3.1 Kant on Nominal and Real Definitions

According to the Logic lectures, a definition is a distinct, complete and precise concept (Log-W, 24: 912–14; Vanzo Reference Vanzo2010: 151; KrV, A727/B755). A concept is distinct if we clearly cognize its marks, i.e. the partial concepts contained in it (Log-W, 24: 847; Vanzo Reference Vanzo2010: 153). A concept is complete or exhaustive if we have cognition of all the marks or partial concepts contained in it (Log-W, 24: 847, 912; Vanzo Reference Vanzo2010: 154). A concept is precise if it is not superfluous, i.e. no mark is contained in another (Log-W, 24: 912–13; Vanzo Reference Vanzo2010: 154). Finally, definitions are original expositions of concepts (A727/B755). As Vanzo explains Kant, if <<human being>> is used to define <<philosopher>>, and <<rational>> is a mark of <<human being>>, an original exposition of <<philosopher>> will mention <<human being>> but not <<rational>>, the latter being a derived (not original) mark from <<human being>> (Vanzo Reference Vanzo2010: 154).

For Kant, definitions are either nominal or real. Nominal definitions are definitions of names or words (Vanzo Reference Vanzo2010: 156). They signify the logical essence of concepts, i.e. the partial concepts analytically contained in a concept (JL, 9: 143). In contrast, real definitions explicate the (real) essence of objects, explicating the (real) inner ground of that which belongs to the possibility of a thing (Vanzo Reference Vanzo2010: 153, 159; Log-W, 24: 919). Real definitions also enumerate all the essential features of an object, the complete essence of a thing, and thus allow us to strictly distinguish objects falling under the definiendum from other objects, something nominal definitions cannot do (Log-W, 24: 919; Vanzo Reference Vanzo2010: 160–1).

As Nunez has shown (Reference Nunez2014: 635), Kant thinks that real definitions consist of two parts: (i) a nominal definition (a definition in terms of genus and differentia, explicating the logical essence of a concept), and (ii) a proof of the possibility of the concept defined. In the Critique, for example, Kant notes that real definitions make concepts (a) distinct, i.e. we know a collection of marks contained in the defined concept (these marks are given by the nominal definition), and (b) contain a mark that secures the objective reality of the defined concept, i.e. the application of the concept to objects of experience (KrV, A241–2, n.).

As we have seen, real definitions prove the (real) possibility of concepts. To conclude this section, we must therefore analyse Kant’s views on possibility. In the pre-critical Prize Essay of 1764, Kant argued that definitions should not constitute the beginning of metaphysical treatises but rather the end point of inquiry. Metaphysics, according to Kant, is based on certain indemonstrable judgements, which provide data about what is immediately certain about one’s object of inquiry, from which further conclusions are then inferred (NTM, 2: 285). These indemonstrable propositions also provide the data from which definitions in the philosophical sciences can ultimately be drawn (281–2). Hence, the indemonstrable propositions provide the data from which the (real) possibility of concepts can be established (ibid.). In metaphysics, Kant argues, we follow the method of Newton, who derived rules governing motion analytically from experience and geometry. Likewise, in metaphysics we proceed from certain inner experiences which provide us with indemonstrable judgements on the basis of which a science of metaphysics can be erected (286).

While Kant relied on Newtonian verification criteria to establish the real possibility of concepts in his Prize Essay, he introduces a novel concept of (real) possibility in the first Critique. There Kant argues that logical possibility (consistency) is not sufficient to establish the real possibility of a concept (KrV, A220/B268). Rather, empirical concepts are really possible, i.e. applicable to objects of experience, because they are borrowed from our experience of objects, whereas a priori concepts are really possible because they express a condition ‘on which experience in general (its form) rests’ (ibid.). For example, the real possibility of the a priori concept of a triangle is not cognized from concepts, but from the fact that ‘this very same formative synthesis by means of which we construct a figure in imagination is entirely identical with that which we exercise in the apprehension of an appearance in order to make a concept of experience of it’ (A224/B271). It is the fact that the a priori concept of a triangle expresses a condition of experience that secures the objective reality of the concept of a triangle: it shows that this concept is applicable to objects of experience and is thus really possible. Insofar as real definitions secure the real possibility of concepts, they show that the concepts are objectively real or applicable to objects of experience.

