Frege on Identity and Identity Statements: 1884/1903

ABSTRACT

In this essay, I first solve solve a conundrum and then deal with criteria of identity, Leibniz's definition of identity and Frege's adoption of it in his (failed) attempt to define the cardinality operator contextually in terms of Hume's Principle in Die Grundlagen der Arithmetik. I argue that Frege could have omitted the intermediate step of tentatively defining the cardinality operator in the context of an equation of the form ‘NxF(x) = NxG(x)'. Frege considers Leibniz's definition of identity to be purely logical, although without saying why it is in line with his logicist project. I argue that the universal criterion of identity that Frege takes from Leibniz's definition and the specific criterion of identity for cardinal numbers embodied in Hume’s Principle (namely equinumerosity) work hand in hand in the tentative contextual definition of the cardinality operator. Yet the interplay between the two criteria is powerless to prevent the emergence of the Julius Caesar problem, let alone suggests how it could be solved. The final explicit definition of the cardinality operator that Frege sets up still rests on the identity criterion of equinumerosity since cardinal numbers are defined as equivalence classes of that relation.

KEYWORDS:

• Hume's Principle
• the contextual definition of the cardinality operator
• the Julius Caesar problem
• all-encompassing domain
• criteria of identity
• the explicit definition of the cardinality operator
• Leibniz's definition of identity
• principium identitatis indiscernibilium

1. Frege's view of identity in the period 1884–1903. Solving an apparent conundrum

In BS, §8, Frege writes:

Now, let

|—A ≡ B

mean: the sign A and the sign B have the same conceptual content, so that we can always substitute B for A and vice versa.

I call this the metalinguistic view of identity.¹ If we frame the stipulation in the terminology that Frege uses after 1890, we can say: with a sentence of the form ‘a = b’ (or ‘A ≡ B’) we express the thought that ‘a’ and ‘b’ have the same reference or corefer. From this we may infer the mutual substitutivity salva veritate of ‘a’ and ‘b’ in all true extensional sentences. Thus identity, metalinguistically construed, is the semantic relation between two signs. By contrast, the objectual view of identity means that identity is a relation in which every object stands to itself and to no other object. Identity is considered a dyadic relation although so understood it never truly relates two objects. One might call this feature anomalous, but I do not think that from a logical point of view it gives rise to serious confusion. If in this connection we take into consideration Wittgenstein's verdict about identity in the Tractatus, we may further distinguish between the standard and a non-standard objectual view of identity. In the Tractatus (Wittgenstein 1961), Wittgenstein holds an objectual but a non-relational view of identity. He argues (5.5301) that identity is not a relation between objects and goes on to write (5.5303): ‘Mentioned in passing, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing’. Wittgenstein may have added: identity is neither a relation in which every object uniquely stands to itself nor a relation between signs. In the logically perfect concept-script that he is envisioning in the Tractatus, he expresses what he calls ‘identity of the object’ by identity of the sign, and difference of objects by difference of the signs (5.53). Accordingly, he dispenses with ‘=’ in his envisaged concept-script but not with identity. In Schirn 2024, I scrutinise Wittgenstein's approach to identity and his dispensation with ‘=’ in a ‘correct’ concept-script.² In an essay not yet published, I argue, as it were in return, that in Frege's GGA the primitive function-name ‘ξ = ζ’ is indispensable for laying the logical foundations of cardinal arithmetic and real analysis.

In the light of the distincion that I drew between the metalinguistic and the objectual view of identity, we may say: If we utter a sentence of the form ‘a = b’ with assertoric force, we acknowledge as true either the thought that ‘a’ and ‘b’ are coreferential or the thought that a is the same as b, depending on how we contrue identity. Frege would say that at the same time we manifest the truth acknowledgement (= the judgement). Note that for him judging is a mental and not a propositional (predicative) act expressed in words, no matter whether it applies to a thought which is expressed by a sentence of ordinary language or by a truth-value name (sentence) of his concept-script. In the concept-script, the mental act of judging is typographically represented by the judgement-stroke.

So much for the preliminaries. If we take a look at Frege's remarks over the post-Begriffsschrift period 1884–1903 on what identity statements express, we may be struck by a perplexing situation. The apparent conundrum occurs as follows. The considerations relating to identity and equations in GLA (1884) and also in FB (1891), which is one of the preludes to GGA and the most obvious, suggest that fairly soon after the publication of BS Frege had abandoned the metalinguistic view of identity and identity statements in favour of an objectual view. Furthermore, (i) in the opening passage of SB, Frege apparently discredits the metalinguistic view of identity on epistemological grounds. In the main text of the opening passage, he does not explicitly state that identity is a relation in which every object stands to itself and to no other object ­but seems to suggest that he holds this view. This is reinforced by the fact that prior to setting out his argument Frege favours an objectual view of identity and identity statements in the first footnote of SB: ‘I use this word [“equality”] in the sense of identity and understand “a = b” to have the sense of “a is the same as b” or “a and b coincide”’. In the footnote, Frege thus seems to anticipate the answer to the questions which he raises at the outset of SB regarding the nature of identity. (ii) In the exposition of the concept-script (GGA I, §1–§50), Frege explicitly construes identity as a relation in which every object uniquely stands to itself and accordingly construes equations as expressing an objectual identity (cf. in particular GGA I, §4, §7, §20, §50).³ He arguably sticks to this view throughout the entire volume. I used the word ‘accordingly’ since nowhere in his work does Frege state that his view of what identity is and his view of what identity statements express fall apart. In other words, a metalinguistic interpretation of true identity statements of the form ‘a = b’ in terms of the relation of coreferentiality holding between ‘a’ and ‘b’ is supposed to go hand in hand with a metalinguistic conception of identity, and an objectual interpretation of identity statements in terms of the coincidence of a and b is supposed to conform to an objectual construal of identity.

However, contrary to Frege's explicit endorsement of an objectual view of identity in GGA I, it could seem that in some of his writings during the period 1884–1903 the metalinguistic interpretation of equations of the form ‘a = b’ in BS is still in force. In GLA and FB, Frege does indeed make some remarks, which, taken at face value, amount to reiterating his BS view of identity and identity statements. At the end of the period 1884–1903, in GGA II (§105 and §138), we encounter again two observations of Frege's on ‘=’ which, on the face of it, suggest that he might have relapsed into the coreferentiality view of identity in BS, although without mentioning any reason for this. Plainly, if there is such a retrogression, it would contravene head-on Frege's view of identity and equations in GGA I.

In what follows, I take a closer look at the apparent discrepancy in the trajectory of Frege's views on identity and what identity statements express in the period 1884–1903. In particular, I examine whether the prima facie incoherence mentioned above can be resolved at least with respect to GGA I and GGA II.

1.1. On some remarks of Frege's on identity statements in the period 1884–1891

If we focus our attention on GLA, we see that Frege most likely construes identity and equations in an objectual way in the course of (a) adopting Leibniz's definition of identity, (b) highlighting the fundamental role of criteria of identity especially in connection with the introduction of abstract objects via abstraction principles,⁴ (c) discussing the Julius Caesar or indeterminacy problem to which an abstraction principle almost inevitably gives rise in a Fregean setting⁵ not only in GLA but also in GGA,⁶ (d) commenting on the character, status and importance of the final explicit definition of the cardinality operator in pursuit of the logicist project, and finally (e) raising the issue of defining the real and complex numbers as extensions of concepts in future work. However, after having dismissed outright a tentatively proposed inductive definition of the natural numbers – on the grounds that it does not permit us to discern 0, 1 and n +1 as self-subsistent objects that can be recognised as the same again – Frege explains (§57): ‘So what we have is an equation, asserting that the expression ‘the cardinal number of Jupiter's moons’ designates the same object as the word “four”.'⁷

In the light of what I just said about Frege's overall treatment of identity in GLA one might regard this remark as a momentary slip of the pen or as a sign of stylistic nonchalance, perhaps just in this one case in GLA concerning identity. However, if we pay attention to what he says about identity and equations in the period 1885–1891, it could seem that the remark I quoted above is not just a flash in the pan. For example, in the lecture FTA (1885) Frege makes an observation which even literally matches his statement in BS, §8 (NS, p. 108): ‘The situation changes radically when one takes these figures to be signs of contents; in that case, the equation states that both signs have the same content. But if no content is present, the equation has no sense’.

