Paul Redding: Conceptual Harmonies: The Origins and Relevance of Hegel’s Logic

Conceptual Harmonies is presented as the first half of a reconstruction of Hegel’s logic aimed at grounding Hegel’s metaphysics (p. xiv). The development of the grounding itself has been sacrificed for the time being in order to provide a novel access to the logic of Hegel’s Science of Logic. Redding attempts to elucidate Hegel’s account of the rationality of the syllogism, a major theme in Hegel’s “Subjective Logic,” through a historical contextualization of the idea of the “middle term.” To this end, he takes an unprecedented journey, at least in Hegel scholarship, through Greek number theory and notions of ratio and proportion, their Neoplatonic elaborations and their echoes in modern geometric algebra and algebraic mathematical logic. More specifically, Redding sets out to trace the logical significance of the syllogism back to a Pythagorean conception of the middle term that Hegel had discovered in Plato.

The starting point of his argument is Hegel’s comparison in his Lectures on the History of Philosophy between Aristotle’s formal logic of the understanding, in which a “geometric” middle term plays a prominent role, and the “most beautiful bond” of rationality spoken of by Plato in Timaeus 31c, in which a combination of Pythagorean means connects two “extremes”:

In the syllogism of the understanding there are two extremes and a middle term, but the extremes have the value of independent characteristics … The first is the singular, the second is the particular, and the third is the universal. This, the understanding’s [form] of the syllogism, is sublated in the Platonic presentation, and the speculative aspect constitutes the syllogism’s properly authentic form and nature; the bond, or middle term, makes the extremes one in the highest degree … The middle has become first and last, and these two extremes become the middle. That is how it first comes about that all of them are of necessity the same, and their unity is constituted in this way. (Hegel 2006–2009, vol. 2, p. 210; cited on pp. 1 & 148)

The recourse to the combination of means – explicit in Timaeus 36a – would make it possible to establish, instead of an endless series of relations of class inclusion (humans are part of mammals, and mammals are part of vertebrates, and vertebrates are part of animals, and so on and so forth), a reciprocal definition of three logically differentiated objects linked by an analogy. Throughout the book, Redding tries to make logical sense of this obscure Platonic idea of a reciprocal definition by means of a combination of means, just as Hegel also tries to make sense of it in the Science of Logic, where he speaks of “the determinacy of the three terms in relation to each other” (Hegel 2010, p. 591; cited on p. 149). Both intend to explain the mutual belonging of the terms involved and their mutual mediation – a connection and “unity” that the simple geometric Aristotelian means cannot provide.

Redding’s hypothesis is that a particular ontology, also of Pythagorean origin, which Plato finally adopted against earlier, standard versions of his theory of forms, lies behind his ideas of the “most beautiful bond” and a complex middle term: the ontology of the determinate one and the indeterminate dyad of “the greater and the smaller” (on this point Redding follows Kenneth Sayre’s interpretation of Plato, which elaborates on Aristotle’s comments in Metaphysics). The middle term of a rational syllogism would participate in the mixed nature of those principles. The main thesis of Conceptual Harmonies is, precisely, that the middle term of a truly rational syllogism is, in Hegel’s rationalization, divided or doubled. Only on that condition, so Redding argues, can it connect universal forms with material singulars.

Such an ontology is also found in Hegel, according to Redding. We may call it dialectical, for there is dialectic in the negation, the determinate negation, of the different values of the different “means.” Discreteness and continuity are indeed combined in Hegel’s account of determinate being, specifically quantity, in the doctrine of Being, despite their obvious incommensurability. However, the approach to such an ontology in Redding’s book is mainly logical, so that the entire discussion of the history of the various means and the notions of proportion, analogy and harmony in the early chapters is put at the service of the interpretation of the logic of the “subjective concept,” to which the last chapters of the book are dedicated.

The best example of the duality of the logical middle term is arguably provided by the syllogisms that make explicit what Hegel calls the “judgment of the concept” (which Hegel considers to be the judgment proper). A singular (e.g., a house) is argued to satisfy a universal standard of evaluation (it is said to be, for example, good) by virtue of certain particularities (because it is so and so constituted) (p. 161). The middle term of this kind of inference, according to Redding, is a concrete singular (this house) and also a particular singular (a house of a certain kind) (p. 202). By introducing the abstraction of some features (p. 194) and thus doubling the middle term, Hegel’s syllogism is able to relate the existent singular to a universal: this house, as a house with certain properties, to goodness. By contrast, Aristotelian syllogisms, in their original formal interpretation, are supposedly incapable of incorporating a truly singular notion, and the singular is not the object of the kind of demonstrative science analyzed in Posterior Analytics. In the well-known example, if Socrates is mortal, it is because men are mortal, and Socrates is nothing but a man – an instance of a man (p. 148) – as far as the argument is concerned.

Redding generalizes this idea of a double middle term in his revision of the figures of judgment and syllogism as discussed in the Science of Logic. The main comparison and affinity here is with William E. Johnson and Charles S. Peirce (pp. 198–206) and their respective accounts of induction and hypothetical reasoning – though many others are explored with other followers of Boole, with whom Hegel shares a sense of the limits of formal calculus (p. 190). Rather than seeing inductive syllogisms and analogies, what Hegel calls “syllogisms of reflection,” as logically defective, Redding argues that all three, Hegel, Johnson and Peirce, think that they should be taken for what they are, productive patterns of reasoning that guide the investigation of relations that Hegel calls inherence and subsumption (p. 198). They would not be flawed, but rather inverted, but correct forms of reasoning, in which what might be a conclusion about a case becomes the premise of a generalization or a hypothesis, as when the mortality of Socrates becomes evidence of the mortality of men. They are figures that manage to bring the singular to bear on arguments in some way, as is necessary for these to have objective value (p. 38). It is in the discussion of inductions and hypotheses that the need to divide the middle term becomes most apparent.

