δihw0, δ0i. Thus, δ0 ∈ {δ0 : hK0, wi °
sel(δ0)}. Now, since α is a proper upper bound for {δ0 : hK0, wi ° sel(δ0)},
α >P δ0. Thus, by Definition 3.7.13, hK,hw0, δ0ii ° P α. Since hw0, δ0i was
arbitrary,hK,hw, δii°DP α.
Proposition 6.3 Let Let P be a standard-sensitive predicate, OP be its
densely ordered set of degrees, hK,hw, δii, K = (W0, R0, V0), W0 ⊆ W ×OP
be a context, hK0, wi ∈vagselbe a constraining selection for hK,hw, δii, and
hS,≺s
pi be a canonical sorites series for P at hK,hw, δiiwith a first elementα
and a last elementβ. hK,hw, δii°D¬P β.
proof: Proved analogously to Proposition 6.2, only using the fact thatβ is a lower bound for{δ0:hK0, wi°sel(δ0)}.
Proposition 6.4 LetP be a standard-sensitive predicate,OP be its densely
ordered set of degrees, hK,hw, δii, K = (W0, R0, V0), W0 ⊆W ×OP be a con-
text, hK0, wi ∈ vagselbe a constraining selection for hK,hw, δii, and hS,≺s pi
be a canonical sorites series for P at hK,hw, δii with a first element αand a last elementβ. hK,hw, δii 6°D∀x, y((P x∧y≺s
px)→P y).
proof: This follows immediately from Propositions 3.6, 6.3 and 6.4.
We are now very close to establishing that canonical sorites series are sorites series by our definition in 3.4. Propositions 6.2 and 6.3, when combined with the peculiarities of Definition 6.1, reveal that canonical sorites series have nearly all the essential features of a sorites series. However, the most important detail is still missing: how does one argue that, despite Proposition 6.4, it should appear to speakers in a context that the context supports the untrue inductive premise
∀x, y((P x∧y≺s
px)→P y) for the canonical sorites serieshS,≺spi? To answer
this will require an answer to Graff 2000’s psychological question, which we will have on the table as soon as we have addressed the epistemological question.
4.6.1
The Epistemological Question
If the inductive premise of the sorites argument is not true, why can we not find any falsifying instances? When faced with our sorites series for “tall,” why can we not swipe our fingers through two men in the series and assert that one is “tall” and the other “not tall?” Why does this idea seem so inherently ridiculous?
Let us suppose for the moment that any actual, concrete sorites series phys- ically realizes some canonical sorites series for the context. The following might provide the answer to our questions.
Theorem 6.5 (Law of Quasi-Boundarylessness) Let P be a standard- sensitive predicate,OP be its densely ordered set of degrees,hK,hw, δii, K =
(W0, R0, V0), W0 ⊆ W ×OP be a context, hK0, wi ∈ vagsel be a constrain- ing selection for hK,hw, δii, and hS,≺s
pi be a canonical sorites series for P at
hK,hw, δiiwith a first element αand a last element β. Ifγ ≺s
p η, then either
hK,hw, δii 6°DP ηor hK,hw, δii 6°D¬P γ.
proof: Letγ ≺s
p η. Now suppose that hK,hw, δii °DP η and hK,hw, δii °
D¬P γ. By the Law of Quasi-Tolerance (Theorem 5.5), then, µ([γ, η])≥f(0), and so by definition ofhS,≺s
pi, it cannot be thatγ ≺spη, contrary to assump-
tion. Thus, eitherhK,hw, δii 6°DP ηorhK,hw, δii 6°D¬P γ.
This Law of Quasi-Boundarylessness states that for any two degrees directly adjacent in a canonical sorites series forP, one cannot assert of one of the pair that it isP and of the other that it is notP. Now, recall our Law of Precise Boundaries (Principle 4.1). By Definition 6.1, any canonical sorites series for P is a well-ordering. The Law of Precise Boundaries, then, states that our drawing a boundary for P within the canonical sorites series — our finding falsifying instances for the inductive premise — requires there being γ, η ∈ S such that γ ≺s
p η, hK,hw, δii° DP η and hK,hw, δii° D¬P γ. Thus, we are
unable to draw a boundary anywhere within the canonical sorites series forP; we can find no falsifying instances.
Let us unpack this idea further. Basically, Principle 4.1 and Theorem 6.5 provide an answer to the epistemological question which trades on the nature of the act of finding falsifying instances for the inductive premise. To find such instances requires our being able to assert of two pointsγ andη adjacent in our sorites ordering that one is P and the other is not. However, it has been a commonplace of dynamic semantics since at least Stalnaker 1978 to hold that the assertion of a proposition within a context requires the context to support the proposition. Given our semantics for the operator D, this simply means that for a speaker to assert within a contexthK,hw, δiithat Ψ requires that hK,hw, δii ° DΨ. Thus, finding falsifying witnesses for the inductive premise requires there beingγ andη such thatγ≺s
pη,hK,hw, δii°DP η, and
hK,hw, δii°D¬P γ, which is just what the Law of Quasi-Tolerance forbids. In other words, the answer which we offer to the epistemological question is similar to that offered by the account proposed in Graff 2000: one cannot find the “boundary” in the sorites series because, wherever one looks, the principles governing vague language prevent one from locating the boundary there. The way in which a standard-sensitive predicatePdepends upon vague selections for its interpretation, combined with the lower bound on the size of vague selections, entails that there is a minimum difference between two objects, below which one can no longer assert of them that one isP and the other not. The construction of a sorites series exploits this fact, and creates a context at which there is a well-ordering in which one cannot draw a boundary for the predicateP. This need not entail, however, that the inductive premise of the sorites argument is true at that context. Why, then, do we think that it’s true?
