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Thinking about Contradictions pp 53–73Cite as

Non-Aristotelian Logic

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Part of the book series: Synthese Library ((SYLI,volume 386))

Abstract

This chapter deals with attempts, contemporary with Vasil’ev’s own, to develop non-Aristotelian logics that present affinities with imaginary logic. Already in Aristotle’s work there are passages that press in the direction of a non-Aristotelian logic, in so far as they show that the syllogism is independent of the principle of contradiction. Some Aristotelian scholars like Heinrich Maier and Isaac Husik had drawn attention to such passages. Husik in particular proposes, on the basis of them and of Herbert Spencer’s philosophy, a hypothetical logic in which the syllogism is independent of the principle of contradiction; judgments are allowed that assert contradictory predicates of the same subject; contradictory objects are subjects of true propositions; and a hypothetical world is assumed, for which such a different logic would be valid. Jan Łukasiewicz was familiar with Maier and Husik’s works. He subjects to rigorous critique the Aristotelian principle of contradiction, claiming that it is uncertain, that it is not a simple, ultimate and necessary principle, and that in relation to contradictory objects it is actually false. Łukasiewicz took the notion of contradictory objects from Meinong, according to whom such objects — which are overdetermined objects of higher order in which a surplus of determinations inheres, amongst which there is a relation of incompatibility — can occur as genuine subjects in true propositions. The chapter concludes with an outline of the controversy between Meinong and Russell, with which Łukasiewicz was thoroughly acquainted, and his proposal of a non-Aristotelian logic in which the principle of contradiction is insignificant.

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Notes

  1. 1.

    Vasil’ev (1911/1989: 126).

  2. 2.

    Vasil’ev (1912: 212 = 1989: 59 [2003: 131]).

  3. 3.

    Cf. Łukasiewicz (1910a/1987: 8).

  4. 4.

    Aristotle , An. post. i 11, 77a10–12. Both the interpolation of Arabic and Roman numerals in Aristotle’s text and the italics, here and in later quotations, are mine.

  5. 5.

    As opposed to traditional logic, in which the subject precedes the predicate, in Aristotle AaC means ‘A belongs to all the C’s,ʼ that is ‘All C are A.’

  6. 6.

    Cf. Ross (1949: 542, ad i 11. 77a10), Bocheński (1970 3: 72), McKirahan (1992: 77), Barnes (1994 2: 145, ad i 11. 77a10); but cf. also Husik (1906: 219) and Łukasiewicz (1910a/1987: 93; 1910b: 32 [1971: 504]).

  7. 7.

    Cf. Aristotle , An. post. i 7, 75b7–14.

  8. 8.

    Cf. Aristotle, An. post. i 10, 76a37–b2; 11, 77a23–24; Metaph. γ 3, 1005a25–27. Cf. also McKirahan (1992: 71–73).

  9. 9.

    A translation of lines 77a15–18 conforming more closely to the interpretation I propose is as follows: “For if you are given (2) something of which it is true to say that it is a man, (2′) even if not-a-man is also true of it, then provided only that it is true to say (1) that man is an animal and not not-an-animal, it will be true to say (3) that Callias, (2″) even if he is not-Callias, is nevertheless an animal, and not not-an-animal.”

  10. 10.

    Aristotle , An. post. i 11, 77a12–21. For a commentary on the passage cf. Ross (1949: 542–543, ad I 11. 77a10–21), Mignucci (1975: 221–237; 2007: 185–187 ad 77a10–21), McKirahan (1992: 76–79), Barnes (1994 2: 145–147, ad i 11. 77a10).

  11. 11.

    Recently, some scholars have hypothesized a paraconsistent approach to the Aristotelian syllogistic. da Costa , Beziau & Bueno (1998: 142–50; cf. also da Costa , Krause & Bueno 2007: 828–829) proposed a paraconsistent interpretation of traditional syllogistic built on the monadic paraconsistent first-order C 1 * logic. Priest (2005: 132) claimed, with reference to An. pr. ii 15, that syllogistic is paraconsistent. Finally, in a more detailed way through the analysis of both An. pr. ii 15 and An. post. i 11, Gomes & D’Ottaviano (2010) showed that Aristotle ’s theory of syllogism is a paraconsistent theory in a broad sense.

