Is Frege a Neo-Kantian Transcendentalist About the Justification for Logical Axioms?

Abstract

The neo-Kantian transcendentalist reading of the epistemic status of logical axioms in Frege argues that he is committed to the neo-Kantian idea that we are epistemically justified in accepting logical axioms because accepting them is necessary for achieving epistemically crucial goals. However, I show that Frege hesitates to be fully committed to neo-Kantian transcendentalism because he struggles to accept the idea that such a teleological reason can constitute an epistemic warrant. This interpretation shows some crucial aspects of his philosophy of logic, such as his understanding of the relationship between the simplicity and the sufficiency of logical systems.

Résumé

La lecture transcendantaliste néo-kantienne du statut épistémique des axiomes logiques chez Frege soutient que ce dernier est attaché à l'idée néo-kantienne selon laquelle nous sommes épistémiquement justifiés d'accepter les axiomes logiques parce que leur acceptation est nécessaire pour atteindre des objectifs épistémiquement cruciaux. Cependant, je montre que Frege hésite à s'engager pleinement dans le transcendantalisme néo-kantien parce qu'il a du mal à accepter l'idée qu'une telle raison téléologique puisse constituer une justification épistémique. Cette interprétation met en évidence certains aspects cruciaux de sa philosophie de la logique, tels que sa compréhension de la relation entre la simplicité et la suffisance des systèmes logiques.

1. The Neo-Kantian Transcendentalist Reading of Frege

This article concerns Gottlob Frege's understanding of the epistemic status of logical laws, which has been a controversial issue among scholars (e.g., Burge, 1992, 1998; Dummett, 1973; Gabriel, 2021; Heck, 2007, 2012; Hutchinson, 2021; Kitcher, 1979; Lockhart, 2016; Pedriali, 2019; Picardi, 2010; Sluga, 1980; Stanley, 1996; Yates, 2021). One of the tasks related to this topic is to figure out Frege's stance on the epistemic status of logical axioms — whether, and how, they can be justified. This article focuses on an interpretation, which I will call the “neo-Kantian transcendentalist (NKT) reading.” The NKT reading ascribes to Frege the teleology-based transcendentalism held by neo-Kantians such as Wilhelm Windelband and Rudolf Hermann Lotze — the claim that accepting propositions as true is justified if accepting those propositions is indispensable for a significant epistemic telos or goal (Gabriel, 2002, 2021; Hutchinson, 2021). This article examines the major arguments for this reading. If my discussion is correct, Frege's attitude toward the neo-Kantian epistemology of logic is more complex and reserved than the NKT reading suggests. Specifically, he accepts that the teleological indispensability suggested by neo-Kantians can be a good reason for taking something to be true while hesitating to accept that it can also be an epistemic warrant.

The common ground I share with proponents of the neo-Kantian reading is that, for Frege, justifying his logical axioms and rules of inference is a crucial task. His lifetime project, logicism, aims to show that the axioms of arithmetic are truths of logic. Frege's strategy to demonstrate this is to prove arithmetical axioms by his logic Begriffsschrift. Specifically, he constructs gapless chains of inferences from the basic laws of Begriffsschrift — i.e., its axioms — to the axioms of arithmetic. This derivation of arithmetical axioms alone is not sufficient for establishing that arithmetic is logic further developed. Frege must establish that the basic laws of Begriffsschrift are logical laws. He has to provide justifications for the basic laws of Begriffsschrift to show that those laws deserve to be regarded as logical axioms. This view of Frege's epistemology of logic and his logicist project is controversial.¹ However, that is at least not an issue between me and the neo-Kantian reading.

The historical and philosophical connections between Frege and neo-Kantianism have been discussed by scholars such as Gottfried Gabriel (1984, 2002, 2013a, 2013b, 2021) and Hans Sluga (1980).² Following them, I assume that Frege knows neo-Kantianism well enough to evaluate its claims. The specific neo-Kantian epistemology of logic, which the NKT reading (Gabriel, 2002, 2021; Hutchinson, 2021) ascribes to Frege, is Lotze's and Windelband's transcendentalism. Windelband, who takes himself to follow Lotze's idea, argues that in order to justify logical axioms, we ought to “show that their validity must be recognized if certain [epistemological] purposes are to be accomplished” (Windelband, 1907, p. 328). Following Windelband, we can put the main claim of neo-Kantian transcendentalism schematically like the following:

(LWT) If accepting a proposition p is indispensable for pursuing an epistemically valuable goal, accepting p as true is justified.

The NKT reading argues that Frege justifies the axioms of Begriffsschrift by appealing to the fact that accepting them is necessary for pursuing an epistemic value that he takes to be significant. Jim Hutchinson and Gabriel provide different interpretations about which epistemic value Frege appeals to in order to justify Begriffsschrift's axioms. Hutchinson (2021, pp. 529–531) argues that the relevant goal is constructing the simplest logical system. Gabriel (2002, p. 47, 2021, pp. 9–11) holds that the relevant goal is truth itself.

Section 2 examines Hutchinson's version. As he argues, building the simplest logical system is a crucial epistemic goal for Frege. Hutchinson's reading, however, has interpretative issues. In a nutshell, Frege has a reason to deny that an axiom's being necessary for the simplest logical system justifies the axiom. Section 3 deals with Gabriel's version of the NKT reading. Frege does believe that logical axioms, such as the law of identity, are indispensable for pursuing truth or judging, and further, that the impossibility of rejecting those axioms is a good teleological reason for accepting them. As we will see, he clearly regards that impossibility as his reason to accept logical axioms. However, Frege hesitates to accept that the impossibility constitutes an epistemic warrant for accepting them. If my interpretation is along the correct lines, Frege is not committed to neo-Kantian transcendentalism about the justification for logical axioms. At the same time, Frege's discussion about neo-Kantian transcendentalism shows that he has a delicate position toward it. Section 4 reveals this sophisticated relationship between Frege and neo-Kantians.

