Intuitionist and Classical Dimensions of Hegel’s Hybrid Logic

Abstract

Hegel interpreters commonly reject attempts to situate Hegel’s logic in relation to modern movements. Appealing to his criticisms of the logic of Verstand or mere understanding with its fixed logical structure, Hegel’s logic, it is pointed out, was a logic of Vernunft or reason—a logic more at home in the thought of Plato and Aristotle than in modern mathematical forms. Contesting this implied dichotomy, it is here argued that the ancient roots of Hegel’s logic, especially as transmitted by late Neopythagorean/Neoplatonic thinkers such as Proclus, gave it many features similar to ones later found in the type of algebraic transformation of Aristotle, started first by Leibniz, reanimated by Boole in the mid-nineteenth century and then developed by others such as C. S. Peirce and Arend Heyting. In particular, the ancient mathematics upon which Hegel had drawn allowed him to anticipate an answer to the criticism that Frege would later aim at Boole, concerning his inability to unite opposed class and propositional calculi. Hegel’s logic would be a hybrid, incorporating features found later in intuitionist and classical logic, but it could be so because of the way he had called upon the mathematics of the ancient Platonist tradition.

KEYWORDS:

• Hegel
• Plato
• logic
• geometry
• intuitionism

Notes

1 The idea of the ‘distinterpretation’ of algebra to a purely syntactic calculus had been proposed by the Cambridge mathematician Duncan Gregory in 1840 (Ewald 1996, 443).

2 See Putnam 1990. There are now numerous challenges to the conventional story of the definitive triumph of the modern classical logic initiated by Frege and Russell. The familiar ‘classical’ and ‘nonclassical’ distinction tends to fracture along many different fault lines, but two that are particularly relevant for Hegel’s logic concern, on the one hand, the different roles given to mathematics between the rival algebra of logic and logicist traditions of the late nineteenth century, and, on the other, the differences between logics focusing on actions (judgments and inferences) and intentional objects (propositions and relations of logical consequence). In relation to the former, see for example Peckhaus 2009 and Grattan Guinness 2004, and in relation to the latter, Prior 1949 and Sundholm 2009. For a wider coverage of the sorts of distinctions found among modern logics see the contributions to Stelzner and Stöckler 2001.

3 Intuitionistic logic would be first developed by Brouwer’s student Arend Heyting in the 1920s.

4 The idea of a hybrid logic of this sort was suggested by Arthur Prior (Blackburn 2006).

5 For example, plotted against the x and y coordinates, a circle of radius 1 unit with its centre at the origin of the co-ordinates could be represented by the equation x² + y² = 1.

6 See, for example, Struik 2011, chs. 2–7. This is because the same cross-ratio relation holds among the angles formed by the lines intersecting at point O. This equivalence of the cross-ratio holding among four collinear points on the one hand and four concurrent lines on the other is an example of the duality between point and lines in projective geometry.

7 See, for example, Gowers 2008. Thus, for any theorem in projective geometry that concerns some complex relation (or relation of relations) among some structured array of points, an equivalent theorem could be found that applied in the same way to an analogously structured array of lines.

8 Russell 1897, v. Russell also acknowledges Whitehead for his own becoming aware of the ‘philosophical importance of projective Geometry’ (Russell 1897, v).

9 See, for example, Wolff 1999 and Lawvere 1991. Grassmann seems to have been much more directly influenced by the idealist philosopher and hermeneutic theorist Friedrich Schleiermacher, and through him, by Friedrich Schlegel (Lewis 1977). Nevertheless, there clearly had been certain parallels in the approaches to logic common to Schleiermacher and Hegel, despite their differences.

10 Perception ‘presents us with various things, with discriminated and differentiate contents’ but ‘what we wish to study here […] is the bare possibility of such diversity’ which can be considered the ‘form of externality’. Russell then asks after the properties that ‘such a form, when studied in abstraction, [must] necessarily possess’ (Russell 1897, 136). He answers that such a form must be essentially relational: ‘In the first place, externality is an essentially relative conception—nothing can be external to itself. […] Hence, when we abstract a form of externality from all material content, and study it in isolation, position will appear, of necessity, as purely relative—a position can have no intrinsic quality’ (136–137).

