rayo, agustin - neo-fregeanism reconsidered.pdf
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Neo-Fregeanism Reconsidered
Agustn Rayo
March 11, 2009
1 Platonism
Mathematical Platonism is the view that mathematical objects exist. Traditional Pla-
tonists believe that one can make sense of a world with no mathematical objects; subtle
Platonists disagree.
The easiest way of getting a handle on traditional Platonism is by imagining a creation
myth. On the first day God created light; by the sixth day, she had created a large and
complex world, including black holes, planets and sea-slugs. But there was something left
to be done. On the seventh day she created mathematical objects. Only thendid she rest.
On this view, it is easy to make sense of a world with no mathematical objects: it is just
like the world we are considering, except that God rested on the seventh day.
The crucial feature of this creation myth is that God needed to do something extra
in order to bring about the existence of mathematical objects: something that wasnt
already in place when she created black holes, planets and sea-slugs. According to subtle
Platonists, this is a mistake. A subtle Platonist believes that for the number of the Fs
to be eight just is for there to be eight planets. So when God created eight planets she
therebymade it the case that the number of the planets was eight. More generally, subtle
For their many helpful comments I am grateful to Roy Cook and Matti Eklund, and to audiences at
the University of Connecticut and the University of St Andrews.
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Platonists believe that there is no sense to be made of a world without numbers. Suppose,
for reductio, that there are no numbers. The subtle Platonist thinks that for the number
of numbers to be zero just is for there to be no numbers. So the number zero must exist
after all, contradicting our assumption.
Essential to the subtle Platonists position is the acceptance ofnon-objectual identity
statements: identity statements in which the identity-predicate takes formulasrather than
singular terms in its argument places. For instance:
For the number of the planets to be eight just isfor there to be eight planets.
[In symbols: #x(Planet(x)) = 8 !8x(Planet(x)).]
This is the sense of identity whereby most of us would wish to claim that for a wedding
to take place just is for someone to get married [i.e. x(Wedding(x) TakesPlace(x))
x(Married(x))], that for something to be hotjust isfor it to have high mean kinetic energy
[i.e. Hot(x)x HMKE(x)], and that for two people to be siblings just isfor them to share a
parent [i.e. Siblings(x, y)x,y z(Parent(z, x)Parent(z, y))]. It is also the sense of identity
whereby some philosophers (but not all) would wish to claim that for there to be a table just
is for there to be some particles arranged table-wise [i.e.x(Table(x)) X(ArrTw(X))].
The non-objectual identity symbol (or its natural-language correlate just is) is
meant to express a form ofidentity. So it had better be reflexive, transitive and symmetric.
In particular, there should be no difference between saying, on the one hand, that for a
wedding to take place just isfor someone to get married and saying, on the other, that for
someone to get married just isfor a wedding to take place.
From this it follows that isnotmeant to capture a relation of metaphysical priority.
If you think that for a wedding to take place just isfor someone to get married, you should
think that there is a feature of reality that can be fully and accurately described by saying
a wedding took place and can also be fully and accurately described by saying someone
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got married. But you have notthereby committed yourself to the view that one of these
descriptions is, in some sense, more fundamental than the other.
First-order identities can be regarded as a special case of non-objectual identities. The
standard first-order identity-statement a= b is equivalent to the conjunction of the non-
objectual identity-statement to be a just isto be b [i.e. x= a x x= b] and a sentence
asserting the existence ofa [i.e. x(x = a)]. (The Kripkean reading of a = b, whereby
a= b can be true at a world in which a fails to exist, is equivalent to x= a x x= b.)
Semi-identitystatements, such as part of what it is to be scarlet is to be red [in symbols:
Scarlet(x)xRed(x)], can also be regarded as a special case of non-objectual identities.
In general, Fx x Gx is equivalent to Fx x (Gx Fx). In light of these connections,
I will use identity-statement in what follows to refer to any statement equivalent to a
sentence of the form (x1, . . . , xn)x1,...,xn (x1, . . . , xn) (n 0).
Identity-statements pervade our pre-theoretic, scientific and philosophical discourse.
Yet they have been given surprisingly little attention in the literature, and are in much
need of elucidation. I think it would be hopeless to attempt an explicit definition of
,not because true and illuminating equivalences couldnt be foundI will suggest some
belowbut because any potential definiens can be expected to contain expressions that
are in at least as much need of elucidation as . The right methodology, it seems to
me, is to explain how our acceptance of identities interacts with the rest of our theorizing,
and use these interconnections to inform our understanding of . (I make no claims
about conceptual priority: the various interconnections I will discuss are as well-placed to
inform our understanding of on the basis of other notions as they are to inform our
understanding of the other notions on the basis of .)
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1.1 Identity
In this section I will suggest four different ways in which our acceptance of identity-statements interacts with the rest of our theorizing. (I have developed these connections in
far greater detail, and addressed some potential objections, in An Account of Possibility
and Towards a Trivialist Account of Mathematics.)
1.1.1 Truth-conditions
A sentences truth-conditions are usefully thought of as imposing a requirement on the
world. Specifically: the requirement that the world be such as it would need to be in
order for the truth-conditions to be satisfied. The truth-conditions of snow is white, for
example, impose the requirement that snow be white, and the truth conditions of grass is
green the requirement that grass be green.
Suppose you think that for something to be composed of water just is for it to be
composed of H2O. Then you should think that what is required of the world in order for
the truth-conditions of the sentence Ais composed of water to be satisfied is precisely what
is required of the world in order for the truth-conditions of the sentence A is composed
of H2O to be satisfied. In particular, you should think that all that is required of the
world in order for the truth-conditions of A is composed of water to be satisfied is for A
to be composed of H2O, and that all that is required of the world in order for the truth
conditions of A is composed of H2O to be satisfied is that A be composed of water.
