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Kant on Infinite and Negative Judgements: Three Interpretations, Six Tests, No Clear Result

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Abstract

In his table of judgements, Kant added infinity as a third quality. An infinite judgement ‘All S are non-P’ is said to differ from the affirmative ‘All S are P’ because it ascribes a negative predicate; and it differs from the negative ‘No S is P’ because it has a richer content. The present paper puts three interpretations of this surplus content to six tests. Among other things, it is examined whether these interpretations marry up with Kant’s solution to the first antinomy, his conception of analyticity and the principle of complete determination. The unpleasant conclusion is that none of the interpretations can make its mark as the most adequate one. It thus remains unclear what Kant had in mind when positioning infinite judgements besides affirmative and negative ones.

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Notes

  1. ‘CPR’ stands for the Critique of Pure Reason, ‘A’ for the page numbers of the first and ‘B’ for the page numbers of the second edition, ‘Prol’ for the Prolegomena and ‘AA’ for the Academy Edition (Akademie-Ausgabe) of Kant’s works. Where available, translations are based on the ones from the Cambridge Edition, albeit corrections were indicated every now and then. More on the prehistory of infinite judgements in Wolfson (1957), Tonelli (1966) and Ishikawa (1990), § 5.

  2. Cf., inter alia, CPR, A 72/B 97; Pölitz Logic, AA 24: 578; Vienna Logic, AA 24: 930.

  3. For Meier (1752b, § 294, 82), Reimarus (1766, § 151, 149) and Ulrich (1772, § 48, 100f.), the class of infinites contains also affirmatives with a negative subject and affirmatives where subject and predicate are negative. Crusius (1747, § 226, 426) includes even negatives with a negative subject and/or predicate.

  4. According to Ishikawa (1990, 33, fn. 13), this is the main goal of Kant’s considerations.

  5. Cf. also Ulrich 1772, § 48, 101; Bolzano 1810, 41; Aebi 1947, 161f.; Bennett 1966, 78; Menne 1982, 160; and Siebel 2014, 202. Keller’s (1876, 27–29) criticism is less far-reaching. He argues that ‘All S are non-P’ boils down to ‘No S is P’ only if the expression in the position of ‘non-P’ is a noun because then the predicate applies to each and every object that is not P. If ‘non-P’ is an adjective, there are things being neither P nor non-P, with the result that ‘All S are non-P’ is not equivalent to ‘No S is P’. The latter is in line with the third interpretation of infinites. Strangely enough, Neuhaus and Scheffler (2001, 47) claim that Kant took ‘All S are non-P’ and ‘No S is P’ to be equivalent.

  6. The authors of the lecture notes Logic Philippi (AA 24: 469), Logic Pöltz (AA 24: 583) and Vienna Logic (AA 24: 928) apparently omitted the negation in ‘Some men are not virtuous’.

  7. For the former conception of a sphere cf. Jäsche Logic, § 8, AA 9: 96; Logic Blomberg, AA 24: 257; Vienna Logic, AA 24: 911. For the latter conception cf. Logic Blomberg, AA 24: 275; Logic Dohna-Wundlacken, AA 24: 760f.; Vienna Logic, AA 24: 925.

  8. Cf. Jäsche Logic, § 49, AA 10: 117; Logic Blomberg, AA 24: 281; Logic Busolt, AA 24: 670f.; Logic Dohna-Wundlacken, AA 24: 770.

  9. Cf. CPR, A 72/B 97; Reflexions on Logic, Refl. 3062 and 3063, AA 16: 635 and 637. Allison (2004, 141) contrasts “the infinite judgement, ‘The soul is not mortal’, with the straightforwardly affirmative, ‘The soul is immortal’ ” because he takes ‘immortal’ not to apply to stones while ‘not mortal’ does. If this were also Kant’s view, the third interpretation of infinites would be badly affected. However, although there are two remarks in the notes on Kant’s logic lectures indicating a distinction between ‘est non mortalis’ and ‘est immortalis’ (AA 24: 764, 930), I doubt that they are conclusive.

  10. Cf. Orenstein 1978, 97; 1999, 403, 406f.; 2000, 523; 2002, 142–144; Flage and Bonnen 1999, 226; Guyer 2006, 76; Hanna 2011, § 2.1.2; Mion 2014, 380. In contrast, van Cleve (1981, 484), Malzkorn (2001, 42) and Berg (2014, 108f.) assume that Kant took all categorical judgements to have existential import.

  11. I am aware of the fact that translating propositions from traditional logic into modern logic is a difficile enterprise. However, I think that the translations are unproblematic in the cases at hand. For example, ‘No S is P’ must clearly not be translated by ‘¬(∃x)(Sx & Px)’ if it is understood as being existentially loaded. But since we are concerned with an interpretation under which negatives have no existential import, this translation seems to be innocuous.

  12. Cf. Jäsche Logic, § 47, AA 10: 116; Aristotle, De Interpretatione, 17b16-19.

  13. Hanna (2001, 130) takes the Kantian extension of a concept to contain not only the actual but also the possible objects satisfying the concept’s criteria. If existential import is understood after this manner, i.e. as meaning that the concept in question must have possible objects at least, then interpretation 1 coincides with interpretation 2.

  14. Vanzo will have recognised that this idea is in conflict with what he wrote in a former paper: “the rule ‘non entis nulla sunt praedicata’ does not apply only to couples of opposed or contradictory judgments, but to every single judgment, be it affirmative, negative, or infinite” (Vanzo 2005, 526). If, in common with infinites, negatives with inconsistent subjects were false, the difference between them and their infinite counterparts could not lie in the fact that only the latter are false when containing an inconsistent subject.

