Achieving determinate reference to the natural-number structure amounts, in the context of model-theory, to the demand that our best theory of arithmetic be categorical, that it “pick out” a unique isomorphism-type.4 If the framework in which such a theory is formulated is that of first-order logic (FOL) this demand cannot be met: the expressive resources of FOL are too limited to rule out non-standard models of arithmetic, i.e., models that satisfy our best first-order description of the natural number structure (say, for concreteness, Peano-Arithmetic (PA)) but that are not isomorphic to N.

The inability of FOL to categorically characterize the natural-number structure runs deep: Button & Walsh (2018: 161) show that no compact logic5 is capable of rendering a theory of arithmetic categorical, for in any such logic it is possible to add non-standard elements (e.g., infinitely large numbers) to the standard model of arithmetic to create an elementarily equivalent non-standard model. Compactness and its close relative, completeness, are highly priced properties of logical systems. The crucial question taken up below concerns whether they are necessary to possess a determinate grasp of a notion.

This, then, is the issue in a nutshell: a framework like that provided by FOL is “too weak” to uniquely pin down the standard model of arithmetic (up to isomorphism) and thus to ensure determinate reference of arithmetical terms. FOL, however, is just an instrument used to formally codify antecedent beliefs of mathematicians about the properties of the mathematical structure N. What the lack of categoricity of PA seems to show is that a first-order formalization of arithmetic is unable to properly capture all of the mathematician’s antecedently held beliefs about that structure, that she believes or knows more about it than what can be adequately expressed in the confines of FOL.

There are two prominent diagnoses of the shortcoming exhibited in the admission of non-standard models and, correspondingly, two ways to address this perceived deficiency. Each motivates one of the enrichments of the formalism of FOL discussed in the next subsection. The first consists in maintaining that we know more about the subsets of N than FOL says we know. This analysis corresponds to a move to SOL with its quantification over subsets and a strengthening of the induction-schema of first-order PA. The second points out that the non-standard models of PA rely on a “misshapen” number sequence and claims that we have a firmer hold on the notion and properties of an ω-sequence (i.e., a progression of elements mirroring the natural number sequence) than FOL admits. This diagnosis naturally corresponds to the use of a formalism—exemplified by L(Q0) below,—that permits the moderate mathematical realist to adequately express more features of such a sequence than can be done in FOL.6

Given the limitations of FOL against the standard framework of model-theory how can the moderate realist claim to unambiguously refer to the natural number structure, the isomorphism-class of N? There are several ways in which she might strengthen or modify FOL to achieve this goal. We will briefly survey the two approaches that we wish to contrast in the following: (i) second-order logic (SOL) and (ii) L(Q0)—FOL extended by the generalized quantifier “there exist (in)finitely many.”7

Adopting a logic more expressive than FOL, such as SOL or L(Q0), appears to resolve the indeterminacy of reference to the natural number structure. Why should the move to a more powerful formal framework be denied to the moderate realist? The worry associated with the use of a logic substantially stronger than FOL is that one is making use of resources one cannot claim to have a firmer hold on than one has on N itself. In other words, one might be accused of invoking notions reference to which is at least as problematic as reference to the natural number structure itself.

Dedekind’s Categoricity Theorem, for example, requires that the semantics of SOL be full. However, in virtue of what can we be certain that the range of the second-order quantifiers is the complete powerset of the domain rather than a (suitably inclusive) subset thereof, that the semantics is full, rather than Henkin? After all, nothing about the formalism itself forces the quantifiers to take on the full interpretation, its use is perfectly consistent with possessing a Henkin semantics. What is needed to guarantee the intended full interpretation—to show that we have secured determinate reference of the second-order quantifiers—is yet another categoricity theorem, a proof that the use of the formalism determines a unique semantics.

But this means that the issue of determinately referring to the natural number structure has not been solved; it has merely been shifted to the level of the semantics of the logic used to characterize it. This is far from resolving the original issue of referential indeterminacy:8

Similar worries apply in the case of L(Q0): since Q0 is indefinable in FOL its semantics involves concepts that go beyond FOL, and the question arises how determinate reference to these is secured. This, coupled with the possibility of non-standard interpretations of the notion of infinity in the background theory (see (Field 1994: 397) and (Parsons 2007: §48)) adds force to the challenge of the model-theoretic sceptic.9

The move to logics stronger than FOL to achieve categoricity therefore seems to only “shift[…] the problem from the identification of postulates characterizing N categorically (’completely’) into the semantics and model-theory of the logic used to state the postulates” (Read 1997: 91), and it is questionable whether any referential determinacy is gained through this. The “doxological challenge” confronting the moderate realist is lifted into the meta-theory—the bump in the carpet has merely been moved. How, then, in light of the possibility of non-standard interpretations of the notions and resources used to provide a categorical characterization of the natural number structure, can the moderate realist claim to achieve a firm grasp of these notions? Call this the revenge challenge for the moderate mathematical realist.