Conversely: suppose you think that what is required of the world in order for the truth-
conditions of A is composed of water to be satisfied is precisely what is required of the
world in order for A is composed of H2O to be satisfied. Then you should think that for
something to be made of water just isfor it to be made of H2O.
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1.1.2 De re intelligibility
Let a story be a set of sentences in some language we understand. I shall assume thatstories are read de re: that every name used by the story is used to say of the names
actualbearer how it is according to the story, and that every predicate used by the story
is used to attribute the property actuallyexpressed by the predicate to characters in the
story. Accordingly, in order for a story that says Hesperus is covered with water to be
true it must be the case that Venus itself is covered with H2O. (I shall ignore names that
are actually empty, such as Sherlock Holmes, and predicates that are actually empty, such
as . . . is composed of phlogiston or . . . is a unicorn.)
Sometimes we describe a story as unintelligible on the grounds that it is too complicated
for us to understand. That is not the notion of unintelligibility I will have in mind here.
I will say that a story is de re unintelligiblefor a subject if her best effort to make sense
of a scenario verifying the story would yield something she regards as incoherent. (De re
intelligibility is the contradictory ofde re unintelligibility.)
Here is an example. Consider a story that says Hesperus is not Phosphorus. My best
effort to make sense of a scenario verifying this story ends in incoherence. For a scenario
in which the story is true would have to be a scenario in which Hesperus itself (i.e. Venus)
fails to be identical with Phosphorus itself (i.e. Venus), and the nonselfidentity of Venus is
something I regard as incoherent. Another example: consider a story that says a wedding
took place, but nobody got married. For a wedding to take place just is for someone to
get married. So a scenario verifying a wedding took place, but nobody got married would
have to be a scenario in which a wedding takes placei.e. a scenario in which someone
gets marriedbut nobody gets married, which is something I regard as incoherent. (Of
course, it would be easy enough to make sense of a scenario in which language is used in
such a way that the expression Hesperus is not Phosphorus or a wedding took place, but
nobody got married is true. But that wont help with the question of whether the original
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story is de reintelligible.)
There is an intimate connection between identity and de re intelligibility: a story is
de reunintelligible for a subject just in case she is in a position to derive something she
regards as incoherent from the result of adding the story to the set of identities she accepts,
using inferences she takes to be logically valid.
Three clarifications: (1) The notion ofde reintelligibility is deeply non-a priori. For
whether or not one takes a story to be de reintelligible will depend on whether one believes
that, say, Hesperus is Phosphorusand this is not the sort of thing that one could come
to know a priori. (2) Perhaps you think there is a sense of intelligibility (distinct from
de re intelligibility) with respect to which it would be right to say that Hesperus is not
Phosphorus is an intelligible story. You could claim, for example, that Hesperus is not
Phosphorus is intelligible in the following sense: it would be verified (in a special sense of
verify) by a scenario in which the first heavenly body to be visible in the evenings is not
the last heavenly body to be visible in the mornings. I myself am skeptical that the needed
notion of verificationwhich is, in effect, an implementation of the idea that our sentenceshave primary intensionsis in good order. But nothing in this paper will turn on my
skepticism. If you think you can understand a notion of intelligibility distinct from de re
intelligibility, thats fine. Just keep in mind that its not the notion under discussion here.
(3) So far we have talked about the intelligibility of stories, but not the intelligibility of
the scenarios that the stories depict. When a scenario is picked out by a story, we shall say
that the scenario is intelligible just in case the story used to pick it out is de reintelligible.
1.1.3 Why-closure
Suppose someone says: I can see that Hesperus is Phosphorus; what I want to understand
is why. It is not just that one wouldnt know how to comply with such a requestone
is unable to make senseof it. The natural reaction is to reject it altogether. One might
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reply, for instance, What do you mean why? Hesperus just isPhosphorus.
Contrast this with a case in which someone says: I can see that the window is broken;
what I want to understand is why, or I can see that the radioactive isotope decayed
at time t rather than a second later; what I want to understand is why, or even I can
see that there is something rather than nothing; what I want to understand is why. In
all three of these cases, one can at least make sense of the request. In the first case, one
may even have a satisfying reply (e.g. The window is broken because it was hit by soccer
ball). In the second case, it is harder to think of a goodreply, but one can at least think
of a bad one (e.g. because God willed it so). Potential replies are even scarcer in the
third case (even bad ones), but one can at least state that there is no good answer to be
given without refusing to make sense the question (e.g. Well, thats just the way things
turned out.). Contrast this with the initial Hesperus/Phosphorus case, where it isnt even
appropriate to say Well, thats just the way things turned out.
Say that a sentence is treated as why-closedjust in case one is unable to make sense
of
Why is it the case that
?
. (Here it is important to stipulate that the question is notto be read as a groundingquestion. In other words: (1) it is not to be read as a request
for justification, as when someone asks Why is Hesperus is Phosphorus? in an effort to
acquire grounds for believing that Hesperus is Phosphorus; (2) it is not to be read as a
request for insight about what the truth of would consist in, as when one reads Why are
elephants mammals? as a request for better understanding what it takes to be a mammal;
and (3) it is assumed not to be a case in which someone is able to see that has the
truth-conditions that it actually has, but does not fully understand why this is so, as when
one reads Why are there infinitely many primes? as a request for a more illuminating
proof of the theorem.)
