2.1 Introduction

In the debate on abstractionism, APs can be conceived of as tools to achieve philosophical goals – whether foundational or not. Philosophical motivations notwithstanding, every abstractionist program relies on APs to interpret significant portions of mathematics. In this sense, the mathematical project comes first, and so we will treat it first. The reader mostly or exclusively concerned with the semantics, epistemology, and ontology of abstraction can skip to the relevant sections.

The most influential project originates with Frege. However, the axiomatic system in his Grundgesetze der Arithmetik (Frege Reference Frege1893/1903) yields paradoxes. Later mathematical investigations into APs have aimed to strike a balance between consistency and mathematical strength. In particular, authors have focused on second-order Peano Arithmetic PA² (especially, Scottish Neologicists such as Crispin Wright and Bob Hale – Section 2.4); real analysis (e.g. Bob Hale and Stewart Shapiro – Section 2.5); set theory (particularly, George Boolos, Roy Cook, and Øystein Linnebo – Section 2.6).Footnote ⁴ All (or most) of such mathematical abstractionist theories are formulated in a language whose background logic is at least second-order – and usually classical. Hence, before presenting those theories, we provide an outline of second-order logic (Section 2.2), and of the inconsistency of Frege’s Grundgesetze (Section 2.3). Sections 2.7 and 2.8 examine two major topics in abstractionism: the invariance of abstraction and the so-called Bad Company problem.

2.2 Second-Order Logic

The language $L$ of second-order logic (SOL) contains, besides the language of first-order logic (FOL), an infinite list of $n$-adic second-order variables $X^n,Y^n,Z^n,\dots$ varying over a second-order domain containing appropriate entities – that is, $n$-adic relations between first-order individuals; Fregean concepts and relations; or sets of ($n$-tuples of) first-order individuals – and existential and universal quantifiers binding second-order variables. The main logical principle of SOL is the Comprehension Axiom schema (CA):

• (CA) $\exists X^n\forall x_1,\dots ,x_n\left(X^n\right(x_1,\dots ,x_n)\leftrightarrow \varphi (x_1,\dots ,x_n\left)\right)$,

where $\varphi$ is any formula of $L$ not containing $X^n$ free. Every instance of CA, obtained by substituting $\varphi$ by any formula of the language of SOL, is an axiom of SOL – that is, CA is an axiom schema.

CA states that there is a relation (or set) $X^n$ such that all the individuals $x_1,\dots ,x_n$ are in the relation $X^n$ (or are members of the $n$-tuples in the set $X^n$) if and only if $x_1,\dots ,x_n$ satisfy $\varphi$. Note that, if $X$ is monadic, that is, it applies to just one individual at a time, $X$ stands for a property, a Fregean concept, or a set of individuals. That $\varphi$ is any formula of $L$ implies that CA is unrestricted.

An interpretation of $L$ consists in appropriate domains, and an assignment function. Let $D_1$ be a nonempty domain of individuals, which first-order variables vary over, and $D_2$ a nonempty domain containing $n$-adic relations between the individuals in $D_1$, which second-order variables vary over, for any positive integer $n$. In terms of sets, $D_2$ contains: sets of individuals of $D_1$; sets of pairs of individuals of $D_1$ (if $X$ is dyadic); sets of triples of individuals of $D_1$ (if $X$ is ternary); and so on. In case $D_2$ contains all such relations (sets), $D_2$ is the so-called powerset of $D_1$.Footnote ⁵ An interpretation of $L$ that verifies every instance of CA is a model of $L$.

In the standard semantics for $L$, and therefore in the standard models of SOL, domain $D_2$ is the powerset of $D_1$. Once $D_1$ is fixed, $D_2$ will also be fixed without further specification. This is not necessarily so in nonstandard models of $L$ – namely Henkin models. In Henkin models, $D_2$ may not contain all $n$-adic relations among individuals of $D_1$ (all subsets of $D_1$), since $D_2$ is just a nonempty set of subsets of $D_1$. Therefore, in Henkin models, it is necessary to fix both $D_1$ and $D_2$ explicitly.

The model-theoretic properties of SOL may vary depending on whether the models considered are standard or not. A difference between standard and nonstandard models of SOL that is significant for the debate on abstraction (see Section 2.3 below) regards the cardinality of $D_2$, in case the first-order domain $D_1$ is infinite. Such a difference is more easily appreciated in terms of Cantor’s theorem, proving that, for any set $A$, the cardinality of its powerset $\wp \left(A\right)$ is larger than the cardinality of $A$. In particular, if $A$ has cardinality $\kappa$ (formally, $|A|=\kappa$), then $|\wp \left(A\right)|=2^{\kappa }$.Footnote ⁶ Since in the standard models of SOL $D_2$ is the powerset of $D_1$, if $D_1$ has cardinality $\kappa$, then $D_2$ has cardinality $2^{\kappa }$ by Cantor’s theorem. Nonstandard models of SOL, on the other hand, are such that, if $D_1$ has infinite cardinality $\kappa$, $D_2$ may have cardinality that is less than $2^{\kappa }$.Footnote ⁷

2.3 Frege’s Grundgesetze der Arithmetik

In Grundgesetze der Arithmetik, Frege relied on an AP embedded in a logical system that, for the sake of simplicity, can be assimilated to a higher-order logic,Footnote ⁸ with the aim of deriving substantial mathematical theories, such as arithmetic, real analysis and, possibly, complex analysis. The infamous AP Basic Law V (BLV) states that the extension of concept $X$ is identical with the extension of concept $Y$ if, and only if, every individual falling under $X$ falls under $Y$ and vice versa:

• (BLV) $\forall X\forall Y(ϵX=ϵY\leftrightarrow \forall x(Xx\leftrightarrow Yx\left)\right)$,Footnote ⁹

where $ϵ$ is the abstraction operator “the extension of.”

The axiomatic system consisting in unrestricted SOL and BLV is inconsistent. This is easily seen by proving so-called Russell’s paradox – in a Fregean spirit, we will talk of concepts and extensions, but the same applies to properties and sets. First of all, the existence of the “Russellian” concept “being the extension of a concept under which that very extension does not fall” is guaranteed by the following instance of (unrestricted) CA:

$\exists X\forall y(Xy\leftrightarrow \exists Y(y=ϵY\wedge \neg Yy\left)\right).$(2.1)

Call such a concept “$R$”. Since it is a theorem of Frege (Reference Frege1893/1903) that for every concept there is the corresponding extension, the extension $ϵR$ must exist.

Let us assume that $ϵR$ falls under $R$: $R\left(ϵR\right)$. Then, $ϵR$ satisfies the formula defining $R$:

$\exists Y(ϵR=ϵY\wedge \neg Y(ϵR\left)\right).$(2.2)

Equation (2.2) implies $ϵR=ϵZ$, for $Z$ instantiating $Y$. By instantiating the left-to-right direction of BLV with $R$ and $Z$

$ϵR=ϵZ\to \forall x(Rx\leftrightarrow Zx),$(2.3)

it must hold that $\forall x(Rx\leftrightarrow Zx)$, stating that every object $x$ falls under $R$ if, and only if, it falls under $Z$. The latter, along with $\neg Z\left(ϵR\right)$, implies that $ϵR$ does not fall under $R$, namely $\neg R\left(ϵR\right)$. Therefore,

$R\left(ϵR\right)\to \neg R\left(ϵR\right).$(2.4)

Let us now assume that $ϵR$ does not fall under $R$: $\neg R\left(ϵR\right)$. By the definition of $R$, the latter implies

$\neg \exists Y(ϵR=ϵY\wedge \neg Y(ϵR\left)\right).$(2.5)

By the usual transformations of logical connectives and quantifiers, it follows that

$\forall Y(ϵR=ϵY\to Y(ϵR\left)\right).$(2.6)

Equation (2.6) implies $ϵR=ϵR\to R\left(ϵR\right)$ by universal instantiation. Since by identity $ϵR=ϵR$, it must hold that $R\left(ϵR\right)$. So,

$\neg R\left(ϵR\right)\to R\left(ϵR\right).$(2.7)

Finally, since if $R\left(ϵR\right)$, then $\neg R\left(ϵR\right)$, and if $\neg R\left(ϵR\right)$, then $R\left(ϵR\right)$, it must be the case that

$R\left(ϵR\right)\leftrightarrow \neg R\left(ϵR\right):contradiction.$(2.8)

Both unrestricted CA and BLV are involved in the derivation of Russell’s paradox in Frege (Reference Frege1893/1903).Footnote ¹⁰ A famous debate between Boolos (Reference Boolos1993) and Dummett (Reference Dummett1994) concerns its origin.