  12. 12.

    Cf. Peirce (1880: CP 3.192–193; W 3, 176–178).

  13. 13.

    Cf. Husik (1906: 217, fn. 1).

  14. 14.

    Maier (1896–1900: ii.2, 238).

  15. 15.

    Cf. Maier (1896–1900: ii.2, 239 fn. 3).

  16. 16.

    Cf. Waitz (1844–1846: ii, 328–329, ad 77 a 10).

  17. 17.

    Husik (1906: 219). The original text by Waitz (1844–1846: ii, 328) runs: “Principium contradictionis quod dicitur in ipsam demonstrationem non assumitur, nisi etiam in conclusione expressum esse debeat.”

  18. 18.

    Husik (1906: 219). The interpolation of Arabic and Roman numerals in Husik’s text, here and in later quotations, is mine.

  19. 19.

    Husik (1906: 219–220).

  20. 20.

    Cf. Husik (1906: 215–216).

  21. 21.

    Spencer (1865/1966: 192; 1873 2: 423–424); cited also in Mill (1872 8/1973–1974: ii, vii, § 5, pp. 278–279; 1872 4/1979: 381 n.).

  22. 22.

    Cf. Spencer (1865/1966: 213).

  23. 23.

    Cf. Spencer (1865/1966: 205, 208).

  24. 24.

    For more details on Mill and Spencer ’s conceptions regarding the principles of contradiction and of excluded middle see Raspa (1999b: 89–97).

  25. 25.

    Husik (1906: 216).

  26. 26.

    Ibid.

  27. 27.

    Husik (1906: 217).

  28. 28.

    Cf. ibid.

  29. 29.

    Łukasiewicz (1910a/1987: 10, 149; 1910b: 16 [1971: 488]).

  30. 30.

    Łukasiewicz (1910a/1987: 11, 149; 1910b: 16–17 [1971: 488]).

  31. 31.

    Łukasiewicz (1910a/1987: 12, 149; 1910b: 18 [1971: 488]).

  32. 32.

    Cf. Łukasiewicz (1910a/1987: 16–18; 1910b: 17 [1971: 489]). On the distinction between equivalence and synonymity see Łukasiewicz (1910a/1987: 15–16).

  33. 33.

    Łukasiewicz (1910a/1987: 18); cf. also Aristotle, Metaph. γ 7, 1011b26–27.

  34. 34.

    Cf. Trendelenburg (1870 3: i, 23, 31–32; ii, 174), Ueberweg (1882 5: § 77, 234–237), Sigwart (1904 3: i, § 23, pp. 188–191 [1895: i, 139–141]).

  35. 35.

    Maier (1896–1900: i, 41–45, esp. p. 42 fn. 1) had indicated, for each of the formulations distinguished by Łukasiewicz , the self-same passages subsequently examined by the latter.

  36. 36.

    Cf. Twardowski (1894 [1977]).

  37. 37.

    Cf. Meinong (1899: GA ii, 381 ff. [1978: 141 ff.]). On the relationship between Meinong’s and Twardowski ’s conceptions regarding the distinction between act, content and object see Raspa (2016: 39 ff.).

  38. 38.

    Here I give only a sketch of Łukasiewicz ’s concept and criticism of the principle of contradiction; for more details on this topic see Raspa (1999a; 1999b: 53 ff., 110 ff., 139 ff., 257 ff.; 2000).

  39. 39.

    Mill (1872 8/1973–1974: ii, vii, § 5, pp. 277–278).

  40. 40.

    Cf. Mill (1872 8/1973–1974: ii, vii, § 5, pp. 278–279; 1872 4/1979: 381 fn. *).

  41. 41.

    Husserl (1900–1901: i, 81 [2001: i, 58]).

  42. 42.

    Cf. Husserl (1900–1901: i, 81–82 [2001: i, 58]).

  43. 43.

