The Wolffian Paradigm and its Discontents

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R. Lanier Anderson

The Wolffian Paradigm and its Discontents: Kant’s Containment Definition of Analyticity in Historical Context 0

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by R. Lanier Anderson (Stanford)*

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Abstract: I defend Kant’s definition of analyticity in terms of concept “containment”, which has engendered widespread skepticism. Kant deployed a clear, technical notion of containment based on ideas standard within traditional logic, notably genus/species hierarchies formed via logical division. Kant’s analytic/synthetic distinction thereby undermines the logico-metaphysical system of Christian Wolff, showing that the Wolffian paradigm lacks the expressive power even to represent essential knowledge, including elementary mathematics, and so cannot provide an adequate system of philosophy. The results clarify the extent to which analyticity sensu Kant can illuminate the problem of a priori knowledge generally.

1. Introduction: Containment Analyticity and the Wolffian Paradigm Kant defines analyticity in terms of concept “containment”: a judgment is analytic just in case “the predicate B belongs to the subject A as something that is (covertly) contained in this concept A” (A 6/B 10)1. Few recent philosophers have been satisfied with the official definition. Many endorse the classical criticism, dating back to Maaß (1789), that the notion of containment is hopelessly obscure, because it must

*0 For comments on earlier versions of this material, my thanks are due to Kit Fine, David Hills, John MacFarlane, Alison Simmons, and Ken Taylor, and audiences at Berkeley and NYU. In its current shape, the paper benefited from helpful criticisms by Michael Friedman, Gary Hatfield, Nadeem Hussain, Paul Lodge, Beatrice Longuenesse, Katherine Preston, Lisa Shabel, and Daniel Sutherland, and from audiences at Villanova, Wisconsin/Milwaukee, the New England Early Modern Philosophy Seminar, and HOPOS 2002. The research was supported by a fellowship at the Stanford Humanities Center, which I gratefully acknowledge. 1 Citations to Kant, Aristotle, and Leibniz use abbreviations noted in the references; other citations identify works by date of publication, sometimes including the date of a relevant earlier edition in square brackets [ ].) Archiv f. Gesch. d. Philosophie 87. Bd., S. 22 –74 © Walter de Gruyter 2005 ISSN 0003-9101

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appeal to contingent, potentially idiosyncratic facts about what one speaker or another happens to associate with a concept.2 Others worry that the containment definition is too narrow to cover all the claims Kant counts as analytic.3 It is unclear, for example, how to extend it to logical truths that are not expressed in categorical (‘S is P’) propositions. Below, I show that still further puzzles arise, once we locate the idea of containment in its eighteenth century logical and metaphysical context. It has therefore been tempting to abandon the containment definition, and define analyticity instead through the principle of contradiction. For Kant himself, though, containment remained central. In his initial introduction of the analytic/synthetic distinction (A 6–7/B 10), the core notion of containment serves to explain the other key features of analyticities (i.e., their status as identical and non-ampliative propositions).4 The definitional priority of what is “thought in” the subject concept is apparent even in Kant’s explicit argument that the principle of contradiction is the “supreme principle of all analytic judgments” (see A 150–2/B 189–91): if the judgment is to be analytic, […] its truth must always be able to be cognized sufficiently in accordance with the principle of contradiction. For the contrary of that which, as a concept, already lies and is thought in the cognition of the object is always correctly denied, while the concept itself must necessarily be affirmed of it, since its opposite would contradict [it]. [A 151/B 190–1]

That is, the principle of contradiction is the “completely sufficient principle of all analytic cognition” (A 151/B 191) because in analyticities the

