Download 1 Defusing Easy Arguments for Numbers Brendan Balcerak Jackson

Defusing Easy Arguments for Numbers
Brendan Balcerak Jackson
I. Easy arguments for numbers
The following pair of sentences (and many like it) pose a familiar puzzle for ontology:
(1) Jupiter has four moons.
(2) The number of moons of Jupiter is four.
Sentence (1) appears to be ontologically innocent, in that its truth does not seem to require the
existence of anything over and above Jupiter and its moons; in particular, it does not appear to
require the existence of the number four. Sentence (2) seems to be semantically equivalent to (1),
just another way of saying the same thing. And yet sentence (2) seems ontologically loaded; it seems
to wear an ontological commitment to numbers on its sleeve. But surely all this cannot be
correct. If it were, we could establish the existence of the number four simply by looking
through a telescope and counting Jupiter’s moons. It can’t be that easy to settle the ontological
question about numbers.
Thomas Hofweber distinguishes various standard ways of responding to the puzzle.
According to what he labels the fictionalist response, we should resolve the puzzle by conceding
that (1) and (2) are not semantically equivalent after all. So characterized, fictionalism is officially
neutral on whether or not numbers exist. But it tends to be endorsed by philosophers who are
independently suspicious of numbers, and so it is often paired with the suggestion that sentence
(2) is merely a useful fiction rather than a literal truth. Under ordinary circumstances (outside the
ontology seminar room) we tend to utter sentences like (2) in a spirit of pretense or makebelieve, as an indirect way of committing ourselves to the literal truth of sentences like (1).1 A
second sort of response is the neo-Fregean one, which takes the appearance of semantic
equivalence between (1) and (2) seriously, and so concludes that (1) is metaphysically loaded after
all. For the neo-Fregean, the paraphrase in (2) merely makes explicit an ontological commitment
that was already implicit in (1). This response, like the fictionalist response, is officially neutral on
the question of whether or not numbers exist. But combining neo-Fregeanism with the rejection
of numbers leads to the unpalatable conclusion that Jupiter does not have four moons after all,
1
For example, see Stephen Yablo (1998) and (2001).
1
and that a great many other everyday claims that we are inclined to accept are false as well. Thus
neo-Fregeanism tends to appeal to those who are ready to accept the existence of numbers.2
Hofweber favors a different way of responding to the puzzle.3 He agrees with the neoFregean that (1) and (2) are semantically equivalent. But he takes the equivalence as grounds for
concluding that (2) is not ontologically loaded after all: despite appearances, the truth of (2) does
not really require the existence of numbers, because it really say no more than what the
ontologically innocent (1) says. We might call this a paraphrastic response, because it is
reminiscent of the once-popular strategy of deflating the ontological commitments of apparently
loaded sentences by offering innocent paraphrases for them. According to this response, easy
arguments fail to establish the existence of numbers, because accepting (2) and its ilk amount to
nothing over and above accepting their innocent paraphrases.4
Like the classic version of the paraphrase strategy, Hofweber’s response to the puzzle
faces what we might call Alston’s challenge. The relation being-a-paraphrase-of is surely symmetric; if
(1) is an adequate paraphrase of (2) then (2) is likewise an adequate paraphrase of (1). Why, then,
should we read the ontological commitments of both sentences off of (1), rather than reading
them off of (2)? Why does the supposed paraphrase relation between (1) and (2) give us reason
to privilege the (apparently) innocent side of the relation when it comes to determining
ontological commitments?5 The inability to answer Alston’s challenge is nowadays regarded by
many as decisive grounds against the classic version of the paraphrase strategy. What is novel
about Hofweber’s version of it is that it marshals independent empirical linguistic evidence for
seeing (2) as inheriting its semantic properties from (1), and hence for seeing the apparent
reference to numbers in (2) merely as a misleading surface appearance. If this is right then
investigation into natural language syntax and semantics provides both a resolution to the puzzle
of easy arguments for ontology, and a response to Alston’s challenge.
Hofweber’s proposal is presented in Section II. In Section III, I argue that it faces several
serious syntactic and semantic problems, and in Section IV I argue that the linguistic evidence
Hofweber adduces in its favor is actually far too slim to be regarded as a satisfactory response to
See, for example, Bob Hale and Crispin Wright (2001) and Crispin Wright (1983). A distinct but related response
accepts easy arguments for abstract objects, but regards the entities whose existence is thereby established to be
merely “pleonastic” or “language-created” entities; see Schiffer (2003).
3 Our focus here will be on Hofweber’s discussion in his (2007). Essentially the same view, with similar supporting
arguments, is offered in Hofweber (2005a) and (2005b).
4 Cf. Quine (1953). Hofweber (2007, p. 4) is dismissive of the classic version of the paraphrase strategy, and he
evidently does not regard his own response to the puzzle as having much in common with it. Nevertheless, his
response shares with the classic paraphrase strategy the idea that the apparent ontological commitments of loaded
sentences like (2) are deflated by virtue of their special semantic relationship to sentences like (1).
5 Alston (1958).
2
2
Alston’s challenge. I conclude in Section V by suggesting that on balance, the linguistic evidence
favors a certain kind of fictionalist response to our initial puzzle about sentences like (1) and (2).