Button and Walsh (2018: 7.9) dispute the very possibility for success of a strategy that adopts a logic stronger than FOL to resolve the indeterminacy of arithmetical reference. The reason is that no logic that is compact will be able to categorically characterize N. This means, they claim, that an “understanding of what consequence amounts to according to any such logic must, then, come from a specification of the formal semantics for that logic” (Button and Walsh 2018: 162). Yet, the notions involved in explicating the semantics of the kinds of logics needed far outstrip the resources of FOL and are thus, against a semantic theory formulated in FOL, “up for reinterpretation” (159). Thus, reference to the notions required to formulate the semantics of the logics needed to categorically characterize N is inevitably as dubious, brute and mysterious as reference to the natural number structure was in the first place. Little has been gained by invoking stronger logics:

The basis of Button & Walsh’s negative assessment of the prospects of a realist strategy making use of a logic stronger than FOL is the insistence on the need for a semantic articulation of such logics. In doing so one will have to invoke mathematical notions determinate reference to which is as much in need of explanation as is reference to the natural number structure itself. The original question thus simply re-emerges “one level up,” for one can ask in virtue of what determinate reference to these objects is achieved (Button and Walsh 2018: 164).

The claim that logics “strong enough to provide categorical theories of arithmetic must be articulated semantically” is based on the non-axiomatizability of these logics that results from their incompactness, coupled with the idea that, to have a determinate grasp of a notion, it must be “completely specified.” There is, however, a different and arguably more natural way to conceive of what it means to have a determinate grasp of a notion. What is required to meet the revenge challenge is the provision of a characterization of the relevant notions of the logic together with a demonstration that this characterization suffices to uniquely fix them, that it, in other words, implicitly defines them. If no alternative interpretation is possible with respect to the provided characterization, the mathematical realist must be taken to have established a mechanism by which determinate reference (and thus the legitimacy of these notions in a description of the natural number structure) is assured.10

Nothing in the idea of a categorical, i.e., determinate, characterization of a notion demands that the logic in which it occurs be completely specified; such completeness is not a necessary condition for its unique determination. Our understanding of a notion and the way it interacts with others might be partial and incomplete, but this might nevertheless suffice to uniquely determine it and thereby guarantee determinate reference. To meet the challenge of the sceptic that characterization has, of course, to be given in terms that are naturalistically acceptable and do not appeal to semantic notions with indeterminate reference. This is what we try to do in the next section.

Our moderate mathematical realist has been engaging in arithmetical reasoning and practice her entire life; first pre-theoretically by counting discrete objects in day-to-day encounters and later much more systematically through her work in a mathematics department. She has now been challenged by the model-theoretic skeptic to justify how her claims can be said to be about the intended structure N, how she can be certain that this was the object she has been talking about all along when doing arithmetic.

During her early studies of arithmetic, e.g. in the context of a high school mathematics class, she established that there is no largest prime number; that, for every prime number n, there is a prime number m that is larger. Asked how many primes there are she concluded that there must be infinitely many, for otherwise there would have to be a prime number n such that there is no larger. However, when trying to express the claim “There are infinitely many natural numbers that are prime” in the language of arithmetic she had been operating with, the language of PA, she finds that this simple-seeming claim can only be expressed in a very cumbersome and unintuitive manner. This state of affairs is rendered even more unsatisfactory by the apparent grammatical parallel between statements of the form “There are n-many objects that have property P” and “There are infinitely many objects that have property P,” the former of which possesses a formalization in FOL that faithfully mirrors its surface grammar, whereas the latter does not.

This is a somewhat unhappy situation, if only because it would be very nice to have a shorthand for saying that there are infinitely many natural numbers n that possess a particular property P, this claim immediately entailing, for example, that there can be no largest natural number m possessing P. It is furthermore a type of claim that our moderate mathematical realist encounters frequently in her arithmetical practice and uses regularly to communicate certain facts about collections of natural numbers without having to resort to cumbersome circumscriptions or paraphrases. To remedy this shortcoming she resolves to introduce an expression filling the linguistic gap discovered.

Note that she does not wish to talk about an object, infinity; she merely wishes to express that there are infinitely many natural numbers that possess a particular property, completely analogous to the claim that there are, say, 397 natural numbers that have a property P.13 She thus does not set out to detail and investigate the properties of the object infinity, on par with the way she characterizes the object 71 (is the 20th prime number; is a permutable prime; is a twin prime, etc.), but, rather, treats the expression “there are infinitely many” as part of the framework used to talk about natural numbers and state properties of these, on par with the grammatically similar expression “there are at least n-many.”14 In other words, she treats the expression as a quantifier of the logical language used to talk about natural numbers, not as something denoting an object in the range of those quantifiers.

