Abstract
Two fundamental rules of reasoning are Universal Generalisation and Existential Instantiation. Applications of these rules involve stipulations (even if only implicitly) such as ‘Let n be an arbitrary number’ or ‘Let John be an arbitrary Frenchman’. Yet the semantics underlying such stipulations are far from clear. What, for example, does ‘n’ refer to following the stipulation that n be an arbitrary number? In this paper, we argue that ‘n’ refers to a number—an ordinary, particular number such as 58 or 2,345,043. Which one? We do not and cannot know, because the reference of ‘n’ is fixed arbitrarily. Underlying this proposal is a more general thesis:
Arbitrary Reference (AR): It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives.
Our aim in this paper is defend AR. In particular, we argue that AR can be used to provide an account of instantial reasoning (one that is better than the prominent alternatives), and we suggest that AR can also figure in offering new solutions to a range of difficult philosophical puzzles.
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Notes
We do not need to always explicitly use the qualification ‘arbitrary’. Given the right context, we may simply use stipulations such as ‘Let n be a natural number’ or ‘Let Pierre be a Frenchman’. We insert the explicit qualification simply to ensure one focuses on the relevant kinds of readings.
‘Knowledge which’ claims are notoriously context sensitive. When we say that we do not know which number n is, we mean that we cannot describe the number in some informative mode of presentation, such as ‘n is 343’. Of course we do know that n is n, that ‘n’ refers to n, that ‘n’ refers to whatever number it refers to, and so forth. This qualification should be kept in mind throughout the article.
Except, of course, in special situations where we know there is only one value that the expression could receive (e.g. in the case of ‘Let n be an arbitrary even prime number’).
For a detailed defence of the claim that reference does not require such special necessary conditions see Hawthorne and Manley (MS).
A tricky question is whether our intentions are sufficient to determine which Frenchman is being referred to. We certainly don’t think that the reference is determined by any informative intentions of the sort ‘I intend ‘Pierre’ to refer to Jacques Chirac’. But it may be that one has an intention that ‘Pierre’ refer to Pierre or that ‘Pierre’ refer to an arbitrary Frenchman, and that these in turn are sufficient to determine the reference of ‘Pierre’. But in so far as our intentions determine the reference in the latter way, there is still an important semantic fact (one concerning the word ‘Pierre’, or the phrase ‘an arbitrary Frenchman’, or analogous phrases in one’s language of thought), a semantic fact that is crucial in determining the reference of ‘Pierre’ and is not itself determined by non-semantic facts.
We allow that the objector interprets ‘use facts’ in a broad enough way so as to include facts about one’s environment or facts about which properties are most natural. It is clear, however, that the objector does not intend count as use facts such semantic or intentional facts as the fact that one uses ‘Pierre’ to refer to, e.g., Jacques Chirac. For a more detailed discussion on how the claim that semantic facts supervene on use facts ought to be interpreted see Kearns and Magidor (forthcoming).
It is worth noting that one could also consider a different interpretation of AR, one according to which facts about arbitrary reference do supervene on use facts. One constraint on developing the theory in this manner is that to ensure that it will allow that not all stipulations of the form ‘Let n be an arbitrary number’ result in n referring to the same number. (Otherwise one will have problems in applying AR to the case of instantial reasoning, in particular to stipulations such as ‘Let n be an arbitrary number and let k be an arbitrary number.) Two additional challenges for this interpretation of the view is to provide an explanation, on the one hand for how reference supervenes on use facts, and on the other hand for why—despite the supervenience claim—we do not and cannot know the referent. One avenue to explore in this context is a brute supervenience view: one according to which the semantic facts supervene on use facts in an entirely unexplanatory manner and are hence unknowable (cf. Williamson 1994; Cameron 2010). But we find this position dialectically inferior to our own: if one is going to allow bruteness concerning the semantic realm into one’s theory, why insist on the supervenience claim, rather than simply postulating brute contingent facts concerning reference, as our own theory does? After all, contingent brute facts seem less offensive than necessary ones. (See also Kearns and Magidor (forthcoming), Sect. 2.3 for a similar argument.) At any rate, at least for the purposes of this article, we have chosen to develop a version AR which denies supervenience.