Taking the contentless numerical figures of formal arithmetic to be signs of contents in proper arithmetic and its logical foundation is clearly the view that Frege adopts. He makes a similar remark in FB (KS, p. 126). He does so independently of his discussion of formal theories of arithmetic and uses his new, terminologically strict notion of reference (Bedeutung) of singular terms instead of that of content first introduced in BS, §8: ‘What is expressed in the equation “2 ^. 2³ + 2 = 18” is that the right-hand combination of signs has the same reference as the left-hand one’. However, this remark is the only one in FB which suggests a metalinguistic view of equations. Observations in which Frege apparently commits himself to an objectual view of identity and equations prevail in FB (cf. KS, pp. 129, 130, 133, 136, 137), as they do in GLA. Here is an example that speaks for itself (KS, p. 130):⁸ ‘[…] so that in “$\begin{array}{c}\text{'}\\ ϵ\end{array}$ (ε² – 4ε) = $\begin{array}{c}\text{'}\\ \alpha \end{array}$(α · (α – 4))” we have the expression for: the first value-range is the same as the second’. Taken on the whole, I think that the scales are clearly tipping towards an objectual view of identity and identity statements in Frege's period 1884–1891.⁹

1.2. Reflections on a couple of remarks on identity statements in GGA II

To conclude section 1, I comment on the two statements in GGA II concerning equations which I indicated earlier.¹⁰ In the course of dismantling the formal theories of arithmetic of Heine and Thomae, Frege observes in GGA I, §105, p. 113: ‘Of course, we ourselves use the equality-sign to express that the reference of the sign-group standing on its left-hand side coincides with the reference of the sign-group standing on its right-hand side’. Taken literally, the remark flies in the face of Frege's objectual conception of identity and his corresponding interpretation of equations in the formal system of GGA. There is hardly any doubt that with his remark he does refer to this system. Thus, the remark might suggest that in GGA II Frege relapsed into his conception of identity statements in BS. Or should we take the remark with a pinch of salt and regard it as a far cry from BS? I tend to favour the second option. Before suggesting my reasons for this, let me mention that with his remark Frege obviously intends to contrast his view that the signs of contentual arithmetic are essentially endowed with references, whereas in formal arithmetic this is not the case. He discusses particular arithmetical equations introduced by Heine and Thomae (cf. GGA II, §105 and §106) to explain the contrast. But the point is nothing new. Frege has made it several times before. He observes that what he says in the quotation above is inapplicable regarding the equation ‘neg(a) = 0’ if its constituent signs are understood in Heine and Thomae's sense, as figures that are manipulated according to certain rules but with no consideration for any meaning or sense.¹¹ Yet Frege could have drawn the contrast at issue by adhering to his objectual view of identity, namely by saying that ‘we ourselves use the equality-sign to express that the number referred to on its left-hand side coincides with (is the same as) the number referred to on its right-hand side’. It comes as no surprise that in GGA II, §107 Frege reaffirms his official objectual view of identity and the corresponding interpretation of equations from GGA I. He writes that in contentual arithmetic the formula ‘2 + ½ = ½ + 2’ says that the sum of 2 and ½ coincides with the sum of ½ and 2. ‘What follows from this for the signs? Evidently, that the sign-group shaped like “2 + ½” may be replaced everywhere by one shaped like “½ + 2”, and conversely’.

The second statement about equations in GGA II that I consider is not concerned with formalist theories of arithmetic but with Dedekind's conception of identity, followed by a comparatively mild critique of his creation of irrational numbers. To set the context, I begin with a quotation from Dedekind 1888, §1:

A thing a is the same as (identical with) b, and b the same as a, if everything that can be thought of a, can also be thought of b, and if everything that holds for b, can also be thought of a. That a and b are only signs or names for one and the same thing is indicated by the sign a = b and likewise by b = a.

What strikes the eye is the fact that Dedekind does not use quotation marks when he mentions certain signs. In the first sentence, he adopts an objectual view of identity, while in the second he formulates his view in metalinguistic terms. I think that for the sake of coherence, Dedekind should have written (if, for the sake of fidelity, we respect his omission of quotation marks): That a is the same as b or that a and b coincide is expressed by the sign a = b and likewise by b = a. In Dedekind's earlier work Stetigkeit und irrationale Zahlen (Dedekind 1872), we find again a metalinguistic formulation of what an equation expresses (§1): If we wish to express that the signs a and b refer to one and the same rational number, we put a = b as well as b = a.

In the light of his exposition in §4 of this work, Dedekind could coherently have formulated this as follows: If we want to express that a is the same rational number as b, we put a = b as well as b = a. In §4, Dedekind is concerned with the relation between any given two cuts (A₁, A₂) and (B₁, B₂) generated by any two numbers α and β. He writes:

If now we compare two such first classes A₁ and B₁ with each other, it may happen that they are completely identical, i.e. that every number contained in A₁ is also contained in B₁, and that every number contained in B₁ is also contained A₁. In this case, A₂ is then necessarily identical with B₂, the two cuts are completely identical, which we indicate in signs by α = β or β = α.

I take this to be a clear statement that despite the metalinguistic formulations quoted above, Dedekind construes identity as an objectual relation just as Frege does in GGA I and probably also in GGA II. We are now prepared to turn to Frege's remark on Dedekind's view of identity in GGA II, §138: ‘Here the sharpness of the distinction between the sign and that which it is supposed to refer to is pleasing and noteworthy, as is the conception of the equality sign which accords exactly with our own’.

It is clear that in GGA II Frege is bound to respect everything that he has laid down in the exposition of the concept-script, including (a) his statement in §4 that identity is a relation in which every object (uniquely) stands to itself, (b) his elucidation of ‘ξ = ζ’ in §7 which underwrites the statement in §4, (c) the development of Basic Law III (in §20) which is in accord with (a) and (b), and (d) his short proofs of the principal laws of ξ = ζ (in §50) which in turn tally with (a) – (c). If in GGA II Frege had favoured a metalinguistic view of identity and equations without adducing good reasons for such a change of mind he would have committed a methodological blunder. Thus, I suggest that despite the two remarks in GGA II quoted above Frege should not be subject to the suspicion that around the turn of the century he silently returned to his Begriffsschrift view of identity and equations. In particular, it is in my opinion indisputable that every equation which occurs in any of the proofs in GGA II – either in the informal or semi-formal part called analysis¹² or in the purely formal part called construction – is understood objectually. Suffice it to give just one instructive example. I naturally take it from Frege's theory of real numbers. More specifically, I take it from the set of conditions which he states in preparation of the definition of the concept positival class and in terms of which he eventually defines this concept (GGA II, §175). Prior to that definition, he had already defined the fundamental notion of a domain of magnitudes (GGA II, §173). It is the definition of the concept positival class which paves the way for defining another key concept of his foundation of real analysis, namely that of a positive class. The latter definition is the penultimate in GGA II, followed by the definition of a function (function-name) that may be called the Archimedian condition.¹³ I think that the following quotation speaks for itself as far as Frege's objectual interpretation of identity in GGA II is concerned.

We now require that the first class belonging to a Relation $\Pi$ coincides with the second class belonging to a Relation K if $\Pi$ and K belong to the same positival class. Thereby it is also stated that the first class belonging to $\Pi$ coincides with the second.

Analogous remarks apply, mutatis mutandis, to all the other proof-analyses in GGA II in which the relation of identity is involved. That Frege endorses an objectual view of identity in the formal proofs he carries out in GGA II seems beyond doubt. Again, it is true that in a couple of places in GGA II he states that an equation of the form ‘a = b’ expresses (the thought) that ‘a’ and ‘b’ refer to the same object. Yet in my view this is perhaps only an infelicitous choice of phrasing or a sign of momentary insouciance. It is probably not intended as a commitment to a metalinguistic view of identity and equations. Similar remarks apply perhaps to Dedekind. In sum, as far as Frege is concerned, I suggest that not only in the first volume of GGA but also in the second he holds that identity is a binary objectual relation and that an equation of the form ‘a = b’ accordingly expresses that a is the same as b. If Frege held any other view in GGA II, this would be a complete enigma to me.

So much for my discussion of Frege's remarks on identity and identity statements in the period 1884–1903 with special emphasis on the apparent tension between some of them. In what follows, I analyse the role and significance of identity and equations in his foundational project in GLA.

2. Criteria of identity in GLA

At the outset of GLA, §62, Frege raises the question of how numbers are to be given to us if we do not have any cognitive access to them through ideas or intuitions. He answers by stating the context principle and then derives from it the task of defining the sense of a sentence in which a number word occurs. He goes on to say that if we use a sign ‘a’ to refer to an object we must be in possession of a criterion of identity which decides in all cases whether b is the same as a, ‘even if it is not always in our power to apply this criterion’. I assume that in GLA Frege considers the first-order domain to be both homogeneous and all-encompassing. By ‘homogeneous’ I mean that he does not distinguish between categories or types of objects. By an all-encompassing first-order domain I mean a domain which includes all objects whatsoever in the universe. Every object of the domain, be it concrete or abstract, can be the argument of every first-level function including first-level concepts and first-level relations.