II. Hofweber’s paraphrastic response
The standard view of sentence (1) is that the expression ‘four’ functions syntactically and
semantically as a quantificational determiner. On this view, sentence (1) has the same syntactic
structure as sentences like the following:
(3)
a. Jupiter has some moons.
b. Jupiter has many moons.
c. Jupiter has few moons.
c. Jupiter has a moon.
On the standard view, the semantic function of ‘some’, ‘many’, ‘few’ and so on is not to pick out
or refer to some entity. (What would be the entity referred to by ‘some’?) Rather, their function
is to indicate something about the size of the sets or collections of entities corresponding to their
arguments. For example, ‘some’ as it occurs in (3a) indicates that the collection of moons of
Jupiter is non-empty, and ‘few’ as it occurs in (3c) indicates that the collection has few members
(perhaps as measured by some contextually salient standard). On this view, ‘four’ as it occurs in
(1) effectively functions as a kind of existential quantifier: it indicates that there are (at least) four
members included in the collection of Jupiter’s moons.6 While this requires that there be moons,
it does not require that there be such a thing as the number four. Hence the standard view
respects the intuition that (1) is ontologically innocent in the relevant sense.
Hofweber accepts the standard view of the syntactic and semantic function of ‘four’ as it
occurs in (1). What is novel is Hofweber’s view of sentences like (2). Following Frege, (2) is
customarily regarded as an identity sentence, with ‘the number of moons of Jupiter’ and ‘four’
functioning as singular terms flanking the identity sign. According to Hofweber, this is wrong.
Rather, (2) is syntactically derived from (1) by a process in which ‘four’ is extracted from its
position as determiner and moved into post-copular position (i.e. after the ‘is’ in (2)). In this new
position, ‘four’ functions syntactically like a noun phrase; this is why the position it occupies is
available to be bound by quantifiers, and to serve as an antecedent for anaphora:
6
Barwise & Cooper (1981); cf. also Chierchia & McConnell-Ginet (2000), chapter 9.
3
(4)
a. There is somethingi that the number of moons of Jupiter is ti.
b. The number of moons of Jupiter is fouri, which ti is my favorite number.
But on Hofweber’s view, ‘four’ nevertheless retains its original semantic function as a determiner.
Thus (2) has exactly the same truth conditions as (1), and hence no more entails the existence of
a number than (1) does.
According to Hofweber, the relationship between (1) and (2) is analogous to the
relationship between (5a), on the one hand, and (5b) and (5c) on the other:
(5)
a. John likes soccer.
b. What John likes is soccer.
c. It is soccer that John likes.
Hofweber sees (5b) and (5c) as also being syntactically derived by a process of movement, in this
case one in which an argument of the verb ‘likes’ in (5a) – either its subject ‘John’ or its direct
object ‘soccer’ – is “extracted” from its original position. In these cases the transformations have
no semantic effect; (5b) and (5c) are truth-conditionally equivalent to (5a). The transformations
rather subserve a pragmatic function, much like the pragmatic function of placing intonational
stress on the word ‘soccer’ in (5a):
(6) John likes SOCCER.
Like (6), (5b) and (5c) evoke a contrast: they suggest that John likes soccer rather than, say,
cricket or rugby. According to Hofweber, the transformation that yields (2) from (1) subserves
an analogous pragmatic function:
(7) Jupiter has FOUR moons.
Like (7), (2) suggests that Jupiter has four moons rather than, say, three moons or five moons.
This observation forms the basis for Hofweber’s central argument for his proposal; we will
examine the argument in Section IV.
It is easy to see how Hofweber’s claims about (1) and (2), and how they relate, resolve
the puzzle with which we began. For Hofweber, the inference from (1) to (2) is unproblematic,
because they have (and are semantically guaranteed to have) precisely the same truth conditions.
4
But the inference cannot be used to establish the existence of numbers, because (2) carries no
more ontological commitment to numbers than (1) does; the occurrence in (2) of expressions
that appear to denote numbers is merely a misleading bit of syntax. Moreover, Hofweber’s
proposal offers a novel reply to Alston’s challenge. It is (1) that reveals the true ontological
commitments of (1) and (2), because (2) is derived from (1) by a syntactic process that is
semantically inert. (2) is nothing more than a stylistic variant of (1) that serves certain pragmatic
purposes, just as (5b) and (5c) are stylistic variants of (5a).
III. Syntactic and semantic problems
Hofweber’s proposal is a novel and elegant way of responding both to our original puzzle and to
Alston’s challenge. But as an empirical hypothesis about the syntax and semantics of sentences
like (1) and (2), it fares quite poorly.
The first problem is simple: sentence (2) cannot be derived simply by “extracting” the
determiner ‘four’ from its position as determiner in (1), as Hofweber suggests. Taken literally,
such an extraction would yield the following:
(8) * Jupiter has ti moons fouri.
The construction in (8) is highly defective. Clearly, some much more subtle and complex process
of syntactic derivation is required to get (2) from (1): the determiner ‘four’ must change its
position relative to the other expressions in (1), as Hofweber suggests; but ‘moons’ and ‘Jupiter’
must also somehow switch their relative positions (from ‘Jupiter … moons’ to ‘… moons …
Jupiter’); the trace left by ‘four’ must somehow be spelled out by inserting ‘the number of’; and
the copular structure (reflected by ‘is’) must somehow be imposed on the sentence. This is a
highly unusual process, to say the least, and Hofweber says nothing at all about the kinds of
syntactic principles and mechanisms that are supposed to account for it. As Frederike Moltmann
(rather delicately) observes, the proposal hypothesizes a series of operations “that would not be
acceptable within current syntactic theory and have no parallel elsewhere.”7
Even if we allow that there is such a series of operations, a second serious problem is
that it is clearly not available for most other quantificational determiners:
7
Moltmann (2013), p. 524.