3.2 Kant on Mathematical and Philosophical Method

I will argue that Kant adhered to the Classical Model and that this explains his views on systematicity. But first I will explain how his acceptance of the Classical Model is compatible with his critique of Wolff’s mathematical method in the first Critique. For it may be thought that Kant’s critique of the mathematical method implies a rejection of an axiomatic idea of science. Kant’s critique of Wolff’s mathematical method is often taken to imply that in philosophy, in contrast to mathematics, it is difficult or impossible to obtain real definitions (Vanzo Reference Vanzo2010: 160, n. 61; Beck Reference Beck1956: 183). If this is the case, how can Kant accept condition (2) of the Classical Model, which holds that non-fundamental concepts in proper sciences are defined in terms of fundamental concepts? In this section, I argue that Kant’s rejection of Wolff’s mathematical method, which should not be identified with the Classical Model, is consistent with adopting the Classical Model, which captures a conception of science shared by both Wolff and Kant (see for similarities between Wolff and Kant on method Gava Reference Gava2018; on Kant’s critique of the mathematical method, see de Jong Reference de Jong1995). With respect to definitions I argue, following Nunez (Reference Nunez2014), that Kant leaves room for the possibility of giving real definitions of philosophical concepts.

In the Doctrine of Method of the first Critique, Kant criticizes Wolff’s mathematical method by arguing that philosophy should not imitate mathematics by beginning with definitions (B758–9). According to Kant, as we have seen, a characteristic of a definition is that it provides a complete or exhaustive description of all the marks of a defined concept (A727/B755). Since the marks that we think in empirical concepts change in the course of empirical investigation, we cannot, if we adopt an analytic method, be certain that we have a complete specification of all the marks of an empirical concept, i.e. a definition. Similarly, if we adopt an analytic method we cannot be certain that we have a complete specification of all the marks of a priori given concepts in philosophy, and hence cannot be certain that we have attained a definition (A727–9/B755–7). Hence, in the philosophical sciences, if we adopt the analytic method, we often only have what Kant calls expositions, i.e. a possibly incomplete set of marks contained in a concept (A729/B757; JL, 9: 143). Note that such expositions follow the model of concepts as specified by the Classical Model: we describe non-fundamental concepts (species) in terms of more fundamental concepts or marks (genera), even though we cannot be certain that our list of marks is complete. In this sense, Kant’s ideas are consistent with the Classical Model. Kant’s point is that in the philosophical sciences we often cannot be sure whether we have obtained a definition. Nevertheless, definitions are an ideal of science. The Jäsche Logik states that definitions are logical perfections ‘that we must seek to attain’ (9: 143). Hence, Kant sees the model of definitions as represented in the Classical Model as an ideal for science, and accepts the model, even if this ideal is difficult to attain. As Kant puts it: attaining definitions in philosophy ‘is fine, but often very difficult’ (KrV, A731/B759 n.).

If definitions are an ideal for science, it should be possible to obtain them. It has been argued that Kant does not think real definitions can be obtained in philosophy (see e.g. Beck Reference Beck1956: 183). However, Tyke Nunez has argued that Kant does think that philosophical concepts allow of real definitions (Nunez Reference Nunez2014; see also de Jong Reference de Jong1995: 265–6). He interprets Kant’s negative remarks concerning the possibility of real definitions in philosophy as referring to philosophy as it was practised before Kant’s Copernican turn (Nunez Reference Nunez2014: 637). According to Nunez, Kant thinks that, although the traditional method of analysis cannot yield definitions of philosophical concepts, Kant adopts a synthetic method in the first Critique, exemplified by the derivation of the categories, that allows for definitions of philosophical concepts (638–44). Remember that Kant took real definitions to consist of (i) a nominal definition and (ii) a mark through which we can secure the objective reality of the defined concept (KrV, A241–2 n.). Nunez argues that Kant provides such definitions of the categories in the Critique. In the chapter on phenomena and noumena, Kant notes that we cannot define the categories without specifying conditions of sensibility that show the objective reality of the categories (A240–1/B300). As Nunez argues (644–8), Kant goes on to give real definitions of some categories (a) by supplying a nominal definition of the category in terms of genus and specific difference, and (b) by specifying the sensible condition or schema that secures the objective reality of the category. For example, Kant nominally defines the category of magnitude as ‘the determination of a thing through which it can be thought how many units are posited in it’ (A242/B300). The schema that secures the objective reality of this category is described as follows: ‘Only this how-many-times is grounded on successive repetition, thus on time and the synthesis (of the homogeneous) in it’ (ibid.). Nunez’s interpretation supports our reading of the model of definitions described in the Classical Model as an ideal for science.

