Draft, August 19, 2007
Arbitrary Reference
Wylie Breckenridge and Ofra Magidor
[Preliminary Draft - please do not circulate]
Let John be an arbitrary Frenchman. Is John French? Is he identical to Tony Blair? Does
he live in Paris? On the face of it, these seem to be intelligible questions, and we take it
that the intuitive answers to them are ‘yes’, ‘no’ and ‘we don’t know’. But how exactly
are we to make sense of such talk of ‘an arbitrary Frenchman’? And what is the semantic
role of ‘John’ in such contexts? In this paper we propose that in contexts such as above
‘John’ is an ordinary proper name, and we use it to refer to a particular Frenchman.
However, we do not and cannot know which Frenchman ‘John’ refers to, because the
reference of ‘John’ was fixed arbitrarily. Thus the intuitive answers to the above
questions are entirely correct: John is a Frenchman. He is not identical to Tony Blair (we
know this because Tony Blair is not a Frenchman and John is). And we do not know
whether or not John lives in Paris (because we do not know which Frenchman John is and
not all Frenchmen live in Paris – if they did, and we knew so, then we would know that
John lives in Paris). More generally we propose and defend the following claim:
Arbitrary Reference (AR): It is possible to fix the reference of names arbitrarily.1 When
we do so, our names refer in the ordinary way, but usually we do not and cannot know to
what they refer.2 .
Maybe split AR into two claims? One about the ability to refer arbitrarily, and one about
our lack of knowledge (an epistemic claim). Just to make clear that the second is an
additional part of what we are claiming (it may have been that we can refer arbitrarily,
but always end up knowing which thing we end up referring to).
In fact, we think this is true not only of names but also for the denotation of any word – e.g. we can fix
which property a predicate denotes arbitrarily. However, for the most of the paper we will focus on the case
of names.
2
We say ‘usually’ because if we fix the reference of ‘a’ to be an arbitrary F where we know there is only
one F, then might know to which object ‘a’ refers.
1
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Whether or not one accepts AR, one might wonder why the project of explaining such
talk of ‘arbitrary objects’ is important. One main reason is that such alleged references to
arbitrary objects plays an important role in a certain kind of reasoning which, following
Kit Fine, we call ‘instantial reasoning’. Here is an example of instantial reasoning.
Suppose we want to show that from the claim that there is someone who loves everyone,
it follows that everyone has someone who loves them. We could reason as follows:
Our claim is that we can refer arbitrarily. We use the example of John the French man to
support this. We use the example of instantial reasoning to argue for a stronger claim
from which AR follows – we argue that we do refer arbitrarily (from which it follows that
we can), the argument being that it gives the best account of instantial reasoning –
inference to the best explanation.
Argument: Assume that there is someone who loves everyone. Let a be a person that
loves everyone. Let b be an arbitrary person. Since a loves everyone, a loves b. So b has
someone who loves them. But b was arbitrary, so it follows that everyone has someone
who loves them.
This argument incorporates two interesting steps. First, we have inferred from the fact
that there is someone is loves everyone, that a loves everyone. Note that we have not
specified who ‘a’ refers to, or who a is (in a sense we have. Maybe say this: there is no
particular person we have in mind for ‘a’ to refer to?). Moreover, we cannot implicitly
assume that a is the person who loves everyone – because it is perfectly consistent with
the assumption in the above argument that there is more than one person who loves
everyone. Rather, what seems to be implicitly assumed is that a is an arbitrary person
who loves everyone. Second, we have inferred from the fact that b - which we explicitly
introduced as an arbitrary person - has someone who loves them that everyone has
someone who loves them. These two steps seem to make use of the informal correlates of
the following two rules of inference:
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Existential Instantiation (EI): From ‘xφ(x)’ deduce ‘φ(a)’ [with some appropriate
restrictions on ‘a’].
Universal Generalization (UG): From ‘φ (a)’ [with appropriate restrictions on ‘a’],
deduce ‘x(φ(x))’.
Arguments that make use of IE or UG (or informal correlates of them) are arguments by
instantial reasoning, and such arguments play a fundamental role in formal and informal
reasoning. But as we have seen, such arguments seem to make both implicit and explicit
uses of stipulations such as ‘Let a be an arbitrary F’ (the implicit stipulations occur in the
case of EI – where it is implicitly assumed that a is an arbitrary φ). Thus, in order to make
sense of such arguments and justify their validity, it is crucial that we make sense of such
talk of arbitrary objects. AR is an important thesis in part because it provides us with a
way (we think the best way) to make sense of such talk.
[Note: maybe take this whole bit out of the official paper?].
The structure of the paper is as follows: In §1, we defend AR against some general
objections and in the process also clarify our view. In §2 we argue that AR can be used to
explain instantial reasoning and justify its validity. Moreover, we argue that our
explanation is superior to other explanations that could be (and have been) given, and
thus that by inference to the best explanation we should accept AR. Finally in §3, we
briefly point out some further potential applications that we hope AR has, but which need
some further development.
§1 Some objections and responses
Not only has AR not been previously defended,3 it seems that the view has not been taken
seriously enough to be explicitly argued against. However, in this section we have
[$$$ Though there are some hints that Berkley held something like this view… Also check
MackieREFS…$$$].
3
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collected some objections that we anticipate might be raised against our view. We hope
that our responses will at least help clarify exactly how the view works.
Objection 1: AR cannot be true because it violates necessary conditions on reference.
