[This is an early draft of a paper which, since being posted, has grown and changed title. Email me if you would like a copy. - TH 9/4/15]
In an interesting paper forthcoming in Phil. Studies, Breckenridge and Magidor argue for this 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.
Their primary argument in favour of AR is that it can be used to give an attractive account of 'instantial reasoning' such as this (their 'Argument 1'):
(1) There is someone x such that for every person y, x loves y [Premise]
(2) Let John be such a person
(3) For every person y, John loves y [Existential Instantiation on 1]
(4) Let Jane be an arbitrary person
(5) John loves Jane [Universal Instantiation on 3]
(6) There is some person x such that x loves Jane [Existential Generalisation on 5]
(7) But since Jane was an arbitrary person, for every person y there is some person x such that x loves y [Universal Generalisation on 6]
I will not attempt to rehearse, or even summarize, their arguments, since they state them well and their paper is freely available on Magidor's website. My purpose here is to give an apparent counterexample to the claim that AR can be used to give an attractive account of instantial reasoning.
The following appears to be a logical truth:
(Unref) If (all unreferred-to objects are white and there is an unreferred-to object), then there is a white object.
(By 'unreferred-to object', I mean an object which is never referred to by anyone or anything.) Here is a quasi-formal argument for (Unref):
(1) All unreferred-to objects are white and there is some unreferred-to object. [Assumption]
(2) All unreferred-to objects are white. [Conjunction Elimination on 1]
(3) There is some unreferred-to object. [Conjunction Elimination on 1]
(4) Let O be such an object.
(5) O is white. [Universal Instantiation on 2]
(6) There is some white object [Existential Generalization on 5]
(Unref) now follows from (1) - (6) by conditional proof.
This seems to be a valid argument. But the theory of instantial reasoning advanced by Breckenridge and Magidor seems to imply that the expression 'O' above refers to an unreferred-to object, which is absurd.
The University of Sydney
Breckenridge, Wylie & Magidor, Ofra (forthcoming). 'Arbitrary reference'. Philosophical Studies.
There is a post about this paper on Ross Cameron's blog here.