There is an intimate connection between identity and why-closure: the sentences one
treats as as why-closed are those one regards as logical consequences of the identity-
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statements one accepts.
1.1.4 Possibility
When one looks at the philosophical literature on possibility (or metaphysical possibility,
as it is sometimes called), one is left with the impression that there are two schools of
thought. According to the metaphysicalschool of thought, there is a gap between intelli-
gibility and possibility. It might be thought to be intelligible, for instance, that Hesperus
fail to be Phosphorus, or that I be a sea-slug. But such rogue scenarios are to be counted
as impossible nonetheless, on the grounds that they are, in some sense, metaphysically
untoward. They violate the metaphysical laws, as it were. (For an example of this sort
of approach, see Kment (2006); for useful discussion, see Rosen (2006).)
This picture strikes me as doubly objectionable. Firstly, I think one should be weary
of a notion of intelligibility according to which it is intelligible that Hesperus fail to be
Phosphorus or that I be a sea-slug. It is certainly not de reintelligibilitynot, at least, on
the assumption that one believes that Hesperus is Phosphorus and that part of what it is tobe me is to be a human and not a sea-slug. The view, presumably, is that is intelligible
is to be cashed out in terms of something along the lines of is not analytic or has a
non-empty primary intension. Notice, however, that each of these proposals relies on the
assumption that one can factor the requirement that a sentences truth-conditions impose
on the world into two parts: an a prioricomponent and a non-a prioricomponent. And I
see no good reason for thinking that this is true in general, even if it seems plausible when
one focuses on toy examples such as Hesperus is Phosphorus.
Secondly, and more importantly, the notion of a metaphysical law strikes me as both
obscure and unhelpful. It strikes me as obscure because I have never been able to find
a use for it outside Deep Metaphysicsmetaphysics of the kind only a metaphysician
would understand (and only a metaphysician would care about). It strikes me as unhelpful
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because it drives a wedge between the notion of possibility and one of its most useful
applications: the representation of content. For in stating a metaphysical law one would
presumably be saying something non-trivial about the way the world is. But if every
possible scenario is a scenario which verifies the metaphysical laws, one cannot hope to
represent the non-trivial requirement that the world must meet in order for a metaphysical
law to hold by dividing up the possibilities into those that meet the requirement and those
that do not.
The right way of thinking about possibility, it seems to me, is by refusing to allow a gap
between possibility and intelligibility: possibility just is de reintelligibility. To see a de re
story as describing a possible world is to take the story to be de re intelligible. By appealing
to de reintelligibility, one is able to steer clear of the analytic/synthetic distinction and
its siblings. One works with a notion which is non-a priori to begin with, rather than
construing necessity as a marriage between a priori truths and non-a priori Kripkean
necessities. The result is that one gets a distinction between necessity and contingency
without having to forego the Quinean insight that conceptual questions cannot be cleanlyseparated from questions of fact.
In addition, the alternative strategy has the advantage of not severing the connection
between possibility and content. To see this, think of a sentences truth-conditions as a
requirement imposed on the world: a requirement that the world be a certain way. Knowing
whether a scenario we regard as intelligible must fail to obtain in order for the requirement
to be met is valuable because it gives us an understanding of how the world would need
to be in order for the requirement to be satisfied. But knowing whether a scenario we
dont regard as intelligible must fail to obtain in order for the requirement to be to be
met is not very helpful. For when one is unable to describe a scenario in a way one finds
intelligible, the claim that the scenario must fail to obtain gives one no understanding
of how the world would need to be in order for the requirement to be satisfied. The
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result is that one can model a sentences truth-conditions as a partition of the space of
scenarios one regards as intelligible. So the notion of possibility that is useful for modeling
truth-conditions is precisely the notion that is correlated with intelligibility. (Earlier I
had described connections between truth-conditions and identity, on the one hand, and
identity and intelligibility, on the other. The present connection between truth-conditions
and intelligibility closes the circle.)
A third advantage of the alternative way of thinking about possibility is that it re-
moves some of the mystery surrounding modal knowledge. For if the stories we regard
as describing a possible world are the stories we regard as de re intelligible, and if the
stories we regard as de reintelligible are the stories we take to be logically consistent with
the identities we accept, then our modal knowledge should be no more mysterious than
our knowledge of identities. And, in some cases at least, our knowledge of identities is
relatively unmysteriouswitness our knowledge of the fact that to be hot just is to have
high mean kinetic energy.
I would like to end this section by mentioning a complication that I have been ignoringto simplify the exposition. In discussing the connections between identity, truth-conditions,
intelligibility, why-closure and possibility, I have tacitly assumed that attention is restricted
to the sorts of sentences that are best formalized in a (non-modal) first-order language. I
am happy to the claim, for example, that one will treat Agustn has a mustache as possibly
true (andde reintelligible) just in case one takes it to be logically consistent with the set of
identities one accepts. But I do not wish to be committed to the claim that one will treat,
e.g. mustached individuals are essentially mustached [in symbols: (x(Mustached(x)
(y(y= x Mustached(x)))))] as possibly true (and intelligiblede re) just in case one
takes it to be logically consistent with the set of identities one accepts. For mustached
individuals are essentially mustached will only be true if different worlds are in agreement
about who has a mustache and who does not, and two worlds can fail to display such
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agreement even if each of them, taken in isolation, is consistent with the set of identities one
accepts. Happily, there is a way of extending the proposal to encompass every sentence of
a first-order modal language. And when the enriched proposal is combined with a suitable
list of identity statements, one gets a non-revisionary list of modal truths, including all
of the standard Kripkean necessities. (The results are made possible by a supervenience
theorem, which shows that by fixing the truth-value of every identity-statement, every
conditional identity statement and every sentence in the language containing no boxes or
diamonds, one fixes the truth-value of every sentence in the modal language. See my An
Account of Possibility for details. )
1.2 The Resulting Picture
In the preceding section I suggested several different ways in which our acceptance of
identity-statements interacts with the rest of our theorizing. In doing so, I hope to have
shown that the notion of non-objectual identity is not a creature of Deep Metaphysics. It
is at the center of our theorizing about content, intelligibility, explanation and possibility.