Kant’s other critical remarks on the mathematical method are also consistent with the Classical Model. In his critique of Wolff’s mathematical method, he says that mathematics has axioms while philosophy does not (KrV, B760–1). However, Kant uses a specific interpretation of the term axiom. Axioms are immediately certain synthetic principles that are intuitive, i.e. that can be constructed in pure intuition (ibid.) Since philosophical principles do not allow of construction, philosophy lacks axioms. However, philosophy does have fundamental principles, as stated by the Classical Model, which are called discursive principles through concepts or acroamata (JL, 9: 110–11; de Jong Reference de Jong1995: 267). These fundamental principles ground the philosophical sciences.

Finally, Kant says that philosophy does not have demonstrations. However, by demonstration he means a constructive proof, as employed in mathematics, which he contrasts with acroamatic (discursive) proofs (KrV, B762). Although philosophy does not make use of constructive proofs, Kant does say that philosophy makes use of discursive proofs. In addition, in line with the Classical Model, he says that cognition must be logically grounded, which means that judgements must be logically derived from higher principles (van den Berg Reference van den Berg2014: 30–4; JL, 9: 51–2; Falkenburg Reference Falkenburg2000: 368–70). Hence, Kant’s critique of the mathematical method in the Doctrine of Method is consistent with adopting the Classical Model.

4.1 Meier and the Classical Model

That Meier endorses the Classical Model becomes clear from his Vernunftlehre (1752). There he notes that concepts are obtained through: experience, abstraction and combination (Meier Reference Meier1752: 416). His account of abstraction states that we obtain more from less fundamental concepts analytically through abstraction until we arrive at fundamental concepts (430–1). We obtain concepts through combination by proceeding synthetically from more simple and more fundamental concepts to more complex and less fundamental concepts (439). Hence, Meier’s account of concepts captures conditions (2a) and (2b) of the Classical Model.

According to Meier, constructing a system of concepts occurs through definitions and logical division. We define lower concepts (species) in terms of their proximate genus and specific difference (Meier Reference Meier1752: 451). Hence, through definitions we systematically relate lower to higher concepts. Constructing a system of concepts also requires that we logically divide higher concepts (genera) into lower concepts (species). This division is guided by the rule that the species must constitute all the concepts contained under the higher concept, i.e. the species must constitute the total extension of the divided concept (437–74; see also Kant at JL, 9: 146–7, discussed in section 4.8).

Meier also distinguishes between indemonstrable judgements, which do not require any demonstration, and demonstrable judgements, which require a demonstration (Meier Reference Meier1752: 514). The indemonstrable judgements are the fundamental judgements on the basis of which we demonstrate non-fundamental judgements in science. Hence, Meier accepted conditions (3a) and (3b) of the Classical Model.

The Classical Model illuminates how, according to Meier, systems of concepts and judgements are constructed. Systems of concepts are constructed by defining non-fundamental concepts on the basis of fundamental concepts, and systems of judgements are constructed by deducing non-fundamental judgements from fundamental judgements. In the next section, we will further explore the connection between axiomatics and systematicity in Meier.

4.2 Wolff and Meier on Mathematical Method: How to Construct Systematic Sciences

In the chapter on method, Meier discusses how we can construct systematic sciences. He stresses the importance of the mathematical method, which Wolff equated with the scientific method (Meier Reference Meier1752: 634). The mathematical method is a variety of the Classical Model, though not identical with it. It states that non-fundamental scientific concepts must be defined in terms of fundamental concepts, and that non-fundamental scientific propositions must be deduced from fundamental propositions (van den Berg and Demarest Reference van den Berg and Demarest2020: 383).

The nature of the mathematical method was explained by Wolff in Die Anfangsgründe aller Mathematischen Wissenschaften (1750 [1999]).Footnote ⁷ Wolff claims that the mathematical method proceeds from definitions to axioms, and from axioms to theorems and problems (5). Definitions, which are either nominal or real, ground axioms. For example, if we define a circle by giving the procedure of moving a straight line around a fixed point, we can infer that all the straight lines that are drawn from the centre to the perimeter are equal to each other (16). This truth is an axiom (Grundsatz). Axioms either show that something is the case, as in the example of equal radii, or that something can be done or constructed, for example that between any two points we can draw a straight line (17). The first type is called axiomata and the second postulata. Since the truth of axiomata and postulata immediately follows from definitions, they do not require proof (ibid.). We derive theorems through strict demonstrations from definitions, axiomata and postulata (21, 25, 27–8; Wolff 1716 [1965]: 501).

