3.2 Schröder’s algebra of absolute terms

Schröder’s exposition in Operationskreis was heavily based on the work of Boole. In An Investigation of the Laws of Thought (Reference Boole1854) (hereinafter, Laws of Thought), Boole developed an algebra of the sum, the product and the difference, i.e., an algebraic theory that established the properties of these three operations by means of equations. The theory can be interpreted in different ways and, specifically, it can be applied both to the logic of classes (to which Boole referred as the ‘logic of primary propositions’) and to sentential logic (i.e., the logic of secondary propositions). The most relevant feature of these applications is that the logic of primary propositions is formally the same as the logic of secondary propositions.Footnote ¹⁵

Schröder continued this equational algebraic system and presented it in Operationskreis. Following Boole, Schröder distinguished between judgements of the first class (Urtheile der ersten Klasse) and judgements of the second class (Urtheile der zweiten Klasse) (Reference Schröder1877, p. 1). Again, a purely algebraic theory, which he called ‘calculus of identity of domains of a manifold’ (Reference Schröder1880, p. 84; 221), can be applied to form, respectively, a calculus of classes or a sentential calculus.

Following his own account of what a universal characteristic should be, Schröder provided a set of basic operations or categories, by means of which the theory could be constructed. These essentially coincide with Boole’s: equality ( $=$ ), sum ( $+$ ), product ( $\cdot $ ), negation ( $_1$ ) and the modules ( $0$ and $1$ ).Footnote ¹⁶ Division and subtraction were defined by means of the basic operations. The objects of these operations were symbolised by letters. All symbols corresponding to the primitive notions received a twofold interpretation depending on the application of the algebra: the calculus of classes or the sentential calculus. The equality symbol ‘ $=$ ’ was interpreted as equality between classes or as a symbol for logical equivalence; ‘ $+$ ’ as union or disjunction; ‘ $\cdot $ ’ as intersection or conjunction; ‘ $1$ ’ as the class of objects of thought under consideration (or Boole’s universe of discourse Schröder, Reference Schröder1877, p. 7) or as “the time segment during which the presuppositions of an investigation to be conducted are satisfied” (Schröder, Reference Schröder1880, p. 87; 224); ‘ $0$ ’ as the empty class or an empty segment of timeFootnote ¹⁷ ; and ‘ $_1$ ’ as complement or negation.Footnote ¹⁸ According to these interpretations, for instance, the formula ‘ $a=a\, +\, b$ ’ can be read as a judgement of the first class or as a judgement of the second class: in contemporary notation, as $a=a\cup b$ or as $\alpha \equiv \alpha \wedge \beta $ (where ‘ $\equiv $ ’ is the symbol of logical equivalence), respectively.

The theory presented in Operationskreis does not have a specific set of reasoning principles—i.e., logical axioms or inference rules—that can be distinguished from the algebraic theorems. Regarding this lack, although Schröder provided what seemed to be an axiomatic presentation in Operationskreis, it could hardly be considered a proper axiomatisation of Boolean logic.Footnote ¹⁹ Peirce advanced towards a true formal axiomatisation of this algebra in ‘On the Algebra of Logic’ (Reference Peirce1880). An essential feature of Peirce’s development is the introduction of the relation of subsumption , by means of which equality can be defined.Footnote ²⁰ This implies that equality was no longer regarded as the simplest logical relation. In fact, Peirce’s incomplete axiomatisation was grounded on the operation. In this sense, his adoption of subsumption as the basic logical operation was a significant advance towards the formulation of a formal theory and had noteworthy implications. On the one hand, the basic operations and the modules could be formally defined; on the other, it paved the way for a nonequational axiomatisation of the logic of secondary propositions. However, Peirce’s account was hindered by the use of both as a copula (i.e., as the connection between subject and predicate in a statement) and as illation (i.e., as the relation between the premises and conclusion in an argument) (Reference Peirce1880, p. 170, fn. 5). The copula was interpreted as the relation of inclusion between classes, whilst the illation was interpreted as the relation of logical consequence and, crucially, also as the conditional. The identification of the copula with the illation shows that Peirce did not distinguish between the principles of reasoning and the algebraic laws; this affected his axiomatisation, since he presented as theorems several logical laws that, to the extent that they deal with notions that do not belong to the algebraic calculus, should be considered axioms.Footnote ²¹

In Vorlesungen Schröder presented a complete formulation of the Boolean algebra of absolute terms, that is, the algebra that deals with simple objects or classes—and not with relationsFootnote ²² : in the first volume, he developed the calculus Peirce had outlined in 1880 and provided an axiomatisation of the calculus of classes; in the second volume, he put forward a sentential calculus by adding an axiom to the first volume’s calculus of classes.Footnote ²³

In 1880 the language of the algebraic theory developed by Boole and refined by Schröder was, in a significant sense, incomplete. Its expressive power could not meet Schröder’s aim, since it revealed important limitations. Schröder recognised one of these shortcomings in his review of Begriffsschrift:

In Laws of Thought (Reference Boole1854, p. 61) Boole introduced the symbol ‘v’, which was meant to represent an indefinite class and was used to translate the particle ‘some’. As Schröder stressed, this symbol presented many problems, one considerable example being the inadequate analysis of particular categorical judgements.

