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R. Lanier Anderson
The Wolffian Paradigm and its Discontents: Kant’s Containment Definition of Analyticity in Histo...
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22
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 ‘<’. A partial ordering is transitive, reflexive, and antisymmetric, like ‘≤’. Thus, the relation of containment itself is a par-
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In light of this partial ordering, containment relations among concepts may be represented as a hierarchy, in which higher concepts stand over the more numerous lower concepts under them. Kant identifies higher and lower concepts as genera and species,14 so containment relations generate a genus/species hierarchy. This permits a treatment of containment according to definite rules of the traditional logic – the rules of logical division: The determination of a concept in regard to everything possible that is contained under it, insofar as things are opposed to one another, i.e., distinct from one another, is called the logical division of the concept. The higher concept is called the divided concept […], the lower concepts the members of the division. [Logic, Ak. 9: 146]
Two rules standardly govern such divisions: 1) the member species must exhaust the sphere of the divided genus, and 2) the species exclude one another, so that none can be predicated of any other.15 That is, divisions are exclusive and exhaustive disjunctions. The division rules suggest a procedure for reconstructing the content of a given concept: start from its genus, and add some differentia that marks out its particular way of having (or being) that genus. The species will then admit of a traditional Aristotelian definition, in terms of its genus and differentia(e).16 The resulting picture promises a clear, non-metaphorical sense of concept containment: higher genera are contained in the lower species formed from them, as reflected in the Aristotelian definitions. To see the picture concretely, consider a concept hierarchy based on ideas familiar from Linnaeus’s (1964 [1735]) work on biological classification:
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tial ordering (containment is transitive, and concepts contain themselves, but containment is antisymmetric – if A contains B and B contains A, then A = B ), whereas ‘higher than’ yields a strict partial ordering of concepts (it is transitive, and a concept is not higher than itself). “The higher concept, in respect to its lower one, is called genus, the lower concept in regard to its higher one species” (Logic, Ak. 9: 96). Kant adds a third rule, not echoed by the other logic books discussed here, which demands that all the species concepts belong under the same genus concept (Logic, Ak. 9: 146). If we think of the division in the standard way, as beginning from the genus, then it goes without saying that there is a common genus for the member species. Perhaps for that reason, other logicians did not state this rule explicitly. ‘Humans are rational animals’ is a definition of this standard type: human beings are defined as animals (genus) of a rational sort (differentia).
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Here, the more specific concepts may be obtained by adding differentiae to the higher ones; e.g., a quadruped is an animal with four limbs and hair, a glirine17 is a quadruped with two front incisors, etc. Thus ‹glirine›, for example, already contains ‹quadruped›, ‹animal›, etc.
The appeal to hierarchies substantially clarifies containment. The division rules insure that every relation between concepts is one of complete inclusion or total exclusion between their contents. Partial, or accidental, overlaps are forbidden by the exclusion rule. Judgments formed by connecting two such concepts are therefore true (or false) by virtue of what the concepts contain. Thus, when we seek the marks contained in a concept, we need not rely on the unconstrained and potentially idiosyncratic intuitions of individuals, as Maaß feared. Instead, the conceptual content can be reconstructed by a division, constrained by explicit rules insuring that its component marks stand in hierarchically nested containment relations. Analysis is then a matter of investigating the objective structure of that concept hierarchy. Kant’s logic of concepts was largely standard, following common treatments of traditional logic in the German-speaking world, which were dominated by the influence of Wolff. Wolff ’s own views are considered below; here I simply note that he and his followers (e.g., Baumgarten, Meier) highlight the logic of concepts, and devote central place within it to the method of conceptual analysis, and the reciprocal procedure of logical division.18 Nor were these ideas confined to the German tradition. Arnauld’s
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In Linnaeus’s classification, the taxon Glires included rodents, and is distinguished by two prominent front incisors in each jaw. See Wolff 1965 [1754], Baumgarten 1973 [1761], Meier 1997 [1752] and 1914 [1752]. See esp., on the importance of the logic of concepts, Wolff 1965 [1754], 110; on the centrality of analysis, Wolff 1965 [1754], 130–2, 136–8, and Meier
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influential Port Royal Logic invokes the same notions about containment. Like Kant, he defines extensions as sums of species concepts (the “inferior terms”, or “subjects”, under an idea), and he appeals to the “comprehension” of ideas, which is what Kant (following the Wolffians) calls “content”. All these logicians convey that notion via talk of what is “contained in” a concept, which Kant appropriated to define analyticity (A 6/B 10). Indeed, ever since Aristotle, containment provided the primary explanatory model for logical relations in all three traditional branches of logic: the theories of the concept, of judgment, and of inference.19 As we have seen, containment provides identity conditions for the theory of concepts, by fixing conceptual contents. Different contents belong to different concepts, and two concepts with the same content are identical.20 The theory of judgment focuses on formal features of the proposition (e.g., quantity, quality) that affect its logical force through effects on what its component concepts contain. Universal judgments connect the predicate to the entire extension of the subject, while particular ones connect only an indeterminate part of the extensions. Affirmative judgments assert that two conceptual extensions overlap; negative judgments that they do not.21
The most significant use of containment, however, comes in the theory of inference. Like judgments, syllogisms were traditionally understood as ways to connect two concepts. But whereas judgment connects
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1914 [1752], 340f., 348–51, 544; and on logical division and concept hierarchies, Baumgarten 1973 [1761], 27–32, and Meier 1914 [1752], 612–18. (Kant taught his logic courses from Meier 1914 [1752].) Similar notions of containment are deployed not only in figures like Crusius 1965 [1747], and Kant, but also in Port Royal (see Arnauld and Nicole 1996 [1683], 39f.) and in the logical papers of Leibniz, though Leibniz departs from the Wolffians on some important matters, including the centrality of logical division. Kant’s logic lectures, and his Logic as edited by Jäsche, use this organization as a starting point (see, e.g., Ak. 24: 904). The arrangement was quite standard – it is present in various forms in Port Royal, Wolff 1965 [1754], Baumgarten 1973 [1761], and Meier 1997 [1752] and 1914 [1752] (in Ak. 16) – and remained so into the early twentieth century. The use of containment to explain the force of inference appears in Aristotle (Pr. Anal. 25b32–26a2, 24b27–31, et passim), and received a standard expression in Port Royal, as well (Arnauld and Nicole 1996 [1683], 162–4). See Meier 1914 [1752], 590f.; Crusius 1965 [1747], 292f.; § 156; and Kant’s logic lectures (Ak. 24: 261, 912). It is because the theory of the concept lays out the basic idea of containment, that the standard practice of beginning logic with the theory of the concept is so natural: the other subfields all depend on the ideas about what concepts contain, first developed in the theory of the concept. See Wolff 1965 [1754], 110. Longuenesse 2001 deploys similar considerations to provide an illuminating analysis of Kant’s logic as a theory of concept subordination. She also connects the picture to distinctive features of the Wolffian version of logic in Longuenesse 1998, 81–106.
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concepts directly, inference compares concepts that lack immediate links, finding an indirect relation through a middle term.22 The standard structure of a syllogism (major premise, minor, conclusion) follows from this understanding of what inference is for. The major premise connects the major term to the middle term, and the minor premise connects the minor term to the middle. The syllogism is correct when the containment relations between the middle and the other two terms show how the contents of the minor and major stand in the relation asserted by the conclusion. Concept containment therefore served as the natural basis for a unified, systematic treatment of the syllogism – a result widely sought by early modern logicians dissatisfied with the intricacies of the mediaeval theory of inference.23 There is one qualification to add to the picture painted so far. Early modern logicians did not universally accept the special role of logical division in the account of containment, which we saw in Kant and the Wolffians. Leibniz, in particular, countenanced concepts whose full analysis would be infinite (and so could not be reconstructed from the top down). In place of division, he envisioned the construction of such contents from elements of his proposed universal characteristic, in ways that need not be constrained by the division rules. In section 4, I explore some motivations for the Wolffian restrictions, but one point needs mention here. I argued that the logical division rules are suffi-
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See FS, where Kant defines a syllogism as a “judgment which is made by means of a mediate characteristic mark” (Ak. 2: 48) – a view echoed in the Critique: “the syllogism is nothing but a judgment mediated by the subsumption of its condition under a universal rule” (A 307/B 364). That is, a syllogism is a connection between two concepts, and is therefore rightly understood as a species of judgment – albeit one that turns on the use of a middle term, or else a special condition governing the universal rule (major premise) through which the conclusion is known. On this notion of a “condition” see Longuenesse 1998, 95–106. Famously, the mediaeval theory distinguished four “figures” of the syllogism (defined by the places of the middle term within the two premises), and identified the “moods”, or patterns of propositional form (with respect to quantity and quality) that preserve validity in each figure. While some general rules governing all syllogisms could be stated, additional rules were needed for particular figures. The apparatus was complicated, which led to the memory-aiding, letter-based system for naming the valid moods (Barbara, Celarent, etc.), and eventually, to the dissatisfaction of the early moderns. Arnauld, for example, complains about the theory’s “uselessness” and triviality, most blatantly in the first edition chapter (later removed) treating the “reduction” of all syllogisms to forms in the first figure (Arnauld and Nicole 1996 [1683], 156f.; see also 135). Another notable example of this tendency is Kant’s own paper “On the False Subtlety of the Four Syllogistic Figures” (FS).
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cient to underwrite a clear notion of containment, but if we were willing to assume a Leibnizian characteristic, the rules would not be necessary to specify pairwise containment relations between concepts. In that framework, one concept contains another if its correct analysis includes the same conjoined string of elementary symbols of the characteristic, regardless of whether the division rules were observed in its formation. Thus, a concept formed by concatenating three primitive concepts, ABC, contains A, B, and C. Notice that it also contains the complex concepts AB, and BC, even though a hierarchy expressing these containment relations, like the one at right, violates the rules of division.24 In these circumstances, the strong reciprocity of content and extension assumed by Kant, which is guaranteed by the exclusion rule in particular, may also be violated (e.g., AB and BC have the same extension, but different contents.)
For the fully general criterion of analyticity sought by Kant, however, there is reason to insist on strong reciprocity and the division rules. If they hold, we know that members of the same division (and their subspecies) exclude one another. If they could be violated, though, we could not infer the incompatibility of two concepts simply from their positions in the hierarchy. Without the exclusion rule, such concepts could always overlap in a common subspecies, so incompatibility could be inferred only from explicitly contradictory marks.25 Judgments about concept exclusion would then have to rest on complete analysis of concepts into ultimate marks, for which there is no clear standard. In typical analytic judgments (e.g., ‘Quadrupeds are animals’), containment relations are not explicit in the compositional syntax of the concepts, as assumed by the Leibnizian characteristic. So in the standard case, the only way to express containment facts is by relative position in a concept hierarchy, which can do the work only by
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For example, it represents non-exclusive divisions, since the two species of B (AB and BC ) have a common subspecies, ABC. The division may not be exhaustive either, because under Leibnizian assumptions, there may be infinitely many elementary concepts, and so there could always be another concept (say, BD) alongside AB and BC in the division of B. Kant’s appreciation that an adequately clear theory of containment required the specification of both conceptual inclusions and conceptual exclusions is marked by his formulation of the “universal rules” about “the logical extension of concepts”, quoted above; they assert not only that what “belongs to” higher concepts also belongs to their lower ones, but also that what contradicts the former contradicts the latter (Logic, Ak. 9: 98).
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adhering to the division rules. In this sense, the rules of division are the minimum sufficient conditions for a precise representation of analyticity. I consider the consequences of these considerations for Wolff and for Leibniz in section 4. Before reaching them, however, we need a more detailed picture of Wolff ’s system. 3. The Wolffian Paradigm Like other standard eighteenth century logics, Wolff ’s Deutsche Logik gives prominent place to containment. It is a key idea in the chapters on concepts and inference, which, as Wolff himself insists, contain the “principle material, on which very much rests” (Wolff 1965 [1754], 110). The method of analysis, by which conceptual contents are made distinct and therefore serviceable for scientific use, also receives special emphasis in Wolff ’s own logic of concepts.26
In addition to the logical importance of containment, the picture of analytic hierarchies bears substantial metaphysical weight in Wolff ’s hands. Wolff posits a world of substances with intrinsic (non-relational) attributes, so a scheme of one-place concepts is a natural fit for his ontology. Beyond this basic point, Wolff grants extraordinary privilege to the syllogism, proposing the ideal that all scientific knowledge ought to be reconstructed in formal inferences. The program makes sense precisely because Wolff thinks the goal of science is to discover the true hierarchy of concepts, whose containment relations would be charted by syllogisms.27 As we saw, the correct hierarchy is supposed to match the concepts in the divine intellect, which God identified as the best compossible set, and then realized in creating the world.28 The system of adequate concepts would thus be a true and fully comprehensive theory of the world so created. Three key elements of Wolffian doctrine show that the rationalist ideal of knowledge from concepts alone must occupy this central place 26
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Concepts are distinct when their marks are individually clear, allowing an articulated, explicit grasp of the concepts’ content: see Wolff 1965 [1754], 126–34, Baumgarten 1973 [1761], 8–12, and Meier 1914 [1752] §§ 115–39 (in Ak., 16: 296–341). See, e.g., the fourth preface to the Deutsche Metaphysik (Wolff 1983 [1751]), esp. § 2, and also, in the body of the same work, passages like § 14 (9), and §§ 177–82 (97–101). The general program is articulated in the Deutsche Logik, discussed below, as well. See Wolff 1983 [1751], 615–16; § 996.