Dealing with the role of criteria of identity in GLA requires that we briefly consider what is generally called ‘the Julius Caesar problem’. With the results of this discussion at hand, I shall then turn – after an interlude in section 3, namely after discussing some special issues concerning Frege's three attempts to define cardinal number – to Leibniz's definition of identity which Frege adopts as his own just after having tentatively proposed, for the sake of illustration, a contextual definition of the direction operator ‘the direction of the straight line a’ (henceforth abbreviated as ‘(CDD)’ – to be transferred, mutatis mutandis, to a tentative contextual definition of the cardinality operator in terms of Hume's Principle (henceforth abbreviated as ‘(CDC)’).

The equivalence of the relation of equinumerosity holding between two concepts F and G and cardinal number identity is Hume's Principle: The cardinal number which belongs to F = the cardinal number which belongs to G if and only if F and G are equinumerous, in symbols: N_xF(x) = N_xG(x) ↔ Eq_x(F(x),G(x)).¹⁴ Recall that in GLA, §62, Frege observes that it might not always be in our power to apply (effectively) a specific criterion of identity to a given range of objects. It is not clear whether in this connection he implicitly appeals to equations, say, of the form ‘N_xF(x) = t’ where ‘t’ is a singular term which has not the form of ‘N_xG(x)’. Note, however, that he does not raise the third and decisive objection to (CDD) in terms of a seemimgly non-sensical example until he has reached §65: By appeal to (CDD), we cannot decide whether England is the same as the Earth's axis. We may call this the England problem. Transferred to (CDC), the problem has widely become known as the Julius Caesar problem: (CDC) is powerless to decide whether, say, the number of planets coincides with Julius Caesar. If Frege had not considered the first-order domain to be all-embracing or at least to include, besides directions and numbers, countries and Roman emperors, neither the England problem nor the Caesar problem could have arisen in the drastic form in which he presents them.

It is unfortunate that in GLA, §62 Frege does not spell out what he means by conceding that it might not always be in our power to apply, say, the identity criterion of equinumerosity between first-level concepts, to cardinal number identity. I presume that in his view this criterion usually fulfils its purpose when it is applied to equations of the form ‘N_xF(x) = N_xG(x)’. Yet Frege was possibly aware that in special cases the determination of the truth-value of an equation of the form ‘N_xF(x) = N_xG(x)’ by appeal to equinumerosity might be beyond the reach of our cognitive capacities. As a matter of fact, Hume's Principle does not always place us in a position to establish the truth-value of such an equation. Take, for example, (q) ‘N_x(x = x) = N_xFCN(x)’, where ‘N_xFCN(x)’ is to abbreviate the predicate ‘finite cardinal number’. As Boolos shows (1987, p. 16), (q) is an undecidable sentence in the formal system FA (Frege Arithmetic), which is standard second-order logic plus Hume's Principle. (q) is true in some models of FA but false in others. Heck (1999, p. 262) mentions further examples of equations of the form ‘N_xF(x) = N_xG(x)’ whose truth-value is undecidable (for us).¹⁵

In the more recent past, quite a few scholars have analysed the Caesar or indeterminacy problem in GLA, though with different results. In this article, I shall not analyse this problem in some detail but confine myself to making a couple of remarks on it. The semi-formal variant of the indeterminacy problem which Frege faces in GGA I, §10 has been investigated to a much lesser extent than the Caesar problem in GLA. The former arises when Frege intends to fix the references of value-range names by means of an informal stipulation couched in a second-order abstraction principle (cf. GGA I, §3). The attempt is only partly successful, but Frege goes on to make further stipulations which seem to empower him to achieve his goal, namely, to endow value-range names with unique references. In §10 (ignoring the considerations in the long footnote to §10), Frege confines himself to considering the question of whether a given value-range is the True or the False. Thus, in GGA he seems to narrow down the first-order domain to the logical objects which he had introduced in §2 and §3. Yet on closer examination appearances turn out to be deceptive. In Schirn 2018, I argue that not only in GLA, but also in GGA the first-order domain is most likely considered to be all-encompassing. Frege's elucidations of the primitive first-level function-names as well as the comments he makes especially on his definition of the application operator in GGA I, §34 seem to confirm this.

Returning to GLA, Frege mentions Julius Caesar only once, namely at the outset of the second, constructive part of the book. In §56, he draws attention to an ‘undecidability problem’ which is supposed to arise from his tentative inductive definition of the finite cardinal numbers in §55 (henceforth abbreviated as ‘(ID)’): ‘we can never decide by means of our definitions whether the number Julius Caesar belongs to a concept, whether that well-known conqueror of Gaul is a number or not’. However, for reasons mentioned below I apply the term ‘the Julius Caesar problem’ only to the third and decisive objection which Frege raises to (CDC). Note that his discussion of the first-order parallelism-direction abstraction must only be transferred, mutatis mutandis, to the second-order equinumerosity–cardinal number abstraction.

Frege's choice of Julius Caesar to illustrate the shortcoming of (ID) as well as his choice of England to show the deficiency of (CDD) is in principle unnecessary, but it is memorable. These bizarre examples were presumably intended to put the spotlight on the problem at issue. Their use suggests that Frege (tacitly) considers the first-order domain to be all-embracing. Yet even if in GLA he had stipulated that the first-order domain is to contain only cardinal numbers, (CDC) would be powerless, at the stage of §66, to determine the truth-value of an equation such as ‘The cardinal number of the eight thousanders of the Himalayas = 1’. It would be unfit to accomplish this at least within the framework of the logicist methodology which Frege deploys in the constructive part of GLA. He defines ‘1’ in terms of the cardinality operator (cf. GLA, §77). The latter must therefore be defined in the first place, and needless to say, the definition must be irreproachable. To all appearances, Frege believes that, unlike (CDC), the explicit definition of the cardinality operator in §68: N_xF(x): =  @φ(Eq_x(φ(x),F(x))) (henceforth abbreviated as ‘(EDC)’) is logically impeccable. Once ‘1’ is defined through ‘N_x(x = 0)’, the equation ‘The cardinal number of the eight thousanders of the Himalayas = 1’ could be transformed into ‘The cardinal number of the eight thousanders of the Himalayas = N_x(x = 0)’ which, by appeal to the right-to-left direction of Hume's Principle, is to be rejected as false since exactly fourteen mountains falls under the concept eight thousanders of the Himalayas.¹⁶ Note that in §73 Frege roughly sketches the proof of the right-to-left direction and points out that the left-to right-direction can be proved in a similar way.¹⁷

Hale and Wright 2001 claim that Frege's criterion of identity for cardinal numbers differs fundamentally from that for human beings (or from that for Roman emperors). They argue that a cross-sortal identity claim such as ‘The number of continents = Julius Caesar' is therefore false. A statement such as ‘The Roman statesman who formed, together with Crassus and Pompey, the political alliance called the first triumvirate = the Roman general who gained military victories in the Gallic Wars, if and only if two special concepts F and G, which are intrinsically related to Julius Caesar, are equinumerous' probably sounds outlandish, if not downright preposterous to the ears of Caesar's admirers. Nevertheless, it could be doubted whether the inference that Hale and Wright draw and which I mentioned above is compelling. In this essay, I do not wish to pursue this issue further, but refer the reader to a more recent discussion of Hale and Wright's position in Boccuni and Woods 2020. The last-mentioned authors present a novel perspective for interpreting cross-sortal identity claims regarding, for example, numbers and persons. The problem of cross-sortal identity claims of the form ‘@1(P) = @2(Q)' is discussed in Cook and Ebert 2005, Ebels-Duggan 2021 and Schirn 2023a. The terms flanking ‘=’ are supposed to have been introduced via two distinct abstraction principles.

3. An objection to what I write in section 2 of this article

The reviewer of my essay raises an objection to one specific aspect of my preceding analysis. They write:

In section 2, the author discusses Frege's criteria of identity in Grundlagen der Arithmetik in light of the Julius Caesar problem. The author argues that ‘The choice of the bizarre example of Julius Caesar to demonstrate the indeterminacy of the cardinality operator is unnecessary,’ because the problem even occurs if the domain is not all-encompassing. This point is certainly correct, but it is also quite evident from Frege's text. In §56, where Frege introduces the Julius Caesar Problem, he explicitly states that an explicit definition of numbers, such as 0, is needed to determine the truth value of an equation.