On the one hand, Boolos blamed it on BLV and its inconsistent requirement that for every concept, there is exactly one extension corresponding to it, which violates Cantor’s theorem.Footnote ¹¹

On the other hand, Dummett (Reference Dummett1991, Reference Dummett1994) blames the contradiction on the impredicativity of CA: Since CA is unrestricted, second-order variables may appear on its right-hand side in the scope of a second-order quantifier, that is, they are bound. The logic underlying Frege (Reference Frege1893/1903) is classical, so it requires that the (first- or second-order) domain be a “definite totality,” given once and for all. Still, by quantifying over the second-order domain, more and more Fregean concepts (properties or sets) can be defined. Hence, the second-order domain grows larger and larger, deeming it no definite totality after all.

Consider for instance the concept “being an extension,” formally introduced by the instance of CA $\exists X\forall x(Xx\leftrightarrow \exists Y(x=ϵY\left)\right)$ – in a language containing the abstraction operator $ϵ$ – and call it “$E$”. Concept $E$ must have an extension ($ϵE$), but then also $ϵE$ must fall under $E$. But if concepts are “identical” just in case exactly the same objects fall under them, then the concept “being an extension” we started from is different from the concept “being an extension” we ended up with. This process never ends, that is, as Dummett argues, there are indefinitely extensible concepts, which are such that “if we can form a definite conception of a totality all of whose members fall under that concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it.”Footnote ¹²

To some extent, both Boolos and Dummett are right: there are consistent axiomatic systems of unrestricted SOL with restricted BLV;Footnote ¹³ but also unrestricted BLV plus restricted (classical) SOL may have models. So, in general, a philosophically meaningful question concerns what the best way out of the paradox is. Broadly speaking, two solutions can be envisaged, each of which modify either SOL or BLV, one way or another.Footnote ¹⁴ However, such solutions to the inconsistency may affect the mathematical strength of the resulting theory. In what follows, we will focus on the fragments of mathematics that can be interpreted by consistent APs, but connections to the solutions to the inconsistency of Frege’s system will emerge.

2.4 Arithmetical Abstraction

Scottish Neologicism (Hale and Wright Reference Hale and Wright2001a, Wright Reference Wright1983) is a radical and very influential solution to the inconsistency of Frege (Reference Frege1893/1903) substituting BLV by a consistent AP strong enough to interpret second-order Peano Arithmetic PA²,Footnote ¹⁵ namely Hume’s Principle

• (HP) $\forall X\forall Y(\#X=\#Y\leftrightarrow X\approx Y)$,

while keeping SOL unrestricted. HP reads informally, “The number of the $X$ s is identical with the number of the $Y$ s if, and only if, $X$ and $Y$ are equinumerous” – where # is the cardinality operator “the number of.” Precisely, the formula “$X\approx Y$” abbreviates the (purely second-order) statement that there is a relation $R$ such that every object falling under $X$ is $R$-related to a unique object falling under $Y$, and for every object falling under $Y$ there’s a unique object falling under $X$ that is $R$-related to it:

($\approx$)

$X\approx Y{\leftrightarrow }_{def}\exists R\left(\forall x\right(X\left(x\right)\to \exists !y\left(Y\right(y)\wedge R(x,y\left)\right))\wedge \forall x(Y\left(x\right)\to \exists !y\left(X\right(y)\wedge R(y,x\left)\right)\left)\right),$

$\exists !x\varphi \left(x\right)$

$\exists x\left(\varphi \right(x)\wedge \forall y(\varphi \left(y\right)\to x=y\left)\right)$

In unrestricted SOL with HP as its sole nonlogical axiom, a formulation of Frege’s definitions of the arithmetical notions necessary to derive PA² can be provided without the need for BLV. The resulting axiomatic system is Frege Arithmetic (FA). Frege Arithmetic has a model in the natural numbers (Boolos Reference Boolos and Thomson1987a) and interprets PA² (Boolos and Heck Reference Boolos, Heck and Schirn1998). The latter result is now known as Frege’s Theorem.

The language of FA consists in the language of SOL augmented by the primitive term-forming operator #, which applies to second-order variables (or constants) – the resulting expressions are singular terms. Of course, every (appropriate) complex formula of such a language corresponds to a concept, since CA is unrestricted. By adopting the notational convention that a concept defined by a formula $\varphi$ is denoted by “$[x.\varphi (x\left)\right]$” Frege’s definitions can be formulated in the language of SOL plus HP:

(Zero)

$0{=}_{def}\#[x.x\ne x]$,

(Predecessor)

$P(x,y){\leftrightarrow }_{def}\exists X\exists u(Xu\wedge y=\#X\wedge x=\#[z.Xz\wedge z\ne u\left]\right)$,

$x$

$y$

$X$

$y$

$x$

(Hereditary)

$Her(X,R){\leftrightarrow }_{def}\forall x,y\left(R\right(x,y)\wedge Xx\to Xy))$,

$R$

$x,y$

$x$

$X$

$y$

$X$

(Ancestral)

$R^{*}(x,y){\leftrightarrow }_{def}\forall X\left(\forall z\right(R(x,z)\to Xz)\wedge Her(X,R)\to Xy)$,

$y$

$X$

$z$

$X$

$R$

$x,z$

$X$

$R$

(Weak Ancestral)

$R^{+}(x,y){\leftrightarrow }_{def}{R}^{*}(x,y)\vee x=y$,

$x,y$

$R$

$R$

$x,y$

$x,y$

(Natural Number)

$Nx{\leftrightarrow }_{def}{P}^{+}(0,x)$,

$x$

$x$

By the underlying SOL, HP, and previous definitions, (appropriate formulations of) second-order Peano axioms are derivable.

Is there any consistent formulation of (SOL plus) BLV that, analogously to HP, interprets arithmetic? The answer may depend upon how large a fragment of arithmetic is at stake. A first group of theories (Table 1; for an explanation of the axioms in Table 1, see Table 2) interprets at most Robinson Arithmetic $Q$, which is weaker than first-order Peano Arithmetic PA:Footnote ¹⁷

Table 1 Consistent subsystems of SOL plus BLV: Robinson Arithmetic

Background logic	Formulation of BLV
FOL	(BLVS) $ϵx.\varphi x=ϵx.\psi x$ $\leftrightarrow \forall x(\varphi x\leftrightarrow \psi x)$	Schroeder-Heister (Reference Schroeder-Heister1987); Parsons (Reference Parsons1987); Burgess (Reference Burgess1998)
(CAP) $\exists X\forall y(Xy\leftrightarrow \varphi y)$	BLV or BLVS	Heck (Reference Heck1996)
(${\Delta }_1^1$-CA) $\forall x(\varphi x\leftrightarrow \psi x)$ $\to \exists Y\forall x(Yx\leftrightarrow \varphi x)$	BLV or BLVS	Wehmeier (Reference Wehmeier1999); Ferreira and Wehmeier (Reference Ferreira and Wehmeier2002)

Table 2 Axioms in Table 1

BLVS	Since FOL lacks second-order quantification, BLV must be schematic – hence the subscript $S$ – with metavariables $\varphi ,\psi$ for formulæ of the underlying language.
CAP	$\varphi$ contains neither $X$ free nor bound second-order variables at all.
	If so restricted, CA is predicative – hence the subscript $P$.
${\Delta }_1^1$-CA	A formula is ${\Sigma }_1^1$ (${\Pi }_1^1$) if it has the form $\exists X\varphi$ ($\forall X\varphi$) and $\varphi$ contains no second-order quantifier. ${\Delta }_1^1$-CA requires that, in order to appear on its right-hand side, a formula $\varphi$ must be at most a ${\Sigma }_1^1$-formula provably equivalent to a ${\Pi }_1^1$-formula $\psi$.

