https://orcid.org/0000-0001-9633-2031
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ABSTRACT
Hegel’s approach to soritical arguments (as well as to paradoxes in general) can be read as a kind of conjunctive paraconsistency: the ‘explosive’ effect of contradictions is avoided by assuming ‘the unity of the opposites’, so that contradictory conjunctions are not simplifiable. The paper reconsiders what Hegel says about the Sorites and justifies the conjunctive interpretation. The first section introduces the analysis, presenting the role of the Einheit Entgegengesetzter for contemporary theories of paradoxes. In the second section, the focus is on Hegel’s interpretation of the Sorites. The third section briefly considers some contemporary accounts of vagueness in the light of Hegel’s view so reconstructed. The last section summarizes the main results and explores some ideas about how to formalize a Hegel-inspired conjunctive view.
Notes
1 Some points of the reconstruction given in Section 2 have been presented in d’Agostini (Citation2008) and Citation2011.
2 See d’Agostini Citation2011, 195–197, and Citation2021, 6856–6859; d’Agostini and Ficara Citation2021, sect. 2.
3 The notion of ‘fusion’ in relevance logic works as a ‘binding operator’ so that ‘p and q’ does not imply ‘p’ (Mares Citation2012); in exact truthmaker semantics the truthmaker for ‘p and q’ is not (always) a truthmaker for ‘p’ or ‘q’ separately (Rodriguez-Pereyra Citation2006, Citation2009; Jago Citation2018); in linguistics, the monumental work of Schein Citation2017 on ‘conjunction reduction’ presents many cases of propositional though irreducible conjunctions. That ‘true contradictions’ involve a similar ‘and’ is the specific hypothesis of conjunctivism in paraconsistency.
4 It is clearly postulated a particular, not strictly ‘dialetheic’, conception of truth: see d’Agostini Citation2021, 6863–6868, and 2023, sect. 6; hereafter: sect. 4.2.
5 By SL1 and SL2 is meant, respectively, the Science of Logic (1812–1816); and the part of the Encyclopedia of the Philosophical Sciences devoted to logic (1817–1830).
6 See on this d’Agostini Citation2008, 203–223. That Hegel ‘attached great importance to paradoxes’ is also stressed by Vieweg Citation2019, 727f.
7 For this notion of ‘paradox’ as compared to other definitions see d’Agostini Citation2009, 19–36.
8 The approach is classical (see sect. 4), so falsity (the truth of negation) and untruth are equivalent.
9 I use inverted commas for mentioned sentences (‘p’); angle brackets for the truthbearers (T〈p〉); no quotation marks for the truthmaker (the fact-state that p).
10 In this respect, Hegel’s solution is a kind of ‘classical glut theory’ (to keep to the taxonomy of Scharp Citation2013, 23).
11 The truth-implying notion of assertion is not universally accepted in the contemporary philosophy of language, but truth-implication is one of the capital features of the dialectical-discussive approach typical of traditional logic: if the proponent asserts/assumes that p, they formally commit themselves to the alethic clause, so that for any asserted ‘p’, ‘T〈p〉’ is to be implied.
12 About the role of stratification in supporting conjunctive paraconsistency see d’Agostini Citation2022.
13 The critical role of vagueness in Hegel’s logic (the ‘indeterminacy’ of natural language) is pointed out in Marconi Citation1979 (18–24) and Nuzzo Citation2010. Some criticisms of both positions can be found in Bordignon Citation2013 (179–180), and further comments in Ficara Citation2021a, 74fn. The perspective I am adopting is focused on the metaphysics underlying Hegel’s logic, more than on his philosophy of language, but it is clear that Hegel’s view of the sorites has a special impact on both (see also Kurtsal Citation2019). In Nuzzo’s reconstruction, the Hegelian account of vagueness (intended as the indeterminacy of natural language) can be interpreted as grounded on the Saussurian difference between langue and parole: the former corresponding to intellectualistic logic (we can say: formal semantic as it is normally conceived), the latter being related to the concrete ‘logic’ of living language and living spirit (Nuzzo Citation2010, 71–78). There are reasons to endorse this interpretation: maybe Hegel’s Vernunftlogik (see hereafter Sect. 2.3) should be called ‘prelogical’ (grounding for logic) more that strictly ‘logical’ in the current standard sense (for this use of ‘prelogical’ see d’Agostini Citation2023).
14 The erotetic approach to the sorites is fairly usual nowadays; see especially the soritical sequence called ‘the Forced March paradox’ (Horgan Citation1994; Priest Citation2019, 142–143).