2. The Basic Laws and the Simplest Logical System

2.1 Hutchinson's NKT Reading via the Simplest Logical System

Hutchinson develops his interpretation through a question about Frege's elucidations of the basic laws of Begriffsschrift in Grundgesetze.³ Here is Frege's elucidation of the law I:

According to §12,

would be the False only if Γ and Δ were the True while Γ was not the True. This is impossible; accordingly

(Frege, 2013, Vol. I §18)⁴

The Begriffsschrift expression

can be compared to a schema of the conditional, i.e., something like “ζ → ξ.”⁵ (Hereafter, let us stick to “→” instead of Begriffsschrift's conditional sign.)

Frege seems to be (informally) proving that the law I is true by appealing to the meaning of the conditional sign he stipulates earlier (Frege, 2013, Vol. I §12). The stipulation amounts to the following:

(mc) “ζ → ξ” refers to the False if “ζ” refers to the True and “ξ” does not refer to the True, and it refers to the True otherwise.⁶

From (mc), it follows that

(ic) “Γ→(Δ→Γ)” refers to the False only if “Γ” refers to the True and does not refer to the True.⁷

Because “Γ” and “Δ” are arbitrary referential names, it follows that

(I) For any “a” and “b,” “a→(b→a)” refers to the True.⁸

Frege provides similar arguments for the basic laws IIa, IIb, III, and IV (and inferential rules).⁹

These proof-like arguments do not align well with Frege's characterization of logical axioms. He writes:

It cannot be required that everything be proven, as this is impossible; but it can be demanded that all propositions appealed to without proof are explicitly declared as such, so that it can be clearly recognized on what the whole structure rests. (Frege, 2013, p. VI)

The basic laws of Begriffsschrift are to be those that lack proofs. “Why does [Frege] offer these arguments” (Hutchinson, 2021, p. 517) that resemble proofs?

What Frege refers to by “proof” is deductive justification. That is why, as Hutchinson correctly points out, a proof should start from thoughts that “are already accepted as true” for Frege (1979, p. 204). I will clarify Frege's idea further in Section 2.2.2. For now, I will take a proof to be an attempt to provide an epistemic warrant for its conclusion by deductive inference. Now, Hutchinson holds that Frege's arguments for the basic laws are not proofs in this sense because the premises of his arguments are not justified. Hutchinson believes that Frege is appealing to the “critical method”:

The basic idea behind the Lotze-Windelband critical method for the justification of axioms is clear enough: if, starting from unjustified presuppositions about a (epistemic) goal, we can establish an inferential connection between an axiom (or our acceptance of it) and the achievability of the goal, that justifies us in accepting the axiom. Moreover, they ascribe to axioms justified in this way the traditional status of “self-evident” and “immediately certain.” (Hutchinson, 2021, p. 524)

Now, Hutchinson needs to answer two questions. First, what is the epistemic goal Frege is pursuing? Second, how is the achievability of this goal inferentially connected to the basic laws?

According to Hutchinson, the epistemic goal Frege is pursuing is to construct the simplest logical system. Frege considers the simplest system a crucial epistemic goal. The simplicity of a logical system is determined by the number of primitive truths — and the inferential rules — the system recognizes as such.¹⁰ In Grundlagen, Frege writes:

The further we pursue these [logical] enquiries, the fewer become the primitive truths to which we reduce everything; and this simplification is in itself a goal worth pursuing. (Frege, 1950, §2)

Also in Grundgesetze, Frege remarks:

It can be demanded that all propositions appealed to without proof are explicitly declared as such […]. One must strive to reduce the number of these fundamental laws as far as possible by proving everything that is provable. (Frege, 2013, p. VI)

If Frege endorses the neo-Kantian method — i.e., attempts to justify the basic laws by appealing to an epistemic goal — it is unsurprising for him to select the simplest logical system as such a goal.¹¹

Why are the basic laws necessary for achieving the simplest logical system? Recall that the premise of Frege's argument for the law I is the meaning stipulation for the conditional sign — (mc) in our formulation. Hutchinson's first claim is that Frege considers accepting (mc), or having a sign whose meaning is stipulated in such a way, to be necessary for achieving the simplest logical system. In “Boole's logical calculus and Begriffsschrift,” Frege writes:

The more primitive signs you introduce, the more axioms you need. But it is a basic principle of science to reduce the number of axioms to the fewest possible. (Frege, 1979, p. 36)

Then, he compares the conditional sign of Begriffsschrift with the identity sign of Boole's system and points out that Begriffsschrift's conditional can perform not only the jobs that can be done by Boole's identity sign but also other necessary jobs that cannot be done by Boole's identity sign (Frege, 1979, pp. 36–37). Thus, Begriffsschrift's conditional sign helps to achieve simplicity.

Recall that the meaning of the conditional sign of Begriffsschrift is stipulated in terms of the truth-values True and False. For Frege, truth-values are objects that serve as sentential references.

The content I called judgeable content. This now splits for me into what I call thought and what I call truth-value. This is a consequence of the distinction between the sense and the reference of a sign. In this instance, the thought is the sense of a sentence and the truth-value is its reference. (Frege, 2013, p. X)

Thus, Frege's stipulation for the sign “→” makes it an expression that refers to a function whose value for any given arguments — not only truth-values but also other objects — is always a truth-value. Why does he provide such a stipulation? As Hutchinson (2021, pp. 529–530) points out, Frege writes:

Only a thorough engagement with the present work can teach how much simpler and more precise everything is made by the introduction of the truth-values. (Frege, 2013, p. X)

Hutchinson concludes that Frege stipulates the meaning of the conditional sign in terms of truth-values in order to simplify his logical system, i.e., to reduce the number of primitive truths.¹²

According to Hutchinson, Frege believes that Begriffsschrift's signs and their references — i.e., objects and functions — “better achieve the cognitive goal of having a logical system that is simple in the relevant sense” (Hutchinson, 2021, p. 530). Further, Hutchinson argues that “because Frege's ultimate characterization of logical laws is in terms of their normative role rather than what they are about, he can consider what it takes to have a maximally simple system of laws” (Hutchinson, 2021, p. 530).¹³ In a nutshell, what Hutchinson suggests seems to be this: Frege has a reliable map for the simplest system, and so he can tell that such a system needs signs similar to Begriffsschrift's logical symbols.

