Numbers and Propositions Versus Nominalists: Yellow Cards for Salmon & Soames
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DOI: 10.1007/s1067001294027
 Cite this article as:
 Urbaniak, R. Erkenn (2012) 77: 381. doi:10.1007/s1067001294027
Abstract
Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higherorder and firstorder logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and noncommitting truth conditions and account for the inferences made without committing themselves to abstract objects. I sketch a modal account of higherorder quantification, on which instead of ranging over sets, higher order quantifiers are used to make (logical) possibility claims about which predicate tokens can be introduced. This approach provides an alternative account of truth conditions for natural language sentences which seem to employ higherorder quantification, thus allowing the nominalist to evade Salmon’s argument. I also show how the nominalist can account for the occurrence of apparently singular abstract terms in certain true statements. I argue that the nominalist can achieve this by, first, dividing singular terms into real singular terms (referring to concrete objects) and only apparent singular terms (called onomatoids), introduced for the sake of brevity and simplicity, and then providing an account of nominalistically acceptable truth conditions of sentences containing onomatoids. I develop such an account in terms of modally interpreted abstraction principles and argue that applying this strategy to Soames’s argument allows the nominalists to defend themselves.
One would hope and perhaps conjecture that the whole general set theory, however beautiful it is, will in the future disappear. With the higher types Platonism begins. The tendencies of Chwistek and others (‘Nominalism’) of speaking only of what can be named are healthy. [Alfred Tarski] ^{1}
1 Introduction
Salmon in his argument against nominalism about numbers (Salmon 2008) and Soames in his argument for the existence of sentence types (Simons 1987; Soames 1999) employ (respectively) higherorder and firstorder logic to model certain natural language inferences. In both cases, the conclusions of those natural language inferences are supposed to carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that.
I argue that this is not a compelling strategy. The nominalist can accept those natural language consequences as long as he provides them with sensible and noncommitting truth conditions and is able to account for the validity of the inferences made without conjuring abstract objects.
In particular, I sketch a modal account of higherorder quantification, on which instead of ranging over sets, higher order quantifiers are used to make possibility claims about which predicate tokens can be introduced. This approach provides an alternative account of truth conditions for natural language sentences which seem to employ higherorder quantification. I also explain how this approach allows the nominalist to undermine Salmon’s argument.
Another phenomenon that needs to be accounted for is the occurrence of apparently singular abstract terms in certain true statements. I argue that the nominalist can explain this fact by, first, dividing singular terms into real singular terms (referring to concrete objects) and only apparent singular terms (called onomatoids), introduced for the sake of brevity and simplicity. Then I provide an account of nominalistically acceptable truth conditions of sentences containing onomatoids. I do so in terms of modally interpreted abstraction principles and argue that applying this strategy to Soames’s argument allows the nominalists to defend themselves.
For most of the paper I focus on the argument about numbers. As I proceed, it becomes clear that the two antinominalistic arguments are similar enough for the considerations to apply, mutatis mutandis, to both of them.
I first describe Salmon’s argument for the existence of numbers and Soames’s argument for the existence of sentence types. Then, I discuss in general terms what I consider the weak point of both arguments and sketch a nominalistic way to use this weakness. In Sect. 6 I describe the nominalist’s response to Salmon’s argument. In Sect. 7 I provide more details of a nominalistic strategy of handling both arguments. In Sect. 8 I explain how it can be used against Soames’s argument.
2 Exhibit One: Salmon on the Existence of Numbers
 [A]
There are exactly two Martian moons.
 [B]
Something is such that it is number two and there are exactly that many Martian moons.
He insists that [B] commits one to the existence of numbers, and thus the nominalist is committed to the existence of numbers as soon as they accept [A].
As Salmon observes, the standard nominalist response to the claim that [B] follows from [A] is that existential generalization introducing a variable replacing ‘two’ in [A] is illegitimate, since this word, as it stands in [A], is an adjective, not a noun.
 [\(\hbox{A}^{\prime}\)]

\(\exists{x} \exists{y} (Mx \wedge My \wedge x \not = y \wedge \forall {z} (Mz \rightarrow x = z \vee y = z))\)
 [1]
\(\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))](M)\)
 [\(\hbox{B}^{\prime}\)]