5
(9)
a. Jupiter has some moons / * The number of moons of Jupiter is some.
b. Jupiter has many moons / * The number of moons of Jupiter is many.
c. Jupiter has few moons / * The number of moons of Jupiter is few.
d. Jupiter has a moon / * The number of moons of Jupiter is a.
Given the orthodox assumption that ‘four’ as it occurs in (1) functions syntactically and
semantically as an ordinary quantificational determiner – an assumption that is crucial for
Hofweber’s response to the puzzle – it is quite mysterious why no other quantificational
determiner can undergo the kind of transformation that Hofweber hypothesizes.
A third problem concerns the hypothesized switch of grammatical category from
determiner to noun phrase that ‘four’ allegedly undergoes. It is far from clear why such a switch
takes place; after all, no such switch takes place with ‘soccer’ in Hofweber’s paradigms in (5),
repeated below:
(5)
a. John likes soccer.
b. What John likes is soccer.
c. It is soccer that John likes.
The expression ‘soccer’ maintains its category as a noun phrase in all of (5a)-(5c). So why doesn’t
‘four’ retain its category as a determiner when it moves into its new position in (2)? One might
be tempted to suggest that ‘four’ switches category because it cannot occur in post-copular
position as a determiner. However, this is entirely ad hoc: if ‘four’ cannot occur in post-copular
position as a determiner then this is a further consideration against the hypothesized movement,
rather than a reason to think that there is a process of movement that somehow forces a change
of syntactic category.
Hofweber notes that standard cleft constructions like ‘It is soccer that John likes’ are not
available for (1):
(10) *It is four which Jupiter has moons.
He takes this as indirect evidence for his proposal: “To extract the determiner from [(1)], one
can’t take recourse to a cleft sentence, but one has to take recourse to [(2)] instead.”8 The
Hofweber (2007), p. 17. Of course, one can take recourse to ‘It is four moons that Jupiter has’. Hofweber does not
consider this possibility.
8
6
reasoning here is doubtful; after all, it is not as though the language is somehow obliged to
provide a grammatical cleft-like construction corresponding to (1). But in any case, this sort of
reasoning is clearly undermined if we are allowed to suppose that expressions can switch
syntactic categories in order to legitimize transformations that would otherwise be
ungrammatical.
A fourth problem is that even if we grant that ‘four’ does switch syntactic category to
become a noun phrase, it is mysterious why it does not also thereby take up the semantic
function of its new category. Consider cases like the following:
(11)
a. Galileo put the telescope on the shelf.
b. Galileo shelved the telescope.
(12)
a. Galileo made the room dark.
b. Galileo darkened the room.
There is some plausibility in taking the pairs in (11) and (12) to be related by a syntactic
derivation, and if so then they are plausibly cases in which the relevant expression undergoes a
switch of syntactic category during the transformation; the noun ‘shelf’ in (11a), for example,
becomes (part of) the verb ‘shelve’ in (11b).9 In these cases, the transformation clearly brings
with it a change of semantic category as well. Indeed, it is hard to see how the sentences could be
intelligible otherwise: if ‘shelved’ in (11b) retained the semantic function of the noun ‘shelf’,
(11b) would be no more interpretable than ‘Galileo shelf telescope’. Hofweber’s proposal leaves
it a mystery why ‘four’ does not undergo a corresponding change of semantic category, and it
leaves it mysterious how (2) manages nevertheless to be an intelligible sentence.
A fifth problem is that Hofweber’s proposal must somehow account, not only for the
apparently referring expression ‘four’, but also for the definite description ‘the number of moons
of Jupiter’ that occurs as subject in (2). Why should we not see it as a genuine denoting
expression whose semantic function is to pick out a number, as on the standard Fregean view?
Hofweber apparently regards ‘the number of’ as a place-holder that functions merely to fill the
gap left by the extracted determiner ‘four’. But once we consider a slightly wider range of cases,
this becomes very hard to maintain:
9
See Ken Hale and Samuel Jay Keyser (1993) for arguments.
7
(13)
a. Jupiter most likely has four moons.
b. The most likely number of moons of Jupiter is four.
(14)
a. Jupiter is expected to have four moons.
b. The expected number of moons of Jupiter is four.
(15)
a. Jupiter has four moons in its orbit.
b. The number of moons in Jupiter’s orbit is four.
The definite descriptions in these cases are clearly not the result of merely inserting ‘the number
of’ into the alleged gap left by the extracted determiner. Hofweber must regard the extra lexical
material – ‘most likely’, ‘expected’ and so on – as somehow making its way into the description
that fills the gap. However, there is no independent reason to think that any such process occurs,
and nothing in Hofweber’s proposal gives us the means to explain how or why it happens in
these cases. We have only the metaphorical image of rearranging expressions (and introducing
whatever extra lexical material is needed) in order to build cleft-like constructions corresponding
to the (a) sentences. From a neutral perspective, it is much more plausible that the (b) sentences
are simply attempts to paraphrase the (a) sentences as closely as possible, using descriptions that
pick out the “correct” number – that is, the number that corresponds to the quantity of moons
indicated in the corresponding (a) sentence. But then we must see the descriptions in the (b)
sentences, and by extension in (2), as ordinary denoting expressions.