It is not clear that what we have seen is enough to support that interpretation; all we can confidently say is that Frege takes his logical symbols to make his system simpler than others.¹⁴ However, let us just ask how such a thought leads to the neo-Kantian justification for the law I. Assume that accepting (mc) is necessary for the simplest logical system. Hutchinson's point seems to be that Frege's argument aims to show that one cannot accept (mc) without accepting the law I because (mc) entails the law I. If our assumption is true, the Lotze-Windelband transcendentalist method implies the following:

(LWT-I) If accepting the basic law I is indispensable for constructing the simplest logical system, accepting the law I as true is justified.¹⁵

If, as Hutchinson argues, Frege is committed to Lotze's and Windelband's methods, he has a reason to view our acceptance of the law I as justified.

2.2 Critical Examination

2.2.1 The Nature of Frege's Arguments for the Basic Laws: Are They Still Proofs?

Again, while Frege takes logical axioms to be incapable of being proven, his arguments for the basic laws (and inferential rules) in Grundgesetze still look like (informal) proofs for them. Hutchinson's NKT reading is an attempt to resolve this potential tension. However, according to the reading, the justification bestowed upon the basic laws consists in their indispensability for achieving the simplest logical system. But the law I enjoys such a justification because it is entailed by (mc) and it is (mc) that is necessary for achieving the goal.¹⁶ Then, Frege's argument for the law I still looks like a proof.

To deal with this issue, Hutchinson introduces the notion of (epistemically) transmuting argument. An argument is transmuting if “it justifies us in accepting its conclusion partly through our accepting claims — such as its premises — that we are not yet justified in accepting” (Hutchinson, 2021, p. 520). Hutchinson claims that Frege's argument for the law I is such a transmuting argument; namely, the argument justifies our acceptance of its conclusion — i.e., the law I — though we are not justified in accepting its premise — i.e., (mc). A transmuting argument is not a proof in the relevant sense, as a proof is supposed to be a deductive justification of a judgement through antecedently justified judgements.

The term “transmuting argument” is not used by Lotze or Windelband. Neither do they explicitly construct the notion of transmuting argument as Hutchinson does. However, Hutchinson maintains that the notion of transmuting argument underlies their critical method. The main idea of the critical method is explicitly formulated by Windelband, who argues that one can justify axioms only by “show[ing] […] that their validity ought to be recognized if certain aims are to be accomplished” (Windelband, 1907, p. 328). Taking g to be a valuable cognitive aim, an axiom A can only be justified by showing:

(AG) Accepting A is necessary for achieving g.¹⁷

Hutchinson comments:

On this view, the only way to justify an axiom is to show that there is a connection between that axiom and a cognitive goal. But the reasoning that shows the connection must proceed from unjustified claims, because the axioms themselves are the “absolute beginning” of justification: whatever claims about goals lie behind them must be unjustified. (Hutchinson, 2021, p. 523)

He argues that “the reasoning that shows […] a connection between [A] and [g],” i.e., the argument to (AG), must have unjustified premises. This is, Hutchinson claims, because if the premises of the argument to (AG) are justified, A cannot be the beginning of proof. According to him, Windelband is committed to this idea:

Accordingly, Windelband never describes the statements about the goal as justified, certain, or valid; they are only “presuppositions.” He refers to the kind of reasoning that connects axioms with goals as “argument” that “shows” something, but never “proof” that “proves” anything, claiming that “it belongs to the concept of an axiom […] to be unprovable.” (Hutchinson, 2021, p. 523)

Hutchinson concludes that Frege's argument for the law I is the argument that accepting the law I is necessary for achieving the simplest logical system. He writes:

Frege thinks his arguments provide such justification [for the basic laws] even though we are not justified in accepting their premises, because of the way they connect the truth of the axioms with a goal: having a simple logical system. (Hutchinson, 2021, p. 531)

First, it is unclear how the idea of transmuting argument can be combined with neo-Kantian transcendentalism. We need to distinguish the reasoning to (AG) from the reasoning from (AG) to A. Hutchinson applies the notion of transmuting argument to the former; he argues that the reasoning to (AG) — “the reasoning that shows a connection between [A] and [g]” — should have “unjustified premises” (Hutchinson, 2021, p. 523). However, neo-Kantians argue that “showing” (Windelband, 1907, p. 328) that (AG) is true is necessary for justifying the logical axiom A. Drawing (AG) from unjustified assumptions is not equivalent to showing its truth in any substantive sense. Furthermore, Hutchinson argues that the reasoning to (AG) should have only unjustified premises because otherwise A would not be the beginning of proof. However, even if the reasoning to (AG) has justified premises, A can still be the beginning of proof if (AG) does not prove A (though it somehow justifies A). If Hutchinson is committed to the claim that the reasoning from (AG) to A is a proof, then he should also apply the notion of transmuting argument to that reasoning. In other words, he should take (AG) to be unjustified because otherwise A would not be the beginning of proof. However, again, the key claim of neo-Kantian transcendentalism is that showing that (AG) is true is necessary for justifying A. Putting forward (AG) as a merely unjustified assumption is not showing that it is true at all. The notion of transmuting argument does not sit well with neo-Kantian transcendentalism.

In fact, understanding neo-Kantians’ comments does not necessitate the concept of transmuting argument. Above all, (AG) does not prove A; it is not a valid principle that if accepting A is necessary for achieving g, A is true.¹⁸ It might be the case that although accepting A is provisionally necessary for starting to pursue g, it may have to be rejected eventually. Or even if attaining g entails the truth of A, we might still lack the capacity to realize g. So, regarding A as the beginning of proof is compatible with affirming that we are justified in accepting A given our justified acceptance of (AG) or of the premises that lead to it. As neo-Kantians say, the argument from (AG) to A is an argument, not a proof. If A is not proven, A can be the beginning of proof:

Because our proving cannot go back infinitely, it must have an absolute beginning, and this must be sought in those which cannot be proved. (Windelband, 1907, p. 322; italics mine)

Thus, A can still be an axiom of a proper logical system even if (AG) is justified.