One can only refer to an object if one stands in the relevant kind of causal connection to
it. When I say ‘Let John be an arbitrary Frenchman’, I do not stand in the right kind of
casual connection to any particular Frenchman, and hence cannot use ‘John’ to refer to
one of them.
Response: First, we should note that we are quite sceptical of such necessary conditions
on reference. We believe that stipulations such as ‘Let ‘Julius’ refer to the actual tallest
spy’ are perfectly acceptable and (at least assuming that there is a unique tallest spy), we
can successfully refer to the tallest spy using ‘Julius’, even though we may not stand in
any interesting casual connection to the tallest spy.
Second, even if we accept such necessary conditions on reference, the objection does not
really threaten AR. All one needs to do is to change the example to one in which there is
a collection of objects for which one does stand in the relevant casual connection to. For
example, suppose there are two cups on the table in front of me, such that I am in able to
refer (according to the objector) to each of the two cups individually. Now let Jake be an
arbitrary cup on the table. According to our view, ‘Jake’ arbitrarily refers to one of the
cups on the table, as is predicted by AR. But objection 1 is not applicable to this case
because the necessary causal conditions for reference are satisfied.
Finally, we should clarify that it does not follow from our view that any stipulation of the
form ‘Let a be an arbitrary F’ successfully results in ‘a’ referring to an object. For a start,
if there are no Fs the stipulation fails, and perhaps this happens also if other conditions
fail to obtain. All that AR requires is that some such stipulations succeed.
Objection 2: If we succeed in fixing reference to a particular Frenchman then something
must determine which Frenchman it picks out. But it is difficult to see what, according to
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AR, determines that. It cannot be the words I use (because I haven’t said anything about
John except that he is an arbitrary Frenchman). Nor can it be my intentions (because there
is no particular Frenchman that I intend to refer to).
Response: We agree that it is neither your words nor your intentions that determine
which Frenchman ‘John’ refers to. In fact, we accept that nothing determines which
Frenchman is picked out. We simply deny the claim that something must determine
which Frenchman is picked out! This may seem like an absurd claim, but it strikes us as
no more absurd then the claim that there are non-deterministic processes in nature – a
claim that as far as we understand is standardly accepted by physicists.
Objection 3: It is standardly accepted that reference supervenes on use. What our words
refer to is determined by the way they are used. But it seems as though according to your
view, reference doesn’t supervene on use. For suppose in the actual world I make the
stipulation ‘Let John be an arbitrary Frenchman’ at time t, and that as it happens ‘John’
picks out Jacques Chirac. There is another possible world w’, which up to time t is an
exact duplicate of the actual world (and thus certainly has the same use facts as in the
actual world), but where in w’, when I utter the same sentence, ‘John’ ends up picking
Nicholas Sarkozy. It thus seems like whatever one counts as use facts, which person
‘John’ refers to does not supervene on use.
Response: That is correct, it is part of our view that reference does not supervene on use
and that the standard assumption that it does is simply wrong. Of course, this does not
mean that use does not play a very important role in determining reference. In the above
case, for example, our use at least ensures that ‘John’ will refer to a Frenchman. And in
general, use might well seriously constrain which things our words could refer to. But on
our view, it does not always completely determine it.
Objection 4: You claim that ‘John’ refers to a particular Frenchman, but that we do not
and cannot know to which Frenchman. But what’s your explanation for the fact that we
do not and cannot know which Frenchman John is?
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Response: Compare this with the case where you are going to flip a coin (assuming that
coin flips are genuinely non-deterministic). You do not and cannot know which side the
coin will land because nothing in the current state of the world determines which side it
will land. Thus no matter how many facts you know about the world (except, of course,
for primitively future facts like that the fact that the coin will land on heads), you cannot
know which side the coin will land. Similarly, according to our view, nothing in the state
of the world prior to your making the stipulation determines which Frenchman is going to
picked out by the stipulation (as we noted above, even all the use facts are not going to
determine this – reference does not supervene on use). Thus no matter how many facts
you know about the world (except, of course, for primitively future facts like that the fact
that ‘John’ will refer to the Frenchman x), you cannot know which Frenchman ‘John’
will pick out.
Admittedly, there is a disanalogy between our case and coin flipping case. For in the coin
flipping case you can at least know the result of the coin flip after it was flipped. Why is
it, then, that we cannot know who ‘John’ refers to after the stipulation is made and the
reference of ‘John’ is determined? It is not obvious how to answer this challenge except
to point out that which person ‘John’ refers to makes no non-semantic difference to the
world. Since we do not have access to primitive semantic facts, we cannot know who
John refers to.
This might be the place to allow for the alternative which I suggested – that it makes a
non-semantic difference which object gets referred to (collapse of a quantum distribution
into a definite state, perhaps partly in the brain). This would not alter AR at all – it would
just make our inability to know a matter of not being able to make the necessary
measurements after the collapse. Maybe that possibility is enough to allay the concerns of
objection 4, without us having to take a stand. (OK, but we need to ask someone more
physicsy…).
Objection 5: You say that ‘we cannot know who ‘John’ refers to. This seems wrong. For
example, what if a being that we know to be omniscient tells us who ‘John’ refers to?
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Response: We certainly think that an omniscient being can (and does) know who ‘John’
refers to. Such a being can presumably have direct access to the primitive semantic facts
(We think there is a fact of the matter, and the omniscient being, by definition, knows all
the facts, so she knows this fact in particular). And it is also conceivable that we can
come to know such facts, e.g. by testimony from an omniscient being. When we said that
we cannot know who ‘John’ refers to, we mean the word ‘can’ to be taken in a sense that
is restricted to possibilities that do not involve scenarios such as testimony from
omniscient beings.