Being part of the logical superstructure for talking about natural numbers she can lay down rules for this quantifier. These state under what conditions we are warranted to infer that there are infinitely many natural numbers that φ (e.g., when there is a natural number that φ’s and for every natural number that φ’s there is a larger one that also φ’s) and express what it means for there to be infinitely many natural numbers that φ, i.e., what we may infer from the claim that there are infinitely many natural numbers that φ (e.g., that there is no largest natural number n that φ’s).

Since the expression “there are infinitely many,” Q0, is regarded as belonging to the logical superstructure for the language of arithmetic, and not as part of the arithmetical lexicon itself, the moderate realist, in a first attempt, settles on a characterization of the central inferences involving the expression Q0 that are free from arithmetic-specific vocabulary and assumptions, i.e., (i) and (ii) below:

where Δφ={∃≥nxφ(x)|n∈N} and ∃≥n is the FOL-definable quantifier “there exist at least n.”15

Note that the moderate realist has, so far, said nothing about the model-theoretic meaning of Q0 and thus, by omission, avoided the challenge of the sceptic. However, for the proof of categoricity to go through she won’t be able to remain silent on this issue. Being pressed by the skeptic to explicitly state under what conditions a statement of the form Q0xφ(x) is true in a model M her reply that M⊨Q0xφ(x) iff M⊨ψ for all ψ∈Δφ is deemed insufficient and rejected on the basis of compositionality considerations. The sceptic demands that the truth conditions be stated explicitly in terms of (properties of) the extension of φ(x) and the moderate realist obliges by providing the following truth-clause for Q0:

Equivalently, she identifies the model-theoretic meaning of Q0 in a model M with domain M with Q0(M)={A⊆M|ω≤|A|} and invokes the standard satisfaction clause for generalized quantifiers.

At this point, the challenge for the moderate realist re-emerges, for the skeptic now asks what it is that ensures her grasp of Q0. How is it that she manages to refer determinately to it, rather than some deviant, unintended Q∗≠Q0? Even worse, if the characterization of Q0 requires determinate reference to ω, as the satisfaction clause suggests, then the categoricity result of the moderate realist begs the question: determinate reference to an ω-sequence has been secured by presupposing ω.

Here, however, the moderate realist has a successful reply: it is the acceptance of the inferences (i) and (ii) that ensures determinate reference to Q0, for Q0 is the unique value for Q0 rendering these inferences valid—there simply is no other, alternative value we could mean when reasoning in accordance with (i) and (ii). In other words, (i) and (ii) uniquely determine Q0—no prior recourse to ω is needed to achieve determinate reference to Q0 and the sceptical challenge thus fails to gain traction.

Let L(Q) be the language of FOL with an added quantifier-symbol Q. Let Q′ and Q∗ be (type-appropriate) interpretations for Q.16 We say that φ is true in a model M under the Q-interpretation, M⊨Qφ, if M⊨φ when Q is interpreted by Q. φ follows from Γ under the Q-interpretation, Γ⊨Qφ, if, for all M, whenever M⊨Qγ for all γ∈Γ, then M⊨Qφ. Then:17

Theorem 1 Suppose that

Then Q′=Q∗.

The theorem establishes that the two patterns of inference (i) and (ii) implicitly define Q. Any two quantifiers interpreting Q and obeying (i) and (ii) necessarily coincide—the inferences constrain the space of admissible interpretations so tightly, there is only a unique denotation that “fits.” Combining this with the fact that Q0 is consistent with (i) and (ii) (see Theorem 2 in the Appendix), this result demonstrates that our hold of Q0 is ensured by (i) and (ii). The inferential practice involving the quantifier “there are infinitely many” is sufficiently rich to ensure that our use of an expression for that quantifier does not admit any alternative, unintended interpretations. Determinate reference to Q0 is thereby guaranteed. Determinate reference to N can then be successfully achieved by means of the theory PA+=PA+{∀x¬Q0y(y<x)}.

Note that the logic of L(Q0), in its intended interpretation, is not recursively axiomatizable. However, inferring according to (i) and (ii) suffices to “fix” a determinate meaning for Q0. Thus, we can possess an implicit, yet full, understanding of Q0 so long as we infer in accordance with (i) and (ii), even in the absence of a complete and recursive axiomatization of the logic of the relevant notion, contra Button & Walsh’s contention (Button and Walsh 2018: 7.9).18 Recursive axiomatizability is thus not a good criterion for possessing a determinate grasp of a notion.19

The defense of a determinate grasp of Q0 on the basis of the inferential patterns (i) and (ii) might, nonetheless, ring hollow to the naturalistically minded realist. The reason for this lies in the infinitary nature of (i) and the accompanying need for naturalistically unacceptable inferential powers required by its utilization.20 Without providing a way of rendering (i) naturalistically digestible the realist must be taken to have transgressed the limits of moderation in the assumption of non-computational inferential capacities. Her defense of Q0 is thus still incomplete. In the next section, we try to fill the argumentative gap that remains to defend the claim that Q0 is, in fact, naturalistically acceptable.21