Let us set aside this issue with Hutchinson's reading of neo-Kantianism. Hutchinson (2021, p. 524) accepts that the lynchpin of the neo-Kantian transcendentalist argument for a logical axiom, or the critical method, is to show that there is a teleological “connection between that axiom and a cognitive goal.” If Frege's argument is a neo-Kantian argument, what we should expect it to establish is that

(F₁) Accepting the law I is necessary for achieving the simplest logical system.

However, Frege's argument is a mere deduction of the law I from the meaning stipulation for “→,” i.e., (mc). What we can establish by this argument is at most that

(F₂) Accepting the law I is necessary for accepting (mc).

Hutchinson applies the idea of transmuting argument directly to Frege's argument for the law I — arguing that the premise of his argument for the law I, i.e., (mc) in our formulation, is unjustified while the law I is justified. Hutchinson seems to think that Frege's argument alone constitutes a neo-Kantian argument. This is a mistake. Frege's argument alone cannot establish (F₁), which must be established by the neo-Kantian argument for the law I. One might say that Frege already accepts that

(F₃) Accepting (mc) is necessary for achieving the simplest logical system.

Surely, if Frege already provides a reason to accept (F₃), his argument that (mc) entails the law I is sufficient to establish (F₁). However, it is still not the case that the argument that (mc) entails the law I alone constitutes the neo-Kantian argument for the law I. If Frege were a neo-Kantian, he would not take his argument to justify the law I single-handedly.

However, let us focus on Hutchinson's claim that Frege's argument from (mc) to the law I is epistemically transmuting. Hutchinson's claim entails that Frege takes (mc) to be unjustified. However, (mc) is Frege's stipulation about the conditional sign's reference, which seems trivially justified. Hutchinson argues that “reflections on how certain objects and concepts would allow for a simpler logical system do not provide evidence in any ordinary way that there really are such objects and functions, or that the claims with which they are introduced are really true” (Hutchinson, 2021, p. 530). Perhaps his point is that we cannot stipulate the existence of truth-values or concepts merely out of their necessities. Fair enough. Still, it is not the case that Frege merely stipulates their existence due to their necessities. He makes philosophical arguments that truth-values are objects in his works such as “On sense and reference” (Frege, 1970, pp. 56–78), “Comments on sense and reference” (Frege, 1979, pp. 118–125), “Introduction to logic” (Frege, 1979, pp. 185–196), and “Logic in mathematics” (Frege, 1979, pp. 203–250). Frege also explains the philosophical points of his object-concept distinction in several places such as “Function and concept” (Frege, 1970, pp. 21–41) and “On concept and object” (Frege, 1970, pp. 42–55).

Moreover, if Hutchinson's intention is to explain why the fact that we need to accept (mc) in order to construct the simplest system cannot justify (mc), he should also admit that (F₁) — the claim that accepting the law I is necessary — cannot justify our acceptance of the law I for the same reason. For the necessity of accepting the law I also does not provide evidence that the law I is true, because the law itself presupposes the existence of the very concepts stipulated by (mc). The necessity of accepting the law I does not provide evidence that such concepts exist. But the whole point of the transcendentalist argument is to justify the law I by appealing to (F₁). Hutchinson's explanation about why Frege does not regard (mc) as justified contradicts neo-Kantian transcendentalism directly.

Hutchinson might retort that taking Frege's arguments for the basic laws to be transmuting is “the only reading that fits with everything Frege says about the axioms, read in the most straightforward way” (Hutchinson, 2021, p. 521). For Frege, logical axioms should be justified. However, logical axioms are not susceptible to proof. At the same time, the only arguments Frege provides for his logical axioms, i.e., the basic laws, appear like proofs. Hutchinson's point is that the only viable way to make these points compatible with each other is to read Frege's arguments for the basic laws as transmuting arguments.

I agree that Frege ought to be able to regard the basic laws as justified. But I disagree that endorsing the notion of transmuting argument is the only way he can do so. First, it is not clear that Frege's arguments for the basic laws can ever play the role of proof. Recall that a proof is a deductive justification, that is, an attempt to deductively justify a truth by appealing to known truths. As Richard Kimberly Heck (2012, pp. 28–29) points out, Frege's arguments are similar to typical Tarskian semantic arguments for the validity of logical laws.¹⁹ When we provide such an argument for a logical law, we appeal to certain “instances” (Heck, 2012, p. 29) of that law; and that is what Frege does in his argument for the law I. Such an argument is of course circular; one who accepts the given instances of the law does so because one already accepts the law or the semantic rationale that underlies the law. That is a case of the Cartesian Circle: “[A]ny justification of a logical law will have to involve reasoning, which will depend for its correctness on the correctness of the inferences employed in it” (Heck, 2012, p. 28). So, a semantic argument for a logical law cannot be a proof in the relevant sense because it endorses something unjustified as a premise. If Frege's arguments for the basic laws are semantic arguments, they are not proofs. If so, there is no reason to regard Frege's arguments as transmuting.²⁰

Second, as far as I can see, the question of how Frege can justify the basic laws of Begriffsschrift has different viable options for answers. For instance, Frege points to logical faculties in several places (Frege, 1979, p. 183, 2013, Vol. II §74). He argues that by our logical faculties, we can grasp logical objects such as value-ranges or the extension of a concept. It is not entirely implausible to argue that Frege takes the basic laws — including the law V about value-ranges — to be justified directly by our logical faculties. That is Tyler Burge's idea, I believe, when he argues that, for Frege, a basic law is such that “understanding it suffices for recognition of its truth” (Burge, 1998, p. 317). Burge takes Frege's arguments for the basic laws in Grundgesetze to be not for proving them but for helping his readers to understand them.

We can even combine Burge's interpretation with Heck's. Tarskian semantic arguments for logical laws merely utilize our tendencies to accept those laws, namely, our intuitions. Say Frege takes the basic laws to be directly justified by our logical faculties. Then, all Frege needs to do to make people realize their truth in a justified way is to make them use their own logical faculties. If his arguments in Grundgesetze are indeed similar to Tarskian semantic arguments, they can do that job because they make us use our own logical faculties that already tend to accept those laws. If that is how Frege understands his arguments in Grundgesetze, we can consistently interpret his comments on logical laws. He can indeed say that logical axioms are justified — directly by our logical faculties.²¹ He can deny that the arguments in Grundgesetze prove the basic laws, saying that he is only trying to make us recognize the truth of those laws by our own logical faculties.²²

It is not my intention to argue for any particular interpretation strongly here. However, it seems clear that there are other viable options to interpret Frege's comments, pointed to by Hutchinson, consistently.