Hofweber takes independent support for the claim that ‘the number of moons of Jupiter’
does not function in (2) as a genuine definite description from the fact that substituting an
indefinite description for it is quite awkward.10
(16) ? A number of moons of Jupiter is four.
But this is not a good test for determining when an expression functions as a genuine definite
description, as is easily seen by cases like the following:
(17)
a. ? A sixteenth president signed the Emancipation Proclamation.
b. ?A winner of the race is Usain Bolt.
c. ? A first day of the week is Monday.
10
Hofweber (2007), pp. 7-8.
8
Clearly we should not conclude that ‘the sixteenth president’, ‘the winner of the race’ and ‘the
first day of the week’ do not function as genuine descriptions in the corresponding definite
sentences. The awkwardness in these cases, and in (16), seems rather to reflect the pragmatic
oddness of using the indefinite when the descriptive material complementing the determiner is
taken (typically or in context) to have a unique satisfier. Notice that there is no awkwardness
when the presumption of uniqueness is absent:
(18)
a. A likely number of moons of Jupiter is four.
b. An expected number of moons of Jupiter is four.
All things considered, then, the case for seeing ‘the number of moons of Jupiter’ as anything
other than an ordinary definite description is really quite thin.11
Where does this leave us? Hofweber’s proposal asks us to recognize an entirely sui generis
and quite complex series of syntactic transformations, transformations that are at the same time
extremely permissive – allowing us to move expressions into and out of apparently any
constituent in the sentence, changing syntactic categories as we go – and extremely restricted in
their application – occurring only with a very specific range of quantificational determiners in
order to yield a specific sort of copular construction. Considered as an empirical proposal in
natural language syntax and semantics, rather than as a solution to a puzzle about ontology,
Hofweber’s hypothesis about the relationship between (1) and (2) has very little to recommend
it.
IV. Loaded sentences as focus constructions
As we noted in Section II, Hofweber sees the hypothesized syntactic transformation that yields
(2) from (1) as subserving a pragmatic purpose. Specifically, he sees it as a mechanism for
indicating pragmatic focus. His evidence for this is a parallel between sentences like (2) and cases
that involve intonational stress, the paradigmatic mechanism for indicating focus. To illustrate,
A further confounding factor in the case of (16) is that there is a reading of ‘a number of Fs’ on which it means
roughly many Fs. On this reading (16) is ungrammatical because of a failure of agreement between the singular
copula and the plural subject. Hofweber is aware of this, but somehow takes it to be evidence that ‘the number of
moons of Jupiter’ is not a genuine definite description in (2), because we do not find a similar singular/plural
alternation with definite/indefinite pairs in general (2007, pp. 7-8). But this is hardly surprising, given that it is not
the case, in general, that there is a reading of ‘an F’ on which it means many Fs.
11
9
compare the following question/answer exchanges, where intonational stress is indicated in allcaps:
(19)
a. What does John like?
b. John likes SOCCER.
c. # JOHN likes soccer.
(20)
a. Who likes soccer?
b. # John likes SOCCER.
c. JOHN likes soccer.
The (b) and (c) sentences in these exchanges are semantically equivalent, and both are equivalent
to the neutral ‘John likes soccer’. And yet (19b), but not (19c), is an appropriate answer to the
question in (19a), while the reverse is the case in (20). (I use ‘#’ to mark pragmatic infelicity.)
This phenomenon is sometimes called question/answer (in)congruence, and Hofweber observes that
sentence (2) triggers question/answer congruence effects just like those illustrated in (19) and
(20):
(21)
a. How many moons does Jupiter have?
b. The number of moons of Jupiter is four.
(22)
a. Which planet has four moons?
b. # The number of moons of Jupiter is four.
Thus Hofweber takes (2) to be a focus construction whose structure places focus on ‘four’, just
as the intonational stress in (20b) places focus on ‘soccer’.
It is clear that Hofweber takes the question/answer congruence data concerning (2) to
lend significant support to his paraphrastic view. What is less clear is why, exactly, it should do
so. It might be tempting to take it as direct support for the existence of a syntactic process
whereby ‘four’ is extracted into “focus position”: perhaps speakers begin with sentence (1), and
then move ‘four’ to the end of the sentence (and rearrange the rest as necessary) in order to
signal that they intend the utterance to address the question of how many moons Jupiter has
(rather than, for example, the question of which planets have four moons). But this is surely not
what Hofweber has in mind. To reason in this way would be to lose sight of the fact that talk of
10
extraction and movement is merely picturesque metaphor; whatever derivational relationship
there is between (1) and (2) is not a matter of speakers making new sentences out of old ones,
and the fact that a speaker chooses (2) rather than (1) to get across a certain focus effect is no
direct evidence that the hypothesized derivational relationship obtains between them.
One line of argument that Hofweber does offer is that the question/answer congruence
data is incompatible with treating (2) as an identity statement, because (he claims) identity
statements do not trigger such effects. But as Berit Brogaard observes, identity statements do
sometimes trigger congruence effects.12
(23)
a. Who is the composer of Tannhäuser?
b. The composer of Tannhäuser is Wagner.
c. WAGNER composed Tannhäuser.
