## Abstract

Kim (Synthese 190(6):1099–1112, 2013) defends a logicist theory of numbers. According to him, numbers are adverbial entities, similar to those denoted by “frequently” and “at 100 mph”. He even introduces new adverbs for numbers: “1-wise”, “2-wise”, and so on. For example, “Fs exist 2-wise” means that there are (at least) two Fs. Kim claims that, because we can derive Dedekind–Peano axioms from his definition of numbers as adverbial entities, it is a new form of logicism. In this paper, I will, however, argue that his theory is vulnerable to an analogue of the so-called Bad Company objection to neo-Fregeanism. This means that we cannot be sure that numbers are actually given to us by Kim’s definition; for, we don’t know whether it is indeed a good definition. So, unless Kim, or somebody else, provides a demarcation criterion between good and bad adverbial definitions, Kim’s theory will remain incomplete.

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## Change history

### 08 March 2018

Unfortunately, there is a typo in the author name. The correct spelling is Namjoong Kim. The author name was updated in the original publication.

## Notes

- 1.
Indeed, whether his theory is a genuine form of logicism is a highly controversial matter. See Sect. 3.

- 2.
In the rest of this paper, we will use “mode \(\phi \)” to refer to an adverbial entity, associated with the adverb \(\phi \). For instance, mode

*frequently*is the adverbial entity denoted by “frequently.” - 3.
You may wonder why the number

*n*of*F*s should be associated with (i) “There are at least*n**F*s” rather than (ii) “There are exactly*n**F*s.” However, he actually thinks that the number*n*, primarily defined as the mode of existence associated with (i), can be used to count the exact number of*F*s as well (2015, Sect. 5). See the next section for details. - 4.
See Kim (2016) for a detailed discussion of adverbial entities other than numbers.

- 5.
- 6.
I cite this reformulation of Frege’s definitions from Kim (2013, p. 1100).

- 7.
This should be distinguished from a related, but distinct problem: how do we decide whether \(\tau =\nu \) or \(\tau \ne \nu ,\) where \(\nu \) is a number word but \(\tau \) is not? See footnote 18 for more details.

- 8.
Actually, Kim also considers the following definition of the number 0: \(\exists _{0}xFx\) (‘

*F*s exist 0-wise’) \(\leftrightarrow \lnot \exists _{1}xFx\). However, he takes the number 1 as the smallest number, “partly for the sake of simplicity of exposition and partly (and relatedly) because cardinal numbers are counting numbers and counting always starts at 1” (2015, p. 121). We adopt the same policy. - 9.
To define \(\ge \), first define \(+\) in the usual recursive way. Next, \(\ge \) can be defined as follows: for any numbers

*m*and*n*, \(m\ge n\) iff \(m=n\) or for some number \(n^{*},\)\(m=n+n*\). - 10.
There are two influential views about the semantics of adverbs. According to the event-based view, if a sentence includes an adverb, then its logical form includes an extra variable quantifying over events and a predicate attached to it (Davidson 1967). For example, “Tom walks rapidly” means “\( \exists x\left( \text{ walks }\left( \text{ Tom },x\right) \, \& \text{ rapid }\left( x\right) \right) \),” where “

*x*” is a variable ranging over events. According to the modal operator view, when a sentence includes an adverb, it also has a hidden quantifier in its logical form; however, the variable quantifies over possible entities, not events (Lewis 1975). For instance, “Possibly, Tom is the winner of the game” means “\(\exists w\left( \text{ Tom } \text{ is } \text{ the } \text{ winner } \text{ at }\, w\right) \),” where “*w*” is a variable ranging over possible worlds. Neither tradition introduces a separate category of adverbial entities. Indeed, Chisholm’s (1957) adverbial view of perception, the most notable discussion of adverbs in the literature, was designed to*avoid ontological commitment*to sense-data as an object (needless to mention adverbial entities). - 11.
He writes: “ALA turns out to be a natural home for arithmetic: the basic concepts of arithmetic can be explicitly defined in the language of ALA, and the Dedekind–Peano axioms can be derived from those definitions by logical means alone” (2015, p. 117).

- 12.
I borrowed this example from Benacerraf (1965).