    A lecture on “Teza Husserla na stosunku logiki do psychologii [Husserl’s Thesis on the Relationship between Logic and Psychology]” held by Łukasiewicz at the Polish Philosophical Society testifies to this (for a short report of the lecture, cf. Łukasiewicz 1904). He speaks more at length about this subject in “Logika a psychologia [Logic and Psychology]” (cf. Łukasiewicz 1907). Cf. also Borkowski and Słupecki (1958: 46–47), Kuderowicz (1988: 142–143), Sobociński (1956: 8–9), and Woleński (1989: 194).

  44. 44.

    Aristotle , Metaph. γ 3, 1005b25–26.

  45. 45.

    Cf. Łukasiewicz (1910a/1987: 30–34; 1910b: 21 and fn. 1–2 [1971: 492–493 and fn. 6–7]).

  46. 46.

    Łukasiewicz’s criticism of Aristotle’s psychological formulation of the principle of contradiction follows another path but reaches the same conclusion. Łukasiewicz takes into account the passages of Metaph. γ 3, 1005b26–32 — read in connection with Int. 14, 23a27–39 — and γ 6, 1011b15–22, which he interprets as two complementary parts of a single attempt conducted by Aristotle to prove the validity of the principle of contradiction even for beliefs. The result achieved by Łukasiewicz (1910a/1987: 19 ff.; 1910b: 18 ff. [1971: 489 ff.]) is that the impossibility for a subject to have contradictory beliefs at the same time is demonstrable only provided that we treat these as if they were sentences for which the alternative true or false is valid. Therefore, the psychological formulation of the principle of contradiction is nothing but a consequence of the logical one. In such a way, Aristotle would fall into that error which is the exact converse of “psychologism in logic,” that is, “logicism in psychology.” However, sentences are not beliefs. The latter are “psychical phenomena” and, as such, are always positive. Consequently, it can never happen that two beliefs are in contradiction like an affirmation and its negation. Such a thing would involve that the same belief should be present and at the same time should not be present in the same mind, but a belief that does not exist cannot be in contradiction with another. In reality, while sentences mean that something is or is not and while they are in a relation of correspondence or of non-correspondence with their own objects or facts, so that they can be true or false, beliefs have a different structure. As psychical phenomena, they do not assert simply that something is or is not but they rather represent an intentional relation with something: without something that is intended, Łukasiewicz says, there is no belief. This intentional relation consists of two parts: the act of belief and the Meinongian objective (see Sect. 6.6). The expression in words or in signs of the second part of the intentional relation is the sentence, which can be true or false, but the first part does not refer to any fact, so we can say that it is neither true nor false. Then, beliefs are not purely logical objects. Cf. Łukasiewicz (1910a/1987: 25, 29–30).

  47. 47.

    Cf. Łukasiewicz (1910a/1987: 49 ff.; 1910b: 22 [1971: 493–494]).

  48. 48.

    Cf. Łukasiewicz (1910a/1987: 57 ff.; 1910b: 23 [1971: 494]).

  49. 49.

    Cf. Łukasiewicz (1910a/1987: 102, 109 ff.; 1910b: 35 [1971: 506–507]).

  50. 50.

    Cf. Raspa (1999b: 80 ff. and passim).

  51. 51.

    By principle of the syllogism Łukasiewicz means, following Couturat (1905: 8), the law of transitivity: ((a → b) ∧ (b → c)) → (a → c); cf. Łukasiewicz (1910a/1987: 155).

  52. 52.

    Cf. Łukasiewicz (1910a/1987: 95, 191–192; 1910b: 32–33 [1971: 504]).

  53. 53.

    According to Łukasiewicz (1910a/1987: 91 and fn. *), Maier did not recognize the fundamental significance this passage has for Aristotle ’s entire system of logic, while Husik, notwithstanding “the correctness of his central idea,” expressed his views in a very imprecise manner. The truth of the matter is that Łukasiewicz is indebted both to Maier and, above all, to Husik.

  54. 54.

    Cf. Łukasiewicz (1910a/1987: 93–95). See also Bocheński (1970 3: 71–72), who shares Łukasiewicz ’s interpretation, and in contrast Zwergel (1972: 21–28) and Seddon (1981: 203–206), who disagree with it.