15 The numeral can also be attached to the predicate-property, but the expressive strategy I propose is more adaptable to the present needs, as it will be clear later.
16 Similar positions are Unger Citation1979 and Wheeler Citation1979, then more recently, and in an almost ‘Hegelian’ inspiration, Priest Citation2003 and Varzi Citation2003 (see Sect. 3 hereafter).
17 Notably, this does not imply trivialism. For Hegel soritical occurrences do not regard Vernunftlogik in general, but only the domain of concepts seen by reason (see Sect. 2.3).
18 That all paradoxical cases present the same problem and deserve the same solution has been proposed by many authors (see Oms Citation2019, 190–191). In particular, Priest Citation2010 especially defends the ‘Principle of Uniform Solution’.
19 Clearly, the premise is ¬Bλn≥10,000.
20 On the ambiguity of ‘simple truth’ see Shapiro Citation2004.
21 See also hereafter: 2.4 and 4.2. In Santos’ interpretation (Santos Citation2019, 294–295) the focus of Hegel’s view is ‘the role played by the repetition’ (295). My interpretation is different. I stress the role of conceptual determination (2.3), and then the underrated unity of the opposites when we deal with the reflexive action of reason (2.4).
22 Hegel’s analysis so involves the old and capital question of the Spinozian determination-negation. The debate on the theme is enormous (see the recent reconstruction of Ficara Citation2021a, 180–181fn), the account here adopted is not so far from the most canonical interpretations. My concern is to stress that what Hegel calls ‘intellectual determination’ is the cut-off point, in the current vocabulary, that is, the point ε at which the soritical sequence of ‘yes’ becomes a sequence of ‘no’. There must be a point of this kind, but the problem with vague predicates is that we do not know which this ε point is (see Sect 3.1).
23 Ficara Citation2021a, 15–18, especially highlights the continuity, in Hegel, of Verstandeslogik and Vernunftlogik. As the former is classical, i.e. submitted to Identity, Non-Contradiction and Excluded Middle, this means that Hegel’s logic is not strictly ‘non-classical’. The logic of speculative reason (Vernunftlogik) is the ‘complete’ picture of logic (intended as the art-science of ‘das Logische’), Verstandeslogik is part of it.
24 It is clear in Hegel what Priest notes with reference to ‘limits’ in general (Priest Citation1995, see especially 249–256), their implying closure (reflexion) as well as transcendence (negation), so releasing instances of the Inclosure Schema α ↔ ¬α.
25 R. Brandom’s reading of Hegel’s logic in terms of semantic holism confirms this line, even if in a different approach, and with a different upshot (Brandom Citation2005 and Citation2014).
26 The meta-logical impact of vagueness in Hegel is the focus of Nuzzo’s reconstruction (Nuzzo Citation2010, 71–78).
27 The tadpole-frog case is presented by Cargile Citation1969.
28 For instance, ‘Pλn’ is true iff |Pλn| ≥ 0.8, and consequently: |Tpλ| = 1 iff Tpλn≥8 and Fgλn≤2 and |Fgλ| = 1 iff Fgλn≥8 and Tpλn≤2.
29 D’Agostini and Ficara Citation2021, Sect. 3. One of the historical sources of dialetheism is namely the Australian Hegelianism of the last century (see Priest and Routley Citation1984 and Ficara Citation2021b). Dialetheism is now a detailed and refined position in philosophical logic, but the Hegelian inspiration is still operating, especially, I think, in the metaphysics and epistemology developed by Graham Priest (see Priest Citation19872, Citation1995, Citation2005, Citation2006 and Citation2014).
30 As Priest here recalls (Priest Citation2019, 140) ‘Being true and false is not a third truth value. It is the possession of two truth values.’
31 It is open opportunity for a subvaluationist evaluation. The connection between subvaluationism and dialetheism is fairly natural.
32 The conceptual approach in Hegel is primary, but it does not disprove as such the propositional (sentential) form of the contradictions that are the subject matter of rational logic (Ficara Citation2021, 80– 81). This happens just because the level in which we discover the contradiction is the rational level, it is the dialectical process whereby a double negation ‘does not produce affirmation, but contradiction’ (Ficara Citation2021, 180).
33 In fact, the move from ‘µ ↔ ¬µ’ to ‘µ ∧ ¬µ’ is not universally accepted, as it requires Excluded Middle or unrestricted application of the T-schema, which are not given in paracomplete logics. But there are reasons to believe that Hegel, in this respect, is a classical logician: EM works, and the Aristotelian T-schema works as well.