Wolff’s mathematical method presents us with an axiomatic method that derives theorems from definitions and axioms through logical proofs. Wolff and Meier take this axiomatic method to be the paradigmatic way of constructing systematic sciences (Hinske Reference Hinske and Funke1990: 159). Having described how Meier’s adherence to an axiomatic ideal for science explains his idea of systematicity, we may now turn to Meier’s discussion of the logical perfections of scientific knowledge. I will show that systematicity can be taken as an ideal for science because it furthers these perfections of knowledge.

4.3 Systematicity and Extensiveness in Meier

According to the perfection of extensiveness, cognition must be applicable to as many objects as possible (Meier Reference Meier1752: 41). The term cognition (Erkenntniß) refers to both concepts and judgements. Hence, the concepts and judgements of a science must be applicable to as many objects as possible.

Systematicity furthers the ideal of extensiveness. According to Meier, systems of concepts are constructed by means of definitions and the logical division of concepts (see section 4.1). The logical division of higher concepts into lower concepts must, as we have seen, satisfy the following rule: (i) the species must constitute the total extension of the divided concept. Condition (i) ensures that a division is complete and that the constructed system of concepts has the widest possible extension. Constructing a system of concepts in accordance with condition (i) thus furthers extensiveness.

4.4 Systematicity and Fruitfulness in Meier

The perfection of fruitfulness concerns the number of consequences that follow from a cognition: if a cognition has many consequences, Meier calls it fruitful (Meier Reference Meier1752: 101). Fruitfulness is a maxim of maximality. It tells us that scientific principles should allow for the derivation of the maximum number of consequences. It is related to the maxim of parsimony, which tells us that sciences should be based on a minimal set of principles. These two maxims combined give rise to the following maxim: in science, we should choose the smallest set of principles (parsimony) that allow us to derive the maximum number of consequences (fruitfulness). This maxim expresses an axiomatic ideal (de Jong and Betti Reference Betti2010), which directs us to derive as many consequences as possible from a minimal number of principles. Insofar as systematic sciences can be viewed as axiomatic sciences, systematic sciences can be taken to further the perfection of fruitfulness.Footnote ⁸

4.5 Systematicity and Truth in Meier

Meier’s account of truth corresponds to condition (4) of the Classical Model. Meier specifies two criteria for truth (Meier Reference Meier1752: 130). The first mark of truth is internal possibility: a true judgement must not be contradictory (131–2). The second mark of truth can be called grounding: a judgement is true if (i) it is a consequence of true grounds and (ii) it is a ground of consequences all of which are true (133). Condition (i) is plausible because we take something to be true on the basis of proofs, which show that something follows from true grounds. Condition (ii) is plausible because of scientific practice, in which we prove the truth of hypotheses by showing that all their consequences are true (134–5).

That systematicity is necessary for knowledge of truths follows from the marks of truth. We can say that a judgement is true if (i) it follows from true grounds and (ii) all its consequences are true. Hence, knowledge of true judgements requires that we systematically order judgements as a deductive interconnection of grounds and consequences.

4.6 Systematicity, Clarity, Distinctness and Completeness in Meier

Other perfections of cognition are the clarity, distinctness and completeness of cognition. These notions can be traced to Wolff, and before Wolff to the views of Descartes and Leibniz. According to Wolff a concept is clear if we can identify the things to which it applies (Wolff 1754 [1978]: 126; van den Berg Reference van den Berg2014: 19). A concept is distinct if we know its marks, i.e. the partial concepts contained within the concept (128; van den Berg Reference van den Berg2014: 19). Finally, a concept is complete if we know all its marks and if the marks we know are sufficient for knowing the things represented by the concept and distinguishing them from other things (129; van den Berg Reference van den Berg2014: 19). Meier adopts similar concepts of clarity, distinctness and completeness (Meier Reference Meier1752: 184, 214, 235).

Distinctness and completeness are perfections of cognition that are improved through establishing systems of cognition. If we construct systems of genera and species, we make the species distinct by explicating the genera that are contained in these species. In turn, we further the completeness of an analysed concept by explicating as many of the marks or more fundamental concepts that are contained in a concept. By constructing systems of concepts we thus improve the distinctness and completeness of cognition.

4.7 Systematicity and Certainty in Meier

Meier’s account of the perfection of certainty corresponds to condition (6) of the Classical Model. Certainty is that which secures that we know a true judgement (Meier Reference Meier1752: 251). In science, according to Meier, we aspire to logical certainty, which is knowledge of a truth through knowledge of its grounds. He distinguishes between exhaustive and inexhaustive certainty (255–6). If we know all of the grounds of a true judgement, as in mathematical demonstrations, we have obtained exhaustive certainty. Establishing a systematic and deductive interconnection of grounds and consequences is necessary for obtaining logical and exhaustive certainty.

