1 Introduction

In 2023 Beall and Ficara present what they call Hegelian conjunctions. A Hegelian conjunction is a true conjunction of contradictory opposites in which the conjuncts, separately taken, are untrue and for which simplification fails. Beall and Ficara propose a semantics based on this idea and hint at some of its logico-philosophical consequences. Although the idea of Hegelian conjunctions is crucial for several reasons (see d’Agostini, 2021 for a comprehensive theory of the role of conjunction in paraconsistency that motivates and inspires the idea of a special Hegelian conjunction), I think that the presentation of Beall & Ficara, 2023 can be expanded, mainly in two directions: first, by giving a closer look at the textual basis used by Beall & Ficara, 2023 to motivate their proposal (the Hegelian fragment called, in the new historico-critical edition of Hegel’s Frühe Schriften, “42”),Footnote 1 and, second, by examining the meaning of the fragment for paraconsistency and dialetheism. In the first section of the paper, I present the content of the fragment by focusing on three concepts introduced by Hegel: the concept of Glauben (believing an antinomy), the concept of Vereinigung (the unification of antinomic opposites), and the concept of Seyn (the independent existence of the antinomic content). In the second section, I focus on the importance of the fragment for paraconsistency and dialetheism by highlighting three main ideas that emerge from the fragment and that are relevant for expanding and deepening the reasoning introduced in Beall and Ficara, 2023. Finally, in the conclusion, I consider the following question: What is the benefit of Hegel for paraconsistency, dialetheism, and for Beall’s work in the specific case? This account is interesting from two points of view: first, for the introduction of the specificity of Beall’s approach into the literature about Hegel’s logic and, second, for promoting unity in the somewhat diffuse contemporary account of true contradictions.

2 Hegel’s fragment on Vereinigung

In a short fragment written by Hegel in Bern and revised in Frankfurt between December 1797 and the beginning of 1798, now numbered, in the new historico-critical edition of Hegel’s Frühe Schriften, “42”, Hegel focuses on the unification [Vereinigung] “through which an antinomy is unified [vereinigt]”.Footnote 2 This is the first Hegelian text explicitly devoted to the connection between the two elements in an antinomy. Accordingly, it deserves closer consideration.

While literature abounds on the role of negation in Hegel’s dialectical logic, there are few works that deal with the “connective” that joins the two elements of a dialectical contradiction.Footnote 3 In this respect, any complete reconstruction of the meaning of unification in dialectics cannot but have Hegel 42 as its first vital theoretical and historical reference point.

In dealing with this text, it is important to keep in mind that these are notes that Hegel took. In them, Hegel addresses the phenomenon that he writes about (Vereinigung) from different angles that are apparently disconnected from each other. Yet this text is extremely fascinating: it is a very short presentation of an ontological, logical and epistemological hypothesis that is at the center of discussions about glut theories and dialetheism. Until now, the fragment has been almost inaccessible, especially in English, though it is a common opinion among all interpreters that its difficulty borders on the enigmatic. It is thus my aim to make these Hegelian notes, and their relevance for logic, more accessible.

‘Vereinigung’, first of all, means unification in German. Instead of using ‘unification’, however, I prefer to retain the German term. As it was used in post-Kantian discussions, especially by Hölderlin, Sinclair and Hegel, the term is evocative of a specific philosophical program, inspired by Hölderlin (1961), deepened by Hegel, and then molded into Hegel’s own idea of the dialectical method (Henrich, 1965–66 and Henrich, 1970, Düsing, 1976, Baum, 1989). Henrich, (1970) especially has studied and recognized the philosophy of Vereinigung in terms of its striking interest for philosophy in general. By contrast, my aim is to focus on its importance for logic.

In Hegel 42, the idea of the unification of antinomic opposites (Vereinigung) as expression of true being, an idea that will constitute one central element of Hegel’s mature concept of dialectics, makes its first appearance. Three basic concepts are at the core of the fragment:

the concept of Glauben (believing)

the concept of Vereinigung (unification)

and the concept of Seyn (being).

I will discuss what Hegel says about each one of them, presenting the German text of the fragment first in italics, followed by my translation.