11 Alan Paterson stresses the influence of Proclus (Paterson 2004/2005, 63), but besides Proclus’ Commentary on Euclid’s Elements, Hegel also possessed significant works by Iamblichus and Nicomachus of Gerasa (Mense 1993, 672).

12 See, in particular, Nicomachus of Gerasa 1926, 284–285 and Proclus 2009, 171, 10–15. The history of projective geometry and the cross-ratio would be reconstructed by the French geometer, Michel Chasles (Chasles 1837).

13 These three means were the geometric, the arithmetic, and the harmonic. The history of this phase of Greek mathematics had been preserved by Pappus of Alexandria, whose writings had been influential in the reintroduction of projective geometry in the seventeenth century.

14 Thus, the harmonic cross-ratio between two pairs of colinear points (A and B) and (P' and P) can be expressed by fractions and said to hold when the double-ratio $\frac{\frac{AP}{PB}}{\frac{A{P}^{\mathrm{\prime }}}{{\mathrm{P}}^{\mathrm{\prime }}\mathrm{B}}}=-1$.

15 Of two terms, a and b, the geometric mean = √ab, the arithmetic mean = $\frac{a+b}{2}$ and the harmonic mean $=\frac{2ab}{a+b}$. The three means are interrelated in complex ways.

16 I have presented the details of these links in Redding 2023.

17 Indeed, the German term for judgment, ‘Urteil’, already suggests such a division, ‘Teilung’, into parts, Teilen.

18 Following Frege, Russell had denied that everyday judgments like Hegel’s ‘the rose is red’ were in fact proper judgments. Without complete propositional content, Russell came to treat such judgments as incapable of truth or falsity. Thus, he treated in this way an example used by the early modal logician, Hugh MacColl, ‘Mrs Brown is not at home’ (Russell 1906). Such an expression was really a type of predicate that would only be capable of truth or falsity if said of a particular point in time, as in ‘Mrs Brown is not at home at time t₁’.

19 See Hegel 2010, 581−587. Such a judgment in Hegel’s account is an explicitly evaluative judgment about a human act or artifact. His examples are ‘this act is good’ and ‘this house is bad’ (Hegel 2010, 583).

20 ‘Hegel actually anticipates an important idea in the newer algebra and vector algebra [of Hermann Grassmann]. […] Hegel […] introduces the concept of absolute value as the concept of a substrate of logical reflection’ (Wolff 1999, 15).

21 Brouwer 1981, 2. At his most extreme, Brouwer seemed to suggest that concepts were relevant only to the linguistic expression of mathematical truths and not the truths themselves.

22 Thus, Whitehead had stated in his Treatise on Universal Algebra that ‘Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning. […] The sole concern of mathematics is the inference of proposition from proposition’ (Whitehead 1898, vi), while Russell, in The Principles of Mathematics, would proclaim pure mathematics to be ‘the class of all propositions of the form ‘p implies q’, where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants’ (Russell 1903, § 1).

23 C.f. ‘The belief in the universal validity of the principle of the excluded third in mathematics [i.e., LEM] is considered by the intuitionists as a phenomenon of the history of civilization of the same kind as the former belief in the rationality of π, or in the rotation of the firmament about the earth’ (Brouwer 1981, 7).

24 Without invoking intuitionism, Robert Stalnaker uses the idea of singular propositions playing to role of witnesses to existential propositions in this way in Stalnaker 2019, ch. 2.

25 Technically, Heyting algebra will have the structure of a ‘semi-lattice’ in contrast to the ‘lattice’ structure of Boolean algebra, this difference being reflected in their different conceptions of negation.

26 In Brouwer’s example, ∼p ⊃ (p ⊃ 1 = 2).

27 In the 1930s Gerhard Gentzen would come to provide a form of logical semantics appropriate to intuitionism. See, for example, Sundholm 2009, 292−8.

28 Such proofs must be of course fallible and be able to be revised under further discoveries about the conditions under which one experiences. Propositions that had previously been assumed and thus without proof or refutation, for example, may come to be shown to be false.

29 On the resistance of the logic of Frege and Russell to reinterpretation, see Goldfarb 1982, 694.