It is worth point out in this context that our claim that one cannot know which Frenchman ‘Pierre’ refers to is not intended to exclude scenarios where an omniscient being knows who Pierre refers to, or where one comes to know this fact by testimony of an omniscient being. When we say that we cannot know who ‘Pierre’ refers to, we mean that we cannot know this in any reasonably ordinary scenario, one that does not involve omniscient beings. See Williamson (1997, p. 926) for a similar qualification regarding his epistemic view of vagueness.
We are assuming here that even in non-deterministic settings there are such facts about the future.
We note that precisely this locution is used in the announcements of the results in the British television show ‘The X factor’.
Cf. the discussion of Fine’s view in Sect. 2.1.3.
Cf. Fine (1985b, p. 127), who suggests similar constraints.
An instantial term is term a, such that in an application of UG we infer ∀xφ(x) from φ(a), or such that in an application of EI we infer φ(a) from ∃xφ(x).
Fine (1985b, especially Chap. 12).
The semi-formal English used in this argument is intended to help keep track of the relative scopes of the quantifiers. We assume that a full syntactic parsing of ordinary English will provide us with similar formal properties.
King (1991) offers a quantificational account that treats the instantial terms themselves as implicit quantifiers, rather than as variables. This point will make no difference to our argument.
Many formal systems are phrased so that the same instantial term cannot act in both capacities. It is worth noting, though, that as long as we are careful to phrase the rules correctly, there should not be a principled difficulty in using a term for both purposes: an instantial term a used in an inference from ∃xFx to Fa, can be treated as an arbitrary F, and generalised over to show that all Fs have a certain property.
That is, unless, the argument is construed so that step 6 follows directly from steps 1 and 4, and step 5 is taken to be completely redundant.
Very roughly this is achieved by interpreting line 5 to say that some French man is tall, rather than merely that someone is tall. But this is actually a gross oversimplification. What King’s system in fact predicts is the following interpretations for line 3, 5, 6 (with ‘F’ standing for French man and ‘T’ for tall):
(3) \( \exists y((\exists x{\text{F}}x \to {\text{F}}y) \wedge {\text{F}}y) \wedge \forall y((\exists x{\text{F}}x \to {\text{F}}y) \to {\text{F}}y) \)
(5) \( \exists y((\exists x{\text{F}}x \to {\text{F}}y) \wedge {\text{T}}y) \wedge \forall y((\exists x{\text{F}}x \to {\text{F}}y) \to {\text{T}}y) \)
(6) \( \exists ((\exists x{\text{F}}x \to y) \wedge {\text{F}}y \wedge {\text{T}}y) \wedge \forall y((\exists x{\text{F}}x \to {\text{F}}y) \to {\text{F}}y \wedge {\text{T}}y) \)
The complexity of these interpretations should already give us serious cause for concern.
Things get even worse with other so called applications of Conjunction Introduction—as we see in the argument from 3 and 5 to 6 in Argument 3 (see footnote 20).
Fine (1985b, p. 134).
We say ‘roughly’ because again the system is more complex than that: line 5 will actually be interpreted as saying that ∃a((∃x∀yLxy → ∀yLay) ∧ ∀bLab) ∧ ∀a((∃x∀yLxy → ∀yLay) → ∀bLab)), and line 6 as saying that ∃a((∃x∀yLxy → ∀yLay) ∧ ∀b∃xLxb) ∧ ∀a((∃x∀yLxy → ∀yLay) → ∀b∃xLxb)). But what is crucial to the argument here are the claims that ∃a∀bLab and ∀b∃xLxb which we get from the first conjuncts of the interpretations of lines 5 and 6 respectively. We thus focus on these claims, and apply similar simplifications in the discussion below.
