The Logical Structure of Dialectic

Abstract

I give a formal model of dialectical progression, as found in Hegel and Marx. The model is outlined in the first half of the paper, and deploys the tools of a formal paraconsistent logic. In the second half, I discuss a number of examples of dialectical progressions to be found in Hegel and Marx, showing how they fit the model.

Keywords:

• Dialectic
• Marx
• Hegel
• dialetheism
• paraconsistent logic

Notes

1 See Priest 1990, Hegel 1931, Priest 2019.

2 Hegel 1969, 440.

3 Hegel 1955, 460.

4 See, further, Priest 1990, Priest 2019.

5 See, e.g. Priest 2006, ch. 5.

6 Of course, for fuller logical exegeses of Hegel, the formal language would need to contain not just monadic predicates, but binary predicates, and maybe even predicates of higher adicity. However, simple monadic predicates will suffice for our purposes.

7 Hegel refers to them thus, occasionally (e.g. Lesser Logic, para 95), though the terminology is, perhaps, more frequently associated with Engels. In Fichtean terminology, the stages are the thesis, antithesis, and synthesis.

8 In some applications of dialectics, it makes more sense to think of $\mathrm{\neg }P$ as a contrary of P, rather than the contradictory. But in this case $Pa\wedge \mathrm{\neg }Pa$ is still a contradiction. (Red and green are contraries; and if something is green, it is not red.)

9 See Priest 1990, sect. 8.

10 One could avoid this by using a different paraconsistent logic. However, a Hegelian dialectical progression requires a new concept to emerge here, so something like the next step is necessary anyway.

11 Alternatively, if we have the operator $\text{^{}}$ in the language, we can simply take $P^{†}x$ to be $\text{^{}}Px=\text{^{}}\mathrm{\neg }Px$. $P^{†}a$ then entails $Pa\wedge \mathrm{\neg }Pa$, though it may not be entailed by it.

12 Actually, this is not necessary for what follows. All we need is that $a\notin {\delta }_4^{-}\left(P^{†}\right)$, but let us keep matters simple.

13 Hegel 1969, 107. Note that all italics in all quotations are the translator's.

14 For an excellent general exposition of Hegel's philosophy, see Taylor 1975.

15 Vol. I, Bk. 1, Sec. 1, Ch. 1. Hegel does, it is true, balk at using non-being instead of nothing. The reason is that the use of explicit negation would build in a formal element to the opposition; and Hegel wishes the opposition, for reasons that are not relevant here, to be a matter of content, not form. However, he points out that using non-being instead, gives exactly the same result: ‘Should it be held more correct to oppose to being, non-being instead of nothing, there would be no objection to this so far as the result in concerned, for in non-being the relation to being is contained; both being and its negation are enunciated in a single term, nothing, as it is in becoming. But we are concerned first of all not with the form of opposition […] but with the abstract immediate negation: nothing, purely on its own account, negation devoid of any relations […]’ (Hegel 1969, 83).

16 Hegel 1969, 105.

17 See, further, Priest 2006, ch. 12.

18 Hegel 1969, 109.

19 Pt. B, Ch. 4, Sec. A.

20 See Taylor 1975, 153 ff.

21 Hegel 1931, 234.

22 Hegel 1931, 229.

23 Marx 1976, 140.

24 Marx 1976, 198.

25 See Katz 1993.

26 The story is well told in Thompson 1980.

27 Marx 1976, 415. Note that the freedom and bondage in question are not in different respects. See Priest 1991, 472 f.

28 Versions of this paper were given at the conference Logic and Politics, at the University of Paderborn, December 2013, the conference Hegel, Analytic Philosophy and Formal Logic, Purdue University, October 2014, and in a seminar in the (online) series, Hegel and Dialetheism, November 2021. Many thanks go to the audiences for their helpful comments, and especially to Leon Geerdink and Stefan Schick.

29 For details of this, see Priest 2006, ch. 5.

30 If we drop the constraint that for all predicates, P, ${\delta }^{+}\left(P\right)\cup {\delta }^{-}\left(P\right)=D$, we have the logic of First Degree Entailment, FDE. If we add the constraint that for all predicates, P, ${\delta }^{+}\left(P\right)\cap {\delta }^{-}\left(P\right)=\mathrm{\varnothing }$, we have classical logic, CL.

31 Or, if the dialectic is finite, for n<3m, for some m>0.

32 One can accommodate the fact that things come into and go out of existence by supposing that there is a monadic existence predicate whose extension/anti-extension varies with n.