Hegel and the Consequentia Mirabilis

Abstract

In this paper I argue that Hegel’s treatment of dialectical inferences, in particular of Plato’s dialectics in the Lectures on the History of Philosophy, belongs to the history of the logical rule that, from Gerolamo Cardano to Bertrand Russell, is known as consequentia mirabilis (CM). In 1906 Russell formalises it as follows: and its correspondent positive form as My paper has two parts. First, I show that dialectical inferences, for Hegel, involve sentences of the form and . Hegel, following Plato, stresses that these inferences are das Wunderbare, the marvellous element of dialectical reasoning. In this sense Hegel’s view belongs to the history of CM from a perspective that is both terminological and logical. Second, I stress the peculiarity of Hegel’s CM. In a classical setting, the conditional is admissible and from we can, by CM, derive , and from , by CM, infer . However, in Hegel’s analysis it is evident that we can neither infer from , nor from . Rather, in dialectical inferences what we can validly infer from and is only .

Notes

1 On the relation between dialectical and classical logic see Gadamer 1976, Düsing 1976, Fulda 1978, 33–69, Kulenkampff 1970, Sarlemijn 1971, Butler 1975, 414–31, Günther 1978, Marconi (ed.) 1979, Priest 1989, 388–415. For a complete and critical appraisal of these and other canonical positions see Marconi 1979 (ed.), 20ff. For an analysis of the link between dialectical, transcendental and ‘common’ (or formal) logic see Nuzzo 1997, 39–82 and Nuzzo 2014, 257–73. The question about the logical meaning of dialectical inferences has gained a new attention in the last 20 years, also thanks to the attempts at reading Hegel in analytical perspective (see among others Horstmann 1984, Stekeler-Weithofer 1992, McDowell 1994, Brandom 2002, Ruggiu and Testa (ed.) 2003, Redding 2007, Nuzzo 2010). See Berto 2005 and Berto 2007, 19–39 for a detailed account of the meaning of dialectical inferences based on Brandom’s inferentialistic holism. See also Ficara 2013, Ficara 2014, Bordignon 2015.

2 Hintikka 1981, 109–10, 110.

3 The history of the interpretations of Plato’s Parmenides is immense. For a detailed philosophical discussion of the reception of Plato’s Parmenides from Plotinus through Hegel to Natorp see Gadamer 1991, 313–27.

4 Hegel 1969ff., Vol. 19, 80–81 English translation Hegel 1892ff., Vol. 2, 57–58.

5 These aspects are differently stressed in the interpretations of Hegel’s dialectics. Löwith 1971, 222 distinguishes the ‘positive abstraction from every determination that is immediately given in order to find the pure thought determinations’ and ‘the negative abstraction from those determinations that pertain essentially to a determination’. Following Hegel’s terminology, he calls the first concrete and the second abstract abstraction, stressing that dialectical thought avoids the latter and pursues the former. That dialectical inferences involve semantic ascent is a characteristic insight of non-classical readings of Hegel’s dialectic. For Marconi 1979, 9–84, 19ff. dialectic requires the application of terms to themselves. In dialectic, the usual restrictions of type do not work. The violation of the restrictions of type on the basis of classical logic (in which the classical rules hold) generates the emergence of antinomies. Findlay 1981, 132–39, 132 observes that dialectical inferences imply what he calls a metabasis, that is:

that genuine passage beyond premises that is also involved in passing from an object language to a meta-language […] and in which [a conclusion is implied] by its premises rather in the sense in which G. E. Moore said that to assert that it is raining is to imply that one believes that it is raining.

Kulenkampff 1970 claims that dialectic implies the attempt at expressing the semantic properties of the language within the language (in Tarksian terms, this means that the language contains its own truth-value).