Objection 6: Even if the stipulation fixes the reference of ‘John’, it cannot fix it to a
particular Frenchman, because one would have achieved exactly the same effect if
instead of saying ‘Let John be an arbitrary Frenchman’, one said ‘Let ‘John’ be a
Frenchman – no particular one’, or even ‘Let John be an arbitrary Frenchman – I mean
no particular one’.
Response: We agree that you would have achieved the same effect if you had used one
of the latter stipulations instead, but we think the effect would still have been that ‘John’
refers to a particular Frenchman. The qualification ‘no particular one’ is a qualification of
the manner of fixing the reference, rather than of the Frenchman referred to. That is to
say, it points out that we are fixing the reference arbitrarily rather than particularly. This
is not uncommon use of ‘particular’ in English. Consider for example the announcer of a
competition result who says ‘In no particular order, the winners are A, B, and C!’. In one
sense, the winners were obviously were given in a particular order: first A, then B, then
C. But in another sense they were not: the order was chosen arbitrarily and not
particularly. Similarly, we suggest, John is a particular Frenchman in the former sense,
but not a particular Frenchman in the latter sense.
Objection 7: Even if the stipulation fixes the reference of ‘John’, it does not fix it to an
ordinary Frenchman, but rather to a special kind of object – an arbitrary Frenchman.4
4
Cf. §2.3 below.
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Response: We are not denying that it is literally true that ‘John’ refers to an arbitrary
Frenchman. But we are denying that this makes John a special kind of object. On our
view, an arbitrary Frenchman is just an ordinary Frenchman that is referred to arbitrarily.
Compare this to the case of making an intentional mistake. It is literally true that the
mistake you make is intentional. But an intentional mistake is not a special kind of
mistake – it’s just a mistake made intentionally. Similarly, suppose I give you a pen as a
gift. The pen in question is then a gift. But this does not make the pen a special kind of
object (a ‘gift-pen’ as opposed to an ordinary pen). A gift is just an ordinary object that
was given in a certain way.
§2. AR and Instantial reasoning (rather than thinking of this as an application of AR,
maybe think of this as an argument for AR – we actually do it, so we can do it (as per my
remarks above). (I think it’s both)
Having responded to some general objections, we now turn to motivate the view by
arguing that it provides the simplest and best explanation of how instantial reasoning
works. [Note: if taking out first section then must insert a bit here about instantial
reasoning]. Consider for example the following mathematical proof, which involves a
typical use of instantial reasoning.
Claim: Every multiple of 4 is even.
Proof: Let n be an arbitrary multiple of 4. By definition of ‘multiple of 4’, x(n=4x). Let
k be a number such that n=4k. Since n=4k=2(2k), n=2(2k). So x(n=2x). So by definition
of ‘even’, n is even. But n was arbitrary, so every multiple of 4 is even.
§2.1 Our explanation
Here is the proof again, annotated according to our view:
 Let n be an arbitrary multiple of 4.
This fixes the reference of ‘n’ to a particular multiple of 4 (perhaps 28).
 By definition of ‘multiple of 4’, x(n=4x).
 Let k be such that n=4k.
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This is an instance of EI. It sets k to be an arbitrary number such that it multiplied by 4
equals n. Note, though, that in this particular case there is only one number which
satisfies the constraint in question, so that we do not really need to appeal to AR to set the
reference of k. Right, and it seems unnatural to think that we fix the reference of ‘k’
arbitrarily above – rather, we let k be the number such that so-and-so. So I’m inclined to
omit this application of AR in the proof. [Not sure I want to – because I think in general
EI is related to arbitrary reference… This is just a limit case].
 Since n=4k=2(2k), n = 2(2k). So x(n=2x). So by definition of ‘even’ n is
 even.
This establishes that the particular number that ‘n’ refers to is even.
 But n was an arbitrary multiple of 4, so every multiple of 4 is even.
This is no doubt the most interesting step of the proof. It involves an application of a
version of UG. Roughly, the version says that if it was proved that an arbitrary F is P then
we can deduce that every F is P. In this case we have proved that n, an arbitrary multiple
of 4, is even and deduce from this that every multiple of 4 is even. The justification for
this move is that since we don’t know which multiple of 4 n is, we can be sure that any
fact about n that we appealed to in the proof is true of any multiple of 4.
Now the obvious objection to this version of UG is that it may seem like, on our view,
UG will come out an invalid rule of inference, i.e. a rule of inference the application of
which can lead us to deduce falsehoods from truths. This seems to be the case because on
our view ‘n’ picks out a particular multiple of 4 and there are some properties of n that
are not shared by all numbers. For example, if n happens to be 28 then it is true that n is a
multiple of 14. But it would be wrong to deduce from that that every multiple of 4 is a
multiple of 14. Thus if UG is the rule that says that if a is an arbitrary F and P(a), then we
can deduce that the every F is P (formally: P(a) [where a is an arbitrary F]├
x(F(x)P(x))), then UG will lead us to infer falsehoods from truths and should be
rejected.
