kant und die berliner aufklärung (akten des ix. internationalen kant-kongresses. bd. i:...

Analytical and Dialectical Oppositions Reconsidered: New Perspectives on Kant's Antinomies

Wolfgang Malzkorn, Bonn

1. Exposition of the Problem

In a well-known passage in the antinomy chapter of the Critique of Pure Reason (= CPR) Kant explains the kind of opposition relation that holds between the statements opposed to one another in each antinomy. Taking the second part of the first antinomy, namely the so-called space-antinomy, as example, he claims:

If I say that as regards space either the world is infinite or it is not infinite (non est infinitus), then if the first proposition is false, its contradictory opposite, the world is not infinite, must be true. Through it I would rule out only an infinite world, without positing another one, namely a finite one. But if it is said that the world is either infinite or finite (not-infinite) then both propositions could be false. For then I regard the world as determined in itself regarding its magnitude, since in the opposed proposition I not only rule out its infinitude, and with it, the whole separate existence of the world, but I also add a determination of the world, as a thing real in itself, which might likewise be false, if, namely, the world were not given at all as a thing in itself, and hence, as regards its magnitude, neither as infinite nor as finite. Permit me to call such an opposition a dialectical opposition, but the contradictory one an analytical opposition. Thus two judgments dialecti-cally opposed to one another could both be false, because one does not merely contradict the other, but says something more than is required for a contradiction. (A 503-4/B 531-2)1

From this passage one could extract the following explications of the relations of analytical opposition and dialectical opposition, respectively.

(Anal. Opp. 1) Two judgments p, q are analytically opposed to one another iff ρ is analytically equivalent to ->q.

(Dial. Opp. 1) Two judgments p, q are dialectically opposed iff there are judgments r, s different from p, q such that ρ analytically implies -iq Λ τ, q analytically implies -φ Λ s, ->q does not analytically imply r and -ip does not analytically imply s.

Note, that it follows from (Dial. Opp. 1) that ρ and q can both be false. This is the case, for instance, if r and s are both false. Note further that r and

1 In quoting from the CPR, I follow the translation by Paul Guyer and Allen Wood, though I shall modify that translation whenever I judge it to be appropriate. Page numbers refer to the first (=A) and second edition (=B).

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3 8 Wolfgang Malzkorn

s are not required to be different judgments. - Although Kant takes the space-antinomy as an example, it is understood from the context, that the relation of dialectical opposition is supposed to apply to all four antinomies.2 However, there are passages from other sources where Kant gives a different account of the opposition relations which hold between the opposed statements in some antinomies. In R 6337, for example, he states that in the third antinomy thesis and antithesis "can both be true since each of them says less than is required for an opposition".3 Moreover this statement is confirmed by Kant's solution of the third (and fourth) antinomy. Therefore, Kant's explication of the concept of dialectical opposition quoted above is not accurate. Can it be improved?

2. An Improved Explication of Dialectical Opposition

As is well known, Kant most often uses the term "antinomy of pure reason" in the singular and talks about the antinomy of pure reason. Within the CPR two explications of that term are given; the first one can be found at A 340/ Β 397-98, the second one at A 407/B 434. It is widely agreed that the latter is more appropriate to the subject as it is developed by Kant than the former, and, though I cannot state my reasons here, I share this opinion.4 According to A 407/B 434 the antinomy of pure reason is "an opposition [...] of the laws of pure reason". Consequently, what we nowadays call the antinomies are particular instantiations or specifications of that opposition of the laws of pure reason.5 Yet, Kant makes it clear beyond any doubt that pure reason is not eo ipso an antinomial faculty; that is, pure reason does not fall victim to antinomies under any possible circumstances whatsoever. Kant describes exactly the particular circumstances under which pure reason naturally and unavoidably runs into antinomies, namely, if reason subscribes to the natural presupposition of transcendental realism. If space and time are taken to be real things or properties of real things rather than mere forms of intuition and if real things are taken to be things in themselves rather than mere appear-ances, then reason falls victim to the antinomies. According to Kant, the antinomies are a particular kind of dialectical illusion that arises under the false presupposition of transcendental realism. Taking these considerations into account, one could try to define the concept of dialectical opposition as follows:

(Dial. Opp. 2) Two judgements p, q are dialectically opposed iff there is a natural (but nonetheless false) presupposition Φ such that, if Φ were true, then ρ and q were analytically opposed.