- 13.
In the first case, we shall use “\(\exists _{n}xFx\)” as the abbreviation of “\(\exists G\left( n\approx \left\{ x|Gx\right\} \subseteq \left\{ x|Fx\right\} \right) \),” where “\(\approx \)” is the one-to-one correspondence operator. In the second case, we can use “\(\exists _{n}xFx\)” to abbreviate “\( \exists R\big (R\left( F,n\right) \& \forall G\big (\big (R\big (G,\left\{ \varnothing \right\} \big )\leftrightarrow \exists yGy\big ) \& \forall m\big (R\big (G,\left\{ m\right\} \big )\leftrightarrow \exists z\exists H\big (R\left( H,m\right) \& \forall z_{1}\big (Hz_{1}\leftrightarrow \big (Gz_{1} \& z\ne z_{1}\big )\big )\big )\big )\big )\big )\).”

- 14.
By “the truth-condition” for a sentence \(\phi _{\tau }\), I mean a biconditional in the form of “\(\phi _{\tau }\) iff \(\ldots \)” Strictly speaking, it is also necessary to show that some instance of \(\phi _{\tau }\) is true, in order to establish the existence of \(\tau \)’s referent. For example, neo-Fregeans establish the existence of the number 0 by using the following instance of Hume’s principle: card \(\left( x\ne x\right) =\) card \(\left( x\ne x\right) \) iff \(\left( x\ne x\right) \thickapprox \left( x\ne x\right) \).

- 15.
Frege writes: “Naturally, no one is going to confuse [Caesar] with the [number zero]; but that is no thanks to our definition of [number]” (1953, \(\S 62\)). This suggests that, if an axiomatic theory is to guarantee the existence of numbers (with the help of the Context Principle), the truth conditions of the relevant identity statements should follow only from its axioms.

- 16.
Strictly speaking, Kim will also have to establish that ($) for each number word,

*some*sentence in which it occurs is true. One way to achieve this will be to prove that \(\exists _{0}x\left( x\ne x\right) \), that \(\exists _{1}n\left( n=0\right) ,\) that \(\exists _{2}n\left( n=0\vee n=1\right) \), and so on, thereby establishing the existence of 0, 1, 2, \(\ldots \) However, I will not assume any specific way to satisfy ($). - 17.
Fine calls this “the disquotationalist solution” to the uniqueness problem (2002, II-4). He discusses two other solutions, but due to the length limitation, I will not discuss them here.

- 18.
Indeed, this is what most other philosopher call “the Julius Caesar problem.” According to MacBride (2003, pp. 128–136), neo-Fregeans have developed two types of solutions. The first type attempts to derive the truth conditions of mixed identity statements from relevant abstraction principles (Wright 1983, pp. 107–117; Hale 1994, pp. 197–200; Hale and Wright 2001b, pp. 367–370). However, such theories tend to counterintuitively differentiate, for example, the natural number 1 from the interger 1, or they fail to distinguish, for instance,

*sui generis*natural numbers from set-theoretical substitutes (Hale 1994, pp. 197–200; Hale and Wright 2001b, pp. 371–380). The second type supposes that objects can be classified into different categories, each governed by a unique abstraction principle. Then, either it is granted that there are transcategorical identities or it is not. In the former case, it is*in principle*impossible to determine the truth value of mixed identity statements, and in the latter case, it follows immediately that the objects in question are numerically different. In either case, the Julius Caesar problem is solved (or dissolved) (Hale and Wright 2001b, pp. 385–396). - 19.
Note that this is equivalent to “Julius Caesar=2 iff \(\perp \),” where \(\perp \) is any contradiction. So (a) not only assigns falsity to “Julius Caesar=2” but it also provides a truth

*condition*. - 20.
Fine considers this option seriously: “We therefore cannot in general expect a solution to the Caesar problem to follow explicitly from the contextual definitions themselves” (2002, II-2).

- 21.
Kim does not specify ALA’s inference rules explicitly. In this paper, we will assume that it has a natural deduction system similar to Lemmon’s (1978).

- 22.
Additionally, ALA has modal operators—accompanied with relevant formation and inference rules—and a concept comprehension rule (2015, pp. 119–120). In this paper, I will not cover them because no issue discussed in this paper hangs upon them. In particular, we will discuss a Russell-style paradox in the next section but its proof does not depend on the mentioned comprehension rule at all.

