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Multiple reference and vague objects


Kilimanjaro is an example of what some philosophers would call a ‘vague object’: it is only roughly 5895 m tall, its weight is not precise and its boundaries are fuzzy because some particles are neither determinately part of it nor determinately not part of it. It has been suggested that this vagueness arises as a result of semantic indecision: it is because we didn’t make up our mind what the expression “Kilimanjaro” applies to that we can truthfully say such things as “It is indeterminate whether this particle is part of Kilimanjaro”. After reviewing some of the limitations of this approach, I will propose an alternative account, based on a new semantic relation—multiple reference—capable of holding in a one-many pattern between a term and several objects in the domain. I will explain how multiple reference works, what differentiates it from plural reference and how it might be used to accommodate at least some aspects of our ordinary discourse about vague objects.

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  1. The observation that epistemicism contradicts some of our pre-theoretic intuitions about ordinary objects does not refute this view, of course. According to Williamson, “the sharp cut-off points for vague terms implied by the epistemic view are in a sense unimaginable, which makes the view counterintuitive without constituting an argument against it” (1997, p. 218).

  2. The method of supervaluations is due to Fraassen (1966) and its application in the context of vagueness to Fine (1975). Following Varzi (2007), I will call “supervaluationism” any view that combines the idea that a vague language admits of several precisifications and the use of the method of supervaluations for coping with the precisifications. Different versions of supervaluationism take different stands on the semantic status of the precifications. According to some versions, they are heuristic devices that have little to do with the actual meaning of the vague expression. According to Lewis’s version of the view, each vague expression is indeterminate in meaning among its precisifications, so the latter can be regarded as admissible interpretations of the former. There is also a version of supervaluationism on which each precisification corresponds to a meaning that the vague expression actually has—this is what Smith (2008) calls “plurivaluationism”. I will focus on Lewis’s version of supervaluationism, but the concerns I will raise apply just as well to the other versions. I will say more about plurivaluationism in footnote 20.

  3. Following Williamson (1994), there has been an ongoing debate about the extent to which supervaluationism requires a revision of classical logic. See Williams (2008) and Jones (2011) for discussion.

  4. Keefe (2000, pp. 186–88) offers a suggestion along these lines, though she focuses more on vague predicates than on vague singular terms. In principle, one could also interpret statements like (1)–(3) metalinguistically, with a truth-predicate in the place of the ‘Determinately’ operator. But the metalinguistic regimentation is less plausible: on the face of it, when we say “Kilimanjaro has no precise number of atoms”, we are using the term “Kilimanjaro”, not mentioning it.

  5. Note that (1**)–(3**) are typical cases of existential statements that come out supervaluationarily true even if none of their substitution instances does. For instance, (1**) comes out supervaluationarily true even if there is no object o for which “Kilimanjaro is identical to o and o has a precise number of atoms” is supervaluationarily true. For discussion of this ‘semantic anomaly’, see Keefe (2000, pp. 181–183).

  6. The supervaluationist’s acceptance of (1**)–(3**) closely parallels her denial of the soritical premise “For no nn grains make a heap but \(n - 1\) grains do not make a heap”, which Keefe describes as “one of the least appealing aspects of [supervaluationism]” (2000, p. 183). According to Keefe, we endorse the premise because we confuse the (wrong) thought that no number of grains makes the difference between a heap and a non-heap with the (right) thought that no number of grains determinately makes the difference between a heap and a non-heap (2000, p. 185).

  7. Even in a supervaluationist setting, (5) and (7) follow from (respectively) (4) and (6) on a substitutional account of quantification. But note that we want the inferences to be valid in natural languages like English and it is unclear whether natural language quantification is interpretable as substitutional. For discussion of the failure of the existential quantifier to commute with the ‘Determinately’ operator, see McGee and McLaughlin (1994, 212).

  8. It has been noticed that these difficulties have repercussions elsewhere, for example when it comes to provide a semantics for vague terms occurring in speech reports (Schiffer 2000) and belief reports (McGee and McLaughlin 2000).