\(\exists {n} [n=\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))]\wedge n(M)]\)
Since nothing other than a single premiss, a definition, and logic is employed [\(\ldots [\hbox{B}^{\prime}\)]] is exactly as it seems: a genuine existentialgeneralization consequence of [[\(\hbox{A}^{\prime}] \ldots\)] The nominalist therefore cannot consistently accept [[\(\hbox{A}^{\prime}\)]] while rejecting [[\(\hbox{B}^{\prime}\)]]. Score one goal for the numbers against their opponents. [\(\ldots\) since] that there are such things as the FregeRussell numbers is provable in higherorder logic alone [\(\ldots\)] [n]umber deniers unintentionally commit themselves, through inconsistency, to there being things of every sort whatsoever, even round squares. Score another goal for the numbers. (Salmon 2008, 1801)
Salmon agrees that the nominalist may simply reject higherorder logic for ontological reasons (he cites Quine as having done so). Still, he insists, it’s enough that the argument sounds convincing for neutral agnostics about abstracta who have no bias “concerning the validity of natural language arguments” [181].
Keeping this argument in mind, let’s take a look at another antinominalistic argument. Both arguments rely on taking certain natural language utterances at face value, and both arguments can be dealt with in a similar manner.
3 Exhibit Two: Soames on the Existence of Sentence Types
 [P1]
\(U_x \not = U_y\)
 [P2]
S _{ x } = S _{ y }
 [C1]
\(\neg (U_x = S_x \wedge U_y = S_y)\)
 [C1a]
\(U_x\not = S_x \vee U_y \not = S_y\)
 [P3]
\(\forall{u, s}[uRs \rightarrow \forall {u^{\prime}} (s=u^{\prime} \rightarrow s=u)]\)
 [C2]
\(U_x \not= S_x \wedge U_y \not = S_y\)
Soames claims that this strategy can be generalized to distinguish propositions from utterances, propositions from sentence types, and propositions expressed by sentences from the meanings of those sentences.^{5} Although, perhaps, some of the steps in such generalizations can be plausibly challenged by a nominalist, I will put this issue aside. Instead, I will argue that the initial argument about sentence types and tokens fails. If Soames’s initial argument successfully generalizes, so does my criticism.^{6}
4 Natural Language, Formalization and Ontological Commitment
There is a difference between natural language phenomena and mathematical theories meant to capture them. It is one thing to accept a certain inference as it is formulated in natural language, it is another to claim that its rendering in a particular formal system preserves ontological commitment and that validity in a formal system of our choice corresponds to the validity of natural language arguments.
These are two separate issues: the formalization might capture valid natural language arguments in extenso without getting the ontological commitment right. Even if we assume that Salmon’s or Soames’s formal apparatus achieves the former (which I grant), it by no means follows that it also does the latter.
There is a difference between accepting the natural language inference from [A] to [B], thinking that the formal framework employed by Salmon or Soames captures in extenso the valid arguments^{7} and assuming that [\(\hbox{A}^{\prime}\)] and [\(\hbox{B}^{\prime}\)] are in all respects adequate renderings of [A] and [B]. Salmon, without any further argument, moves from the claim that the nominalist should make the first two moves to the claim that they also should make the last one.
If the nominalist can account for the validity of the natural language argument (which involves what appears to be higherorder quantification) without relying on Salmon’s rendering, they won’t have to be committed to whatever [\(\hbox{B}^{\prime}\)] seems committed to. The nominalist, however, needs to tell a sensible story about the natural language argument from [A] to [B], and one of my main goals is to sketch such a story.
This point applies to Soames as well. He also makes his formalization do the heavylifting. Once you interpret sentenceterms as real singular terms and the sameness predicate as applicable only to genuine singular terms, you’ve fixed the outcome: accepting the claim that x and y made different utterances but uttered literally the same sentence will, in this rendering, force you to take sentence types to be objects distinct from the tokens. But if the nominalist can say a sensible story about the behavior of what prima facie seems like singular terms without taking them to be referring singular terms, they don’t have to be committed to whatever Soames’s formalization is committed to. Another goal of this paper is to sketch such a story.
The antinominalist might try to prevent this move by claiming that Salmon’s and Soames’s renderings are default interpretations of the relevant natural language sentences, and the onus probandi of showing their inadequacy belongs to the nominalist. One problem is that this strategy will lead nowhere. The antinominalist will insist on and the nominalist will deny such claims. There seems to be no clear procedure for deciding what the default formalization is, and there are no obvious reasons to think that what’s default is right.^{8} Another issue is that even among logicians and philosophers of language there is no univocally accepted approach to abstract terms and the platonist interpretation of abstract terms in natural language is not commonly accepted.^{9} For this reason, I will not devote any more attention to this issue and move on to sketching philosophical stories that nominalists can develop to account for the validity of the relevant natural language arguments.
5 Choice of Strategy
Let’s grant that the nominalist should accept the natural language inference from [A] to [B]. This is not only because an unbiased observer might still buy into Salmon’s argument, but also because arguments whose higherorder formal renderings are valid are abundant in mathematics, and a nominalist who would deny their correctness would set themselves against a legion of working mathematical theories (Lewis 1991). Putting the issue of making nominalistic sense of all mathematical theories, I will focus on sketching a way the nominalist can accept the natural language inference discussed by Salmon without ontological guilt.