In general, if CA is restricted to ${\Sigma }_1^1$- or ${\Pi }_1^1$-formulæ, CA plus BLV in either form is inconsistent. It is not known whether there are intermediate consistent ${\Delta }_1^1$-CA and ${\Sigma }_1^1$/${\Pi }_1^1$-CA.

Though not at all trivial, Robinson’s $Q$ is still a rather small fragment of arithmetic, especially compared to Frege’s original goal. In order to interpret larger parts of arithmetic by BLV, the theories in Table 1 have to be revised.

In Antonelli and May (Reference Antonelli and May2005), an axiom system containing unrestricted CA and restricted BLV that interprets PA is presented, where numbers are conceived of as concepts of objects. In order to define cardinals as equivalence classes of equinumerous concepts, extensions (value-ranges) are needed: (concept) $N$ is a number if, and only if, there is a concept $X$ such that an object $x$ falls under $N$ if, and only if, $x$ is the value-range of a concept equinumerous to $X$. Such an object is called a witness of $N$. A conditional formulation of BLV is provided:

$\forall X\forall Y\forall x\forall y\left(VR\right(X,x)\wedge VR(Y,y)\to (\forall z(Xz\leftrightarrow Yz)\leftrightarrow x=y\left)\right),$(2.9)

where $VR(X,x)$ means “$x$ is the value-range of $X$.” (2.9) does not guarantee that value-ranges exist. To correct this, Antonelli and May’s (Reference Antonelli and May2005) system contains also a further axiom:

$\forall x(\varphi x\to \exists X(Nn\left(X\right)\wedge Wtn(X,x)\left)\right)\to \exists xVR(\varphi ,x),$(2.10)

where $Nn$ is the predicate “being a natural number,” and $Wtn$ is the relation of $x$ being a witness of $X$. This axiom guarantees that, for any formula $\varphi$ applying exclusively to witnesses of natural numbers, there is a value-range.Footnote ¹⁸

Consistent systems as the ones in Ferreira (Reference Ferreira and Tahiri2018), Boccuni (Reference Boccuni2010, Reference Boccuni2013) in Table 3 (for an explanation of the axioms in Table 3, see Table 4) interpret PA² by adding a further round of higher-order variables (and comprehension axioms thereof) to SOL, and calibrating the restrictions on BLV or BLV_S. The language of the axiomatic theory in Ferreira (Reference Ferreira and Tahiri2018) consists in: impredicative second-order variables $X,Y,Z,\dots$ varying over impredicative concepts; predicative second-order variables $X,Y,Z,\dots$ varying over predicatively definable concepts; and the variable-binding operator $ϵ$. The language of the axiomatic theory in Boccuni (Reference Boccuni2010, Reference Boccuni2013) adds to regular second-order quantification quantifiers binding plural variables $xx,yy,zz,\dots$ governed by (unrestricted) plural logic,Footnote ¹⁹ and the functional operator $ϵ$.

Table 3 Consistent subsystems of SOL plus BLV: PA²

Background logic	Formulation of BLV
(CAImp) $\exists X\forall y(Xy\leftrightarrow \varphi y)$	(BLVS) $ϵx.\varphi x=ϵx.\psi x$	Ferreira (Reference Ferreira and Tahiri2018)
(CAP) $\exists X\forall y(Xy\leftrightarrow \psi y)$	$\leftrightarrow \forall x(\varphi x\leftrightarrow \psi x)$
(CAPL) $\exists xx\forall y(y\prec xx\leftrightarrow \varphi y)$	(BLV) $\forall X\forall Y(ϵX=ϵY$	Boccuni (Reference Boccuni2010, Reference Boccuni2013)
(CAP) $\exists X\forall y(Xy\leftrightarrow \psi y)$	$\leftrightarrow \forall x(Xx\leftrightarrow Yx))$

Table 4 Axioms in Table 3

CAImp	$\varphi$ is unrestricted.
CAP	$\psi$ does not contain either $X$ free or bound predicative second-order variables or impredicative second-order variables at all.
BLVS	No formula in the scope of $ϵ$ contains impredicative second-order variables at all.
CAPL	$\varphi$ is unrestricted.
CAP	$X$ varies over Fregean concepts, and $\psi$ contains neither bound second-order variables nor free plural variables.
BLV	$X,Y$ are concepts definable by CAP.

Ferreira’s (Reference Ferreira and Tahiri2018) system not only interprets PA², but it does so by FA. Boccuni (Reference Boccuni2010, Reference Boccuni2013) interprets PA², but because of the restrictions imposed on BLV, it falls short of providing the definitions needed to recover FA.Footnote ²⁰

2.5 Real Number Abstraction

A further mathematical theory abstractionists have been interested in interpreting via abstraction is real analysis.Footnote ²¹ Frege himself devoted Part III of the second volume of Grundgesetze to a theory of ratios of magnitudes, which was Frege’s way of conceiving of real numbers and which he left unfinished in the wake of Russell’s paradox.Footnote ²²

More recently, further attempts to recover real analysis by abstraction are investigated in, for example, Shapiro (Reference Shapiro2000), Hale (Reference Hale2000), Roeper (Reference Roeper2020), and Boccuni and Panza (Reference Boccuni and Panza2022). We will consider Hale’s and Shapiro’s proposals.

Shapiro (Reference Shapiro2000) provides a piecemeal reconstruction of abstractionist real analysis, starting from natural numbers as introduced by HP, and proceeding by several APs. First, integers as abstract objects are defined over differences between pairs of natural numbers $a,b,c,d$:

• (DIF) $INT<a,b>=INT<c,d>\leftrightarrow (a+d)=(b+c)$.Footnote ²³

Addition and multiplication for integers can be defined by the underlying unrestricted SOL.

Secondly, quotients of pairs of integers $m,n,p,q$ are introduced by the AP

• (QUOT) $Q<m,n>=Q<p,q>\leftrightarrow \left(\right(n=0\wedge q=0)\vee (n\ne 0\wedge q\ne 0\wedge m\times q=n\times p\left)\right)$.

Rational numbers are quotients $Q<m,n>$ for $n\ne 0$.Footnote ²⁴

Finally, by defining addition and multiplication for rationals, and the relation “less than” ($\le$), an AP for cuts is introduced:

• (CUT) $\forall P\forall Q\left(C\right(P)=C(Q)\leftrightarrow \forall r(P\le r\leftrightarrow Q\le r\left)\right)$,

where $P\le r$ holds between a concept $P$ of rationals and a rational $r$ if, and only if, $r$ is an upper bound of $P$. Say that $P$ is bounded, if $P\le r$ holds. A real number is a cut $C\left(P\right)$, for $P$ bounded and nonempty. Shapiro’s (Reference Shapiro2000) reconstruction delivers nondenumerably many cuts forming a totally ordered and Dedekind-complete field – that is, real analysis is recovered. Still, such a construction has a “decidedly structural feel” – Ebert and Rossberg (Reference Ebert, Rossberg, Ebert and Rossberg2016, p. 14, italics in the original).

Hale (Reference Hale2000, Reference Hale2002, Reference Hale2005) provides a reconstruction of real analysis, which is more Fregean in spirit, since Hale’s proposal is to define real numbers as abstract objects defined over pairs of magnitudes.Footnote ²⁵ The AP

• (RATIO)

where $Rat$ is an abstraction operator mapping ordered pairs of magnitudes $a,b,c,d$ to objects (ratios of quantities, i.e. reals), and $m,n$ are positive integers, states that the ratio of $a$ to $b$ is the same as that of $c$ to $d$ if, and only if, any positive integer multiple of $a$ is equal to, greater than, or less than any positive integer multiple of $b$ if, and only if, the same holds of the corresponding multiples of $c$ and $d$.

A few issues arise from Hale’s reconstruction: For example, does the resulting theory imply the existence of nondenumerably many magnitudes, in order to deliver the existence of nondenumerably many reals?Footnote ²⁶ Is the equivalence relation on the right-hand side of RATIO definable in purely logical terms?

