Hegel’s Interpretation of the Liar Paradox

Abstract

In his Lectures on the History of Philosophy, Hegel develops a subtle analysis of Megarian paradoxes: the Liar, the Veiled Man and the Sorites. In this paper, we focus on Hegel's interpretation of the Liar. We note that in Hegel's treatment there are positive suggestions for a new analysis of the paradox. Faced with the Liar's sentence ‘µ’ that says ‘“µ” is false’, Hegel's idea is that the conjunction ‘µ and not µ’ is to be held true, but the two conjuncts ‘µ’ and ‘not µ’, separately taken, are untrue. Other parts of Hegel's work confirm the idea that a true contradiction for him is made of two untrue sentences, which is an interesting but unusual account of inconsistency. In this paper, we explore the plausibility of the hypothesis with the lens of the contemporary theories of paradoxes. We compare Hegel's view with standard dialetheism, and we present Hegel's idea of truth which underlies the theory.

Acknowledgements

We owe a special debt of gratitude to Jc Beall. He has given decisive contributions to the development of the theory here presented. The idea of ‘Hegelian conjunction’ is largely based on Beall and Ficara 2014, the first attempt to draw the consequent semantics. Evidently, all responsibility for errors or unclarity is up to us. For helpful comments we are grateful to Michela Bordignon, Volker Peckhaus, Bridget Pupillo, Guy Watts, Juliette Weyand, as well as to two anonymous referees.

Notes

1 We focus our discussion mainly on Hegel’s Lectures on the History of Philosophy, with some minor reference to other writings. It is a shared view in the tradition of Hegel Studies that the text not only reflects the original version of Hegel’s Lectures (conceived and held for the first time in Jena in 1805/1806, then re-proposed with minor changes in Heidelberg in 1816/1817, before being given regularly in Berlin between 1819 and 1830) but are also a fundamental reference point for anyone interested in understanding Hegel’s thought. The Lectures, as highlighted in Gadamer 1976, are a true laboratory for Hegel himself, who articulated his conception through the interpretation of both classical and modern authors. A similar judgement on the role of the Vorlesungen is given by Düsing 1990, 169–191; Pöggeler 1990, 42–64; Riedel (ed.) 1990; Schäfer 2001.

2 The idea is not so implausible as it may first appear. If anything, conjunction is not a simple operator. A number of studies in linguistics and philosophy of language show that its behaviour is not always ruled by the classical equivalence ‘p and q’ is true iff ‘p’ is true and ‘q’ is true. This is specified for instance in Humberstone 2011, 658–661, and in the monumental work of Schein 2017 about conjunction reduction; but also in truthmaker metaphysics and semantics (Rodriguez-Pereyra 2006 and Jago 2018), in relevance logic (Read 1988 and Mares 2012), in the epistemology of paradoxes (see Sorensen 2018), in non-adjunctive approaches to paraconsistency (Varzi 2004). More information about all this will be given later on (see also d'Agostini 2021).

3 Sparse hints of a conjunctive view (variously intended) are to be found in Priest 2006, 11; Beall 2006a, 102–114; Kabay 2010; Cobreros et al. 2015, 375–393; Estrada Gonzales 2017. Details about the theory, with reference to Hegel and/or in general, are to be found in d'Agostini 2008b, 2014, 2021; Beall and Ficara 2014; Ficara 2021. The label conjunctive paraconsistency is adopted by Ripley (2015, 771–772) to mean the property of a consequence relation so that A $\wedge \mathrm{\neg }$A $\models$ B does not hold (for wffs A, B $\in$ L), as opposed to collective paraconsistency, whereby A, $\mathrm{\neg }$A $\models$ B does not hold (see also Barrio and Da Ré 2018). We can use the same label for a position similar to the one endorsed by Hegel, because in a Hegel-sympathetic conjunctivism, non-explosion is to be guaranteed by the joined and inseparable occurrence of A and $\mathrm{\neg }$A. Some more details in d'Agostini 2021.