2.2.2 The Relationship Between a Logical System's Simplicity and Sufficiency in Frege

There is evidence to suggest that Frege would reject the idea that the objective of achieving the simplest logical system could justify the basic laws of his logic — even if he were to embrace the Lotze-Windelband transcendentalist method. My argument draws on Frege's understanding of how the simplicity of a system relates to its sufficiency. He states:

The fundamental principle of reducing the number of primitive laws as far as possible wouldn't be fully satisfied without demonstrating that the few that remain are sufficient. (Frege, 1979, p. 37)

Therefore, according to Frege, we cannot confirm that a logical system is the simplest, without proving that it is sufficient. A system's simplicity entails its sufficiency. A sufficient logical system, as Frege articulates, is one in which we “can manage throughout with [its] basic laws” (Frege, 1979, p. 38), implying that its axioms and inferential rules are competent enough to facilitate necessary logical inferences. Frege notes that the second and third sections of his book Begriffsschrift (Frege, 1972) are dedicated to demonstrating the sufficiency of his logical system. In those sections, Frege develops a step-by-step derivation of a sentence, “which […] is indispensable to arithmetic although it is one that commands little attention, being regarded as self-evident” (Frege, 1979, p. 38). The sentence articulates the ancestral relation:

If a series is formed by first applying a many-one operation to an object, and then applying it successively to its own results, and if in this series two objects follow one and the same object, then the first follows the second in the series, or vice versa, or the two objects are identical. (Frege, 1979, p. 38)

Frege deduces this statement from “the definitions of the concepts of succession in a series, and of many-oneness by means of [the] primitive laws” (Frege, 1979, p. 39) of his Begriffsschrift. Thus, the logical tasks Frege performs to show that his logical system is sufficient are essentially inferential tasks. This shows that Frege is committed to the following claim:

(SS) If a logical system is the simplest, it is sufficient for inferential tasks.

Frege asserts that performing such tasks “could not render this [i.e., that Begriffsschrift is sufficient] more than probable” (Frege, 1979, p. 38). Indeed, even if he shows that his system can handle certain inferential tasks, the system may not be able to handle others. Without a principled method to decide how much a logical system should be able to do to be sufficient, Frege would need to revisit the same question repeatedly — perhaps even after he thinks he proves the axioms of arithmetic based on the basic laws in Grundgesetze. However, he never provides any principled criteria for deciding the sufficiency of a logical system. What Frege does to establish the sufficiency of his logical system is complete his logicist project. In the Foreword of Grundgesetze, Frege writes:

One must strive to reduce the number of these fundamental laws as far as possible by proving everything that is provable. […] This ideal I believe I have now essentially achieved. (Frege, 2013, p. VI; italics mine)

It is evident that Frege adds this Foreword after completing the manuscript for Volume I of Grundgesetze that contains the proof of arithmetical axioms (with natural numbers). He believes that the ideal of simplicity is achieved, only after (a good chunk of) his logicist project is completed. That is because Frege considers the logicist project as a demonstration of Begriffsschrift's sufficiency that is necessary for its being the simplest, and such a demonstration only renders it probable that the system is sufficient. If providing such a demonstration is the only method he uses to show that his system is sufficient, he is committed to the following claim:

(IT) One can demonstrate that a logical system is sufficient only by applying the system to inferential tasks.

Assume that both (SS) and (IT) are true. Let “L” refer to a logical system. Say that we attempt to prove that

(Ax) The axioms of L are indispensable components of the simplest logical system.

Let “L*” be a general name that refers to any logical system that contains at least the axioms of L as its own axiom. To prove (Ax) is to prove that one of L*'s will ultimately be recognized as the simplest. By (SS), to prove the latter, we should prove that one of L*'s will ultimately be recognized as sufficient. However, by (IT), we can only demonstrate the sufficiency of a system by applying it to inferential tasks. It is clear that by this method of demonstrating sufficiency, we are unable to prove that any one L* will indeed be sufficient. For any given L*, in principle, an inferential task might emerge that requires additional axioms to be incorporated. More importantly, this method of demonstration cannot prove that any one of L* would be sufficient. For it does not preclude the possibility of an inferential task requiring a logical system that can do all the jobs L* can do and is not an L*. If one is committed to (SS) and (IT), one cannot prove (Ax).

Surely, one can still argue that (Ax) is true by showing that while possessing the least number of axioms, L can cope with inferential tasks as well as, or better than, other logical systems. Unless a better logical system is proposed, the claim that L is the simplest logical system will stand. For one who is committed to (SS) and (IT), this is the only way to validate (Ax). Therefore, if Frege hopes to establish that the basic laws of Begriffsschrift are necessary for the simplest logical system, he should validate the simplicity of Begriffsschrift by performing inferential tasks such as the logicist project.

For Frege, inferring is justifying:

We justify a judgement either by going back to truths that have been recognized already or without having recourse to other judgements. Only the first case, inference, is the concern of Logic. (Frege, 1979, p. 175)²³

The act of inference as such is of course governed by the norm of soundness:

[…] [I]n presenting an inference, one must state the premises with assertoric force, for the truth of the premises is essential to the correctness of the inference. (Gabriel et al., 1980, p. 79)

Therefore, when one infers a conclusion from premises, one should be committed to the truth of those premises:

From false premises nothing at all can be concluded. A mere thought, which is not recognized as true, cannot be a premise. Only after a thought has been recognized by me as true, can it be a premise for me. (Gabriel et al., 1980, p. 182)

According to this understanding of inference, if one infers derivative truths from certain logical laws, then one is already in a position where one can justifiably recognize the truth of those laws.²⁴

Now, consider the following claim:

(Ax-BS) The basic laws of Begriffsschrift are necessary for the simplest logical system.