- 23.
See Kim (2015, p. 1109).

- 24.
In ALA, numerical predicates were allowed to take numerical variables as parameters. Since we have added non-numerical modes of existence to the domain, it is now perfectly natural to allow the same predicates to take non-numerical adverbial variables as well. As a result, an adverbial variable can occur in both the parametric and adverbial positions of the same sentence in ALA

_{+}. - 25.
Conversely, an identity statement is syntactically okay as long as it is not cross-sortal.

- 26.
Observe that the right-hand side is well-formed in ALA

_{+}. See footnotes 24 and 25. - 27.
In the rest of this paper, we will use “pme,” “the paradoxical mode of existence,” and “mode

*paradoxically*,” as synonyms. - 28.
See Cook (2009) for an up-to-date list.

- 29.
Proof: Suppose that \(\xi \) is \(\beta \)-conservative. Hence, there is no sentence \(\phi \) in ALA such that (i) \(\not \models \phi \) but (ii) \(\xi \models \phi ^{\beta }\). However, let \(\phi \) be any contingent sentence. So \(\phi \) satisfies (i). Thus, \(\xi \) is consistent; for, if \(\xi \) were inconsistent, then \(\phi \) would satisfy both (i) and (ii). Done.

- 30.
This means that

*SUCC*is a second-order formula satisfied by all and only models*M*such that the size of*M*’s domain is a successor cardinal. For brevity, I will not discuss exactly how to construct such a formula. - 31.
Analogous to footnote 30.

- 32.
I owe this point to an anonymous reviewer.

- 33.
When applied to ALA

_{+},*T*will assign a value in \(D_{adv}\times \wp \left( D_{obj}\cup D_{adv}\right) \) to \(R\left( \cdot ,\cdot \right) \). Also, we will deal with languages expanding ALA_{+}by adding adverbial constants and/or a monadic functor “\('\)”; in such a case,*T*will assign a value in \(D_{adv}\) to an adverbial constant and/or a value in \(\wp \left( D_{adv}\right) ^{2}\) to “\('\).” - 34.
According to the semantics of ALA

_{+}, the range of an individual variable’s quantification covers adverbial entities as well as objects. - 35.
Here,

*a*plays the role of a generic cardinal for all infinite sets. - 36.
It suffices to show that (a) \( M^{*}\models \forall F\left( \exists G\left[ R\left( 1,G\right) \& \forall x\left( Gx\leftrightarrow Fx\right) \right] \leftrightarrow \exists xFx\right) \) and that (b) \( M^{*}\models \forall F\left( R\left( n',F\right) \leftrightarrow \exists x\left( Fx \& \exists G\left[ R\left( n,G\right) \& \forall y\left( Gy\leftrightarrow \left( y\ne x \& Fy\right) \right) \right] \right) \right) \). Due to the space limitation, I omit the detailed steps to prove these facts, but observe that (a) holds just when (c) the von Neumann cardinality of

*F*s is at least \(1_{vn}\) iff there is an*F*, and that (b) holds exactly when (d) the von Neumann cardinality of*F*s is at least \(n+1_{vn}\) iff there is an*x*such that*x*is an*F*and the von Neumann cardinality of \( y\ne x \& Fy\) is at least*n*. Clearly, (c) and (d) are both true. - 37.
This proof is essentially the same as Weir’s proof of (HP)’s conservativeness except that it has been modified for Kim’s adverbial definition of numbers (Weir 2003, pp. 24–25)

- 38.
For the proof of the conservativeness of (Def. of pme

_{4}), let \( \chi =\forall P\left[ \exists _{\text{ pme }_{4}}vPv\leftrightarrow \left( LIM\vee \exists v\left( Pv \& \lnot \exists _{v}u\left( u=v\right) \right) \right) \right] \) and \( \beta =\lnot \forall P\big [\exists _{w}vPv\leftrightarrow \big (LIM\vee \exists v\big (Pv \& \lnot \exists _{v}u\big (u=v\big )\big )\big )\big ]\). - 39.
This step is possible because, for any

*D*, we can find a successor cardinal \(\kappa \ge \text{ card }\left( D\right) \). Similarly for a limit cardinal.