3 Believing [Glauben]

Glauben is sometimes also translated as “faith”. However, I prefer to use “believing” for the reason that this maintains the openness of the German Glauben, which can be used to refer to faith, but also has the general meaning of believing. Hegel thus defines Glauben or believing as follows:

  1. a)

    Glauben ist die Art, wie das vereinigte wodurch eine Antinomie vereinigt ist, in unserer Vorstellung vorhanden ist

Believing is the way in which the unified through which an antinomy is unified is present in our representation (Hegel 42, 10).

This means that there is a “unified”, i.e. something through which the two terms of an antinomy are joined together, and also that believing is the way in which we have, in our representation, this thing in front of us. One example comes from the Kantian antinomies, which Hegel must have had in mind when he was writing.Footnote 4 Antinomies for Kant are couples of valid arguments with conclusions one of which is the negation of the other, e.g. “matter is continuous” and “matter is not continuous”. In Hegel 42, Hegel presents a somewhat striking view: the idea that believing implies unifying antinomic opposites and is, therefore, belief in an antinomic content. One can find a similar idea in other texts. For example, in the Lectures on the History of Philosophy, Hegel states that, in the concept of matter, two opposite determinations, continuity and punctuality, are truly unified in one. The true belief here is belief in the antinomic content as true, i.e. belief in matter’s being both continuous and not continuous.

4 Unification [Vereinigung]

On Vereinigung, Hegel writes:

  1. b)

    Die Vereinigung ist die Tätigkeit; diese Tätigkeit reflektiert als Objekt ist das Geglaubte

Vereinigung is activity; this activity grasped as an object is what is believed (Hegel 42, 10).

The unification of opposites is a procedure or (subjective) activity; we perform it by unifying the opposites. If we reflect on that unification, we “hypostasize” it and it becomes an object that stands against or in front of us and in which we believe. In other words, the thing that is unified and through which the two conclusions of an antinomy are joined together is an activity of our own. For example, in thinking about the antinomic content of matter, we bring the antinomic properties of “being continuous” and “not being continuous” together. The antinomic content is the “activity grasped as an object”, i.e. an object that corresponds to this active bringing together of opposites and that stands before us.

The next passage explains the relationship between unification and the opposites.

  1. c)

    Das widerstreitende kann als widerstreitende erkannt werden nur dadurch, dass schon vereinigt worden ist. Die Vereinigung ist der Maßstab, an dem die Vergleichung geschieht, an welchem die entgegengesetzten, als solche, als unbefriedigte erscheinen.

That which contradicts can be known as contradictory only if it has already been unified. Unification is the criterion on whose basis comparison occurs, the criterion that lets us see the opposites, as such, as unsatisfied/insufficient (unbefriedigte) (Hegel 42, 10).

To illustrate, take the case of “matter is continuous” and “matter is not continuous”. One of these opposites, matter’s continuity, can be recognized as an opposite only if it has already been unified with its negation, matter’s punctuality, and vice versa. Hence, Vereinigung is that through which comparison occurs – it is the condition for the possibility of assessing opposites as opposites or, as Hegel puts it, the criterion [Maßstab] for doing so.Footnote 5 Through this comparison, we see that the opposites as such [die entgegengesetzten, als solche], i.e. on their own, emerge as “unsatisfied” or insufficient [als unbefriedigte erscheinen]. They are not able to express what is, namely, the full content at stake: the unity of the opposites in one concept.

  1. d)

    Wenn man zeigt, dass die entgegengesetzten beschränkten als solche nicht bestehen können, dass sie sich aufheben müssten, dass sie also um möglich zu sein, eine Vereinigung voraussetzen, dann hat man bewiesen, dass sie vereinigt werden müssen, dass die Vereinigung sein soll […] aber dass die Vereinigung ist wird nicht bewiesen, sondern geglaubt […] beweisen heißt die Abhängigkeit aufzeigen […] die Vereinigung ist das unabhängige.

If we show that the limited opposites cannot subsist as such, that they must overcome themselves (aufheben), and hence that to be possible they presuppose a unification, then we have proved that they must be unified, that their unification ought to be […] however, that this unification is, is not proved, but rather believed […] to prove means to show dependence […] this unification is the independent (Hegel 42, 10).

The antinomic opposites cannot subsist on their own, for they “overcome themselves” (sich aufheben).Footnote 6 If we take the first opposite, matter’s continuity (A), we can show that it cannot subsist on its own, i.e. be taken as true, for the concept of matter also implies non-continuity; thus, the first opposite overcomes itself and turns into its negation. However, if we take the other opposite, matter’s punctuality (~ A), we can also show that it cannot be taken as true, for the concept of matter also implies non-punctuality; the second opposite thereby overcomes or negates itself as well.