As a consequence of what Schröder affirmed in the above quote, not all concepts could be expressed by means of a set of categories of Boolean algebra of absolute terms as of 1880. This explains Schröder’s evaluation of this theory in his review of Begriffsschrift. Specifically, the Boolean algebra of absolute terms neither provided an adequate analysis of particular categorical statements, nor possessed the means to express relational concepts, nor possessed quantification. From 1880 all categorical judgements could be successfully analysed after the introduction of the notion of subsumption by Peirce. Schröder added quantification to the calculus of propositions in the second volume of Vorlesungen.Footnote ²⁴ Even so, a substantial number of concepts—those whose definition requires the use of relational concepts—could not be defined in the algebra of absolute terms. This means that many judgements could not be expressed by means of this algebra. In summary, the language of the algebra of absolute terms cannot be taken as a lingua characterica.

3.3 Concept-script from Schröder’s perspective

Even though the Boolean algebra upon which Schröder’s work was based in 1880 presented several expressive shortcomings, Schröder’s critique of Frege’s concept-script insisted upon its failure as a fulfilment of a lingua characterica. In fact, in the Begriffsschrift review Schröder deemed the concept-script to be a mere calculus ratiocinator:

The crucial point in Schröder’s diagnosis is that he saw no vocabulary of basic notions in Begriffsschrift. This means that the language of this formal system, lacking symbols for the categories, is unable to define all complex concepts by means of a few. This is, in fact, quite accurate, since besides the logical symbols, the language of the concept-script only has letters, which cannot be considered categories in any sense.Footnote ²⁶

All in all, Schröder considered the concept-script to be a calculus of judgements (Reference Schröder1880, p. 87; 224). In this sense he accused Frege of having created only a calculus ratiocinator. Even in this limited context he did not see that Frege had gone beyond Boolean logic. According to Schröder, Frege failed to provide a language rich enough to be able to define the complex notions of a discipline. Nevertheless, as I shall explain in §5.1, the concept-script, by means of its logical symbols—in particular, its theory of quantification—can successfully express all logical relations that help to build the derived notions of any scientific subject matter. Schröder did not appreciate that the concept-script had expressive resources that Boolean logic lacked. Moreover, Schröder failed to recognise the presence of a formal system in Begriffsschrift, that is, of a set of principles of reasoning (basic laws and inference rules) by means of which all formal steps in a proof could be rendered explicit. According to Schröder’s 1877–1880 conception of a calculus ratiocinator, the principles of reasoning could be incorporated in the calculus as transformation rules based on equalities. The concept-script, seen as a calculus of judgements, was for him a mathematical calculus (see Footnote 10). In this sense, the fact that the concept-script was adequate for being of assistance in proofs was unnoticed by Schröder.

This assessment of Begriffsschrift exposes the fact that Schröder undervalued its second chapter and, moreover, that he completely dismissed its third chapter. In chapter II of Begriffsschrift, not only the propositional apparatus of the concept-script is developed, but also its basic law regulating the use, in the calculus, of the notion of generality. In fact, Frege obtained in this second chapter the formal theorems required to conduct the proofs of the third. Schröder spurned Frege’s unnecessarily complex notation and the generality of the results of chapter III (Reference Schröder1880, pp. 92–93; 230–231), while he failed to recognise that Frege succeeded in justifying that the proof of some theorems that were instrumental in arithmetic did not rely on intuition. The lack of a careful study of the final chapters of Begriffsschrift could have caused Schröder’s partial evaluation.

3.4 Schröder’s algebra of relative terms

Peirce began the development of the logic of relatives as early as (Reference Peirce1870). He made two essential steps for the algebraic development of logic and, in particular, for the progress of the algebra of relatives in his papers, ‘The Logic of Relatives’ (Reference Peirce1883a) and ‘On the Algebra of Logic: A Contribution to the Philosophy of Notation’ (Reference Peirce1885), namely the introduction of indices as individual variables and the interpretation of the generalised product $\prod $ and the generalised sum $\sum $ as quantifiers.Footnote ²⁷