Based on an idea of Feferman (2015), it was argued in (Bonnay and Speitel 2021) that the fact that Q0 can be uniquely determined by a consequence relation including the inferences (i) and (ii) is sufficient to render it a logical notion. Here I do not wish to go as far, for what is at issue for the moderate realist is not the logical standing of Q0, but the question as to its naturalistic acceptability on the basis of the inferential patterns (i) and (ii). Nonetheless, on the current approach the quantifier “there are infinitely many” is considered part of the logical superstructure used to formulate a (referentially determinate) theory of arithmetic. Our grasp of the notions that form part of this superstructure is mediated through our inferential uses of them. The crucial question thus becomes: can the sort of inferential use required to fix the meaning of Q0 be made naturalistically acceptable in the contexts relevant for the moderate realist?

The central point of the above defense of using a stronger logic to categorically characterize N was that to possess a determinate grasp of the quantifier Q0 it is not necessary to have a prior grasp of, or access to, an ω-sequence given independently of the use of Q0. Rather, accepting certain inferences governing the “intuitive,” pre-theoretical behaviour of Q0 ensures that reference is thereby uniquely fixed. The referent thus determined must be the intended interpretation of Q0, because it is the only interpretation consistent with its inferential characterization. Note that the inferences laid down for the quantifier Q0 are not grounded in properties of an independently assumed object, such as ω, but rather concern the need for a logico-linguistic device to be able to state properties of certain progressions and collections of natural numbers.

However, even if the direct presupposition of ω could be avoided through recourse to inferential patterns (i) and (ii), this only shifts the challenge. To make the approach feasible for the moderate realist it must be established that (i) and (ii) constitute a naturalistically acceptable way of fixing the reference of Q0. And the infinitary nature of (i) raises serious doubts about this assumption.

For while the infinity of the set of inferences of type (ii) seems unproblematic—in every particular case the actual inference is finite and thus, in principle, performable—this is not the case for inferences of type (i). Here it appears in principle impossible for cognitively finite agents like us to perform the inference as such performance would require one to go through infinitely many premises.22 To maintain that an inference must be performable without assuming non-computational deductive abilities amounts, in the current context, to a compactness-demand: whatever can be inferred on the basis of a premise-set Γ should already be inferable on the basis of a finite premise-set Σ⊆Γ in order for the inference from Γ to its conclusion to be performable. Only such compact inferences ought to count as feasible and are thus naturalistically acceptable.

Apart from arguing that performing infinitary inferences simply is possible under certain circumstances the moderate realist could at this stage point out that it need not be recognized, or even be recognizable, that a particular inference is being performed in order to reason according to it. Determinate reference, she might claim, is achieved by inferring in accordance with inferential patterns of the form (i), i.e., by not violating the pattern in one’s reasoning, rather than by actually performing the inference. However, such a defense is bound to leave the case for the determinacy of arithmetic on the basis of adopting L(Q0) in a worse state than the moderate realist might hope. Moreover, a more robust strategy is available to her.

For she can point out that the relevant contexts for performing an inference of form (i) are, for her purposes, arithmetical contexts. In these contexts, she can legitimately avail herself of arithmetical vocabulary in stating inferential patterns involving Q0. In particular, she can replace the infinitary inference (i) by the, less general but finite, inference23

where LOφ(x)(R) is the sentence that says that R is a left-minimal, right-unbounded, strict linear order of the φ’s.24 (i) and (iii) are equivalent over models with appropriate signatures (see Theorem 3 in the Appendix), and the assumption of (ii) + (iii) thus also suffices to uniquely determine Q0. Hence, in the context of the language of arithmetic the moderate realist is able to determine the intended meaning of Q0 in a naturalistically acceptable fashion.25

A further revenge objection looms large, for in the compact characterization of Q0 via (ii) + (iii) extensive use was made of arithmetical concepts (e.g., linear orders). This, the model-theoretic sceptic complains, is illegitimate as it amounts to using notions and concepts in determinately fixing the meaning of Q0 whose very determinacy, ultimately, depends on Q0. We can here fruitfully distinguish between the positions of the naïve and the refined sceptic. The naïve sceptic holds that the concepts used in achieving determinate reference to N must be completely understood/understandable in all their consequences. This, I argued above, is unnecessary to possess a determinate grasp.