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within his system. I have already suggested the importance of the analysis of concepts and of inference, both of which are elaborated below. In addition, Wolff claims that the principle of contradiction is the basis of all knowledge. It is the first idea introduced in the Deutsche Metaphysik’s chapter “On the First Grounds of our Cognition” (Wolff 1983 [1751], 6–8; §§ 10–12), and Wolff immediately makes very strong claims for its fruitfulness: “not only do inferences have their certainty from it, but it is also through it that a proposition that we experience is placed beyond all doubt” (Wolff 1983 [1751], 6; § 10). Wolff does appear to qualify this extraordinary claim, by counting experience as an irreplaceable source of cognitions, and treating the principle of sufficient reason as a separate basis for knowledge.29 For him, however, both these alternate sources rest ultimately on the principle of contradiction: The certainty of reason therefore grounds itself in the certainty of inferences. But I have already shown in the [Deutsche Logik] that the certainty of inferences depends on the ground of the principle of contradiction (§ 10). Since now experience, too, ultimately has it [the principle of contradiction] to thank for its certainty (§§ 10, 330), so all certainty of cognition derives from it. [Wolff 1983 [1751], 239; § 391] That is, experiences rest directly on the principle of contradiction, and claims of sufficient reason are to be made out as formal inferences, which depend on the same principle.30
It is perhaps surprising that Wolff attempts to subsume experiences immediately under the principle of contradiction, arguing that once we have them, “it would be impossible for us to think that [what we experience] was not, while it is” (Wolff 1983 [1751], 6; §10). His defense of the idea that experiences are thereby known on the basis of the principle of 29
30
Wolff specially praises Leibniz for recognizing the separate importance of the sufficient reason principle (Wolff 1983 [1751], 17; § 30), and it is crucial for Wolff ’s metaphysics, since it is the principle of God’s choice of the best possible world, by which alone a particular set of possible essences is picked out (from among all the possibilities resident in God’s intellect) for realization (‘Erfüllung’) as actual things (Wolff 1983 [1751], 588–608; §§ 951–86). Inferences rest on the principle of contradiction because if they failed, “one would have to accept that something could at once be and not be” (Wolff 1965 [1754], 165). This follows from underlying facts about the containment of genera in their species. For example, in order to deny the syllogism ‘All bodies are extended; The planets are bodies; Therefore the planets are extended’, one would have to admit (with the minor) that planets are bodies, and yet deny the conclusion that they are extended, which amounts (given the major) to denying they are bodies. So the planets would be bodies, and not be bodies – hence the contradiction.
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contradiction is circular, as is his attempted derivation of the principle of sufficient reason from that of contradiction.31 For our purposes, however, the failure of these arguments is less important than Wolff ’s conviction that all such knowledge could be derived from the principle of contradiction alone. Such a claim does sound extravagant, given current philosophical common sense. Some recent philosophers have resisted attributing it to Wolff at all, in light of his willingness to grant experience an important role in concept formation (Wolff 1965 [1754], 123–5, 134–5; see esp. ch. 1, §§ 5–6), and even in philosophical knowledge itself.32 Wolff admits that philosophy must be “constantly joined” with historical knowledge of matters of fact, which supplies an indispensable basis for the philosophical knowledge of reasons (Wolff 1963 [1728], 15, §26). Nevertheless, close attention reveals that Wolff ’s concessions to the role of experience are narrow, so that Kant (Prol. 270) and contemporary Wolffians were correct to see him as genuinely committed to the sufficiency of the principle of contradiction asserted in the Deut-
31
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Wolff claims that “experience ultimately has [the principle of contradiction] to thank for its certainty” (Wolff 1983 [1751], 239; § 391, my italics). But the contradiction he derives depends on a prior assumption of such certainty. He starts from a particular experience (e.g., of the clock striking six), and assumes (for reductio) that the experience might be false. The argument has this form: Assume that p; Now assume (for reductio) that not-p; But we already assumed that p, so with the reductio assumption it follows that p and not-p, which is impossible; Therefore the reductio assumption must be false; So, (certainly) p. But this argument simply misses the point. Someone who doubts an experience does not suggest that we imagine an experiential content that “was not, while it is” (Wolff 1983 [1751], 6; § 10) – e.g., that we represent the content, “the clock strikes six, and at the same time does not strike six.” Rather, the thought is simply that the clock might not have struck six. A contradiction arises only if we assume all along that the original experience is certain, so that such a possibility is ruled out. But then the certainty is not derived from Wolff ’s proof, but presupposed by it. Wolff ’s (1983 [1751], 17–18; §§ 30–1) proof of the principle of sufficient reason is similarly circular. Wolff assumes two objects, A and B, with the same essence. The key claim is that if the sufficient reason principle could fail, then something could arise in A, but not B, if we substituted B for A in the same circumstances. But then A and B would not be the same, contrary to the hypothesis. So failure of the principle is impossible. It is true that sufficient reason fails iff A could change without any ground, and that in the imagined scenario there is nothing (i.e., no ground) insuring a parallel alteration in B. But does it follow that A and B could not then be the same? It follows only if sameness guarantees the same behavior, because there must be a reason explaining any difference between the two. That is, Wolff ’s proof simply assumes the principle of sufficient reason itself. For a helpful account of Wolff ’s reliance on empirical claims in philosophy, see Kuehn 1997, whose strong emphasis on the point (esp. 229–36) suggests that he might be among those to resist the highly rationalistic reading of Wolff I defend below.
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sche Metaphysik, § 10. The role of historical knowledge in philosophy is just to provide initial premises for demonstrations33, and if they are not to compromise philosophy’s “complete certitude” as a demonstrative science, even those premises must be “firm and immutable” certainties (Wolff 1963 [1728], 18–19, 60; §§34–5, 117). That is, they must be immediate instances of the principle of contradiction.34 Nor, Wolff insists, will the role of empirical concepts “disturb the connection of truths in philosophy” (Wolff 1963 [1728], 19, §36), as long as such concepts can be rendered distinct. Distinctness is supposed to help because it leads to results which we nowadays would take to be discoveries of additional empirical facts about an object, but which Wolff instead understands (strikingly!) as disclosing conceptual containment relations – i.e., strictly demonstrative inferential connections among conceptual marks. Thus, Wolff insists that when a telescope reveals that the Milky Way is an aggregate of stars, the experience has just uncovered marks that were contained in the concept ‹Milky Way›, but which we were previously unable to distinguish (Wolff 1965 [1754], 134–5).35 Thus, Wolff confidently asserts that empirical information is needed primarily in the order of discovery, so that its only role in the completed order of the philosophical system is to establish immediately certain first premises for demonstrations. At earlier stages in the development of a science, of course, our ignorance of the true system may force us to make more extensive appeals to experience, resulting in a hybrid form of knowledge: “He who confirms philosophical theses by experiments and observations, which he knows historically but cannot demonstrate, attains a certain intermediate level between historical and philosophical knowledge” (Wolff 1963 [1728], 31, §54). The goal, however, is to discharge merely empirical assumptions in favor of genuinely philosophical (i.e., demonstrative) knowledge of the reasons grounding the facts. For example, Wolff insists that “in rational psychology we derive a priori from a unique concept of the human soul all the things which are observed a posteriori to pertain to the soul [in empirical psychology]” (Wolff 1963 [1728], 57, § 112).36 It might still be objected that such a final system is unattainable, and that an 33
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If the role of historical knowledge exceeds this limit, then the result is not genuine philosophical knowledge, but a hybrid “intermediate” form (Wolff 1963 [1728], 31, § 54). Recall, Wolff explicitly insists that it is “through it [the principle of contradiction] that […] experience is placed beyond all doubt” (Wolff 1983 [1751], 6; § 10, my italics), and again that “experience, too, ultimately has it [the principle of contradiction] to thank for its certainty (§§ 10, 330), so all certainty of cognition derives from it” (Wolff 1983 [1751], 239; § 391). According to Wolff, then, any experience that is certain enough to contribute to genuine philosophy is so because it rests directly on the principle of contradiction. Thanks to an anonymous reviewer for this journal for comments that pressed me to be clearer about the ideas in this paragraph. Cf. Wolff 1963 [1728], 55f. (§ 110) for similar, though slightly more diffident, claims about rational physics. Wolff also describes (63f., § 121) a logical strategy for discharging empirical assumptions. In general, he insists, empirical knowledge allows us to connect the concept of the predicate to that of the subject, but it
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empirically grounded approximation is enough for us. But while Wolff clearly insists that “no one man can accomplish this work of philosophy” (Wolff 1963 [1728], 44, § 86), it is not at all clear that it outstrips all combined efforts of finite minds. On the contrary, he repeatedly insists that “there need be no fear that philosophy as we have defined it is impossible”, and “we can gain possession of philosophy as we have defined it above” (Wolff 1963 [1728], 20, 21; §§ 37, 38). Thus, Wolff ’s stance is more radically rationalistic than is sometimes maintained. While experience plays a role in concept formation and in the advancement of science, ultimately all truth rests on containment and is logical in character – a fact reflected in the fully demonstrative order of justification proper to the completed system of philosophy.37
This commitment to the logical character of truth harmonizes with the two Wolffian views about the shape of philosophy introduced above. First, analysis is a very prominent philosophical method. For Wolff, philosophy is complete understanding of things from the grounds of their possibility (Wolff 1965 [1754], 115, 110). Complete understanding, in turn, comes from “adequate” concepts, for which we not only know the marks, but know them distinctly, in that we explicitly represent their component marks, and the marks of those marks, and so forth (Wolff 1965 [1754], 129–31). Analysis delivers such adequate concepts, by making explicit the interconnections among the nested series of marks (e.g., by locating the component concepts in a hierarchy). The systematic hierarchy then offers the wanted knowledge from grounds, since a higher concept, taken together with the differentia, provides a ground for its lower species.
Second, the same picture informs the unusual privilege Wolff asserts for syllogistic inference. Since, for Wolff, genuine science is knowledge from grounds, theoretical argumentation should be reconstructed as a series of syllogisms, which reveal the grounds of their conclusions. Mathematical knowledge was the model, and the Deutsche Logik gives a specific example by recasting Euclid’s proof of the angle-sum property for triangles in syllogistic form (Wolff 1965 [1754], 173–5; see section 5, below). In support of this priority for inference, Wolff not only offers reasons parallel to those later made familiar by Frege – e.g., that formal reconstruction can expose undefended assumptions and facilitate corrections (Wolff 1965 [1754], 178–9, 243) – but further, he claims that the syllogistic is the actual
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does not fix whether the connection is a matter of direct containment in the definition of the subject, or whether it is restricted by a certain additional condition. Genuine philosophical knowledge settles this question, specifies the relevant condition, and thereby transforms the initial claim into a genuine containment truth. (The predicate is contained in the concept of the subject-plus-condition.) The conceptual character of truth itself for Wolff comes out clearly in his characterizations of truth in the Latin Logic. For discussion, see Longuenesse 1998, 95–7.