4 For Hegel’s general view about Megarian paradoxes, see d'Agostini 2008b, some theses of which are revised and developed in Sections 1 and 2 of the present article.

5 Hegel Werke 18, 518/Hegel 1892ff., vol. I, 452. We quote Hegel’s Werkausgabe (see Hegel 1969ff) as Hegel Werke, followed by the volume and page number. If the English translation is available, we also give the page number of the translation. If it is not, the translation is ours and we only give the page (and volume) number of the German text.

6 Diogenes Laertius, Book II, 108 (see Diogenes Laertius 2018). The term ‘paradoxon’ was not used in classical Greek debates. It appeared, seemingly for the first time, as a noun in Aristotle’s Sophistical Refutations (Ch. 12, 172–173), and Aristotle visibly refers here to the arguments of those ‘dialecticians’ who lead people to contradict themselves (eis paradoxon agein); thereby it is not difficult to identify the exponents of the Megarian school. Further details are to be found in Boger 1993. Hegel himself does not use ‘paradoxes’, but rather ‘sophisms’, or ‘elenchi’. In the Science of Logic he refers to the Megarian elenchi, writing ‘they are familiar under the names of “the bald” and “the heap”. These elenchi are, according to Aristotle’s explanation, ways in which one is compelled to say the opposite of what one had previously asserted’ (Hegel Werke 5, 397/Hegel 1969, 335).

7 See, with some differences, Quine 1962; Sainsbury 1987; Rescher 2001; Olin 2003; Clark 2002; Cook 2013. Such a definition is discussed by Clark 2002, 132–135 and Sorensen 2003, 104–110. A slightly different definition (favoured by Schiffer 2003, Lycan 2010 and Paseau 2013) requires a paradox to be ‘a set of incompatible propositions each one of which enjoys some plausibility on its own’ (Schiffer 2003, 5). The discussion is reconstructed in d'Agostini 2009, 19–42. In the German logical tradition, the meaning of ‘paradox’ differs slightly from the one most commonly used in Anglo-American debates. On discussions about the meaning of ‘paradox’ in Germany circa 1900, see Peckhaus 2000, 501ff.

8 Nowadays such a definition is adopted by Sorensen 2003, xii: ‘a question […] that suspends us between too many good answers’.

9 In pre-modern times, the erotetic form was often used to express what is now expressed by monological arguments (see Hintikka 2007, 19–24); so, to understand theories of paradoxes in the history of logic, it might be advisable to take into account the peculiarity of this approach (Sorensen 2003). Otherwise, the dialogical approach (in discussive or erotetic form) is not incompatible with the current monological treatment of logical themes; as Dutilh Novaes (2020, Ch. 6 and 7) stresses, it survives in the modern conception of deduction.

10 On the connection between refutation (elenchos) and shame (elencheie), see Lesher 2002, 19–35.

11 For this definition and its consequences for a general theory of paradoxes, see d'Agostini 2009.

12 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

13 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

14 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

15 Hegel Werke 18, 528/Hegel 1892ff., vol. I, 458–459.

16 ‘The simplicity of the truth is thus grasped as the principle. With us this appears in the form of making such statements as that one of the opposites is true, the other false; that a statement is either true or not true; that an object cannot have two opposite predicates’ (Hegel Werke 18, 528/Hegel 1892ff., vol. I, 458–459).

17 Hegel Werke 18, 523–529 and 536f/Hegel 1892ff., vol. I, 454–459 and 466f.

18 Hegel Werke 18, 526–527/Hegel 1892ff., vol. I, 457. That Hegel ‘attached great importance to paradoxes’ is stressed by Vieweg 2019, 727, footnote 86.