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Versions of this charge are very widespread: Allison 1973, 42–5, offers helpful discussion of Maaß’s argument. See also, e.g., Beth 1956/7, 374; Beck 1965, 77–80; Bennett 1966, 7; Brittan 1978, 13–20; Allison 1983, 73–5; Kitcher 1990, 13, 27; and to some extent, Parsons 1992, 75. (Longuenesse 1998, 275f., et passim, is a notable exception to this tradition.) Later scholars in the interpretive line (e.g., Kitcher 1990, 27) often echo the famous criticisms of analyticity by Quine (1961 [1953]), as well as the argument due to Maaß. See, e.g., Shin 1997; Van Cleve 1999, 19–21. Kant first defines analyticities in terms of containment (A 6/B 10, quoted above), and goes on to write that “Analytic judgments are thus those in which the connection of the predicate is thought through identity. […] One could also call [them] judgments of clarification […] since through the predicate [they] do not add anything to the concept of the subject” (A 7/B 10; first and last italics mine). For an interesting recent discussion which emphasizes the definition in terms of the principle of identity/contradiction, by contrast to the focus on containment defended here, see Proops, forthcoming.

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predicate is “thought in” the subject. Here, then, the class of analyticities is identified by the containment criterion, and the principle of contradiction just accounts for their truth, in that the opposite of any containment analyticity is a contradiction. So on Kant’s view, analyticities are true by contradiction. Note, however, that some further argument would be required to show the converse – that every truth following from the principle of contradiction is analytic. Kant offers no such argument, leaving containment as the key defining mark of analyticity.5 I will argue that containment has such priority for Kant – and rightly – due to its salience for the logico-metaphysical paradigm his analytic/synthetic distinction was designed to rebut – the system of Christian Wolff and his followers.6 Clear indications of this target can be seen in the paragraph just preceding Kant’s introduction of the distinction (A 3–6/B 7–10). There, Kant argues that previous philosophers failed to appreciate the real problems about a priori knowledge for two main reasons: first, the “splendid example” (A 4/B 8) of mathematics enticed them to overestimate the power of reason; and second, “perhaps the greatest part of the business of our reason consists in analyses of concepts” (A 5/B 9; my ital.), which are in fact unproblematic. This combination points directly to Wolff, who treated mathematics as the prototype for the rest of knowledge, and exploited the analysis of concepts as a central tactic in his project of reforming science to accord with the math-

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Kant never explicitly addresses the possibility that truths of general logic, based on the principle of contradiction, might not turn on containment, and gives no argument to rule out that possibility. As we shall see, there are such truths, and so the two definitions of analyticity (in terms of containment and in terms of the principle of contradiction) come apart. In section 4 below, I suggest reasons for Kant to prefer the containment definition, as he does in the quoted passages. Here I simply note that such a position is consistent with Kant’s thesis that the principle of contradiction is the supreme principle of analytic judgments: it could be the principle of all analyticities, and simultaneously explain additional truths as well. Indeed, Kant insists that the principle is just as binding on synthetic truths as on analytic ones (B 14, A 150–1/B 189–90). Various doctrines of Leibniz, Wolff, and their German rationalist followers played a central and often underappreciated role in shaping Kant’s thought, as we have been reminded by significant recent scholarship, including work by Longuenesse (1998, 2001) and Laywine (1995), among others. The present study follows Longuenesse in highlighting the importance of the traditional logic for understanding core metaphysical views in Kant and his predecessors.

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ematical model.7 Wolffianism, then, is the paradigm “dogmatic” philosophy that gets impeached by the Kantian critique when it insists on the importance of synthetic judgments, and thereby recasts the problem of a priori knowledge (see B xxxv–vii, A 855/B 883, B 23–4).

The key feature of Wolff ’s philosophy, for present purposes, is its aspiration to uncover the rational structure of the world by revealing every truth as a conceptual one.8 For Wolff, God creates the world by identifying the best possible system of conceptual essences, and then realizing them. The goal of inquiry is therefore to find the adequate concepts, which are delivered by analysis, through which we render concepts fully distinct (Wolff 1965 [1754], 128–35). Analyses trace the logical relations among concepts by revealing what constituent “marks” they contain, so the logic of concepts turns out to be a study of their containment relations. Since mathematical arguments best exemplify the logical clarity he wants, Wolff proposes to reformulate all theoretical reasoning, including empirical science, in the same strict logical form. In such a system, every truth would be analytic, in the sense of Kant’s containment definition. Given this context, we can begin to see the dialectical force of making an analytic/synthetic distinction. Kant’s complaint against Wolff is that while some few judgments are true by analytic containment, almost all propositions of genuine cognitive interest – truths of experience, of natural science, and tellingly, also Wolff ’s paradigmatic mathematical judgments – are not even capable of expression within a Wolffian system, because they are synthetic, not analytic. If Kant’s distinction is sound, then Wolff ’s ideal of a purely conceptual system of metaphysics is unsustainable.