Our response to this objection is that the version of UG we accept is different than the
one just rejected. On the version we wish to adopt UG says (roughly) that if a is an
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arbitrary F, and if it was proved that P(a), we can deduce that every F is a (formally: if ├
P(a) [where a is an arbitrary F], then├ x(F(x)P(x)). The justification for our version
of the rule is that in our proof of P(a), the only facts about a that one is able to appeal to
are facts which are true of all Fs.5 The use of rules of inference of this form is not
unfamiliar in logic. For example all normal modal logic systems (i.e. all modal logics that
extensions of the system K), one gets the rule of necessitation according to which if ├ p
then├ □p. Now it is important that the rule does not say: p ├ □p. There are many
instances for which p is true, but □p is false, and thus adopting the latter rule would lead
us to infer falsehoods from truths. But the rule of necessitation as it is actually stated is
completely acceptable: if p is provable, then p is not only true but is a logical truth. And
if p is a logical truth then it is a necessary truth, and hence □p is true.
It is natural to think of ignorance as a kind of epistemic deficiency. But the above
discussion shows that sometimes ignorance can give you an epistemic advantage. The
fact that we do not which number n is helps us to ensure that everything we prove about n
will be also true of all multiples of 4. Thus our ignorance helps us ignore those facts
which in this context ought to be ignored. There are probably other cases where
ignorance provides us with this power. Suppose for example that you are on an
admissions committee for a university, and you want to ensure that you assess the
candidates only on the basis of their qualifications and not on the basis of their race,
gender, or economic background. One way to achieve this might be to try as hard as you
can to not let the facts you know about their race, gender, or background to factor into
your decision. However, this is hard to achieve: as much as you try, these facts can
subconsciously influence your judgment. A better way to achieve this is to simply make
sure the application files you consider do not contain any information about the
5
One worry with this claim is the following. Let n is arbitrary number. There are some facts, which we do
know about n that are not true of all numbers – namely semantic facts. For example, we know that n has a
name, and we know that n has been referred to. But not all numbers have a name and not all numbers have
been referred to. Doesn’t our version of UG entail that since we know that n has been referred to then we
are entitled to deduce that every number has been referred to? Our response to this worry is that knowing a
claim is different than logically proving it. Roughly, in order to prove it, one would need to establish that it
is true in every possible situation. It is true that if n is actually an arbitrary number, then n is actually
referred to. But it not true if n is an arbitrary number in a possible situation s, then n is refereed to in s.
[$$say more about this…$].
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candidates’ race, gender, or background. If you are simply ignorant of these facts, you
can be sure that you are ignoring them in your decision making process. (A similar point
can be made with respect to double blind tests). It is often acknowledged that there are
sometimes practical advantages in ignorance (e.g. ignorance of certain facts might make
you happier). But ignorance can sometimes have epistemic benefits too.
It seems then, that by appealing to AR, we are able to give a fairly straightforward
semantic story of how proofs using instantial reasoning work. But that would not be a
very convincing argument in favour of AR if it turned out that there were equally good
(or even better) explanations, which do not appeal to AR. In the remainder of this section
we briefly mention some alternative explanations that could be given to instantial
reasoning, and show why they face serious problems and complications.
[$$ note: the following two sub-sections are particularly rough, so please bear with us
…$$$]
[reference Fine’s chapter in these sections].
§2.2 Non referential views
The obvious way to oppose our view is to claim that not only does ‘n’ in the above proof
not refer to a particular multiple of 4 - it does not refer at all. Here are various possible
versions of this view, and some of the reasons we think they should be rejected.
Theory 1: Not only does ‘n’ in the above proof fail to refer, but all the statements in the
proof that involve ‘n’ are meaningless. The steps in the above proof involve a
meaningless manipulation of symbols that is merely of instrumental value for proving
meaningful and true statements such as ‘Every multiple of 4 is even’.
Problem:
Our main objection to this view is that if the meaningless manipulation of symbols
always gives us correct results, then we would like some account for why this is so.
Compare this to the case of a deductive formal system such as predicate calculus. It is not
unreasonable to claim that statements in predicate calculus are simply meaningless
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manipulations of symbols. However, we also have some semantic account for why this
manipulation of symbols gives us correct results. For example, the deduction rule that lets
us infer P from PQ is justified by the apparatus of giving semantic interpretations to
formulas of predicate calculus, and by the semantic claim that for any interpretation I to
the formulas of predicate calculus, if P is true relative to I, then PQ is true relative to I.
(In short, it is shown by the soundness theorems for Predicate Calculus). In contrast
theory 1 (at least as it stands) does not provide a notion of a semantic interpretation to
statements containing apparent reference to arbitrary objects, and hence does not give any
account for why instantial reasoning is valid.6
Theory 2: ‘n’ in the above proof should be treated as a variable, and all the formulas
containing n should be treated as open formulas.
Problem: The main problem with theory 2 is that if some steps in the above proof are
open formulas, than according to the standard semantics for predicate calculus all such
steps are truth-valueless. But this seems wrong. Intuitively, we can talk of each step as
being true or false, correct or incorrect. Moreover, we think of the steps of a proof as
being valid because they preserve truth, and of the conclusion of a proof as being true
because it follows from true premises using valid steps. But if some of the steps in the
proof are truth-valueless, then it is unclear how the notion of validity applies, and why we
can conclude that the conclusion of the argument is true.7
Theory 3: ‘n’ in the above proof should be treated as a variable. However, instead of
treating all formulas containing n as open we can treat the proof as implicitly starting
One suggestion would be to treat the above proof as a kind of proof schema, where ‘n’ is treated as a
schematic letter which could be replaced for a name of an actual multiple of 4, yielding a proof that the
multiple of 4 is even. The worry with this suggestion is that it is not clear why it warrants the last step,
namely the claim that every multiple of 4 is even. After all, some numbers do not have a name (really? If I
asked you to give an example you’d end up using a name for the number, which shows that it does have a
name and hence is not an example) so one cannot use an instance of the proof schema to show that they are
even. [$$say more about this$$$]. [don’t agree with W’s comment, but the problem is that every number
could be given a name and then we can follow the schema… Note that Fine doesn’t have knock down
objections to this.. .Though problem with EI – cannot interpret ‘Fa’ as true for any ‘a’ (maybe for any F?)
