Abstract
In his latest book Aboutness, Stephen Yablo has proposed a new ‘easy road’ nominalist strategy: instead of engaging in the hard work of paraphrasing a scientific theory which presupposes numbers in a nominalistically acceptable way, nominalists are, according to Yablo, entitled to accept the theory as true, while rejecting the existence of numbers, if from the theory’s content the presupposition that there are numbers can be subtracted away, yielding thus a number-free content remainder. Perfect extricability, i.e. extricability in every possible world, of the presupposition that there are numbers from any content apparently involving them is, in Yablo’s view, sufficient to make the existence of numbers moot. In this paper I will argue that perfect extricability fails as a criterion of ontological mootness.
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Notes
According to Gendler Szabó (2017, pp. 787–791), Yablo’s conception of a sentence’s subject matter—of what the sentence speaks about—marks a significant departure from and an improvement of the Fregean theory of meaning: the subject matter of a sentence, as Yablo understands it, is, in Szabó’s view, a much better candidate for its Bedeutung than its truth value.
It can be regarded as his third: the first being fictionalism about numbers (Yablo 2001, 2005) and the second his account of non-catastrophic presupposition failure (Yablo 2006, 2009); all the three are of the ‘easy’ stripe. Yablo seems to consider his most recent defence of nominalism the most general one, as he believes that the others may be obtained from it as special cases (2014, pp. 149–151, 165–177). He does not claim that there are no numbers, but that their existence is moot; I will, however, regard his agnosticism about numbers as a weak form of nominalism. The issue is rather terminological: what is undeniable is that he would not accept the existence of numbers, both in his older and in his more recent works. To justify my terminological choice, I will note that in (Yablo 2012) he himself considers his agnosticism as a version of the easy road nominalism.
Quine famously claimed that we are committed to entities quantified over in our best scientific theory, and that we “remain so committed at least until we devise some way of so paraphrasing the statement as to show that the seeming reference to [these entities] on the part of our bound variable was an avoidable manner of speaking” (1961, p. 13). It is the easy road nominalists’ objective to show that we are not bound to a universe ‘overpopulated’ with abstract entities, and need not be cast out of the far preferable ‘desert landscapes’ simply for want of a paraphrase.
To illustrate this type of indispensability, Yablo cites the well-known example of Putnam’s (1975, pp. 295–297): a cubical peg fifteen–sixteenth of an inch high cannot pass through the round hole one inch in diameter because fifteen–sixteenth times the square root of two is greater than one: any explanation which would not be purely geometrical but would take into account the microphysical structure of the peg would not be appropriate, as it would not be general enough.
Yablo has already envisaged the following objection, inspired by Wittgenstein’s investigations of the ‘logic of colour concepts’ (2007): if it were somehow discovered that crimson is, say, the only glowing shade of red, it seems that (4), after all, could be subtracted from (5), leaving
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The tomato is of glowing colour
as the remainder. Yablo rejects this objection and claims that if the tomato were, for instance, dull green, it would not be crimson because it is not red, and not because it is not of glowing colour (2014, pp. 154–155).
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Kripke (2003, p. 8) is cautious not to attribute the skeptical paradox that seems to threaten the very possibility of meaning straightforwardly to Wittgenstein; however, exegetical matters need not concern us here.
Since the truth value of the remainder R is defined for every P-world—i.e. it coincides with the value of C in that world—R is a partial function from the set of possible worlds to the set of truth values, and in that sense can be said to exist for any pair of P and C; if R is, furthermore, a total function, P is then perfectly extricable from C. As Yablo puts it, “when B is intuitively inextricable from A, a proposition A − B still exists, just don’t try evaluating it at (too many) worlds where B fails” (2014, p. 147).
The content P → C is, of course, the weakest content that fills the gap between P and C, but it is not independent of P, since it has a truthmaker which forces P to be false, namely ¬P (see Yablo 2014, pp. 183–184).