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This method of analysis of concepts served as the terminological inspiration for Kant’s analytic/synthetic distinction, which applies to containment relations among concepts in a judgment. This Kantian sense of ‘analytic’ and ‘synthetic’ is related, but not identical, to the older distinction between analytic and synthetic methods of procedure in demonstrative science. A full account of the different terms must await another occasion, but to avoid misunderstanding, it is important to note here that Wolff makes tactical use of the method of analysis of concepts in the service of an overall strategy in philosophy that is synthetic in the older sense. Such a synthetic procedure generates the philosophical system “from the top down”, starting from the simplest and most general results and establishing more specific claims on the basis of those principles. Longuenesse 1998, 95–7, demonstrates Wolff ’s commitment to the conceptual character of all true judgment as such.

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The notion of containment analyticity thus lies at the heart of Kant’s indictment of the metaphysics of his day. I aim to elucidate the antiWolffian argument just sketched by clarifying the notion of containment on which it depends. The first step is to locate containment in the context of traditional logic (section 2). Far from a subjective or psychologistic notion, containment had a technical meaning, constrained by specific logical rules. Next, I show that containment truth played an essential role in Wolff ’s attempted logical reconstructions of scientific knowledge (section 3). Wolff ’s strong claims on behalf of containment truth generate serious puzzles, which I explore in section 4. Some of these worries are fatal for the Wolffian system, but not for the more modest appeal to containment that figures in Kantian analyticity. From this vantage, Kant’s insistence on the expressive limits of analyticity could be seen as a way to save the traditional notion of containment by emphasizing its restricted scope. I then address some details of Kant’s anti-Wolffian brief. Kant’s most direct attack claims that mathematical truth, Wolff ’s central case, is synthetic, not analytic. Elsewhere, I have explored Kant’s charge in the case of arithmetic;9 here (section 5), I extend the argument to geometry, where Kant makes the most direct contact with Wolff ’s own discussion. Finally (section 6), I consider judgments of empirical science, which Wolff surprisingly treats as containment truths. Kant is clearly right (contra Wolff ) that empirical claims are typically synthetic. Nevertheless, empirical science does include analyticities, in the specific sense of the containment definition, and Kant’s account of them is unsatisfactory. Given his conceptions of analyticity and empirical concept formation, I will argue, Kant was not entitled to conclude that all analyticities are a priori. This last result will be disappointing to those who look to analyticity for a general account of the a priori, but Kant himself was keen to insist, again contra Wolff, that analyticity cannot explain a priori knowledge in general. In that sense, the revision to Kant’s view is a friendly amendment. I will even suggest that it clarifies Kant’s views on the relation between the a priori and the empirical, and thereby illuminates the general question of a priori knowledge that motivated the critical departure from Wolffianism.

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See Anderson, forthcoming, esp. section 3.

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2. Concept Containment: the Standard Picture I turn first to the idea of containment as it figured in the traditional early modern logic used by Kant and his predecessors. Kant deploys two related notions of containment, which he ties to the “content” and “extension” of concepts. These notions are intensionally conceived: the content of a concept is the group of intensional concepts (marks) that it “contains in” itself as components, whereas the concept’s logical extension comprises the group of lower, or more specific, concepts it “contains under” itself (Logic, Ak. 9: 95–7, esp. §§ 8, 9, 11).10 The two types of logical containment are strongly reciprocal. By this I mean, first, that everything in the extension of a concept, A, contains A as part of its content, and conversely, everything included in the content of A covers A as part of its extension. Second, Kant holds that In regard to the logical extension of concepts, the following universal rules hold: 1. What belongs to or contradicts higher concepts also belongs to or contradicts all lower concepts that are contained under those higher ones; and 2. conversely: What belongs to or contradicts all lower concepts also belongs to or contradicts their higher concept. [Logic, Ak. 9: 98]