The objection prompted me to reflect again, from a slightly different angle, on some of the issues that are involved in Frege's definitional strategy regarding cardinal number in GLA, §55, §65, §68, §72, §74, §77 and §84. However, I respectfully disagree with the claim that it is quite evident from GLA, §56 that Frege would still face a version of the Caesar problem if he restricted the first-domain in his exposition to cardinal numbers. There is no direct clue in §56 that Frege was aware of this. He does not consider an example such as, say, ‘The number of planets = 9’. (The second objection that Frege raises to (ID) is that we cannot prove by means of (ID) that a must equal b if ‘$N_x^a$F(x) ∧ $N_x^b$F(x)’ holds. The objection presupposes what has yet to be proved, namely that cardinal numbers are objects.) Moreover, contrary to what the reviewer asserts, in §56 Frege does not state that an explicit definition of cardinal numbers is required in order to determine the truth-value of an equation. It is not clear which type(s) of equation the reviewer has in mind. Do they mean to include or invoke equations in which the equality-sign is flanked by a numerical term that purports to refer to a cardinal number and a term which does not purport to refer to a cardinal number but rather purports to refer to a person such as ‘Julius Caesar’? Be that as it may, a possible way out of the difficulties to which in Frege's view (ID) gives rise is passed over in silence in §56.¹⁸ There are still a few hurdles for Frege to overcome before he can be sure that the requirements for framing explicit definitions of ‘0’, ‘1’ and ‘∞₁’ have been met in the spirit of his logicism.

The situation concerning Frege's three attempts to define cardinal number is not as straightforward as it might appear at first glance. Instead, it is, in my view, relatively complex. In what follows, I try to make this clear in broad outline by drawing attention to some interrelated aspects in Frege's development of his logicist project in GLA. At the same time, I seize the opportunity, indirectly prompted by the reviewer, to consider an issue which I seem to have overlooked in my previous research on GLA. It concerns the insight that Frege might have dispensed with (CDC) without any significant loss for stressing the key points of his logicist programme.

To begin with: On closer examination, the Julius Caesar problem that Frege introduces and analyses in GLA, §56 is spurious. He raises a genuine Caesar problem only in §66 if we transfer, mutatis mutandis, the case of directions to that of cardinal numbers as he implicitly suggests. The diagnosis that Frege presents of the failure of (ID) in the light of the presupposition that cardinal numbers are objects is negative across the board and, therefore, does not offer a constructive way out of the quandary. Strictly speaking, it is also inconsistent in one important respect.

If Frege considered numbers to be second-level concepts, as on the face of it (ID) might suggest, he would be committing a kind of type error by asking whether Julius Caesar is a number. In that case, the concept of cardinal number would have to be a third-level concept under which cardinal numbers qua second-level concepts fall. Yet Julius Caesar is an object. Thus, when Frege asserts that by means of (ID) we can never decide whether Julius Caesar is a number or not, he seems to presuppose that the concept of cardinal number is of first level. Strictly speaking, (ID) does not permit the formulation of the Julius Caesar objection, because the objection presupposes that numbers are objects, and (ID) arguably does not define the natural numbers as objects, nor was it intended to do this. Frege's remarks at the end of §56 leave no doubt about that, whence my charge of inconsistency in his line of argument to which I alluded above. This point becomes even more obvious when we rephrase the definienda of (ID) as ‘There are exactly 0 Fs’, ‘There is exactly one F’, etc. What Frege actually defines in the first two clauses of (ID) are the numerically definite quantifiers in the guise of the second-level predicates ‘the number 0 belongs to the concept φ’ (‘$N_x^0$φ(x)’) and ‘the number 1 belongs to the concept φ’ (‘$N_x^1$φ(x)’), which sound somewhat strange to modern ears. I conclude, then, that the Caesar objection in GLA, §56 patently misses its mark and, therefore, has no impact on Frege's project of introducing and defining cardinal numbers as logical objects.

Let us imagine a reader of GLA who has followed Frege's exposition until §57 without glancing at the ensuing sections. The reader might feel unable to confidently predict how Frege intends to proceed in his outline of the logicist project after having jettisoned (ID) as inappropriate for his purposes. However, the critical remarks in §56 regarding (ID) suggest at least that the individual cardinal numbers had unerringly to be defined as identifiable and re-identifiable objects, not as second-level concepts. Obviously, the attempt to set up explicit definitions of ‘0’ and ‘1’ in terms of the cardinality operator in a section following §56 more or less directly would be bound to fail: the reference of the cardinality operator has not yet been fixed. Frege's diagnosis of the failure of (ID) may suggest that defining ‘0’ and ‘1’ in a methodologically correct way requires several well-ordered steps, in the first place the introduction of a specific criterion of identity for cardinal numbers in general, i.e. a means of grasping them and of recognising them again as the same if they are given to us in different ways. However, the imagined reader of GLA who has arrived at §57, may not have an inkling that Frege will present the requisite criterion of identity in the outfit of a contextual definition of the cardinality operator. There is nothing in the foregoing investigation that indicates such a choice.

I for one think that in pursuit of his foundational strategy in GLA Frege could have omitted altogether the intermediate step of tentatively defining the cardinality operator in the context of an equation of the form ‘N_xF(x) = N_xG(x)’. The modified strategy could have been carried out as follows: (1) a specific criterion of identity for cardinal numbers is provided: the one-to-one correspondence of the objects falling under F with those falling under G seems to be the ideal candidate; (2) the identity criterion is installed as the right-to-left direction in an abstraction principle governing cardinal numbers which yields Hume's Principle; in contrast to Frege's strategy, in my scenario Hume's Principle is directly introduced as a theorem, not as a definition; (3) the Julius Caesar problem to which Hume's Principle gives rise is stated;¹⁹ its possible solution has to be analysed, regardless of whether it is clad in the garb of (CDC) or, as in my scenario, directly presented as a theorem that had to be proved once the cardinality operator was defined in a methodologically sound manner. In my thought experiment, I disregard, for the sake of simplicity, the fact that (EDC) rests on Frege's questionable assumption that one knows what the extension of a concept is. It is precisely for this reason that the supposed problem-solving power of the explicit definition of the cardinality operator in GLA, §68 might after all turn out to be an illusion. In saying this, I have the alleged solution to the Julius Caesar problem in mind that Frege seems to associate with (EDC). So much on the situation that he faces, either in his own or in my slightly modified scenario, after having dismissed (ID).

If Frege had succeeded in solving the Julius Caesar problem arising from (CDC) by means of an additional stipulation without invoking extensions of concepts, an explicit definition of the cardinality operator may have turned out to be unnecessary within the bounds of the logicist project that he outlines in GLA for cardinal arithmetic (cf. Schirn 2010, pp. 49 f.). Clearly, to avoid jeopardising the supposed logical nature of cardinal numbers, Frege would have to make sure that the additional (non-definitional) stipulation rests neither on intuition nor on experience. He might have been able to pursue this strategy for two reasons: (a) (CDC) is not rejected on the grounds that it offends against any previously stated principle of definition,²⁰ and (b) equinumerosity is definable in second-order logic. No more and no less would be required by Frege's own lights to uphold the claim that the contextual definition defines the cardinality operator in purely logical terms. The definition of cardinal numbers as extensions of concepts (equivalence classes) would be removed. Note that clad in a contextual definition, Hume's Principle need not be a truth of logic. Yet once its initial status as a definition has been abandoned by Frege, it must be established as a truth of logic if it to play a key role in the logical foundation of cardinal arithmetic. In pursuit of the potential no-class strategy that I am considering, the individual numbers qua objects as well as the first-level concept of cardinal number could be defined in terms of the cardinality operator. This agrees with what Frege actually does, albeit by appeal to (EDC) in the line-up of the cardinal number definitions that he tentatively presents in GLA, §55, §65 and authoritatively in §68.

4. Leibniz's definition of identity and Frege's adoption of it in GLA

In what follows, I consider Leibniz's definition of identity and the role which this definition plays for Frege when he sets up (CDC) in the course of outlining his logicist programme. In GLA, §63, Frege writes:

Our purpose is to form the content of a judgement which can be understood as an equation such that each side of this equation is a number. We therefore do not want to define identity specially for this case, but by means of the already familiar concept of identity to arrive at that which is to be regarded as identical.

In order to implement this strategy, Frege adopts Leibniz's definition of identity as his own. Unfortunately, he does not explain it on the background of Leibniz's logic. Nor does he say much about its place within the logic underlying GLA. Explanations of this kind would have been useful since the way in which Leibniz's definition should be understood and is probably understood by Frege cannot be taken for granted. So let us first take a look at the definition and the context in which it appears in Leibniz's logic.

Leibniz states his definition of identity in his fragment ‘Non inelegans specimen demonstrandi in abstractis’ (see Leibniz 1875–1890, VII, p. 228; cf. p. 236; see also Leibniz 1903, pp. 362 f.). The definiens is framed in terms of the mutual substitutivity of A and B salva veritate, and the complete definition is as follows:

Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint A et B, et A ingrediatur aliquam propositionem veram, et ibi in aliquo loco ipsius A pro ipso substiuendo B fiat novo propositio seque itidem vera, idque semper succedat in quancunque tali propositione, A et B dicuntur esse eadem; et contra, si eadem sint A et B, procedet substitutio quan dixi.