The refined sceptic, on the other hand, assumes that the notions we use to characterize the natural number structure must be fully determinate before we can use them towards this goal. This, I think, is equally mistaken. The determinacy of a notion, and the determinacy of theories making use of that notion, can develop in tandem, mutually precisifying and fixing each other through their interaction. We might have to use a moderate amount of first-order describable arithmetic to state essential properties of the quantifier “there are finitely/infinitely many” which we can then, in turn, use to characterize the (models of the) theory we applied to describe those properties. Each component by itself might inherit indeterminacy from the other. In conjunction, however, they mutually ensure determinate reference:26 “we needn’t first secure the determinacy of a concept before we use it in reasoning: if that were required, reasoning could never get started” (Field 2001: 342).

In using the theory of linear orders to describe the conditions under which we might infer a sentence involving Q0 we are leaving the particular kind of linear order we have in mind underdetermined. Despite this underdetermination, it suffices to fix the intended meaning of Q0 which we then use to specify the nature of the relevant linear order further. The objection thus, once again, relies on additional, unjustified assumptions regarding the process of fixing reference.

The question of the model-theoretic skeptic is “[w]hat are the limits of the semantic facts that our inferential practice might determine?” (Field 2001: 339). The answer provided, by invoking non-standard models, is that those limits fall short of determining all relevant facts concerning the notion of finitude. Field claims that this is due to—artificially and unnecessarily—restricting ourselves to the inferential practice of pure mathematics, thereby ruling out further resources we have at our disposal for fixing the content of concepts. Just as there are non-formal causal or phenomenal constraints that help us fix the reference of “planetary body” or “cat,” he maintains that we can avail ourselves of additional constraints on permitted models, thereby ruling out some unintended candidates.

In the case of mathematics Field advocates for broadening our conception of the inferential practice we take to contribute to fixing reference of mathematical expressions. Relevant additional constraints on permitted models could stem from physical applications of the mathematical apparatus.27 Given that observational or, more broadly, non-mathematical scientific vocabulary can be fixed by non-mathematical constraints, that unintended models for these notions can be ruled out on the basis of mechanisms that are not subject to a revenge-application of the “just-more-theory”-manoeuvre, we are thus supplied with additional means to restrict the extensions that mathematical vocabulary used in the formulation of theories may take on.

More concretely, the hope is that “if the physical world is as we typically think it is, our physical beliefs are enough to determine the extension of ‘finite’” (Field 1994: 416). The precise nature of the physical assumptions Field considers for fixing the extension of “finitely many” is unimportant for present purposes.28 The “key to the argument … is the assumption that the physical world provides an example of a physical ω-sequence that can be determinately singled out” (Field 2001: 341). Since access to this ω-sequence proceeds via physical vocabulary, which is not subject to the same referential indeterminacy affecting mathematical vocabulary, it constrains reference of our mathematical vocabulary in a non-objectionable way: “the constraints on the physical vocabulary determine a privileged class of interpretations in which ‘finite’ determinately stands for what it should” (Field 1994: 417). Expanding the inferential practice to be taken into account in fixing the notion of “finite” to include applications of mathematics to physical theory thus yields an inferential practice rich enough to rule out non-standard interpretations of the quantifier “there are finitely many.”

Our approach has much in common with Field’s, including the idea that knowledge of concepts such as “(in)finitely many” is mediated through our inferential use of them. It nonetheless differs in an important respect: Field leaves the inferential practice of pure mathematics and incorporates inferential practices of the (applied) sciences that use mathematics. The approach of §3, while potentially having to move beyond purely logical practices,29 remains squarely within the inferential practice of mathematics itself, with no recourse to physical or observational notions of any kind.

This makes a difference: while Field’s approach requires a prior grasp of a particular and special physical object—an instantiated ω-sequence on the basis of which we are able to obtain a determinate notion of “finite set” by means of which we can then appropriately state the semantics of the quantifier “there are finitely many”—the approach pursued above deems the existence of a (partial) characterization of the inferential behaviour of the quantifier “there are infinitely many” sufficient to determine its meaning.30

Thus, on Field’s account, to achieve determinacy of reference for arithmetical notions we have to accept several strong empirical assumptions concerning the infinitude of space or time.31 It is this dependence on empirical fact—Field’s cosmological assumption concerning the structure of the world (Field 2001: 340)—that has been most frequently criticized:

“[U]sing the physical world to explain mathematical determinacy” risks getting “the intuitive conceptual priority … backwards” (Warren and Waxman 2020: 490), thereby leaving the determinacy of the notion “finitely many” in an unstable state.32 The account proposed in §3 offers an alternative to fixing the meaning of the expression “there are infinitely many” that introduces new inferential patterns to take into account, but does not rely on a prior physical basis for its determinacy.33

The by far most popular response to the non-categoricity of PA consists in a move to a second-order framework and to consider PA₂ instead. Dedekind’s Categoricity Theorem then ensures the categoricity of arithmetic. However, just as popular as this line of response to the underdetermination of arithmetic are criticisms of approaches utilizing second-order quantification to guarantee determinacy (see (Weston 1976), (Field 2001), (Shapiro 2012), (Meadows 2013)). The main objection to the use of SOL in establishing the categoricity of PA₂ relates to the assumption of a full semantics for second-order logic. For here the model-theoretic skeptic asks in virtue of what the proponent of SOL can claim to be using a full rather than a Henkin-semantics (see §2.3).