This sheds some interesting light on the Kripke–Wittgenstein paradox: if Yablo is right, determining what following the same rule on a novel occasion amounts to becomes only a special case of solving the problem of finding an unstated premise in an enthymematic argument. In his more recent article, Yablo (2016) relates indicative conditionals to content extraction: in his view, evaluating an indicative conditional A → B reduces to determining the truth value of the ‘incremental content’ of B over A, which is nothing else but the remainder one gets when A is subtracted from B. Perfect extricability and content extraction in general are no doubt useful conceptual tools for addressing various philosophical problems; in this paper I am criticising only the way Yablo utilises them in defending nominalism.
(12), of course, presupposes both (11) and
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There is a king of France,
which Yablo discusses in the cited passage. I will focus on (11); nevertheless, my points are equally applicable to (13): both presuppositions are prefectly extricable from (12), only with different remainders.
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If any other man ruled France as well, he would have to be equal in power and title with the French king, and hence would have to be a king himself, but then (11) would not be true. The rules of monarchy prevent there being a reigning queen—or a queen regnant—if the country already has a king; that is why in monarchies ruled by a queen, her spouse usually has the title of prince or duke. A queen consort, on the other hand, is married to a king, but she does not reign the country.
It might be objected that (18) is unsuitable for truthmaker of (11) → (~ 12), since it claims that ∀x(x is not bald ∨ x does not rule France) and truthmakers are not supposed to be disjunctive (Yablo 2014, p. 75). However, the objection can be met in a way already envisaged by Yablo (2014, pp. 65–67) in discussing Hempel’s paradox of confirmaton: if (18) is barred from truthmaking, what makes (11) → (~ 12) true in a world w in which there are bald people, none of whom rule France, is the exhaustive conjuntion of contents, each claiming for a particular bald individual in w that it does not rule France; of course, the truhmakers in that case will vary across these worlds.
Yablo has never actually provided proof that the presupposition that there are numbers is everywhere perfectly extricable. He seems to assume that this claim is completely unproblematic. However, my aim in this paper is not to contest this claim, but to show that even if it were the case, it nonetheless would not be sufficient for authorising nominalist conclusions.
It might be objected that perfect extricability’s nonperformance on ‘the king of France’ test hinders only agnosticism about numbers, but leaves nominalism proper unharmed. The counterexample with the king of France, so the objection goes, only shows that perfect extricability is not—contrary to what Yablo claims—a satisfactory criterion of ontological mootness, but it does not preclude perfect extricability from being a sufficient condition of nonexistence: the presupposition that there is a unique king of France and the presupposition that there are numbers are both perfectly extricable, yet this is completely in accord with nominalism proper for it precisely shows that there (determinately) are no numbers, just as there (determinately) is no (unique) king of France. However, the objection is invalid. It can be demonstrated, in an utterly analogous way, that the presupposition
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There is a unique king of Spain,
engendered by an occurence of the definite description ‘the king of Spain’ in a sentence, is perfectly extricable from any content ostensibly about the Spanish monarch. If perfect extricability were a sufficient condition of nonexistence, it would follow that there is no unique king of Spain, which is false: Spain is presently ruled by King Felipe VI, who ascended the throne in 2014, and by him only. Nominalism proper thus fares no better than agnosticism about numbers: perfect extricability of the presupposition that there are numbers is sufficient neither for establishing that the ontological status of numbers is moot nor for establishing that there are no numbers.
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Acknowledgements
My research has been supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (179067) and by the University of Rijeka, Croatia (uniri-human-18-239). I would like to thank Miloš Arsenijević and Timothy Williamson for their valuable comments on an earlier version of this paper, and to the anonymous reviewers, who greatly helped me accomplish the final version. The paper was presented at the conferences in Dubrovnik (Mind, World and Action 2015), Timisoara (Topics in Analytic Philosophy 2017) and Bratislava (Reasoning and Analytic Methods 2017)—I am grateful to audiences there for their constructive feedback.
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Jandrić, A. Content Extraction, Ontological Mootness and Nominalism: Difficulties on the Easy Road. Erkenn 87, 2329–2341 (2022). https://doi.org/10.1007/s10670-020-00304-6
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DOI: https://doi.org/10.1007/s10670-020-00304-6