Leibniz does not explicitly distinguish here between A qua object and ‘A’ qua sign that refers to A. Trivially, two signs ‘A’ and ‘B’ cannot be said to be identical, but we can express that they corefer or are coreferential.²¹ Just as little can an object A be identical with an object B if A and B are two objects. It seems that Leibniz was perfectly aware of this when in connection with his principium identitatis indiscernibilium (henceforth PII) he observes: ‘positing two indiscernible objects means placing the same object under two names’ (Leibniz 1875–1890, vol. VII, p. 372).²²

Leibniz grounds PII by invoking his principum rationis sufficientis (cf. Leibniz 1875–1890, vol. VII, p. 301; Leibniz 1903, pp. 25, 515; Leibniz 1948, pp. 13, 267 f., 287). The latter principle, to which he also refers as principium nobilissimum, principium generalissimum or fundamental axiom (cf. Leibniz 1875–1890, vol. III, p. 530, vol. VI, p. 602, vol. VII, p. 301), says that no sentence can prove to be true unless ‘there is a sufficient rationale why it is so and not different’, although the rationale is not always known to us, in short: Nihil est sine ratione. Leibniz points out that the rationale for the truth of a sentence lies in the intrinsic connection of the subject with the predicate. As I said above, according to PII, there cannot be two objects which differ only numerically from one another (cf. Leibniz 1960, p. 476). If a and b are two objects, PII says that there is at least one internal property which belongs to a but not to b, or likewise in the words of Leibniz: which is contained in the complete concept of a but not in that of b (cf. Leibniz 1875–1890, vol. VI, p. 608). If two objects a and b were indiscernible, we would not dispose of a principle of individuation for them. But that would contravene the supreme principles of human reason.²³ This is how Leibniz argues for his view that there cannot be two indiscernible objects, neither in the realm of the a priori sciences arithmetic and geometry nor in nature (cf. Leibniz 1960, p. 475; Leibniz 1903, pp. 261, 362).

Leibniz's definition of identity is intimately related to PII but it must not be confused with PII. In any case, I assume that he intends to define the objectual identity of a with b and actually does. By appeal to PII, one can in fact show that in his logic the mutual substitutivity of two singular terms ‘a’ and ‘b’ in true (extensional) sentences salva veritate²⁴ amounts to the claim that a and b have all their internal properties in common and hence are indistinguishable from one and another. The plural form of speech is here almost unavoidable and should be taken cum grano salis: a and b are supposed to be one and the same object. In symbols, Leibniz's definition of identity can be rendered as follows: $a=b:=\mathrm{\forall f}\left(f\left(a\right)\leftrightarrow f\left(b\right)\right)\text{Df}$ In words: a is identical with b if and only if in every true (extensional) sentence in which ‘a’ occurs, ‘b’ can be substituted salva veritate for ‘a’. An alternative reading is: a = b if and only if for all internal properties f: f(a) ↔ f(b)).²⁵

For the sake of accuracy and probably in the spirit of Leibniz, I have added the term ‘extensional’ in parentheses. There is some textual evidence that Leibniz was aware of the logical difference between extensional and non-extensional sentences. In one place (Leibniz 1960, p. 475; cf. also Leibniz 1903, p. 261), he writes that the two equivalent terms ‘Petrus’ and ‘the apostle who denied Christ’ can be substituted one for the other salva veritate in all sentences, unless we have what he calls a ‘reflexive’ language use. Judging from the context, I presume that he has here primarily sentential contexts in mind where the mutual substitutivity salva veritate of coreferential terms may break down, for example, in a modal context such as ‘It is necessary that p’.

Since according to Leibniz's remarks in connection with PII the complete concept of an object a is equivalent to the ‘conjunction’ of all internal properties of a, the definiens of his definition of identity could also be rendered as follows: if any property f is a component of the complete concept Δ of a, then it is also a component of the complete concept Γ of b, and vice versa. Since ‘f’ is a variable, Δ and Γ must ‘coincide’ if a = b. It is possible that in GLA Frege would have endorsed Leibniz's talk of the complete concept of an object, had he explicitly considered it.²⁶ Note, however, that in some parts of his work Leibniz applies the relation of identity also to concepts.²⁷ With one, possibly undesirable, exception Frege does not. After 1891, he regards the relation of mutual subordination of first-level concepts, that is, their coextensiveness, as the direct analogue of identity (cf. NS, p. 132).

I assume that Frege and also Russell followed Leibniz in regarding the definition of identity as supplying a criterion of identity for abstract and spatio-temporal objects alike.²⁸ I further assume that Frege and Russell would have agreed that a viable definition of ‘=’ should meet this generality constraint. According to Frege's line of argument in GLA, it is the adoption of Leibniz's definition of ‘=’ that enables him to frame contextual definitions in terms of abstraction principles, in each case providing a criterion of identity (= a specific equivalence relation) for the abstracta. Thus, the universal criterion of identity taken from Leibniz's definition and the specific criterion of identity for the cardinals in Hume's Principle (= equinumerosity) are supposed to work hand in hand in (CDC). Although in GLA the relationship of dependence between a local and a gobal identity criterion is considered a prerequisite not only for the formal correctness but also for the logicist viability of (CDC) – both criteria must be purely logical – it is powerless to prevent the emergence of the Julius Caesar problem, let alone suggests how it could be solved. From Frege's point of view, the requirement of relying on a specific, purely logical identity criterion for cardinal numbers not only applies to (CDC) which delivers the requisite criterion so to speak free of charge. The requirement must also be met when Frege comes to set up (EDC) in GLA, §68. In contrast to (CDC), (EDC) does not overtly display a criterion of identity for cardinal numbers but it nonetheless crucially rests on such a criterion. For Frege, the choice once again falls on equinumerosity. The sign ‘=’ has disappeared on the left-hand side of (EDC) and with it the appeal to Leibniz's definition of identity. Yet ‘=’ reappears in the equivalence that follows immediately from the explicit definition, namely in ‘@φ(Eq_x(φ(x),F(x))) = @φ(Eq_x(φ(x),G(x))) ↔ N_xF(x) = N_xG(x)’.

Frege chooses equinumerosity for important reasons, even though it may not represent the only legitimate choice. He defines cardinal numbers as equivalence classes of equinumerosity but in principle might have defined them in a different way in accordance with the requirements of a logical foundation of cardinal arithmetic. To be sure, he only could have chosen a different definition if it had allowed him to impose specific constraints on it, such as the derivability of Hume's Principle from it in due consideration of the application of the natural numbers in counting. Strictly speaking, Frege defines equinumerosity only with some delay in §72 as the one-to-one correlation of the objects falling under F with those falling under G. But it would be unfair to blame him for disregarding the principle that the meaning of the definiens – here of (EDC) – must be fixed prior to laying down the definition.²⁹ In §68, just before setting up (EDC), Frege already points out that equinumerosity is to be understood and defined as one-to-one correlation, and we might as well grant him this concession.³⁰ A concise account of Frege's treatment of (EDC) in relation to Hume's Principle is to be found in Heck 2019, pp. 527–533.

When I said that in (CDC) the identity criterion taken from Leibniz and the identity criterion of equinumerosity collaborate, I did not mean to suggest that in those cases in which the terms flanking ‘=’ are not both of the form ‘N_xG(x)’ the global identity criterion steps in for the local identity criterion. It is true that if the identity criterion ∀f(f(a) ↔ f(b)) is applied to the equation, say, ‘The number of planets = Julius Caesar’ it yields straightaway the value ‘false’, whereas the criterion of equinumerosity fails us in this case, at least at the stage when Frege tentatively sets up (CDC). Yet although in his view (CDC) rests on the prior definition of ‘=’ in purely logical terms – and ideally ought to rest also on the prior definition of equinumerosity – it is the criterion of equinumerosity on the basis of which the truth-value of any equation not only of the form ‘N_xF(x) = N_xG(x)’, but also of the form ‘N_xF(x) = t’ ought to be decidable. As I mentioned in section 3, in the latter case, an appropriate additional stipulation in combination with (CDC) rather than (EDC) might have enabled Frege to solve the Julius Caesar problem regarding the introduction of cardinal numbers. I presume, however, that regarding a satisfactory solution to the Caesar problem in GLA, he knew his metes and bounds at that stage of his foundational project.

Frege emphasises that all the laws of identity are contained in what he calls universal substitutability. From this it follows for him that in order to justify (CDD) he has to demonstrate that the term ‘D(b)’ can be substituted for the term ‘D(a)’ in all true (extensional) statements S in which ‘D(a)’ occurs and thereby always ‘S[D(a)] ↔ S[D(b)]’ holds if line a is parallel to line b. Since regarding ‘D(a)’ and ‘D(b)’ we are, in the context that Frege provides, only familiar with the fundamental case of direction identities, it suffices to show that ‘D(b)’ can be substituted for ‘D(a)’ salva veritate in such identities or in statements which contain a direction identity as a constituent statement. For all supplementary (partial) definitions of the direction operator qua constituent of statements about directions other than equations substitutibility salva veritate must be preserved. Again, the case which Frege makes for the abstraction principle governing direction identity has to be transmitted, mutatis mutandis, to the abstraction principle governing cardinal number identity.