The aim of the present section is to make some brief remarks about the philosophical
picture that emerges from these theoretical connections.
In theorizing about the world, we do three things at once. First, we develop a language
within which to formulate theoretical questions. Second, we endorse a family of identity-
statements, and thereby classify the questions that can be formulated in the language into
those we take to be worth answering and those we take to be pointless. (For instance, by
accepting to be composed of water just isto be composed of H2O, we resolve that why
does this portion of water contain hydrogen? is not the sort of question that is worth
answering; had we instead accepted to be composed of water just is to be an odorless,
colorless liquid with thus-and-such properties, we would have taken why does this portion
of water contain hydrogen? to be an interesting question, deserving explanation.) Third,
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we set forth theoretical claims that address the questions we take to be worth answering.
(For instance, having accepted to be composed of water just isto be composed of H2O,
we set forth theoretical claims as an answer to the question why is this portion of water
an odorless, colorless liquid with thus-and-such properties?.)
In light of the connection between identity and possibility, when one endorses a set of
identity-statements one is, in effect, endorsing a space of possible worlds. And in light of the
connection between possibility and intelligibility, a space of possible words can be regarded
as a framework for ascertaining the limits of intelligibility. In other words: by endorsing
a set of identity-statements one is, in effect, endorsing a framework for ascertaining the
limits of intelligibility.
It is against the background of such a framework that it makes sense to distinguish
between factual sentencesi.e. sentences that are verified by some intelligible scenarios
but not othersand necessary truthsi.e. sentences that are verified in every intelligible
scenario. What one gets is a post-Quinean revival of the position that Carnap advanced
in Empiricism, Semantics and Ontology. Carnaps frameworks, like ours, are devices fordistinguishing between factual sentences and necessary truths. The big difference is that
whereas Carnaps distinction was tied up with the analytic/synthetic distinction, ours is
not.
(Is there an objective fact of the matter about which choice of identity-statements
i.e. which choice of frameworkis correct? Nothing in the paper will turn on the answer
to this question. My own view, for what its worth, is that there is not. All one can
say is that different choices are more or less amenable to the development of a successful
articulation of ones methods of inquiry, given the way the world is. The empirical facts
make it the case that acceptance of to be composed of water just is to be composed of
H2O leads to particularly fruitful theorizing, and one has strong reasons to accept it on
this basis. But there is no more to be said on its behalf. And when it comes to other
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identity-statements, the empirical pressures are even milder. For example, whether it is a
good idea to accept to be a reptile is to have a certain lineage rather than to be a reptile is
to have a certain phenotype might to a large extent depend on the purposes at hand. But,
to insist, nothing in this paper turns on whether one favors a pragmatic attitude towards
identity-statements. And if you think there is an objective fact of the matter about which
identities are true you can use it to convert the subject-relative characterizations of de
reintelligibility, why-closure and possibility that I set forth in the preceding section into
objective ones: a story is de re intelligible just in case it is logically consistent with the
true semi-identity statements; is why-closed just in case it is a logical consequence of the
true semi-identities; a story describes a possible world just in case it is de reintelligible.)
1.3 Back to Platonism
I introduced the difference between traditional and subtle Platonism by saying that whereas
traditional Platonists believe that one can make sense of a world with no mathematical
objects, subtle Platonists think one cannot. But if one is prepared to identify possibility
and intelligibility, as I suggested in section 1.1.4, one can also state the difference as
follows: traditional Platonists believe that mathematical objects exist contingently; subtle
Platonists believe that they exist necessarily.
This way of characterizing the debate might strike you as suspect. If so, it may be
because you have a different conception of possibility in mindsomething akin to the
metaphysical conception of possibility I described in section 1.1.4. (See also Rosen (2006).)
You may be thinking of necessary existence as a special kind of metaphysical status. It is,
of course, open to traditional Platonists to claim that numbers enjoy necessary existence
in this sense. But notice that in doing so they would be engaging in Deep Metaphysics.
For whether or not an object enjoys this special metaphysical status is the sort of question
that only a metaphysician would be in a position to ascertain, and only a metaphysician
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would care about.
In contrast, the claim that numbers exist necessarily in the sense of necessity that is
at issue here has consequences far beyond metaphysics. It delivers the result that the
existence of numbers is no longer a factual question: nothing is required of the world in
order for the truth-conditions of a truth of pure mathematics to be satisfied. Satisfaction
is guaranteed by the framework with respect to which we have chosen to carry out our
theoretical investigation of the world. And this, in turn, has non-trivial epistemological
consequences.
Non-factuality, in our sense, does not entail a priori knowability. (To be composed
of water just isto be composed of H2O will count as non-facutal even though it is not
knowable a priori.) But there is still room for mathematical truths to enjoy a privileged
epistemological status. For in adopting a frameworki.e. in adopting a family of identity-
statementsone makes decisions that are closely tied to empirical considerations, but
also decisions that are more organizational in nature. This is because a delicate balance
must be struck. If one accepts too many identities, one will be committed to treating asunintelligible scenarios that might have been useful in theorizing about the world. If one
accepts too few, one opens the door to a larger range of intelligible scenarios, all of them
candidates for truth. In discriminating amongst these scenarios one will have to explain
why one favors the ones one favors. And although the relevant explanations could lead to
fruitful theorizing, they could also prove burdensome.