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(albeit implicit) path through which we arrive at results like the anglesum property in the first place. Since syllogisms exploit containment relations among concepts, such inference-based inquiry serves to chart the containment structure proper to a regimented scientific concept hierarchy. In all three cases, then, (i.e., for analysis, inference, and the principle of contradiction), Wolff ’s core commitments make sense in the context of a vision of the true system of philosophy as an adequate genus/species hierarchy. But how do we attain such a purely logical system of concepts, particularly when so much of science is patently empirical? Strikingly, Wolff is just as keen to insist on syllogistic reconstruction for empirical arguments as for mathematics. Parallel to his logical recasting of the angle-sum proof, for instance, he offers a syllogistic restatement for an empirical argument that air has an expansive force (Wolff 1965 [1754], 176–8). As we saw, direct experiences are supposed to be immediately certain by the principle of contradiction, so Wolff allows them to enter such demonstrations as unprovable premises, much the way definitions would. Such views are unlikely to impress many philosophers post-Hume. Experiences (e.g., of a balloon that contains air expanding inside a vacuum jar) are singular, whereas the conclusions Wolff seeks (e.g., ‘Air has an expansive force’) must be general. The resulting problem of induction is especially troubling for Wolff, given that all reasoning is supposed to follow rules of syllogistic, which can infer from universal to particular, but never the other way round. Wolff ’s response is instructive, albeit unsuccessful. His empirical reconstructions contain no distinct step generalizing from particular experience to universal claim (e.g., from the air-filled balloon expanding inside the vacuum jar to ‘Air expands when resistance is removed’). Instead, the experience is represented by a singular proposition (‘The air expands inside the balloon, when the vacuum jar is pumped out’), and the syllogism exploits the standard rule that treats singular judgments as universal for purposes of inference, since they apply the predicate to the whole extension of the subject.38 Wolff can therefore seem to derive 38
Consider, for example, the syllogism, Socrates is mortal; Every truly wise Athenian is Socrates; Therefore, all truly wise Athenians are mortal. Under rules of syllogistic inference, one can draw a universal conclusion only if both premises are universal. Our inference can be valid, therefore, only because the major premise is taken as a universal, even though it is singular. And such
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a universal conclusion without violating the inference rules. Unfortunately, the sort of inference he needs is something like this: What expands when resistance is removed has expansive force; The air expands inside the balloon, when resistance (the air in the vacuum jar) is removed; Therefore, the air has expansive force.
Here the argument moves from a minor premise about the air inside the balloon in this experiment to a conclusion about the air (in general) having expansive force. Thus, Wolff infers the result by equivocating on the term ‘the air’, which is singular in the minor, and universal in the conclusion. Wolff apparently failed to notice the flaw, perhaps because for him what is essential to the minor premise is not its singular use of the subject concept, but that it finds the right concept to capture the experience in the first place.39 That concept identifies the appropriate “condition” of the predicate’s application to the subject, thereby allowing the experience to fall in neatly as a minor premise which reveals the containment relation between the major and minor (see sec. 4 below). Again, Wolff holds that experience can lead us to the distinct, even adequate, concepts proper to a stable scientific hierarchy (Wolff 1965 [1754], 134–5). Thus, determining the content of concepts is not simply a matter of introspection or stipulation, but a process of discovery, in which experience helps correct and refine our concepts, revealing what they contain, and ultimately rendering them adequate to the true order of essences resident in God’s intellect.
Inquiry, then, pursues the correct hierarchy of concepts, adjusted to reflect the results of analysis and experience, and ordered by their inferential (containment) relations. The hierarchy makes explicit the logical relations among marks that were contained implicitly in the confused concepts with which we begin our quest for genuine science. If we construct it correctly, it encodes containment analyticities. On this picture, moreover, all the functions Leibniz attributed to his char-
39
treatment is legitimate, since the singular proposition asserts the predicate of the entire subject, just as a universal proposition would do. Wolff attempts to exploit this feature of syllogistic inference to get from the singular premises that capture an experience to universal conclusions, but in doing so, he commits a fallacy of equivocation, described in the text below. Wolff insists that generalizing from a singular experience to a universal proposition is “very easy”, as long as close attention is paid to the circumstances attending the experience, so that the empirical concept we form of the thing experienced captures its essence, thereby identifying the condition that makes the judgment a containment truth (Wolff 1965 [1754], 189).
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acteristic make sense together.40 Wolff ’s hierarchy of concepts would be a universal language of communication, but also a formal logic, and a tool of discovery. First, it would be a system of concepts, not words, so it offers a common basis for communication: indeed, eighteenth century concept systems based on logical division, like the Linnaean hierarchy cited above, typically had the explicit aim of improved scientific communication. Second, as long as the logical division rules were followed, a calculus based on the categorical syllogism, or on Leibniz’s “general theory of containment” (NE 486), could be developed to provide the logical syntax of the language. And finally, the point of theoretical inquiry would be to discover the hierarchy that cut the world at its conceptual joints – the one that identified the system of compossible essences God picked out as best. Therefore, the concepts in the hierarchy would have content and refer to objects, because they correspond to possibilities that have been realized by God.
4. Logical Puzzles Elegance promotes success, and the Wolffian paradigm just sketched became a dominant vision of the proper shape of philosophy within Kant’s intellectual world. Kant’s core argument against the paradigm criticizes its expressive power. Analytic judgment cannot represent truths of mathematics and experience, so any system of science must be essentially synthetic, contra Wolff. Prior to such arguments, however, there is room for doubt within the framework of traditional logic about the notions of containment and hierarchies, which are crucial not only to Wolff ’s view, but also to Kant’s argument about the limits of analyticity. In this section, I focus on two puzzles: 1) the restricted applicability of the containment definition, and 2) whether hierarchies can make a place for concepts other than genera and species. These two concerns are unified by a plausible worry that the division rules are just too restrictive, so that the division-based account presented here illegitimately impoverishes the notion of containment truth. The first issue arises because the containment definition applies directly only to categorical judgments, of the form ‘S is P’, apparently
40
See Wolff 1983 [1751], 324, §§ 179–81 for a suggestion in this direction.
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overlooking more complex propositions, such as those involving truth functions or relational concepts. It is well known that traditional logic lacked dedicated devices for representing relations, creating serious expressive limitations.41 In my view, such limits are fatal for the Wolffian program, but in a way that actually supports Kant’s view. I have argued elsewhere that logical truths turning essentially on relations should be seen as synthetic, not analytic, sensu Kant, so the containment definition need not (indeed, should not) accommodate them.42
By contrast, inferences like modus ponens (given ‘If p then q’ and ‘p’, infer ‘q’) clearly belong within Kant’s formal general logic, which is analytic (or so he suggests; see A 65–6/B 90–1, A 76/B 102, A 151–4/B 190–3).43 This will be troubling for the containment definition if some such inferences cannot be explained through concept containment. Kant is ill positioned to resist the worry, because he categorizes syllogisms not by the classical distinction among the four “categorical” figures, but via a three sided separation of categorical inferences from “hypothetical” and “disjunctive” ones, based on their distinctive (categorical, hypothetical, and disjunctive) major premises. Kant complains that traditional accounts ignore this distinction, treating the “extraordinary” hypothetical and disjunctive forms as parasitic on the “ordinary” cat41
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Russell, in particular, never tired of making this point against the traditional logic. For a concise version of his point, directed against Leibniz, see Russell 1989 [1937], 12–15. Friedman 1992a, 52–135 offers a more detailed and historically nuanced discussion, exploring implications for Kant’s philosophy of mathematics. In particular, Friedman provides an illuminating treatment of limits on the power of traditional logic which are exposed by contrast to the expressive resources available to modern logic through quantifier nesting in propositions involving relations. See Anderson, forthcoming, section 4. The first cited passage characterizes analysis as the “logical treatment of concepts in general” (A 65–6/B 90–1), and the second insists that “general logic abstracts from all content of cognition […] in order to transform [its representations] into concepts analytically” (A 76/B 102). Such hints that general logic is analytic fit well with Kant’s suggestions that his analytic/synthetic distinction tracks his characteristic distinction between the merely logical and the real. In the note at A 596/B 624, for example, Kant indicates that merely logical possibility turns on analytic relations among concepts, whereas real possibility requires synthesis; and perhaps most telling, in the Metaphysik Mongrovius Kant distinguishes between the logical relation of ground and consequence, which is analytic, and the real ground/consequence relation, which is synthetic (Ak. 29: 820; see also A 279/B 335, A 721/B 749). Finally, Kant counts the principle of analytic judgments (the principle of contradiction) as part of logic (A 151/B 190), and asserts that general logic cannot explain synthetic judgments, as opposed to analytic ones (A 154/B 193). But for a qualification, see note 49, below. (Thanks to Béatrice Longuenesse for discussion.)
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egorical syllogism (Logic, Ak. 9: 122; B 140–1). But concept containment was a plausible explanatory principle for all inference because of the privilege usually granted to the categorical case, where inference turns on containment relations among terms. In Kant, by contrast, modus ponens falls under the basic rule of the hypothetical syllogism: “The consequence from the ground to the grounded, and from the negation of the grounded to the negation of the ground, is valid” (Logic, Ak. 9: 129). Kant explains this rule not via containment, but by the abstract principle (governing all inference types) that the major gives a rule subject to some condition, the minor asserts that the condition holds, and the conclusion infers that the rule applies. (In our case, the rule is the hypothetical, ‘if p then q’, which gets the minor, ‘p’, leading to the conclusion, ‘q’.) Longuenesse (1998, 93–106) shows that Kant inherited talk of the “condition” of inferences from Wolff44, and in some cases, the role of condition could be filled by a containing middle term. For instance, if I seek to infer ‘Caius is mortal’, the concept ‹human› indicates the needed universal rule (‘All humans are mortal’). I thereby treat ‹human› (a middle term) as the condition, which both holds for my conclusion, and allows it to be subsumed (via containment relations) under the rule (see A 322/B 378). But nothing in Kant’s inference rule guarantees such a containment-based treatment. Consider the syllogism ‘If there is perfect justice, then obstinate evil will be punished; there is perfect justice; so obstinate evil will be punished’; surely it falls under Kant’s general rule, but here the condition is the judgment ‘There is perfect justice,’ which shares no concept with the conclusion, and so it seems the inference does not follow by containment among terms.45
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There is a closely related notion of the condition of a judgment, as well, which Kant inherits from Wolff. In categorical judgments, for example, the subject concept is said to be the condition of the applicability of the predicate. (In this respect, the modern practice of formalizing propositions of the form ‘All A are B’ by means of the conditional would not be wholly foreign to Wolff and Kant.) See Longuenesse 1998, 95–104 (esp. n. 53) for illuminating discussion. At least, the containment relation is not made explicit, as we might expect in logical inference. Whatever logical relation there is seems to obtain not among the terms, but between the two judgments as wholes; e.g., it is unclear what containment relation holds between ‹perfect justice› and ‹obstinate evil› or ‹punished›, though perhaps the state of affairs of there being perfect justice somehow entails or involves the punishment of the obstinately evil. Kant himself even notes that in hypothetical inference there is no middle term, and suggests that it is therefore not a genuine inference of reason, or syllogism (Logic, Ak. 9: 129). Longuenesse (unpublished) rightly notes that the truth of a hypothetical judgment for Kant depends on whether the consequence relation it asserts in fact holds, and is not a simple function of the truth values of antecedent and consequent, as in our material conditional (see Ak. 9: 105–6). But as I argue in the text below, Kant is not entitled to assume that every such true consequence is a logical one, in the narrow sense of turning on containment relations among terms. Where they are not, hypothetical inference will not be containment analytic.
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We therefore face a dilemma: If hypothetical inferences like modus ponens are not reducible to categorical ones, and need not turn on containment, then either they are non-analytic or the containment definition fails. Kant did not see this dilemma, but we can identify the reasons. In practice, he typically conceived of hypothetical and disjunctive judgments in terms of a restricted normal form – the hypothetical form is ‘If A is B, then A is C,’ and disjunctive judgment follows ‘Either A is B or A is C.’46 Now, concept containment threatened to fail as an account of “extraordinary” syllogisms because their validity might turn on a consequence relation among judgments, not a logical relation among their concepts. But if we restrict attention to the privileged forms, direct relations between concepts again become central. In normal form, the component judgments of the major have a common subject (A), so the conditional or disjunction effectively obtains among the predicate concepts (B and C ). Among concepts, moreover, the ground/consequence relation was typically viewed in terms of containment (a higher concept grounds its lower species; Logic Ak. 9: 96), and the disjunctive relation just is logical division. Thus, normal form extraordinary inferences appear to rest on concept containment after all.47 Kant’s logic lectures offer an example: ‘If the soul is not composite, it is not perishable; The soul is not composite; Therefore, the soul is not perishable.’ As Kant noted, the major conditional is supposed to be true because one can infer (based on containment) that “What is not composite is not perishable” (Ak. 24: 763). Thus, the soundness of the syllogism depends on the fact that ‹composite› is contained in ‹perishable›.
Unfortunately, the appeal to normal form is insufficient to obviate our dilemma. Despite Kant’s own normal form conceptions of hypothetical and disjunctive judgment, his explicit rules for extraordinary inferences are broad enough to cover
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Inferences in normal form would then be ‘If A is B, then A is C; but A is B; therefore A is C’ (hypothetical), and ‘A is either B or C; but A is not C; therefore A is B’ (disjunctive). I say Kant “typically” conceived the matter this way because a few of his examples of such judgments are not in normal form, notably the hypothetical ‘If there is perfect justice, then obstinate evil will be punished’, cited above, which Kant gives at A 73/B 98. Wolff, in particular, is committed to the claim that they do. He insists that all extraordinary inferences can be reduced to normal categorical ones (and thence to categorical inferences in the first figure) simply by identifying the appropriate concepts (those in the conclusion, and the right middle term relating them). See Wolff 1965 [1754], 170f., 168. This turns out not to be true, for reasons given in the text below.