(24)
a. Who is Wagner?
b. # The composer of Tannhäuser is Wagner.
c. # WAGNER composed Tannhäuser.
The (b) sentence in these examples appears to be a paradigmatic identity sentence (and
Hofweber agrees that it is) and yet it exhibits congruence effects exactly analogous to the
corresponding (c) cases of intonational stress.13 And in any case, to resolve the ontological puzzle
we need to determine, not just whether (2) is an identity sentence, but whether either or both of
‘four’ and ‘the number of moons of Jupiter’ functions semantically as singular terms in (2). The
mere fact that (2) exhibits question/answer congruence effects does nothing to show that they
do not. To see this we have merely to note that ‘Wagner’ and ‘the composer of Tannhäuser’ both
clearly function as ontologically-committing singular terms in (23b)/(24b).
Brogaard (2007).
Hofweber (2007, pp. 19-20) discusses these cases, and sketches an alternative non-structural explanation of the
question/answer congruence effect exhibited by (24b) in terms of topic rather than focus; he suggests that the
length and complexity of the description in the subject position of (24b), along with the lack of conversational
salience of the object it denotes, makes it difficult to mark ‘Wagner’ as the topic of the sentence. But this reply
concedes that intonation-neutral question/answer congruence effects cannot be used as a straightforward diagnostic
for structural focus; structural focus is one explanation among others for such effects. Thus we cannot conclude
that the intonation-neutral question/answer congruence effects exhibited by (2) are due to structural focus, as
Hofweber does, until we have ruled out other possible explanations – including explanations that appeal to subtle
morphological and contextual (and perhaps other) features of the case, as does Hofweber’s explanation of the
effects in (24b). Later in this section I discuss several ways in which (2) turns out not behave like a focus
construction when we look beyond question/answer congruence; this provides reason to suspect that the
question/answer congruence effects are ultimately not best explained in terms of structural focus.
12
13
11
What is needed is some argument from question/answer congruence effects to the
conclusion that (2) has the same truth conditions as (1). We might try to provide such an
argument as follows. First, note that there is no truth-conditional difference between the focusneutral sentence ‘John likes soccer’ and its stressed counterpart ‘John likes SOCCER’. The
effect of intonational stress is not to change the meaning of the sentence, but merely to change
its communicative function. The same holds for ‘Jupiter has four moons’ and ‘Jupiter has FOUR
moons’. The question/answer congruence effects exhibited by (2), however, suggest that (2)
functions just like ‘Jupiter has FOUR moons’: what this sentence does with intonational stress,
(2) does with syntactic structure. Perhaps, then, the change of syntactic structure from (1) to (2)
likewise merely results in a change of communicative function, rather than a change of meaning,
so that (2) retains the same ontologically innocent truth conditions as (1).
In effect, the line of argument just sketched asks us to infer from the fact that two
sentences differ in communicative function, to the conclusion that they have the same truth
conditions. But this is clearly an invalid inference: if anything, a difference in communicative
function is prima facie evidence against semantic equivalence, since we would naturally expect
sentences with different meanings to be capable of serving different communicative functions.
(Just think of all the sentences that are not appropriate answers to the question, ‘Which planet
has four moons?’ and notice how many of them mean something different than (1).) If we take it
for granted that (1) and (2) are semantically equivalent, then the syntactic differences between
them might well contribute to an explanation of the difference between them in terms of
question/answer congruence effects. But noting that there are such differences does not lend
any weight to the hypothesis that they are semantically equivalent sentences that merely differ in
syntactic structure. Thus the congruence data provides no grounds for the paraphrastic view
over the fictionalist view.14 And notice that even if we grant that (1) and (2) are equivalent, the
congruence data does not allow us to conclude that the truth conditions of both (1) and (2) must
be the ontologically innocent ones that the paraphrastic response maintains. The neo-Fregean
can happily agree that (1) and (2) differ in their pragmatic function, just as Hofweber claims, and
yet insist that both have ontologically loaded truth conditions.
Still, it is hard to escape the thought that the paraphrastic view must derive some kind of
support from the observation that the communicative function of (2) mirrors that of ‘Jupiter has
Hofweber (2007, pp. 9-10) argues that seeing (2) as a focus construction solves what he calls the “puzzle of
extravagance” – basically, the question of why natural languages like English should systematically have two distinct
ways of saying how many moons Jupiter has. But posing the puzzle in this way clearly presupposes that (1) and (2)
are semantically equivalent. It is perhaps also worth noting that the sort of extravagance Hofweber finds puzzling is
rampant in natural language: witness ‘Brutus stabbed Caesar’ and ‘Caesar was stabbed by Brutus’, ‘Mary taught John
arithmetic’ and ‘Mary taught arithmetic to John’ and (infinitely) many others.
14
12
FOUR moons’. It is therefore important to see that the pragmatic similarities between the two
sentences are actually quite limited.
One significant difference is that typical uses of intonational stress typically generate
conversational implicatures, whereas the use of sentences like (2) does not. For example,
suppose that two astronomers are discussing which planets to observe in order to test their
theory about planets with large numbers of moons. They might have a dialogue like the
following:
(25)
a. No planet in our solar system has enough moons.
b. Well, Jupiter has FOUR moons.