In RPA, Frege surprisingly points out that he construes Leibniz's explanation of identity not as a definition. What made him change his mind? Frege writes (KS, p. 184): ‘Since every definition is an equation, identity itself cannot be defined. Leibniz's explanation could be called a principle that brings out the nature of the relation of identity, and as such it is of fundamental importance’. The argument does not deliver what it promises. Frege speaks of the nature of the relation of identity without specifying it and without making any further comment on Leibniz's definition of identity and his adoption of it ten years earlier in GLA. As I pointed out earlier, in the light of the first footnote of SB and especially of GGA I, §4, §5, §7, §20, §50 we may firmly assume that in the period in which Frege wrote the review he construed identity as a relation which every object bears to itself and to no other object. However, it is not clear whether in GGA he considered it mandatory to introduce identity as a primitive function for the sole reason that every definition is an equation and that therefore identity is (allegedly) indefinable. Frege might also have thought that in order to fix the references of value-range names he had to make sure that identity is assigned a key role among the primitive first-level functions of his system. Or he even might have had another (or an additional) reason to introduce identity as a primitive function. I leave this question to the reader's attention and end with a brief summary of some of the points I have made in this essay.

5. Summary

1. At the outset of my essay, I analysed a conundrum that seems to arise from some of Frege's remarks on equations in the period 1884–1903. I argued that Frege most likely held an objectual view of identity and identity statements throughout the entire period 1884–1903.

2. In section 2, I examined the nature and role of criteria of identity in GLA. I argued that if Frege had not considered the first-order domain to be all-embracing, the Julius Caesar problem could hardly have arisen in the drastic form in which he presents them. Yet even if Frege had expressly narrowed down the first-order domain to cardinal numbers, the problem of the indeterminacy of the reference of the cardinality operator would have emerged from (CDC). The choice of the example of Julius Caesar to demonstrate the referential indeterminacy of the cardinality operator is unnecessary. At the stage of GLA, §66, the truth-value even of a statement such as ‘The cardinal number of the eight thousanders of the Himalayas = 1’ could not have been determined by appeal to the identity criterion of equinumerosity. I further argue that Frege was possibly aware that in special cases the determination of the truth–value of an equation of the form ‘N_xF(x) = N_xG(x)’ by invoking the touchstone of equinumerosity might be beyond the reach of our cognitive capacities.

3. In section 3, I made some comments on Frege's tentative inductive definition of the natural numbers in GLA, §55. I argued that this definition is spurious and that the genuine Julius Caesar arises only from (CDC). Nonetheless, Frege could have omitted the intermediate step of tentatively defining the cardinality operator in the context of an equation of the form ‘N_xF(x) = N_xG(x)’. He could have done this without any significant loss for highlighting the key points of his logicist programme. In particular, he could have introduced Hume's Principle directly as a key theorem of cardinal arithmetic that had to be proved once the cardinality operator had been defined irreproachably. At the end of section 3, I argued that if Frege had succeeded in solving the Julius Caesar problem arising from (CDC) by means of an appropriate additional stipulation without appeal to extensions of concepts, an explicit definition of the cardinality operator may have proved to be unnecessary in pursuit of his logical foundation of cardinal arithmetic.

4. In section 4, I argued that the global criterion of identity taken from Leibniz's definition of identity (= coincidence of all internal properties of a and b) and the local criterion of identity for cardinal numbers embodied in Hume's Principle (= equinumerosity) are supposed to cooperate in the tentative contextual definition of the cardinality operator. Frege does not say this explicitly but the way he proceeds suggests this. However, the supposed cooperation between these two criteria does not mean that in those cases in which the terms flanking ‘=’ are not both of the form ‘N_xG(x)’ the global identity criterion steps in for the local identity criterion. As far as the explicit definition of the cardinality operator is concerned, it still rests on equinumerosity, although not directly as (CDC). The sign ‘=’ has disappeared on the left-hand side of the explicit definition and with it the appeal to Leibniz's definition of identity. Yet ‘=’ reappears in the equivalence that follows immediately from (EDC), namely in ‘@φ(Eq_x(φ(x),F(x))) = @φ(Eq_x(φ(x),G(x))) ↔ N_xF(x) = N_xG(x)’.

Acknowledgements

I am grateful to Daniel Mook and an anonymous reviewer for valuable suggestions for improving my essay. Special thanks go also to the editor of this journal, Volker Peckhaus, for his invaluable help regarding the preparation of my manuscript for publication.

Notes

1 Frege does not use quotation marks when he mentions the signs ‘A’ and ‘B’, a ‘no go’ in his mature period.

2 On Wittgenstein's elimination of ‘=’ in a correct concept-script see also Rogers and Wehmeier 2012 and Lampert and Säbel 2021.

3 On Frege's view of identity and of the cognitive value of logical equations in GGA I, see Schirn 2023b.

4 Frege does not use the term ‘abstraction’ when he is concerned with what we nowadays call Fregean abstraction. He probably thought that due to his rejection of what he considered misguided methods of abstraction (cf. GLA, §21, §34, §45, §49–§51; KS, pp. 164 f., 186 ff., 214 ff., 240–261, 324 ff.; NS, pp. 75–80) the term ‘abstraction’ had acquired a negative meaning, at least for himself. When he comments on Cantor's method of obtaining the ordinal or cardinal number of a set via a single or a double act of abstraction (KS, pp. 164 f.; cf. NS, pp. 75–80), he says that the verb ‘to abstract’ is a psychological expression and should be avoided in mathematics.

5 A Fregean abstraction principle is of the form ‘(Q(a) = Q(b)) = (R_eq(a, b))’. ‘Q’ is a singular term-forming operator, a and b are free variables of the appropriate type, ranging over the members of a given domain, and ‘R_eq’ is the sign for an equivalence relation holding between the values of a and b.

6 In GGA I, we encounter a formal variant of the Caesar problem in GLA. In §10, Frege discusses the problem that by appeal to his informal stipulation in §3 concerning value-range names one cannot decide whether an object which is not given to us as a value-range – such as the True or the False denoted by a truth-value name (sentence) – is a value-range. Heck (cf. Heck 2012) appropriately calls the metalinguistic stipulation in GGA I, §3 the ‘Initial Stipulation’. It is couched in a second-order abstraction principle. In GGA I, §20, Frege obtains Basic Law V from the Initial Stipulation by (a) transforming the stipulative mode of the latter into the assertoric mode of the former, (b) converting the stipulated coreferentiality of two metalinguistic sentences into an objectual identity (= truth-value identity) and (c) by fitting out the objectual identity with a formal guise. See the discussion of the referential indeterminacy of value-range names in a broader context in Schirn 2018, 2019, 2023a and 2023b.

7 In GLA, Frege does not yet distinguish terminologically between the sense (Sinn) and the reference (Bedeutung) of an expression. But as in BS, §8, he can explain the difference of the cognitive value of a true statement of the form ‘a = b’ and that of a (true) statement of the form ‘a= a’ in semantic terms. In this essay, I confine myself to saying about the nature and role of identity in GLA only what has to be said. The fact that in GLA Frege did not yet make the terminological distinction between sense and reference is of minor importance for my analysis.

8 In an undated letter to Peano (probably written around 1896), Frege remarks: ‘As far as I am concerned, I take identity, complete coincidence, to be the meaning [Bedeutung] of the equals sign and in definitions at least there seems to be no other possible meaning’ (WB, p. 195). The term ‘reference’ may also be a correct translation of Frege's use of ‘Bedeutung’ in this context, but I think that the choice of ‘meaning’ is here slightly more appropriate.

9 Thau and Kaplan 2001 argue – unconvincingly in my view – that Frege did not abandon his Begriffsschrift view of identity and identity statements; see the critique of their view in Heck 2003. See in this context also May 2001 and May 2012.

10 In BS, §8, Frege underscores that the introduction of a sign for identity of content necessarily produces a bifurcation in the meaning of all signs. After BS, this is no longer an issue to which he draws attention. This does not mean that in his mature period Frege thought that his earlier thesis about the introduction of ‘≡’ was completely useless. But at least after 1891 the idea of a bifurcation in the meaning of all signs seems to have lost the importance for him which he had attached to it in BS. On Frege's theory of identity in BS see Pardey and Wehmeier 2019. To all readers who cannot read GGA in the original, I recommend to use the outstanding English translation by P. Ebert and M. Rossberg of GGA, referred to in the ‘References’ at the end of this article as ‘BLA’. It is a milestone in Frege research. Analogous remarks apply to Ebert and Rossberg 2019.

