Abstract
A theory of vagueness gives a model of vague language and of reasoning within the language. Among the models that have been offered are Degree Theorists’ numerical models that assign values between 0 and 1 to sentences, rather than simply modelling sentences as true or false. In this paper, I ask whether we can benefit from employing a rich, well-understood numerical framework, while ignoring those aspects of it that impute a level of mathematical precision that is not present in the modelled phenomenon of vagueness. Can we ignore apparent implications for the phenomena by pointing out that it is “just a model” and that the unwanted features are mere artefacts? I explore the distinction between representors and artefacts and criticise the strategy of appealing to features as mere artefacts in defence of a theory. I focus largely on theories using numerical resources, but also consider other, related theories and strategies, including theories appealing to non-linear structures.
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Notes
Cook (2002, pp. 245–246).
Shapiro suggests a more substantive role for sharpenings where the interpretations that are quantified over “represent a possible state of a conversation among competent speakers with vague predicates” (2006, p. 69). This rests on a substantive and controversial philosophical contextualist theory about how vague predicates function; I claim that no such role is needed for precisifications.
There is no difference in verity so small that it is never representative, however, as Cook shows (p. 244). So, for example, small differences in verity of the predication of “tall” to consecutive members of a sorites series will always be representative, though we can make the difference in height between them as small as we like.
Thanks to an anonymous referee for this suggestion.
The thought that small differences within an assignment can be representative does not help here: on any model, “Tek is tall” must get a higher value than “Tim is tall” if Tek is slightly taller than Tim, but a range of different assignments preserving that relation will be acceptable.
The problem also affects the very claims that Smith uses to motivate his appeal to a degree theory. According to “the closeness picture of vague predicates”, “closeness of x and y in F-relevant respects makes for closeness of ‘Fx’ and ‘Fy’ in respect of truth.” (2008, p. 146). But if such closeness in truth-value in all the acceptable assignments only means that we can “talk as if” they are close in respect of truth, then we cannot truly say that the condition is satisfied or that vagueness is successfully captured even in his own terms.
A many-valued supervaluationist view of this type was defended in Sanford (1993).
I would argue that the many-valued supervaluationist theory is inferior to the classical supervaluationist view, but I will not pursue that argument here. This paper is specifically concerned with the appeal to artefacts in modelling vagueness and this issue is unlikely to be crucial to the comparison between these two theories.
Zardini’s theory involves certain other very striking features, such as the denial of the transitivity of validity. I won’t enter into these issues here.
Weatherson maintains that the truer than relation is implicitly defined by its role in the false theory according to which it is linear (i.e. a standard degree theory, which he calls M). We can understand that theory, so, he claims, we grasp the truer than relation. Implicit definition by a false theory is fine, he observes, since “we know what phlogiston and ether mean because of their role in some false theories” (p. 51). The analogy does not work, however, since we understand phlogiston but know that it applies to nothing. If we were to admit that the theory was false but still seek phlogiston in the real world, we might be guided by the false theory up to a point, but we could not take it as defined by that theory since nothing can fulfil the required criteria. If there is a question as to whether we understand a non-linear truer than, it won’t do to point out that we can understand a different relation without the questionable feature.
I use “more intelligent than” as a non-linear comparative for the purposes of comparison with “truer than”. We might also wonder whether Weatherson’s framework can successfully model truer than relations with two sentences predicating a multi-dimensional predicate such as “intelligent”. If the different dimensions of variation represented by different predicates blocks comparability, might the different dimensions relevant to a single predicate also block it? If so, the account would only represent “a is intelligent” as more true than “b is intelligent” in a very limited range of the cases where a and b are borderline intelligent and a is intuitively more intelligent than b.
This is not yet to rule out the possibility of a very different sort of theory that also adopts a non-linear structure but that is unlike Weatherson’s in various respects. It is far from clear how such a theory could work, however, and there is no space to consider the options here.
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Acknowledgments
This paper began as a short reply to a paper by Dorothy Edgington at the Princeton Conference on Philosophical Logic, organised by Delia Graff Fara; many thanks to Dorothy and Delia for this great opportunity. I gave later versions of the longer paper at a Barcelona Workshop on Vagueness and Metaphysics, at a Workshop on Vagueness and Self-reference in Lisbon, at the 4th Cambridge Graduate Conference on the Philosophy of Maths and Logic, at the colloquium of the Logic & Language research group of the ILLC at the University of Amsterdam and at a departmental seminar at the University of Nottingham. I am very grateful to audiences on all these occasions. I am also very grateful to Dominic Gregory and a very helpful referee for Philosophical Studies for objections, questions and suggestions. And I acknowledge the ‘Borderlineness and Tolerance’ project, of which I am a part (ref. FFI2010-16984, MICINN, funded by the Government of Spain).
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Keefe, R. Modelling vagueness: what can we ignore?. Philos Stud 161, 453–470 (2012). https://doi.org/10.1007/s11098-011-9750-1
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DOI: https://doi.org/10.1007/s11098-011-9750-1