The first thing to observe is that going merely substitutional about higherorder quantification doesn’t seem viable for the nominalist. If they think that the admissible substituents, predicate tokens, are finite sequences over a finite alphabet and only actually existing things can be quantified over, they run out of tokens when they try to emulate higherorder quantification over infinite domains.^{10}
Another stab would be to go inferentialist about higherorder quantification in the vein of (Rossberg and Cohnitz 2009). On this approach, the meaning of logical constants and the logic itself are specified by means of purely syntactic rules of inference. All there is to secondorder logic is the notion of secondorder derivability and standard rules for handling Boolean connectives, identity and first and secondorder quantifiers. While Rossberg and Cohnitz convincingly argue that the resources used on this approach are nominalistically acceptable, I choose not to employ this strategy.
A wider discussion of inferentialism is beyond the scope of this paper, but my dissatisfaction with the inferentialist approach in this case stems roughly from the following intuition. Natural language claims containing prima facie higherorder quantification not only occur in inferences, but also can be true or false. The task of the nominalist is not only to explain how such claims can be successfully used in inferences, but also to account for this ability. Some account of how such prima facie higherorder contingent claims are related to the world is needed. Perhaps the inferentialist could insist that such contingent claims are true in virtue of being derivable from some firstorder contingent claims. I’m not sure how successful this strategy would be, especially in light of independence phenomena. I cannot pursue these issues in this paper and I find a more direct account of such a relation between sentences and the world preferable to the one which proceeds via derivation.
One way out of this quandary for the nominalist is to go modal. In this setting, the second order ‘For some F, ϕ’ reads ‘it is possible to introduce a predicate token F such that ϕ’ rather than ‘there is a set F such that ϕ’. Thus, the general strategy is to take natural language claims about which predicate tokens it is possible to introduce as intuitively clear, to accept the modality involved as primitive. Then, the truthconditions of natural language claims containing prima facie higherorder quantification are to be given in terms of such “nameability” claims, and the validity of prima facie higherorder natural language arguments is to be explained in terms of truth preservation modulo such translation.
This deals with the problem faced by going merely substitutional. Given a predicate token P and a set A say that P captures A iff P is true of all and only those objects which are in A. While there might be not enough predicate tokens to mimic quantification over all subsets of an infinite domain (it is not true that for each subset there exists a predicate token which captures it), there are sufficiently many ways predicates could be to do the job.
One important intuition about nameability, which will allow us to avoid paradoxes, is that the approach is typed. The initial domain of extralinguistic objects (which is the domain of firstorder quantification) doesn’t change with the introduction of new predicate tokens. Predicate tokens are at specific levels and can apply to objects one level below only. (Typing can be weakened to obtain something more resembling a cumulative hierarchy. I will briefly discuss this move at some point.)
A crucial move here is the assumption that the possibility involved is sufficiently clear and can be taken as a primitive. One can challenge it by requesting a more formal truth theory. This is a tricky challenge, because once the nominalist tries to accommodate this request, the truth theory is bound to employ some settheoretic apparatus, and the nominalist is asked to explain why this apparatus should be available to them. But the claim that a formal truth theory is needed should not be accepted without argument. For the sake of brevity and simplicity, in this paper I will simply assume that a formal semantics is not needed and the nameabilityspeak is clear enough. However, a more detailed formal approach is developed in Urbaniak (2010). Notice that even if the philosophical viability of the nominalist approach described here needs further defence, sketching the approach suffices as a challenge to Salmon or Soames: as long as they do not provide further reasons to prefer their frameworks over the nominalist reading, their initial arguments are not satisfactory. Let’s now see in more detail how this approach allows the nominalist to handle the arguments in question.
6 The Nominalist Story About Moving from [A] to [B]
The symbols of the formal language are: standard brackets, Boolean connectives (\(\neg, \wedge, \vee, \rightarrow, \equiv\)), firstorder individual variables (\(x, y, x_1, \ldots\)), secondorder predicate variables (\(F, G, \ldots\)), thirdorder predicate variables (\(F^3, G^3,\ldots\)), first, second and third order identity (=, = _{2}, = _{3}), lambda operator (λ), firstorder quantifiers (∃, ∀), modal secondorder and thirdorder existential and universal quantifiers (\(\Upsigma, \Uppi, \Upsigma^3, \Uppi^3\)). The formation rules are standard.^{11}
The language is devised to speak of extralinguistic objects and possible predicate tokens. Firstorder identity and Boolean connectives are understood in a standard way. Higherorder identity is coextensiveness (e.g. P = _{2} Q is true just in case predicate tokens associated with P and Q apply to the same things). The firstorder quantifiers have their standard meaning: they quantify over the initial domain of extralinguistic objects. The secondorder quantifier ‘\(\Upsigma F\)’ is read as ‘it is possible to introduce a firstorder predicate token such that’ (the universal ‘\(\Uppi F\)’ is understood analogously and so are thirdorder quantifiers). The lambda operator constructs a thirdorder predicate: ‘\(\lambda F[\ldots F \ldots](K)\)’ is interpreted as “the predicate token K could salva veritate serve the role of F in ‘\(\ldots F \ldots\)’”, which boils down to the deflated reading of ‘\(\ldots K\ldots\)’. ‘\(\lambda F[\ldots F \ldots]=_3K^3\)’ reads ‘for any \(H,\,\ldots H \ldots\) just in case K ^{3}(H)’.
 [\(\hbox{A}^{\prime}\)]