11 Thomae points out that despite the close kinship that arithmetic qua calculation game bears to chess, we must draw an important distinction between the two. The rules of chess are characterized by arbitrariness, while the set of rules for arithmetic is such ‘that by means of simple axioms the numbers can be referred to as intuitable manifolds and can thus make important contributions to our knowledge of nature’ (Thomae 1898, quoted after Frege, GGA II, §88). Frege (GGA II, §89) correctly observes that the distinction drawn by Thomae arises only when we are concerned with applications of arithmetic and, hence, when we abandon the ground of formal arithmetic. Thomae's formal arithmetic no more contains theorems and proofs than does the game of chess. Yet the former differs from the latter inasmuch as new figures along with new rules can always be introduced in it, whereas in chess everything is fixed from the beginning and thus remains unchanged (cf. GGA II, §93).

12 Some analyses which precede the formal proofs (the constructions) contain formulas including a concept-script definition of a new function-name. The steps that lead to the definition are always explained in more or less detail.

13 Frege does not introduce a specific term for this dyadic function. For the sake of brevity, he defines the corresponding simple name in GGA II, §199 (see the detailed explanation of the definition of this function-name in Cook 2013, pp. A–41 f.). Frege needs this function-name in order to prove what he calls the Archimedian Axiom (see GGA II, p. 191). ‘If two Relations belong to the same positive class, then there is a multiple of the one that is not less than the other.’ A sustainable definition of the concept positive class requires, Frege thinks, a certain productive detour. The irrational can only be obtained as a limit. Yet a definition of the notion of limit requires a relation of the smaller to the greater, which Frege wishes to define with the help of the concept positive class. In order to address the central question ‘When is a class a positive class?’, the wider concept positival class must first be defined. Equipped with the latter Frege can define the notion of upper limit, and with this notion at hand he arrives at the concept positive class. Frege introduces the concept positival class by stating five conditions to be embodied in the definition of this concept. The first two are: ‘What belongs to a positival class must be a Relation such that both it and its converse are single-valued. The Relation composed of such a Relation and its converse, being a null magnitude, must not belong to the positival class’ (GGA II, §175). By adding three further conditions he arrives at the definition of the concept of a positival class. It is the most complex definition of all twenty-seven definitions which Frege sets up in GGA I and II, followed by the slightly less complex definition of the concept of a positive class. A class of Relations designated by ‘$\Sigma$’ is a positive class if and only if the class is a positival class, is not bounded from below, and every proper sub-class of the class named by ‘$\Sigma$’ that is downwards closed relative to some Relation has a limit (cf. Cook 2013, p. A–41). Cook's comments on all GGA-definitions (Cook 2013, pp. A–28–A–42) are very useful. As to the nature and role of some of Frege's definitions in GGA, see also the interesting discussions in Cook and Ebert 2016 and Kremer 2019.

14 Note that this formulation of Hume's Principle is a schematic one. Here its two sides are (closed) sentences, that is, ‘F and G are used as schematic letters for monadic first-level predicates, not as variables for first-level concepts.

15 In GLA, Frege highlights the cognitive value of equations. In doing so, he has perhaps mainly mathematical equations in mind. He writes (§67): ‘The multiple and significant applicability of equations rests rather on the fact that one can recognize something as the same again although it is given in a different way.’ This dictum about the fruitfulness of true equations of the form ‘a = b’, in which ‘a’ and ‘b’. present the same object in different ways, stands in sharp contrast to Wittgenstein's appraisal (in the Tractatus) of the nature, status and role of equations. From Frege's point of view, the fruitful and effective application of a concept-script is of paramount importance for its acceptability. Designing a concept-script only for its own sake without pursuing the purpose of applying it in the sciences may degenerate into sterile l’art pour l’art. Recall in this context the famous dictum that Frege states in the course of demolishing the mathematical formalism of Heine and Thomae in GGA II, §91: ‘Now it is applicability alone which elevates arithmetic above a game to the rank of a science. Applicability thus necessarily belongs to it.’ Much earlier, in GLA, §87, Frege wrote: ‘The laws of number […] are not really applicable to external things; they are not laws of nature. They are, however, applicable to judgements that hold of things in the external world: they are laws of laws of nature.’ Regarding Frege's application constraint see the analyses in Sereni 2019 and Panza and Sereni 2019.

16 Frege defines: n + 1: = N_x(x = 0 ∨ … ∨ x = n). So, every finite cardinal number can be defined as the number which belongs to a concept under which all its predecessors fall. For the sake of brevity, I did not spell out the corresponding definition for the cardinal number 14.

17 On Frege's proof of the the right-to-left direction (Theorem 32) and the contraposition of the left-to-right direction (Theorem 49) of Hume's Principle in GGA I, §54–§65 and §68–§69 see Heck 2012, pp. 173–179 and May and Wehmeier 2018.

18 See also the critical comments on (ID) in Dummett 1991, chapter 9.

19 In considering a way out of the Julius Caesar or indeterminacy problem arising from (CDD), Frege points out in GLA, §67 that one should not presuppose that an object could be given only in one single way. Plainly, if the syntactic rules of a formal system were such that we may refer to a cardinal number only by using a numerical term of the form ‘N_xG(x)’, the Caesar problem for cardinal numbers would not have arisen at all in such a system.

20 This would not apply in GGA.

21 In some places, Leibniz speaks also of identical or coincident terms, but what he has in mind is their equivalence, as a quotation from his study ‘Specimen calculi universalis’ shows: ‘Termini aequivalentes sunt, quibus res significantur eadem, ut triangulum et trilaterum’ (Leibniz 1903, p. 240). See also Leibniz 1903, p. 363 where he explains the coincidence of statements in terms of their mutual substitutivity salva veritate. A related remark can be found in his study ‘Characteristica geometrica’ (Leibniz 1849–1863, vol. V, p. 150).

22 See in this connection the remarks in Wittgenstein 1961, 5.5303 and Wittgenstein 1969a, §215 and §216, pp. 385 f.

23 However, it seems logically possible to construct in thought a symmetrical universe in which a and b share all their internal properties and nevertheless differ from one another numerically. Whitehead and Russell 1910 write (p. 57): ‘It should be observed that by indiscernibles he [Leibniz] cannot have meant two objects which agree as to all their properties, for one of the properties of x is to be identical with x, and therefore this property would necessarily belong to y if x and y agreed in all their properties.’ PII has been criticized with various arguments, including the following: (1) PII cannot be justified by saying: to a belong at least two properties – namely the property of being identical with a and the property of being different from b – which cannot be predicated of b. For the assertions (i) that a but not b has the property of being identical with a and (ii) that a but not b has the property of being different from b do not prove anything but only reiterate the hypothesis that a and b are different. (2) PII is a tautology since it merely states that different objects are different. (3) Since it is logically possible to construct a symmetric universe in which two objects share all their internal properties and nonetheless differ numerically from one another, PII does not hold with logical necessity. In the argument, it is assumed that there is only the symmetric model world and not an asymmetric observer outside this world. The stronger version of the objection is as follows: Since one can consistently construct a symmetric universe in thought, PII is patently false. For more on identity and indiscernibility in the context of ante rem structuralism see Shapiro 2008.

24 Leibniz (1903, p. 363) defines the ‘coincidence’ (that is, logical equivalence) of two statements likewise through their mutual substitutivity salva veritate. ‘Coincidere dico enuntiationes, si una alteri substitui potest salva veritate, seu quae se reciprocè inferunt.’

25 According to Quine 1970, §5, p. 63, identity can be defined in what he calls a standard language (cf. §2). Suppose that the language contains one one-place predicate ‘A’, two two-place predicates ‘B'’ and ‘C’ and one three-place predicate ‘D’. ‘x = y’ can then be defined through Ax ↔ Ay ∧ ∀z(Βzy ↔ Βyz ∧ Bxz ↔ Byz ∧ Czx ↔ Czy ∧ Cxz ↔ Cyz ∧ ∀z'[Dzz'x ↔ Dzz'y ∧ Dzxz' ↔ Dzyz' ∧ Dxzz' ↔ Dyzz']). According to this definition, ‘x = y’ says that the objects x and y are indistinguishable with respect to the predicates ‘A’, ‘B’, ‘C’, and ‘D’, and also with respect to their relations to other objects z, z’, insofar as those relations are expressed in simple sentences. As to Ramsey's definition of identity (1931, p. 53), I find it hard to make heads or tails of it. He defines identity as follows: ‘So (ϕ_e).ϕ_ex is a tautology if x = y, a contradiction if x ≠ y. Hence it can suitably be taken as the definition of x = y. x = y is a function in extension of two variables. Its value is [a] tautology when x and y have the same value, [a] contradiction when x, y have different values.’ Ramsey probably designed this definition exclusively for equations of a formal language that he was envisioning under the influence of Whitehead and Russell 1910 and Wittgenstein's programmatic ideas in the Tractatus. Ramsey's theory of identity is severely criticized by Wittgenstein in his so-called middle period; cf. Wittgenstein 1964, pp. 141 ff., Wittgenstein 1967, pp. 189 ff. and Wittgenstein 1969b, pp. 315 ff.