2 Cf. CPR A 5 0 5 - 6 / B 533-34. - I am thus opposing J. E. Llewelyn (1964), p. 171. 3 Kants gesammelte Schriften, vol. 18, p. 658 (my translation); cf. also R 5962, op. cit., p. 4 0 4 4 Cf. Ν. Hinske (1966), p. 488, Ν. Hinske (1971), columns 393-94, and W. Malzkorn (1999),

chapter 2.1.1. 5 Unfortunately, I cannot go into the details here; I have to refer the reader to W. Malzkorn

(1999), chapter 2.1.


Analytical and Dialectical Oppositions Reconsidered 39

Contrary to (Dial. Opp. 1 ) this definition refers to the concept of analytical opposition and thus presupposes an adequate understanding of that concept. Is that concept adequately defined by (Anal. Opp. 1)? Consider the second antinomy. The main statement of the thesis is:

(T2) Every composite substance in the world consists of simple parts.6

The corresponding statement of the antithesis is:

(A2) No composite thing (=substance, W.M.) in the world consists of simple parts.7

Obviously, (T2) is a general affirmative or A-proposition, whereas (A2) is a general negative or Ε-proposition. Generally, two judgments of this form (that have the same subject and predicate terms) are not contradictory but rather opposed to one another as contraries. So, what shall we say about the relation between (T2) and (A2)? Are they just contrary opposites? Or, do we have to look for a suitable presupposition Φ such that, ίίΦ were true, (T2) and (A2) would be contradictory opposites? - There are two presuppositions each of which would do the job:

(i) if some composite substance in the world consists of simple parts, then every composite substance in the world consists of simple parts;

(ii) if some composite substance in the world does not consist of simple parts, then no composite substance in the world consists of simple parts.

However, neither of these statements comes even close to the kind of presupposition Kant had in mind. Each of them excludes hybrid mereological structures (i. e. mereologies where some substances consist of simple parts and some do not); but why should a transcendental realist presuppose that there are no hybrid mereological structures, when this is yet to be proved (in the proofs of (T2) or (A2), respectively)? And whatever Kant is up to in his examples of the smelling bodies in CPR A 503/Β 531 or in his example of the four-cornered circles in Prolegomena, § 52b, it has nothing to do with either (i) or (ii). Therefore, I suggest taking (T2) and (A2) as contrary statements and modifying the explication of the concept of analytical opposition to include contradictory as well as contrary opposites.

(Anal. Opp. 2) Two judgements p, q are analytically opposed iff ρ => ->q is analytically true.

Moreover, this definition agrees perfectly with Kant's explications of the concept of opposition in his lectures on logic.8 According to the lecture manuscripts on logic (and some of Kant's handwritten notes), a proper oppo-

6 Cf. CPR A 4 3 4 / B 462 7 Cf. CPR A 4 3 5 / B 4 6 3 8 Cf. Logic Blomberg, in Kants gesammelte Schriften, vol. 24, pp. 281-82 ; Logic Pölitz, op.

cit., p. 584; Logic Dohna-Wundlacken, op. cit., p. 770. - A different account is given, for example, by J. E. Llewelyn (1964), p. 171, and M. Wolff (1981), pp. 41-61.


40 Wolfgang Malzkorn

sition is either a contradictory or a contrary opposition. The third kind of opposition included in the traditional square of opposition, i. e. subcontrary opposition, is not a proper opposition according to Kant. As I shall show later, the contrary opposition between (T2) and (A2) depends on a presupposition, so that it can be qualified as a dialectical opposition in the sense explained in (Dial. Opp. 2). However, before I turn to the discussion of particular antino-mies, I have to say more about the concept of presupposition underlying (Dial. Opp. 2).