7
Another problem with all the views that construe ‘n’ as a variable is that we are hoping to show that the
best way to explain how the semantics of variables works is by appeal to AR, and thus appeal to variables
here just shifts the same issue again. However, for the moment we leave this issue aside.
6
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with a universal quantifier of the form ‘for every multiple of four, n’, and all subsequent
formulas as being in the scope of this quantifier.
Problem: One problem with theory 3 is that although it would make the proof as a whole
truth-valuable, it does not make the individual steps of it truth-valuable. But that seems
wrong. We can talk of individual steps as being true or false, correct or incorrect. In
particular, we might tell someone that the first three steps of their proof were correct,
while the fourth was incorrect.
Another problem with theory 3 is that it is unclear how exactly this theory is supposed to
work. What is the connective that connects all the formulas in the scope of the quantifier?
If it’s simply a conjunction then the proof will be interpreted as having the following
structure: ‘For every multiple of 4 n, n is a multiple of 4 and x(n=4x)….and n is even)’.
But this seems wrong – isn’t each step supposed to be established from the previous steps
rather than just stated as part of a long conjunction? More plausibly, we might try to go
for a more complicated combination of a conjunction of implications: ‘For every multiple
of 4 n, n is a multiple of 4 and if n is a multiple of 4 then x(n=4x) and if x(n=4x)
then… and if x(n=2x) then n is even’. But this doesn’t establish the conclusion without
first, showing why this conjunction of implications is correct and second, some further
step which shows that the long conjunction entails that every multiple of 4 is even. Thus
according to the suggested structure, some steps in the proof should be construed as
material implications while others should be construed as deductions. The full version of
the proof would look roughly as follows.
(1) ├ n such that n is a multiple of 4 (n is a multiple of 4 and if n is a multiple if… and
if… then n is even).
(2) n such that n is a multiple of 4 (n is a multiple of 4 and if n is a multiple of 4
then…and… then n is even) ├ n such that n is a multiple of 4.
(3)├ n such that n is a multiple of 4.
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But it is much more natural to construe all the steps as involving deduction rather than
implication, and the proof as having something like the following structure:8
(1) ├ n is a multiple of 4
(2) n is a multiple of 4 ├ x(n=4x)
(3) ├ x(n=4x)
(4) x(n=4x) ├ n=4k
…
(j) x(n=2x) ├ n is even.
(j+1) ├ n is even.
(j+2) ├ every multiple of 4 is even.
As the deduction theorem shows, deduction is closely related to material implication. But
the very fact that we need the deduction theorem, also shows that ‘A├B’ is not the same
claim as ‘├AB’. We are not arguing here that theory 3 cannot provide us with some
sound reconstruction of the proof, but rather that this reconstruction is not very faithful to
what seems to be the structure of the proof (it is at least less faithful than the
reconstruction provided by AR).
Our final problem with theory 3 is that it is harder to see how the theory can be
accommodated to explain cases of EI. But most writers on this topic (including us) have
assumed that EI and UG are highly related, and that however we want to understand talk
of arbitrary objects, both inference rules are explained by appeal to such talk – UG
involves inferring from the fact that an arbitrary F is P that all Fs are P, and EI involves
inferring from the fact that there exists an x such that P(x) to the fact that P(a), where a is
an arbitrary P. Theory 3 does not seem to do justice to this analogy. [Say more about this
– see Fine’s book… The biggest problem is the interaction of the two…].
8
Admittedly, the move from (j+1) to (j+2) is not completely in intuitive (see discussion of UG above), but
still we expect the step to involve deduction rather than implication.
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Theory 4: ‘n’ in the above proof should be treated as a variable, and each of the formulas
containing n should be taken as implicitly starting with the universal quantifier ‘For every
multiple of four n’. maybe use general quantifier notation?: [every n: n is a multiple of 4]
Problem: One advantage of theory of 4 over any of theories 1-3 is that according to
theory 4, each step (better: sentence/formula?) in the proof has a truth-value. However,
theory 4 faces problems of its own. First, it is not clear what the role of stipulations such
as ‘Let n be an arbitrary number’ is according to theory 4. One might think this
stipulation indicates that all the universal quantifiers over n in the proof range over
numbers. But that cannot be right, for we can have embedded stipulations such as ‘Let n
be arbitrary number, and let m be an arbitrary number greater than n’. If n is simply a
variable then it is unclear what we have specified m as ranging over. (And surely we
would not want the second stipulation to be bound by the universal quantifier ‘For all
numbers n’)(this needs a bit of fleshing out – it seems natural to think that we have
specified m as ranging over numbers greater than n, so we need to add some comments
about why this is no good). Second, it is not clear what the role of the last step of the
proof is: if the penultimate step of the proof ‘n is even’ is actually elliptical for ‘For every
multiple of 4 n, n is even’ then it is essentially identical to the last step (‘every multiple of
4 is even’), so it is unclear why we need to repeat this statement in the last step (Yes. In
the last step it feels like we are adding to the proof, drawing a conclusion, not just
reiterating or emphasising the previous step). Finally, it is not clear how theory 3 can
account for ‘splitting proofs’, i.e. proofs that are split into several cases. Suppose that we
want to prove that every number has certain property P. We might construct a proof with
the following structure: ‘Let n be an arbitrary number. Suppose that n is prime. … It
follows that n is P. Suppose that n is composite. … It follows that n is P. So n is P. So
every number is P’ (would be helpful to use an actual example here). [Maybe replace
splitting proofs with any proof that has a supposition – e.g. proofs by conditional proof!