We mentioned Shapiro’s abstractionist reconstruction of reals is taken to have a “structural feel,” as opposed to Hale’s, which is more Fregean in spirit. In order to adjudicate between such different conceptions of real numbers, the so-called Frege’s (applicability) Constraint (FC) has been invoked. According to FC:

A structural conception of the reals such as Shapiro’s is based on Dedekind’s cuts,Footnote ²⁸ as opposed to an abstractionist construction according to which reals are ratios of magnitudes. But, if the main applications of real numbers are for measuring quantities, such as, for example, masses, temperatures, or lengths, reals as Dedekind’s cuts fail FC, whereas abstractionist definitions such as the one provided by RATIO satisfy applicability, since RATIO has the applications to magnitudes built in its right-hand side.Footnote ^29, Footnote ³⁰

2.6 Set Abstraction

So far, we have seen how much arithmetic and real analysis can be recovered via different APs and some formulation of SOL. It is time we address a further worry, which was not in Frege’s purview, but is indeed a concern for many current abstractionist programs: How much of contemporary set theory, that is, Zermelo-Fraenkel set theory augmented with the axiom of choice (ZFC), can be interpreted by abstraction?

ZFC is an axiomatic set theory usually expressed in FOL with identity, augmented by the primitive nonlogical constant $\in$ (for set membership), and whose intended interpretation is a nonempty collection of sets.

ZFC is taken to capture the iterative conception of sets: roughly, by axiomatically postulating the existence of some basic sets, more and more sets are formed through stages by the iterated application of operations among sets available in the set-theoretic universe. This procedure gives rise to a cumulative hierarchy of sets that is open-ended: there is no final stage of set formation. It is now customary to call such a hierarchy “$V$.”Footnote ³¹

The axioms of ZFC are:

• (Extensionality) $\forall x\forall y(x=y\leftrightarrow \forall z(z\in x\leftrightarrow z\in y\left)\right)$,

stating that any sets $x,y$ are identical just in case they contain exactly the same members;

• (Separation) $\forall z\exists x\forall y(y\in x\leftrightarrow y\in z\wedge \varphi y)$,

where $\varphi$ does not contain $x$ free, stating that, for any given set $z$, there is a set $x$ containing all individuals $y$ that are members of $z$ and satisfy the formula $\varphi$;

• (Empty Set) $\exists x\forall y(y\in x\leftrightarrow y\ne y)$,

stating the existence of the empty set. Since Extensionality guarantees that there is a unique empty set, the individual constant “$\varnothing$” can be explicitly defined in the language of ZFC;

• (Pairing) $\forall x\forall y\exists z\forall u(u\in z\leftrightarrow u=x\vee u=y)$,

stating that there is a set containing two elements;Footnote ³²

• (Foundation) $\forall x\left(\exists y\right(y\in x)\to \exists y(y\in x\wedge \neg \exists z(z\in y\wedge z\in x)\left)\right)$,

stating that every nonempty set $x$ contains an individual $y$ sharing no elements with $x$;Footnote ³³

• (Union) $\forall x\exists y\forall z(z\in y\leftrightarrow \exists u(z\in u\wedge u\in x\left)\right)$,

stating that, for every set $x$ of sets $u$, there is a set $y$ containing the members of the members of $x$;Footnote ³⁴

• (Powerset) $\forall x\exists y\forall z(z\in y\leftrightarrow z\subseteq x)$,

stating that, for every set $x$, there is a set $y$ containing all the subsets of $x$, where the notion of subset is explicitly definable as $x\subseteq y{\leftrightarrow }_{def}\forall z(z\in x\to z\in y)$. The powerset $y$ of $x$ can be defined as: $\wp \left(x\right)=y{\leftrightarrow }_{def}\forall z(z\in y\leftrightarrow z\subseteq x)$;

• (Infinity) $\exists x(\varnothing \in x\wedge \forall y(y\in x\to y\cup \left\{y\right\}\in x\left)\right)$,

stating the existence of at least a set containing denumerably many members;Footnote ³⁵

• (Replacement) $\forall u\exists !w\varphi (u,w)\to \forall z\exists x\forall y(y\in x\leftrightarrow \exists w(w\in z\wedge \varphi (w,y)\left)\right)$,

stating that, if a formula $\varphi$ relates each set $u$ to a unique $w$, then starting from any set $z$ in the hierarchy, another set $x$ can be formed by replacing all members of $z$ by other individuals according to $\varphi$;

• (Choice) $\forall x\left(\forall y\right(y\in x\to \exists z(z\in y))\wedge \forall y\forall z(y\in x\wedge z\in x\wedge y\ne z\to \neg \exists u(u\in z\wedge u\in y))\to \exists w\forall y(y\in x\to \exists !z(z\in y\wedge z\in w)\left)\right)$,

stating that, for every set $x$ of pairwise-disjoint nonempty sets $y,z$, there exists a set $w$ containing exactly one element from each set in $x$.

Possibly, the most renowned (consistent) AP interpreting at least some of ZFC is the restriction of BLV Boolos calls “New V” – see Boolos (Reference Boolos1989, Reference Boolos1987b):

• (New V) $\forall X\forall Y(ϵX=ϵY\leftrightarrow (\left(Big\right(X)\wedge Big(Y\left)\right)\vee \forall x(Xx\leftrightarrow Yx)\left)\right)$,

where $Big$ is the property of concepts “being equinumerous with the universal concept $x=x$.” Boolos’s suggestion is based on the set-theoretic conception of the limitation of size, according to which in order to avoid contradiction, sets in the set-theoretic universe cannot be too large. The limitation of size view, along with the iterative conception, is taken to be incorporated in ZFC.

The violation of the limitation of size constraint is provided as a possible explanation of set-theoretic paradoxes in Cantor’s naїve set theory.Footnote ³⁶ Besides Extensionality, the latter amounts to the axiom schema:

• (Naїve Comprehension) $\exists x\forall y(y\in x\leftrightarrow \varphi y)$ – $\varphi$ does not contain $x$ free,

stating the existence of a set corresponding to any formula $\varphi$.

Naїve Comprehension is inconsistent. By plugging the condition $y=y$ in the right-hand side of Naїve Comprehension, there must exist the universal set $u$, that is, the set containing every individual – including itself, since also $u$ is self-identical. Now, recall Cantor’s theorem: For any set $x$, the cardinality of its powerset $\wp \left(x\right)$ is larger than the cardinality of $x$. The theorem is proved by showing that there is no injective function mapping each subset of $x$ to the elements of $x$. The universal set $u$ will contain every set, including all of its subsets. Hence, $u$ must contain enough elements for each of its subsets to be mapped into. If so, the cardinality of $\wp$($u$) cannot be larger than $u$’s, which contradicts Cantor’s theorem. So, the universal set $u$ does not exist. This is the so-called Cantor’s paradox.

The latter (as well as other paradoxes such as that of Russell, the Burali–Forti paradox concerning the nonexistence of the largest ordinal number, and the contradiction ensuing from the assumption of the existence of the largest cardinal number) is (at least partially) blamed on the fact that Naїve Comprehension allows for the existence of sets that are “too big,” Limiting the size of the existing sets is one of the strategies (and philosophical underpinnings) of contemporary set theory.

Besides having models, Boolos’s New V implies that “small” concepts have the same extension just in case they are co-extensional; but in case concepts $X,Y$ are Big, their extensions are identical, even if $X,Y$ are not co-extensional.Footnote ³⁷ Interestingly, PA² and some amount of ZFC are derivable from New V. The notion of set can be explicitly defined in Boolos’s setting:

• (Set) $Set\left(x\right){\leftrightarrow }_{def}\exists X(x=ϵX\wedge \neg Big(X\left)\right)$,

meaning that $x$ is a set just in case there’s a concept $X$ whose extension is $x$ and $X$ is small. Set membership can also be explicitly defined: $x\in y{\leftrightarrow }_{def}\exists X(y=ϵX\wedge Xx)$. By such definitions, New V proves Extensionality, Empty Set, Pairing, Separation, and Replacement. It also proves restricted versions of Union and Foundation.Footnote ³⁸ New V entails neither Powerset nor Infinity. The latter result means that by New V and unrestricted SOL, it cannot be determined whether the concept of being a natural number is big. Finally, in general, New V and similar APs have very different models, that is, both well-founded and non-well-founded.Footnote ³⁹

Boolos’s result falls short of interpreting all of ZFC in a consistent system of BLV. Cook (Reference Cook2003) proposes an extension of New V, starting from an AP for ordinal numbers, namely the Size-Restricted Ordinal Abstraction Principle (SOAP),Footnote ⁴⁰ so that all of ZFC is interpretable in a consistent system of BLV.