Logic and mathematics play an important role in finding the right balance between
these competing considerations. Consider, for example, the decision to treat p p
as a logical truth (i.e. the decision to accept every identity-statement of the form for pto
be the case just is for p to be the case). A friend of intuitionistic logic, who denies
the logicality of p p, thinks it might be worthwhile to ask why it is the case that
p even if you fully understand why it is not the case that p. In the best case scenario,
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making room for an answer will lead to fruitful theorizing. But things may not go that
well. One might come to see the newfound conceptual space between a sentence and its
double negation as a pointless distraction, demanding explanations in places where there
is nothing fruitful to be said.
The result is that even if none of the decisions one makes in adopting a family of
identity-statements is wholly independent of empirical considerations, some decisions are
more closely tied to empirical considerations than others. And when it comes to identity-
statements corresponding to logic and mathematics, one would expect the focus to be less
on particular empirical matters and more on questions of framework-organization. So there
is room for a picture whereby an epistemically responsible subject can believe that numbers
exist even if her belief isnt grounded very directly in any sort of empirical investigation.
2 Neo-Fregeanism
Humes Principle is the following sentence:
FG(#x(F(x)) = #x(G(x)) F(x)x G(x))
[Read: the number of the Fs equals the number of the Gs just in case the Fs
are in one-one correspondence with the Gs.]
Neo-Fregeanism is the view that when Humes Principle is set forth as an implicit definition
of #x(F(x)), one gets the following two results: (1) the truth of Humes Principle is know-
ablea priori, and (2) the referents of number-terms constitute a realm of mind-independent
objects which is such as to render mathematical Platonism true. (Neo-Fregeanism was first
proposed in Wright (1983), and has since been championed by Bob Hale, Crispin Wright
and others. For a collection of relevant essays, see Hale and Wright (2001).)
Just like one can distinguish between two different varieties of mathematical Platonism,
one can distinguish between two different varieties of neo-Fregeanism. Traditional and
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a thought can be split up in many ways, so that now one thing, now another, appears as
subject or predicate (Frege (1892) p. 199), it is natural to interpret him as embracing the
identity-statement:
For the concept square root of 4 to be realized just isfor there to be at least
one square root of 4.
And when he claims, in Grundlagen 64, that in treating the judgement line a is parallel
to line b as an identity, so as to obtain the direction of line ais identical to the direction
of line b, we carve up the content in a way different from the original way, it is natural
to interpret him as embracing the identity-statement:
For the direction of line a to equal the direction of line b just is for a and b to
be parallel.
In both instances, Frege puts the point in terms of content-recarving, rather than as an
identity-statement. But, as emphasized above, ones views about truth-conditions are
tightly correlated with the identities one accepts.
Neo-Fregeans are evidently sympathetic towards Freges views on content-recarving.
(See, for instance, Wright (1997).) And even though talk of content-recarving has become
less prevalent in recent years, with more of the emphasis on implicit definitions, a version
of neo-Fregeanism rooted in subtle Platonism is clearly on the cards. In the remainder
of the paper I will argue that such an interpretation of the program would be decidedly
advantageous.
2.1 Objects
Suppose you introduce the verb to tableize into your language, and accept for it to
tableize just isfor there to be a table (where the it in it tableizes is assumed to play
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the same dummy role as the it in it is raining). Then you will think that it tableizes
and there is a table have the same truth-conditions. In each case, what is required in
order for the truth-conditions to be satisfied is that there be a table (equivalently: that
it tableize). So you will think thatfor the purposes of stating that there is a table
object-talk isoptional. One can state that there is a table by employing a quantifier that
binds singular term positionsas in there is a tablebut also by employing an essentially
different syntactic structureas in it tableizes.
If object-talk is optional, what is the point of giving it a place in our language? Ac-
cording to the non-metaphysical view, as I shall call it, the answer is compositionality.
A language involving object-talkthat is, a language including singular terms and quan-
tifiers binding singular term positionsis attractive because it enables one to give a recur-
sive specification of truth-conditions for a class of sentences rich in expressive power. But
there is not much more to be said on its behalf. If one could construct a language that
never indulged in object-talk, and was able to do so without sacrificing compositionality
or expressive power, there would be no immediate reason to think it inferior to our own.Whether or not we choose to adopt it should turn entirely on matters of convenience. (For
an example of such a language, and illuminating discussion, see Burgess (2005).)
Proponents of the non-metaphysical view believe that it takes very little for a singular
term to be in good order. All it takes is a compositional specification of truth-conditions
for whichever sentences involving the term one wishes to make available for use. The reason
there is nothing more that needs to be done is that there was nothing special about using
singular terms to begin with. In setting forth a language, all we wanted was the ability to
express a suitably rich range of truth-conditions. If we happened to carry out this aim by
bringing in singular terms, it was because they supplied a convenient way of specifying the
right range of truth-conditions, not because they had some further virtue.