Every multiple of 4 is such that if it is also a multiple of 3 then it’s a multiple of 12]. But
now consider how this proof is interpreted according to theory 4. The two suppositions
are interpreted as ‘Every number is prime’ and ‘Every number is composite’ respectively.
Since each of the suppositions is trivially inconsistent, it is easy to derive any claim from
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these suppositions. But clearly, we do not wish theory 4 to entail that for every property
P, every number is P. On our referential account splitting proofs are straightforward: it is
correct to infer from the claims that (i) n is either prime or composite; (ii) if n is prime
then P(n); (iii) if n is composite than P(n); that n is P. But according to theory 4, what is
involved is an inference from the claims that (i) every number is either prime or
composite; (ii) if every number n is prime then every number is P; (iii) if every number is
composite then every number is P; that every number is P. This inference is clearly
invalid and thus theory 4 needs to give some alternative construal of splitting proofs.
[Add something about Fine’s problems with applying EI to this theory…]. [Add
something about combining the wide scope and narrow scope strategy – e.g. using wide
scope only in suppositions – the problem is divorcing suppositions from assertions..].
[Also add something about mixed strategies, particularly King’s view… The view should
at least be subjected to the complaint about the last step being a repetition and the fact
that the view is very complicated].
§2.3 Alternative referential views
Another alternative to our view agrees with us that ‘n’ in the above proof refers to an
object, but denies that it refers to a particular multiple of 4. By far the most well-defended
and worked out view of this sort is Kit Fine’s view.9
According to Kit Fine, stipulations such as ‘Let n be an arbitrary number’ fix ‘n’ to refer
to a special kind of object – an ‘arbitrary number’, which is a distinct object from any
particular number. Roughly, an arbitrary number has all and only the properties that are
shared by all (particular) numbers. And in general, each arbitrary object has a ‘range of
values’ that is associated with it, and such that its properties are determined by the
properties of the values in the range.
Problems: Fine’s work contains an impressive defence of his view, with many intricate
details. We cannot do the view justice in such a short space. Nevertheless, we would like
to point out to several problems and deficiencies in Fine’s view: [maybe rephrase this
sentence]
9
REFS to Fine (two papers and book).
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First, as Fine himself points out, the view that arbitrary numbers have all and only the
properties that are shared by all numbers cannot quite be correct. For example, all
particular numbers have the property of being non-arbitrary, but arbitrary numbers do not
have this property. Similarly, all particular numbers have the property of being particular,
but arbitrary numbers do not have this property. To avoid this problem Fine distinguishes
between what he calls ‘generic conditions’ and ‘classical conditions’. Generic conditions
are properties such as ‘being a number’ or ‘being either odd or even’ – properties that
hold of an arbitrary number if and only if they hold of every number. Classical conditions
are properties such as ‘being non-arbitrary’ which do satisfy this principle. However,
Fine gives no account of what makes a condition generic, except for the fact that such a
condition applies to an arbitrary object if and only if it applies to all of the objects in its
‘value range’. Perhaps this alleged circularity is not a devastating objection to Fine’s
view, but the distinction between generic and classical conditions makes his theory at
least more complex and less elegant than one would wish for.
Second, consider generic conditions such as ‘being a number’. Suppose we ask how
many numbers are there between one and ten. The intuitive answer is that there are ten
such numbers. It seems that on Fine’s view, however, the answer ought to be 11: for in
addition to the ten particular numbers between one to ten, there is also the arbitrary
number between one to ten.10 A similar problem arises from the fact that according to
Fine all arbitrary objects are abstract objects. But since every person is concrete, and
‘being concrete’ is the kind of condition that Fine classifies as generic, it would seem that
the arbitrary person has to be concrete rather than abstract.
Thirdly, Fine has a problem with the question of multiple arbitrary numbers. In some
places he accepts that there can be more than one arbitrary number.11 This is problematic
because Fine then owes us an explanation of how stipulations such as ‘Let n be an
In some places Fine seems to say that the sense in which conditions such as ‘being a number’ are
interpreted as generic, it is simply part of meaning of this predicate that as applied to an arbitrary object o it
is true if and only if all the objects in o’s range are numbers. But this position seems highly unsatisfactory:
the predicate ‘is a number’ in English already has a sense, and that sense does not seem to involve such
generic paraphrases. [Probably best to move this to main text – this position is more emphasized in the
book].
11
[See Book formal details!]
10
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arbitrary number’ manage to fix the reference of ‘n’ to one rather than another arbitrary
number. In particular, if it turns out that such an explanation has to appeal to AR, then
AR ought to be accepted. But once AR is granted, one might as well use it to explain
instantial reasoning in the first place, without the need to appeal to Fine’s theory.
