Arbitrary Reference

(with Ofra Magidor)

    A quiz

  1. Let Marie be an arbitrary French woman.
  2. The view

  3. Ofra and I are defending the following view:

    We can and sometimes do refer to things arbitrarily

    Suppose I have two dogs in front of me. I can refer to each of them particularly. I can also refer to one of them arbitrarily:

  4. When I do, I succeed in referring to one of the two dogs. But I do not and cannot know which.
  5. An explanation of the quiz

  6. We propose that in the opening line 'Marie' gets its reference fixed arbitrarily to a particular French woman, although we do not know which. We do know some things about her (she is French, she is not a man), but we do not know whether she lives in Paris. This explains why we respond the way we do.
  7. Knowledge failure

  8. That we do not and cannot know to which things 'Marie' and 'Fido' refer is not all that strange - the same can be true of particular reference:
  9. Brute facts

  10. What is strange is the way in which these reference facts come about. In the Fido case, there seems to be no non-semantic fact that determines which dog I have referred to (unlike in the die rolling cases). At least we can't think of any such fact, so we propose that there isn't one. That 'Fido' refers to the dog that it does is a brute semantic fact. We take this to be the most problematic feature of the view.
  11. Conditions

  12. Under what conditions can we refer arbitrarily? Here is a sufficient condition: If I can refer to each of a number of things particularly, then I can refer to one of them arbitrarily. So arbitrary reference is no more difficult than particular reference.
  13. It might even be easier. There are two numbers x such that x2 + 1 = 0. Suppose we want to name one of them 'i' - how can we do it? There is no non-haecceitistic property that one has but the other does not, so we cannot use these to uniquely pick out one and name it 'i'. There might be a haecceitistic property that one has but the other does not (such as being identical to i), but we cannot express these properties until after we have referred to the two numbers. We propose that this is a case in which we cannot refer to either of the two numbers particularly, but we can refer to one of them arbitrarily. If so, this is a case in which arbitrary reference is easier than particular reference.
  14. Why think that the view is true?

  15. Because:
  16. Instantial reasoning

  17. We propose that we refer arbitrarily every time we employ instantial reasoning. Suppose I want to argue that if every dog is cute and every dog is fluffy then every dog is cute and fluffy. Here is one way to do it, using instantial reasoning:
      The dog argument
    1. Suppose that every dog is cute and every dog is fluffy.
    2. So every dog is cute.
    3. And every dog is fluffy.
    4. Let Fido be some dog.
    5. Then Fido is cute.
    6. And Fido is fluffy.
    7. So Fido is cute and fluffy.
    8. So every dog is cute and fluffy.
    9. So if every dog is cute and every dog is fluffy then every dog is cute and fluffy.

    Question: what role is 'Fido' playing in this argument, and how are the steps in the argument justified? We argue that arbitrary reference gives the best explanation.

  18. Our explanation

  19. We propose that 'Fido' names a particular dog by having its reference fixed arbitrarily in line 4, and that the steps are justified as follows:
    1. Suppose that every dog is cute and every dog is fluffy.
    2. Every dog is cute. (From 1, by simplification)
    3. Every dog is fluffy. (From 1, by simplification)
    4. Let Fido be some dog. ('Fido' now refers to a particular dog)
    5. Fido is cute. (From 2 and 4, by universal instantiation)
    6. Fido is fluffy. (From 3 and 4, by universal instantiation)
    7. Fido is cute and fluffy. (From 5 and 6)
    8. Every dog is cute and fluffy. (From 7, and the fact that in the derivation of 7 we have not appealed to anything about Fido that is not true of every dog)
    9. So if every dog is cute and every dog is fluffy then every dog is cute and fluffy. (From 1-8, by conditional proof)
  20. Note. We might have given a 'without loss of generality' (WLOG) argument instead - replace line 4 with, "Without loss of generality, consider my dog Fido." The argument would still be a good one. The explanation for why a WLOG argument works carries across, we propose, to cases of instantial reasoning.
  21. Alternative explanations

  22. What are the alternatives? Here are the main ones (that we know of):
  23. 'Fido' is a meaningless symbol

  24. ... and we use it because it helps us move from truths to truths (compare: formal deductive systems).
  25. Problems:
  26. 'Fido' is an unbound variable

  27. ... and the sentences in which it occurs are open formulae.
  28. Problems:
  29. 'Fido' is a variable, implicitly bound at the sentential level

  30. (Jeff King has a sophisticated version of this view - his 'context dependent quantifier' view.) We can make the dog argument more explicit as follows:
    1. Suppose that every dog is cute and every dog is fluffy.
    2. Every dog is cute.
    3. Every dog is fluffy.
    4. Let x range over dogs.
    5. For every dog x: x is cute.
    6. For every dog x: x is fluffy.
    7. For every dog x: x is cute and fluffy.
    8. Every dog is cute and fluffy.
    9. So if every dog is cute and every dog is fluffy then every dog is cute and fluffy.
  31. Problems:
  32. 'Fido' refers to a dog but does so indeterminately

  33. 'Fido' has had its reference fixed only partially in line 4 of the dog argument - it remains to be 'precisified'. A sentence like 'Fido is cute' is true iff it is true relative to every precisification of 'Fido' (i.e. every way of fixing its reference to a particular dog); it is false iff it is false relative to every precisification of 'Fido'; it is neither true nor false otherwise.
  34. Problems:
  35. 'Fido' refers to a special kind of thing - the arbitrary dog

  36. (Kit Fine). The arbitrary dog is something in addition to all dogs. Is it a dog? Yes, on a generic reading of 'is a dog', but not on the classical reading (or else we get problems with counting). On the generic reading it has exactly those properties that are had by all dogs.
  37. Problems:
  38. Future work

  39. The view generalizes: we can and sometimes do fix the content of an expression arbitrarily. In particular, we can fix the content of a predicate arbitrarily. We are developing a view of vagueness according to which 'heap' (for example) had its content fixed arbitrarily to a certain (sharply defined) property. Thus 'heap' has a sharp cut-off point, but we do not and cannot know where it is. This gives an epistemicist view of vagueness, one that explains our ignorance in a way that differs from Williamson's.