These rules entail that concepts with the same extension also have the same content, and vice-versa. Not only must any two such concepts include the same marks “belonging to” their contents or extensions, but also they must each exclude the very same marks, which “contradict” the content or extension. (Concepts sharing the same content and extension are thus equivalent: Kant calls them “convertible” or “reciprocal” [Wechselbegriffe]; Logic, Ak. 9: 98, also Ak. 24: 261, 755, 912). In this sense, conceptual content and logical extension cannot come apart: any difference in content entails a difference in logical extension, and conversely. Kant then orders concepts as higher and lower based on their reciprocal containment relations. Consider three passages from the Logic: 10

Concepts also must have non-logical extensions, for Kant, as is clear from the discussion of concept formation in the Logic (Ak. 9: 91–5, esp. §§3–7). Not only other concepts, but also intuitions and objects of experience, may be said to “fall under” concepts in this non-logical sense. For just that reason, non-logical extensions are not subject to the narrow restrictions on logical extensions, and so they afford our cognitions much greater expressive power, and become crucial to Kant’s explanation of the possibility of synthetic judgment. The point is discussed below, where we encounter cognitions that transcend the limits of analyticity (and purely logical extensions). On the contrast between logical and non-logical extension, see Longuenesse 1998, 50, 47, and Anderson, forthcoming, n. 28.

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R. Lanier Anderson The content and extension of a concept stand in inverse relation […]. The more a concept contains under itself, namely, the less it contains in itself, and conversely. [Ak. 9: 95] Concepts are called higher (conceptus superiores) insofar as they have other concepts under themselves, which in relation to them are called lower concepts. A mark of a mark – a remote mark – is a higher concept […]. [Ak. 9: 96] The lower concept is not contained in the higher, for it contains more in itself than does the higher one; it is contained under it […]. Furthermore, one concept is not broader than another because it contains more under itself – for one cannot know that – but rather insofar as it contains under itself the other concept, and besides this still more. [Ak. 9: 98]

One concept is higher than another just in case (1) the lower concept is contained under (i.e., in the extension of) the higher, and reciprocally (2) the higher concept is contained in the lower, as a mark. The more a concept contains in itself, the lower and more specific it is, and the less it applies to, or contains under itself. Conversely, the less content a concept includes in itself, the higher and more abstract it is, and the more it has under itself.

Crucially, however, as the last quoted sentence indicates, there is no absolute sense to judgments of the amount contained in (or under) a concept.11 We can specify how much one concept contains only relative to some other, and even then only if the two stand in a direct containment relation. For example, if ‹gold› (along with other concepts like ‹iron›, ‹copper›, etc.) is contained under ‹metal›, then ‹metal› is broader, or higher, than ‹gold›; it contains more under and less in itself than ‹gold›.12 By contrast, if neither of two concepts is contained in the other, then it is simply not determinate which is higher. Thus, the ordering of concepts from higher to lower is not connected, or total (not every pair of concepts stands in a determinate relation of higher to lower). It is a strict partial ordering.13 11

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The amount contained cannot be settled by appeal to the number of the concept’s marks, because the marks are themselves intensional concepts, which carry more or less content. This aspect of Kant’s view entails that there are no fundamental, elementary marks out of which all conceptual content is constructed; if there were such conceptual elements, presumably each complex concept would contain a definite number of them, and there would be an absolute measure of the amount it contained. This implication separates Kant from Leibniz, who did hope for an adequate universal characteristic of such conceptual elements. Angle brackets (‹ ›) indicate the mention of a concept. This fact is noted by Sutherland, forthcoming. A strict partial ordering is one that is transitive and irreflexive, like ‘