Perhaps for this reasoning, in other places Fine insists that there is only one (independent)
arbitrary number.12 This raises several issues. For a start, if there is only one arbitrary
number, why doesn’t the stipulation say ‘Let n be the arbitrary number’? More
importantly, this introduces serious complication to embedded introductions of arbitrary
objects such as ‘Let n be an arbitrary number, and let m be a arbitrary number that is
distinct from n’. According to Fine (on this version?), n and m cannot simply be two
distinct arbitrary numbers. Rather, he distinguishes between ‘dependent’ and
‘independent’ arbitrary objects. Thus while ‘n’ picks out the one and only independent
arbitrary number, ‘m’ picks out a dependent arbitrary number, one that is dependent on n
in a particular way. Roughly, the idea is that m and n cannot be assigned the same value
simultaneously. However, the notion of arbitrary objects being assigned values is highly
problematic: the initial thought that arbitrary objects are associated with a range of values
seemed innocuous enough. But it is not clear in what sense an arbitrary object is actually
assigned one particular value from its range. [Maybe delete this because it is clarified in
book!]. But it is not clear that this move solves all of Fine’s problems. Consider a
stipulation such as ‘Let m, n be arbitrary numbers’. At a first pass Fine can try to construe
m as an independent arbitrary number and n as one that is dependant on m (though with a
vacuous dependency constraint). But this would entail that mn – which clearly cannot be
inferred from the stipulation. To solve this Fine proposes that in this case m, n refers to
(respectively) the first and second elements of the arbitrary ordered-pair. [ref!]. But this is
problematic if n is introduced (in the same manner) much later in the proof than m was
introduced (this would mean that m has to retroactively fix its reference this way) –and
moreover it is not clear that this would help, since on the face of it the first and second
elements of the arbitrary ordered-pair must be distinct. (explain why – these are not just
numbers! They are individuated intentionally). .
12
REFS (paper, first chapter of book).
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And in general, the need to talk about value assignments for arbitrary objects, and the
notion of dependant and independent arbitrary objects complicates Fine’s view and
makes it further from ordinary intuitive reasoning. [Whole paragraph needs to be
rephrased a bit..].
Finally, note that Fine’s view forces him to reject classical logic. The arbitrary number,
for example, has the property of being either even or odd (because every particular
number has the property of being either even or odd). But the arbitrary number is not
even (because some particular numbers are not even) and the arbitrary number is not odd.
Thus P or Q can be true in spite of the fact that P is not true and Q is not true. Fine is
happy to endorse this consequence (partially because he thinks classical logic should
anyhow be rejected in light of the phenomenon of vagueness), but we certainly see this
rejection as a high price for the view to pay.
Fine’s view has many interact details and we cannot do them justice in this brief
discussion. But we hope that our brief discussion has at least convinced the reader that
Fine’s view employs an incredibly complicated apparatus. Even if Fine’s view can
achieve a formally consistent account of arbitrary objects, it seems that the account is
going to be a lot more convoluted and less straightforward than the account suggested by
AR.
§3 Further Applications (instances? – these are three more cases in which we refer
arbitrarily)
In the previous section we motivated AR by arguing that it provides the best explanation
for how instantial reasoning works. In this section, we briefly note three further problems
that AR might be able to solve, although how exactly AR applies to these problems is a
question that requires further work.
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1. Benacerraf’s problem: A problem which was raised by the philosopher of
mathematics Paul Benacerraf is the following.13 Take two adequate set theoretic
construals of ordered pairs. According to one the order pair <a,b> is the set {a, {b}} and
according the second the order pair <a,b> is the set {{a,b}, a}. Both construals are
equally adequate: they each capture the only feature that is important for us in order pairs,
namely that <a,b>=<c,d> if and only if a=c and b=d. But which one of them is the
ordered pair? According to Benacerraf, all possible answers are unsatisfactory. If we say
that both construals are correct, then since <a,b>=<a,b> it would follow that {a,
{b}}={{a,b}, a}, which is clearly wrong. If we say that, for example, the first construal is
correct but the second is incorrect, that would be completely arbitrary since both
construals equally serve the purpose for which we introduced the notion of ‘ordered
pairs’ for. And if we say that neither construal is correct, then it seems like we simply
postpone the problem. For whatever construal is correct – i.e. whatever ordered pairs
actually are, we may ask why they are this rather than one of the equally adequate set
theoretic construals just described. As structuralist in the philosophy of mathematics
might put it, all there is to the notion of an ordered pair is the structural property that
<a,b>=<c,d> if and only if a=b and c=d, and thus we should not prefer one collection of
objects which exemplify this structure over another as being ‘the true ordered pairs’.
We suggest that AR provides us with a new way to respond to Benacerraf’s problem: of
all the possible collections of objects that exhibit the structural property of ordered pairs,
the collection that ‘ordered pairs’ picks out is fixed arbitrarily. (Thus we could introduce
the notion of an order pair by a stipulation like: ‘Let ‘order pairs’ be an arbitrary
collection of objects that exhibit the relevant structural property…’). This means that
there is a fact of the matter whether or not <a,b>={a, {b}}, but we do not and cannot
know this fact. Benacerraf is correct in thinking that the choice of construal is arbitrary,
but is incorrect in thinking this prevents ordered pairs from referring to one construal
rather than another. Our solution to Benacerraf’s problem also manages to account for the
structuralist intuition that the only feature of ordered pairs that is important is their
Benacerraf’s original example involves natural numbers rather than ordered pairs. (We present the
ordered pairs version for simplicity).