19 Hegel Werke 18, 536/Hegel 1892ff., vol. I, 466.

20 In the Megarian philosophy of language—as Hegel reconstructs it, especially focusing on Stilpo—a proposition is composed ‘by a subject plus a predicate’, and if one looks at the ‘pure subject’ and the ‘pure predicate’, one finds two different universal objects: their connection in the proposition is inexplicable and conveys contradiction. For instance, the pure concept of ‘Socrates’ is perfectly identical to itself; and the universal ‘running’, or ‘staying’, are identical to themselves, too. What is the meaning of ‘Socrates runs’ / ‘Socrates stays’? How can Socrates be modified by ‘running’ or ‘staying’, and ‘running’ or ‘staying’ by Socrates? On this topic, see especially Hegel’s account of the Veiled man (Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461).

21 By ‘conceptual’ (begrifflich), Hegel means ‘concerning the universal’ (das Allgemeine). For Hegel, universals/concepts are predicates expressing the properties of objects and their relations to each other. See Hegel Werke 5, 26/Hegel 1969, 36–37. The question about the meaning of Hegel’s concept of ‘concept’ in relation to analytical philosophy is complex; what interests us here is to highlight that the conceptual realm for Hegel is not a mental domain (as it is well explained, in relation to analytical philosophy, by Brandom 2014—see also Nuzzo 2010, d'Agostini 2003, Ficara 2021). We are also interested in stressing the connection between the conceptual perspective and paradoxical arguments (as it is reconstructed by d'Agostini 2008b, 206–207).

22 For Priest and Berto 2008: ‘With a little bit of rubdown, even the current debate on logical paradoxes may be viewed as a ramification and formal specification of the Kant-Hegel dialectics’.

23 Hegel Werke 18, 529/Hegel 1892ff., vol. I, 459. As a marginal note, we can observe that the Liar was actually intended in the tradition as a liar (pseudomenos), which is slightly different from how we are used to considering it nowadays. The interesting difference is that the contradiction is also de se (about what the speaker believes) and not only de re. We have developed the discussion about ‘the Liar as a Liar’ in d'Agostini and Ficara 2020, 256–259.

24 Hegel Werke 18, 529/Hegel 1892ff., vol. I, 459.

25 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

26 Hegel Werke 18, 529–530/Hegel 1892ff., vol. I, 460.

27 Hegel Werke 18, 529–530/Hegel 1892ff., vol. I, 460.

28 In this sense, there might be reasons to believe that Hegel is some sort of ‘semantic dialetheist’ (see Ficara 2015, 53).

29 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

30 Hegel Werke 18, 530/Hegel 1892ff., vol. I, 460–461.

31 Hegel Werke 18, 530/Hegel 1892ff., vol. I, 460–461.

32 Hegel here compares the case to suicide, noting that, similarly, the death penalty must and cannot be applied in case of a person who tries to kill herself: in this case, death must and cannot be a punishment (Hegel Werke 18, 530/Hegel 1892ff., vol. I, 460–461).

33 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

34 Hegel Werke 18, 531/Hegel 1892ff., vol. I, 461.

35 In this respect (as in other respects, see Havas 1983, 63ff.) Hegel is a classical logician: he believes that, ultimately, gaps and gluts are the same thing. Hegel also stresses here the functional equivalence of closure (self-predication, self-definition, circularity, etc.) and presuppositions or pro-sentential aspects of assertions: he sees that (as in Menedemus’ case) they can contradict the content, thus preventing the use of simple truth. The structure of the Liar, as it is interpreted by Hegel, is in accordance (Section 2.2.2).

36 Hegel Werke 18, 528/Hegel 1892ff., vol. I, 458–459.

37 The triadic pattern of dialectical thought is a conception that has raised hefty controversies, beginning in the first years after Hegel’s death and lasting into the present. See, for instance, Adorno 1966, in which the dialectical process is likened to a dancing procession—the spring procession in the German town of Echternach, in which the same steps (one forward and two backwards) are laboriously repeated, producing minimal or no advancement.

