Abstract
It might be thought that vagueness precludes the possibility of classical conceptual analysis and, thus, that the classical or definitional view of the nature of complex concepts is incorrect. The present paper argues that classical analysis can be had for concepts expressed by vague language since (1) all of the general theories of vagueness are compatible with the thesis that all complex concepts have classical analyses and also that (2) the meaning of vague expressions can be analyzed by having the degree of vagueness of a given analysandum be “mapped” onto the vagueness of an analysans.
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Notes
There is one sort of view that I will not consider in detail here: One might hold that language indeed includes vague expressions, but that what is expressed by language is not vague. That is, one might hold that the predicate “is short” is vague, but what is expressed by “is short” (namely the concept of being short) is not vague. This would be a welcome thesis for those attempting to give a theory of concepts that avoids problems involving vagueness, but defending it looks to be a tall order. Hence, I set this strategy aside.
Williamson’s (1994, 2) general criterion of vagueness is merely that the extension of such vague expressions is unclear. This is simple enough, even though it runs together the issue of whether such a lack of clarity is merely a matter of ignorance (as the epistemic view holds) or a matter of the boundaries of that expression’s extension being metaphysically imprecise (as supervaluationism and multi-valued theories hold).
Sorites paradoxes can be generated for other sorts of vague expressions as well. For instance, the verb “runs” looks vague since an appropriate sorites series would be a series of runners beginning with someone who is clearly running (at speed s, say). Her neighbor is moving at a millionth of a mile per hour less than s, and this would not seem to make a difference to whether she is running or not. Now, her neighbor is moving at two millionths of a mile per hour less than s, and this does not make a difference with respect to whether “runs” applies either, etc. But given that the series is long enough, it would seem that we could infer that the last member of the series would be running, yet if the series has a million members, the last member’s speed would be zero (a clear case where “runs” would not apply). The same sort of series would seem to generate a paradox for the adverbial expression “runs slowly” as well: use the same sort of series, yet let s be clearly a slow running speed and let each successive member of the series be running a millionth of a mile per hour faster than the preceding member, etc.
Does the conditional go the other way around? That is, are all vague expressions such that a sorites paradox may be generated for it? I will consider the answer here to be “yes,” but see K&S (1999) for a more extended discussion of the proper criterion of vagueness.
As for the other options, one might try to either deny (P1) or just accept the conclusion (C). Some theorists on vagueness (e.g., Unger 1979; Dummett 1999) take this kind of line and infer that the predicate employed in the sorites argument is simply incoherent (for discussion, see K&S 1999, 10, 12–14; see also Williamson 1994, Chapter 6). The strategy strikes me as a last resort. If we take it, then given the pervasiveness of vagueness, the concession is that almost all language is incoherent. If other accounts of vagueness allow for coherent propositions expressed in terms of vague expressions, then those options should be pursued first. There is also the danger of the argument undercutting itself: if most all language is incoherent, then the very argument aimed at showing most language to be incoherent would likely be given in incoherent terms too. But arguments given in incoherent terms strike me as unsound, as it surely does to many others too.
The “respect” here is that the tolerance principle is not denied—the principle is not false, on the three-valued approach.
Alternatively, one might leave a proposition p as having no truth-value and redefine the connectives accordingly. This kind of view might just be what three-valued theorists have in mind when they allow for propositions to take an indefinite “truth-value.” For example, one finds Tye (1999) saying that “The third value here is, strictly speaking, not a truth-value at all but rather a truth-value gap (281).” So, for my purposes here, I treat such strategies as being on a par with that discussed in the text. Still another option is to treat some propositions as being both true and false, and as such hold that truth-value gluts are possible for some propositions. I set this last option aside.
Tye also considers other versions of the sorites argument that do not employ a series of conditionals, but here, the focus is just on one version of the argument to illustrate Tye’s strategy.
According to Tye’s three-valued table for the logical connectives, biconditionals are indefinite if either or both components take the value indefinite. So at least the substitution instances for classical analyses involving concepts expressed by vague predicates will turn out indefinite on some cases. Therefore, classical analyses themselves take the value indefinite. But what will this mean with respect to the requirement that classical analyses are necessary? One could take them to be necessarily indefinite, or indefinitely necessary, it seems. This would need to be explored further in order to see for sure whether classical analysis is really possible on a three-valued system like Tye’s, at least with logical consequence defined as a truth-preserving relation.
It might indeed look alarming that all classical analyses involving concepts expressed by vague expressions will turn out indefinite in truth-value. But one should note that on Tye’s approach, there will be many propositions that will be indefinite in truth-value (including every predication of a vague predicate of something that is a borderline case, for example, as well as apparent tautologies like if p, then p). And given that on Tye’s view there will be some metalinguistic propositions that will turn out indefinite in truth-value (e.g., the proposition that every sentence is true, false, or indefinite (K&S 1999, 48)), perhaps the consequence for classical analyses is not quite as unpalatable as it appeared at first sight.
The degree theorist might still allow the argument to be valid, but on a different notion of validity than preservation of degree of truth. Sainsbury (1995, 42) suggests that validity on a degree theory might be construed like this: “[T]he conclusion of a valid argument cannot have a greater degree of falsehood [where the degree of falsehood of P is equal to 1 minus the numerical truth-value of P] than the sum of the degrees of falsehood of its premises.” If this notion of validity is chosen, then the degree theorist gets to take the sorites argument as valid, but still not accept all of its premises.
Among them might include the following. A consequence of the above degree-theoretic account of classical analysis is that whatever assignment of numerical truth-values is given for propositions of the form x is F, x is F 1 ,…, and x is F n , it must be the same assignment across all possible worlds. If this cannot be the case, then it seems there cannot be any classical analyses (since classical analyses are to be necessary truths). But it does seem as if whatever the actual degree of vagueness is for some predicate “is F” (reflected by its corresponding assignment of numerical truth-values), “is F” will have the same degree of vagueness across all possible worlds. For a difference in degree of vagueness between “is F” and “is G” looks sufficient for “is F” and “is G” to be different predicates, whether those predicates are in the same world or in different ones.
Laurence and Margolis also include such a requirement for analyzing vague terms in terms of other vague terms: “[F]uzziness or vagueness needn’t prohibit a definitional analysis of a concept, so long as the analysis is fuzzy or vague to exactly the same extent that the concept is (1999, 24; italics added).”
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Earl, D. Vague Analysis. Int Ontology Metaphysics 11, 223–233 (2010). https://doi.org/10.1007/s12133-010-0070-2
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DOI: https://doi.org/10.1007/s12133-010-0070-2