Following Peirce, in the third volume of Vorlesungen Schröder expanded the language and the existing calculus of the algebra of absolute terms in order to build a system that provided an adequate logical analysis of relative terms. This expanded system’s set of categories is the result of adding specific notions of the algebra of relative terms to the existing basic notions of the algebra of absolute terms. Each identical operation (that is, each operation that deals with absolute terms) has a relative counterpart. Accordingly, the primitive notions of this expanded set are the following:Footnote ²⁸ identity ( $=$ ) and relative identity ( $1$ ’); product ( $\cdot $ ), generalised product ( $\prod $ ), and relative product ( $;$ ); negation ( $\overline {\phantom {a}}$ ) and conversion ( $\breve {\phantom {a}}$ ). The basic vocabulary, composed of the symbols for the categories, is enlarged in order to express derived notions in the language. These are sum ( $+$ ), generalised sum ( $\sum $ ) and relative sum $(\oplus)$ ; subsumption ; and the modules ( $0$ , $1$ and $0$ ’).Footnote ²⁹

While the relative operations $(\oplus$ , $;$ and $\breve {\phantom {a}}$ ) are only applied to relatives, the identical operations ( $=$ , $\cdot $ , $\prod $ , $+$ , $\sum $ , $\overline {\phantom {a}}$ and can be interpreted either as operations between relatives or as operations between coefficients (see Footnote 27). A noteworthy example is that of the symbols ‘ $\prod $ ’ and ‘ $\sum $ ’, which can respectively be read as the universal and existential quantifiers when they are applied to coefficients—and not to relative terms. According to this reading, it is possible to quantify over individuals or relatives. This circumstance brings into consideration the two different domains over which $\prod $ and $\sum $ take values. The domain of individuals is denoted by $1^1$ and is called the ‘first-order domain’ (Denkbereich der ersten Ordnung) (Schröder, Reference Schröder1895, p. 5). The second-order domain (Denkbereich der zweiten Ordnung) is the sum of all individual relatives and is denoted by $1^2$ (Schröder, Reference Schröder1895, p. 10). Individual relatives are just classes whose unique element is an ordered pair of elements of the first-order domain.Footnote ³⁰ Thus the relatives are the result of an identical sum of individual relatives.

Departing from these elements, Schröder obtained an algebraic theory, the theory of relatives, which contains as subtheories the calculus of classes and the sentential calculus that had been axiomatised in the first two volumes of Vorlesungen. The theory of relatives was never axiomatised by Schröder: he just offered in the third volume of Vorlesungen a set of postulates (Reference Schröder1895, p. 17; pp. 22–35), which essentially consists in a collection of stipulations (Festsetzungen) that establish the meaning of the basic operations and informal clarifications that amend them.Footnote ³¹ In particular, Schröder provided two stipulations that define $\sum $ and $\prod $ as identical sums and products of relatives, respectively (Reference Schröder1895, pp. 35–36). These stipulations relied on the interpretation of $\sum $ and $\prod $ as sums and products of coefficients, i.e., as quantifiers, which in turn were not defined by Schröder. He offered instead some explanations of the expressions ‘some’ and ‘any’ that quantified expressions obey (Reference Schröder1895, pp. 36–37). Schröder acknowledged that these explanations were not real axioms, and thus that they did not belong properly to the theory of relatives—but then, the laws of quantification, which depended on these explanations, could not be proved in general in the theory of relatives.Footnote ³²

In fact, Schröder did not try to single out the formal apparatus of the theory of relatives. Even though this theory contained a logical language, Schröder never isolated it. Similarly, he did not distinguish the logical principles from the algebraic laws; he only provided a list of stipulations that regulate the meaning of the components of the language.Footnote ³³ Crucially, Schröder did not establish a complete system of rules of inference by means of which the reasoning performed in the theory of relatives could be justified. In the second volume of Vorlesungen, he discussed the validity of some traditional modes of inference (Reference Schröder1891, pp. 256–276), but he never developed a system of rules of inference by means of which the proofs could be fully regimented.Footnote ³⁴ The theory of relatives was developed with the same kind of reasoning that is used in mathematical theories, where the reasoning principles are not explicit. Unsurprisingly, the lack of a formal apparatus for the theory of relatives was of no concern to Schröder.

Schröder pursued the development of the properties of the algebra of relatives, which is an abstract structure constituted of the aforementioned basic operations. Parallel to this interest was Schröder’s belief that all mathematical objects could be viewed as relatives. As he put it in ‘On Pasigraphy’:

This claim not only implies that all mathematical notions can be seen as relatives, but also that there is no need to deal with absolute terms at all. In fact, Schröder was convinced that each statement of the theory of relatives—containing indices, symbols for relatives, identical and relative operations—could be rephrased so as to obtain an equivalent expression in a restricted language which refers only to relatives and binary operations; the resulting expression would thus contain no indices or unitary predicates and all operations would be interpreted as operations between relatives. Schröder referred to this process as a compression or condensation (Reference Schröder1895, pp. 550–551).Footnote ³⁵ The resulting system—which I shall call the ‘calculus of relatives’ —is thus the product of eliminating in the theory of relatives all reference to individuals by means of the condensation of formulas.Footnote ³⁶

Schröder’s pasigraphic project for the reduction of any mathematical theory, and in particular the theory of relatives, to the calculus of relatives can help one understand why he showed no interest in the axiomatisation of the theory of relatives or in formalising any mathematical theory. Schröder offered on many occasions examples of the generality and the expressive power of the calculus of relatives, by means of which several mathematical theories could be expressed. In the third volume of Vorlesungen, he translated Dedekind’s theory of chains into the calculus of relatives (Reference Schröder1895, pp. 346–387).Footnote ³⁷ In this context, he declared:

Moreover, in ‘On Pasigraphy’ (Reference Schröder1899, pp. 54–59) Schröder provided definitions of some of the most basic notions and facts in arithmetic expressed exclusively in the language of the calculus of relatives.