  15. For the following reasons, I count Ishikawa among the scholars assigning this understanding to Kant. First, he identifies the contrast between ‘is P’ and ‘is non-P’ with the “real opposition” Kant promotes in his early essay Attempt to Introduce the Concept of Negative Magnitudes into Philosophy (1763). According to Kant, such a real opposition holds between ‘Tugend’ (virtue) and ‘Untugend’ (negative virtue) because both presuppose a being with knowledge of a corresponding law or at least conscience (AA 2: 182). Secondly, Ishikawa seems to read ‘non-red’ along the lines of ‘not red but of a different colour’, entailing that it is applicable only to coloured objects.

  16. Cf. CPR, A 503–507/B 530–535; Refl. 5962, AA 18: 403f.; and Gram (1967), 501.

  17. Cf. Gram (1967), 499. Along the same lines, van Cleve (1981, 484) argues that the inadmissible condition of the second antinomy is not that matter exists but rather that matter exists as a thing in itself.

  18. One of the reviewers worried that the tests based on analyticity may not have the force I ascribe to them. For while “in analytic judgements the relation to the object is irrelevant because in them only the relation between concepts needs to be considered”, “the whole notion of infinite judgements belongs to the transcendental, not general, logic”, which does not abstract from this relation. I understand this to mean that ‘All fairies are female’ does not threaten the claim that positive judgements have existential import because, as an analytic judgement, it is about concepts (cf. Gram 1980, 179), the consequence being that its truth does not require that there are fairies but only that the concept of fairies exists. However, Vanzo (2014, 215f.) convincingly argued that Kant did not take analytic judgements to relate to concepts but to the objects they are about at first glance.

  19. Cf. CPR, A 258f./B 314, A 593f./B 621f.; see also Siebel (2011), 94f.

  20. Über eine Entdeckung, nach der alle neue Kritik der reinen Vernunft durch eine ältere entbehrlich gemacht werden soll, AA 8: 235. In a footnote, Vanzo (2014, 230; my emph.) says of the cited sentence that it “only implies that the analytic judgement ‘all bodies are extended’ would be true even if no bodies currently existed”. The reason for this interpretation is presumably that Vanzo’s translation of “sie selbst mögen nun existiren oder nicht” is “whether they exist now or not” (my emph.). But the German word ‘nun’ does not determine time in this context but has the function of emphasising the surrounding statement.

  21. This difficulty was overlooked by Vanzo (2014). He argues against the thesis that Kant burdened affirmatives with existential import by referring to analytic affirmatives with empty subjects without recognising that analytic affirmatives with inconsistent subjects undermine his own thesis that Kant burdened affirmatives with consistency import.

  22. Stuhlmann-Laeisz (1976, 75f.) prefers the stronger reading. Early Bolzano might have borrowed this notion from Kant. In the Allgemeine Mathesis, we read: “If A cannot be B, the expression […] A cum B does not denote a concept but is just a mere assemblage of words. […] Thus, in my opinion, […] the words […] ‘a figure of two straight lines’ do not express a concept.” (Bolzano 1810, 30) In his later writings, Bolzano is fine with calling contradictory combinations concepts (cf. Bolzano 1837, § 67, 304; 1830–35, 48, 51–53).

  23. Cf. Marc-Wogau (1952), 145; Proops (2005), 591; and Neuhaus and Scheffler (2001), 49.

  24. Ulrich (1772, 229, 251f.), Feder (1783, 230) and Bolzano (1851, § 26, 38) understood it in this way. Since Rohs (1978, 171) does not accept predicate negation, he lets the principle state that every predicate either applies or does not apply to every object.

  25. Cf. Reflexions on Logic, Refl. 3063 and 3065, AA 16: 637 and 639; Logic Pölitz, AA 24: 577; Vienna Logic, AA 24: 929.

  26. One of the reviewers suggested that ‘The S is non-P’ means that the S falls “within a non-denumerably infinite extension consisting of all the not-Ps and everything else whatsoever, i.e., any old thing, whether phenomenal or noumenal”. By contrast, ‘The S is not P’ means that the S falls “within a finite or denumerably infinite extension consisting of all and only things, phenomenal or noumenal, to which the predicate P determinately fails to apply”. I am not sure that I fully understand these interpretations. As to negative judgements, first, what is the function of the adjunct ‘determinately’? Are there things to which an unnegated predicate fails to apply, but only indeterminately? Secondly, why should the set of things to which the unnegated predicate (determinately) fails to apply never be non-denumerably infinite? As to infinite judgements, first, are “the not-Ps” the things mentioned in the characterisation of negative judgements, i.e. the things failing to be P? Secondly, is “everything else whatsoever” to be understood literally, such that, say, the extension of ‘non-mortal’ contains mortal things, too? The main problem, however, is that these interpretations are meant to entail that ‘The S is non-P’ can be true while ‘The S is not P’ is false. This is in conflict with Kant’s claim that the former has a richer content than the latter.

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Acknowledgements

I would like to thank Sonja Schierbaum and Mika Perälä for organising an inspiring workshop and editing this special issue. Furthermore, I would like to thank the reviewers of Topoi whose comments were very helpful.

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    Mark Siebel

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Siebel, M. Kant on Infinite and Negative Judgements: Three Interpretations, Six Tests, No Clear Result. Topoi 39, 699–713 (2020). https://doi.org/10.1007/s11245-017-9476-6

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