4.6.2
The Psychological Question
Due to the Law of Quasi-Boundarylessness, I cannot imagine a situation in which I could say of two men adjacent in a sorites series for “tall” that one is tall and the other is not tall. Now, usually if I cannot imagine a situation in which I would say that some P is not aQ, common sense dictates to me that “all P’s areQ” is true. But what if, in the case of the sorites series, common sense runs afoul of a strange subtlety in our language?
Picture the following scenario: our practices for verifying the truth of sen- tences in our language are rational simply because they have been discovered in the past to be widely successful. That is, our belief that a universal statement, if false, should admit the discovery of falsifying instances is not based on any
a priori knowledge of the semantics of universals, but on the fact, discovered at the dawn of prehistory, that in nearly all cases it is beneficial to consider a universal sentence true if no falsifying instances can be found. Perhaps this belief was of such use to our early ancestors that it even found its way into our innate “common sense” concerning the use of language. That belief, like many of our common sense beliefs, need not be true, only extremely useful. What if the sorites argument is, like the M¨uller-Lyer illusion, a “trick” whereby our deeply-held common sense beliefs are suddenly caught with their pants down.
If the inductive premise of the sorites isn’t true, why does common sense compel us to believe it? The answer is that we have adopted a common-sense view of the way in which the semantics of universals interacts with the act of verification, one which the sorites argument interestingly reveals to be incor- rect. We are inclined to incorrectly believe the inductive premise because in most everyday cases, in all normal contexts, the inaccuracy of a universal claim entails that it is in principle possible to find objects which one may assert fal- sify the claim. When one introduces a sorites series, however, one unwittingly manipulates the context to create a singularly bizarre and rare occurrence: a context in which a universal claim may be not be true but it is in principle impossible to find any objects which falsify the claim. This is the real sense in which the sorites paradox witnesses a “breakdown” in the language-game of vague predicates. It is not, however, a case in which the principles governing vague predicates have brought us into inconsistency, but one in which our deep- seated beliefs about how the semantics of universal claims connect with human action become challenged.
This response to the psychological question now permits us to argue that all canonical sorites series should be sorites series by Concept 3.4.
Theorem 6.6 (Law of Sorites Susceptibility) IfP is a standard-sensitive context dependent predicate, then P is sorites susceptible.
proof: Let P be a standard-sensitive predicate, and let OP be its densely
ordered set of degrees. By Definition 3.3, it is sufficient to show that there is a sorites series forP at some context. So, lethK,hw, δii,K= (W0, R0, V0),W0⊆
W×OP be a context, andhS,≺s
pibe a canonical sorites series forPathK,hw, δii
with a first elementαand a last elementβ. The Law of Quasi-Boundarylessness states that for any two degreesγ, η∈OP such thatγ≺s
pη, eitherhK,hw, δii 6°
DP ηor hK,hw, δii 6°D¬P γ. Thus, the speakers athK,hw, δiiwill not be able to assert of any suchγ, ηthatP η and¬P γ, and so their common sense beliefs regarding the relationship between the truth of universals and the ability to find falsifying instances compel them to believe that at the contexthK,hw, δii the sentence ∀x, y((P x∧y ≺s
p x)→ P y) is true. That is, to the speakers at the
contexthK,hw, δii, it appears thathK,hw, δii°D∀x, y((P x∧y≺s
px)→P y).
Concept 3.4, Proposition 6.2, and Proposition 6.3 now imply that hS,≺s piis a
sorites series forP at hK,hw, δii.
4.6.3
Interim Summary
Let us pause for the moment and take note of where we stand. Thus far, we have explained our analysis of the sorites paradox, and shown that this analysis can be conjoined with our semantic theory to predict that all standard-sensitive predicates are sorites susceptible.
The resolution which we offer to the sorites paradox is that we believe the inductive premise of the sorites argument to be true because we cannot find any falsifying instances, and we cannot find any falsifying instances because doing so would require our making assertions which our semantic theory says can’t happen. LetPbe a standard-sensitive predicate. Since there is a lower bound on the sizes of vague selections, there is a minimum difference between two objects such that one may assert of one of the pair that it’sP and of the other that it’s not. Our perspective on the sorites argument is that it exploits this minimum of difference to create a well-ordering of objects within which one cannot draw a precise boundary forP. At this point, one’s common sense beliefs regarding the relationship between the truth of universals and the act of finding falsifying instances kicks in and forces one to the belief that the universal sentence is true, even though it’s not. The argument in our proof of Theorem 6.6 shows that this perspective on what is “going on” in the sorites paradox allows our theory to predict that all standard-sensitive predicates are sorites susceptible.
However, our theory has yet to predict that all sorites susceptible predicates are standard-sensitive. Furthermore, we have conspicuously neglected to men- tion the existence of a difficult variation of the sorites, one which our analysis has yet to face. In the last two sections of this thesis, we address both these matters. We begin, in the next section, with the problem of the “higher-order sorites”.