The utterance of (25b) conversationally implicates that four moons is (probably) enough to test
the theory. But we do not get this implicature if we use (2) in place of (25b), at least not unless
we illicitly place intonational stress on ‘four’:
(26)
a. No planet in our solar system has enough moons.
b. Well, the number of moons of Jupiter is four.
Another striking difference concerns the use of focus-sensitive particles like ‘even’, ‘also’ and ‘in
particular’, which interact with focused constituents to trigger presuppositions or implicatures.
(27) Jupiter even has FOUR moons.
A typical utterance of (27) carries the presupposition, roughly, that Jupiter was expected to have
significantly fewer moons; in a context where everyone expected Jupiter to have four or more
moons, an utterance of (27) seems quite odd. Contrast this with the following:
(28)
a. The number of moons of Jupiter is even four.
b. Even the number of moons of Jupiter is four.
The presupposition in (27) is missing here. In fact, the sentences in (28) are quite awkward
unless we place intonational stress on some constituent. It is entirely unclear why this should be
so if – as Hofweber maintains – ‘four’ is focused in (28) just by virtue of its syntactic position,
even in the absence of intonational stress. Or consider the following:
13
(29)
a. Jupiter also has FOUR moons.
b. The number of moons of Jupiter is also four.
c. Also the number of moons of Jupiter is four.
(29a) suggests (quite oddly) that Jupiter has some number of moons in addition to four. But it is
very hard to hear any such suggestion in (29b) or (29c); if anything, they suggest that something
else besides the moons of Jupiter numbers four. Again, this is inexplicable if the syntactic
position of ‘four’ in (29b) and (29c) is just an alternative way of indicating focus.
Undoubtedly, Hofweber’s congruence data indicates one respect in which sentence (2)
and ‘Jupiter has FOUR moons’ share a communicative function. But in many other respects, the
focus-related communicative functions of the two types of sentences are very different. This can
hardly be regarded as a compelling basis for seeing some deep relationship between the two –
and in particular, for seeing (2) as somehow being a mere stylistic variant of ‘Jupiter has FOUR
moons’. Hofweber’s observations about question/answer congruence effects are far too slim a
basis on which to rest the paraphrastic view of the relationship between (1) and (2).
V. Indifference
As we saw in Section II, the orthodox syntax and semantics for sentences like (1) vindicates the
intuition that they carry no ontological commitment to numbers. The objections of the previous
two sections bring out some serious problems for Hofweber’s attempts to explain away the
appearance that (2) does carry ontological commitment. In the absence of further argument,
then, we have good empirical grounds for concluding that (1) and (2) are not semantically
equivalent after all.15
In fact, we should already recognize that (1) and (2) are not semantically equivalent, for
reasons that have nothing to do with ontological worries about numbers. As it turns out, Jupiter
actually has somewhere between sixty-two and sixty-four moons. 16 This fact is perfectly
compatible with the truth of (1), whose truth conditions only require that Jupiter have at least
A different strategy for avoiding ontological commitment with (2), which we have not considered here, is to treat
it as a so-called specificational sentence in which the expression ‘the number of moons of Jupiter’ actually has the
semantic function of a concealed question. On this view, (2) means roughly, the answer to the question of how many moons
Jupiter has is that Jupiter has four moons. This view is subjected to powerful critique in Brogaard (2007) and Felka
(manuscript).
16 The imprecision here reflects uncertainty among astronomers about what to classify as a moon.
15
14
four moons. This is why there is nothing incoherent or contradictory about uttering the
following sequence:
(30) Jupiter has four moons. In fact, it has sixty-two.
But it is incompatible with the truth of (2), which requires Jupiter to have exactly four moons:
(31) The number of moons of Jupiter is four. In fact, it’s sixty-two.
The second sentence in (31) can only be understood as a retraction of what is asserted by the
first sentence. Or consider the following argument:
(32)
a. The number of moons of Jupiter is four.
b. The number of moons of Saturn is four.
c. Therefore, Jupiter and Saturn have the same number of moons.
This is clearly a valid argument. But its conclusion is false, and so one or both of (32a) and (32b)
must be false, even though Jupiter and Saturn both have at least four moons. Thus Jupiter must
have exactly four moons for (2) to be true, and (1) and (2) are not semantically equivalent after
all.17 The fact that we tend to treat them as interchangeable in ordinary contexts requires some
other sort of explanation.18
Hofweber seems to assume that any such explanation must rely on a very robust notion
of fiction or pretense or metaphor; he sets fictionalism aside with the remark that “to really hold
this fictionalist line one must believe that fiction goes very deep in our ordinary discourse.”19 But
strictly speaking, someone who is a fictionalist in Hofweber’s sense need not hold any such
The argument here does not rule out the possibility that (2) is semantically equivalent to ‘Jupiter has exactly four
moons’. But it is even less plausible to suppose that (2) is syntactically derived from ‘Jupiter has exactly four moons’
than that it is syntactically derived from (1). What process would account for the “deletion” of ‘exactly’ from (1)?