To substantiate this claim, Frege should apply the basic laws of Begriffsschrift to inferential tasks. Namely, he should derive derivative truths from those laws. If Frege does so and believes — like neo-Kantians — that we can be justified in accepting those laws, he considers himself to be already in a position that can justifiably recognize their truth. If so, Frege cannot maintain that the justifications for those basic laws consist in (Ax-BS). (Ax-BS) is something he can demonstrate by making those inferences, not a claim that he is justified in accepting before he performs those inferences. Therefore, even if Frege is committed to the Lotze-Windelband teleological transcendentalism, the goal that justifies the basic laws cannot be the simplest system.

3. The Basic Laws and the Pursuit of Truth

3.1 Gabriel's NKT Reading via Truth

According to Gabriel (2002, 2021), Frege justifies the basic laws of Begriffsschrift by arguing that accepting those laws is necessary for pursuing truth. Gabriel highlights the following passage from the Foreword of Grundgesetze:

As to the question, why and with what right we acknowledge a logical law to be true, logic can respond only by reducing it to other logical laws. Where this is not possible, it can give no answer. Stepping outside logic, one can say: our nature and external circumstances compel us to judge, and when we judge we cannot discard this law — of identity, for example — but have to acknowledge it if we do not want to lead our thinking into confusion and in the end abandon judgement altogether. (Frege, 2013, p. XVII)

The passage addresses the question of with what right we acknowledge a logical axiom to be true, i.e., how we can justify our acceptance of a logical axiom.²⁵ Frege introduces a possible answer. I will call this answer, i.e., the claim after the colon in the above quotation, “IR (impossibility of rejecting).” (IR) contains the claim that accepting the law of identity — for example — is necessary for pursuing the goal of judgement. Taking the goal of thinking — or judging, in Frege's case — to be truth, Gabriel (2002, p. 47) regards (IR) as going hand-in-hand with Windelband's neo-Kantianism: “Logic can say to everyone: you want truth? Then remember, you must recognize the validity of these [axioms], if your desire is ever to be satisfied” (Windelband, 1907, p. 330).

Gabriel argues that Frege accepts (IR) and that his acceptance of it shows that he is a neo-Kantian transcendentalist. However, we need to be a little bit careful here because the above passage immediately precedes the following:

I neither want to dispute nor to endorse this opinion, but merely note that what we have here is not a logical conclusion. What is offered here is not a ground of being true but of our taking to be true. (Frege, 2013, p. XVII)

Frege is expressing his hesitation to accept (IR). However, Gabriel (2002, p. 47) argues that the rest of the paragraph that includes the above passages clarifies that Frege accepts (IR). Frege writes:

And further: this impossibility, to which we are subject, of rejecting the law does not prevent us from supposing beings who do so; but it does prevent us from supposing that such beings do so rightly; and it prevents us, moreover, from doubting whether it is we or they who are right. At least this is true of myself. (Frege, 2013, p. XVII; italics mine)

By “this impossibility of rejecting the law [of identity]” Frege should be referring to the impossibility given in (IR). Frege states that the impossibility prevents him from doubting that he rightly accepts the law of identity, which clearly shows that Frege is committed to (IR). Gabriel further interprets it as Frege's acknowledgement of the neo-Kantian transcendentalist claim that (IR) justifies the logical axiom. Pointing to the above passages, Gabriel writes:

Nevertheless, Windelband demands, and Frege develops, an argument for why we must accept the law of logic. […] Though Frege “neither dispute[s] nor support[s] this view,” he in fact accepts it […]. (Gabriel, 2002, p. 47)²⁶

As the evidence that Frege accepts the view, Gabriel refers to Frege's above statement: “[The impossibility] prevents us, moreover, from doubting whether it is we or they who are right” (Frege, 2013, p. XVII). Then, why does Frege express hesitation? Gabriel (2021, p. 9) argues that Frege is emphasizing that what is presented is not a logical justification: “What we have here is not a logical conclusion” (Frege, 2013, p. XVII). Because Frege is in the middle of dealing with logical issues, he prefers not to distract his readers with non-logical issues.

3.2 Criticism of Gabriel's NKT Reading

The way Gabriel handles Frege's hesitation in the paragraph we are analyzing is not satisfactory. Gabriel's claim is that the view that Frege states he suspends his judgement about in the paragraph is in fact endorsed by him in the same paragraph. Saying that he does not intend to divert his readers from logical issues does not resolve that apparent inconsistency. If we can develop a reading of the paragraph in which Frege does not endorse the view he declares that he suspends his judgement about, that would be a more desirable option.

Let us first ask what “this opinion” is about which Frege suspends his judgement. One candidate is of course (IR). However, it is clear enough that Frege is committed to (IR); he is claiming that the impossibility contained in (IR) prevents us — including himself — from suspecting that one can correctly reject the law of identity.²⁷ Thus, if Frege referred to (IR) by “this opinion,” he would be plainly inconsistent within a single paragraph.

Another candidate for what “this opinion” refers to is suggested by Gabriel: the neo-Kantian transcendentalist claim that

(ER) What (IR) offers constitutes a good epistemic reason to accept the law of identity.

To regard “this opinion” as referring to (ER) is plausible. Frege is presenting (IR) as a possible answer to the question of how we can justify our judgement that logical axioms — e.g., the law of identity — are true. Judgements are epistemic actions of recognizing truth. So, the justification Frege is looking for must be of an epistemic kind; he is looking for an epistemic reason to accept the law of identity. Thus, it is only natural to take “this opinion” to refer to the position that puts forth (IR) as such a good epistemic reason for accepting the logical law, i.e., (ER).

As we have seen, Gabriel argues that Frege in fact endorses (ER) by stating that the impossibility of rejecting the law of identity prevents him from thinking that one can correctly deny the law. Frege does acknowledge (IR) as a reason that he does not doubt the law of identity, and that is the core issue. If Frege were thereby expressing his commitment to (ER), again, he would be plainly inconsistent within a single paragraph. If we hope to read the paragraph in question consistently, we need to develop a reading in which Frege takes (IR) to be a good reason for accepting the law of identity while suspending judgement about (ER), i.e., hesitating to take (IR) to be an epistemic reason for doing so. Such a reading will show that Frege's position about neo-Kantian transcendentalism is more complicated than Gabriel suggests.