Proponents of themetaphysical view, in contrast, believe that that object-talk is subject
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to a further constraint: there needs to be a certain kind of correspondence between the
semantic structure of our sentences and the metaphysical structure of reality. In partic-
ular, they presuppose the following: (1) there is a particular carving of reality into objects
which is more apt, in some metaphysical sense, than any potential rivalthe one and only
carving that is in accord with realitys true metaphysical structure; (2) to each legitimate
singular term there must correspond an object carved out by this metaphysical structure;
and (3) satisfaction of the truth-conditions of a sentence of the form P(t1, . . . , tn) re-
quires that the objects paired witht1, . . . , tn bear to each other the property expressed by
P. (For an example of this sort of view, see Sider (typescript).)
A consequence of the metaphysical view is that one cannot accept for it to tableize just
isfor there to be a table. (Since it tableizes and there is a table have different semantic
structures, there cant be a single feature of reality they are both accurate descriptions of
when it is presupposed that correspondence with the one and only structure of reality is a
precondition for accuracy.) For similar reasons, one cannot accept for some things to be
arranged tablewisejust is
for there to be a table, or for the property of tablehood to be in-stantiatedjust isfor there to be a table, or, indeed, any identity statement where
and are atomic sentences with different semantic structures. Accordingly, proponents
of the metaphysical view takes themselves to be in a position to make distinctions that
a proponent of the non-metaphysical view would be unable to understand. For instance,
they might take themselves to be in a position to make sense of a scenario in which some
things are arranged tablewise but there is no table.
I am ignoring a complication. Friends of the metaphysical view might favor a moderate
version of the proposal according to which only a subset of our discourse is subject to the
constraint that there be a correspondence between semantic structure and the metaphysical
structure of reality. One could claim, for example, that the constraint only applies when
one is in the ontology room. Accordingly, friends of the moderate view would be free to
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accept a version of, e.g. for there to be some things arranged tablewise just is for there
to be a table by arguing that only there are some things arranged tablewise is to be
understood in an ontology-room spirit. Ignoring this version of the view allows me to
simplify the exposition with no loss in generality: everything I will say about the non-
moderate view applies to the moderate view when attention is restricted to ontology-room
discourse.
The difference between the non-metaphysical and the metaphysical view is subtle but
important. First, the metaphysical view indulges in Deep Metaphysics where the non-
metaphysical view does not. For the metaphysical structure of reality is a paradigm exam-
ple of the sort of thing only a metaphysician could understand, and only a metaphysician
would care about. (For arguments to the contrary, see Sider (typescript).) Second, and
more importantly for our purposes, the non-metaphysical view leaves room for meaning-
lessness where the metaphysical view does not. Suppose it is agreed on all sides that the
singular terms t1 and t2 are both in good order, each of them figuring meaningfully in
sentences with well-defined truth-conditions. A proponent of the metaphysical view is, onthe face of it, committed to the claim that it must be possible to meaningfully ask whether
t1= t2 is true. For she believes that each oft1 and t2 is paired with one of the objects
carved out by the metaphysical structure of reality. So the question whether t1= t2 is
true can be cashed out as the question whether t1 and t2 are paired with the same such
object. (Friends of the metaphysical view could, of course, deny that this is a meaning-
ful question. But such a move would come at a cost, since it it would make it hard to
understand what is meant by carving reality into objects.)
For a proponent of the non-metaphysical view, in contrast, there is no tension between
thinking that t1 and t2 figure meaningfully in sentences with well-defined truth conditions
and denying that one has asked a meaningful question when one asks whether t1= t2is
true. For according to the non-metaphysical view, all it takes for a singular term to be in
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good order is for there to be a compositional specification of truth-conditions for whichever
sentences involving the term one wishes to make available for use. And there is no reason
one couldnt have a compositional specification of truth-conditions for a large range of
sentences involving t1 and t2 without thereby specifying truth-conditions for t1= t2.
Arithmetic is a case in point. The subtle Platonist has a straightforward way of spec-
ifying a compositional assignment of truth-conditions to arithmetical sentences. (See ap-
pendix A.) On this assignment, every arithmetical sentence a non-philosopher would care
about gets well-definied truth-conditions, as does every sentence in the non-arithmetical
fragment of the language, but no truth-conditions are supplied for mixed identity-statements,
such as the number of the planets = Julius Caesar. And for good reason: there is no
natural way of extending the relevant semantic clauses to cover these cases.
Proponents of the metaphysical view will claim that something important has been left
out. For in the absence of well-defined truth-conditions for the number of the planets =
Julius Caesar, it is unclear which of the objects carved out by the metaphysical structure
of reality has been paired with the number of the planets. But proponents of the non-metaphysical view would disagree: it is simply a mistake to think that such pairings are
necessary to render a singular term meaningful.
If the non-metaphysical view is rightyou might be tempted to askin what sense
are we committed to the existence of numbers when we say that the number of the planets
is eight? In the only sense of existence and numbers that non-metaphysicians are able
to understand, say I. In order for the truth-conditions of the number of the planets is eight
to be satisfied the number of the planets must be eight, and in order for the number of the
planets to be eight there have to be numbers. What else could it take for someone who
asserts the number of the planets is eight to be committed to the existence of numbers?
Of course, the subtle Platonist will make a further claim. She will claim that it is also
true that all it takes for the truth-conditions of the number of the planets is eight to be
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satisfied is that there be eight planets. To think that this contradicts what was said in
the preceding paragraph is to fail to take seriously the idea that for the number of the
planets to be eight just is for there to be eight planets. It is true that the number of
the planets is eight carries commitment to numbers, but it is also true that commitment
to numbers is no commitment at all. (If you have trouble getting your head around this,
consider the following. The subtle Platonist accepts For the number of the planets to be
zero just isfor there to be no planets. She also accepts For the number of the planets to
be distinct from zero just isfor there to be planets. Putting the two together, she accepts
For the number of planets to be either identical to or distinct from zero just isfor there
to be some planets or none; equivalently: For the number of the planets to exist just is
for there to be some planets or none. But there are be some planets or none has trivial
truth-conditionstruth-conditions whose non-satisfaction would be unintelligibleso the
number of the planets exists must also have trivial truth-conditions.)