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even cases like modus ponens using our material conditional, for which containment provides no explanation. Such inferences would still have looked like general logic to Kant, yet they cannot be analytic by the containment definition. More seriously, even if appeal to normal form allowed us to limit attention to hypothetical relations among concepts, it is simply not true that all such relations rest on analytic containment. Many must be synthetic, and for deeply Kantian reasons. Consider (most obviously), ‘If something is an event, then it has a cause’. The claim could serve perfectly well as major premise for hypothetical syllogisms, but if the analytic/synthetic distinction is to have any point, their conclusions cannot follow because ‹cause› is contained in ‹event›. Kant’s entire program rests on the insight that the connection of those concepts is synthetic. So inferences from such a major would not be containment-analytic; if they are analytic at all, it must be because the form of hypothetical inference preserves analyticity even where that cannot be explained via containment.48
The dilemma is therefore real: either hypothetical inferences of general logic are non-analytic, or analyticity is not containment. But the result is not fatal for the containment definition. Kant should simply choose the first horn of the dilemma. After all, analyticity is supposed to be conceptual truth, and these inferences threaten to be non-analytic precisely because they do not rest on the logic of the concepts. I do not deny that Kant may have found the result discomfiting, since we now face synthetic truths not only in mathematics, but even in general logic, which he thought was analytic.49 From a modern point of view, how48
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The key point is this: the essential logical relation that warrants hypothetical inference is the consequence asserted between the antecedent and consequent in the major premise (see note 44), and that relation need not depend on containment relations among the concepts included in those constituent judgments. Consider the example, ‘If the sun shines on a stone, it becomes warm; The sun is shining on this stone (in my garden); therefore, it will become warm’. The inference is good logically, but the consequence asserted in the major is not a logical but a real (i.e., synthetic) ground/consequence relation (cf. Metaphysik Mongrovius, Ak. 29: 821). It is that synthetic consequence that underwrites the inference. Admittedly, I do use containment to subsume the stone in my garden under ‹all stones› so that it can serve in both ‘The sun shines on this stone’ and ‘This stone becomes warm’. But the inference turns on the consequence relation holding between those two propositions, which does not depend on any containment among ‹stone›, ‹sun shining›, and ‹becoming warm›. Rather, it is introduced into the logic by the assumption of the major premise. Thanks to Béatrice Longuenesse for discussion. In one sense, though, Kant might even be thought to have anticipated this result. While he often thinks of analytic relations of concepts as characteristically “logical” (see note 42, above), at other times, he treats the analytic/synthetic distinction as separate from logic. Perhaps the clearest case occurs in the Prolegomena: “Judgments may have any origin whatsoever, or be constituted in whatever manner according to their logical form, and yet there is nonetheless a distinction
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ever, this is less surprising than it might have been to Kant. Many truths of our formal logic, especially in polyadic quantification theory, count as synthetic, given Kant’s version of the distinction.50 The right conclusion to draw, then, is that the expressive power of analytic judgment is even weaker than Kant himself imagined: analyticities express only a fragment of general logic.51
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between them according to their content, by dint of which they are either merely explicative […] or ampliative […]; the first may be called analytic judgments, the second synthetic” (Prol. 266; cf. OD Ak. 8: 239). Here Kant indicates that many basic logical notions cut across the analytic/synthetic distinction: for example, all the different logical types of judgment (e.g., categorical, hypothetical, disjunctive) may be used to express either analytic or synthetic judgments. Recall, the need for an amendment to the doctrine of the containment-analyticity of general logic arose precisely because hypothetical judgments may express synthetic truths in this way. In such cases the consequence asserted (which can then be used as the primary inferential link in a hypothetical syllogism) is a relation of real consequence, which does not turn on containment among the component concepts. Nevertheless, once the consequence itself is assumed as a premise, the inference using it is perfectly logical, in the sense of general logic. This result can even be reconciled with the comments cited in note 43, according to which analyticity marks the boundary of the logical in one key sense. We should simply read the Prolegomena passage, with its focus on the judgment’s content, as noting that the truth or falsity of a judgment (as determined under the rules of assertoric force proper to its logical form, whichever one) may arise either from strictly logical relations among its conceptual components (in which case the judgment is analytic), or from a non-logical connection among them (in which case the judgment is synthetic, and expresses a real, and not merely logical, relation). Thanks to Beatrice Longuenesse for references and discussion. Jaakko Hintikka has made one version of this point in an influential series of papers (e.g., Hintikka 1965, 1967, 1968, and 1969) arguing that logically valid arguments using the strategy of existential instantiation should be viewed as synthetic. Friedman’s (1992a) point about the expressive limitations of logics without a true theory of relations could also be seen in this light. This conclusion receives further discussion in Anderson, forthcoming. In addition, an anomaly in the Critique’s initial introduction of the analytic/synthetic distinction provides indirect confirmation that analyticity sensu Kant should be limited to a fragment of general logic. In offering the containment definition, Kant notes that he will “consider only affirmative judgments, since the application to negative ones is easy” (A 6/B 10). While true, this remark is surprising: the extension that would be well worth extra discussion here involves not negative categorical judgments, but hypothetical and disjunctive judgments – the important and special nature of which Kant himself is at pains to emphasize in many other contexts. In fact, as we have seen, the containment definition does not extend to all truths of general logic involving such judgments, so Kant’s anomalous neglect of the hypothetical and disjunctive cases is actually responsive to the underlying logical situation, though he seems not to have been fully aware of the point.
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Our second puzzle is that not all the “marks” included within concepts are happily viewed as genera and species, and the other types seem to have no home in a hierarchy. In the hierarchies described above, only genus/species concepts occupy nodes, but traditional logic acknowledged five universals, or concept-types (genera, species, differentiae, propria, and common accidents). If hierarchies cannot provide definite locations for the three further types, then perhaps logical division fails to offer a comprehensive explanation of conceptual content (and containment). In that event, it would be no wonder that analytic hierarchies cannot represent all the truths Leibniz and Wolff claimed to be conceptual, and Kant’s claim that analyticity is expressively weak would be substantially less interesting than it first seemed. Because differentiae are essential for logical division, it especially important to account for them, but that very circumstance points toward a solution: differentiae can simply be introduced into concept hierarchies. As de Jong (1995, 623–7) indicates, the work can be done within the porphyrian trees used to represent logical division. His example is a defining tree for the concept ‹human›, via division of the concept ‹substance›:
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Such trees make explicit the role of differentiae that was implicit in the Linnaean hierarchy sketched above, by inserting them (‹corporea›, ‹animatum›, etc.) into the tree. Since division works by adding differentiae to a genus, it can always be represented this way.52 Note that the differentiae entering porphyrian trees need not be simple, like the syntactical elements of Leibnizian characteristic. On the contrary, some will carry complex content capable of articulation. As Arnauld (1996 [1683], 41) points out, not only concepts of kinds or substances, but also many concepts that would serve as differentiae – e.g., concepts of modes like shape – can themselves be ordered as higher and lower concepts. Thus, contents deployed as differentiae in one tree, may admit of analyses made explicit in others, where they appear at the primary nodes. In de Jong’s tree, ‹corporeal› is a differentia, defining ‹body› as a species of ‹substance›, but its content could be articulated in another tree, e.g., by locating it as a subspecies under ‹divisible›, ‹extended›, etc., and as a genus covering the various modes of bodily existence. The same extensions permit accommodation of certain propria, in addition to differentiae.53 Propria are necessary attributes linked to the genus and differentia defining a species, and in the strict case, they apply to all and only the lower concepts under the species. In some instances, analyticity will be the right explanation of their necessity, and the system of extended porphyrian trees clarifies the relevant containment relation. For example, in our supplemental tree representing the content of the differentia ‹corporeal›, we will find ‹extended› contained in (above) it, so the expansion of de Jong’s tree by this supplement represents ‹extended› as an analytic proprium of ‹body›. In general, concepts contained in the differentia defining a species are contained in the species itself, and this can be made explicit by extending the tree. Not all propria fit this model, however. Most obviously for Kant, synthetic a priori truths give rise to propria: e.g., the angle-sum property is a strict proprium of ‹triangle› – all and only triangles have it – but it is neither a genus above ‹triangle› nor is it contained in the differentia ‹three sided›. The property is not contained in ‹triangle› at all, but synthetically linked to it. Some logicians recognized further kinds of propria as well, more loosely tied to the defining terms than the “strict” cases al-
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The appeal to trees including differentiae reveals how deeply the division-based account of containment depends on the traditional, intensionalist understanding of concepts. The tree does not explain the contents of the differentiae it deploys. It assumes them, so as to clarify genus and species concepts, which can be explicated as sums of differentiating marks only because the marks have intensional content already. Since logical division simply articulates the containment relations among intrinsically intensional concepts, Kant’s notion of containment will not satisfy Quinean urges for a global reduction of intensional notions to non-intensional ones; see Anderson, forthcoming, section 1 B. for discussion. Thanks to Alison Simmons for discussion.
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ready considered. For most of the further cases, the relation to the target concept will not be analytic.54 The verdict on propria is mixed, then: some can be represented in an analytic hierarchy, and some cannot. This is embarrassing for the Wolffian paradigm, which makes all truth conceptual, and I know of no serious effort in the Wolffian tradition to address the issue.55 From Kant’s point of view, however, the result is no indictment of the division account of containment. There is an obvious reason many propria cannot be located in a hierarchy: they are synthetic. If we are willing to admit an analytic/synthetic distinction, the limitations of porphyrian trees become a merit; precisely those limits permit perspicuous representation of which propria are analytically related to a given concept, and which are not, thereby marking off analytic from synthetic properties.56 The final class of universals, common accidents, calls for similar treatment. They apply to some, but not all, things under the target species (this makes them accidents), and apply to other things as well (therefore, they are “common”). They occur in “accidental predications,” which take the target species as subject, and assert an accident of it: e.g., ‘Some humans are ill’; ‘Some learned people are virtuous’. In such judgments, the extensions under the two concepts overlap (this makes the predication true), but neither is wholly contained within the other (this makes it accidental). Such partial overlaps violate the exclusion rule for division, so accidents can have no place in an analytic hierarchy.57 Again, for Kant this is not a problem, but an asset of the system of concept hierarchies. Accidental predications like ‘Some learned people are virtuous’ are synthetic, and the inability to represent them in hierarchies shows why.
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In Port Royal, for example, the additional cases include propria belonging to the concept soli sed non omni (which apply to only, but not all, things under the species), those applying omni sed non soli (which apply to the whole extension of the species, but other extensions as well), and even some pertaining omni et soli sed non semper (which apply to all and only the species, but not always); see Arnauld and Nicole 1996 [1683], 43. In my view, the last class, in particular, would be better understood as accidents; Arnauld’s example is gray hair, as a proprium of only (sic!) humans. In particular, Eberhard’s attacks on Kant in their famous controversy seem strikingly insensitive to the difficulties. See Allison 1973 for texts and helpful discussion. Again, Kant was keen to make this distinction against Eberhard. See OD (Ak. 8: 229–31), in Allison 1973, 141–3. See Logic Ak. 9: 103, for Kant’s argument that such accidental predications cannot have “rational form”. Again, the only kind of overlap permitted in an analytic hierarchy is the complete inclusion of one extension in the other. Given the containment definition, this is in order, since only in that case does the truth of the judgment turn on what the concepts contain within themselves (as opposed to what happens to fall under them, in their non-logical extensions, which determines truth in the case of accidents).