38 A similar treatment of the Liar, in terms of two reductions to the absurd, is proposed in d'Agostini 2009.

39 Hegel is fundamentally a classical logician, in the sense that, for him, untrue means false (see Havas 1983, 63ff.). Nowadays, the ‘untrue’ formula of the Liar is called the ‘revenge Liar’, as it takes revenge against truth-value gap theories, whereby ‘µ’ would be neither true nor false.

40 In displayed formulas we use the angle brackets for truthbearers, inverted commas for mention, and no quotation mark for states of affairs. For instance: S believes that 〈p〉 is true, she accepts ‘p’ and believes that p subsists.

41 The terminology may vary, depending on the aims or levels of the analysis: ‘factivity’ may be called disquotation, or release, or transparency, or descendant T, or T-Out; ‘ubiquity’ stands for self-assertion, or capture, or T-In, or ascendant T. For these different expressions see Beall 2007, 1–30; and Scharp 2013.

42 It is what Field 2008, 8 calls ‘The Central Argument from Equivalence to Contradiction’.

43 The consequentia mirabilis is a rule which Hegel knew well, as it is the formal version of the ancient elenchos, and both are some of the acknowledged roots of Hegel’s dialectics (see Ficara 2018).

44 The argument that draws the contradiction from the Liar’s sentence can be presented in a variety of ways. In some versions, there is no reference to the T-schema; in other versions, there is no biconditional (see Beall 2014 and Beall et al. 2017). We do not consider these positions here; we are interested in the biconditional version of the argument (typical of Priest 1987 and 1995) because it is the closest to what Hegel says (see Section 2.1).

45 See Priest and Routley 1984 and Priest 1989, 388–415 (see also Priest 1987, Ch. 12). In recent years, the binomial dialectic/dialetheism has been the subject of renewed interest: see among others: Berto 2005, 173ff.; Schick 2010; d'Agostini 2008a and 2014; Ficara 2013 and 2021; Bordignon 2014a; Priest 2019 and typescript; Moss 2020.

46 In his ‘Survey of the definitions of dialetheism’, Martin (2014, 225–228) lists twenty-six definitions, of which nine are given in terms of D1, seven in terms of D2, and the others in terms of both D1 and D2 or otherwise. Altogether, we could say that D1 gives the minimal distinctive feature. As Priest clearly states: ‘the fact that we are faced with, or even forced into operating with, information that is inconsistent, does not, of course, mean that that information is true. The view that it may be is dialetheism […] A dialetheist is therefore a person who holds that some contradictions are true’ (Priest 2007, 131). In this sense, we feel authorized to say that the Hegelian conjunctive view is a kind of ‘anomalous’ dialetheism.

47 There is a wide and variously oriented literature that shows the possible failures of CT, as we have suggested in Note 2. In particular, as Schein (2017, 97) stresses, the idea that non-reducible ‘ands’ are not properly conjunctions is an ‘Aristotelian fallacy’ which we ought to forget. Notably, in an exact truthmaker semantics, the left-to-right direction of CT does not hold (see Rodriguez-Pereyra 2006, 957–982 and 2009, 428–442; and developments in Jago 2018, Ch. 2), which (especially for Rodriguez Pereyra) reveals a difference between the metaphysical and the logical (classical) conceptions ofentailment.

48 In the literature about inconsistency, there is a variety of non-adjunctive positions; see, to begin with: Jaśkowski 1948 and 1999, and Rescher and Manor 1970–71, Lewis 1982, Varzi 2004. For a detailed account, see Berto 2007, 131–150.

49 For further details see d'Agostini 2021. For now, consider that the (non-simplifiable) conjunctive ‘and’ is content-sensitive: it ‘binds’ two state descriptions so that one cannot be stated (assumed or asserted) without the other, and this is due to their mutually self-defeating contents. As we will note later, the notion of ‘fusion’ proposed by Mares (2012, 7–21) in the framework of a ‘situational semantics’ (not far from Hegel’s ‘natural logic’) could be adapted to the conjunctivist idea.