The success of the project of using SOL to achieve determinate reference to the natural number structure thus stands and falls with the ability to ensure that the semantics for the second-order quantifiers is full. And the crucial question here is: in virtue of what is it that the proponent of SOL can reasonably claim that her second-order quantifiers range over the full power-set of the domain, rather than an inferentially indistinguishable subset thereof.34

Murzi and Topey (2021) defend the claim that the proponent of SOL is indeed using a full, rather than a Henkin-semantics. Their core contention is that it is not the concrete rules and axioms of any particular formal theory that fix the meaning of the expressions occurring in them, but that it is the dispositions to infer underlying those rules which ground their determinate semantics.

On their account, the dispositions to infer in accordance with a particular set of rules are not exhausted by concrete rules for particular languages or language-fragments. Rather, the particular rules are accepted because they are manifestations of these dispositions when considering a concrete language. The fact that rules are grounded in dispositions to infer means that we are “disposed to accept all instances of those rules irrespective of how we expand our language” (Murzi and Topey 2021: 3). This, in turn, means that it is best to understand rules not as particular collections of their instances with respect to a concrete vocabulary but, rather, as what Humberstone (2011) calls language-transcendent, i.e., as functions from languages to collections of instances over these languages, or, following McGee (1997; 2000; 2006), as open-ended, i.e., as continuing to hold unrestrictedly under expansions of the language. It is the open-endedness of rules, grounded in language-transcendent inferential dispositions, that fixes the full semantics of the second order quantifiers: “the open-endedness of our inferential dispositions … suffices to yield full second-order logic” (Murzi and Topey 2021: 1).

Dispositions to infer remain undisturbed by the language in which the concrete inferences take place. The robustness under expansions or switchings of the language constitutes the open-endedness of rules of inference manifesting these dispositions. A rule is open-ended if it continues to hold, if it remains valid, under the addition of new vocabulary to the language. To account for the extreme generality of dispositions to infer the semantics of the expressions occurring (non-schematically) in the rules of inference must be able to guarantee the continued validity of these rules no matter the precise language in which they are instantiated. This affects, in particular, the rules for the second-order quantifiers. Assuming that every object in the domain of discourse of a model is in-principle nameable forces the range of the quantifiers to be maximal, thereby ensuring a full semantics.

Let us reconstruct the argument in a bit more detail: Because our dispositions to infer are unrestricted, rules of inference must be open-ended – they must continue to apply irrespective of any particular language. Assuming that “no item is in principle unnameable in any language that expands our own, guarantees that the logical rules hold universally: for any item whatsoever, there exists an expansion of our language in which there’s a term naming that item, and the logical rules hold in all such expansions” (Murzi and Topey 2021: 19).35 The open-endedness of the rules, combined with the in-principle nameability of every object in the domain, guarantees that the interpretations of the quantifiers (first- and second-order) must be permutation-invariant. Generalizing the result by (Bonnay and Westerståhl 2016) that permutation-invariance forces the first-order quantifiers to take on their standard interpretation to the second-order case,36 Murzi & Topey then establish that the only semantics consistent with the open-ended understanding of the rules for the second-order quantifiers is a full semantics.37 Reasoning in accordance with our dispositions to infer thereby secures a full semantics for second-order logic and rules out Henkin-semantics as viable alternative—the indeterminacy of second-order semantics is thus removed.38

Murzi & Topey propose an attractive strategy to secure a full semantics for the second-order quantifiers and, thereby, determinate reference to the natural number structure. They provide a diagnosis of the indeterminacy of the second-order quantifiers that is continuous with a phenomenon already found in the first-order case and propose a unified mechanism for fixing the semantics for the expressions of FOL and SOL.

Both their and our diagnosis of the mistake in the sceptic’s argument maintains that the sceptic fails to appropriately capture what is necessary to possess a determinate grasp of a notion (cf. (Murzi and Topey 2021: 26)). Similarly, they say that “speakers (and theorists) need not grasp the complex mathematical concepts involved in the standard semantics for SOL in order for it to be guaranteed that SOL is correctly interpreted by that semantics” (Murzi and Topey 2021: 26). It is sufficient to infer in accordance with the inferences characteristic of a notion, or to exhibit the right kinds of dispositions to infer, to ensure that the notion in question possesses its intended meaning.

