Naive Abstraction

Let us use again the functor ‘the baldness status of a man with ξ hairs’ in order to categorize numbers of hairs of hairy and less hairy men in a soritical series according to the men’s baldness. Let us also assume this time that the categorization is so fine-grained as to encompass many categories: the baldness status of 0-baldness (enjoyed by all and only those numbers such that a man with one of those numbers of hairs has roughly the same number of hairs as a man with 0 hairs), the baldness status of 1-baldness (enjoyed by all and only those numbers such that a man with one of those numbers of hairs has roughly the same number of hairs as a man with 1 hair), the baldness status of 2-baldness (enjoyed by all and only those numbers such that a man with one of those numbers of hairs has roughly the same number of hairs as a man with 2 hairs). . . the baldness status of 1,000,000-baldness (enjoyed by all and only those numbers such that a man with one of those numbers of hairs has roughly the same number of hairs as a man with 1,000,000 hairs). Note that we are still in the dark about the identities of the objects (baldness status) just listed, and so, while the list is certainly complete, it may well be redundant.

We assume that the relation a-man-with-x-hairs-has-roughly-the-same- number-of-hairs-as-a-man-with-y-hairs is an equivalence relation (reflexive, symmetric and transitive):

(REFLhairs) A man with xhairs has roughly the same number of hairs as a man with x hairs;

(SYMhairs) If a man withxhairs has roughly the same number of hairs as a man y hairs, then a man withy hairs has roughly the same number of hairs as a man with x hairs;

(TRANShairs) If both a man with x hairs has roughly the same number of hairs as a man with y hairs and a man with y hairs has roughly the

same number of hairs as a man with z hairs, then a man with x hairs has roughly the same number of hairs as a man with z hairs.

It is crucial to note that ‘A man with ξ0 hairs has roughly the same

number of hairs as a man withξ1 hairs’ is vague, and so a tolerant logic must

be used when reasoning with it. Note that, under some plausible additional assumptions, (TRANShairs) amounts in effect to a tolerance principle for ‘A man with ξ0 hairs has roughly the same number of hairs as a man with ξ1

hairs’. For we can also assume the two following necessary and sufficient conditions for the application of the predicate:

(NEChairs) A man with x hairs has roughly the same number of hairs as a man with y hairs only if the absolute value of the difference of x with

y is ≤10;

(SUFFhairs) A man with xhairs has roughly the same number of hairs as a man with y hairs if the absolute value of the difference of x with y is ≤1.

Consider then the premises:

(11) A man with 0 hairs does not have roughly the same number of hairs as a man with with 1,000,000 hairs;

(12) A man withihairs has roughly the same number of hairs as a man with

i+ 1 hairs.

(11) and (12) are validated by (NEChairs) and (SUFFhairs) respectively.

However, from (12) we have that a man with 0 hairs has roughly the same number of hairs as a man with 1 hair. From (12) we also have that a man with 1 hair has roughly the same number of hairs as a man with 2 hairs, which, together with (TRANShairs) and the previous lemma that a man with 0 hairs has roughly the same number of hairs as a man with 1 hair, yields

that a man with 0 hairs has roughly the same number of hairs as a man with 2 hairs. With another 999,997 structurally identical arguments, we reach the conclusion that a man with 0 hairs has roughly the same number of hairs as a man with 999,999 hairs. From (12) we also have that a man with 999,999 hairs has roughly the same number of hairs as a man with 1,000,000 hairs, which, together with the previous lemma that a man with 0 hairs has roughly the same number of hairs as a man with 999,999 hairs, yields that a man with 0 hairs has roughly the same number of hairs as a man with 1,000,000 hairs. It would then seem that the contradictory of (11) follows simply from (12) and (TRANShairs). Fortunately, it doesn’t, as the reasoning just rehearsed implicitly appeals to (Tl_{). Within a tolerant framework, our (very plausible)}

assumptions about ‘A man with ξ0 hairs has roughly the same number of

hairs as a man with ξ1 hairs’ are consistent.

The lesson is that, within a tolerant framework, the transitivity of a relation must be sharply distinguished from its chain transitivity (to the best of my knowledge, Parikh [1983], p. 247 has been the first to draw this very important distinction). In the classical first-order theory of relations, the transitivity of R:

(TRANSR) For everyx, y, z, if both x Rsy and y Rs z, then x Rs z.

implies its finite chain transitivity:32

(CTRANSRf) For every x0,x1, x2. . .xi, if x0 Rs x1 and x1 Rsx2 and x2 Rs

x3. . . and xi−1 Rs xi, then x0 Rs xi.

This is not so if the background logic is weakened so as to exhibit suitable failures of transitivity, as is the case in every tolerant logic.

32_{An infinitary version of chain transitivity would be:}

(CTRANSRi) For everyX,x,y, ifX is well-ordered byR, xis the minimum element of

We do get a hint about the identities of baldness status by considering that, given the current fine-grained understanding of the categorization in- duced by ‘the baldness status ofξ’, the following abstraction principle strikes us as a conceptual truth:

(ABS) The baldness status of a man withihairs is the same as the baldness status of a man with j hairs iff, [a man with i hairs has roughly the same number of hairs as a man with j hairs].

The relation mentioned on the right-hand side is an equivalence relation, and so (ABS) at least avoids the immediate incoherence of abstracting on a relation which is not an equivalence relation. Note however that such an incoherence threatens only on the controversial assumption that identity is itself an equivalence relation—in particular, that it is transitive. Such an assumption holds in any tolerant logic—such as CT1_{—accepting (id}⇒

2 ) (and

so validating (Tm_{)), but fails for weaker tolerant logics. We will stick to it}

given its attractiveness, and show how a first-order naive theory incorporating (ABS) is consistent in CT1 (see Shapiro [2006], pp. 165–89 for a stimulating discussion of naive abstraction principles in a transitive framework).