If we show this, we have proof of the fact that the opposites have to be unified, that their unification has to be [sein soll].Footnote 7 We have thereby proved the necessity of the contradiction. Hegel also states that, while proof is a procedure that shows how something necessarily depends on something else, e.g. that each opposite depends on its unification with the other, “Vereinigung is what is independent” and “that Vereinigung is, is not proved but believed”. We have a proof that Vereinigung sein soll (ought to be), as the opposites cannot subsist on their own. However, we do not have a proof of the existence of Vereinigung, or the fact that Vereinigung is. This fact becomes the object of belief.

If Hegel has already shown the epistemological (Glauben: believing the Vereinigung) and the logical (Beweis: demonstrating the necessity of the Vereinigung) bases of Vereinigung, we can now introduce a metaphysical one: Being, or the existence of Vereinigung. At this point, however, a question that is important for both philological and logical reasons should be addressed. That Vereinigung is independent, and hence “not proved” is normally read as meaning that it is not logical and not conceptual (Düsing, 1976, 51). More specifically, Düsing (1976, 51) stresses that Hegel, in this fragment, has not yet developed his mature dialectic, which implies that the unification of the antinomic opposites is a logical operation. The Vereinigung Hegel writes about in Hegel 42 pertains, at least for Düsing, to the sphere of feeling. In this sense, Düsing (1976, 53) highlights the primacy of the certainty of faith (Glaubensgewissheit) over demonstration: Vereinigung is not proved, but is instead felt/believed. Yet, this “non logical” interpretation would go against the definition of Vereingung as the unification through which the two elements of an antinomy are unified. Moreover, Düsing’s interpretation conflicts with Hegel’s identification of unification with the copula “is” in a judgement, as indicated in the following paragraphs, especially f, and also with being as what is presupposed and that to which we refer when we formulate a judgement, as Hegel proposes in g.

In this light, as will become clearer in the analysis of the following paragraphs, my suggestion is that Hegel’s statement “Vereinigung is what is independent” is simply a sign of Hegel’s realism. What Hegel means with this statement is that Vereinigung is what we assert as true and that in which we believe. As such, it expresses what exists independently from us, in a realist sense of “true”. The statement “that Vereinigung is, is not proved” thus means the following in my account: while that Vereinigung must be is the result of the proof through which we show that each opposite cannot subsist without being unified with the other, that Vereinigung is instead expresses something that we believe to be true, i.e. in the sense that it exists independently from us.

  1. e)

    Das in Rücksicht auf diese entgegengesetzte Unabhängige kann in anderer Rücksicht wieder ein abhängiges entgegengesetztes sein; und dann muss wieder zur neuen Vereinigung fortgeschritten werden, die jetzt wieder das Geglaubte ist.

That which is independent with respect to these opposites can in turn be a dependent opposite in some other respect; and then once more a new Vereinigung must be sought, which now becomes that which is believed in.

We see here a first hint at the later Hegelian idea of the dialectical process. To go back to the example used to illustrate d), matter’s continuity (C) implies matter’s punctuality (not-C) and vice versa; hence, the unification of continuity and punctuality (C and not-C) must be. Let us call the unification of C and not-C “C*”. C* becomes the object of our true belief – C* thus stands, from one point of view, for the unification of C and not-C. However, from another point of view, it is possible to set C* against something, namely, against not-C*. e) describes the passage or, as in Hegel’s later terminology, the dialectical development from the antinomic opposites, to their unification, to a new object (here: C*), and then to another opposition (C* against not-C*) with another unification (C* and not-C*).Footnote 8

5 Being (Seyn)

On the connection between Vereinigung and being Hegel writes:

  1. f)

    Vereinigung und Seyn sind gleichbedeutend; in jedem Satz drückt das Bindewort: ist, die Vereinigung des Subjekts und Prädikat aus – ein Seyn.

Vereinigung and being have the same meaning. In every sentence [Satz] the connecting word “is” expresses the unification of subject and predicate – a being (Hegel 42, 10).

The copula “is” connects the subject with the predicate in a sentence and expresses the unification of subject and predicate that is one and the same as “a being” [ein Seyn].