\( \exists {x} \exists {y} (Mx \wedge My \wedge x \not = y \wedge \forall {z} (Mz \rightarrow x = z \vee y = z))\)
 [1]

\(\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))](M)\)
 [B^{ N }]

\(\Upsigma^3 G^3 [G^3=_3\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))]\wedge G^3(M)] \)

To start with, ‘\(\Upsigma^3 G^3\)’ says that it is possible to introduce a predicate token G ^{3}.

The first conjunct in the scope of the quantifierboils down to saying that the thirdorder predicates on both sides of = _{3} apply to the same lowerorder predicates:$$ G^3=_3\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))] $$$$ \forall {H} [G^3(H)\equiv \lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))](H)] $$

For any \(H,\,\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))](H)\) boils down to \(\exists {x} \exists {y} (Hx \wedge Hy \wedge x \not = y \wedge \forall {z} (Hz \rightarrow x = z \vee y = z))\) which simply says that H applies to exactly two objects.

The last conjunct says that G ^{3} applies to M.
As far as commitment goes, this is just a longwinded way of saying:It is possible to introduce a thirdorder predicate token G ^{3}, such that G ^{3} would apply to the same predicate tokens as the lambda expression which applies to a predicate just in case this predicate applies to exactly two objects, and such G ^{3} would apply to M.
Which seems like an elaborate way of saying:It is possible to introduce a predicate token G ^{3}, which would apply to all and only those lowerorder predicate tokes which apply to exactly two objects, and such G ^{3} would apply to M.
M applies to exactly two objects.
Thus, prima facie, [B^{ N }] is noncommitting, for it only says that it is possible to introduce a certain predicatetoken satisfying a certain condition, rather than saying that a certain set exists.
What bearing does all this have on whether [B] follows from [A]? Well, on this nominalist reading [B], if it is taken to be committing and interpreted as [\(\hbox{B}^{\prime}\)], doesn’t follow from [A]. Yet, the nominalist will insist, our usual willingness to accept [B] is justified because [\(\hbox{A}^{\prime}\)] is a correct rendering of [A] and entails [B^{ N }] which translated back to natural language yields [B]. Salmon argues that since [B] follows from [A] and [B] is ontologically committing, so is [A]. Our nominalist rather suggests that since [B] follows from [A] and [A] is noncommitting, nor is [B].
Perhaps things are not as they seem and there are reasons to think the approach sketched in this section is not nominalistically acceptable.^{12} But even if this is the case, Salmon’s original argument needs to be supplemented with an argument against the nominalistic acceptability of alternative accounts of higherorder quantification.^{13} ^{,} ^{14}
7 Numerical “Singular” Terms and Abstraction
The nominalist should be able to explain how numerical terms can function in true sentences not only as adjectives, but seemingly also as singular terms. One way to start is to say that since there are no abstract objects (or at least, since it is preferable to account for the truth of mathematical statements without appealing to abstract objects), apparently singular abstract terms don’t refer: terms that pretend to refer to such objects are introduced for brevity and simplicity of our discourse. On philosophical grounds, such nominalists divide prima facie singular terms into those that really refer to objects and those which only pretend to do that and syntactically behave like singular terms, but in fact don’t refer to anything (let’s call the latter onomatoids).^{15} Thus, such a nominalist says that most of those sentences in which no abstract terms occur have literal truthconditions, whereas truthconditions of sentences containing abstract terms are more elaborate and don’t require realism about the reference of such terms.
 [HP]

