Diagrammatic and Modal Dimensions of the Syllogisms of Hegel and Peirce
Round 1
Reviewer 1 Report
The paper is about Hegel’s account of the syllogism, its roots in ancient (especially Platonic and Aristotelian) philosophy and its meaning for the history of the subsequent logic. The question about the meaning of Hegel’s concept of logic and its relation to logic as we understand and practice it today is still source of a certain embarrassment. A full comprehension of the specifically Hegelian input to this tradition is still a desideratum. In this perspective, this paper is an important contribution to overcoming this state of regrettable separation between Hegel’s logic and the tradition of the history of logic. It manages to clarify thorny questions that are the source of controversies in the reception of Hegel – the question about the connection between logic and mathematics, the difference between judgements of inherence and judgements of subsumption among others – to illuminate some of the roots of Hegel’s account in ancient philosophy and to explain its multiple connections to more recent views. It is an important reference point for anyone interested in understanding what is Hegel’s logic, and what its meaning for the discipline we call “logic” today is. The materials presented in the paper are very rich, and the operation of a mediation between the two traditions very complex. In this light, some key concepts and arguments could profit from a more detailed explanation.
-- The author states on the “ratio of powers” (256-258) that “Hegel’s discussion of this ratio is far from clear, but we would expect the construction of squares as having a role, the Greek word for power, dynamis, having been taken from the word for square.” Does this mean that the expression Hegel uses is “ratio of powers” (please add the German expression too) and that the Greek word he refers to is the Greek word that stands for square? A short explanation as well as mentioning the German original expression would make this important point clear.
-- The author writes about the difference between the doubled and the simple middle term and states (333-336): “what Hegel says about Plato’s beautiful bond in the Lectures on the History of Philosophy, where he insists that in relation to the ratios structuring Plato’s cosmic animal, its three-dimensionality required a “doubled middle term”, “gedoppelt Mitte” [7] (p. 211) for the unification of its parts, and in contrast with which he had suggested the middle term of Aristotle’s formal syllogism to be simple or univocal.”
In the passage in Hegel’s LHP about Plato’s Timaeus Hegel first distinguishes between the (Platonic) genuinely speculative) syllogism (Vernunftschluss) and the Verstandesschluss, then he writes that the speculative syllogism (Vernunftschluss) involves a triplicity while “in nature” we have a quadruplicity or a doubled middle (eine gebrochene Mitte, eine gedoppelte Mitte). Here (similarly to his Habilitationsthese) he seems to stress the difference between the lex mentis and the lex naturae (whereby Plato’s doubled middle term would be expression of the lex naturae) and the Please explain how this fits with your interpretation (aimed at highlighting the role of the double middle term for the history of the subsequent logic).
In sum, I recommend publication after a minimal revision along the lines suggested above.
Author Response
It was gratifying to receive the positive comments. Two very pertinent points are raised which I have addressed as follows.
Issue 1: “The author states on the “ratio of powers” (256-258) that “Hegel’s discussion of this ratio is far from clear, but we would expect the construction of squares as having a role, the Greek word for power, dynamis, having been taken from the word for square.” Does this mean that the expression Hegel uses is “ratio of powers” (please add the German expression too) and that the Greek word he refers to is the Greek word that stands for square? A short explanation as well as mentioning the German original expression would make this important point clear.”
I’m grateful for this being raised as the original text was clearly too brief and schematic in this regard and was in need of being filled out. I have expanded on the paragraphs in question on page 7 as follows.
"Hegel’s inverse ratios draw upon Euclid’s definition 12 of Book V, which states that “inverse ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent” [34]. That is, if a : b :: c : d, then a : c :: b : d. As Book V, believed to have been written by Eudoxus, is concerned with ratios of continuous magnitudes, Hegel’s transition from direct to inverse ratios would appear to coincide with that between discrete and continuous magnitudes. We might expect, then, that in line with his idea of the two stages of negation involved in Aufhebung, features of both direct and inverse ratio should be found as somehow redetermined in his ratio of powers (Potenzenverhältniss).