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Thus, Kant’s containment definition can accommodate the five traditional universals, either by locating them in extended hierarchies, or by showing that their relation to the target species is synthetic, so that they should be excluded from an analytic hierarchy. By contrast, the Wolffian paradigm faces major difficulties. Our result thus militates in favor of Kant’s analytic/synthetic distinction, as the best way to maintain the advantages of clarity and generality derived from the division account of containment. One could still try to save the containment theory for all truth, however, by rejecting the restrictive division rules themselves. Recall that Leibniz reached specific concepts not by dividing a summum genus, but by purely syntactic combinations (using conjunction and negation) applied to elementary symbols of his characteristic, unconstrained by exclusion or exhaustiveness rules. By rejecting those constraints, conceptual truth à la Leibniz promises greater expressive power than the Wolffian system criticized by Kant. Nevertheless, Kant’s target is not simply a “straw man”. The added power comes at a cost, which helps explain the broad adoption of Wolffian restrictions within German rationalism. First, the division rules provide crucial checks on intuitive judgments about containment. Absent the rules, particular containment claims can seem arbitrary, and the general notion threatens to degenerate into the metaphor mocked by Quine.58 Wolff and Kant echo such concerns, when they insist (contra Leibniz) that analysis must terminate in a highest genus, at least in part because its division allows a privileged, fully explicit specification of conceptual contents, presenting each as the conjunction of a definite, finite set of marks.59 Further, the explanatory force of containment is threatened altogether without the Wolffian restrictions. If the analysis revealing an alleged containment cannot be completed in principle60, then in an important sense, containment no longer explains the truth in question. In particular, truth no longer admits of illuminating logical explication in terms of a Leibnizian “general theory of containment” (see NE 486). As we have seen, a calculus based on concept containment can plausibly pretend to sufficiency only if inferences can be reduced to normal form as categorical syllogisms. But in such syllogisms, nothing follows from particular premises alone. Universal premises are always needed, and they express relations of complete inclusion or exclusion of one concept by another – the very conceptual relations distinctively
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See Quine 1961, 20. By contrast, in the Leibnizian framework, the analysis for some concepts could never be complete (they require “infinite analysis”), and so analysis could not always deliver clear answers to questions about containment. In addition, as Adams 1994, 66f., points out, the strictly syntactic character of Leibniz’s account has counterintuitive consequences: since the syntax is all that matters, simply exchanging two elementary concepts – substituting one for the other at every occurrence in the system of philosophy – would make no difference. It can be difficult to see how such a system could bear the metaphysical weight imposed by the Wolffian paradigm. According to the Leibnizian picture, even God knows such truths not by completing the analysis, but via direct intuitive insight.
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captured by analytic hierarchies. Thus, every such inference depends on at least one conceptual relation that is analytic in the strict, division-based sense. For this reason, containment’s explanatory work in fact depends on conceptual relations of the restricted sort envisioned by the division account of analyticity. German rationalism therefore faces a trade-off: the added expressive power potentially gained by relaxing the division rules goes along with a loss of explanatory power in the notion of containment itself. Note, however, that a parallel trade-off affects the Wolffian paradigm: the clarity gained for the general notion of containment through the division rules greatly restricts the power of concepts subjected to those rules. The point can be seen most clearly in the case of propria. Above, I treated ‹extended› as an analytic proprium of ‹body›, contained in the differentia ‹corporeal›. ‹Extended› was thus supposed to apply to all and only bodies. There is a puzzle about the ‘only’ side, however: ordinarily, ‹extended› is not restricted to bodies, but applies to other things like geometrical objects and durations, which do not have the differentia ‹corporeal›. To make ‹extended› a strict, “all and only” proprium, we must restrict its sense. One obvious way to indicate the needed qualification would be to index all concepts contained above ‹corporeal› in the supplemental tree, so that in effect, we attribute to ‹body› only corporeal extension, divisibility, etc. Broader considerations also mandate such indexing for the Wolffian paradigm. Without some such system, violations of the exclusion rule would occur whenever we introduced differentiae with shared content into excluded parts of a tree. For example, absent indices, we could not use ‹red› as a differentia for both ‹cardinal› and ‹poppy› without creating a partial overlap and violating the exclusion rule. The cost of such indices is substantial, however. Once the marks of a differentia must be indexed in this way, they no longer express that there is something common to things falling under the different parts of the tree. Our concepts themselves turn out to be much less contentful, and much less explanatory, than they seemed.61
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Analogous issues exert pressure in favor of systems of indexing even within the less restrictive Leibnizian framework – especially in connection with the containment theory of truth. A complete concept can contain everything true of a particular substance only with indices. For example, the predicate ‹famous philosopher› can be contained in the concept ‹Immanuel Kant› only subject to some system for temporal indexing, which makes it clear that Kant has the property post-1781. See Mates 1972 and Adams 1994, 71–4, for discussion of costs and benefits of different ways to handle the indexing. It becomes clear from their discussions, as well, that Leibniz must deploy something that has the effect of indexing claims involving the existence of actual individuals to the world that is best. Leibniz himself preferred to build this information into the content of the complete concepts themselves, in a way parallel to what I propose in the text for the Wolffian treatment of propria. It is this approach that permits (or forces!) Leibniz to infer from the concept containment theory of truth to the denial of transworld identity for individuals. See Adams 1994, 71–4, for an illuminating discussion.
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This is troubling for the Wolffian paradigm and its ideal of conceptual truth, but again, for Kant it is to be expected. The point of the analytic/synthetic distinction was to identify the limits of analytic judgment. The trade-offs may therefore serve as a final confirmation of the general moral: the logico-metaphysical puzzles plaguing containment can be resolved, but only by acknowledging stringent restrictions on the work we expect analyticity to do. At this point, with a clearer notion of containment in hand, we can specify the nature of those limits more closely by turning to an instance of Kant’s deepest argument exploiting the restrictions imposed by the division rules – the argument for the syntheticity of elementary mathematics.
5. The Syntheticity of Geometry and the Failure of Wolff’s Reconstruction Program Kant’s claim that mathematical truth is synthetic, not analytic, directly rejects the logical and metaphysical presumptions of the Wolffian paradigm. Wolff claimed to reconstruct all knowledge in strictly syllogistic form, and mathematics was to be the paradigm for the rest of science in this respect. If Kant is right that it is synthetic, then Wolffian reconstructions turn out to be impossible even in the best case, and the entire program is doomed. Kant’s view in philosophy of mathematics thus turns out to be central to his overall critique of contemporary metaphysics. We saw in section 4 that analytic containment is expressively weak: it captures only a fragment of modern first-order logic, and indeed, only a fragment even of general logic as Kant understood it. Kant’s negative argument that mathematics cannot be based on mere concepts turns on these expressive limitations of containment truth.62 Two features of containment prove especially salient for the argument as I will present it here: the strong reciprocity of conceptual content and logical extension, and the character of the ordering of concepts by containment. Strong reciprocity, recall, entails that concepts with the same logical extension also have the same content, and vice-versa. That is, content and extension cannot come apart. The restriction that terms with overlapping extensions must also share the same content is precisely what guarantees that judgments connecting such terms are conceptual truths.63 62
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In taking Kant’s argument to focus on limits of the logic available to him, the reading I offer follows in the tradition of groundbreaking work by Michael Friedman (1985, 1992a). There are two possible cases. If the extensions overlap entirely, then the conceptual contents are likewise wholly identical. If the extensions overlap in part, then one must be fully contained within the other. In that case, reciprocity guarantees
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The partial ordering imposed on concepts by their containment relations is also limited. Conceptual contents do enter into part/whole relations, since marks are construed as partial contents contained in lower concepts. In that sense, a concept contains “more” than its partial marks. But there are no simple marks or standard units of conceptual content against which we could measure how much a given concept contains, and thereby order concepts independently, as a basis for determining their containment relations. Instead, we must first determine the specific containment relations among concepts, and only then establish the partial ordering, based on the judgments of containment. So the ordering permits comparisons of how much concepts contain (in contents or extensions), but these are meaningful only where one concept is wholly contained in the other, so that the concepts compared are identical (at least in part): “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).64 Concepts can be equivalent, or equal, then, only if they are fully identical: to be equal, they have to contain the “same amount,” but such “amounts” are comparable only by virtue of shared marks. Adding the reciprocity constraint to the mix, it follows that concepts sharing all marks in their logical extensions, must also share the same content, and vice-versa. So in the domain of conceptual truth, to be equal is to be strictly identical. Thus, within the restrictions of analyticity, no sense can be given to equivalent but non-identical contents or logical extensions.65 We can now see why analytic hierarchies lack the power to express mathematics. Even very simple mathematical truths turn crucially on the equality (in quantity) of different things, but the constraints governing concept containment block the equivalence of non-identicals. In a
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that the lower concept will contain the higher one, so their contents will be the same in part. Either way, the judgment is analytic. Jäsche based this section on R 2886 (Ak. 16: 560–1), where Kant adds an example: the concepts ‹human› and ‹metal› are not comparable – one “cannot know” which contains more, since they do not contain one another. The only meaningful judgments of that sort compare a concept to one of its own marks. Then we can say that the concept contains more, since it contains the mark itself, plus some additional content. This is to be expected, given the restrictions on analyticity identified in section 2. If two concepts could be equal but non-identical as a matter of conceptual truth, it would violate the exclusion rule: non-identical extensions would be available only in excluded parts of the tree, but to express their equivalence as a conceptual truth would demand that they come under the same concepts.
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letter to Johann Schultz, Kant argues for the non-analyticity of arithmetic very much along these lines: I can make a concept of just the same magnitude through various kinds of combination and separation […]. So, I can attain the determination of one and the same magnitude = 8, through 3 + 5, 12 – 4, 2 · 4, or 23. But in my thought 3 + 5, the thought 2 · 4 was not contained at all; just as little, therefore, was the concept of 8 contained, which has the same value as both. [Ak. 10: 555]
In analytic definitions like ‘Bachelors are unmarried men’, the predicate and subject have the same content – the same conceptual marks – and therefore the same extension. By contrast, ‘3 + 5 = 8’ is true even though ‹3 + 5› and ‹8› do not have the same content, because the concepts nevertheless “determine the same object” (viz., the magnitude 8) in their extensions.66 Thus, conceptual content and extension come apart, violating the reciprocity demanded of analytic conceptual relations. The failure of reciprocity is fully transparent in an equation like ‘3 + 5 = 2 · 4’, where it is obviously wrong to attribute the same content to the concepts.67 Since the terms are not conceptually identical, the equivalence can emerge only under the concepts, in the magnitude they apply to, and the judgment cannot rest on concept containment. To explain such truths, we must postulate non-logical extensions, which, unlike logical extensions, need not be strictly reciprocal with conceptual content, and can therefore be used to express synthetic truth. In fact, as Sutherland notes68, the need to express the equality of non-identicals is as widespread and essential within elementary mathematics as the notion of quantity, or magnitude, itself. Since such claims 66
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To determine a concept, for Kant, is always to transform it, qua general representation, into a more specific representation, by choosing some “route down” through a hierarchy articulating the various, specific ways of having that general concept. Thus, all concepts, qua general, are determinable, but only objects or intuitions are fully determined (see A 571–2/B 599–600). So when Kant here insists that the various concepts connected in an equation (e.g., ‹3 + 5›, ‹2 · 4›, ‹8›) “determine the same object” (Ak. 10: 555), that just means that they overlap in their extensions (i.e., what falls under them). The two concepts involve different operations on different numbers, and so share no common content. Kant returns to the same idea in restating his key argument later on in the letter: “Supposing it [‘3 + 4 = 7’] were an analytic judgment, then I would have to think exactly the same thing by 3 + 4 as by 7, and the judgment would only make me more clearly conscious of my thought. Since now 12 – 5 = 7 yields a number = 7, by which I actually think just the same thing which I previously thought by 3 + 4, so […] when I think 3 and 4, I would at the same time think 12 and 5, which is contrary to consciousness” [Ak. 10: 556]. See Sutherland, unpublished.
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exceed the limits of analyticity, it is fair to say, as Kant does, that the very notion of magnitude is not fully expressible through concept containment alone: “the possibility of continuous magnitudes, indeed even of magnitudes in general, is never clear from the concepts themselves, since the concepts of them are all synthetic” (A 224/B 271).69 If anything, the need to represent equal but non-identical magnitudes is even more obvious in geometry than it is in arithmetic. When we consider how such claims were actually sustained in geometrical science as Kant knew it, it will become clear why concept containment cannot do the needed work, and so must fail to capture basic argumentative patterns of elementary geometry, including the very argument Wolff cited as the paradigm for his syllogistic reconstruction program. The centrality of comparisons equating non-identical things is apparent from the very first propositions of Euclid, Book I. The well known Prop. I, 1 demonstrates a general procedure for constructing an equilateral triangle. The proof strategy constructs two circles on a given line segment, AB – each with AB as radius, and one of the two endpoints as center. Joining each endpoint to a point where the circles intersect, C, yields a triangle, which must be equilateral: AB and AC are equal, since they are radii of the same circle, and likewise for AB and BC. Two things (AC and BC ) that are equal to the same thing (AB ) are equal to one another, so the three different sides are all equal. Euclid immediately extends the result (in Prop. I, 2) to obtain a general procedure for generating a new line segment equal to any arbitrary line segment, from an endpoint not on the line. The proof begins by connecting the given point, A, to one endpoint of the given line, BC. This allows us (following Euclid I, 1) to construct an equilateral triangle, ABD, on the new line, AB. The proof then constructs two circles, one (FCG ) with center B and radius BC, the other (GEH) with the center at D and radius DG, formed by extending the side DB until it meets the first circle, FCG, as in the figure. At that point, we are in a 69
For insightful discussion, see Sutherland, unpublished, who shows how intuition as Kant understands it makes up for this lack, and allows the explicit representation of quantity. The crucial idea is that intuition can represent a strictly homogenous manifold, capable of marking numerical difference without any specific (i.e., conceptual) difference. Such different but strictly homogenous manifolds then provide a basis for mathematical judgments: they can be equal but nonidentical, they can be combined to yield more of the same, etc.