Recently, an alternative view concerning APs that interprets set theory, that is, dynamic abstraction in Linnebo (Reference Linnebo2010, Reference Linnebo2013, Reference Linnebo, Ebert and Rossberg2016, Reference Linnebo2018), has been proposed. Its philosophical underpinning is as follows. Suppose APs are philosophically conceived of primarily as principles determining identity and distinctness facts about abstracts. If so, first-order quantification on the right-hand side of APs should presuppose the identity of the individuals it varies over, but those are exactly the objects APs are supposed to individuate in the first place. In this respect, such APs are first-orderly impredicative – see, for example, Linnebo (Reference Linnebo, Ebert and Rossberg2016).

Dummett (Reference Dummett1991) argued that only first-orderly predicative APs do not presuppose that the objects introduced on the left are already available for quantification, and thus that there is philosophical motivation to prefer first-orderly predicative APs over first-orderly impredicative ones. However, predicative APs are typically mathematically weak.Footnote ⁴¹ According to Dummett, abstractionists are therefore left with the dilemma of choosing between mathematically fruitful but philosophically unsound impredicative abstraction, and philosophically motivated but mathematically dry predicative abstraction.

The aforementioned concern can motivate alternative conceptions of abstraction. In this regard, Linnebo (Reference Linnebo2010, Reference Linnebo2013, Reference Linnebo, Ebert and Rossberg2016, Reference Linnebo2018) provides the philosophical motivation and formal tools for such an alternative framework – which we will further investigate in Section 4: In Linnebo’s dynamic abstraction, the individuation of BLV-extensions proceeds in stages (via stepwise extensions of the interpretation of first-order quantifiers), starting from an ontology of “old” objects, whose identities are already established and which the equivalence relation of BLV applies to, and providing individuation conditions for “new” objects (i.e. extensions) whose identity is dependent on the objects in the previous interpretation. In such a dynamic setting, predicative abstraction is iterated over larger and larger domains. On the one hand, this approach retains the philosophical motivation for first-order predicativity. On the other, it restores the mathematical strength of (predicative) APs.

The formal setting is constituted by a (modal) plural logic and a plural formulation of BLV. In this setting, the interaction between old and new objects is cashed out in terms of the modal operators $\square \varphi$ and $◊\varphi$ – which nevertheless are not conceived of as expressing metaphysical modality, but rather as “no matter how the domain of abstract objects is extended, it will remain the case that $\varphi$,” and “the domain of abstract objects can be extended so that it is the case that $\varphi$,” respectively. In order to spell out exactly how modality and abstraction interact, Linnebo factorizes abstraction in an existence principle and an extensionality principle. The former is

• (Potential Collapse) $\square \forall xx◊\exists y\left(EXT\right(xx,y\left)\right)$,

which postulates that, no matter how the domain is extended, for every plurality $xx$, the domain can be extended so that there is the extension of $xx$. Furthermore,

• (Extensionality) $EXT(xx,x)\wedge EXT(yy,y)\to (x=y\leftrightarrow \square \forall z(z\prec xx\leftrightarrow z\prec yy\left)\right)$

posits that if $x,y$ are the extensions of $xx,yy$ respectively, they are identical just in case $xx,yy$ are coextensive, no matter how the domain is extended. At the same time, the underlying plural logic is also modalized:

$◊\exists xx\square \forall y(y\prec xx\leftrightarrow {\varphi }^{◊}y),$(2.11)

where ${\varphi }^{◊}y$ is the “potentialist translation” of $\varphi$, that is, the result of replacing each ordinary quantifier in $\varphi$ with the corresponding modalized quantifier. Still, given the background assumption that pluralities are modally rigid (pluralities cannot gain members and, if $x\prec xx$, then necessarily so), the modal plural comprehension axiom has to be restricted – hence, for example, the alleged plurality corresponding to the formula “$x=x$” would not be admitted since it would not be modally rigid, provided the expansion of the domain.Footnote ⁴² Notably, all (potentialist translations of) axioms of ZFC except Infinity and Replacement follow. For Infinity and Replacement to follow as well, further assumptions are needed.Footnote ⁴³

On the philosophical side, Linnebo’s approach shares Aristotle’s two fundamental insights in the philosophy of mathematics: First, the mathematical universe is never complete and it is always possible to individuate new mathematical objects (Linnebo and Shapiro Reference Linnebo and Shapiro2017); second, these objects depend for their existence and their properties on nonmathematical entities – see Section 4.2.3.

2.7 Invariance

Famously, Frege conjectured BLV to be a logical principle.Footnote ⁴⁴ After all, one of his main aims was to derive the basic laws of arithmetic from logic and definitions alone. Of course, given Russell’s paradox, BLV is not logical at all. But the mere consistency of an AP is not sufficient for its logicality: Even consistent APs such as, for example, HP seem hardly logical, since they may imply the existence of denumerably many individuals.Footnote ⁴⁵

What the logicality criteria are supposed to be is a complex issue.Footnote ⁴⁶ Still, there’s agreement that one of the (necessary) conditions for logicality is topic neutrality:

Topic neutrality is often formally cashed out in terms of invariance under permutations of the logical expressions, namely their insensitivity to the particular identities of the objects they vary over – where, generally, a permutation $\pi$ is a one-one function mapping a domain onto itself.Footnote ⁴⁸

Starting from Fine (Reference Fine2002), the notion of invariance has been investigated as concerns APs. As Antonelli (Reference Antonelli2010b) points out, there are three senses in which invariance may be investigated in this respect: the invariance of the abstraction operator; the invariance of the equivalence relation; the invariance of the overall principle. There are very few invariant abstraction operators, and they are rather uninteresting mathematically. Furthermore, the invariance of the overall AP follows from the invariance of its equivalence relation; therefore, Antonelli (Reference Antonelli2010b) focuses on the latter: Given a relation $R$ on a second-order domain that is closed under permutations,

1. (1) $R$ is internally invariant if, and only if, for any permutation $\pi$, $R(X,Y)$ if, and only if, $R\left(\pi \right[X],\pi [Y\left]\right)$;

2. (2) $R$ is doubly invariant if, for any pair ${\pi }_1,{\pi }_2$ of permutations, $R(X,Y)$ if, and only if, $R\left({\pi }_1\right[X],{\pi }_2[Y\left]\right)$;

3. (3) $R$ is (simply) invariant if, and only if, $R(X,\pi [X\left]\right)$ holds for any permutation $\pi$.

Double and simple invariance are logically equivalent; simple invariance implies internal invariance.

The equinumerosity relation appearing on the right-hand side of HP is simply invariant. According to Antonelli (Reference Antonelli2010b), what the simple invariance of equinumerosity shows is that the very notion of cardinality can be deemed logical by those supporting the view that invariance is a necessary condition for logicality, or at the very least the invariance of equinumerosity can play a role in the acceptability of APs (e.g. Fine Reference Fine2002) and thus in the debate on the so-called Bad Company problem – see Section 2.8. The alleged logicality of equinumerosity does not imply the logical nature of cardinal numbers. In this sense, the logicality of cardinality supports a deflationist view of cardinal numbers: Since HP is (simply) invariant, it is insensitive to the underlying nature of the objects in the domains it permutes over; so “[a]nything at all – even ordinary objects – can play the role of these abstracta, as long as the choice respects the equivalence relation.”Footnote ⁴⁹ Later, Cook (Reference Cook2017) refines the notions of invariance presented in Antonelli (Reference Antonelli2010b) and Fine (Reference Fine2002) by investigating doubly internal invariance,Footnote ⁵⁰ and shows that HP is the most fine-grained AP satisfying it.Footnote ⁵¹

Neologicists claim that HP is analytic albeit not a logical truth. In §3 of Grundlagen, Frege famously claims that a sentence is analytic if “in carrying out [its proof] we come only on general logical laws and definitions.” Frege’s notion of analyticity must be distinguished from Kant’s, according to which a judgment is analytic if the predicate is already contained in the subject.