As is well known, Kant takes transcendental realism to be the crucial presupposition being in question. He solves the antinomy of pure reason by arguing for the failure of transcendental realism and replacing transcendental realism by transcendental idealism. In the case of the mathematical antino-mies, Kant even claims that the replacement of transcendental realism by transcendental idealism renders the respective antinomial statements false. So, one might assume that the presupposition in question is not only a sufficient condition for the analytical opposition of the respective antinomial statements, but also a necessary condition for the truth of those statements. But, this would be mistaken as Kant's solution of the dynamical antinomies shows. In the case of the dynamical antinomies, Kant claims that only the replacement of transcendental realism by transcendental idealism makes the joint truth of both respective antinomial statements possible. Consequently, the presupposition in question cannot be a necessary condition for the truth of the antinomial statements. Yet, one might argue instead that the presuppo-sition in question is a necessary condition for the provability of the antinomial statements. This is indeed true according to Kant and it should be considered in the explication of the concept of dialectical opposition, since, on the one hand, it strengthens the ties between presupposition and opposition relations and, on the other hand, it leads the way to a correct analysis of the proofs of the respective opposites. Consequently, we end up with the following defini-tion:

(Dial. Opp. 2') Two judgments p, q are dialectically opposed iff there is a naturally assumed (but nonetheless false) presupposition Φ such that (a) ρ and q are provable only with respect to Φ and (b), if Φ were true, then ρ and q would be analytically op-posed.

3. Dialectical opposition and the antinomies

Despite the improved analysis of the concept of dialectical opposition, one might still wonder whether or how an analytical opposition between two judgments can depend on the truth or falsity of a particular presupposition. Note, however, that the presupposition in question could be a semantical presupposition. Indeed, the crucial presupposition, Kant had in mind, i. e. transcendental realism, can be conceived as a semantical presupposition, and



so can its opposite, i. e. transcendental idealism. The latter can be taken to imply the following semantical claims:

(I) the universe of discourse must be divided into two classes: the class of intuitively given objects (appearances) and the complementary class of so-called things in themselves; and

(II) predicates of time determinations, predicates of spatial determinations, and the categories of the understanding may be applied only to intui-tively given objects, in order to yield propositions capable of being true.

In contrary, transcendental realism can be taken to imply the following semantical claim:

(III) all (physical) objects are things in themselves and subject to time determinations and spatial determinations.

Let me now turn to a brief discussion of the particular antinomies. I shall start with the first one. Within the antinomial statements of that antinomy (thesis and antithesis) we find mutatis mutandis two pairs of propositions opposed to one another.9 The first one is:

(T i l ) The world has a beginning in time. (All) The world has no beginning in time.

The second one is:

(T12) The world is enclosed in boundaries in space. (A12) The world is not enclosed in boundaries in space.

Note, that the words "finite" and "infinite" do not occur in Kant's formu-lation of the thesis. As regards the antithesis, the infiniteness of the world as regards time and space is merely a consequence drawn from (All) and (A12), respectively. Therefore, contrary to several other commentators, I do not take claims about the finiteness or infiniteness of the world as major claims of the thesis or antithesis. It can be misleading to do so, as I shall argue below. For, one might follow Kant's example for dialectical opposites in A 503-4/B 531-32 (quoted above) and present, roughly, the following logical analysis of thesis and antithesis:

(Thesis) Determined magnitude as regards time/space (the world) λ Finite as regards time/space (the world),

(Antithesis) Determined magnitude as regards time/space (the world) λ Infinite as regards time/space (the world).