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## Acknowledgements

I am grateful to the audience of my talk in the 2014 Summer Meeting of the Korean Society of Analytic Philosophy for their helpful comments and inspiring questions. Thanks to two anonymous reviewers for pointing out many potential errors, and for encouraging me to discuss such core issues as the Context Principle and conservativeness. Special thanks go to Roy Cook and Joongol Kim for their invaluable advice and encouragement. Finally, this work would’ve been very difficult without Ilho Park’s help.

## Author information

## Additional information

The original version of this article was revised: Unfortunately, there is a typo in the author name. The correct spelling is Namjoong Kim.

## Appendix

### Appendix

For the proofs below, we need to use the techniques of model theory. Unfortunately, the standard model theory for the second-order logic does not apply to ALA_{+} directly, because the truth-conditions of some sentences of ALA_{+} have not yet been defined. To overcome this difficulty, we shall regard any formula in the form of \(\exists _{\mu }\delta \phi _{\delta }\) as the abbreviation of \( \exists \psi \left( R\left( \mu ,\psi \right) \& \forall \delta \left( \psi \delta \leftrightarrow \phi _{\delta }\right) \right) \), where \(\delta \) is an individual/adverbial variable, \(\psi \) is an individual/adverbial predicate variable, \(\phi _{\delta }\) is an open formula with a free occurrence of \(\delta \), and \(R\left( \cdot ,\cdot \right) \) is a special cross-sortal predicate whose first parameter is an adverbial term and the second parameter is an individual/adverbial predicate term. In particular, “*R* ” will be regarded as a constant predicate, not as a free variable. As such, an ALA_{+}-sentence in its unabbreviated form does not really include a quantifier with a subscripted adverbial term in it. Even with this interpretation, ALA_{+} differs from the standard second-order logic because it is many-sorted. Thus, we will identify a model *M* with \(\left\langle D_{obj},D_{adv},T\right\rangle \), where \(D_{obj}\) is the domain of ordinary objects, \(D_{adv}\) is that of adverbial entities, and *T* is a function mapping non-logical constants to values in the appropriate domain.^{Footnote 33} For any such model \(M=\left\langle D_{obj},D_{adv},T\right\rangle \), we define an *M*-assignment to be the assignment of a value in \(D_{obj}\cup D_{adv}\) to each individual variable,^{Footnote 34} of a value in \(D_{adv}\) to each adverbial variable, and of a value in \(P_{1}\times \cdots \times P_{n}\) to each *n*-adic predicate variable such that, for any \(k\in \left\{ 1,\ldots ,n\right\} \), either the *k*-th parameter is an individual term and \(P_{k}=\wp \left( D_{obj}\cup D_{adv}\right) \), or it is an adverbial term and \(P_{k}=\wp \left( D_{adv}\right) \). Despite the obvious differences, it is possible to use the soundness theorem of the standard second-order logic to establish the soundness of ALA_{+}.

### A. Consistency of (Def. of pme_{1}) and (Def. of pme_{2})

Show the consistency of (Def. of pme_{1}) and (Def. of pme_{2}). Without losing any generality, we only prove the consistency of

(Def. of \(\text{ pme }_{1}\)) \( \exists _{\text{ pme }_{1}}vPv\leftrightarrow \left( INF\vee \exists v\left( Pv \& \lnot \exists _{v}u\left( u=v\right) \right) \right) \).

### Proof

It suffices to show the consistency of

(Def. of pme

_{1}^{#}) \( \exists P_{1}\left[ R\left( \text{ pme }_{1},P_{1}\right) \& \forall v\left( P_{1}v\leftrightarrow Pv\right) \right] \leftrightarrow \left( INF\vee \zeta \right) \),