For example, when we say “Aristotle is the founder of traditional logic” or “the cat is on the mat”, we use the copula “is” to unify the subject, “Aristotle”, with the predicate, “being the founder of traditional logic”, and thus to express a being, i.e. an individual that instantiates a property, such as being the founder of traditional logic or being on the mat.

Hegel follows the logical tradition of his time, according to which a judgement is the unification of the subject with the predicate through the copula “is”. Nonetheless, the form of Vereinigung that he defines is peculiar.Footnote 9

6 The relevance of Hegel 42 for paraconsistency

In sum,

  1. (a).

    tells us that the Vereinigungen Hegel has in mind, so-called “Hegelian conjunctions”, are true unifications of contradictories. (a) shows that Vereinigung is a relation between terms or sentences in which one is the contradictory opposite of the other, e.g. the unification of two conclusions of antinomic arguments such as “matter is continuous” and “matter is not continuous”, or “humans are free” and “humans are not free”. (a) also tells us that the unification of the contradictories is the object of a true belief.

  2. (b).

    is about the epistemology and metaphysics of Verenigungen. A Vereinigung is the activity we perform when we unify antinomic opposites, an activity which is then grasped as the object of a true belief.

  3. (c).

    says that the unification is the norm or criterion for assessing the truth of the opposites.

  4. (d).

    tells us that Vereinigungen are necessary and express something that “is independent”. In other words, they stand for contradictions that are true in a realist meaning of “true”. In (d), Hegel presents the “proof” of the necessity of Vereinigung, which consists in showing that the possibility condition of each opposite is its unification with the other. The proof is, according to Hegel’s intentions, deductive, and its conclusion thus necessary, a valid argument whose conclusion is truly contradictory.

  5. (e).

    is a first description of the dialectical process, which highlights that the unification of opposites is “something else” other than the opposites and, as such, has its own opposite.

  6. (f).

    presents the unification in logico-linguistic terms – more specifically, as the copula is in a judgement.

I cannot examine these logico-philosophical insights in detail here. I will thus focus on three points that are relevant for expanding and deepening the views presented in Beall & Ficara, 2023. These are:

  1. i.

    Vereinigung is a unification of antinomic opposites that is true;

  2. ii.

    Vereinigung is an operation or activity we perform and, at the same time, the expression of a reality that exists independently of us;

  3. iii.

    Vereinigung is the possibility condition of the truth of the opposites; the opposites are, by themselves, insufficient (Beall & Ficara, 2023 reads this point in terms of how simplification fails, as the opposites are not true and their unification is, nonetheless, true).

Hence, we have, in short, three insights that emerge from the consideration of Hegel 42 and that are relevant for paraconsistency:

  1. i.

    the admission of true contradictions

  2. ii.

    the assumption of a realist sense of “true”

  3. iii.

    the idea of a unification, a conjunction of contradictories, for which simplification does not work.

The third aspect lets us see that Hegel is a paraconsistent philosopher in the sense of conjunctive paraconsistency (d’Agostini, 2021).Footnote 10 Conjunctive paraconsistency is the perspective according to which there might be non-explosive true contradictions yet contradictory propositions cannot be considered separately true. This perspective invalidates the classical logical principle called explosion by blocking simplification. The term “explosion” is used today with reference to the law known since the Middle Ages as ex contradictione (or ex falso) sequitur quodlibet (ECQ). The law is also called the Pseudo- Scotus Law because it was discussed in a text wrongly attributed to Duns Scotus (In universam logicam quaestiones).Footnote 11 ECQ states that from a contradiction everything follows. Its proof in the original Latin formulation is:

6.1 ECQ

  1. 1.

    Sortes est et Sortes non est

  2. 2.

    Sortes est

  3. 3.

    Sortes est vel homo est asinus

  4. 4.

    Sortes non es

  5. 5.

    Homo est asinus

The proof shows that from the contradiction Sortes est et Sortes non est we can logically derive any sentence. The admission of one contradiction produces explosion, i.e. trivialism, or the view that everything is true, and the loss of every logical constraint. Each step from 1. to 5. is motivated by classical laws: simplification explains the passage from 1. to 2., disjunction introduction the one from 2. to 3., and, in the steps from 3. to 5., simplification (again from 1.) and disjunctive syllogism (from 3. and 4.).