These two senses of analyticity must moreover be distinguished from the contemporary notions of metaphysical and epistemic analyticity (Boghossian Reference Boghossian1996):

• a statement is metaphysically analytic if it is true purely in virtue of the meaning of some of its component expressions;

• a statement is epistemically analytic if grasping its meaning is sufficient to have a justified belief that the content expressed by that statement is true.

Hale and Wright (Reference Hale and Wright2001a) claim that even though HP is analytic neither in Kant’s nor in Frege’s sense,Footnote ⁵² it can still be deemed analytic in the sense that it is “determinative of the concept it thereby serves to explain” (p. 14).

It is worth noting that a statement can be analytic in Hale and Wright’s sense without being metaphysically or epistemically analytic. In particular, a statement can be analytic in the Neologicist sense (that is, the implicit definition of a concept) without being metaphysically analytic (that is, true in virtue of meaning).

Hale and Wright (Reference Hale, Wright, Boghossian and Peacocke2000) argue that HP is epistemically analytic, that is, can be known a priori (see Section 5.3). Neologicists claim moreover that PA² is analytic in Frege’s sense, that is, its axioms can be derived by an implicit definition and SOL (see Section 5.3.2). In the most recent iterations of the abstractionist program, analyticity has been dropped (Linnebo Reference Linnebo2018, p. 3) even though the claim that APs can provide implicit definitions is retained (Rayo Reference Rayo2013, p. 187; Linnebo Reference Linnebo2018, p. xiii).

2.8 The Bad Company Problem

Many APs, even those that are consistent, are unacceptable because they are incompatible with other principles with the same form (Dummett Reference Dummett1991, Reference Dummett and Schirn1998; Boolos Reference Boolos and Heck1998a; Weir Reference Weir2003).Footnote ⁵³

An example is Wright’s (Reference Wright and Schirn1998) Nuisance Principle (NP), which states that two concepts have the same “nuisance” if their difference is finite:Footnote ⁵⁴

• (NP) $\left(\forall X\right)\left(\forall Y\right)\left(\nu \right(X)=\nu (Y)\leftrightarrow (\text{Fin}(X\wedge \neg Y)\wedge \text{Fin}(\neg X\wedge Y)\left)\right)$.

NP has only finite models. Therefore, it cannot be satisfied jointly with HP, which, to the contrary, has models that are at least denumerable.Footnote ⁵⁵ As remarked by Linnebo (Reference Linnebo2009b, p. 324), “attractive principles like Hume’s Principle are surrounded by bad companions.” The Bad Company problem consists in sorting out acceptable principles from unacceptable ones.

Unacceptable principles seem easy to come by. Many pairwise inconsistent abstractions take the form of “Distraction principles” (Weir Reference Weir2003, p. 17), namely, APs whose abstraction function behaves like the function characterized by BLV unless both $X$ and $Y$ satisfy some second-order formula $\Phi$:

• (D) $\left(\forall X\right)\left(\forall Y\right)\left(\sigma \right(X)=\sigma (Y)\leftrightarrow (\left(\Phi \right(X)\wedge \Phi (Y\left)\right)\vee \forall x(Xx\leftrightarrow Yy)\left)\right)$.

At the same time, New V and other mathematically promising principles are instances of (D). The problem consists in selecting all and only the acceptable Distractions while leaving out the principles that are incompatible with them.

Over the years, several authors have formulated a plethora of increasingly strong criteria for acceptable abstraction (cf. e.g. Cook (Reference Cook2021a, Reference Cook, Boccuni and Sereni2021b), and Cook and Linnebo (Reference Cook and Linnebo2018) for recent overviews of the criteria and their rationale). Here, we will simply list the main ones.

Criterion 3 (Semantic Field-Conservativeness) An AP is acceptable only if it is semantically Field-conservative (FCON), that is, for any theory $T$ to which AP can be consistently added and for any sentence $\varphi$:

$\{T+AP\}\models {\varphi }^{\neg @}\text{only if}T\models \varphi ,$

where ${\varphi }^{\neg @}$ is the result of restricting the quantifiers of $\varphi$ to objects that satisfy the formula $\neg \exists F(x={\Sigma }_{AP}(F\left)\right)$, that is, are not abstracts of AP.Footnote ⁵⁶

The mutual relations between UNB, FCON, IRN, and SSTB are proved in Cook and Linnebo (Reference Cook and Linnebo2018). Cook and Linnebo also argue that a solution to the Bad Company problem requires the combination of strong stability with two other criteria:

Criterion 7 (Monotonicity) An AP is acceptable only if it is monotonic (MONO), that is, its equivalence relation $Eq$ is intrinsic, that is, for any $X$ and $Y$,

$Eq(F,G)\text{iff}E{q}^{F\cup G}(F,G),$

where $E{q}^{F\cup G}$ is the result of restricting all the quantifiers of $Eq$ to objects that fall under either $X$ or $Y$.

The relations among combinations of SSTB, HSTB, and MONO are proved in Cook (Reference Cook, Boccuni and Sereni2021b). Both Fine (Reference Fine, Gendler and Hawthorne2005) and Cook (Reference Cook2017) explore invariance as a condition on acceptable abstraction.

The Bad Company problem gets worse, however, when we consider the mathematical strength of acceptable APs: The APs that would be most suited for recovering mathematical theories other than arithmetic are often unacceptable. An example is George Boolos’s New V. New V is sufficient to recover a remarkable portion of ZFC; however, it is not conservative, since it implies that the universe is well-ordered (Shapiro and Weir Reference Shapiro and Weir2000, pp. 304–309). In general, Uzquiano (Reference Uzquiano2009) shows that no AP that complies with SSTB can recover ZFC. As noted by Studd (Reference Studd2016, pp. 595–596), philosophical abstractionism seems to face a dilemma: Either the criteria introduced are permissive enough to include promising cases of abstraction such as New V, but, then, the Bad Company problem resurfaces, or those criteria are restrictive enough to avoid it, but, then, New V and other promising principles are ruled out.

3.1 Introduction

According to Frege and Scottish Neologicists (Section 2.4), expressions of the form “$\#X$” are singular terms. There is no general agreement, however, on whether expressions appearing on the left-hand side of APs are indeed such. These considerations often rely on the analysis of natural language. For instance, in the natural language, number words like, for example, “four” can take on different roles. In this section, we will focus particularly on numerical expressions, since most of the relevant literature concerns number words. Specifically, we will survey the main positions concerning their semantic role: the substantival view, ascribing singulartermhood to number words (Frege; Scottish Neologicism; Rayo; Linnebo – Section 3.2); the adjectival reading, which regards number words as modifiers of nouns (Hofweber; Moltmann – Section 3.3.1); the quantificational perspective, conceiving of number words as numerical quantifiers (Hodes – Section 3.3.2). Other things being equal, similar considerations may apply to expressions of the form “$\Sigma \alpha$” appearing on the left-hand side of APs in a more general abstractionist perspective.

3.2 The Substantival View

Consider the following statement:

1. (1) Jupiter has four moons.

Statement (1) can be provided with a reading that Frege subscribed to and Dummett (Reference Dummett1991, p. 99) calls substantival. According to this reading, (1) can be paraphrased as

1. (2) The number of Jupiter’s moons is four,

whose logical form is that of an identity statement between the referent of the singular term “the number of Jupiter’s moons” and the referent of the singular term “four.”Footnote ⁵⁷

It is fair to say that Frege’s reading of expressions of the form “the number of Jupiter’s moons,” “4,” and more in general abstract terms of the form “$\Sigma \alpha$” is tightly connected to his view concerning the basic logical categories of function and argument – see, for example, Frege (Reference Frege, Bauer-Mengelberg and van Heijenoort1879, Reference Frege and by P. Geach in Geach1891, Reference Frege1893/1903), and Cook (Reference Cook, Zalta and Nodelman2023) for an overview.

In Frege’s intentions, a logically perfect language, like the one exposed in Frege (Reference Frege, Bauer-Mengelberg and van Heijenoort1879, Reference Frege1893/1903), was to be devoted to several aims, among which was formulating the axioms and inference rules of logic, in order to provide explicit definitions of the mathematical notions necessary to derive the basic laws of arithmetic from logical principles and definitions alone – as well as the basic laws of real and, possibly, complex analysis. At its core, such a language relied on the fundamental logical distinction between function and argument, with which Frege replaced Aristotle’s dichotomy between subject and predicate in the logical analysis of language.