where \(\zeta \) is “\( \exists v\left( Pv \& \lnot \exists P_{1}\left( R\left( v,P_{1}\right) \& \forall u\left( P_{1}u\leftrightarrow u=v\right) \right) \right) \).” Consider any sets \(D_{obj}\) and \(D_{adv}\) such that \(\text{ card }\left( D_{obj}\cup D_{adv}\right) \) is infinite. Hence, \(\left\langle D_{obj},D_{adv},\cdot \right\rangle \models INF\). So, it suffices to find a function *T* such that \(\left\langle D_{obj},D_{adv},T\right\rangle \) satisfies the left-hand side. Consider any *T* such that \(T\left( \text{ pme }_{1}\right) \in D_{adv}\) and \(T\left( R\right) =\left\{ \left\langle T\left( \text{ pme }_{1}\right) ,x\right\rangle |x\in \wp \left( D_{adv}\right) \right\} \). Since \(\left\langle T\left( \text{ pme }_{1}\right) ,\cdot \right\rangle \in T\left( R\right) \), \( \left\langle D_{obj},D_{adv},T\right\rangle \models \exists P_{1}\left[ R\left( \text{ pme }_{1},P_{1}\right) \& \forall v\left( P_{1}v\leftrightarrow Pv\right) \right] \). Thus, (Def. of pme_{1}^{#}) has a model. By the soundness of ALA_{+}, (Def. of pme_{1}^{#}) is consistent. Done. \(\square \)

### B. Conservativeness of Kim’s definition

Prove that

(\(\beta -\text {CONS}_{\xi }\)) for any set \(\Gamma \) of formulas and for any formula \(\phi \) in ALA

_{+}, if \(\Gamma \not \models \phi \), then \(\Gamma ^{\beta },\xi \not \models \phi ^{\beta }\)

where \( \xi =\left[ \forall F\left( \exists _{1}xFx\leftrightarrow \exists xFx\right) \& \forall F\left( \exists _{n'}xFx\leftrightarrow \exists x\left( Fx \& \exists _{n}y\left( y\ne x \& Fy\right) \right) \right) \right] \) and \( \beta =\left( n\ne 1 \& \forall m\left( n\ne m'\right) \right) \).

### Proof

Suppose that \(\Gamma \not \models \phi \). Thus, there exists a model \(M=\left\langle D_{obj},D_{adv},T\right\rangle \) such that for any \(\psi \in \Gamma \), \(M\models \psi \), but \(M\not \models \phi \). So, there is an *M*-assignment *s* such that, for any \(\psi \in \Gamma \), \(M,s\models \psi \) but \(M,s\models \lnot \phi \). Let *N* be any \(\omega \)-sized set disjoint with \(D_{obj}\cup D_{adv}\) and with \(\wp \left( D_{obj}\cup D_{adv}\right) \). Let \(D_{adv}^{*}=D_{adv}\cup N\). Choose an \(a\in N\), and then let \(N^{*}=N-\left\{ a\right\} \). Let \(N_{vn}\) be the set of strictly positive von Neumann numbers \(1_{vn}\) (\(=_{df}\left\{ \varnothing \right\} \)), \(2_{vn}\)(\(=_{df}\left\{ \varnothing ,\left\{ \varnothing \right\} \right\} \)), \(\ldots \) Define \(+\) as the usual addition operation on von Neumann numbers, \(\text{ card }\left( \cdot \right) \) as the usual cardinality operation measuring the size of a set by a von Neumann number, and \(\le \) as the usual predicate expressing inequality between von Neumann numbers. There exists a bijection *b* mapping \(N_{vn}\) onto \(N^{*}\). We define \(g:D_{adv}^{*}\rightarrow D_{adv}^{*}\) as follows: for any \(x\in D_{adv}\), if \(x\in N^{*}\), then \(g\left( x\right) =b\left( b^{-1}\left( x\right) +1_{vn}\right) \); otherwise, \(g\left( x\right) =a\).^{Footnote 35} Now we construct a new model \(M^{*}=\left\langle D_{obj},D_{adv}^{*},T^{*}\right\rangle \) such that \(T^{*}\left( 1\right) =b\left( 1_{vn}\right) ,\)\(T^{*}\left( R\right) =\)\(\{\left\langle x,S\right\rangle \in D_{adv}^{*}\times \wp \left( D_{obj}\cup D_{adv}\right) |\,\text{ if } \, S\, \text{ is } \text{ finite, } \, x=b\left( y\right) \, \text{ for } \text{ some }\, y\le \text{ card }\left( S\right) ;\)\(\text{ otherwise, }\, x=a\}\), and \(T^{*}\left( '\right) =g\). Given that \(T^{*}\) assigns these values to “1,” “*R*,” and “\('\),” it is easy to show that \(M^{*}\models \xi \).^{Footnote 36} Observe that *s* is an \(M^{*}\)-assignment, too. Using mathematical induction on the complexity of formulas, we can show that, for any \(\psi \in \Gamma \cup \left\{ \lnot \phi \right\} \), if \(M,s\models \psi \), then \(M^{*},s\models \psi ^{\beta }\). Thus, for any \(\psi \in \Gamma ^{\beta }\), \(M^{*},s\models \psi \), but \(M^{*},s\models \left( \lnot \phi \right) ^{\beta }\). Since all \(\psi \in \Gamma \cup \left\{ \lnot \phi \right\} \) are closed, it follows that, for all \(\psi \in \Gamma ^{\beta }\), \(M^{*}\models \psi \), but \(M^{*}\not \models \phi ^{\beta }\). Therefore, \(\Gamma ^{\beta },\xi \not \models \phi ^{\beta }\). Done.^{Footnote 37}\(\square \)