The forms of judgment and syllogism are said to act as “phases” or “cycles” towards a demonstration based on a scientific redefinition of the concepts employed. In the idea of a succession of forms of abstraction and concreteness in different phases, an antecedent of classical pragmatist positions is discernible.^[1] There would be an “inferential commerce” between premises and conclusions, a double directionality, in a developed, complete rational syllogism (p. 162). A final cycle of redefinition is already announced in the transition from the “judgment of necessity” to the “judgment of the concept”. In Redding’s interpretation, there is here a final concretization (and “resemanticization”) of the subject of judgment, which passes from being a particular singular, of a particular class, to being a (possible) paradigmatic singular, a singular that becomes a “measure” (p. 210). Normative aspects are thus involved in the “judgment of the concept” that were barely implicit in the previous figures. The question is now whether something conforms to a norm, better or worse, more or less, and not what it is. This gives the judgment of the concept a characteristic reflexivity: the judge and the judgment judge as much as they are judged, since reality, in paradigmatic cases, measures the judgment.

Instead of a “global isomorphism” between judgments and facts, Redding speaks of a “local homomorphism” (p. 49) between inherence and subsumption (p. 151), i.e., between Hegel’s “judgment of existence” (“This rose is red”) and Hegel’s “judgment of reflection” (“This rose is a red one, or is of a red color”) – judgments with rather opposite logical functions, but which are, according to Redding, “homologous” (pp. 163, 195) because of their structural equivalence. This homomorphism or homology is treated as a kind of equivalence that does not imply identity (p. 195), but rather “identity in difference.” Redding argues that these homologies do not properly reflect or mirror the world, but that we orient ourselves by establishing them, ultimately relying on notions that are “essentially contested” (the phrase is borrowed from W. B. Gallie, see pp. 163, 209), although paradigmatically instantiated in particular objects. The consequence of this contestability is made explicit in the penultimate chapter of the book. The logic of human reasoning about “values” could not be reduced to or explained by the kind of judgments characteristic of the positive sciences (p. 208) – it is rather the other way around.

Redding, in my view, rightly links this pragmatic conception of predication and inference to a resistance to the Cartesian arithmetization of geometry, i.e., to “analysis.” In this regard, he is able to present his reconstruction of Hegel’s logic as an alternative to Brandom’s (p. 207). Instead of entering into dialogue with classical analytic geometry and Frege and Russell, his reconstruction engages with projective and nonmetric geometry and nineteenth-century Leibnizian logic. Although their ultimate intentions are comparable, since for both Redding and Brandom logic makes explicit what we do when applying concepts and then reasoning about their application, the former’s emphasis on the incommensurability of logical terms is original and an exegetical advance with distinctive epistemological and logical implications.

However, if we consider the wider context of Hegel’s logic, I would say that Redding is, by and large, building on strictly Brandomian premises. He assumes, like Brandom and Pippin, that the concept has a fundamentally subjective nature – which he sometimes expresses indirectly by celebrating the Copernican turn. And he also assumes that in the figures of Hegel’s “formal” concept the universal, the particular and the singular are explained and articulated through and through. Accordingly, what follows the section on “subjectivity” in the Science of Logic can, in his view, only be a confirmation of Hegel’s “formal” logic (see the final endnote to ch. 9). Indeed, the concept is supposedly objectified in the cycles of determination analyzed in the discussion of the “subjective concept.”

Is it really so, as Redding claims? When Hegel speaks later in the Science of Logic of syllogisms, such as the syllogism of external purpose or the syllogism of action, when the “means” adopts the form of a mechanical process or a teleologically oriented activity (and eventually an action), is the hypothetical and evaluative character of the logical universals and the homomorphic (or homological) quality of the relations between terms confirmed? And when he later speaks of teleology being the truth of mechanism, in the Objectivity section, is it then confirmed that the application of non-normative concepts, concepts that we might call “mechanical,” depends on the logic of evaluative concepts? And when he later discusses the logical idea of life, is it then confirmed that natural kinds are concrete universals without normative connotations, that evaluation and “the greater or lesser” are reserved for contestable concepts concerning intentional actions and human artifacts? And above all, when he later discusses the logical idea of cognition, is it then confirmed that evaluative concepts are indeed essentially contested? Is this what Hegel thinks of, for example, the good and the true? If, as I believe, these questions are not answered in the section on the “formal concept” of Hegel’s Logic, then Redding’s introduction to Hegel’s metaphysics is only part of the necessary logical introduction that it deserves.

Nevertheless, the fact that many questions remain open at the end of the book does not mean that we have not made progress, or even that we have not taken some essential and critical steps towards such a metaphysics. The idea of a split middle term, in which the discrete and the continuous are dialectically intertwined, seems to me to be an idea of remarkable importance, even if it is insufficiently developed in this “first” introductory argument. At this point, I insist, Redding’s study breaks neatly with the non-dialectical readings of Pippin and Brandom, in line with other recent and independent contemporary attempts – notably those of Ficara (2021) and Moss (2023) – to revive Hegelian logic by tracing it back to its Platonic roots. The argument of Conceptual Harmonies is thus a further, if not definitive, sign that the debate on Hegel’s metaphysics is finally changing, as Kreines predicted almost twenty years ago.