50 Such an approach is typical of what Hegel called elsewhere ‘natural logic’: a perspective focused on the concrete contexts, cases or situations in which people reason, dialogue, discuss, and accept or reject theses (Hegel Werke 5, 20–27/Hegel 1969, 31–37).

51 As long as we admit that an asserter is someone who is formally intended to believe what she says, an asserter that p is someone who is intended to believe that ‘p’ is true. There are controversies about the B-implication or T-implication of assertives (see Marsili 2018, 638–648); suffice it to say that the logical evaluation of assertions implies believed truth as a formal requirement (which does not exclude that the speaker might lie).

52 This approach is fairly common in classical and contemporary treatments of the paradox. In essence, it corresponds to the idea that Liar-like contradictions do require complex predicates or concepts, related to properties whose stratification (iteration, diagonalization, indefinite extensibility etc.) may produce (or seem to produce) the joined occurrence of what, in principle, cannot jointly occur. For similar developments of Tarski’s aetiology, see for instance Dummett 1993; Simmons 1993 and more recently Field 2008. Scharp 2013 focuses in particular on the inconsistency of truth, but also other concepts have similar properties of transparency and ubiquity. Cook 2013 adopts a Curryfied version of the paradox, so that ‘µ’ is intended to say something like ‘Tµ → $\mathrm{\neg }$Tµ’.

53 On Hegel’s view on LNC and LEM, see Stekeler-Weithofer 1992, 23ff. We basically agree with Stekeler’s diagnosis. For Stekeler-Weithofer, Hegel does not question the validity of LNC and LEM, but rather criticises ‘the formalistic assumption according to which these principles hold in general’.

54 The first proposal of a conjunctive model for contradictions has been given by Beall and Ficara 2014, with specific reference to Hegel’s notion of Vereinigung. d'Agostini 2021 advances an interpretation of FDE with a special gap for overdetermination instead of the value ‘both’.

55 On the notion of fusion in relevance logic, see also Read 1988, new edition 2010, 36–42; Mares 2012. Priest 2013 presents a slightly different account.

56 For a more exhaustive account see d'Agostini 2021, 12–15.

57 This implies some interesting consequences for the Hegelian notion of negation. In case of true contradictions, one would say that negation is ‘on holiday’ (see Beall 2006b); it does not have any exclusionary property, while it preserves its feature of excluding the opposite when accompanied by truth. Clearly, this postulates a difference between negation as ‘border operator’ and falsity, i.e. the truth of negation (d'Agostini 2021).

58 On this subject see Ficara 2021, 84–122. In this section some theses already suggested in Beall and Ficara 2014 and Ficara 2021 are revised and further developed.

59 During his Habilitationsprüfung, the exam required to qualify for a professorship in Germany, and in which course candidates defend several philosophical theses before an academic committee, Hegel argued for 12 theses, among them for the view that contradictio est regula veri. For Rosenkranz 1868, 135ff., the theses are a list of the cornerstones of Hegel’s philosophical thought, of all the points Hegel never ceased to defend.

60 According to Nuzzo 2011, 91–105, in the preface to the Phenomenology of Spirit Hegel lays out a conception of truth that is the very frame of the Science of Logic and, more generally, of his ‘mature’ views on logic.

61 Among the most recent accounts in this line, see especially Pinkard 2003, 119–134; Brandom 2005, 131–161; Nuzzo 2011, 91–105.

62 The coherentist account of das Wahre ist das Ganze is typical of British Hegelianism at the turn of the twentieth century. It is to be found in Joachim 1906, 72–79, and more recently in Brandom 2005, 131–161.

63 As we will show in what follows, Hegel uses the expressions ‘principle of scepticism’ (das Prinzip des Skeptizismus) in the Skeptizismusaufsatz and ‘principle of completion’ (das Prinzip der Vervollständigung) in the Differenzschrift. For some further details about these two principles, see Ficara 2021, 174f.