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position to construct the line AE – provably equal to the given BC – by producing DA until it intersects circle GEH. The argument notes that BG is equal to BC, since they are radii of the same circle, and DG equals DE for the same reason. Since the triangle DBA is equilateral, DA and DB must be equal. But since DG and DE are equal, the remainders left when the equals DA and DB are removed from them (viz., AE and BG ) are likewise equal. And now, since AE is equal to BG, and BG to BC, AE is equal to the given BC. These two propositions are crucial for what follows in plane geometry precisely because they permit the construction, for any line, of another line equal to it, and thereby comparison of different but equivalent magnitudes, of the sort we saw to raise problems for analytic containment. (The Euclidean geometer deploys such comparisons in order to measure one figure by another – a crucial device by which the relations of geometrical equality (congruence) and inequality are established.) The proofs exhibit two basic strategies for showing the equivalence of non-identicals, important throughout elementary geometry. First, in Euclid I, 1, the sides of the triangle are equivalent because they are parts of the same whole whose common relation to the whole entails their equality: e.g., AB and AC are radii of the circle A. Prop. I, 2 exploits the same idea (e.g., in comparing BC to BG ), but it also uses a second approach, which infers the equality of geometrical magnitudes because they are the complements remaining after the removal of a common part (or equal parts), from two equal wholes: thus, since AE and BG are the complements which, along with the equal DA and DB, go to make up the equal DE and DG, they must themselves be equals. In short, the proofs establish the equivalence of non-identicals by identifying magnitudes as equal parts of the same whole, or as complementary parts of equal wholes. Both kinds of judgment violate the strictures of analytic containment. Containment cannot underwrite judgments about equal parts of the same whole for reasons similar to those we saw in arithmetic. From one perspective, in fact, the two cases appear as mirror images, and the reci-
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procity of content and extension is again the key constraint. In ‘3 + 5 = 2 · 4’, content and extension came apart because two conceptually different terms had the same extension. In the present arguments, reciprocity fails because we confront different magnitudes (e.g., AB, AC) that are conceptually indistinguishable.70 Conceptually speaking, both are just radii of the circle on A, but for the proof they must nonetheless be distinct (different sides of the triangle). Thus, the judgment of equality assigns different extensions to the same conceptual content, violating reciprocity. Kant insists that such non-identical equals cannot be differentiated by conceptual means, but only via intuition. Missing that point, in fact, was the key failing of Leibniz’s claim of the identity of indiscernibles: Since he had before his eyes solely their concepts, and not their position in intuition […] he extended his principle of indiscernibles, which holds merely of the concepts of things in general, to the objects of the senses […] [But] places are entirely indifferent with regard to the inner determinations of the things, [so] a place = b can just as readily accept a thing that is fully equal and similar to another in a place = a, as it could if the former were ever so internally different […] Thus that putative law […] is simply an analytical rule of comparison of things through mere concepts. [A 271–2/B 327–8; my italics]
Note that Leibniz’s principle would be perfectly correct for a system of conceptual truths, since as we saw, within such a system equivalents must be actually identical, on pain of violating reciprocity. But, Kant insists, we can in fact distinguish “fully equal and similar things”, like the line segments AB and AC, by locating them at different (intuitively represented) places. So Leibniz’s principle, and with it the ideal of a fully sufficient system of strictly analytic truths, fails. At first this argument seems too quick. It seems to leave the dialectic at an impasse: Kant asserts that we can represent non-conceptual, merely numerical distinction between “fully equal and similar things” via intuition, but Wolffians (and Leibnizians) might counter that really we cannot, since they would insist (as Kant himself complains) that intuitions are just confused concepts. In particular, Wolffians would surely contend simply that there must be some conceptual differentiation more specific than ‹radius of circle A›, which allows us to distinguish the radii AB and AC despite their sharing a common magnitude. Just here, though, Kant’s negative argument that mathematical truth is non-conceptual gains real bite, for it shows not only that we can rep-
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Or at least, the Leibnizian/Wolffian philosophy of mathematics needs to claim that they are conceptually equivalent. For discussion see below, and especially note 71.
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resent merely numerically distinct equals, but that we must if mathematical knowledge is to be possible. The argument can be stated as a dilemma. Either AB and AC are distinguished by conceptual marks alone, or not. If not, they are conceptually equivalent, but then the Wolffian must count them as fully identical, since concepts with the same content must also have the same extension. In that case the proof, which rests on their distinctness, will not go through. But if they are distinguishable by conceptual marks, then their equivalence as established in the proof is not a matter of concepts alone. For from a conceptual point of view (it is now granted), they are not equivalent.71 Either way, then, the mathematical truth does not depend on what the concepts contain; it is synthetic. Again, Leibniz was right to insist, with his principle of indiscernibles, that the only equivalence relation available in a 71
The Leibnizian/Wolffian philosophy of mathematics could of course fall back to insist – indeed, with more plausibility – that the concepts in question are identical only in part. On that view, the conceptual descriptions of AB and AC would be distinguishable by some relatively more specific conceptual marks (e.g., meeting the circumference at different points), even though they would overlap in others (e.g., they fall under the common genus ‹radius of circle A›, as well as, presumably, under the same generic magnitude concept). But such partial identity turns out not to be sufficient for the purposes of geometrical reasoning, where the crucial comparison will arise not from locating different species under the same genus (i.e., in the area of the partial identity), but precisely where the concepts differ and the partial identity runs out. Thus, in Euclid I, 1 we identify the different species of ‹radius of circle A› (viz., AB and AC ) only on the way toward a more illuminating comparison between AC and BC, the radius of the circle centered on B. In the geometrical argument, AC can be equated with BC not because of any conceptual equivalence between them, or between the concepts ‹radius of circle A› and ‹radius of circle B›, but because each of them is equal to the same thing (AB ), albeit each precisely in the respect in which they differ from one another. (AC equals AB qua radii of circle A; BC equals AB qua radii of circle B.) That is, the whole proof turns on the clever construction which brings it about that the different concepts ‹radius of circle A› and ‹radius of circle B› overlap in extension, in much the way we saw to be prohibited for analytic relations among concepts by the strong reciprocity of conceptual content and extension, when we were considering the arithmetic example of ‘3 + 5 = 2 · 4’. The failure of reciprocity here indicates that merely partial identity of the concepts is insufficient to underwrite all the substitutions that are needed (and licensed) within the geometrical reasoning. Thus, there remains an important sense in which the Leibnizian is right to conclude that if truth is to be strictly conceptual, then equality must just be identity; as a result, the recourse to partial identity does not gain anything. (Thanks to Michael Friedman for extremely penetrating comments and discussion on this point, though I suspect he will not be fully persuaded by my solution to the puzzle, and he is not responsible for the remaining shortcomings of my argument here.)
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system of purely conceptual truth is identity itself. If all truth were conceptual, therefore, the equivalence of non-identicals could not be represented as true. On such a basis, Kant rightly maintains, elementary mathematics would be impossible. The second kind of equality judgment – about complementary parts of equal wholes – also represents non-identical magnitudes as equal, so the Wolffian faces the same dilemma there. In addition, such judgments exhibit further features of the diagrammatic reasoning typical in plane geometry which themselves conflict with requirements for analyticity. To judge of equality and inequality, the Euclidean geometer exploits spatial overlaps among objects depicted in diagrams, so as to compare unknown magnitudes to known ones. This is what makes diagrammatic reasoning essential to the demonstrations in their Euclidean form. As Lisa Shabel has shown in detail72, such comparisons rest not on finegrained “eyeballing” of a diagram’s relative magnitudes, but on obvious part/whole relations among the overlapping figures. Thus, returning to Euclid I, 2, we note that DB and BG are complementary proper parts making up the segment DG. The same two segments, though, also form a side of the triangle DBA, and a radius of the circle FCG, and in connection with those figures, they have known relations to other magnitudes crucial for the proof (DB = DA as sides of an equilateral triangle, and BG = BC as radii of a common circle). Thus, it is the partial overlap of the circle GEH, the triangle DBA and the circle FCG that allows us to infer the wanted equality of AE and BC – as radii of GEH, DG is equal to DE (= DA+AE ); But DG is formed from DB (the side of DBA) and BG (the radius of FGC); So since DB = DA and BG = BC, AE must be equal to the given BC. We can express the result in terms of concepts by noting that the complement formed from the radius of circle GEH by removing a side of the equilateral triangle DBA must be equal to a radius of the circle FCG. The argument for this result, though, depends
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See Shabel 1998.
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not on any analysis of the contents of those concepts, but on the way the two circles and the triangle overlap in the segment BG. The power of Euclid I, 2 for the rest of plane geometry rests on its role in the same pattern of diagrammatic reasoning. Its result allows us to place a segment equal to a given line segment at an arbitrary point in the space of construction, thereby permitting measurement of one figure by another. Again, it is crucial to such reasoning that the representations it uses (e.g., of the triangle DBA and the circles GEH and FCG ) overlap in part, but not entirely. That sort of partial overlap is just what the division rules prevent in the analytic relations among concepts. Thus, mathematical reasoning of this sort cannot be a matter of tracing analytic containment relations, because the representations used in the argument cannot be constrained by the rules that govern purely conceptual contents. Kant is therefore right to insist that the reasoning must deploy intuitions, not just analytically related concepts. In proposing the syllogistic reconstruction of all mathematics, Wolff claims just the opposite. Given the centrality of concept containment to categorical syllogisms, and Wolff ’s belief (1965 [1754], 168, 170–1) that every inference is equivalent to a normal (first figure) categorical syllogism, the program amounts to charting containment relations in an analytic hierarchy. Not surprisingly, Wolff ’s key example of such reconstruction in the Deutsche Logik – the angle-sum theorem for triangles – receives Kant’s explicit attention, and was indeed one of his most frequent examples of the syntheticity of mathematics. Wolff ’s reconstruction is vulnerable to both of the criticisms just explored. Wolff ’s (1965 [1754], 173–5) proposal takes earlier Euclidean theorems as major premises. Crucially, Euclid I, 29 shows that alternate angles formed by cutting two parallels are equal; so Wolff constructs a triangle with a line through the apex parallel to the base, from which to draw minor premises. He then proceeds syllogistically as follows: All alternate angles on parallel lines are equal; aBAC and aABD are alternate angles on parallel lines; Therefore, aBAC and aABD are equal.
After a parallel argument that aACB is equal to aCBE, Wolff uses Euclid I, 13 as a major, again together with the figure, to obtain the syllogism:
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All the angles standing on a line from one point are together equal to two right angles; aABD, aABC, and aCBE stand on a line from one point; Therefore, aABD, aABC, and aCBE are together equal to two right angles.
Adding syllogisms based on his first three conclusions, and the major “Equal angles can be substituted for equals without altering the magnitude” (Wolff 1965 [1754], 174), Wolff carries out the substitution, and infers that the angles of the triangle together equal two right angles. One immediate problem with this argument tracks the issue about induction that plagued Wolff ’s reconstruction of empirical scientific reasoning. Here, too, the inference cannot reach the needed conclusion by strictly syllogistic means. Once Wolff introduces minor terms with singular reference, based on the diagram, he cannot use syllogisms to get back to a universal conclusion.73 Wolff glosses over the problem by leaving his conclusion in the form of the singular statement about the angles of VABC. The reconstruction program as a whole, however, requires the generalization to ‘all triangles’. After all, the further proofs that depend on Prop. I, 32 will need a universal major premise, just as the present sorites depended on universal majors from earlier theorems. If those premises had been particular or singular, the present sorites would not have been valid.
That formal flaw hints at the underlying problems already identified above. The role of the diagram and the singular minor premises in the geometrical reasoning is not incidental, designed merely to facilitate reference to certain features of our concepts ‹triangle›, ‹parallel›, etc. On the contrary, the diagram is crucial, in that it represents distinct figures, falling under different concepts, and shows them overlapping in space. Only those relations among the representations of the figures’ parts allow the argument to represent non-identical magnitudes (e.g., aBAC and aABD), and then demonstrate their equivalence. If we were limited to concepts alone, as we saw, the only terms that could fall under Wolff ’s last major premise permitting substitution of equals would be identical. By way of contrast, consider Kant’s treatment of the proof: Give a philosopher the concept of a triangle, and let him try to find out in his way how the sum of its angles might be related to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and in it the concept of equally
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From a somewhat different point of view, Hintikka 1967 and 1969, following Beth 1953/4, 1956/7, also emphasizes the importance of the singularity of representations used crucially in mathematical reasoning in general, and Euclidean argument in particular.