  9. The idea of using Leibniz’s law to show that a vague objects is distinct from any of its precise counterparts was first discussed by Evans (1978).

  10. The fallacy consists in thinking that since “It is indeterminate whether Sparky is part of Kilimanjaro” is true, there must be an x such that it is indeterminate whether Sparky is part of it. On the supervaluationist account, there is no such x (this is why the inference from (4) to (5) fails). So no such x can be shown to be distinct from \(\hbox {K}_{1}, \hbox {K}_{2}, \hbox {K}_{3}\ldots \) using Leibniz’s Law. See Varzi (2001) for discussion.

  11. Saying that vague objects are part of our ordinary conception of the world is not saying that vague identity is part of our ordinary conception of the world. Pace Evans (1978), the existence of vague objects does not immediately require matters of identity to be vague. For discussion of this point, see Williamson 1994, pp. 255–256.

  12. The idea of a fact or truth requiring ‘no more of the world’ than another fact or truth can be found in Thomasson (2007) and deRosset (2015). One natural way of glossing this idea is in terms of grounding: roughly, a fact requires nothing more of the world than another fact if full grounds for the latter are also full grounds for the former (Thomasson (2007, p. 16) offers a suggestion along these lines, but she speaks of ‘sufficient truth-makers’ instead of ‘full grounds’). Sider (2015) discusses alternative ways of precisifying the slogan ‘nothing over and above’, though he focuses mainly on its applications in mereology.

  13. I take this to be the case with respect to at least one way of using the term “the Beatles” in English, but clearly nothing crucial hinges on the correctness of this particular analysis. Other plausible (though no less controversial) examples of plural terms include the locution “Russell and Whitehead” as it occurs in “Russell and Whitehead wrote Principia Mathematica” or the pronoun “they” as it occurs in “They carried the piano upstairs”.

  14. See also Linnebo (2014) for a definition along similar lines.

  15. It is a well-known fact that a sentence like “The Beatles wrote a song” is ambiguous between a reading on which the Beatles wrote a song together and a reading on which each of the Beatles wrote a song on his own (Lasersohn 1995 offers a helpful discussion of this topic as well as compelling reasons for locating the source of the ambiguity in the predicate). But that is not the point here. The point is, rather, that the first reading of the sentence (i.e. the non-distributive one) neither requires nor rules that each the Beatles wrote a song on his own. It is in this sense that the non-distributive reading allows, but does not mandate the distribution.

  16. For a discussion of multiple location, see Hudson (2005).

  17. One non-distributive notion of denotation that Oliver and Smiley explicitly define (2013, p. 99) is the notion of maximal denotation:

    \(\bullet \)       ‘b’ maximally denotes \(\hbox {x, y, z,}\ldots \) if and only if all the things that ‘b’ denotes are among \(\hbox {x, y, z}\ldots .\) I note in passing that exact denotation and maximal denotation are not to be equated with one another. If a term ‘b’ maximally denotes x, there is nothing else apart from x that ‘b’ denotes. But if a term exactly denotes x, that does not mean that there are no other things that the term denotes—in fact, in the case of multiple terms what happens is precisely that the exact denotation of the term does not coincide with its maximal denotation.

  18. As I suggested in Sect. 3, for Kilimanjaro to be nothing over and above \(\hbox {K}_{1}, \hbox {K}_{2}, \hbox {K}_{3}\ldots \) is for facts involving Kilimanjaro to require no more of the world than facts involving \(\hbox {K}_{1}, \hbox {K}_{2}, \hbox {K}_{3}\ldots \) , where this notion can naturally be glossed in terms of grounding (see footnote 12). I take it to be relatively uncontroversial that, whenever ‘a’ refers to \(\hbox {a}_{1}, \hbox {a}_{2}, \hbox {a}_{3}{\ldots }\) (either multiply or plurally), full grounds for facts involving \(\hbox {a}_{1}, \hbox {a}_{2}, \hbox {a}_{3}{\ldots }\) will also be full grounds for facts involving a.