Nonetheless, one might wonder how innocent the notion of open-endedness, which does much of the work in Murzi & Topey’s argument, really is. In particular, is the invocation of this notion tantamount to presupposing an understanding of full second-order quantification? If so, little is gained by its use in defending a full semantics for SOL. But even if not, there is the worry that it invites a “just more theory”-type revenge objection, in that by talking about possible extensions of a language we are (implicitly) quantifying over all possible languages, where this kind of quantification is as much in need of explanation as second-order quantification itself. I don’t believe this objection has as much force as some of its proponents have taken it to possess. The crucial idea, when it comes to open-endedness, is to reason in accordance with a rule come what may. This “come what may”-part, however, appears to place far fewer demands on our grasp of the notion of open-endedness than does quantification over all possible languages. What we need to know and understand is the general shape a language can take, but this falls short of requiring a grasp of the totality of all possible languages.39

There is, however, a further worry concerning the need for open-ended logical principles over and above open-ended mathematical principles. This line of criticism can be interpreted as questioning the need for a move to a second-order framework, plagued by all the usual worries regarding its ontological commitments, logical status, etc., given that its advantages,—determinate mathematical reference,—can already be had in a first-order setting. The central insight here is that the first-order induction-schema, properly understood as open-ended, already suffices to achieve categoricity without requiring the additional use of second-order quantifiers. For, properly understood, the schema says that induction holds for all properties on the natural numbers. Its limitations stem from the restrictive interpretation of “property” in the usual setting of first-order PA and an understanding of the schema as a meta-language abbreviation for an infinite set of object-language sentences. This, however, arguably misrepresents its intended scope and meaning. Thus, the unrestricted nature of induction might be better captured by directly using schematic variables in the object language.40 The use of such object-language schematic expressions, the argument goes, best expresses the generality and open-endedness of induction without incurring the ontological commitments that go along with explicitly quantifying over these variables.41

The schematic interpretation of the induction schema, as a way of expressing its unrestrictedness explicitly in the object-language itself, is sufficient to establish the categoricity of arithmetic in the free-variable fragment of second-order logic, i.e., SOL without second-order quantifiers but permitting the occurrence of second-order variables.42 Not only is the assumption of second-order quantifiers ranging over the full powerset of the domain therefore not needed to achieve categoricity, many of the (perceived) disadvantages of the use of full second-order logic in the characterization of the natural number structure can thus also seemingly be avoided: a proper implementation of open-endedness in the form of the schematic interpretation of the induction schema yields categoricity “on the cheap.”

Leaving the ontological costs of schematic induction as against second-order quantified induction undecided, one advantage that has been attributed to the free-variable fragment of second-order logic is that it steers clear of issues pertaining to the non-robust nature of the power-set operation that features so prominently in the full semantics of the second-order quantifiers. If, as has often been claimed, we are lacking a clear understanding of the notion “all subsets of” when it comes to infinite domains, i.e., of the power-set operation, this lack of clarity immediately affects our understanding of full second-order quantification itself. Note that the open-endedness defense advanced by Murzi & Topey is of no help here, as we can grant that whatever interpretation our second-order quantifiers must have if our dispositions to infer are unrestricted must be full. Yet, since it is indeterminate what it means for an interpretation to be full, due to the non-robust nature of the set-theoretic operation used to spell out the notion of fullness, the fact that the appropriate semantics for the second-order quantifiers must be a full semantics does little to remove the indeterminacy inherited through this “background indeterminacy” (see (Lavine 1994: 237/238)).

None of this seems to affect the understanding of the schematic version of induction:43

It is not difficult to see that the free-variable fragment of second-order logic is equivalent to only allowing initial universal quantifiers over second-order variables (see (Shapiro 1991: 246–250) and (Corcoran 1980)). For this reason it has been described as a mere notational variant of a fragment of SOL, affected by the same sorts of problems pertaining to the semantics, ontological commitment, and indeterminacy of SOL.44 Whatever the precise relation between SOL and its free-variable fragment, however, their difference prompts some questions for the way Murzi & Topey wish to secure the determinacy of arithmetical reference.

The question is why we should want to invoke second-order quantification at all. Given the success of the free-variable fragment, why not agree with the dispositional story concerning the open-endedness of genuine arithmetical claims (i.e., induction), but stop there? What need is there to invoke further logical notions when the open-endedness of the mathematical claims of the theory under consideration suffice to determinately fix reference? Why, in other words, assume more than is needed for the goal at hand? This point is not so much an indication of a shortcoming of the position, but a request for a justification of the additional logical machinery employed. It is a significant request, because the invocation of that machinery under its preferred interpretation brings with it an interesting further type of indeterminacy, seemingly absent in the free-variable fragment: the non-robustness of the power-set operation needed to spell out the intended semantics. Let me elaborate further on this point.

While the indeterminacy of the background theory of models, engendered by the possibility of non-standard models of set-theory, affects all model-theoretic responses to the skeptical challenge, the indeterminacy “cuts deeper” in the case of second-order logic. The existence of Löwenheim-Skolem models of the background set-theory affects the determinacy of the entire model-theoretic apparatus that provides the space of meanings for the notions under consideration. But there are further types of indeterminacy affecting specifically second-order quantification. I will address the former type of indeterminacy first.