### C. Conservativeness of (Def. of pme\(_{3}\)) and (Def. of pme\(_{4}\))

Show the conservativeness of (Def. of pme_{3}) and (Def. of pme_{4}). Without losing any generality, we only prove that

(\(\beta -\text {CONS}_{\chi }\)) for any set \(\Gamma \) of formulas and for any formula \(\phi \) in ALA

_{+}, if \(\Gamma \not \models \phi \), then \(\Gamma ^{\beta },\chi \not \models \phi ^{\beta }\),

where \( \chi =\forall P\left[ \exists _{\text{ pme }_{3}}vPv\leftrightarrow \left( SUCC\vee \exists v\left( Pv \& \lnot \exists _{v}u\left( u=v\right) \right) \right) \right] \) and \( \beta =\lnot \forall P\left[ \exists _{w}vPv\leftrightarrow \left( SUCC\vee \exists v\left( Pv \& \lnot \exists _{v}u\left( u=v\right) \right) \right) \right] \).^{Footnote 38}

### Proof

Suppose that \(\Gamma \not \models \phi \). Thus, there exists a model \(M=\left\langle D_{obj},D_{adv},T\right\rangle \) such that for any \(\psi \in \Gamma \), \(M\models \psi \), but \(M\not \models \phi \). Thus, there is an *M*-assignment *s* such that, for all \(\psi \in \Gamma \), \(M,s\models \psi \), but \(M,s\models \lnot \phi \). Let *S* be any set disjoint with \(D_{obj}\cup D_{adv}\) and with \(\wp \left( D_{obj}\cup D_{adv}\right) \) such that (i) \(\text{ card }\left( D_{obj}\cup D_{adv}\cup S\right) \) is a successor cardinal.^{Footnote 39} We construct a new model \(M^{*}=\left\langle D_{obj},D_{adv}^{*},T^{*}\right\rangle \) such that (ii) \(D_{adv}^{*}=D_{adv}\cup S\), (iii) \(T^{*}\left( \text{ pme }_{3}\right) \in S\) and (iv) \(T^{*}\left( R\right) =\left\{ x|x\in S\times \wp \left( D_{obj}\cup D_{adv}^{*}\right) \right\} \). By (iii) and (iv), for any \(M^{*}\)-assignment \(s^{*}\), \( M^{*},s^{*}\models \exists P_{1}\left[ R\left( \text{ pme }_{3},P_{1}\right) \& \forall v\left( P_{1}v\leftrightarrow Pv\right) \right] \), that is, \(M^{*},s^{*}\models \exists _{\text{ pme }_{3}}vPv\). By (i) and (ii), \(M^{*}\models SUCC\). Hence, \(M^{*}\models \chi \). The rest of the proof is similar to in Appendix B. Therefore, \(\Gamma ^{\beta },\chi \not \models \phi ^{\beta }\). Done. \(\square \)

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### Cite this article

Kim, N. Bad company objection to Joongol Kim’s adverbial theory of numbers.
*Synthese* **196, **3389–3407 (2019). https://doi.org/10.1007/s11229-017-1602-x

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### Keywords

- Number
- Logicism
- Frege
- Adverb
- Modes of existence
- Bad company
- Embarrassment of riches