4.8 Kant on Systematicity as a Logical Perfection

Having described Meier’s views on systematicity, the Classical Model and the logical perfections, we may now turn to Kant. Kant describes systematicity as a logical perfection (JL, 9: 139–40).Footnote ⁹ The ideal of systematicity is related to the procedure of constructing systems of concepts. Kant states that a system of concepts depends on the ‘distinctness of concepts both in regard to what is contained in them and in respect of what is contained under them’ (ibid.). As was the case for Meier (section 4.1), this means that a system of concepts is constructed by specifying both the intension of concepts through definitions and the extension of concepts through logical division (Longuenesse Reference Longuenesse and Wolfe1998: 150–1; JL, 9: 140). In the following, I describe how definitions and logical division give rise to a system of concepts.

Kant distinguishes between synthetic and analytic definitions and between nominal and real definitions (JL, 9: 141–4). Synthetic nominal definitions are made through the arbitrary combination of concepts. We can, for example, give a nominal synthetic definition of the concept ‘square’ by combining the concepts ‘four-sided’, ‘equilateral’ and ‘rectangle’ (Log-D, 24: 757). In this way, we define less fundamental concepts (species) in terms of more fundamental concepts (genera and differentia), proceeding synthetically from the simpler, more fundamental to the complex, less fundamental concepts.

Synthetic real definitions are, as we have seen (section 3.1), based on synthetic nominal definitions. They consist of (i) a nominal definition and (ii) a proof of the real possibility of the defined concept. In mathematics, we prove the possibility of the defined concept through a constructive procedure that shows how an object is possible (JL, 9: 141). For example, we can define a circle by showing that it can be constructed by letting a straight line move around a fixed point. This construction is based on a synthetic nominal definition proceeding from genus and differentia to species, e.g. the nominal definition of a circle as a (curved) line (genus) whose points are all equidistant from a centre point (differentia) (see Nunez Reference Nunez2014: 633). The same is true for definitions of the categories. They consist of (a) a nominal definition of the category in terms of genus and specific difference, and (b) a specification of the schema that secures the objective reality of the category (Nunez Reference Nunez2014: 644–8).

To conclude: all synthetic real definitions are based on synthetic nominal definitions. This means that when giving synthetic real definitions we define complex, less fundamental concepts in terms of simpler and more fundamental ones and we proceed synthetically from simple, fundamental to complex, less fundamental concepts. Kant’s views on synthetic definitions thus capture conditions (2a) and (2b) of the Classical Model. Through defining concepts in terms of more fundamental ones, we construct a system of concepts.

Analytic definitions are constructed by analysing given concepts and making these concepts distinct and complete (JL, 9: 142). Using this method we proceed analytically from the more complex, less fundamental to more simple and fundamental concepts, and define less fundamental in terms of more fundamental concepts by means of genus and specific difference. Kant’s views on analytic definitions thus likewise capture conditions (2a) and (2b) of the Classical Model. As we have seen, however, Kant is sceptical about obtaining analytic definitions of empirical and a priori philosophical concepts.

Definitions bring about a system of concepts by specifying their intension. According to Kant, as was the case for Meier (section 4.1), the construction of a system of concepts also requires the determination of the extension of concepts through logical division (JL, 9: 146–7). Through logical division we obtain lower concepts (species) from higher concepts (genera). In the Jäsche Logik, Kant notes that the logical division of concepts must satisfy the following rules: the species (a) exclude each other; (b) belong under one higher concept; and, again like Meier, (c) constitute the total extension of the concept that is divided (van den Berg Reference van den Berg2014: 22; Anderson Reference Anderson2005: 29; JL, 9: 146–7). By following these rules, we systematically determine the extension of a concept.

Kant’s views on the systematicity of judgements are articulated in the doctrine of proof of the Jäsche Logik. A proof consists of three parts: (i) conclusion, (ii) ground of proof, (iii) the way in which (i) follows from (ii) (Log-D, 24: 748; Zinkstok Reference Zinkstok2013: 96–7). Through proofs we systematically relate grounds to their consequences. Proofs must terminate in fundamental principles (JL, 9: 71). Hence, Kant’s doctrine of proof captures conditions (3a) and (3b) of the Classical Model. Through providing proofs from fundamental principles we construct systems of judgements. In the next section, we discuss the similarities between Meier’s and Kant’s accounts of the logical perfections of knowledge.

