Elaborar una propuesta metodológica, para la estructuración de un Sistema de

In our discussion of the sorites paradox, we have so far spoken only of the “classical” sorites argument, which operates on a particular vague predicateP from the lexicon of the language. However, recall from our discussion in Section

3.2.2 that for any vague predicate Φ, the predicate “is a borderline case for Φ” is also vague. We may immediately perceive, then, that for anyn∈N, a sorites argument may be constructed for the predicate “is ann-th order borderline case for P.” The way this realization is often informally put is that, just as there could be no two men with only an inch difference in their height, one of whom was tall and the other not tall, there could be no pair of men with, say, only a half-inch difference in height, one of whom was a borderline case for “tall”, and the other not, nor a pair of men with only a quarter-inch difference in height, one of whom was a borderline borderline case for “tall” and the other not,. . . and so on. Strictly speaking, these are all just versions of the same, old sorites argument, but philosophers sometimes speak of the latter as revealing “higher- order” sorites paradoxes, since they operate on predicates denoting higher-order vagueness classes.

We would like in this section to argue that the general form of analysis offered in the previous section for the classical sorites paradox can likewise be applied to the higher-order sorites paradoxes. That is, we would like not only to offer resolutions to these paradoxes, but also to correctly predict that for alln≥1, the predicateIn_{P is sorites susceptible. Unfortunately, this matter is still largely a}

topic for future research. Given the simplicity of our formal representations of vague selections, we cannot yet make our arguments regarding the higher-order sorites as precise as those laid out in the last section for the classical paradox. Nevertheless, the following remarks should make clear to the reader how the intended extension is to proceed.

The basic observation informing our approach to the higher-order sorites is that the analysis of Section 4.6 rests only on the presence of a lower bound on the size of the set ofsel(x)-points in the vague selection. This, when combined with the second Vagueness Principle, establishes the crucial lower bound on the distance between two degreesα, β such that one could in a context beDP and the other D¬P. Similarly, then, one should be able to use the lower bound on the size of vague selections to deduce for any n ≥1 a lower bound on the size of the set of In_{sel(x)-points, one which could then establish a minimum}

distance between two degrees such that one could in a context be DIn_{P and}

the otherD¬In_{P. Such a minimum distance would thus permit in any context}

the construction of a canonical sorites series for the predicateIn_{P. Our analysis}

of the classical sorites would then apply straightforwardly to sorites series for all higher order vagueness predicatesIn_{P, and we would correctly predict that}

all these predicates are sorites-susceptible.

Again, although we cannot argue as precisely as we did in previous sections, the following considerations should indicate to the reader how a finer, more so- phisticated representation of vague selections might lead to a developed account of the higher-order sorites. Suppose that we have a vague selection of points along a dense linear ordering.

¾ -

sel

Isel_I2_sel Isel

I2_{sel I}2_{sel I}2_sel

Suppose, moreover, thatαandβare two points in the ordering such that for this vague selectionDIn_{sel(α) andD¬I}n_{sel(β). SinceDI}n_{sel(α), we may conclude}

that In_{sel(α) and for nom∈Nis it true thatI}m_In_{sel(α). Thusαis in some}

interval ofIn_{sel points, and it is not in any of the intervals ofI}m_In_selpoints.

We may picture this as follows.

• α ¾ - In In+1 _In+1 In+2 In+2 _In+2 _In+2

Similarly, from the fact thatD¬In_{sel(β), we may conclude that¬I}n_sel(β)

and for nom∈Nis it true thatIm_¬In_{sel(β). Recall that by our semantics for}

I,IΦ holds iffI¬Φ does. Thus,β is not included in anyIn_{sel interval, and nor}

does it lie in any of the intervals of Im_In_{sel points. We conclude, then, that}

the points αandβ must be separated by some interval γ containing for every m∈Na subinterval ofIm_In_{sel(α) points. That is, we know that the following}

picture depicts the least possible distance betweenαandβ.

• α • β ¾ - In In+1 _In+1 γ In+2 In+2 _In+2 _In+2

Now, following our discussion in Section 4.5.2, we conclude that each subin- terval withinγmust have a minimal size. Thus, the intervalγseparatingαand βmust have a minimal size. Let²be the lower bound forµ(γ). Since [α, β]⊃γ, µ([α, β])≥µ(γ)≥². Therefore, we have deduced an²∈Rsuch that DIn_selα

andD¬In_{selβ impliesµ([α, β])≥².}

The graphic underneath Proposition 5.3 depicts the intuitive relationship within a context hK,hw, δii between the various vagueness orders for some standard-sensitive predicate P in hK,hw, δii and the uncertainty orders of a constraining selection for hK,hw, δii. A more sophisticated formalization of vague selections should be able to capture this relationship in full, but accord- ing to our depiction in the graphic, all DIn_{sel degrees in the vague selection}

correspond toDIn+1_{P degrees in the context, and all D¬I}n_{sel degrees corre-}

spond toD¬In+1_{P degrees. Thus, our lower bound² on the distance between} DIn_{sel andD¬I}n_{sel degrees becomes a lower bound on the distance between}

DIn+1_{P and D¬I}n+1_{P degrees. We conclude, then, that for all n≥ 1 there} exists an²∈Rsuch thatDIn_{P αandD¬I}n_{P β impliesµ([α, β])≥².}

Now, given this value², it should again be possible to construct a canonical sorites series, this time for the predicateIn_{P. LethK,hw, δiibe any context. By}

the nature of vague selections and Proposition 5.3, we are guaranteed the exis- tence of pointsα, βsuch thathK,hw, δii°DIn_{P αandhK,hw, δii°D¬I}n_{P β.}

Clearly, there exists some well-ordering ≺s

p of degrees from OP such that this

α is its first member, β is its last member, and all points γ, ηadjacent in the ordering are such that µ([γ, η])< ². By reasoning which should be familiar to the reader, it will thus be impossible for any speakers in hK,hw, δii to assert of degrees adjacent in this ordering that one is In_{P and the other ¬I}n_{P. If}

one accepts the analysis of the sorites propounded in the last section, it follows that speakers inhK,hw, δiishould be compelled then to believe that the context supports the untrue inductive premise ∀x, y((In_{P x∧y ≺}s

p x)→ InP y), thus

creating the existence of a sorites paradox for the predicateIn_P.

Therefore, this envisioned analysis, when fully developed, would correctly predict that for all n ∈ N and standard-sensitive predicates P, the predicate In_{P is susceptible to a sorites paradox. Moreover, it would also expand our}

resolution of the classical sorites to its higher-order cousins.