The question is why Frege hesitates to accept (ER). To answer, we first need to analyze the nature of what (IR) offers:

(a) Our nature and external circumstances force us to judge, and (b) when we judge we cannot discard this law — of identity, for example — but have to acknowledge it if we do not want to lead our thinking into confusion and in the end abandon judgement altogether. (Frege, 2013, p. X)

(IR) is a conjunction of (a) and (b). Focus on (b), the claim that we cannot reject the law if we do not want to abandon judgement entirely. Frege is committed to (b), which is equivalent to the following:

(b*) We cannot reject the law of identity without abandoning judgement altogether.

Frege ascribes the property of irrejectability without abandoning judgement to logical laws in general. He firmly believes — before he encounters Russell's Paradox — that arithmetical laws are logical laws. In Grundlagen, Frege points out that we can “assume the contrary of some one or other of the [Euclidean] geometrical axioms without involving ourselves in any self-contradictions” (Frege, 1950, §14). We can even deduce things from these assumptions, and doing so is “not useless by no means” (Frege, 1950, §14). However, arithmetical axioms are different:

Can the same be said of the fundamental propositions of the science of number? Here, we have only to try denying any one of them, and complete confusion ensues. Even to think at all seems no longer possible. (Frege, 1950, §14)²⁸

If we try to deny an arithmetical axiom, we end up abandoning judging altogether. If Frege ascribes the relevant kind of irrejectability to derivative logical laws such as arithmetical axioms, he will do so to a more fundamental one. Frege is committed to (b*), and thus (b).

Note that (b) — or (b*) — alone does not provide a reason that we should accept the law of identity as true. If we do not intend to pursue the goal of judgement further, we can simply reject the law of identity. However, (IR) contains (a), i.e., the claim that we are compelled to pursue judgement. If there is no choice for us other than pursuing judgement, we do not intend to abandon judgement. Given (b), our only rational choice is to accept the law of identity as true. Therefore, only when (a) and (b) are combined do we come to have an answer to the question of “why and with what right we acknowledge [the law of identity] to be true” (Frege, 2013, p. XVII).

In a nutshell, the point of (IR) is that because we aim to make judgements and we cannot do so without accepting the law of identity, we should accept the law. The justification (IR) provides for the law of identity is thus fundamentally an instrumental reason to accept it.²⁹ This is the essential nature of the teleological justifications neo-Kantian transcendentalists suggest for fundamental laws: “[Y]ou want truth? […] [Y]ou must recognize the validity of these [axioms], if your desire is ever to be satisfied” (Windelband, 1907, p. 330; italics mine). Such an instrumental reason to accept the law of identity may be a good reason to do so. In general, if G is a goal we ought to achieve and we cannot do so without doing M, then we have a reason to do M. If our pursuit of truth necessitates accepting the law of identity, we have a reason to accept the truth of the law of identity, as long as we ought to achieve the goal of truth. In other words, in such a case, we are instrumentally rational in accepting the law's truth.

Nevertheless, such an instrumental reason for accepting the law of identity, which is provided by (IR), does not constitute a reason that the law is true. If (IR) were true, our choice to accept the law would still count as rational even if the law turned out to be false; accepting it would be our last bet in any event. Frege captures this property of the instrumental justification provided by (IR) when he states the following:

What is offered [in (IR)] is not a ground of being true but of our taking to be true. (Frege, 2013, p. XVII)

That is, Frege is correctly pointing out that the instrumental reason provided by (IR) is not an evidential warrant for the law of identity, i.e., a piece of truth-conducive information that supports its truth. The instrumental reason applies whether or not the law is true, and thus it does not reveal anything about whether the law is true.

Recall (ER):

(ER) What (IR) offers constitutes a good epistemic reason to accept the law of identity.

We can now explain why he hesitates to accept it. In the paragraph, Frege is dealing with the question of what can serve as epistemic warrants for logical axioms. In the traditional epistemology, “epistemic warrant” means “evidential warrant.” (IR) only provides a teleological reason to accept the law of identity, and (ER) considers such a non-evidential reason as an epistemic warrant. Frege could not reconcile the notion of non-evidential epistemic warrant. That is why, expressing his reluctance to accept “this opinion,” i.e., (ER), Frege emphasizes that it is not an evidential warrant, although he takes it to be his reason to accept the law of identity.

We should distinguish (ER) not only from (IR) but also from the following:

(GR) What (IR) offers constitutes a good reason to accept the law of identity.

When Frege states that (IR), the teleological irrejectability of the law of identity, prevents him from doubting that he correctly accepts it, his commitment to (GR) is clear. But Gabriel further insists that Frege's acceptance of (IR) shows his commitment to (ER). This claim is unwarranted. First, commitment to (GR) does not imply commitment to (ER). Frege's endorsement of (IR) as his reason not to doubt the law of identity is insufficient to establish his commitment to (ER). Second, if Gabriel's reading were correct, there is no proper candidate for what Frege refers to by “this opinion.” If Gabriel is right, Frege himself accepts (IR) as his epistemic reason to accept the law. If so, what can he possibly be reluctant to accept? Third, Frege provides a clear reason to hesitate to accept (ER): (IR) does not give us an evidential warrant. It is (ER) that Frege refers to by “this opinion” and is reluctant to fully endorse. He is reluctant to be a full-blooded neo-Kantian transcendentalist like Lotze and Windelband.

4. Why Frege Is Sympathetic to Neo-Kantianism

According to the NKT reading, Frege is committed to Lotze's and Windelband's neo-Kantian transcendentalism, which regards fundamental propositions like logical axioms as justified teleologically. We have seen two versions of the NKT reading. Neither version is supported by the text.

The paragraph from the Foreword of Grundgesetze shows that Frege's position with respect to neo-Kantian transcendentalism is delicate. He is sympathetic to it; he admits that the teleological reason to accept the law of identity is a good reason to do so. He even takes himself to be subject to it. But Frege hesitates to be fully committed to the position because he is not sure that purely teleological reasons can be considered epistemic warrants.³⁰ The paragraph makes it clear that Frege is not a neo-Kantian transcendentalist.