According to this analysis, one might then conclude that the first conjunct of each claim is the false presupposition in question. - 1 argue that this general schema of analysis is quite correct, but that one should pay more attention to Kant's actual formulation of the antinomial statements. There are many difficulties connected with the analysis just presented which I cannot point

9 Cf. CPR A 4 2 6 / Β 4 5 4 and A 4 2 7 / B 4 5 5


42 Wolfgang Malzkorn

out here.101 suggest the following analysis of (T i l ) and (All) instead. In a first step, define two predicates, namely, "has a beginning in time" and "has no beginning in time" in the following way:

(Def. beginning) χ has a beginning in time iff χ exists in time and there is a time t where χ has existed such that for all times t' earlier than t, χ has not existed at t',

(Def. no beginning) χ has no beginning in time iff χ exists in time and for any time t where χ has existed there is an earlier time t' such that χ has already existed at t'.

In the second step then analyse (Ti l ) and (All) as singular propositions in which these predicates are applied to the world. The predicates are defined in a way such that (T i l ) and (All) fulfil the following conditions:

(a) (T i l ) implies that the world exists in time, but states more than that; (b) (All) implies that the world exists in time, but states more than that; (c) if it were true that the world exists in time, (T i l ) and (All) would be

contrary opposites; (d) neither (TI 1 ) nor (Al 1 ) could be proved without presupposing that the

world exists in time; (e) if it is false that the world exists in time, (T i l ) and (All) are both false;

and, finally, (f) on the basis of the semantical claims of transcendental idelism ((I) and

(II)) a strong case can be made against the presupposition that the world exists in time.

An analogous analysis could be given for (T12) and (A12); but I have to stop here, in order to turn to the second antinomy.11

As I have already said above, the main statement of the thesis of the second antinomy is a general affirmative statement, while the corresponding state-ment of the antithesis is a general negative statement. Consequently, we have a pair of statements one of which has the logical form All A's are B's, whereas the other one has the form No A is Β.

As is well known, two statements of this form are contrary opposites, that is, they can jointly be false, but they cannot jointly be true. However, the opposition relation vanishes, if one translates such statements into the lan-guage of formal logic like it is sometimes done, taking Vx(Ax Bx) as translation of All A's are B's and Vx(Ax z> -¡Bx) as translation of No A is B. The reason is that this translation does not preserve the existential import of a general (affirmative or negative) statement in traditional logic.12 Thus, the opposition between the opposed statements in Kant's second antinomy de-

10 I would like to refer the reader to chapters 2.2 and 3.1 of W. Malzkorn (1999). 11 A similar analysis of the logical structure of the first antinomy is given in Niko Strobach's

interesting paper in this same volume. 12 M. Wolff (1995), pp. 281-292 , has claimed that according to traditional logic including Kant

only affirmative statements have existential import, whereas negative statements do not have existential import. I have refuted this claim in W. Malzkorn (1997), p. 201.



pends on their existential import. (T2) and (A2) are only opposed to one another, if there are composite substances in the world. Is there any connec-tion between this (logical) observation and the crucial presupposition Kant had in mind? - According to what I call the standard interpretation of Kant's second antinomy the answer is negative. That interpretation follows Kant's suggestion in the example of the smelling bodies in CPRA 503/B 531 thatthe presupposition in question is somehow connected with the predicate terms of the respective statements, rather than with the subject term. Note, however, that Kant's example of the four-cornered circle in Prolegomena, § 52b, points into a different direction. Following that example, I argue that the presuppo-sition in question is the proposition that there are substances in the world which are actually composed of other actual substances. In addition to some explicit remarks of Kant which can be cited in favor of this interpretation,13

it fulfils the following conditions:

(a) (T2) implies that there are substances in the world which are actually composed of other actual substances;

(b) (A2) implies that there are substances in the world which are actually composed of other actual substances;

(c) if it were true that there are substances in the world which are actually composed of other actual substances, (T2) and (A2) would be contrary opposites;

(d) neither (T2) nor (A2) could be proved without presupposing that there are substances in the world which are actually composed of other actual substances;

(e) if it is false that there are substances in the world which are actually composed of other actual substances, then (T2) and (A2) are both false; and, finally,

(f) on the basis of the sematical claims of transcendental idealism ((I) and (II)) a strong case can be made against the presupposition that there are substances in the world which are actually composed of other actual substances.