Perhaps you think that even if the subtle Platonist is right and there is a single feature
of reality that is fully and accurate described by both the number of the plants is eightand there are exactly eight planets, one of these descriptions is more fundamental then
the other: one of them carves reality at the joints. Believe this if you wish, but keep
in mind that the distinction you seek to make is a creature of Deep Metaphysics. You
have, in effect, endorsed what I earlier called the moderate version of the metaphysical
view whereby one of the two descriptions picks out objects that are carved up by the
metaphysical structure of reality, and is thereby fit for the ontology room, and the other
hooks up with the world in a way that has lesser metaphysical pedigree, and is therefore
not fit for the ontology room.
The moral I hope to draw from the discussion this section is that subtle Platonists
have reason to adopt the non-metaphysical view of singular-terms. This goes, in par-
ticular, for subtle neo-Fregeansneo-Fregeans who favor subtle Platonism. But once one
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embraces the non-metaphysical view, there is no pressure for thinking that a mixed identity-
statement such as the number of the planets = Julius Caesar should have well-defined
truth-conditions. So there is no reason to thinkin spite of what Frege suggests in 66
of the Grundlagen and what proponents of neo-Fregeanism tend to presupposethat a
characterization of the concept of number will be unacceptable unless it settles the truth-
value of mixed identity-statements. In short: if you are a subtle neo-Fregean, you should
be resolute about it, and stop worrying about mixed identities.
2.2 Abstraction Principles
I have never been able to understand why a traditionalneo-Fregean would think it impor-
tant to use Humes Princple to characterize the meaning of arithmetical vocabulary, instead
of using, e.g. the (second-order) Dedekind Axioms. In the case ofsubtleneo-Fregeans, on
the other hand, I can see a motivation. Humes Principle, and abstraction principles more
generally, might be thought to be important because they are seen as capturing the dif-
ference between setting forth a mere quantified biconditional as an implicit definition of
mathematical terms:
(f() =f() R(, ))
and setting forth the corresponding identity-statement:
f() =f(),R(, ).
And this is clearly an important difference. Only the latter delivers subtle Platonism,
and only the latter promises to deliver an account for the special epistemic status of
mathematical truths.
If thats whats intended, however, it seems to me that there are better ways of doing
the job. Consider the case of Humes Principle. In order to convince oneself that it succeeds
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in characterizing the meaning of our arithmetical vocabulary, one needs to prove a certain
kind of completeness result: one needs to show that setting forth Humes Principle as an
implicit definition of numerical terms is enough to pin down the truth-conditions of every
arithmetical sentence one wishes to have available for use. This, in turn, involves proving
the following two results:
1. Given suitable definitions, every true sentence in the language of pure arithmetic is
a logical consequence of Humes Principle.
[This result has come to be known as Freges Theorem. It was originally proved in
Frege (1893 and 1903), by making what Heck (1993) identified as a non-essential use
of Basic Law V. That the result could be established without using Basic Law V was
noted in Parsons (1965) and proved in Wright (1983).]
2. Given suitable definitions, every true sentence in the (two-sorted) language of applied
arithmetic is a logical consequence of Humes Principle, together with the set of true
sentences containing no arithmetical vocabulary.
Both of these results are true, but neither of them is trivial.
Now suppose that one characterizes the meaning of arithmetical vocabulary by using
the compositional semantics in appendix A. The assignment of truth-conditions one gets
is exactly the same as the one one would get by setting forth the identity-statement corre-
sponding to Humes Principle as an implicit definition of arithmetical termsbut with the
advantage that establishing completeness, in the relevant sense, turns out to be absolutely
straightforward. One can simply read it off from ones semantic clauses. (In fact, the
easiest method I know of for proving the second of the two results above, and thereby es-
tablishing the completeness of Humes Principle, proceeds via the appendix A semantics.)
None of this should come as a surprise. When one uses the method in the appendix to
specify truth-conditions for arithmetical sentences one avails oneself of all the advantages
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of a compositional semantics. Not so when one sets forth Humes Principle as an implicit
definition.
There is, however, a much more important reason for preferring the method in the
appendix over specifications of truth-conditions based on abstraction principles. Neo-
Fregeans have found it difficult to identify abstraction principles that can do for set-
theory what Humes Principle does for arithmetic. (For a selection of relevant literature,
see Cook (2007).) But if one waives the requirement that the meaning-fixation work be done
by abstraction principles, the difficulties vanish. The subtle Platonist has a straightforward
way of specifying a compositional assignment of truth-conditions to set-theretic sentences.
(See appendix B for details.) As in the case of arithmetic, one gets the result that every
consequence of the standard set-theoretic axioms has well-defined truth-conditions. And,
as in the case of arithmetic, one gets subtle Platonism. In particular, it is a consequence of
the semantics that for the set of Romans to contain Julius Caesar just is for Julius Caesar
to be a Roman.
Perhaps there is some other reason to insist on using abstraction principles to fix themeanings of set-theoretic terms. But if the motivation is simply to secure subtle Platonism,
it seems to me that neo-Fregeans would do better by using a compositional semantics
instead.