26 In GLA, §53, Frege writes: ‘By properties which are predicated of a concept I naturally do not understand the characteristic marks which make up the concept. These are properties of the things which fall under the concept, not of the concept.’ In BG (KS, p. 175), Frege refers to these earlier remarks and writes: ‘If the object $\Gamma$ has the properties $\Phi$, X and $\Psi$, I may combine them into $\Omega$, so that it is the same if I say that $\Gamma$ has the property $\Omega$, or, that $\Gamma$ has the properties $\Phi$, X and $\Psi$. I then call $\Phi$, X and $\Psi$ characteristic marks of the concept $\Omega$, and, at the same time, properties of $\Gamma$.’

27 Leibniz designed an arithmetical semantics only for syllogistic logic, not for the more extensive calculus universalis of a general logic of concepts. In this semantics, he assigns pairs of numbers to the concept constants and interprets the operators belonging to the logic of concepts by appeal to certain arithmetical operations. As contentual elements of the calculus universalis he has: (a) concept constants A, B, C, … ; (b) the operators of negation ‘non’ and of conceptual conjunction AB; (c) the relations of inclusion and identity as applied to concepts and their negations: ⊂, $\text{⧸}\subset$, =, ≠ (rendered as ‘est’, ‘non est’, ‘sunt idem’ or ‘eadem sunt’, ‘diversa sunt’) as well as the conceptual operator M(A) (‘A est ens’, ‘A est possibile’) which is designed to single out in the set of concepts the consistent ones. In his opus ‘Generales Inquisitiones de Analysi Notionum et Veritatum’ of 1686 (see Leibniz 1903, pp. 356–399), Leibniz provides an axiomatization of the calculus universalis which is isomorphic to standard set-theoretic algebra.

28 See Whitehead and Russell 1910, *13. The authors use ‘⊃’ instead of ‘$\equiv$’ in the definiens of their definition of ‘x = y’ and thus dispense with the second partial condition related to ‘$=$’: (*13.01): x = y.  = : (ϕ):ϕ ! x. ⊃. ϕ ! y Df. ‘It is to be understood that the sign “=” and the letters “Df” are to be regarded as together forming one symbol. The sign “=” without the letters “Df” will have a different meaning’ (Whitehead and Russell 1910, p. 11). The definition of ‘$=$’ states that x and y are identical if every predicative function satisfied by x is also satisfied by y. The limitation to predicative functions is due to Russell's theory of types which includes a hierarchy of functions and propositions. However, Whitehead and Russell argue, by appeal to the axiom of reducibility, that the scope of their definition of identity matches the scope that a definition of ‘x = y’ would have, if it were permitted to include all functions, irrespective of their type. More specifically, they argue as follows: We can assume that if x is the same as y and is true, ϕ(y) is also true, irrespective of the type of the function, since this statement must hold for any function. However, one cannot claim, conversely, that if with all values of ϕ, ϕ(x) implies ϕ(y), then x and y are identical. The rationale is that the phrase ‘all values of ϕ’ is only admissible if it is restricted to functions of one order, that is, either to predicates or to second-order functions or to functions of any other specific order. As a consequence of this, one obtains so to speak a hierarchy of different degrees of identity: ‘all predicates of x apply to y’, ‘all second-order properties of x belong to y’, etc. (Frege would deny that an object has any second-order property or falls under a second-level concept.) Each of these statements implies all its predecessors; for example, if all second-order properties of x belong to y, then all predicates of x belong to y. Yet we cannot, without invoking an axiom, claim conversely that if all the predicates of x belong to y, then all the second-order properties of x must also belong to y. Hence, without appeal to the requisite axiom we cannot be sure that x is identical with y if the same predicates belong to x and y. Whitehead and Russell believe that PII, as they understand the principle, can provide the requisite axiom to find a way out of this impasse. In the Tractatus (5.5302), Wittgenstein raises the following objection to the definition of identity in Whitehead and Russell 1910: ‘Russell's definition of “=” is inadequate, beause according to it one cannot say that two objects have all their properties in common. (Even if this sentence is never true, it still has sense.)’ Yet according to Russell there are no two objects that have all their (predicative) properties in common, although he does not say that this logically impossible. Why should it be possible to say, by appeal, to his definition of identity, that two objects have all their properties in common? Russell just defines ‘x = y’ through the formal equivalent of ‘x and y have all their (predicative) properties in common’, but the variables ‘x’ and ‘y’ are understood to take always the same value if the condition for x and y on the right-hand side of the definitional equation is fulfilled. Thus, any two terms ‘a’ and ‘b’ in place of ‘x’ and ‘y’ are supposed to refer to one and the same object. If to say that two objects are identical is nonsense, then, in consideration of Russell's definition of ‘=’, Wittgenstein would have to concede that it is likewise nonsense to say that two objects have all their properties in common. I do not see why Russell's definition should be replaced by a definition according to which one can sensibly say, though only falsely, that two objects have all their properties in common. In brief, it seems to me that Wittgenstein's inadequacy charge is beside the point. In the wake of Wittgenstein, Waismann (1936) also criticized Russell's definition of identity. However, Waismann's critique is not free of misinterpretation and terminological inconsistency. Waismann erroneously assumes that it is a consequence of Russell's definition of ‘=’ that the sentence ‘a and b have all their properties in common’ boils down to a tautology and, hence, does not express an empirical fact (cf. p. 61). The argument that he adduces for this is as follows: If the sentence ‘a and b have all their properties in common’ is to say that a and b are not two things but one thing, then it is tantamount to saying that ‘a’ and ‘b’ are synonymous. Yet if ‘a’ and ‘b’ are synonymous, then the sentence ‘a and b have all their properties in common’ says exactly the same as the sentence ‘a and a have all their properties in common’. First, the inference from the assumption that if an identity statement of the form ‘a = b’ is to express that a and b are one and the same thing, then this amounts to saying that ‘a’ and ‘b’ are synonymous, is invalid. Second, contrary to what Waismann insinuates (p. 61), Russell usually does not use ‘=’ between two proper names when he considers identity statements which in his view possess relevant cognitive value. He argues that the statement ‘Scott is the author of Waverly’, for example, does not say the same as ‘Scott is Scott’ and therefore is not a tautology. It is true, however, that in Russell 1918, p. 246, Russell characterizes ‘Scott is Sir Walter’ as a tautology, if the two names are used as names: ‘So if I say that “Scott is Sir Walter”, using these two names as names, neither “Scott” nor “Sir Walter” occurs in what I am asserting, but only the person who has these names, and thus what I am asserting is a pure tautology.’

29 In GGA, Frege is in the comfortable position to dispose of two criteria of identity for cardinal numbers, namely (a) the relation of coextensiveness contained in Basic Law V and (b) the relation of equinumerosity embodied in Theorem 32, the right-to-left direction of Hume’s Principle. Criterion (a) is tighter woven than criterion (b). Coextensive first-level concepts (functions) are equinumerous, but the converse does not hold generally.

30 In his so-called middle period beginning approximately around 1930, Wittgenstein launches a head-on attack on one of the key pillars of Frege and Russell's logicist projects: their explicit definition of the cardinal number of a concept or class in terms of an equivalence class of the relation of equinumerosity between first-level concepts (in Frege's GLA) or between value-ranges of monadic first-level functions (in GGA) or in terms of an equivalence class of what Russell – deviating from Frege's terminology – calls the relation of similarity between classes (in Russell 1903 and in Whitehead and Russell 1910). Although Frege's theory of value-ranges differs significantly from Russell's class theory in both Russell 1903 and Whitehead and Russell 1910, Wittgenstein tends to lump them together in some places. In his middle period, Wittgenstein raises also objections to related views of his two fellow logicians, for example, to their conceptions of logical abstraction in general and of equinumerosity or similarity in particular, as well as to Frege's view of ascriptions of number (Zahlangaben) (cf. Wittgenstein 1964, 1967, 1969b). In some places at least, the discussion strikes me as jejune and nebulous. The global objection is that the entire logic of Frege and Russell rests on a confusion of concept and form. On the face of it, the objection sounds grandiloquent and pretentious. In any event, Wittgenstein spares himself the trouble of advancing a clear and persuasive argument for his claims.