Slightly oversimplifying, we can take Frege’s functions to be denoted by predicates with at least a free variable (in a contemporary notation), and Frege’s arguments to be placeholders for names of objects (e.g. individual constants in contemporary formal logic). According to Frege, a function is “unsaturated” (i.e. it has to take names as arguments in order to have a (truth-)value), whereas names of objects are “saturated” (i.e. nothing is required for them to have values, if any). Therefore, when asking what logical categories expressions in

1. (3) $2+3=5$

belong to, and considering the expression “2,” either “2” stands for a function that has to be saturated in order to have a value, or no saturation is needed at all. Both logico-linguistic analysis and cautionary anti-empiricist views (such as J. S. Mill’s, which Frege disputed in e.g. Frege Reference Frege1884) prompted Frege to argue that “2” is a name of a (self-subsistent) object.

In this respect, Frege’s substantival reading of (1) as (2) is coherent both with his distinction between function and argument, and his view of numbers as objects. Given his foundational aims, equations like (3) were of the utmost importance to Frege, and his substantival reading accounts for them: The logical form of (3) consists in the application of a function symbol (“+”) to two singular terms (“2” and “3”), so that the number “2 + 3” denotes is the same number (=) as the number denoted by “5.”

Since number words (and numerals) are singular terms standing for self-subsistent (abstract) objects (i.e. the numbers), and since we have no direct access to abstract objects, an issue may arise as to how the reference of number words (and numerals) is fixed. In this respect, Frege attributes a crucial role to his Context Principle, which cautions, against psychologistic inclinations, to “never ask for the meaning of a word in isolation, but only in the context of a proposition.”Footnote ⁵⁸ Particularly relevant to the aim of reference fixing of number words in general are identity statements like (2) and (3). Still, as Frege (Reference Frege1884, §62) famously points out:

In particular, the meaning of identities such as (2) has to be established, with no use of the expressions “the number of Jupiter’s moons” or “four” at all. Therefore,

In this respect, the truth-conditions of an AP for numbers (and Fregean abstract objects in general) are expected to deliver contextually the reference of the singular terms on its left-hand side.Footnote ⁶⁰ At this point, Frege discusses what later would be known as HP, if only to discard it because of the so-called Caesar Problem,Footnote ⁶¹ and provide an explicit definition of numbers as equivalence classes closed under the equinumerosity relation – which was doomed to failure due to Russell’s paradox.Footnote ⁶²

In a rather Fregean vein, Wright (Reference Wright1983) and Hale and Wright (Reference Hale and Wright2001a) rely on the Context Principle in order to bestow meaning upon expressions like “$\#X$.” In particular, in Hale and Wright’s view, the proposition supposed to contextually fix the reference of “$\#X$” is HP.

That is not enough, though, to guarantee that expressions like “$\#X$” are indeed singular terms to begin with. In order to establish this, Scottish Neologicists rely on several claims.

First, they propose syntactic criteria to distinguish singular terms from other expressions. In particular, those criteria amount to an inferential test and an Aristotelian test, both tracing back to Dummett (Reference Dummett1973), according to which genuine singular terms display different inferential behavior than other expressions (especially, quantifiers), and singular terms, unlike predicates, have no contradictory expressions (i.e. it makes no sense to use “not (Mary)” in a sentence, but it does to use “not (smart)”), respectively.Footnote ⁶³

Second, once these criteria are in place, Scottish Neologicists rely on the Syntactic Priority Thesis (SPT) and HP. According to SPT, the truth of claims involving numerical terms in singular term position suffices to guarantee that they refer to objects. Hence, since HP contains expressions of the form “$\#X$” that are singular terms by the syntactic criteria aforementioned and because they appear in singular term position (e.g. they flank the identity symbol), its truth is sufficient to justify the attribution of objectual reference to the singular terms on its left-hand side.

Precisely, by its surface syntax and the criteria for singulartermhood, “$\#X$” is indeed a singular term. Moreover, the right-hand side of HP is a truth of pure SOL (see Definition 2.4 in Section 2.4). Since HP is a material biconditional, from the truth of HP and its right-hand side, the identity statements on its left-hand side are also true. Since the latter are true claims containing numerical terms in singular term position, those terms are singular terms that do refer to objects (SPT).Footnote ⁶⁴

A similar reliance on the syntactic role of abstract terms and their appearance in (atomic) true sentences for their referentiality to be secured is in Rayo’s (Reference Rayo2013) compositionalism. Consider an uninterpreted first-order language $L$, and the assignment of truth-conditions to $L$’s sentences. Of course, such an assignment has to satisfy logical entailment,Footnote ⁶⁵ but otherwise we can pick any assignment. According to compositionalism, if the sentences of $L$ are interpreted in this way, for a singular term $t$ of $L$ to be referential (i.e. to refer to an object in the world), all that is needed is that the world satisfies “the truth-conditions that were assigned to ⌜ $\exists x(x=t)$⌝ (or some inferential analogue).”Footnote ⁶⁶

Linnebo (Reference Linnebo2018) labels both the Neologicist metasemantic view of reference fixing and Rayo’s compositionalism ultra-thin conceptions, by which in general “an expression refers provided that, first, it has all the appropriate syntactic and inferential characteristics of a singular term, and second, the expression figures in appropriate true sentences.”Footnote ⁶⁷ He then contrasts both views with his thin conception, according to which, for the reference of (singular) abstract terms such as “$\Sigma \alpha$” to be fixed, APs as identity criteria for appropriate kinds of objects are sufficient. The process of reference fixing proceeds in stages. We start from a domain of “old” objects, reference to which is already accomplished. Then, an AP is introduced: Its right-hand side contains first-order quantifiers restricted to the old objects; and singular terms on its left-hand side refer to “new” objects.Footnote ⁶⁸ Reference to the latter is accomplished via identity and distinctness facts based on an old ontology of individuals – which Linnebo (Reference Linnebo2018, p. 141) labels semantic nonreductionism.Footnote ^69, Footnote ⁷⁰

3.3 Nonsubstantival Views

In this section, we will present two main alternatives to the substantival view: the view by which (most) number words are adjectives, and consequently stand for a varied landscape of higher-order entities (Section 3.3.1); and the view according to which number words are quantificational expressions (Section 3.3.2).

3.4 The Caesar Problem

As mentioned in Section 3.2, Frege discarded HP as a contextual definition of the natural numbers because of the so-called Caesar Problem (CP). In §67, Frege quickly raises an objection to his own attempt to define directions using an AP:

HP is similarly incapable of establishing the truth-value of statements like, for example, “Caesar =  $\#X$.”Footnote ⁸⁹

CP generalizes over APs. As Cook and Ebert (Reference Cook and Ebert2005, p. 122) point out, APs “fix the truth conditions for identity statements regarding abstracts (i.e. the truth conditions of the identity on the left-hand side of the abstraction principle) in virtue of the equivalence relation on the right-hand side.” However, generally APs stay silent as for the truth-conditions of “mixed” identity statements of the form

$\Sigma \alpha =q,$(3.1)

where “$q$” is not a term of the form “$\Sigma \alpha$.”Footnote ⁹⁰

CP is particularly worrisome for Neologicism. If CP is not solved, Neologicists cannot argue that HP’s truth contextually bestows reference upon number-terms appearing on its left-hand side, and therefore, by SPT and the criteria for singulartermhood, HP-terms are singular terms denoting objects. As Wright (Reference Wright and Miller2020, p. 306) puts it,

In Section 4.3, we’ll see how Neologicists propose to solve CP by metaphysical principles. For the remainder of this section, let us focus on two other solutions to CP: conventionalism and deflationism.

The guiding idea of the former is that, like the truth of APs is putatively fixed by stipulation (see Section 5.3), further stipulations can determine the truth or falsehood of “mixed” identities. Conventionalism is a “piecemeal” approach (Studd Reference Studd2023, p. 237): At any given moment, linguistic stipulation can determine some, but possibly not all, cross-sortal identities left open by APs. Versions of conventionalism are defended also by Rayo (Reference Rayo2013, pp. 80–81) and Linnebo (Reference Linnebo2018, p. 160).