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A A Semantics for the Language of Arithmetic
Consider a two-sorted first-order language with identity, L. It containsarithmetical vari-
ables (n1, n2, . . .), constants (0) and function-letters (S, + and ), andnon-arithmetical
variables (x1, x2, . . .), constants (Caesar and Earth) and predicate-letters (Human(. . . )
and Planet(. . . )). In addition, L has been enriched with the function-letter #v(. . .) which
takes a first-order predicate in its single argument-place to form a first-order arithmetical
term (as in #x1(Planet(x1)), which is read the number of the planets).
If is a variable assignment and w is a world, truth and denotation in Lrelative to
and w can be characterized as follows:
Denotation of arithmetical terms:
1. ,w(ni) =(ni)
2. ,w(0) = 0
3. ,w(S(t)) =,w(t) + 1
4. ,w((t1+ t2)) =,w(t1) + ,w(t2)
5. ,w((t1 t2)) =,w(t1) ,w(t2)
6. ,w(#xi((xi), nj)) = the number ofzs such that Sat((xi), z/xi, w)
7. ,w(#ni((ni), nj)) = the number ofms such that Sat((ni), m/ni, w)
Denotation of non-arithmetical terms:
1. ,w(xi) =(xi)
2. ,w(Caesar) = Julius Caesar
3. ,w(Earth) = the planet Earth
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Satisfaction:
Where
[]w
is read
it is true at w that
,
1. Sat(t1= t2,,w) ,w(t1) =,w(t2) (for t1, t2 arithmetical terms)
2. Sat(t1= t2,,w) [,w(t1) =,w(t2)]w (for t1, t2 non-arithmetical terms)
3. Sat(Human(t),,w) [,w(t) is human]w (for t a non-arithmetical term)
4. Sat(Planet(t),,w) [,w(t) is a planet]w (for ta non-arithmetical term)
5. Sat(ni ,,w) there is a number m such that Sat(, m/ni, w)
6. Sat(xi ,,w) there is a zsuch that ([y(y= z)]w Sat(, z/xi, w))
7. Sat( ,,w) Sat(,,w) Sat(,,w)
8. Sat(,,w) Sat(,,w)
Philosophical comment:
As argued in the main text, a sentences truth-conditions can be modeled by a set of
intelligible scenarios. Accordingly, when w is taken to range over intelligible scenarios
as is intended abovethe characterization of truth at w yields a specification of truth-
conditions for every sentence in the language.
The proposed semantics makes full use of arithmetical vocabulary. So it can only be
used to explain the truth-conditions of arithmetical sentences to someone who is already
in a position to use arithmetical vocabulary. What then is the point of the exercise? How
is it an improvement over a homophonic specification of truth-conditions?
The first thing to note is that the present proposal has consequences that a homophonic
specification lacks. Conspicuously, a homophonic specification would be compatible with
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both traditional and subtle Platonism, but the present proposal is only compatible with
subtle Platonism (since it entails that there is no intelligible scenario with no numbers).
And unlike a homophonic specification, the present proposal can be used to go from the
assumption that every sentence in the purely mathematical fragment of ones metalanguage
is either true or false to the assumption that every sentence in the purely mathematical
fragment of the object-language is either necessarily true or necessarily false.
The reason the present proposal has non-trivial consequences is that mathematical vo-
cabulary never occurs within the scope of [. . .]w. So even though mathematical vocabulary
is used to specify the satisfaction clauses, theworldsin the range ofw can be characterized
entirely in non-mathematical terms. As a result, the formal semantics establishes a con-
nection between mathematical and non-mathematical descriptions of the world. It entails,
for example, that the intelligible scenarios at which the number of the planets is eight are
precisely the intelligible scenarios at which there are eight planetsfrom which it follows
that for the number of the planets to be eight just isfor there to be eight planets.
Why not make this connection more explicit still? Why not specify a translation-function that maps each arithmetical sentence to a sentence such that (i) contains
no mathematical vocabulary and (ii) the sublte Platonist would accept ? Al-
though doing so would certainly be desirable, it is provably beyond our reach: one can
show that no such function is finitely specifiable. (See Rayo (2008) for details.)
Technical comments:
1. Throughout the definition of dentation and satisfaction I assume that the range of
the metalinguistic variables includes merely possible objects. This is a device to
simplify the exposition, and could be avoided by appeal to the technique described
in Rayo (2008) and Rayo (typescript b).
2. In clause 5 of the definition of satisfaction I assume that number includes infinite
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numbers. This is done in order to ensure that there are no empty terms in the
language. If one wanted to restrict ones attention to the natural numbers one could
do so by working in a free logic.
B A Semantics for the Language of Set-Theory
Consider a two-sorted first-order language with identity, L. It contains set-theoreticvari-
ables (1, 2, . . .), and non-set-theoretic variables (x1, x2, . . .), and predicate-letters
(Human(. . . ) and Planet(. . . )). If is a variable assignment and w is a world, truth in
L relative to and w can be characterized as follows:
1. Sat(xi= xj,,w) [(xi) =(xj)]w
2. Sat(xi j,,w) (xi) (j)
3. Sat(i j,,w) (i) (j)
4. Sat(Human(xi),,w) [(xi) is human]w
5. Sat(Planet(xi),,w) [(xi) is a planet]w
6. Sat(i ,,w) there is a set such that (Goodw() Sat(, /xi, w))
7. Sat(xi ,,w) there is a zsuch that ([y(y= z)]w Sat(, z/xi, w))
8. Sat( ,,w) Sat(,,w) Sat(,,w)
9. Sat(,,w) Sat(,,w)
where a set is goodw just in case it occurs at some stage of the iterative hierarchy built up
from objectsxsuch that [Urelement(x)]w. (For philosophical and technical comments, see
appendix A.)
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