The number of Fs is the same as the number of Gs if and only if some R establishes the equinumerosity of Fs and Gs.
For heuristic purposes and to avoid lengthy informal description, here’s a slightly more formal account of how this approach can be developed. First, since the nominalist doesn’t believe in concepts, instead of using Fregean comprehension with concepts, they rather speak of possible predicates, along the lines outlined above.
Definition 1
\(\langle D, P, \Updelta\rangle\) is an infinite naming structure (INS), if D is an infinite domain of objects, \(\Updelta \subseteq P \times D\), and for any \(A\subseteq D\), there is a \(\sigma \in P\) such that \(\forall {x\in D} [x\in A \equiv \Updelta(\sigma,x)]\).^{17} □
We are putting the issue of actual infinity aside and don’t worry about the assumption that the domain of objects is infinite.^{18} P is the intended assembly of possible predicate tokens over D satisfying comprehension and \(\Updelta\) is the ‘satisfaction’ or ‘true predication’ relation.^{19}
 [2]
\(\langle D, P, \Updelta, v\rangle \models Fx \,\hbox{ iff } \Updelta(v(F),v(x))\)
 [3]
\(\langle D, P, \Updelta, v\rangle \models \Upsigma {F}\phi \,\hbox{ iff } \langle D, P, \Updelta, v^{\prime}\rangle \, \hbox{where} \, v^{\prime} \,\hbox{is\,an} \, F\hbox{variant\,of} \,v.\)
So the satisfaction of a quantified phrase depends on whether certain possible predicate tokens are available.
 [2]
Fx is satisfied (given an association of predicate variables with predicate tokens and of individual variables with individuals) iff the predicate token (associated with) F applies to the object associated with x.
 [3]
\(\Upsigma F\phi\) is satisfied (given an association of predicate variables with predicate tokens and of individual variables with individuals) iff it is possible to associate F with a possible predicate token (leaving the association of other variables unchanged) so that ϕ comes out satisfied.
Now (since we are heading to a nominalist interpretation of the language of arithmetic) a slightly more formal description of what [HP] does “on top of” INS:
Definition 2
An HPmodel is an INS \(\langle D, P, \Updelta\rangle\) extended with a set \(N_M=\{N(\sigma) \mid \sigma \in P\}\) and an equivalence relation ≈ on N _{ M } such that for all σ, τ in P, [N(σ) ≈ N(τ) ≡ σ ∼ τ] where ∼ is the equinumerosity relation between possible predicate tokens. □
N can be seen as belonging to the model and being a function from P onto N _{ M }. Equally well one can simply speak of P and N _{ M } being just sets such that the conditions built into the definition are satisfied. The equivalence relation is the “identification” relation between apparent singular terms, and the definition above can be easily seen as embracing Hume’s Principle. We also say that N(τ) is the successor of N(σ) (i.e. S(σ, τ)) iff τ is equinumerous to a possible predicate that applies to all those objects which σ applies to, and one more object).^{21}

If \(\sigma\in P,\,\neg (\exists {x\in D} )\Updelta(\sigma,x)\), then N(σ) is a natural numerical term (NNT) and all elements of N _{ M } ∪ N _{ A } ≈related to it are NNTs.

If σ is an NNT and S(σ, τ), τ is an NNT.

Nothing else is an NNT.

Associate individual constants with elements of N _{ M }. This is straightforward: ‘0’ is associated with N(τ) for some unsatisfied possible predicate token τ.^{22}

Associate St for any term τ with the successor of the NNT associated with t.

Interpret identity in the language of PA _{1} as ≈, the “fake identity” relation.

Understand the Boolean connectives, addition and multiplication in the standard manner.^{23}

Interpret quantification in PA _{1} as ranging over N _{ M }.
 [SS0 + SS0 = S0 + SSS0]

The possible NNT associated with ‘SS0+SS0’ bears ≈ to the possible NNT associated with ‘S0+SSS0’.
 [∃ x 0 + x = SSS0]

It is possible to introduce an NNT σ such that the NNT associated with ‘0’ addedwithinmodel to σ yields a possible NNT which bears ≈ to the NNT associated with ‘SSS0’.
On the account described above, identities between numerical terms (which syntactically behave like singular terms) can be true, but are not grounded in the existence of numbers. We could consider a language in which also mixed identities (identities whose one argument is a real singular term and another argument is an onomatoid) are wellformed. Such mixed identities would always come out false, because no real object is the referent of an abstract term on this view.
 [B]

Something is such that it is number two and there are exactly that many Martian moons.
 [\(\hbox{B}^{\prime}\)]

