Hegelian Conjunction, Hegelian Contradiction

Abstract

Understanding what is the strictly formal import of Hegel's view is something on which few analytic philosophers have seen time worth spending. Still in our view there is enough in Hegel's work to suggest that a formal account of his dialectial process might be profitable both for better understanding Hegel's ideas and for introducing a new sort of glutty logic. The focus of our analysis is Hegel's Conjunction, which, in our view, is the crucial dialectical operation, on which all others are ultimately based.

KEYWORDS:

• Conjunction
• contradiction
• Hegel's dialectics
• true contradictions

Acknowledgments

For precious help during the preparation of this paper and for extremely useful comments on one of its versions we are deeply grateful to Franca d'Agostini.

Notes

1 It is not our aim in this paper to focus on the question about the kind of opposition at stake in Hegel's dialectics. We simply take for granted that Hegel's dialectic involves a view of opposition as contradiction, and a genuine critique of the law of non-contradiction. One of us has already strongly argued for this view in recent work. See Ficara 2021.

2 The new glutty logic/the new way of being glutty implies a specific understanding of glut as ‘true contradiction’ but not as ‘sentence that is both true and false’. For more details on the difference between the two understandings of glut see D'Agostini 2021, sections 1.1–1.5. For further developments of this new way of being glut theorist see also D'Agostini 2021.

3 See D'Agostini and Ficara 2021 for the analysis of Hegel's interpretation of the Liar from the point of view of conjunctive paraconsistency.

4 We quote Hegel's Werkausgabe (see Hegel 1969) as Hegel Werke, followed by the volume and page number. Hegel Werke 18, 529.

5 Here, we use & for a generic conjunction symbol; we will devote specific symbols below to indicate standard conjunction and our target ‘Hegelian conjunction’.

6 In his early writings, Hegel formulates a specific theory about the link joining the two terms of an antinomy – he calls this connective Vereinigung. This use and the underlying concept is confirmed in his later works. The German term Vereinigung means, literally, ‘unification’. In Hegel's use Vereinigung has not only logical, but also epistemological and theological implications. In what follows, we focus on the logical meaning, and translate Vereinigung as (hegelian) conjunction (whereby we distinguish hegelian conjunction from standard logical conjunction). See for more historiographical background on Vereinigung as well as for a slightly different interpretation of its semantics Ficara 2021, 173f. In D'Agostini and Ficara 2021 we examine the connections between the theory of Vereinigung and Hegel's notion of the completeness of truth.

7 Hegel Werke 1, 251.

8 Hegel Werke 2, 26.

9 Hegel 1991, §§ 69–71 refers to the ‘three sides of das Logische’. We refer to them as ‘stages’ or also thesis antithesis synthesis (as in Priest 2013, 3).

10 Hegel explains the move from s1 to s2 in terms of reflection, which generates the negation of the proposition at s1; however, our aim here is not to find a formal equivalent for reflection but just to give a formal account of ‘Hegelian conjunction’.

11 The given information – that the closure relation is just classical logic extended with the rule that $⋏$ and ∧ are entirely equivalent – is sufficient to specify the next topic, which is stage three (see § 3.3); but the following is provided for those unfamiliar with the idea of ‘classical closure’. We note that for iterating the process, the closure operator for early stages is different than on the initial ‘first cycle’ (first triple), so to speak. See § 5.4.

12 In familiar parlance, we treat truth/falsity in a model for atomics as ‘super-truth/falsity’ (with respect to all relevant points, in this case, the closures of early stages).

13 See for more details about the role of the failure of simplification for paraconsistency D'Agostini 2021.

14 In Ficara 2021 and D'Agostini and Ficara 2021 an interpretation of Hegelian conjunction close to the one given in this paper is defended and further developed.

15 We change Havas' notation to be in line with our usage in this paper.

16 Whether this failure promises anything interesting with respect to familiar curry paradoxes, which are sometimes blocked via deviant conjunction behavior (requiring adjustment to structural contraction rules) might be worth exploring, but we leave it aside here.

17 See on this D'Agostini 2021, sections 1.3, 1.4 and 1.5.

18 In fact, Priest 1989 uses this as reason to think that Hegel espouses a simple subclassical logic in which excluded middle is not only true but also false (though strictly speaking, on Priest's proposal excluded middle winds up being simply valid, not both valid and invalid, but we leave this for another venue).

19 Note that, except at ‘classically filtered’ early stages, negation itself is not a ‘dualizing’ logical operator in the way that it is when one ignores Hegelian conjunction.

20 We are grateful to Graham Priest, who advanced this objection in conversation, and who in Priest 1989 subscribes to the given pragmatic reading in his own glut-theoretic reading of Hegel (one of what we're calling ‘standard’ glut-theoretic readings, whereby Hegel takes the conjuncts of an Hegelian contradiction to each be gluts, both true and false on their own).

21 We are grateful to Sam Wheeler for prompting our focus on this question, and especially to Dave Ripley for enjoyable and fruitful conversation on the topic.