Hegel’s discussion of this ratio is far from clear, but we would expect the construction of squares to have a role. The Greek word for power, dynamis, had been taken from the word for square, [29] (pt. 1.2)—an etymology reflected in the fact that in modern languages such as English and German, a number multiplied by itself is described as “squared” (in German, “quadriert”) or as raised to the “power” (“Potenz”) of 2. The Greeks, however, had interpreted such arithmetical “squares” geometrically, identifying the square of a number n (n2) with the area of a square constructed on a line segment of length n. This introduced the problem of incommensurability among magnitudes, as it was realized that the length of sides of squares with certain areas, for example, an area of 3 square units, were incommensurable with the length of the sides of a square with area 1 square unit (in modern terms, that the number √3 is irrational).
Hegel was critical of the tendency in modern thought, exemplified, for example, by Descartes’ analytic geometry as well as modern calculus, to disregard these “qualitative” differences by reducing continuous magnitudes to discrete numerical values. The ratio of powers was meant to represent a quantum in which this incommensurability among different kinds of magnitudes is aufgehoben rather than ignored. Thus, he describes the ratio of powers as a quantum that posits itself “as self-identical in its otherness”. In “determining its own movement of self-surpassing”, this ratio “has come to be a being-for-itself” and is “posited in the potency of having returned into itself; it is immediately itself and also its otherness” [12] (p. 278). Hegel’s verbiage may be opaque, but there is a clear sense of some type of relation that brings otherwise incommensurable magnitudes into some form of unity. His descriptions, I suggest, can be illuminated by Plato’s “beautiful bond” in which he took so much interest."
So, making the linguistic point in question has allowed me to elaborate on the general significance of the issue of the incommensurability of discrete and continuous magnitudes for Hegel. In line with the Greek approach, Hegel insists on the significance of complex relations of incommensurability holding among such magnitudes, refusing the modern approach which seeks to identify the continuum with the sequence of “real” numbers, and reduce geometry to arithmetic.
Issue 2: "In the passage in Hegel’s LHP about Plato’s Timaeus Hegel first distinguishes between the (Platonic) genuinely speculative) syllogism (Vernunftschluss) and the Verstandesschluss, then he writes that the speculative syllogism (Vernunftschluss) involves a triplicity while “in nature” we have a quadruplicity or a doubled middle (eine gebrochene Mitte, eine gedoppelte Mitte). Here (similarly to his Habilitationsthese) he seems to stress the difference between the lex mentis and the lex naturae (whereby Plato’s doubled middle term would be expression of the lex naturae) and the Please explain how this fits with your interpretation (aimed at highlighting the role of the double middle term for the history of the subsequent logic)."
This is also a great question but one not so easy to address in any comprehensive way in this context as it bears on questions of Hegel’s broader philosophical doctrines (metaphysical, theological, historical) which are rather beyond the scope of the paper. My response has therefore been to construe the criticism of Aristotle’s formal logic as presupposing the issue that, for Aristotle, logic was fundamentally an organon for the investigation of the empirical world—that is, broadly, “nature”. It is because of this that Plato’s “split” middle term, found in the syllogism in its concretized “natural” form in the Timaeus, could be relevant to a criticism of Aristotle. I have expanded paragraphs on page 3 to bring this out more clearly. These now read:
"In this paper I pursue parallels between the ways that Hegel and, later and seemingly independently, Charles Sanders Peirce, would attempt to modify Aristotle’s syllogistic logic in ways that exploited issues of order, direction and spatial orientation in quasi-diagrammatic ways. Like many later commentators on Aristotle’s syllogistic logic, Hegel understood it to be based on Plato’s speculative thought. However, Aristotle had practiced philosophy “as a thoughtful observer of the world who attends to all aspects of the universe” [7] (p. 232), and so from this perspective, judgments and syllogisms were primarily intended to be adequate to the world as observed. In relation to this, Hegel would note how Plato’s own syllogism, when understood as itself instantiated in nature as in Timaeus, exhibited an important difference to Aristotle’s. While the middle terms of Plato’s syllogisms were described as doubled or “split”, those of Aristotle’s formal syllogism were single and univocal [7] (p. 210). Hegel would thus modify Aristotle’s in an attempt to restore its “speculative” or “rational” Platonic foundations.