\(\exists {n} [n=\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))]\wedge n(M)]\)
 [B^{ N }]

\(\Upsigma^3 G^3 [G^3=_3\lambda F[\exists {x} \exists {y} (Fx \wedge Fy \wedge x \not = y \wedge \wedge \forall {z} (Fz \rightarrow x = z \vee y = z))]\wedge G^3(M)]\)
 [B^{ S }]

∃ x [x = SS(0) ∧ N(Martian Moon) = x]
Then we have an identity between numerical terms, so we continue:It is possible to introduce a natural numerical term\(\ldots\)
(This easily boils down to saying: ‘It is possible to introduce a natural numerical term which corresponds to predicate which applies to exactly two objects’). The last bit of [B^{ S }] requires that this possible numerical term should bear ≈ to the numerical term corresponding to ‘is a Martian moon’. So we continue:\(\ldots\) which bears ≈ (the fake identity introduced by [HP]) to the natural numerical term corresponding to a predicate which applies to exactly two objects\(\ldots\)
After a rather straightforward simplification this says:\(\ldots\) and this newly introduced NNT bears ≈ also to the NNT ‘N(Martian Moon)’.
which, I submit, commitmentwise, says nothing more than ‘there are exactly two Martian moons’.It is possible to introduce a natural numerical term corresponding to predicates satisfied by exactly two objects, and such numerical term would also correspond to the predicate ‘is a Martian moon’.
One worry about this approach might be that it is typed, and (if the system is extended to other higherorders) there are separate numbers at each level, so that there is no unique number two etc. While this is true for the typed approach (which I followed for the sake of simplicity), the issue does not arise if a cumulative hierarchy is used instead. On this approach, introducing predicate tokens proceeds in stages, pretty much mimicking the (initial steps of) the iterative hierarchy of sets. First there are extralinguistic individuals and predicate tokens can be introduced which apply to those individuals. Then, predicate tokens can be introduced which apply to such firstorder predicate tokens or extralinguistic individuals, and so on. (It is important that the hierarchy is rather cumulative than typed: predicates can have “mixed” reference.) After each level of predicates the numerical onomatoids are expanded and associated with equinumerous predicates from below (even if those predicates belong to different levels). On this approach, the number of Martian moons is the same as the number of ‘x = 0 ∨ x = 1’, and so on. More details on the cumulative approach are available in Urbaniak (2010).
8 Nominalistic Abstraction and Sentence Types
The strategy employed to account for the behavior of numerical singular terms can be generalized. In particular, the nominalist can take terms referring to utterances to be real singular terms but interpret “sentence type” terms as onomatoids introduced by means of modally interpreted abstraction principles.
 [ST]

