Bad company objection to Joongol Kim’s adverbial theory of numbers

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.

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.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,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₊.

1.1 A. Consistency of (Def. of pme₁) and (Def. of pme₂)

Show the consistency of (Def. of pme₁) and (Def. of pme₂). 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 pme₁^#) $\exists P_{1}\left[ R\left( \text{ pme }_{1},P_{1}\right) \& \forall v\left( P_{1}v\leftrightarrow Pv\right) \right] \leftrightarrow \left( INF\vee \zeta \right)$,

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 pme₁^#) has a model. By the soundness of ALA₊, (Def. of pme₁^#) is consistent. Done. $\square$

1.2 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, $g\left( x\right) =a$.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 $M^{*}\models \xi$.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.³⁷$\square$

1.3 C. Conservativeness of (Def. of pme$_{3}$) and (Def. of pme$_{4}$)

Show the conservativeness of (Def. of pme₃) and (Def. of pme₄). 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 $\beta =\lnot \forall P\left[ \exists _{w}vPv\leftrightarrow \left( SUCC\vee \exists v\left( Pv \& \lnot \exists _{v}u\left( u=v\right) \right) \right) \right]$.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.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. $\square$