According to deflationism, as in, for example, Antonelli (Reference Antonelli2010a,Reference Antonellib), the role of APs is to provide an “inflationary thrust” on the first-order domain by “giv[ing] a lower bound on the cardinality of the domain of objects, relative to the size of the class of all concepts, taken modulo a given equivalence relation.”Footnote ⁹¹ According to Antonelli, APs do not provide information about the nature of these objects; by contrast, anything can be an abstract object, provided that it belongs to a domain with the appropriate cardinality. This proves to be a dissolution of CP, since, for example, $\#X$ is identical to Caesar in some models of HP and distinct from Caesar in other models.Footnote ⁹²

4.1 Introduction

Traditionally, APs are defended along with realist ontologies (in particular, Platonism). Platonism is typically characterized by two claims:Footnote ⁹³ There exists a realm of abstract objects to which mathematical language refers and that is described by our mathematical theories; and these objects exist and have their properties independently of mathematicians and their thoughts, language, and practices. We will first show how abstractionists defend the existence claim (Sections 4.2–4.2.2). We will then consider various ways in which the independence claim can be cashed out (Section 4.2.3).Footnote ⁹⁴ Finally, in Section 4.3, we consider the view that APs characterize the intrinsic nature of the objects that these principles introduce.

4.2 The Existence of Mathematical Objects

As seen in Section 3.2, the Neologicist defense of Platonism is based on three premisses.Footnote ⁹⁵ The first premise concerns the logical form of the left-hand side of APs:

1. (1) The numerical terms that appear on the left of APs are singular terms.

Premise (1) is syntactic because abstractionists claim that it is possible to identify singular terms independently of any consideration of the semantic role of these expressions – see Section 3.2.

The second premise expresses a sufficient condition for singular terms to refer:

1. (2) A singular term $t$ refers, and hence the object to which it refers exists, if $t$ appears in a true (extensional) sentential context, and specifically in true identity statements.

Premise (2) should be uncontroversial: Given the standard semantic clauses for first-order logic, “$t=t^{\prime }$” is satisfied by a structure $M=⟨D,I⟩$ if, and only if, $I\left(t\right)=I\left(t^{\prime }\right)$.Footnote ⁹⁶

Finally, the third premise asserts that there are true instances of the right-hand side of HP:

1. (3) There are true instances of the right-hand side of APs.

These instances include, for example, statements of the equinumerosity of a concept with itself.Footnote ⁹⁷

Since there are true instances of its right-hand side, and HP is a material biconditional, then there are also true instances of the left-hand side of HP. Combined with (1), it follows that there are true identity statements between singular terms. And since the left-hand side meets the condition specified in (2) for ensuring that a singular term refers to an object that exists, then numerical terms refer to objects, and so there are numbers.Footnote ⁹⁸

The logical assumptions of this argument consist in the second-order comprehension schema CA, which states that there is a concept $X$ corresponding to any formula $\varphi \left(x\right)$ (with $x$ that occurs free in $\varphi$). For example, to show that $0$ exists, Neologicists consider a concept whose extension is (necessarily) empty, say, the concept of being not self-identical. The argument also assumes classical logic.Footnote ⁹⁹ The only nonlogical assumption of the argument is Hume’s Principle. HP states that statements that feature numerical terms are materially equivalent to statements that do not feature these terms.Footnote ¹⁰⁰

The abstractionist ontology of mathematics faces, however, the following problem of the origin. Assume that APs can be stipulated without presupposing the existence of mathematical objects. Let’s also assume that pure higher-order logic is ontologically neutral, as Neologicists do (Wright Reference Wright, Leng, Paseau and Potter2007; Hale Reference Hale2019), rather than “set theory in sheep’s clothing” (Quine Reference Quine1970, chapter 5). Abstractionists must explain how the ontological commitment of mathematical theories originates if both SOL and AP are not committed to the existence of objects of a particular sort. For example, if the right-hand side of HP is not committed to the existence of numbers, then its left-hand side is not committed to the existence of numbers either. Vice versa, if the left of HP does carry a commitment to numbers, then by contrapositive reasoning, the right-hand side carries the exact same commitment. But since some true instances of the right of HP are purely second-order logical truths, then SOL is not ontologically innocent.

The different responses to this problem are tabled in Figure 1. First, we can distinguish between asymmetric and symmetric conceptions of abstraction (see Linnebo Reference Linnebo2018, §1.7). According to the symmetric conception, the two sides of APs have the same ontological commitments. Among these positions, we can distinguish between those that adopt the Quinean criterion according to which the ontological commitment of a theory consists in what must lie in the range of its (first-order) quantifiers in order for the statements of that theory to be true, and those that reject this criterion.Footnote ¹⁰¹ According to the asymmetric conceptions, by contrast, the commitments of the left of an AP exceeds those of its right.Footnote ¹⁰² We will consider these positions proceeding from the bottom to the top of Figure 1.

Figure 1 Responses to the origin problem.

4.2.2 Content Recarving

The non-Quinean criterion of ontological commitment is based on the idea that the two sides of APs share the same content.

A version of this idea was introduced by Frege himself, who in §64 of Grundlagen (p. 71) claimed that the concept Direction can be introduced by taking the right-hand side of the corresponding AP “as an identity”:

The judgment “line $a$ is parallel to line $b$”, or, using symbols, “$a//b$”, can be taken as an identity. If we do this, we obtain the concept of direction, and say: “the direction of line $a$ is identical with the direction of line $b$”. Thus we replace the symbol $//$ by the more generic symbol =, through removing what is specific in the content of the former and dividing it between $a$ and $b$. We carve up the content in a way different from the original way, and this yields us a new concept.Footnote ¹⁰⁶

Frege’s metaphor was followed, most notably, by Hale and Wright, who claim that the left-hand side of an instance of HP corresponds to a “reconceptualization” of its right-hand side (Wright Reference Wright1999, p. 312). This proposal has been criticized by Fine (Reference Fine2002) and by Potter and Smiley (Reference Potter and Smiley2001, Reference Potter and Smiley2002).Footnote ¹⁰⁷ More recently, Fregean semantics has been defended by Rayo (Reference Rayo2013).Footnote ¹⁰⁸

Rayo calls his position trivialism. Specifically, Trivialist Platonism (about finite cardinals) is the view that all the instances of the following schema are (trivially)Footnote ¹⁰⁹ true:

Numbers: For the number of the $X$’s to be $n$ just is for there to be $n$ $X$’s.

HP entails that for any finite $n$, $\#\left(X\right)=n\leftrightarrow {\exists }_{n}x\left(X\right(X\left)\right)$. Under plausible assumptions about the logic of the “just is”-operator, Numbers follows from this version of HP (where $\equiv$ is the “just is”-operator):

$\#X=\#Y\equiv X\approx Y.$

Rayo claims that “just is”-statements such as Numbers are “no difference” statements: For example, there is no difference between the number of the dinosaurs being zero and there being no dinosaurs (Rayo Reference Rayo2013 p. 4).

The nondifference claim is understood in terms of sameness of facts: Rayo claims that the two sides of Numbers provide “[a] full and accurate description of the same feature of reality” (Rayo Reference Rayo2013, p. 6). Rayo regiments his claim in terms of (sameness of) truth-conditions. This notion is explicated, in turn, in terms of (the identical) demands that the truth of two sentences make on the world (Rayo Reference Rayo2013, p. 52). Finally, Rayo claims that the sentence $\varphi$ has the same ontological commitment as $\psi$ if $\varphi \equiv \psi$. This notion of ontological commitment differs from the Quinean one in that sentences with different logical forms can nonetheless make the same demands on the world (Rayo Reference Rayo2007).

Let’s give an example. Suppose that we do accept that the two sides of the direction principle have the same ontological commitment (in Rayo’s sense). Now consider a particular line $a$. Since we accept this “just is”-statement, we accept also that for the direction of $a$ to be self-identical, and hence for that direction to exist, just is for $a$ to be parallel to itself. Therefore, the direction of $a$ exists, and so there are directions; however, the existence of directions does not require more from the world than what is required for two lines to be parallel.Footnote ¹¹⁰