It is thus clear that Schröder aimed at a reduction in terms of the calculus of relatives, that is, at a definition of the basic notions of a mathematical theory in terms of relatives and operations between them, and a reformulation of the axioms and fundamental laws of the theory by means of the language of the calculus of relatives. This reduction should be distinguished from a formalisation. The latter requires a formal language and involves the disassociation of the nonlogical constants of the resulting expressions from the meaning of the primitive symbols of the formalised theory. Schröder never used the calculus of relatives for significant formalisations of mathematical theories.Footnote ³⁸ Although Schröder had at his disposal all the technical elements required to isolate a formal language from the language of the calculus of relatives, he never attempted such a move. The possibility of attributing different interpretations to the nonlogical constants of a formal language was alien to his aim, but such attribution is instrumental in the formalisation of mathematical theories and the adoption of a model-theoretic point of view from which metalogical questions, such as the categoricity of a theory, can be addressed.Footnote ³⁹

To the extent that the calculus of relatives is successful in reducing all mathematical concepts by means of the notions of relatives, this algebraic system can be seen as a pasigraphy and hence as a successful realisation of a Leibnizian lingua characterica. Furthermore, Schröder was convinced that the calculus of relatives was a calculus ratiocinator. This was based on two assumptions. He first assumed that the use of principles of reasoning essentially amounted to the application of identities that consist in rules of transformation of formulas assisted with propositional rules. These purported principles of reasoning were restricted to the calculus of relatives. This could not be seen by Schröder as a limitation, for his second assumption, as we have seen, was that that (almost) everything can be reduced to a relative.Footnote ⁴⁰ Of course, this does not mean that Schröder defended this account when he wrote his review of Begriffsschrift; in 1880 he had not yet developed the calculus of relatives and, in fact, he did not even have a satisfactory axiomatisation of either the logic of classes or the sentential logic.

4.1 Evaluation of available languages

Frege’s notions of an adequate lingua characterica and calculus ratiocinator are tied to the development of his formal system. In some of the papers written between 1880 and 1882 Frege used these concepts as tools for an evaluation of what a formal system should be and subsequently in his defence of the concept-script. In fact, in ‘Booles rechnende Logik und die Begriffsschrift’ (Reference Frege1881) he analysed in some detail the expressive capabilities a language must have to be considered a lingua characterica.

First, Frege shared Leibniz’s and Schröder’s rejection of natural language as a suitable tool for scientific means (Reference Frege1881, pp. 13–14; 12–13). In his view, this language allows for imprecisions both in the expression of conceptual relations and in the use of implicit presuppositions in proofs. Following Frege’s example, two expressions which share a morphological structure, such as ‘Berggipfel’ and ‘Baumriese’ (‘mountain top’ and ‘giant tree’, respectively), can express different conceptual structures (Reference Frege1881, p. 13; 13). Conceptual expressions are not constructed in natural language in such a way that they reflect the structure of concepts. In fact, while the concept ‘Baumriese’ consists in the intersection of the concepts ‘to be a tree’ and ‘to be giant’, the concept ‘Berggipfel’ is only subordinated to the concept ‘to be a top’.Footnote ⁴¹ In ‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’ (Reference Frege1882c, p. 108; 84), Frege also alluded to the ambiguity of words; he mentioned the noun ‘horse’, which can be used to refer either to an individual, or to the species, or to a concept.

Second, Frege discussed the adequacy of mathematical languages as good candidates for use as a lingua characterica. Mathematical languages have a regimented syntax and a concise and univocal way of referring to basic concepts, which are essential features for the avoidance of ambiguities. However, these languages and, in particular, the language of arithmetic, lack those formal resources that are indispensable for the expression of the logical relations that help to define new concepts. In Frege’s words:

As Frege stated (Reference Frege1881, p. 30; 27), many complex concepts in arithmetic cannot be defined exclusively by means of arithmetical language, so they have to be incorporated in the theory with the assistance of natural language. The language of arithmetic’s lack of expressive power is thus mended at the price of the imprecision and ambiguity of natural language. Moreover, the language of arithmetic has no way of formally conducting a proof and, again, must be complemented with natural language in drawing inferences.Footnote ⁴² The fact that arithmetical proofs are spelled out using natural language means that there is no guarantee that their formal steps are fully explicit and, therefore, the correctness of the proof cannot be rigorously warranted.