4.9 Kant on Systematicity and the Perfections of Fruitfulness, Extensiveness and Truth

Kant discusses Meier’s account of the perfections in his logic lectures and accepts most of them (Falkenburg Reference Falkenburg2000: 355–66). However, in his lectures he does not explicitly relate these perfections to systematicity. Kant does relate the idea of systematicity to the perfections in the first Critique, where, as Falkenburg notes (Reference Falkenburg, Fulda and Stolzenberg2001: 312), Kant states that the goal of the systematic unity of knowledge amounts to the goal of satisfying logical perfections of knowledge (A838/B866). If we consider Kant’s remarks on systematicity in the first Critique, we find connections between the ideal of systematicity and the ideals of fruitfulness and, by implication, extensiveness and truth. Kant argues that through the systematic ordering of cognition, we achieve unity alongside the greatest possible extension (A644/B673). How a system furthers unity alongside maximal extension becomes clear if we discuss Kant’s remarks on systematicity in relation to the perfection of fruitfulness.

In our discussion of fruitfulness, we have introduced the following maxim: science should be based on the smallest set of principles (parsimony) that allow us to derive the maximum number of consequences (fruitfulness). Kant articulates this maxim in the section ‘On the regulative use of the ideas of pure reason’, where he discusses the maxims of genera and specification (see Guyer Reference Guyer1990: 21–34; 2003: 282; Falkenburg Reference Falkenburg2000: 382–5 discusses these maxims and also the logical perfections in this context). There he notes that systematic sciences are brought about by following what he calls the maxim of genera, according to which we subsume species under genera, then subsume these under higher genera, and so forth until we reach a highest genus (KrV, A651–2/B679–80). Applied to systems of judgements, this maxim expresses the idea that sciences should be based on a minimum number of principles. However, Kant notes that in science we should also follow the maxim of species, which directs us to maximally specify each species into subspecies, these subspecies into further subspecies, and so forth (A654–5/B682–3). Applied to systems of judgements, this maxim tells us that in science we need to choose principles that allow for the derivation of the most consequences. It is by following these two maxims that we establish systematic sciences (A658/B686). Hence, systematic sciences follow the maxim that we should choose the smallest set of principles (parsimony) that allow us to derive the maximum number of consequences (fruitfulness). This analysis also suggests that Kant adopted the perfection of extensiveness, and that he took systematicity to further the ideal of extensiveness, because his maxim of species directs us to continuously divide species into subspecies, thus ensuring that our cognition is applicable to as many objects as possible.

There are also similarities between Meier’s and Kant’s remarks on truth. Kant argues that systematicity is necessary for cognizing truths. Without systematic unity we have ‘no sufficient mark of empirical truth’ (KrV, A651/B679). Remarks such as these have led to much debate. Allison (Reference Allison and Sedgwick2000 82: 84–6) argues, following Ginsborg (Reference Ginsborg1990: 171–92), that systematicity is necessary for the formation of determinate empirical concepts, and thus a necessary condition for knowledge of empirical truths. This may be true, but we can explain Kant’s ideas in a simpler fashion if we take into account the position of Meier.

According to Meier, as we have seen, we know that a cognition is true if it is (i) a consequence of true grounds and (ii) a ground of consequences all of which are true. Hence, a systematic interconnection of grounds and consequences is necessary to cognize truths. In the Jäsche Logik, Kant puts forward a similar theory, arguing that a true cognition (x) has (true) grounds from which it follows and (y) does not have false consequences (JL, 9: 51). With respect to (y), Kant notes that ‘from the truth of the consequence we may infer the truth of the cognition as ground, but only negatively: if one false consequence flows from a cognition, then the cognition itself is false’ (JL, 9: 52). Hence, in line with Meier’s condition (ii), we may infer from the truth of all consequences to the truth of the ground, although Kant notes that one cannot know the totality of consequences of a ground and that we can therefore only use this criterion to establish hypothetical or probable truths (ibid.). In the first Critique, Kant relates Meier’s condition (ii) to what he calls the hypothetical use of reason. If we employ the hypothetical use of reason, we assume a universal problematically, and test whether several particular consequences follow from it. If these consequences follow from the rule, then the universality of the rule is inferred, although, again, we can only approximate universality through this mode of inference (A 646–7/B674–5).