64 Hegel Werke 2, 229.

65 Hegel Werke 2, 229.

66 Croce 1906, Ch. IV.

67 Hegel Werke 3, 14–15/Hegel 1977, 4.

68 Hegel Werke 3, 24/Hegel 1977, 11. These passages are the source of innumerable interpretations. See, among the most recent works: Schnädelbach 1993; Pinkard 2003; Koch et al. 2003; Puntel 2005; Nuzzo 2011, 91–105; Miolli 2016; Ficara 2021.

69 On the role of this fragment in the development of Hegel’s dialectical method, see Pöggeler 1981, 42–45 and Düsing 2012, 11–114, among others.

70 Hegel Werke 1, 251.

71 Hints at the semantics of conjunction in dialectical contradictions along these lines can be found in Wetter 1958; Havas 1981, 257–264; Priest 1989, 388–415; Beall and Ficara 2014; Bordignon 2014b; Ficara 2021. In particular, Bordignon 2014b, 87–88 develops a position similar to ours, writing that ‘from the expression of the truth of Becoming […] it is not possible to deduce the truth of the single conjuncts [Being and Nothing]’. She also claims that ‘in the linguistic expression of the contradictory nature of Becoming, the truth of p (Being and Nothing are the same) does not imply the falsity of not-p (Being and Nothing are not the same): p and not-p are both true, and true in their unity’ [our translation]. In our account (see also Beall and Ficara 2014) the conjuncts are false/untrue.

72 Hegel Werke 2, 26.

73 Hegel Werke 18, 44–45/Hegel 1892ff., vol. I, 26.

74 This dualism is especially relevant for Hegel’s analysis of the sorites (see d'Agostini 2008b, 212–214).

75 Hegel Werke 18, 45/Hegel 1892ff., vol. I, 26.

76 Details about this and other crucial theses of Hegel’s conception of truth and logic can be found in Ficara 2021. We do not expand on the consequent semantics; some preliminary guidelines have been given in Beall and Ficara 2014 and d'Agostini 2021, 24–27.

77 In the traditional version: $\begin{array}{lll}1& \mathrm{p}\wedge \mathrm{\neg }\mathrm{p}& \mathrm{A}\\ 2& \mathrm{p}& \mathrm{S}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}.1\\ 3& \mathrm{p}\vee \text{β}& \vee \mathrm{I}3\\ 4& \mathrm{\neg }\mathrm{p}& \mathrm{S}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}.1\\ 5& \text{β}& \mathrm{D}\mathrm{S}3,4\end{array}$

78 On the difficulty of ‘ruling out’ for dialetheists, see especially Berto 2007, 282–288 and 2008. Notably, insofar as they are committed to D2 and CT, dialetheists assume that truth distributes over the conjunction. So, by sticking to the T-schema and the usual meaning of conjunction and negation, dialetheists are led to admit that a true contradiction is true and false. Interestingly, this means the Law of Non-Contradiction holds unrestrictedly, because for any true ‘p $\wedge$ $\mathrm{\neg }$p’, ‘$\mathrm{\neg }$(p $\wedge \mathrm{\neg }$p)’ is true too. As especially stressed by Field (2005, 24–27) in this way dialetheism encounters some difficulties, as it cannot clearly distinguish its own position from all those positions that claim that ‘µ and not µ’ is false, or is neither true nor false, or each of its terms is untrue, or only one is true.

79 There are hints of a similar position in Aristotle’s conception of antiphasis (see Ramirez 2017). An interesting example is given by Christology after the Calcedonian council, whereby Christ’s ‘divine humanity’ is interpreted in terms of ‘two in one’. The idea of a paraconsistent Christology (with some suggestions about possible developments in the conjunctive line) is presented by Beall 2021.