To summarise, a language which realises Frege’s notion of a lingua characterica has to fulfil both a material and a formal requirement. On the one hand, the language must render the basic concepts and the operations between them in a rigorous and unambiguous way—just like arithmetical language but, significantly, unlike contemporary formal languages, whose nonlogical symbols do not refer to any specific notion. On the other hand, obtaining complex notions requires the language to possess the resources necessary for the expression of statements in which those notions are defined. The calculus—seen as a calculus ratiocinator—is expected to rigorously justify how a judgement is obtained from other judgements in a proof while making explicit all principles of reasoning and implicit presuppositions. According to this account, the resulting language cannot be purely abstract; from Frege’s point of view, the judgements of a lingua characterica must have a definite and specific meaning.

4.2 Boolean logic from Frege’s perspective

Frege’s particular understanding of how the language and the calculus of a logical system should be is a perspective which allows us to substantiate his criticism of the Boolean algebra of logic. The main focus of Frege’s diagnosis is, in his view, the inability of the language of Boolean logic to express content.Footnote ⁴³ In 1880–1882 the Boolean algebra of logic was a theory that studied the properties of the operations of the calculus of classes, which are common to those of the sentential calculus. The subject matter of the Boolean logic of classes consists of operations between concepts taken extensionally, that is, operations between classes that correspond to extensions of concepts. In this sense, algebraic logicians dealt with classes in an abstract sense, even though specific concepts were always mentioned in their examples. Boolean operations between concepts are not relative to any specific extension; they are meant to be applied generally. This is a crucial point, since it is the main reason for Frege’s claim that Boolean logic is a mere calculus ratiocinator.

Therefore, according to Frege, Boolean logic should be considered ‘abstract logic’; as he put it in ‘Über den Zweck der Begriffsschrift’:

The simple operations employed by Boolean logicians as categories cannot really serve to define all concepts in a scientific discipline; many concepts do not simply come from the intersection or the union of two previously available concepts.Footnote ⁴⁴ Besides, the equational perspective again hinders to a great extent the expressive capabilities of the language. In this context, an adequate way of rendering quantification, as well as the propositional connectives, is essential. After the contributions of Peirce to the algebra of logic in (Reference Peirce1883a) and (Reference Peirce1885), these shortcomings were largely solved, at least as a matter of possibility.

Frege also considered the inadequacy of Boolean logic as a calculus ratiocinator. His idea of the rigourisation of a scientific proof not only involved the unambiguous and precise expression of the contents of premises and conclusion; he also referred, on the one hand, to the demand that all principles of reasoning and every formal step must be made explicit; and, on the other hand, to the fact that intuition should not play any part in any formal step (Frege, Reference Frege1881, p. 36; 32). In fact, neither Boolean logic nor Schröder’s algebra of absolute terms—as of 1882—possessed a proper formal system; these systems were not axiomatised and, in particular, they lacked a set of inference rules with which arithmetical reasoning could be fully regimented.Footnote ⁴⁵ As we saw in §3.4, Schröder never produced a true calculus ratiocinator.

Frege’s critique of the Boolean algebra of logic went beyond the limitations of its language and calculus. He also questioned its nature as a unified system of logic and, specifically, the relationship between the logic of classes and the sentential logic. In ‘Über den Zweck der Begriffsschrift’ he expressed himself in the following way:

Since the logic of classes and sentential logic are different applications of the same algebraic system, there should be a way of relating the statements of these two logics—primary and secondary propositions, respectively. Boole tried to relate these two kinds of propositions by appealing to what was—according to Frege—an unfortunate reduction of secondary to primary propositions. In contrast, Frege did not divide his logic into two parts that only diverge on the interpretation of the symbols, but instead provided a unique system of logic. The underlying reason for these divergent strategies is a different evaluation of the main object of logic: “[f]or in Aristotle, as in Boole, the logically primitive activity is the formation of concepts by abstraction, and judgement and inference enter in through an immediate or indirect comparison of concepts via their extensions” (Frege, Reference Frege1881, p. 16; 15). As is well known, Frege rejected the idea that judgements come from the combination of concepts (Reference Frege1881, pp. 16–17; 14–16 and Reference Frege1882b, pp. 100–101; 94) and defended, on the contrary, the view that concepts are obtained through the analysis of judgements. This novel perspective, which broke with tradition, freed logic from the constraints of operations between concepts—which had been deemed insufficient by Frege—and put the focus on the fruitfulness of the decomposition of statements, which were articulated by means of connectives and quantification.Footnote ⁴⁶

In summary, Frege’s criticism of Boolean logic is thus based on four interrelated elements: the fact that its terms do not refer to the notions of specific disciplines, but to abstract classes; its inadequacy for expressing the logical relations that constitute the complex notions of a discipline; the lack of a formal apparatus needed for rigorously and precisely conducting a scientific proof; and the artificial reduction of secondary propositions to primary propositions. In contrast, the overcoming of these shortcomings is one of the leading motivations for the construction of the Begriffsschrift concept-script.