The hypothetical use of reason thus tests whether the consequences that follow from a rule are true and infers to the universality of the rule itself. As such, the hypothetical use of reason illustrates Meier’s condition (ii). Since we know truths on the basis of a systematic interconnection of grounds and consequences, it is no surprise that Kant claims that systematicity is necessary for knowing (empirical) truths. The basic idea adopted by Meier and Kant is that we can only know truths (whether apodictic or probable) through inferential relations, and this presupposes that cognitions are systematically ordered.

We have seen that, for both Meier and Kant, the ideal of systematicity and the logical perfections of knowledge are intimately related. A properly systematic science satisfies the logical perfections of knowledge, which all sciences should satisfy to the highest extent. Hence, it is no surprise that Meier and Kant took systematicity to be an ideal for science.

5.1 Lambert on Scientific Knowledge, Completeness and the New Mathematical Method

In his Neues Organon (1764), Lambert distinguishes between common knowledge and scientific knowledge (see also Watkins Reference Watkins, Dyck and Wunderlich2018: 181–4). Scientific knowledge is characterized by dependency relations, whereas common knowledge is not (Lambert Reference Lambert1764: 389). Lambert construes sciences as deductive systems in which consequences are deduced from fundamental principles (394–5). Hence, he construes sciences as axiomatic deductive systems (Wolters Reference Wolters1980: 51, 55). Lambert states that scientific judgements are regarded as a systematic whole, in which every judgement is related to others (390). Hence, an axiomatic science, in which non-fundamental judgements are demonstrated on the basis of fundamental ones, satisfies Lambert’s criteria for a system of judgements.

According to Lambert, systematicity is the defining characteristic of scientific knowledge (Sturm Reference Sturm2009: 141). Lambert also argues that completeness is a characteristic of systematic sciences. Sturm (142) notes that completeness for Lambert involves having a principle that guides research. In the following, I will extend Sturm’s analysis and explain the notion of completeness.

In his Architectonic, Lambert states that completeness is a perfection of the foundational sciences (Lambert Reference Lambert1771: 29). In his Neues Organon, he notes that a scientific doctrine is complete if we can specify how all of its parts are to be treated (1764: 75). As an example of a complete science, Lambert mentions trigonometry (Reference Lambert1764: 406; cf. 1771: 13). The example of trigonometry illustrates Lambert’s notion of completeness. In his Mathematisches Lexicon (1716), Wolff defined trigonometry as a science that allows one to derive from three given parts of a triangle the other three parts. Wolff thinks that all problems in trigonometry reduce to a limited number of cases: (i) if two sides are given and one angle, to find the other two angles and the third side, (ii) if two angles are given and one side, to find the other two sides and the remaining angle, (iii) if three sides are given, to find the three angles, and (iv) in spherical triangles, if three angles are given, to find the three sides (Wolff 1716 [1965], 1428–9).

The cases above describe all the (types of) problems of trigonometry and the way in which they are to be treated. It is this feature that Lambert associates with completeness: it is the enumeration (Abzahlung) of all cases (Fälle) and the specification of the rules (Regeln) in accordance with which these cases are to be treated (traktirt) that makes a doctrine complete (1764: 75). This matches Lambert’s general idea of a system as a whole with parts that are related to each other (Waibel Reference Waibel, Stolzenberg and Rudolph2007: 53). Interestingly, Lambert also remarks that, since the realm of truths is infinite, completeness is often an ideal that we can only approximate (Reference Lambert1771: 30–1). In the next section, we will see that Kant argues, like Lambert, that the completeness of a systematic science consists in being able to say how all of its parts are to be ordered. In addition, like Lambert, he argues that the completeness of a science is an ideal that we can only approximate.

If we turn our attention to Lambert’s Anlage zur Architectonic (1771), it becomes clear that he rejected Wolff’s mathematical method. According to Lambert, as Heis and several other interpreters have shown (Heis Reference Heis2014: 616–17; Dunlop Reference Dunlop2009: esp. 48–54; Laywine Reference Laywine, Domski and Dickson2010: 114–21; Wolters Reference Wolters1980: 51–4), not every concept can be defined, since some concepts are simple. Because simple concepts cannot be defined, propositions that contain only simple concepts cannot be inferred from definitions. Hence, pace Wolff, postulates and axioms, which contain simple concepts, cannot be inferred from definitions. Rather, the possibility of every concept that is defined needs to be proven on the basis of postulates or axioms, which show how a certain concept is possible or consistent (Heis Reference Heis2014: 616–17; on Lambert’s postulates, see Laywine Reference Laywine, Domski and Dickson2010: 114–21; Dunlop Reference Dunlop2009: 52–62; Wolters Reference Wolters1980: 88–95; Wellmann Reference Wellmann2017: 142–6). In short, definitions do not ground axioms and postulates, but the converse is true: axioms and postulates ground the possibility of definitions (Lambert Reference Lambert1771: 10–12). This new mathematical method distinguishes Lambert’s conception of axiomatics from that of Wolff.