In any first-order framework Löwenheim-Skolem models are as unsatisfactory as they are unavoidable. They create an inescapable level of indeterminacy for any defensible realist position as conceived in the context of this paper.45 Consider, for example, the second-order quantifiers. Provided they are understood in terms of a set-theoretic semantics and we somehow make sure that a correct interpretation of them is a full semantics this still does not completely remove any possible indeterminacy. This is because what it means for an interpretation to be full might differ between different models of the background set-theory used to spell out the notion of a full interpretation. The existence of non-standard models of first-order ZFC, however, is guaranteed by the Löwenheim-Skolem Theorems. As a consequence, the notions needed to spell out the content of “full interpretation” and, as a result, that notion itself, remain indeterminate.46 All Dedekind’s Categoricity Theorem is able to provide is thus an assurance that once we have fixed an interpretation of the background set-theory, and thereby settled the meaning of, for example, “all subsets,” all models of the target-theory will be isomorphic. Analogous considerations apply to the notion “infinitely many.”

Assuming the standard model of set-theory outright is certainly much less warranted than assuming the standard model of arithmetic. However, making some moderate assumptions about the shape of acceptable models of the background set-theory helps reduce the unconstrained indeterminacy caused by Löwenheim-Skolem type phenomena significantly. These assumptions amount to supposing that the models of set-theory we are dealing with are transitive models.47 Call these acceptable models of the background theory.48 What we remain ignorant about when dealing with an acceptable model concerns the true extent of the set-theoretic universe.

A notion that is insensitive to the extent of the set-theoretic universe, i.e., that is such that it does not change its meaning depending on the actual extent of the universe of sets, is an absolute notion (see, e.g., (Jech 2003: 163)). A non-absolute notion is one whose meaning depends on the extent of the underlying universe of sets in a particularly crucial way: it changes its meaning over fragments of the set-theoretic universe depending on how those fragments are expanded or restricted—its meaning is intrinsically unstable. Here we are specifically concerned with notions absolute among acceptable models of ZFC.

While a property such as being finite is absolute (among transitive models of ZFC) (see, e.g., (Kunen 2011: 123)), several set-theoretic notions crucial to the full semantics of SOL, such as, e.g., the power-set operation, are not. SOL’s demands on the background universe of sets are thus much more substantial.49 This has consequences for the set-theoretic robustness of second-order consequence more generally.50 Note that the instability of non-absolute notions is a very different sort of indeterminacy than the one that arises due to the existence of non-standard models of the background set-theory which, in some sense, cuts deeper. The latter is a result of the inability to rule out deviant models of the background set-theory due to expressive limitations of first-order languages. The former shows that even if the notions “is a set” and “∈” are not thoroughly mis-interpreted, as long as their interpretation is left partial, indeterminacy can arise. Thus, even among acceptable models of the background set-theory non-absolute notions are affected by indeterminacy.

The finite/infinite distinction turns out to be set-theoretically robust in this sense: the notion of finitude is absolute (among transitive models of ZFC). This means it is, to a certain degree, independent of the real, underdetermined set-theoretic universe in that transitive models of set-theory will not disagree on which sets to count as finite. No matter what transitive model of set-theory we are operating in, then, i.e., no matter the actual extent of the set-theoretic universe, the notion “finitely many” will not admit multiple, divergent interpretations in them. What is underdetermined is what is the actual extent of the universe of sets, but not the meaning of “finitely many” in it, whatever it may be. The intended semantics for second-order logic, on the other hand, is deeply affected by this additional, intermediary type of indeterminacy and it is unclear how to rule it out short of simply postulating a primitive understanding of what it means to be a full semantics.

This demonstrates that the indeterminacy of the notions of SOL is very different from the indeterminacy of the quantifier Q0: In the case of the notions at issue in L(Q0) indeterminacy enters through first-order weaknesses of the background theory. The crucial notions of SOL, however, are not only vulnerable to an object-theory level indeterminacy which manifests itself through the possibility of Henkin-models,51 but, additionally, to an “intermediate” indeterminacy which stems from variability in the meaning of crucial notions, depending on the extent of the underlying set-theoretic universe. The meaning of the second-order quantifiers ∀X and ∃X is thus inherently more unstable, and therefore more indeterminate, than that of the quantifier Q0.

Within any particular model of the background set-theory a categoricity theorem ensures that the notions under consideration possess a unique interpretation. Here, the account of §3, as well as the alternatives considered above, succeed in fixing notions required for determinate reference to the natural numbers. Nonetheless, I believe the fact that second-order notions are affected by an additional type of indeterminacy provides substantial reason to prefer a first-order response to the skeptical challenge.