5.1 Application of the concept-script

The logic of Begriffsschrift, the concept-script, is Frege’s attempt to fulfil his ideal of both a lingua characterica and a calculus ratiocinator. As I shall explain in §5.3, one of Frege’s main motivations in writing Begriffsschrift was, in fact, the desire to construct the formal structure necessary to complement scientific languages—one that is capable of being used for the rigourisation of arithmetic.

In many historical studies one point persistently ignored is that what Frege meant as a realisation of this ideal was the use of the concept-script as a tool for scientific disciplines.Footnote ⁴⁷ This use is, nevertheless, so essential that it shaped the nature of the concept-script. As a formal system, the concept-script has a set of basic laws and rules of inference. However, in Begriffsschrift Frege neither introduced any nonlogical constant nor defined the notion of an atomic formula. Specifically, the language of the concept-script, as it was presented in Begriffsschrift, lacks individual constants and predicate symbols. In fact, the only nonlogical symbols of the language presented in Begriffsschrift are letters, which express generality.Footnote ⁴⁸ In Chapter II of Begriffsschrift Frege exemplified the use of a language which, whilst being of use in expressing general logical truths, does not have the means to render any specific meaning. After all, no arithmetical theorem or definition can be expressed exclusively by means of letters (in contemporary terms, variables). The omission both of nonlogical constants and of the notion of atomic formula should not be seen as epochal slips, since they are perfectly consistent with Frege’s aim. The concept-script was meant to be applied to a specific discipline and offer all the tools required for the rigorous definition of complex notions and the reconstruction of the proofs of the discipline. As the first step of such an application, the discipline provided the symbols needed to refer to its categories and derived notions. Accordingly, the language of the concept-script lacked a vocabulary of primitive concepts: it acquired the nonlogical symbols from the subject matter to which it was applied. Arithmetic was for Frege a paradigmatic discipline to which the concept-script could be applied. In his words:

It is thus easy to understand why there is no definition of the notion of an atomic formula in Begriffsschrift. The lack of nonlogical constants (i.e., individual constants, predicate and relation symbols) was seen by Frege as a significant advantage, since it enabled the concept-script to be applied to a wide variety of possible subject matters and provide a symbolisation of their statements. This symbolisation was related to the formalisation which contemporary formal languages perform, particularly in the sense that they do not involve a reduction of a theory to a more general theory. However, Frege’s symbolisation and a formalisation showed some significant differences. The language of the concept-script is not a formal language in the contemporary sense: Frege stated explicitly and insistently that when the concept-script is applied to arithmetic, he did indeed want to preserve the arithmetical symbols and their meaning.Footnote ⁴⁹ In this sense, the symbols of the symbolised discipline are not replaced with nonlogical constants, but are kept in such a way that the resulting statements do express a content.

Frege devoted the second half of ‘Booles rechnende Logik und die Begriffsschrift’ (Reference Frege1881) to exemplify the ability of the concept-script to serve as a tool for a scientific discourse. He saw it as a distinctive advantage in comparison with the expressive capabilities of Boolean logic. For instance, he offered the following example:

“The real function $\varPhi (x)$ is continuous at $x=A$ ; that is, given any positive non-zero number $\mathfrak {n}$ , there is a positive non-zero $\mathfrak {g}$ such that any number $\mathfrak {d}$ lying between $+\mathfrak {g}$ and $-\mathfrak {g}$ satisfies the inequality $-\mathfrak {n}\leqq \varPhi (A+\mathfrak {d})-\varPhi (A)\leqq \mathfrak {n}$

I have assumed here that the signs $<,>,\leqq $ mark the expressions they stand between as real numbers.” (Frege, Reference Frege1881, pp. 26–27; 24)

In this example, atomic expressions of arithmetic remain intact. In particular, Frege did not replace arithmetical terms with nonlogical constants. The concept-script provided a rigorous means to express generality and also the logical relations that link these atomic expressions—in this case, conditional and negation.