I now turn to the third antinomy. Considering Kant's claim in R 6337 that in the third antinomy thesis and antithesis "can both be true since each of them says less than is required for an opposition" (see quotation above), one could think that the antinomial statements of the third antinomy are subcontrary opposites in the sense of the traditional square of opposition. However, Kant does not present a pair of a particular affirmative (I-)statement and a particular negative (O-)statement. Although the thesis can be taken as a particular negative statement, the antithesis must then be taken as the corresponding general affirmative (A-)statement:

Cf. CPR A 5 0 5 / B 533; Metaphysical Foundations of Natural Science, Kants gesammelte Schriften,voi. 4,pp. 5 0 6 - 7 ; R 5310, Kants gesammelte Schriften, vol. 18, p. 1 5 0 ; a n d R 5 6 5 0 , op. cit., p. 299. Cf. also W. Malzkorn (1998) and W. Malzkorn (1999), chapter 3.2.


4 4 Wolfgang Malzkorn

(T3) Some occurences of causality in the world are not occurences of causal-ity according to the laws of nature.

(A3) All occurences of causality in the world are occurences of causality according to the laws of nature.

Therefore, over one and the same undivided universe of discourse, or, in other words under the semantical presupposition of transcendental realism (III), the thesis and the antithesis are contradictory statements. If, however, the semantical claims of transcendental idealism ((I) and (II)) are true, and if the antithesis is restricted to the realm of appearances while the thesis is allowed to range over the whole universe of discourse (appearances and things in themselves), then the opposition vanishes and both antinomial statements can jointly be true.

So much for the third antinomy. Since an analysis of the fourth antinomy is much more difficult and complicated without providing any further insight as regards the current subject, I shall skip that antinomy and stop here.14

References

Falkenburg, Β. (2000): Kants Kosmologie. Die wissenschaftliche Revolution der Naturphilosophie im 18. Jahrhundert; Frankfurt/M. 2000

Hinske, N. (1970): Kants Weg zur Transzendentalphilosophie. Der dreißigjährige Kant; Stuttgart-Berlin-Köln-Mainz 1970

Hinske, N. (1971): Entry "Antinomie" in: Historisches Wörterbuch der Philosophie, ed. by J. Ritter; Darmstadt 1971-, vol.1, columns 393-396

Kant, I.: Critique of Pure Reason, first edition Riga 1781, second edition Riga 1787, transi, and ed. by P. Guyer and A. Wood; Cambridge University Press 1998

Kant, I: Kants gesammelte Schriften, ed. by the Königlich Preußischen Akademie der Wissen-schaften, from vol. 23 by the Akademie der Wissenschaften der DDR; Berlin 1900

Llewelyn, J.E. (1964): "Dialectical and Analytical Opposites"; Kant-Studien 54 (1964) 171-174 Malzkorn, W. (1997): "Rezension von M.Wolff" (1995); Philosophisches Jahrbuch 104 (1997)

196-202 Malzkorn, W. (1998): "Kant über die Teilbarkeit der Materie"; Kant-Studien 89 (1998) Malzkorn, W. (1999): Kants Kosmologie-Kritik. Eine formale Analyse der Antinomienlehre·,

Berlin-New York 1999 (= Kant-Studien Ergänzungshefte, Band 134) Strobach, N. (2000): "Qualifizierte Negation als Schlüssel zum Verständnis der 1. Antinomie in

Kants KrV"; abgedruckt in den vorliegenden Kongressakten, Seite 147. Wolff, M. (1981): Der Begriff des Widerspruchs. Eine Studie zur Dialektik Kants und Hegels;

Königstein/Ts. 1981 Wolff, M. (1995): Die Vollständigkeit der kantischen Urteilstafel. Mit einem Essay über Freges

Begriffsschrift·, Frankfurt/M. 1995

14 I would like to refer the reader to chapters 2.5 and 3.4 of W. Malzkorn (1999).


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