6 Hegel 1969ff., Vol. 19, 82, English translation Hegel 1892ff., Vol. 2, 59–60.

7 Hegel 1969ff., Vol. 19, 82, English translation Hegel 1892ff., Vol. 2, 59–60.

8 The passages on Plato’s dialectics in Hegel’s Lectures on the History of Philosophy confirm, in my view, the reading of Hegel’s notion of negation in standard propositional terms I have proposed in Ficara 2014, 29–38. The question about possible formal counterparts of Hegel’s notion of negation is still a controversial one in the literature. Puntel 1996, 131–65 claims that Hegel’s conception of negation is ‘not intelligible’. Among the classical works on Hegel’s concept of negation see Henrich 1978, as well as Düsing 2012, Chapter 1, Cortella 1995, Landucci 1978, Koch 1999, 1–29, Perelda 2003, Redding 2007, Chapter 3, Viellard-Baron 2013, 46–68, Pippin 2014, 87–110. Among the works on the link between Hegel’s negation and the non-classical logical tradition the essays collected in Marconi (ed.) 1979 are worth mentioning. Berto 2005, 284ff. examines the relationship between holistic inferentialism and Hegel’s dialectics, focusing in particular on the meaning of dialectical negation as material incompatibility. In Ficara 2014, 29–38 I highlight the continuity, but also the difference, between Hegel’s determinate negation and glutty negation. Wolff 1986, 107–28 highlights that the background of Hegel’s notion of negativity is the Kantian distinction between logical and real opposition. On Hegel’s and Kant’s notion of negation see also more extensively Wolff 1981.

9 For the history of consequentia mirabilis see Bellissima and Pagli 1996. An analysis of the philosophical meaning of CM-arguments can be found in D’Agostini 2002, D’Agostini 2009, Chapter 10.

10 Bellissima and Pagli 1996, 7.

11 See Bellissima and Pagli 1996, 11–12.

12 Theaetetus 170, XXII, 219–20 (translation from Hamilton and Cairns 1961).

13 Aristotle Metaphysics Gamma IV, 8 1012 b 12–22 (translation from Barnes 1984).

14 Questiones disputatae de veritate, Qu. I, Art. 5 (2) (translation from Mulligan 1952). See also Bellissima and Pagli 1996, 202. Centrone 2012, 127–57, shows that Bolzano’s arguments against Lambert’s critique of CM are also an important step in the history of CM.

15 Russell 1906, 159–202.

16 On the differences (as well as the debate about the differences) between CM and reductio arguments see Bellissima and Pagli 1996, 153.

17 As Kneale 1957, 62–66, 62 writes: ‘Like Aristotle, the Stoics used argumentation of this kind as a weapon against skepticism, for example, to confute those who said there was no proof, and it may be that through St Augustine’s Si fallor sum they inspired Descartes’ Cogito ergo sum’. See also Bellissima and Pagli 1996, 127.

18 The question about the irrefutability of both truth and philosophy is at the core of Hegel’s analysis and critique of ancient scepticism. See Hegel 1969ff., Vol. 2, 229ff.; Hegel 1969ff., Vol. 3, 159ff.; Hegel 1969ff., Vol. 19, 358ff.

19 Hegel 1969ff., Vol. 19, 82 English translation Hegel 1892ff., Vol. 2, 59–60.

20 I take this as formal expression of ‘the set of all sets which are not members of themselves is member of the set of all sets which are not members of themselves’.

21 See Priest and Routley 1984, 92–93. I agree with Priest and Routley 1984 about the formal closeness between dialectical and paradoxical inferences: both are sound arguments conveying a true contradiction. However, I stress that Hegel has a different view of the conclusion of a paradoxical inference than the one endorsed by Priest and Routley in 1984.

22 See for example Sainsbury 2009, Chapter 6.2. The connection is explicitly addressed and explained by D’Agostini 2002 and D’Agostini 2009, Chapters 10 and 11. It is also hinted at by Bellissima and Pagli 1996, 151ff.

23 A possible way to express the Liar paradox (L) formally is:

(L)

L: L is false (this is the formal expression of the Liar sentence ‘this sentence is false’)

The Liar sentence immediately implies:

L T’L’ (where ‘T’ is the truth predicate, and falsity is the negation of truth)

If we assume the truth-schema:

Taa

And apply it to L, we have

T’L’T’L’. The reconstruction is taken from Cook 2009, 171. The passage from L: L is false [formally expressed as: (L) ‘FL’] to LL is usually explained via the capture- and release-behaviour of the truth predicate in the T-schema (and the definition of falsity).

24 D’Agostini 2009, 140–41.

25 D’Agostini 2009, 113–14.