Although Lambert adopted a new mathematical method, he accepted the axiomatic conception of science articulated by the Classical Model. In the Anlage zur Architectonic, Lambert specifies the simple concepts that ground the so-called foundational sciences (1771: 46; Wellman Reference Wellmann2017: 141–2). For example, Lambert argues that the concept of identity, together with the concepts of force and solidity, form the basis for the foundational science called calculus quantitatum (47). Similarly, the fundamental concept of extension, together with the concept of unity, constitutes the basis of geometry (51). In this way, Lambert considers relations between simple concepts and specifies which combinations of concepts form the basis of which science. Lambert’s simple concepts ground non-fundamental concepts insofar as non-fundamental concepts are composed of simple ones (Reference Lambert1764: 420–2). As Lambert puts the point, if we analyse composite concepts (zusammengesetzte Begriffe) we arrive at simple concepts (einfache Begriffe), which provide the foundation of all knowledge (ibid.). Hence, Lambert accepts conditions (2a) and (2b) of the Classical Model. Moreover, through combining simple concepts Lambert obtains fundamental propositions of sciences, since axioms and postulates consist of combinations of simple concepts (1771: 20). From these fundamental propositions, we derive non-fundamental ones, as stipulated by conditions (3a) and (3b) of the Classical Model. Hence, Lambert adopts the Classical Model and this model elucidates his conception of systematic science.

5.2 Kant on the Completeness of Systematic Sciences

Kant follows Lambert by arguing that systematic sciences are complete and that completeness is an ideal for science. In the first Critique he notes that a system is based on an idea of the whole that precedes and determines each part (and hence is complete):

The nature of this idea of the whole is obscure. Zinkstok (Reference Zinkstok2013: 95) argues that Kant took this idea to be an idea of the highest genus of a Porphyrian tree. This is partly correct. Kant often construes the highest genus of a system of concepts as a transcendental idea that functions as a focus imaginarius when unifying our cognition. These ideas specify a certain domain of investigation, and thus allow us to distinguish a particular science from another science in terms of their different domains. For this reason, Kant claims that through transcendental ideas the domain (Umfang) of a science is determined (Rauscher Reference Rauscher and Guyer2010: 296).

However (as an anonymous referee has stressed), it is not clear how a transcendental idea confers unity among the parts of a science. In our discussion of Lambert, we have seen that he took complete sciences to be sciences in which all the parts are specified and we have rules in accordance with which to treat these parts. Kant adopts a similar idea, insofar as he argues that a systematic and complete science is a science based on an idea of the whole, which means that we have a priori rules that allow us to relate all the parts of a science. It is for this reason that Kant claims that through the idea of the whole ‘the position of the parts with respect to each other is determined a priori’ (KrV, A 832/B 861).

An example from natural history illustrates the idea of a whole. (Here I follow van den Berg Reference van den Berg2014: 23–4, which draws on Müller-Wille Reference Müller-Wille2007: 546–7; Anderson Reference Anderson2005; Oittinen Reference Oittinen and Onnasch2009: 63–9. On the impact of Kant’s natural history and theory of race on his account of systematicity, see Sandford Reference Sandford2018.) In Linnaeus’ classification of plants, he divides the plant realm into classes by specifying the number of stamens, and into orders by specifying the number of pistils. We provide definitions of classes by specifying the genus ‘plant’ and by specifying the number of stamens (‘with one stamen’, ‘with two stamens’, etc.). We further specify the classes and provide definitions of orders by specifying a particular class as genus and by specifying the number of pistils (‘with one pistil’, ‘with two pistils’, etc.). We thus have a priori rules that govern the construction of a system of concepts and that determine the relation between these concepts. This is Kant’s point when he says that a system is based on an idea of the whole that determines the place of each part: we must, as was stressed by Lambert, have rules that specify how we are to treat and relate the parts of a science. For Kant, moreover, a complete empirical specification of genera and species in natural history (a ‘finished natural history’) is a regulative ideal. Hence, like Lambert, Kant saw completeness in this sense as an ideal for science that we can only approximate.