Frege’s aim was not only to symbolise—in the explained sense—arithmetical laws, but to provide all the formal resources required to reconstruct arithmetical proofs in purely formal terms. He conceived the application of the concept-script as the combination of two different elements: on the one hand, the incorporation of the syntactic resources of the concept-script to obtain a fully symbolised language and, on the other, the use of its calculus to render explicit all the formal steps in proofs. In ‘Booles rechnende Logik und die Begriffsschrift’ Frege offered a complete reconstruction of the proof of an arithmetical theorem—stating that the sum of two multiples of a number is in its turn a multiple of that number—using the formal resources of the concept-script (Reference Frege1881, pp. 30–36; 27–32).Footnote ⁵⁰ To aim at a formal reconstruction of a proof means, in particular, that all its steps have to be made explicit and expressed by means of the logical formalism. Since some of these steps are purely logical—for instance, proceeding from a generalised statement to that particular case that is relevant in an inference—the concept-script has to provide a way to fill them. This is achieved by means of logical laws. Frege incorporated four logical laws in the above proof—two as premises and two as reasoning principles. These logical laws take the form of concept-script propositions, which cannot be directly applied to a nonlogical proof, since they only contain letters and logical symbols. Therefore, no proper proposition of the concept-script can be incorporated as a premise in the reconstruction of an arithmetical proof; one of its possible applications is used instead. These applications are obtained by means of substitutions: some of the letters occurring in a particular proposition of the concept-script are replaced with the relevant complex expressions, which are typically a combination of concept-script symbols and arithmetical symbols. In ‘Über den Zweck der Begriffsschrift’, while defending himself against Schröder’s attack concerning the two-dimensional nature of concept-script symbolism, Frege explicitly mentioned how propositions of this formal system were incorporated into arithmetical proofs:

To sum up, while the language of the concept-script can be used to formulate statements which express in a rigorous way the relationship that binds the notions of a scientific discipline together, its calculus makes it possible to reflect the inferential relations between the judgements of the discipline. The calculus provides the laws of reasoning—in the form of instances of concept-script propositions and inference rules—that enable one to rigorously conduct a proof of a theorem of a given discipline by making explicit all the formal steps. Since this calculus renders every step in a proof, it allows the elimination of natural language in inferences.

5.2 Concept-script and abstract logic

In ‘Über den Zweck der Begriffsschrift’, Frege claimed: “I did not wish to present an abstract logic in formulas, but to express a content through written symbols in a more precise and perspicuous way than is possible with words” (Reference Frege1882b, pp. 90–91). Frege’s insistence on the need to express content can be understood in the light of the application of the concept-script and its use for the symbolisation of mathematical theories. This particular use demands the development of a formal system which contains all the logical principles and rules by means of which the inferences of a subject matter can be conducted. The concept-script is such a system, that is, a formal system with a set of basic laws and a set of inference rules that allow the proof of logical laws. In this sense, taken in isolation, the formal system of Begriffsschrift is an abstract logic:

Frege defended himself from Schröder’s critique of the concept-script by claiming that, even though the establishment of the basic elements of the concept-script is an inescapable task, this is just one aspect of his whole project.

A development of logic that goes beyond an application to a scientific discipline—such as Frege’s explained use of the concept-script—involves the discovery and formulation of the laws that establish the meaning of the logical notions. A paradigmatic example of this development would be the formulation of laws that characterise the relations between connectives and also quantifiers. Boole, Peirce and Schröder did formulate such laws.Footnote ⁵¹ In contrast, although he had all the expressive and logical means to do so, Frege did not systematically study the relations between connectives in Begriffsschrift.

Actually, one detail that shows Frege’s indifference towards abstract logic is the fact that in Begriffsschrift he only obtained those logical laws that are strictly indispensable for the proofs in chapter III.Footnote ⁵² He did not analyse logical laws by themselves as if logic were its own subject matter.

This circumstance is particularly patent in Frege’s treatment of the Felapton and Fesapo syllogism modes in chapter II of Begriffsschrift. From a logical perspective, these two modes are distinguished because they correspond to different forms of reasoning; they belong to different syllogistic figures, as the order of the terms in the major premise shows:

However, Frege applied these two modes in the following way: let b be an arbitrary individual with a certain property:

Strictly speaking, this argument cannot be taken as a Felapton or Fesapo, because its premises are singular. The significant aspect is the fact that the premises ‘No M is P’ and ‘No P is M’ are not distinguished, since they are applied as ‘b is not f’. However, the distinction between the two major premises of Felapton and Fesapo is of logical interest: in contemporary notation, these two judgements are formalised as ‘ $\neg \exists x(Mx\wedge Px)$ ’ and ‘ $\neg \exists x(Px\wedge Mx)$ ’ respectively, and in virtue of the commutativity of the conjunction one can justifiably claim that they share a meaning. This logical property is of no interest to Frege’s account; it is enough to have one single symbolisation for both premises and, thus, for both arguments:

“We see how this judgement

(59)

replaces one mode of inference, namely Felapton or Fesapo, which are not differentiated here since no subject is distinguished.” (Frege, Reference Frege1879b, sec. 22, p. 51; 163)

In sum, although it is clear that Frege needed to develop a formal system in Begriffsschrift, it would be wrong to understand that he only aimed at obtaining an abstract logic, that is, that he intended to construct a formal system in the contemporary sense.