13
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structural properties: this is why we use these and only these features to fix the reference
arbitrarily, and although ordered pairs have some properties other than their structural
properties we do not an cannot know anything about their non-structural properties. On
the other hand our account manages to avoid some of the main problems that
structuralism faces, namely questions regarding the nature and ontological status of
structures. [footnote about why we refer to the same thing… though problem with
disconnected tribe…].
2. Referring to indiscernibles: There are various cases where we are presented with two
objects such that there is no non-de re property that one has but the other does not, and
yet we seem to be able to refer to each of the two objects. One such case is Max Black’s
two spheres case (footnote a reference). Suppose there is a universe that contains only
two qualitatively identical spheres. Call one of them ‘A’ and the other ‘B’. In so far as
‘A’ successfully refers to one of the spheres, we have managed to fix its reference
without standing in any casual connection to it, or providing any description that
distinguishes it from the other sphere. How is this possible? We are able to explain this
using AR: ‘A’ picks out one of the spheres arbitrarily.
One might worry that this is a fictional example, and hence ‘A’ and ‘B’ do not refer to
the any spheres (at best they refer to fictional spheres). 14 But there are also non-fictional
examples of the same phenomenon. Mathematicians claim that there are two square roots
to –1: i and –i. But which of the roots does ‘i’ pick out and which one does ‘-i' pick out?
The problem is that there are no (non-de re) mathematical properties that distinguish the
two. That is, if one were to switch (? How does one switch numbers?) i and –i and
correspondingly every complex number a+bi with a-bi, one would get a structure that is
exactly isomorphic to the actual complex numbers. Again, AR provides a solution: ‘i’
arbitrarily refers to one of the two square roots of –1 and ‘–i' refers to the other. (What if
mathematicians, when they were fixing the reference of ‘i’, thought that there was just
one number x such that x2+1=0, so did not take themselves to be referring arbitrarily
14
Though if David Lewis is correct that there are spatially and causally disconnected universes to our own,
then for all we know there actually is a spatially and casually disconnected universe that contains exactly
two qualitatively identical spheres – we have no reason to think not
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when they fixed the reference of ‘i’. Would that mean that the reference would fail, or
would it still proceed arbitrarily anyway. The issue here, I guess, is how much speaker
intentions constrain the process. Maybe this is also the point to mention your idea that
perhaps reference is always arbitrary, albeit sometimes the set of possible referents has
just one member (in which case we know what the referent is.) [I thought that once
people conceded there was a root to –1 they immediately accepted both, but maybe I am
wrong about this. But the general point about intentions is interesting…].
3. Vagueness: We propose that AR might also provide a new account of vagueness.
Consider the word ‘tall’. We are sympathetic to Williamson’s view according to which
‘tall’ picks out a determinate property and that statements of the form ‘X is tall’ are either
true or false (assuming no reference failures for ‘X’ and other unrelated problems).
However, we find some of the details of Williamson’s view dissatisfying. First,
Williamson claims that which property ‘tall’ picks out supervenes on the use of the word,
but that we nonetheless do not and cannot know which property ‘tall’ picks out. This
seems somewhat puzzling: if the meaning of ‘tall’ supervenes on its use then why can’t
linguists (perhaps subject to a lot of research) work out its meaning? Second, there seems
to be something puzzling about the thought that a possibly small set of use facts can
completely determine the boundaries of a word. Consider for example the phrase
‘Wylie’s room’. Plausibly, this phrase has been used on a very limited set of occasions.
Still, according to Williamson this small set of use facts is sufficient to determine
whether, for example, a point 1mm past the outer edge of Wylie’s door frame is part of
his room or not. (what about the very first use – are there enough facts in that case to fix a
reference?) [Maybe reference my paper with Stephen?].
We suggest that AR can provide us with an alternative theory of vagueness that benefits
from the advantages of Williamson’s theory (e.g. retaining classical logic) but avoids
some of the problems of his view. According this suggestion ‘tall’ arbitrarily refers to one
of a range of acceptable properties. It follows by AR that ‘tall’ refers to a particular
determinate property but we do not and cannot know which one. This provides a response
to the problems just raised. First, our explanation for why we do not and cannot know
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which property ‘tall’ picks out is different than that of Williamson. On our view we do
not know this fact precisely because which property ‘tall’ picks out does not supervene
on use, and hence no amount of linguistic investigation of the use facts is going to help.
Second, we are in a better position to explain why a small number of use facts are
sufficient to determine which object exactly ‘Wylie’s room’ picks out, for since the
reference of ‘Wylie’s room’ is fixed arbitrarily the use facts are not the only thing playing
a role in the determination of reference. (Similarly, with ‘Let ‘John’ be a Frenchman’ our
very minimal use of ‘John’ was sufficient to fix reference to a particular Frenchman).
Many people have resisted epistemicism about vagueness because they feel that
postulating that a vague word has one particular meaning rather than another is
completely arbitrary. AR allows us to endorse a version of epistemicism while doing
justice to this intuition. It is indeed the case that any choice for the precise meaning of a
vague word is arbitrary – a vague word gets to refer to its particular meaning via the
mechanism of arbitrary reference.15
§4 Conclusion
AR might seem like an initially implausible claim. However, we hope to have shown that
the claim has considerable explanatory power: it can be used to respond to some of the
most fundamental puzzles in philosophy. If such explanations are convincing, this should
give us serious reasons to think that in spite of its initial implausibility, AR is true.
15
There are difficult questions about how to account for higher order vagueness. We have some
preliminary thoughts about this issue, but they require more work…
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