In his reading of the Timaeus, Hegel linked Plato’s dual or split middle term to the three-dimensionality of the cosmos, as encoded in the Pythagorean tetraktys which, among other things, was meant to represent the spatial dimensions of the universe: one dot representing a point, two dots representing the two points that define a line, three, those defining a plane, and four, those defining a volume. But the Pythagoreans had organized numerical quantities into ratios and proportions, or ratios of ratios, such that three points of a plane, for example, were to be thought of as a pair of ratios sharing a common “middle term” that divided the “extremes”. As in the fourth row of the tetraktys, corresponding to the three dimensions of space, the extremes were mediated by two, not one, middle terms [8] (pp. 55-63; 109-115). Effectively, increasing the number of “middle terms” for Hegel increased the dimensionality of the “logical space” inhabited by judgments."
It is surely correct that there are complex background issues unaddressed here concerning the relation of the “triplicity” of the Vernunftschluss itself to its quadruplicity when applied to nature. While I have definite thoughts here, these are complex issues that bear on diverse aspects of Hegel’s metaphysics that would take the discussion in a direction away from the focus of the text and difficult to integrate into it. But perhaps it is relevant in this context to gesture towards the direction I would like to pursue this issue somewhere else.
First, I’m not convinced that the “speculative syllogism” being referred to in its “triplicity” in the passages from Hegel’s comments on the Timaeus are actually directed to Plato’s Vernunftschluss (although from the comments I’m not exactly sure what paragraphs are being referred to). With reference to the “absolute Schluss” in the paragraph starting in Vorlesungen über die Geschichte der Philosophie, vol. 3, p. 40, line 150, I tend to suspect that Hegel is here speaking in the “first person” and so alluding to his own version of the Schuss rather than Plato’s. In a theological register, Hegel was concerned with the trinary relation found in the Christian doctrine of the trinity to the four-fold structure of nature in Pythagoreanism. Indeed, his concerns with these in his Habilitationsthese as mentioned seem to accord broadly with this issue having been raised by Baader only a few years before in the work I refer to on page 2, who was clearly trying to relate the more pantheistic version found in Schelling to his own overtly Christian approach.
Earlier in the Vorlesungen (p. 17), Hegel comments on the difficulty facing the ancient philosophers, clearly including Plato, for grasping the logic of the transition from the ideal to the corporeal realm. Thus, with Plato there is a type of “fall” into nature of the soul, but there understood as a type of “external” event. As with his interpretation of ancient philosophy in general, Hegel seems to think that an adequate account here presupposes an elaboration of the categories involved that was distinctly “modern”, in a broad sense as initiated in the religious sphere by Christianity. There, Christ similarly “falls” into nature, but such an event now explicitly construed as built into the very concept of the triune god in question. However, this is all still in the theological register of “Vorstellungen” and, for Hegel (and most modern readers), in need of ultimate translation into conceptual form.
All this certainly needs to be done to make an adequate case for Hegel’s logic, but pursuing it here, I suspect, will likely strain the reader’s patience and be seen as extraneous to the specific case being made in the paper. So, in short, while very grateful for this thoughtful and pertinent question and the broaching of the more general issues that lie behind it, I am reluctant to pursue these broader aspects much further in this context. I hope it is agreed that the more constrained response as suggested above here, at least for this context, is adequate.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The paper is interesting and contributes to the understanding of the difficult topic which is Hegel's Science of Logic. I think that the contribution is precise and well done. Nothing more to say.
Author Response
I am grateful for this review and glad to read that the article was found interesting and helpful.
Reviewer 3 Report
I always had problems with understanding Hegel, but the authors made it clear to me that Hegel's logic should be taken seriously if you go beyond first-order classical predicate logic.
Author Response
I am grateful for this review and am encouraged to read that it made a case for taking Hegel's logic seriously when logic is construed beyond the scope of classical first-order predicate calculus.