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That is, the geometer does not rest with relations of (full or partial) identity among ‹triangle› and other concepts, but instead constructs a triangle, which can be extended to form further figures of known relation to the magnitude of a right angle. This is part of a constructive proof strategy. The whole idea is to create explicit representations of the two terms of the problem (the angles of the triangle, and the magnitude of the right angle), and bring them into a relation in space. Thus, extending the side AC to E produces a diagram in which one angle of the triangle (aACB ) overlaps with a figure of known relation to the magnitude of a right angle (viz., the adjacent angles formed on AE from C ). The proof then turns on this overlap, for the geometer proceeds by comparing the non-overlapping parts – dividing the external angle aBCE by the line CD (parallel to AB ), so as to exploit Euclid I, 29 to show aABC equal to aBCD, and aBAC to aDCE. Since the argument turns on partial overlaps among these representations, they cannot be mere stand-ins for concepts in relations of analytic containment. Rather, they are intuitions, whose mutual relations are not confined by the division rules. Using the resulting flexibility, the proof can establish a judgment of the equality of nonidenticals (the internal angles of the triangle ABC, the complementary angles formed on AE at C ). As we saw, no such judgment could be true by containment, since it would violate the reciprocity of content and extension. The angle-sum property, then, is essentially synthetic. The very argument Wolff took as paradigmatic is in fact a counterexample to the possibility of his reconstruction program.
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6. Empirical Concepts and Revisable Analyticity We have seen two main ways that Kant’s analytic/synthetic distinction highlights the poverty of containment truth. First, by restricting the scope of analyticity, the distinction offers the most promising resolution for the logical puzzles about containment canvassed in section 4. Second, Kant shows that elementary mathematics is synthetic, in the sense of the containment definition. Since no credible system of knowledge could do without these truths, Wolff ’s reconstruction program cannot be sustained, and the underlying ideal of science as an analytic hierarchy adequate to the essences in God’s intellect must therefore be abandoned. The limits on analyticity pose a central problem of Kant’s philosophy – explaining how synthetic judgments are possible. A key feature of his solution emerges already in his philosophy of mathematics. Synthetic judgments link concepts that are not contained within one another by identifying overlaps of their non-logical extensions. Information about such overlaps comes from intuitions, which present “in concreto” the objects falling under the concepts. Since the contents of intuitions are not fixed by their places in a hierarchy, they are not constrained by intermediate species concepts subject to division rules, and can carry the complex information needed for synthetic truths. The point has importance beyond mathematics. Empirical concept formation, too, depends on synthesis of intuitions. The combination of intuitive contents given in experience is a precondition for its basic activities – comparison, reflection, and abstraction: to form empirical concepts, we compare the collected concrete contents, determine what they have in common (via reflection), and then abstract away from their differences.74 For Kant, then, while our intellectual faculties do still shape scientific theory, e.g., by identifying systematic and inferential links among concepts, they cannot determine the full shape of the system of science, since the conceptual contents are filled in by intuition, not fixed via top-down division alone.
This view is a serious departure from the Wolffian paradigm, in which experience merely helps to reveal parts of the genus/species hier74
The role of synthesis as a precondition of concept formation is one basis for Kant’s claim that all analysis of concepts presupposes synthesis (A 77–8/B 103, B 133–4n). On concept formation, see Kant’s Logic (Ak. 9: 94f.), and for a detailed treatment of Kant’s theory of comparison, reflection, and abstraction, see Longuenesse 1998, 107–30. The doctrine of comparison, reflection, and abstraction is one of two main topics in the theory of concepts in Kant’s Logic. The other is the discussion of containment, conceptual content and extension, and higher and lower concepts – treated in section 2. Thus, the two main parts of Kant’s theory of the concept focus on the primary bases for synthetic and for analytic relations among concepts, respectively.
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archy, the structure of which is outlined a priori. By contrast, Kant demotes the global analytic hierarchy to the status of a regulative ideal, a focus imaginarius, useful for improving our theories’ systematicity, but ultimately unrealizable (A 644/B 672, A 653–7/B 681–5). The point is a deep one. For Kant, the Wolffian hierarchy of fully adequate concepts is not merely so far unreached by our incomplete science; it is necessarily unreachable, since much of our knowledge is synthetic. Despite the demotion, however, analytic hierarchies remain a theoretical goal for Kant, even in empirical science. They are useful precisely because they identify a clear class of logical relations among concepts, and thereby help to systematize knowledge involving those concepts. When Wolff claimed to discover the marks contained in a concept by experience, his mistake was not the thought that empirical concepts can stand in analytic relations, but the stronger thesis that all deliverances of experience must be constrained by the logical form proper to concept containment.
Some analyticities, then, deploy empirical concepts, and are, to that extent, discovered with the aid of experience. Kant clearly acknowledges such truths (see Prol., Ak. 4: 267). For him, though, they are invariably a priori, despite the role of empirical concepts: “Judgments of experience, as such, are all synthetic. For it would be absurd to ground an analytic judgment on experience, since I do not need to go beyond my concept at all” (A 7/B 12). In my view, Kant’s inference here is simply mistaken. To see how, consider again the model analytic hierarchy I presented above, based on the first edition of Linnaeus (1964 [1735]):
Obviously, subsequent zoology prompted extensive revisions to Linnaeus’s original classification. Significant corrections were made already by Linnaeus himself. In the tenth edition of Systema Naturae (1956 [1758]), he introduced the term ‘Mammalia’ in place of ‘Quadrupedia,’ when he recognized the Cetacea (whales) as belonging to the group, resulting in the following classification:
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Once the whales were classed in the same group with the (former) quadrupeds, the concept ‹quadruped›, which includes the mark ‹fourlimbed›, was no longer appropriate, so Linnaeus deploys the new concept, ‹mammal›. Clearly, this theory change is rooted in experience. Still, the revision touches the content of the concepts, not merely empirical matters of fact. Telling similarities between whales and other mammals had been recognized since ancient times, and were probably known to Linnaeus at the time of the 1735 edition.75 So what Linnaeus changed in 1758 was not so much the body of empirical knowledge about the whales, as certain judgments about the relative importance of, and the internal relations among, different empirical marks built into the taxonomic concepts. Thus, Linnaeus’ introduction of ‹Mammalia› is a case of the empirical revision of analyticities: in the 1758 hierarchy, ‘Whales are mammals’ counted as analytic, whereas in 1735, ‘Whales are fish’ was analytic, and the whales were excluded from the quadrupeds. As Philip Kitcher notes76, such cases call for a distinction between two kinds of theory revision – loosely, factual belief change and conceptual change. In belief change, new evidence forces us to exchange the belief that p (e.g., that proteins can-
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Similarities of whales to other mammals were known already to Aristotle, but he classified them in a group of their own. In the late seventeenth century John Ray, while making some concessions (in notes) to the common opinion that the whales are fishes, had recognized them in his official classification table as belonging with the quadrupeds, on the basis of similarities in physiology and pulmonary anatomy (Raven 1986, 379). Already in the first edition Linnaeus approvingly cites Ray’s work in his discussion of the animal kingdom (1964 [1735], 26). In the first edition classification of fishes, however, Linnaeus declined to develop his own system, instead following the results of his countryman, Petrus Artedi, who classed whales, dolphins, etc. with the fishes. When Linnaeus revisited the animal kingdom in more detail for the tenth edition of Systema Naturae, he came over to Ray’s view, and thereafter the new category ‹Mammalia›, and the classification of the whales as mammals, became standard. See Kitcher 1982, 222–9.
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not act as sole agents of infection) for a denial of p (e.g., prions are infectious agents), but the content of the key concepts (‹protein›) remains stable. By contrast, Linnaeus’s reclassification of the whales is conceptual, precisely in that it alters the analyticities that follow from his system. A distinctive pattern in the adjustment of truth values separates such conceptual revision from ordinary belief change. It is true that some claims of the 1735 system, like ‘Whales are fish’, are simply reassigned from true to false. For others, though, the change is more complicated. ‘Glirines are quadrupeds,’ for example, is not simply false in the new 1758 system.77 Rather, it is unwarranted because it deploys a discredited concept. To explain the pressure to revise such judgments, therefore, we need to postulate a distinct class of conceptual revisions, separate from ordinary changes of belief about empirical facts. It counts as an added selling point of the division account of analyticity developed here, that it affords explicit criteria to make this contrast distinct, and thereby to clarify the implications of a given (or proposed) change of belief.
It might be thought that conceding this distinctive type of conceptual change provides comfort to the official Kantian view rejected above, that analyticities are really a priori, even when they deploy empirical concepts. Kant could insist that such judgments are at least conditionally independent of experience – i.e., a priori, given those concepts. It could then be admitted that analyticities are revisable, but only along with the concepts, via distinctively conceptual change.78 I think this reply underestimates the seriousness of the issues raised by the revisability of empirical concepts, and also the tensions between Kant’s official view on analyticity and his own insights about the nature of the empirical concepts needed to convey synthetic truth. First, note that in defending the apriority of analyticities, Kant explicitly insists they permit us to “become conscious of the necessity of the judgment, which experience could never teach” (A 7/B 12, my italics). That is, on the official view, analyticities “could not be otherwise” (B 3), which sits uncomfortably with their empirical revisability. The proposal under consideration can reconcile apriority (and hence necessity) with revisability only by claiming that the content of all the empirical concepts remains strictly fixed across the theory change, so that the revision strikes not what content a concept has, but which concept is being deployed, and therefore what judgment is actually being asserted. Under 77
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Perhaps it is even right to say that such judgments are correct, in some suitably bracketed or qualified way – they are “true in the old system” in something like the way it is “true in the fiction” that Emma Bovary acquired unrealistic ideas about life from romance novels. Thanks to Béatrice Longuenesse and Nadeem Hussain for discussions exploring objections of this sort.
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those assumptions, we could interpret both ‘Whales are fish’ (sensu Linnaeus 1735) and ‘Whales are mammals’ (sensu Linnaeus 1758) as true and analytic a priori. The natural idea that these judgments conflict, however, would be a fallacy of homonymy, since ‘whale’ does not refer to the same concept in the two cases. On this picture, we do not so much revise the judgment ‘Whales are fish’ (sensu Linnaeus 1735) itself, which remains true; we just stop asserting that judgment, along with others using the 1735 concepts, in favor of judgments using the 1758 concepts. To my mind, such a view does insufficient justice to the conflict between ‘Whales are fish’ and ‘Whales are mammals’. Indeed, the view is too Wolffian, in its insistence that whatever we discover about the content of concepts must be strictly confined by the constraints of analyticity. Once we take seriously the task of explaining synthetic knowledge, I think we are quickly forced to concede the possibility of synthetic grounds for revising the contents of empirical concepts, based on reasons that Kant himself appreciated. Conceptual revisions like Linnaeus’s do not just exchange one global concept hierarchy for another, or even one overall concept of whales for another. The revision is localized on the taxonomic judgments tied to the reclassification of the Cetacea. Since these judgments do exploit analytic relations among conceptual marks, what the concepts contain is implicated in the change. Still, the modification should not be understood as replacing one concept with another, but as a revision of concepts that preserve their identity though the change. Kant suggests just such a conception of the revision of empirical concepts, in support of his argument that they cannot be definitively defined: One makes use of certain marks only as long as they are sufficient for making distinctions; new observations, however, take some away and add some, and therefore the concept never remains within secure boundaries. And in any case, what would be the point of defining such a concept? – since when, e.g., water and its properties are under discussion, one will not stop at what is intended by the word “water” but rather advance to experiments, and the word, with the few marks that are attached to it, is to constitute only a designation and not a concept of the thing. [A 728/B 756]
Empirical concepts like ‹water›, then, must be identified in terms of their “designation”, and not merely their content as articulated in one or another concept hierarchy. Such concepts designate a definite non-logical extension, by reference to which the identity of the concept may be preserved, even when “new observations” lead us to “take some [marks]
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away and add some” to the concept’s content and logical extension. Precisely this “externalist” aspect of the semantics of empirical concepts allows them to serve as a basis for synthetic judgments: facts about the non-logical extension are unconstrained by the conceptual content, and that independence enables them to force revisions in it.79 But now, since the “designation” is available to guarantee the concept’s identity across such theory change, it is meaningful to speak of genuine alterations in the content of the same concept. Analyticities themselves thereby become vulnerable to experience in a real sense that is hard to reconcile with the strict apriority and necessity Kant officially attributes to them. Kitcher points out a philosophically significant irony in this neighborhood80: Kant actually shares with Quine the thesis that analyticity is not sufficient to provide a full explanation of a priori knowledge. That is why he demands a special account of the synthetic a priori. If the last argument is correct, we can say in addition that analyticity is not even a sufficient condition of strict apriority for particular judgments. What establishes analyticity is a definite pattern of logical relations among a concept’s component marks, and whether such a pattern obtains within a given class of concepts can be (in part) an empirical question. Indeed, in light of the massive reclassification of organisms underway in current work toward a phylogenetic systematics, the Linnaean case is a particularly good example of the potential for dramatic change even in conceptual structures possessing the right logical form to support analyticities.81 Kant should therefore have recognized empirically revisable 79
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As Putnam 1975 shows, such externalism can illuminate the complicated interplay between empirical and conceptual/theoretical contributions to the meaning of a term. As I argued in the text, even dramatic changes of belief touching core marks of a theoretical term are often implausibly counted as changes of meaning alone (cf. Putnam’s discussions of ‘water’ and ‘electron’), precisely because the term’s content depends in part on what it designates, or, for Kant, on its relation to experience. See Kitcher 1982. The phylogenetic program proposes major revisions in judgments that look like analyticities (e.g., ‘Sharks are fish’), in response to molecular evidence about the actual evolutionary history of organisms. For example, in one version, phylogenetic classification of vertebrates would abandon the Linnaean category ‘Pisces’, in favor of the taxa Chondrichthyes (cartilaginous vertebrates), in which the sharks find their place, and Osteichthyes (bony vertebrates), which includes not only bony fishes, but also birds, mammals, reptiles, etc. From a more basic point of view, however, phylogenetic systematics can be seen as insisting on taxonomic concepts that are amenable to analytic containment, in the sense developed here. In the years leading up to the emergence of phylogen-
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analyticities. Where experience suggests a deep enough problem with a scheme of classification whose structure supports analytic conceptual relations, we revise the content of the concepts, and thereby the analyticities.