The sentence uttered by x is the same as the sentence uttered by y if and only if the utterances made by x and y are sufficiently similar not to make a linguistic difference.
This abstraction principle, on the nominalistic reading, tells us that given that x and y make two utterances, onomatoids ‘S _{ x }’ and ‘S _{ y }’ can be introduced. [ST] gives us also truthconditions for formulas in which sentenceterms occur: mixed identities are false by default and the wellformed claim ‘S _{ x } = S _{ y }’ is true just in case U _{ x } and U _{ y } remain in an appropriate equivalence relation.
There is an important difference between Soames and the nominalist. Soames seems to think that it is enough to establish [C2] to accept the existence of sentence types. Our nominalist doesn’t think the truth of [C2] carries commitment to such objects. Soames would accept, as a consequence of [P2], ‘∃x x = S _{ y }’. The nominalist, on the other hand, would accept ‘\(\Upsigma x x=S_y\)’ instead, where \(\Upsigma x\) means ‘it is possible to introduce a sentencetype term x such that’ and identity is understood as in [ST]. For the nominalist, [P2] is true not in virtue of ‘S _{ x }’ and ‘S _{ y }’ referring to one and the same object, but rather because other, noncommitting truth conditions of [P2] (determined by the righthand side of [ST]) are satisfied. And just because [C2] tells us that sentences are not to be identified with utterances it doesn’t follow that sentence terms are genuine singular terms.
Perhaps, the relevant notion of sufficient similarity employed in [ST] requires more explication and perhaps its nominalistic acceptability needs to be additionally defended. On the other hand, claiming that shape and sound similarity recognition, available even to not very well developed animals, clearly requires the existence of abstract objects is a bit of a stretch. ^{26} Whatever your view on the nominalistic acceptability of [ST] is, issues like these have to be seriously considered before taking sentencetype terms to be genuine singular terms.
9 Final Remarks
Salmon’s and Soames’s arguments rely on specific formal interpretations of certain natural language sentences. They take those interpretations to be a guide to the ontological commitment of natural language sentences in question. Even if their formal interpretation captures the natural language arguments adequately, it does not follow that it also gets the ontological commitments right. I argued that the nominalist can provide an alternative account of the validity of the arguments in question, on which the conclusions do not have the commitments Salmon and Soames would like them to have. The conclusions are true, but for quite different reasons.
The differences between philosophers sympathetic to Salmon’s and Soames’s arguments and the kind of nominalist we were talking about are deeper than it might initially seem and often result from certain metatheoretic convictions. The former are inclined to think that most of philosophical questions about existence are trivial and can be answered just by looking at what natural language existence claims people accept. The latter tend to believe that apart from a weak interpretation of quantifiers (or a weak notion of an object) on which mathematical existential claims come out true (and on which, presumably, existence claims about correctly introduced onomatoids come out true), a more robust quantifier or notion of an object is available, and answering what exists in this robust sense is not easy. Arguing for the plausibility of one of these metatheoretic approaches or finding a sensible alternative to both of them is a challenge I will not address in this paper. ^{27}
As cited by Carnap, folder RC 0901609 from the Carnap archive. Translated by Mancosu (2005, 334).
The argument isn’t of course extremely new, but I’m not interested in identifying the first person who formulated it. Suffice it to say, Soames’s formulation is commendable for its clarity.
He is not explicit about how R should be read, but the context indicates that ‘xRy’ means ‘x is an utterance of y’.
Soames’s formulation is semiformalized, I’m giving a slightly more formalized version. Also, it is worth emphasizing that the variables involved have to be interpreted as twosorted, s ranging over sentences and u ranging over utterances. Otherwise, [P3] on its own would entail that any sentence is identical with all its utterances. For suppose uRs. By [P3] we get: \(\forall {u^{\prime}} (s=u^{\prime} \rightarrow s=u)\). If we are allowed to “mix” variables, by universal instantiation this yield \(s=s \rightarrow s=u\). But the antecedent is trivially satisfied, so s = u. Rather, the premise says that a sentence is identical to all its utterances if it is identical to any utterance at all.
“The reason for considering this argument in some detail is that it generalizes. For example, to distinguish statements (propositions) from utterances, imagine a case in which x and y assert the same proposition. Since their utterances are different but their statements are the same, either x’s utterance is not identical with what x stated or y’s utterance is not identical with what y stated or both. To complete the argument, one may note the implausibility of claiming that the statement asserted by one of the speakers was a sequence of sounds (marks) that the speaker did not produce. [\(\ldots\)] The same arguments can be made for distinguishing statements from sentences. Just as the same statement can be made by different utterances, so utterances of different sentences can result in assertion of the same statement [\(\ldots\)] Although there is a close relationship between the meanings of sentences and the propositions they express, the two cannot always be identified. To see why, we need to consider two different kids of cases: those in which utterances of sentences with different meanings express the same proposition and those in which utterances of sentences with the same meaning express different propositions.” (Soames 1999).
Perhaps a nominalist can be happy with reconstructing sentence types as equivalence classes of sentence tokens. In what follows, however, I will not assume that a nominalist is happy with classes, whether they are equivalence classes or not.
That is, it correctly diagnoses valid natural language arguments modulo representation in the formal system.
Suppose even that the objectual reading is the default one. Still, it might have become default in virtue of its capturing the arguments in extenso and its simplicity, despite its ignored ontological inadequacy.
Let’s skip a rather long history of nominalism before the XXth century and ignore the fact that from a larger perspective firstorder and higherorder logics were developed quite recently and so by no means are tools that naturally come to mind. Even among prominent representatives of modern logic naive trust in the reference of apparently abstract terms hasn’t been that common.