As I explained in the previous section, the concept-script, unlike Boolean logic, was created as a general tool for the development of a discipline such as arithmetic. Its logical symbols and letters were expected to be combined with the nonlogical symbols of the language of this discipline, which made it possible to incorporate the propositions of Begriffsschrift as laws of reasoning in the proofs of the discipline. Before this application, the formulas of the concept-script, as Frege put it in ‘Über den Zweck der Begriffsschrift’, were “actually only empty schemata” (Reference Frege1882b, p. 103; 97) and its letters were expected to be replaced with relevant combinations of symbols in different contexts. In this way, the calculus—seen as an abstract formal apparatus—could be organically related to the lingua characterica that resulted from the combination of the linguistic resources of the concept-script and the atomic statements of a given discipline.

5.3 Concept-script as lingua characterica and logicism

Some historical studies claim that the core of Frege’s aim of constructing a lingua characterica is the realisation of his logicist project by means of the 1879 concept-script.Footnote ⁵³ In this sense, Frege’s defence of the concept-script against Boolean logic would be closely connected with a vindication of a logicist programme announced in Begriffsschrift.

Frege explicitly addressed the claim that the concept-script was the basis of his fulfilment of the notion of lingua characterica. In order to argue for this claim, he showed that this formal system could be applied to scientific disciplines and, in particular, to arithmetic. However, in doing so, he did not consider the logicist thesis. Quite the contrary, the application of the concept-script—as Frege envisioned it—is incompatible in several ways with a full endorsement of the logicist programme.Footnote ⁵⁴

In the Preface of Begriffsschrift Frege associated the construction of the concept-script with a twofold goal. The main task of this formal system was to establish rigorous foundations for some propositions that are instrumental in arithmetic and prove that their justification does not need to appeal to intuition (Frege, Reference Frege1879b, p. 8; 104). However, Frege also put forward a methodological goal: he aimed at the construction of a formal structure appropriate to complement scientific languages. This goal was associated with Leibniz’s attempt to create a characteristica universalis:

Immediately after this passage Frege expressed his conviction that the concept-script could be successfully applied to geometry, pure kinematics, mechanics and physics as a whole (Reference Frege1879b, p. xii; 106). Considering these examples and particularly what has been discussed in §5.1, the use of the concept-script as the basis of a lingua characterica should be seen as an application, not as a reduction. Frege stressed on several occasions his intention to combine the logical symbols and the letters of the concept-script with the atomic statements of arithmetic. He also conceived the reconstruction of arithmetical proofs as the supplementation of arithmetical laws with those laws of reasoning required to render explicit every formal step.

Frege substantiated and exemplified the application of the concept-script, that is, its use as a basis of a lingua characterica, in the papers written after the publication of Begriffsschrift, between 1879 and 1882.Footnote ⁵⁵ His exposition in these papers does not contribute to the deployment of the logicist thesis. Frege did not provide a single definition of an arithmetical notion exclusively by means of logical notions; on the contrary, he included explicit definitions of arithmetical concepts (Reference Frege1879a, pp. 990–93; 205–208 and Reference Frege1881, pp. 23–29; 21–27), which only relied on simpler arithmetical concepts left undefined. Therefore, according to Frege’s exposition, arithmetic retained its basic notions and consequently its domain of specific objects, relations and operations. In this regard, the generality expressed by the letters occurring in the formulation of arithmetical laws using the formal resources of the concept-script is always limited to the arithmetical domain.Footnote ⁵⁶ Therefore, arithmetical laws were not treated as logical laws in any sense; their validity was considered only in an arithmetical context.

Moreover, in the sole proof of an arithmetical law that can be found in the papers written between 1879 and 1882—the proof of the theorem that “the sum of two multiples of a number is in its turn a multiple of that number” (Reference Frege1881, pp. 30–36; 27–32)—Frege used two arithmetical laws as premises:

“The numbers whose multiples are to be considered are subject to no conditions other than that the following addition theorems:

hold for them (…). Of the theorems of pure logic we principally require that introduced as (84) on p. 65 of the Begriffsschrift (…). In addition we need the formula (4) which is introduced as (96) on p. 71 of the Begriffsschrift.” (Frege, Reference Frege1881, pp. 30–31; 27–28)

Frege even distinguished the quoted arithmetical laws from the “theorems of pure thought” he needed as logical laws in the proof. In consequence, this inference cannot be taken as a proof that an arithmetical law is, in fact, logical; regarding this inference, Frege only set forth formal demands and stressed that “[p]recision and rigour are the prime aims of the concept-script” (Reference Frege1881, p. 36; 32).

All in all, I am not denying here that the concept-script could be tied to the development of the logicist thesis; I argue against the claim that to serve as the vehicle of this thesis was its sole and constitutive function. Disregarding the intuitions that Frege might have had concerning the logical nature of arithmetical truths,Footnote ⁵⁷ Frege’s account of the application of his concept-script, which was essential in the controversy he maintained with Schröder and in his articulation of the notion of lingua characterica, was independent of the logicist programme.