At the same time, the passage reveals why Frege is sympathetic to the Lotze-Windelband style transcendentalism. The passage appears in the middle of Frege's extended criticisms against psychologism about logic. Frege's main antagonist is Benno Erdmann, who makes philological contributions to the neo-Kantian movement in Germany.³¹ It is not a coincidence that Frege mentions (IR) — which contains the neo-Kantian transcendentalist idea — when he criticizes Erdmann's psychologism. In the paragraph, Frege attempts to establish that neo-Kantianism, which is supposed to constitute Erdmann's philosophical background, in fact does not support his logical psychologism.

According to Frege, the main issue that divides him from psychologistic logicians is the nature of truth. While psychologistic logicians take truth to be subjective, Frege takes it to be objective. Earlier in the Foreword of Grundgesetze, he writes:

[The nature of truth] is what the psychological logicians conflate. Thus, in the first volume of his Logik, pp. 272 to 275, Mr B. Erdmann equates truth with general validity, grounding the latter on general certainty regarding the object judged, and this in turn on general consensus amongst those judging. And so, in the end, truth is reduced to being taken to be true by individuals. In opposition to this, I can only say: being true is different from being taken to be true […] and is in no way reducible to it. (Frege, 2013, p. XV)

Because truth is objective, any law that states what is true, i.e., any descriptive law, “can be conceived as prescriptive” in the sense that everybody “should think in accordance with it” (Frege, 2013, p. XV) if they hope to pursue truth. Logical laws are basically descriptive laws, i.e., laws of being true. Thus, if something is a logical law, everyone should judge according to it. Even those whose judgements do not seem to collapse although they deny logical laws ought to judge according to those laws. They are erroneously opposing those laws. But psychological logicians cannot say so. This is Frege's main complaint against them:

What, however, if beings were even found whose laws of thought directly contradicted ours […]? The psychological logicians could only accept this and say: for them those laws hold, for us these. I would say: here we have a hitherto unknown kind of madness. (Frege, 2013, p. XVI)

Let us return to the latter half of the paragraph that contains (IR):

[(IR), i.e., the neo-Kantian justification for the laws of logic] […] prevents us from supposing that such beings do so rightly […]. At least this is true of myself. [1] If others dare in the same breath to both acknowledge a law and doubt it, then that seems to me to be an attempt to jump out of one's own skin against which I can only urgently warn. [2] Whoever has once acknowledged a law of being true has thereby also acknowledged a law that prescribes what ought to be judged, wherever, whenever and by whomsoever the judgement may be made. (Frege, 2013, p. XVII)

By [1] and [2], Frege explains why (IR) prevents us from suspecting that one correctly denies the law of identity. [2] affirms the objectivity of truth. If we take a person to be correct in acknowledging the truth of a “law” that goes against the law of identity, then we should also take such a “law” to be true, which means that we should reject the law of identity. As [1] — which even psychological logicians would willingly accept — states, we should not accept and reject the law of identity at the same time. So, if (IR) provides a good reason to take the law of identity to be true, it thereby provides a good reason to regard a person who denies it as doing so incorrectly. What Frege shows here is that accepting (IR) as a reason to accept the law of identity does not amount to denying that truth is objective. If psychological logicians take neo-Kantian claims like (IR) to provide grounds for their position, that is a mistake. Frege is arguing that neo-Kantianism cannot be a source of psychologism about logic by showing that the former is compatible with the objectivity of truth. As we have seen, what divides Frege from psychological logicians is the objectivity of truth. This reading of the above passage is strongly supported by Frege's comment that immediately follows the above passage:

Surveying the whole matter, it seems to me that different conceptions of truth lie at the source of the dispute. For me, truth is something objective, independent of the judging subject, for psychological logicians, it is not. (Frege, 2013, p. XVII)

Frege's complaint against Erdmann seems to be that although Erdmann is philosophically grounded in neo-Kantianism, he mistakenly takes it to be a source of his logical psychologism. As Frederik Beiser (2014) and Scott Edgar (2008) point out, neo-Kantianism is rather strongly anti-psychologistic, and German philosophers around the end of the 19^th century, where neo-Kantianism is predominant, take anti-psychologism to be an orthodoxy. For Frege, to be an anti-psychological logician is to accept the objectivity of truth. Indeed, as Jeremy Heis (2018, §2.2, §3.2) describes, neo-Kantians strongly support the objectivity of truth.³² I have assumed that the claims about the alleged historical connection between Frege and neo-Kantians are true. My suggested reading shows that Frege has quite a deep insight into neo-Kantianism. So, it is probable that he also knows that neo-Kantians fully embrace the objectivity of truth. If so, Frege would take neo-Kantianism to be a legitimate anti-psychologistic position that accepts the objectivity of truth. Neo-Kantianism is his crucial ally in the epistemology of logic.

5. Conclusion

In this article, I have examined the NKT reading of the epistemic status of logical axioms in Frege. The NKT reading argues that Frege is committed to the idea that logical axioms are justified by their teleological indispensability for valuable epistemic aims. There are two versions of the NKT reading. One version, suggested by Hutchinson, argues that Frege takes his basic laws to be justified because they are necessary for achieving the simplest logical system. Hutchinson regards Frege's arguments for the basic laws in Grundgesetze as his neo-Kantian transcendentalist argument. However, interpreting those arguments in that way is not viable. Moreover, because Frege takes a system's simplicity to be associated with its sufficiency, he would not think that the goal of the simplest logical system can justify logical axioms, even if he were a neo-Kantian transcendentalist. The other version of the NKT reading suggested by Gabriel argues — pointing to a passage in the Forword of Grundgesetze — that Frege considers the basic laws as justified because accepting them is necessary for the pursuit of truth. However, Gabriel's reading is not entirely charitable in that it makes Frege inconsistent in a single paragraph. What I have shown is that, in a consistent reading of the passage in question, Frege genuinely hesitates to accept neo-Kantian transcendentalism because he cannot fully accommodate the idea that the teleological, non-evidential indispensability can constitute an epistemic warrant.

I believe that Frege's historical and philosophical connection with neo-Kantianism is now well established. However, if my discussion is along the right lines, Frege's position toward neo-Kantianism is not outright accommodation. The philosophical relationship between Frege and neo-Kantianism thus requires further examination.