(An analogous point holds for the suggestion that (1) is semantically equivalent to ‘The number of moons of Jupiter
is at least four’.) Alternatively, one might suggest that (1) is semantically ambiguous between an at least four and an
exactly four reading, and that the argument here only shows that (2) is not equivalent to (1) on the latter reading. One
problem with this suggestion is that if (1) really is semantically ambiguous in the way suggested, this is presumably
due to a lexical ambiguity in the determiner ‘four’; but then Hofweber’s proposal should predict a corresponding
ambiguity in (2) that we do not in fact find. And in any case, it is not very plausible that (1) is semantically
ambiguous: the exactly four reading of (1) is better regarded as an instance of Gricean scalar implicature of the sort
that is reliably associated with existential quantifiers in general. (Thanks to Diego Tajer and an anonymous referee
from Linguistics & Philosophy for discussion.)
18 The fact that (1) and (2) are not equivalent is also a problem for the treatment of (2) as a specificational sentence
(see footnote 15), since that treatment likewise predicts that (2) should be true as long as Jupiter as at least four
moons.
19 Hofweber (2007), p. 5 n. 10
17
15
thing. It is enough to note that in many ordinary conversational contexts, we tend to treat as
interchangeable sentences that are – in a rough, purpose-relative sense – close enough in truth
conditions, whether or not they are strictly equivalent. We simply bracket or ignore the
difference in truth conditions, and set aside as irrelevant certain possible circumstances that
would invalidate the inference from one to the other. For example, in most ordinary contexts we
are inclined to treat the following two sentences interchangeably:
(33)
a. Galileo likes Copernicus.
b. The person Galileo likes is Copernicus.
In so doing, however, we are implicitly bracketing circumstances in which Copernicus is a dog or
a robot, or – more exotically – circumstances in which an eliminativist metaphysics about
persons in general is true. This is often a perfectly reasonable thing to do: it might not matter to
us that (33a) and (33b) are not strictly equivalent, if the reasons the equivalence fails are not
currently of interest to us. Evidently we sometimes do this with (1) and (2) as well, bracketing
circumstances in which Jupiter has at least four moons but not exactly four moons. This can
happen quite automatically, without involving any deliberate pretense, and so it can easily go
unnoticed – as, indeed, it evidently has by nearly everyone who has considered pairs like (1) and
(2) in the context of ontological debates.
Developing this rough suggestion into a systematic explanation of the interchangeability
of pairs like (33a) and (33b) and (1) and (2) must be left as a project for further research. I
conclude with two observations that indicate what I take to be a promising direction for such
research to take.
The first observation is that the phenomenon at work here is closely related to the
phenomenon that Matti Eklund calls indifference: “with respect to much that we say or imply we
do not commit ourselves either to its literal truth or to its truth in any fiction; we are, simply,
non-committed.”
20
Eklund’s characterization of indifference begins from the familiar
observation that when a speaker assertively utters a declarative sentence S, she ordinarily intends
her assertion to rule out certain possibilities as ones that do not actually obtain. But it need not
always be the case that the possibilities the speaker intends to rule out are all and only those that
are inconsistent with the literal truth of S. This is so even when we would not regard the speaker
as speaking figuratively or non-literally. For example, in a conversation about whether the others
Eklund (2005), p. 558. Harry Frankfurt, talking about essentially the same phenomenon, describes it as a matter of
the speaker offering “a description of a certain state of affairs without genuinely submitting to the constraints which
the endeavor to provide an accurate representation of reality imposes” (2005), p. 8.
20
16
at a party we are attending are enjoying themselves, the speaker might see someone standing
across the room holding a glass of clear liquid and assertively utter the following:
(34) The man drinking water is happy.
In this context, the speaker obviously intends to rule out various possibilities about the mood of
the person she sees, namely those that are inconsistent with her being happy. But she might not
intend to rule out possibilities in which the man is drinking something other than water – say,
vodka – or possibilities in which the man is merely pretending to drink. It is not that she is
speaking figuratively or engaging in pretense when she describes the man as drinking water. It is
simply that, given the topic of conversation, it is beside the point whether or not these
possibilities obtain.
A pressing question for Eklund’s notion of indifference is the question of how it is
determined which possibilities a speaker rules out with an utterance, and in particular how this is
related to the literal meaning of the utterance. Surely the latter must place some constraints on the
former: one cannot use any given sentence to rule out whatever set of possibilities one likes. My
own view is that a satisfactory account must make reference to the idea that a conversation is
structured, in part, by the question or questions that the participants jointly take to be under
discussion, and that literal meaning functions as a guide for the participants to help identify the
speaker’s intended answer to the question currently under discussion.21 Such an account has
obvious potential application to the case of (2). It might well be the case that utterances of (2) in
ordinary contexts are used in order to rule out possibilities in which Jupiter has fewer or more
than four moons, and not in order to rule out the possibility that there are no numbers, even
though the literal truth of (2) is inconsistent with this possibility. The existence or non-existence
of numbers is simply orthogonal to the questions under discussion in many ordinary
conversations in which (2) is uttered. In fact, given Hofweber’s observation that (2) is like
‘Jupiter has FOUR moons’ in being an appropriate answer to the question of how many moons
Jupiter has, we should expect (2) to have a standard use in contexts in which the speaker is
indifferent to its ontological implications.
The second observation is that indifference can be easily extended to yield a promising
explanation of the apparent interchangeability of (1) and (2). In most ordinary contexts, an
utterance of (1) is used to rule out the possibility that Jupiter has fewer than four moons. In fact,
Cf. Roberts (1996); Balcerak Jackson (forthcoming) explores the relationship between literal meaning and the
question under discussion as it pertains to the effort to distinguish genuine from merely verbal disagreement.