  19. Notice that, in order to accommodate the possibility of multiply referring terms, the definition of distributivity I offered in the last section has to be amended by replacing “among” with “in or among”.

  20. The recognition that distributivity may fail in the presence of multiple reference marks a major difference between the approach I am advocating in this paper and the view Smith (2008) calls “plurivaluationism”. Plurivaluationists treat expressions like “Kilimanjaro” as having multiple semantic values, but since they deal with semantic multiplicity in the same way in which other supervaluationists deal with semantic indecision (i.e. using the method of supervaluations) their account shares the same limitations as any other form of supervaluationism. In addition, the application of the method of supervalutions to sentences involving multiple terms raises difficulties of its own: given the prima facie analogy between plural terms and multiple ones, it is not entirely clear what justifies the application of the method to sentences involving the latter, but not to sentences involving the former.

  21. Scha (1981) makes exactly this move.

  22. Oliver and Smiley offer many examples of multi-valued functions, from the square root of (both 2 and \(-\)2 are square roots of 4) to the husbands of (seven distinct individuals are the husbands of Elizabeth Taylor). As an alternative to using a multi-valued valuation function, one could decide to assign each multiple term a single set having several singletons of elements of D as members. However, I find it somewhat inelegant to assign multiple terms semantic values of a different kind than singular and plural terms. Better to treat singular, plural and multiple terms uniformly—among other things, this allows us to see singularity as a limit case of both plurality and multiplicity.

  23. See, among others, Landman (1989a, b) and Schwarzschild (1996).

  24. For simplicity of exposition, I follow Oliver and Smiley (2013) in using just one function that assigns values to both terms and variables, instead of distinguishing a valuation function for terms and a variable assignment function for variables.

  25. The same can be said of the predicates ‘true of’ and ‘hold of’ in the semantics Oliver and Smiley offer for plural logic (see Oliver and Smiley 2013, pp. 96, 146 and 217). Notice that, since ‘hold of’ is non-distributive, it cannot be defined in terms of distributive notions such as membership—instead, it is best seen as a primitive. We could avoid having a primitive non-distributive predicate in the meta-language (and instead state the truth-condition for predication in terms of membership) if we decided to assign each multiple term a single set having several singletons of elements of D as members. But there are good reasons not to do that: see footnote 22.

  26. Notice that there is nothing special about “ part of...” here. Presumably the application-conditions of “ the weight value of...” and “ the boundary of...” to multiple-terms are also indeterminate. As a result, whenever “Kilimanjaro does not have a precise weight” and “Kilimanjaro has fuzzy boundaries” come out true, “n is the weight value of Kilimanjaro” and “b is the boundary of Kilimanjaro” come out indeterminate for any n and b that correspond (respectively) to the weight value and boundary of some but not all of the things in Kilimanjaro.

  27. See, for instance, Boolos (1984).


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Thanks to Donald Baxter, Philipp Blum, Cian Dorr, Kit Fine, Manuel García-Carpintero, Martin Lipman, Matthew McKeever, Bryan Pickel, Giulia Pravato, Carlos Romero, Sven Rosenkranz, Moritz Schulz, Stewart Shapiro, Achille Varzi, Elia Zardini and two anonymous referees for helpful comments on the ideas in this paper. I am also grateful to the participants of the 65th Eidos Meeting in Ligerz, the LOGOS Seminar in Barcelona, the 8th Arché Graduate Conference in St Andrews, the 1st UNAM-IIFs Philosophy Graduate Conference in Mexico City and the Phlox Research Seminar in Hamburg. The research leading to this paper has received funding from the Swiss National Science Foundation Sinergia Project ‘Grounding - Metaphysics, Science, and Logic’ (Project 147685).

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Correspondence to Giovanni Merlo.

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Merlo, G. Multiple reference and vague objects. Synthese 194, 2645–2666 (2017).

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  • Vague objects
  • Supervaluationism
  • Plural reference
  • Multiple reference