Typically, paraconsistent logicians try to prevent explosion by working on the admissibility of disjunctive syllogism, e.g. in the step to 5 from 3. and 4. The dismissal of disjunctive syllogism is highly controversial, even among paraconsistent logicians.Footnote 12 Conjunctive paraconsistency, however, acknowledges a way to stop explosion that is much less demanding. Admitting that, in a true contradiction, the two conjuncts are inseparable means that the step from 1. to 2. becomes invalid. Without this step, the argument for explosion hence fails.

If the first insight shows that Hegel’s theory of Vereinigung is logically and conceptually related to paraconsistency, and especially conjunctive paraconsistency, the first and second insight together allow us to see that Hegel is a dialetheist in terms of the dialetheism presented in Norman et al., 1989, xx. What such dialetheists are interested in are paraconsistent theories that are true in a realist sense of true, according to which there are true contradictions and they are made true by something, the world, that exists independently of us. I have highlighted this insight as the specific content of (d).

On this basis, and by way of conclusion, we can now consider both the question of the relevance of Beall’s contribution to the literature on Hegel’s logic, as well as that of the relevance of Hegel for paraconsistency, dialetheism and, in particular, Beall’s own position.

7 Conclusion

The reason for the particular interest in Beall & Ficara, 2023 within the literature about Hegel’s logic thus consists in its focus on Vereinigung and in the attempt to grasp what Hegel says on it from a logical point of view. The strange nature of the Hegelian dialectical conjunction has been addressed by various authors (Wetter, 1958, Havas, 1981, Priest, 1989, Bordignon, 2014, d’Agostini & Ficara, 2022, et al.). The specificity of Beall & Ficara, 2023 is, in this respect, its focus on Vereinigung and the idea that any attempt to understand what the form of Hegel’s dialectics is must come to terms with it as the fundamental dialectical operation that is even more fundamental than negation.Footnote 13

That the confrontation with Hegel’s dialectics and the tradition of Hegelianism have been fruitful for the birth and self-definition of paraconsistent logics has already been noted (Batens, 1986, 161, Batens, 1999, Da Costa, 2016, Ficara, 2021). However, the question now becomes: is it still reasonable, particularly for a paraconsistent logician such as Beall, to read Hegel, or even to engage in dialogue with the tradition of Hegelianism?

The reason reading Hegel should be interesting for a paraconsistent logician committed to the truth of at least some contradictions is that Hegel’s reflections on contradictions and their logic is the result of an organic metascientific, metaphilosophical, and metaphysical view, of which Hegel 42 contains only hints. However, the fragment is rooted in a philosophical context in which contradictions begin to play a central role and an organic account of them begins to become possible. For Kant, there are necessary contradictions (even though we cannot derive any objective/true knowledge from them) and their genesis is clearly identified: the necessary contradictions are those produced by pure reason (thought thinking about itself). For Fichte, contradiction further becomes programmatically assumed as the method of philosophy. For Schelling, the contradiction is assumed as a constitutive element of the construction of the world. Finally, for Hegel, contradiction is the method of philosophy insofar as it is a constitutive element of reality. Accordingly, dialectics, as the logic of contradictions, becomes for him the logic of philosophy insofar as the actual development of reality is dialectical, i.e. involves contradictions.Footnote 14

In Hegel’s work, and particularly in Hegel 42, one can find a specification of the logic of Vereinigung that is the product of an awareness about the origins, the problem, and the possibility conditions of necessary and true contradictions. Grasping this logic would thus not be a mere academic exercise or a fun game. Instead, it could be the path towards answering some crucial metascientific, metaphilosophical, and metaphysical questions. In short, Hegel presents a view about why it is inevitable to admit contradictions, as well as a compelling and strikingly unified account of the nature and genesis of irreducible contradictions and of the necessity to admit them as true. In Hegel 42, there are only hints at the epistemological and metaphysical reasons for developing a logic of Vereinigung. However, Hegel’s work in the writings that follow this fragment is devoted to the definition and clarification of this genesis and the reasons for it. Traditionally, Hegel’s work counts as obscure and not completely comprehensible for us, especially insofar as we work in analytic philosophy, for whose birth and development, as is well known, a certain obliteration of dialectics has been constitutive. Hegel 42, although crucial for grasping the logic of dialectical contradictions, also counts as almost inaccessible and is regarded by all interpreters as deeply enigmatic. My aim here has thus been to pave the way for rendering Hegel’s organically unified account, or at least some of its parts, more accessible to us by presenting this fragment’s content in more detail.