Just to give a few examples: Frege didn’t settle with Hume’s Principle, didn’t take natural numbers to be objects sui generis, but tried to define them as composed objects whose nature was by no means immediately obvious to generations of mathematicians rather familiar with arithmetic. Dedekind and structuralists also put forward a rather elaborate theory of how the language of arithmetic works, rather avoiding simplistic objectual interpretations of natural number terms. Russell decided not to take class terms to refer to anything and developed his noclass theory (Whitehead and Russell 1910, vol. I, Introduction, ch. III, section (2), p. 75). Tadeusz Kotarbiński, Tarski’s teacher, had an even more general view about singular terms. For him, the only genuine singular terms were those singular terms which referred to spatiotemporal objects and the truth conditions of sentences containing other apparent singular terms were to be reduced to sentences containing only genuine singular terms (Kotarbiński 1929). More recently, Boolos’s (1998) approach to plural quantification suggests that plural quantification in natural language does not commit one to the existence of sets just because standardsecond order translations of certain true sentences do. (Boolos himself was rather concerned with plural quantifiers in set theory, so he assumed the existence of sets; still, he didn’t think that it is plural quantification that carries ontological commitment.)
In general, various nominalistic approaches are being recently developed (Chihara 1990; Hellman 1989; Lewis 1991; Goodman 1966; Goodman and Leonard 1940; Simons 1987; Martin 1988, 1992) and to claim that the platonist reading of mathematical statements is the default reading that almost everyone accepts (even only among logicians, philosophers, and perhaps mathematicians) is to ignore evidence.
Rossberg and Cohnitz (2009) describe a certain interesting strategy of avoiding this problem while still going substitutional. Roughly, the idea is that all the needed substituents are there, in the world. They exist as proof or tokenshaped configurations of, say, particles, yet to be discovered or pointed at. My worry about this strategy is that, slight eccentricity of this proposal aside, application to infinite domains still requires assuming that there are uncountably many such configurations. This suggests going modal and speaking about possible configurations or the possibility of world containing more matter than it currently does; but if that’s the case, going modal without the additional eccentricity of the proposal might be preferred.
While I mentioned that the approach is rather cumulative than typed, I here follow the standard first and secondorder typing, mostly for the sake of simplicity and because this yields a system most similar to the one used in the original argument.
A more detailed discussion of the nominalistic acceptability of the theoretical devices employed in this strategy lies beyond the scope of this paper.
One might feel tempted to think that introduction of predicates itself is not nominalistically acceptable, because each predicate has to have a set at its denotation. Thinking that natural language predicates are best construed as referring to sets is just a bias. Equally well one can think of predicates as applying directly to (perhaps multiple) objects (and thirdorder predicate tokens to apply to multiple secondorder predicate tokens).
The approach, so far, is similar to that of (Chihara 1990) who takes existential quantifiers to be constructibility quantifiers (thus, existential higherorder quantifier is read ‘it is possible to construct a formula’). Some differences arise when we look at more formal details (Urbaniak 2010), especially when cumulative approach is chosen over the typed one. More important differences arise when abstraction principles are brought in. These are the subject of sections to follow.
This approach is inspired by certain ideas of Kotarbiński (1929), a Polish logician and Tarski’s teacher, who held this view and dubbed such apparent singular terms onomatoids.
Thus, to introduce equinumerosity we have to talk also about possible binary secondorder predicate tokens which apply to two objects in a certain order. Then, given a fixed domain, F and G are equinumerous iff it is possible to introduce a binary predicate token R such that (1) for any x with Fx there is a y with Rxy and Gy, (2) for any y with Gy there is an x with Fx and Rxy, (3) if Fx, Fz, Gy, Rxy and Rzy, then x = z and (4) if Gx, Gz, Fy, Ryx and Ryz, then x = z.
A more nominalistically kosher description would go: Consider a situation in which there are infinitely many objects and for any selection of those a predicate token can be introduced which applies exactly to them.
One worry about this approach is whether a nominalist can legitimately use semantics formulated in settheoretic terms. Again, this issue lies beyond the scope of current considerations, but the main moves described in this paper can be described without the settheoretic apparatus, in the spirit of footnote 17. It is also worth mentioning that even Quine didn’t think that using firstorder logic commits one to sets just because the semantics of firstorder logic is settheoretic.
An Fvariant of an assignment is an assignment which differs from the original at most in what it assigns to F.
Minimal HPmodels allow us to introduce numerical terms of a very specific form: ‘the number of σ’. However, nothing prohibits us from introducing other numerical terms; they don’t have to be all of the form ‘N(σ)’. Such an extension can be easily accounted for by a slight modification of the notion of an HP model. For the sake of simplicity, we skip it.
That such predicate token is possible follows from the comprehension principle built into the notion of INS.
Given that NNT are an ωstructure ordered by the (transitive closure of) successor relation, this is pretty straightforward.
‘Added within the model’—this just refers to the function on N _{ M } which does the job of addition in the ω sequence generated by N _{ M } and the successor relation.
Would we also require that for any bunch of sufficiently similar sounds or shapes there is an abstract objects (abstract coconutlook, abstract dogbark, etc.) in virtue of which they are similar? If yes, the doctrine seems to be neither parsimonious nor helpful. Parsimony considerations aside, no one doing research on how certain animals recognize, say, the sound of a dog, seriously tries to solve the problem by proposing that there is an abstract object, the ultimate bark in whose barkhood all barks participate and that it is in virtue of accessing this abstract object and comparing the sounds they hear against it, squirrels know that a dog is nearby.
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