7. Conclusion: Apriority and Containment Analyticity This departure from Kantian orthodoxy on the apriority of analyticity sheds light on the problem of apriority in Kant more generally. I have argued elsewhere (following Friedman82), that Kant’s account of causality treats all causal laws (including specific, empirical ones) as a priori “in a sense”.83 But of course, empirical laws are revisable in the face of experience, and so cannot be strictly a priori. The case of revisable analyticities is parallel in a highly illuminating way. In both instances, the key source of revisability resides in the empirical concepts themselves. A causal law is empirically revisable, according to Kant, because we have no a priori “concept” identifying the causally salient “ac-
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etic techniques, it became widely recognized that phenetic taxa concepts descended from Linnaeus did not in fact obey the rules of division, and would have to be understood as “cluster concepts”, or family resemblance concepts of some sort (Hull 1964, 1965). In one sense, this was not surprising, since the Darwinian revolution overthrew the Linnaean conception of species as fixed essences whose interrelations depended on logical rules alone. Nevertheless, Darwin’s own conviction was that the evolutionary approach need not dramatically upset the taxonomic status quo, and this turns out to be reasonable. For the genealogical relations at the heart of taxonomy from a Darwinian point of view (Darwin 1859, 486) have the same hierarchical logical structure exemplified by the traditional Linnaean classification. In this context, the phylogenetic emphasis on monophyletic taxa (i.e., groups containing all and only the descendants of a given ancestor population) can be seen as a rigorous reinstatement of the rules of division, since monophyletic groups display the same nested hierarchical structure imposed by the rules of division in the traditional logic of concepts (see esp. de Queiroz and Donoghue 1988; also de Queiroz 1988, de Queiroz and Gauthier 1994). Controversy over the species concept might then be understood as (in part) a question of the theoretical importance of analyticity. Insisting on monophyletic species (with Donoghue 1985) preserves the analyticity of relations among taxonomic categories, by strictly following actual descent relations, which form a nested hierarchy. By contrast, recognizing species as non-monophyletic lowest taxa sacrifices that theoretical virtue, in favor of other aims better served by reticulated or “cluster”-type species concepts (e.g., limiting the number of recognized species). See Friedman 1992a, 1992b. See Anderson 2002.
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tual forces” (A 207/B 252) that link two lawfully connected states. We have only empirical concepts of the forces, and potential revision would strike precisely those concepts. Likewise, the empirical concept in a revisable analyticity may misidentify the systematically salient marks for scientific classification, and need revision. Nevertheless, in judgments of both sorts, the crucial link forging the unity of the judgment (the claim of a necessary causal connection, or of a containment relation) is supposed to be made by the understanding a priori. Kant envisions that the relevant empirical concepts (e.g., ‹floating boats› and ‹moving downstream›; or ‹human› and ‹animal›) fill in the a priori form of judgment. Thus, empirical revision reaches only the content of the concepts, and the way they more closely specify the basic connection between cause and effect, or genus and species. It does not touch the basic form of experience itself, which is abstractly represented in these judgment forms. Still, in a corrected Kantianism, the departure from the Wolffian paradigm for a priori metaphysics must be even more radical than Kant himself appreciated. Not only are most of the claims Wolff counted as containment truths not in fact analyticities, but further, it is not even true that all the remaining analyticities count as strictly a priori knowledge. Analyticities involving empirical concepts should have a kind of mixed status; they are simultaneously parasitic on the a priori (albeit merely regulative) structure of the analytic concept hierarchy, and dependent on the specific, empirically determined content of their concepts. The result is that they are revisable, but only through a special kind of theory change. In this respect, their status is analogous to the “relative a priori” proposed by later Kantians who sought to preserve Kant’s insight into the special role of constructive principles in science, but were reluctant to follow his claims to the unique, unrevisable truth of the Newtonian laws of motion and Euclidean geometry.84 Strikingly, almost all analyticities should be granted only this “relativized a priori” status, the only exceptions being those that deploy pure a priori concepts. Indeed, even within that narrow class, most relations of genuine theoretical importance (e.g., mathematical relations, the metaphysical relation of cause and effect, etc.) remain synthetic, and the analyticities in the neighborhood are of interest primarily because 84
The notion of a relativized a priori was proposed already by neo-Kantians near the turn of the twentieth century, and the idea has recently been revived in a thought-provoking form by the present-day neo-Kantian, Michael Friedman (2001).
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they clarify concepts from which we can derive synthetic insights. This conclusion can serve as a final reminder of Kant’s central moral – that analytic truth in his sense is surprisingly weak, and cannot be expected to provide the key to the system of philosophy, even in its a priori parts.
Works by Kant Citations to Kant use the pagination of the standard Akademie edition (Ak.), cited below, except those to the Critique of Pure Reason, which use the standard (A/B) format to refer to the page numbers of the first (=A) and second (=B) editions. Works are identified by the abbreviations given in parentheses. Kant, I. 1992a. Theoretical Philosophy, 1755–1770. Transl. and ed. D. Walford and R. Meerbote. Cambridge. –. 1992b. Lectures on Logic. Transl. J. M. Young. Cambridge. –. (Ak.) 1900ff. Kants gesammelte Schriften. Ed. Königlich Preussische Akademie der Wissenschaften. Berlin. –. (A/B) 1998. Critique of Pure Reason. Transl. and ed. P. Guyer and A. Wood. Cambridge. –. (FS) “The False Subtlety of the Four Syllogistic Figures”. In Kant 1992a, 85–105. –. (ID) On the Form and Principles of the Sensible and the Intelligible World (the “Inaugural Dissertation”). In Kant 1992a, 373–416. –. (Logic). Immanuel Kant’s Logic: a Manual for Lectures, ed. G. B. Jäsche. In Kant 1992b, 517–640. –. (OD) 1973. “On a Discovery According to which Any New Critique of Pure Reason Has Been Made Superfluous by an Earlier One”. Transl. and ed. H. Allison. In Allison 1973, 107–160. –. (Prol.) 1997. Prolegomena to Any Future Metaphysics that may be Able to Come Forward as a Science. Transl. and ed. G. Hatfield. Cambridge. Works by Others Adams, R. M. 1994. Leibniz: Determinist, Theist, Idealist. Oxford. Allison, H. 1973. The Kant-Eberhard Controversy. Baltimore, MD. –. 1983. Kant’s Transcendental Idealism. New Haven, CT. Anderson, R. L. 2002. “Kant on the Apriority of Causal Laws”. In History of Philosophy of Science: New Trends and Perspectives. Ed. M. Heidelberger and F. Stadler. Dordrecht: 67–80. –. forthcoming. “It Adds Up After All: Kant’s Philosophy of Arithmetic in Light of the Traditional Logic”. Philosophy and Phenomenological Research. Aristotle. (Pr. Anal.) 1941. Prior Analytics. Transl. A. J. Jenkinson. In Basic Works of Aristotle. Ed. R. McKeon. New York: 65–107. Arnauld, A. and Nicole, P. 1996 [1683]. Logic, or the Art of Thinking, Containing, besides Common Rules, Several New Observations Appropriate for Forming Judgments (the “Port Royal Logic”). Transl. J. V. Buroker. Cambridge.
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Baumgarten, A.G. 1973 [1761]. Acroasis Logica in Christianum L. B. de Wolff. Hildesheim. Beck, L. W. 1965. Studies in the Philosophy of Kant. Indianapolis, IN. –. 1969. Early German Philosophy: Kant and his Predecessors. Cambridge, MA. Bennett, J. 1966. Kant’s Analytic. Cambridge. Beth, E. W. 1953/4. “Kants Einteilung der Urteile in analytische und synthetische”. Algemeen Nederlands Tijdschrift voor Wijsbegeerte en Psychologie 46: 253–64. –. 1956/7. “Über Lockes ‘allgemeines Dreieck’”. Kant-Studien 48: 361–80. Brittan, G. G. 1978. Kant’s Theory of Science. Princeton, NJ. Crusius, C. A. 1965 [1747]. Weg zur Gewissheit und Zuverlässigkeit der menschlichen Erkenntnis. Ed. G. Tonelli. Hildesheim. Darwin, C. 1859. On the Origin of Species by means of Natural Selection, or, The Preservation of Favoured Races in the Struggle for Life. London. de Jong, W. R. 1995. “Kant’s Analytic Judgments and the Traditional Theory of Concepts”. Journal of the History of Philosophy 33: 613–41. de Queiroz, K. 1988. “Replacement of an Essentialistic Perspective on Taxonomic Definitions as Exemplified by the Definition of ‘Mammalia’”. Systematic Biology 43: 497–510. de Queiroz, K. and Donoghue M. 1988. “Phylogenetic Systematics and the Species Problem”. Cladistics 4: 317–38. de Queiroz, K. and Gauthier, J. 1994. “Toward a Phylogenetic System of Biological Nomenclature”. Trends in Ecology and Evolution 9: 27–31. Donoghue, M. 1985. “A Critique of the Biological Species Concept and Recommendations for a Phylogenetic Alternative”. Bryologist 88: 172–81. Friedman, M. 1992a. Kant and the Exact Sciences. Cambridge, MA. –. 1992b. “Causal Laws”. In Guyer 1992, 161–199. –. 2001. Dynamics of Reason. Stanford, CA. Guyer, P. (ed.) 1992. The Cambridge Companion to Kant. Cambridge. Hintikka, J. 1965. “Are Logical Truths Analytic?”. Philosophical Review 74: 178–203. –. 1967. “Kant on the Mathematical Method”. The Monist 51: 352–75. –. 1968. “Are Mathematical Truths Synthetic A Priori?”. Journal of Philosophy 65: 640–51. –. 1969. “On Kant’s Notion of Intuition (Anschauung)”. In Kant’s First Critique. Ed. T. Penelhum and J. J. Macintosh. Belmont, CA: 38–53. Hull, D. 1964/5. “The Effect of Essentialism on Taxonomy – Two Thousand Years of Stasis”. The British Journal for the Philosophy of Science (I) 15: 314–26, and (II) 16: 1–18. Kitcher, Patricia. 1990. Kant’s Transcendental Psychology. Oxford. Kitcher, Philip. 1982. “How Kant Almost Wrote ‘Two Dogmas of Empiricism’”. In Essays on Kant’s Critique of Pure Reason. Ed. J. N. Mohanty. Norman, OK: 217–250. Kuehn, M. 1997. “The Wolffian Background of Kant’s Transcendental Deduction”. In Logic and the Workings of the Mind: the Logic of Ideas and Faculty Psychology in Early Modern Philosophy. Ed. P. Easton. Atascadero, CA: 229–250.
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