21
17
because of the familiar scalar implicature effects triggered by existential quantifiers like ‘four’, an
utterance of (1) is very often used to rule out the possibility that Jupiter has anything other than
exactly four moons. This is no doubt part of the reason we so easily overlook the fact noted
above, that (1) cannot actually semantically entail (2). If speakers are furthermore typically
indifferent to the ontological implications of utterances of (2), as our first observation suggests,
then we should expect that speakers often use utterances of (2) to rule out just the same
possibilities as would utterances of (1). At least across a significant range of contexts, whatever
conversational purpose is achieved by uttering (1) would be just as well achieved by uttering (2),
and vice versa. Thus even though (1) and (2) are not semantically equivalent, it is no surprise that
we tend to regard them as interchangeable; this is simply a reflection of our tendency to be
indifferent towards the ontological possibilities that (2) rules out but (1) does not.
More would need to be said in order for an account along the lines indicated to be
adequate. One promising feature of it, however, is that it does not rely on specific claims about
the syntax, semantics or pragmatics of numerical determiners like ‘four’ or definite descriptions
like ‘the number of moons of Jupiter’. Thus it generalizes straightforwardly to other cases in
which apparently interchangeable pairs of sentences generate easy arguments for ontological
conclusions, such as the following:
(35)
a. Rolf is furry.
b. Rolf has the property of being furry.
(36)
a. Rolf is a dog.
b. It is a fact that Rolf is a dog.
As Hofweber notes, pairs like these pose puzzles analogous to the puzzle posed by (1) and (2):
how can it be so easy to establish the existence of such controversial entities as properties and
facts? The apparent interchangeability of these pairs can be given a pragmatic explanation in
terms of indifference in just the same way as the apparent interchangeability of (1) and (2).
Unlike active pretense and fictionalizing, indifference certainly is pervasive in everyday discourse.
Perhaps our tendency to treat ontologically loaded sentences as a mere paraphrases of their
innocent counterparts is simply a further illustration of this phenomenon.22
In preparing this paper I benefited enormously from discussions with Magdalena Balcerak Jackson, Matti Eklund,
Katharina Felka, Kit Fine, Thomas Hofweber, Hannes Leitgeb, Frederike Moltmann, Benjamin Schnieder, Diego
Tajer, Robert Schwartzkoppf and Ede Zimmermann. I am also grateful to audiences for helpful discussion at the
Semantics and Philosophy in Europe 5 conference at the University of Turin, Italy and at the Munich Center for
Mathematical Philosophy. The research for this paper was supported by a grant from the Deutsche
Forschungsgemeinschaft.
22
18
References
Alston, William P. 1958. Ontological commitments. Philosophical Studies v. 9, pp. 8-17.
Balcerak Jackson, Brendan. Forthcoming. Verbal disputes and substantiveness. Erkenntnis.
Barwise, Jon & Cooper, Robin. 1981. Generalized quantifiers in natural language. Linguistics &
Philosophy v. 4, pp. 159-219.
Brogaard, Berit. 2007. Number words and ontological commitment. The Philosophical Quarterly v.
57, pp. 1-20.
Chierchia, Gennaro & McConnell-Ginet, Sally. 2000. Meaning and Grammar: An Introduction to
Semantics (2nd edition). Cambridge: MIT Press.
Eklund, Matti. 2005. Fiction, indifference, and ontology. Philosophy and Phenomenological Research v.
71, pp. 557-579.
Felka, Katharina. Manuscript. Number words in specificational sentences.
Frankfurt, Harry. 2005. On Bullshit. Princeton: Princeton University Press.
Hale, Bob and Wright, Crispin. 2001. The Reason’s Proper Study: Towards a Neo-Fregean Philosophy of
Mathematics. New York: Oxford University Press.
Hale, Ken and Keyser, Samuel Jay. 1993. On argument structure and the lexical expression of
syntactic relations. In Ken Hale and Samuel Jay Keyser (eds.), The View from Building 20: Essays in
Linguistics in Honor of Sylvain Bromberger. Cambridge: MIT Press.
19
Hofweber, Thomas. 2007. Innocent statements and their metaphysically loaded counterparts.
Philosopher’s Imprint v. 7, pp. 1-33.
Hofweber, Thomas. 2005a. A puzzle about ontology. Noûs v. 39, pp. 256-283.
Hofweber, Thomas. 2005b. Number determiners, numbers, and arithmetic. The Philosophical
Review v. 114, pp. 179-225.
Moltmann, Frederike. 2013. Reference to numbers in natural language. Philosophical Studies v. 162,
pp. 499-536.
Roberts, Craige. 1996. Information structure in discourse: towards an integrated formal theory of
pragmatics. Ohio State Working Papers in Linguistics, volume 49: Papers in Semantics, pp. 91-136.
Schiffer, Stephen. 2003. The Things We Mean. New York: Oxford University Press.
Quine, W.V. 1953. On what there is. Review of Metaphysics v. 2, pp. 21-38.
Wright, Stephen. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University
Press.
Yablo, Stephen. 2001. Go figure: a path through fictionalism. Midwest Studies in Philosophy v. 25,
pp. 72-102.
Yablo, Stephen. 1998. Does ontology rest on a mistake? Aristotelian Society Supplementary Volume
72, pp. 229-283.
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