We are fully aware that King is in no way committed to saying that the original claim has the same syntactic structure as that of the complex interpretation. But even so, the semantic complexity of the interpretations is in itself worrisome.
Fine (1985b, p. 14).
Note also that since the counting problem seems to generalize, Fine would need to argue that pretty much every predicate is ambiguous in this manner.
See Fine (1985b, p. 18). Fine later relaxes this condition (ibid. p. 35), but claims he does this only because “for certain technical purposes, a smoother theory is obtained”.
Cf. Sect. 3.2.
This seems to be the solution favoured by Fine (1985b, p. 19).
Fine (1985b, p. 101).
Fine (1985b, pp. 75–80).
Fine (1985b, p. 41).
Fine (1985b, p. 11).
Our account is in some ways very close to that involved in systems of Hilbert’s Epsilon Calculus though as far as we know Hilbert was not particularly concerned with the metaphysical underpinnings of the epsilon operator. The idea that instantial terms associated with EI refer to particular objects appears in Mackie (1958, pp. 30–31), though it is not clear that Mackie is sensitive the problem of having more than one object potentially satisfying the existential quantifier, and to the metaphysical implications of such cases. Finally, Fine (1985b, pp. 136–138) discusses (and criticises) the view that instantial terms refer to ordinary particular objects, though the account he has in mind seems to be one where the agent is aware of which particular object is being referred to, rather than an account based on AR.
Fine (1985b, p. 71).
One class of exceptions is that we can, in a sense, demonstrate that Jane has been named, referred to, referred to by us, and so forth: all properties that Jane may not share with all persons. But this does not seem to us a grave problem: there is indeed no temptation to apply UG to these special cases.
Fine (1985b, p. 75).
Fine also observes this peculiarity and thinks it needs to be explained, but offers a different explanation than ours—one involving his distinction between occupied and vacant objects (see Fine 1985b, p. 80).
Benacerraf (1965). (Benacerraf uses a different example then the one we present above, but the problem is basically the same.)
One interesting question about our account is whether it entails that we all refer to the same object by ‘〈a,b〉’. It is fairly easy to work out our account so that at least different speakers of the same community use the term with the same reference: after all, it may be that the reference is fixed arbitrarily once and for all for the community, and then used by everyone parasitically on the initial reference fixing, as is standard in the framework of the casual theory of reference. It may be harder to show that a different community with similar ‘order pair’ practices refers by ‘〈a,b〉’ to exactly the same object as we do, but we do not see this as terribly worrying.
It is worth noting that the actual world might well contain examples of this kind. For examples, according to some interpretations of quantum field theory, there are in fact cases of distinct yet completely qualitatively identical particles.
This is not to say that one cannot try to provide an account of how such statements can be truthful even if ‘i’ fails to refer. The point here is that such an account will be needed, and that one cannot simply be content by saying that this is a case of reference failure.
See especially Williamson (1994).
See Kearns and Magidor (2008) for a detailed argument.
One difficult question, however, is how the AR proposal would address the problem of higher-order vagueness. We leave this issue for further work.
Further potential applications of AR that we have not discussed here include the problem of partially defined predicates, the problem of inscrutability of reference, the semantics of variables, the semantics of pronouns, and the semantics of conditionals.
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Acknowledgments
We are grateful to audiences at Cornell University, CSMN Oslo, Macquarie University, and the University of Oxford, as well as to Ross Cameron, John Hawthorne, Stephen Kearns, Jeff King, Vann McGee, Moritz Schulz, Stewart Shapiro, Nick Smith, and Robbie Williams for helpful discussions.
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Breckenridge, W., Magidor, O. Arbitrary reference. Philos Stud 158, 377–400 (2012). https://doi.org/10.1007/s11098-010-9676-z
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DOI: https://doi.org/10.1007/s11098-010-9676-z