  55. 55.

    Cf. Łukasiewicz (1910a/1987: 96); but cf. also Peirce (1878: CP 5.403; W 3, 266–267), whom we have already alluded to (see Sect. 2.3) and to whom we shall return presently (see Sect. 5.4 fn. 61 and Sect. 5.6).

  56. 56.

    Cf. Meinong (1899: GA ii, 386 [1978: 144]).

  57. 57.

    Meinong (1904: GA ii, 489 [1960: 82]).

  58. 58.

    Meinong (1904: GA ii, 494 [1960: 86]).

  59. 59.

    On this controversy, only touched on in passing here, cf. Griffin & Jacquette (eds., 2009). A careful reconstruction of the dispute is provided by Farrell Smith (1985). I have dealt with it in Raspa (1995/1996: 181 ff.; 1999b: 247 ff.), where the relevant literature on the topic is listed.

  60. 60.

    Cf. Russell (1905b/1973: 105).

  61. 61.

    Cf. Russell (1905b/1973: 115–116). On the notion of existence here presupposed see Russell (1905a/1973: 98–99).

  62. 62.

    Cf. Meinong (1900: GA i, 470; 1906: GA v, 389).

  63. 63.

    Cf. Kant (1781 1–17872: A 151 = B 190–191 [1998: 280]). Cf. also Raspa (2015: 137).

  64. 64.

    Ernst Mally (GA i, 494, Zusatz 17) also observed that an object which possesses the only determinations to be a mountain and to be golden, and for the rest it is ontologically incomplete in every other respect, cannot exist or be real.

  65. 65.

    Russell (1905b/1973: 117).

  66. 66.

    Cf. Meinong (1915: GA vi, 180).

  67. 67.

    Cf. Łukasiewicz (1910a/1987: 110 and fn.; 1910b: 35 and fn. 1 [1971: 506 and fn. 14]).

  68. 68.

    Cf. Łukasiewicz (1957 2: 12, 46–47, 73–74).

  69. 69.

    Cf. Łukasiewicz (1957 2: 46, 88).

  70. 70.

    Cf. Łukasiewicz (1958 2 [1963: 67–68]; 1937 [1970: 243, 248]). Cf. also Sobociński (1956: 11 ff.) and Jordan (1963: 13). In 1910, Łukasiewicz (1910a/1987: 8, 9) used the term ‘metalogical,’ but not in the sense that it later acquired and still retains in mathematical logic today.

  71. 71.

    Cf. Łukasiewicz (1910a/1987: 95–101); a glance at the same argument can be detected also in Łukasiewicz (1910b: 33). For further details cf. Raspa (1999a: 76 ff.; 1999b: 262 ff.).

  72. 72.

    Cf. Łukasiewic z (1910c [1987]; 1913: 32–33).

  73. 73.

    Cf. Łukasiewicz (1920 [1970: 87–81]; 1930/1988: 107–109 [1970: 164–166]; 1961: 125 [1970: 126]).

  74. 74.

    Cf. Jaśkowski (1948 [1969]). In his brief historical survey of the problem followed by the exposition of the known solutions, Jaśkowski (1948: 57 ff. [1969: 143 ff.]) ignores Vasil’ev’s point of view. Independently of Jaśkowski , Newton C. A. da Costa (1963; 1964a; 1964b; 1964c; 1974) also began to study inconsistent and non-trivial systems. Cf. also D’Ottaviano (1990b: 20 ff.).

  75. 75.

    Cf. Vasil’ev (1910: 8 = 1989: 17).

  76. 76.

    Cf. Vasil’ev (1912–1913a: 80 = 1989: 122 [1993: 350]).

  77. 77.

    Cf. Mally (1909).

  78. 78.

    Itelson ’s point of view is referred to in Couturat (1904: 1038–1039): “celle-ci est la science des êtres, des objets existants, tandis que la Logique est la science de tous les objets réels ou non, possibles ou impossibles, abstraction faite de leur existence (De rebus omnibus et de quibusdam aliis). Ainsi la Logique est délivrée de toutes les difficultés d’ordre métaphysique; elle n’a pas à s’occuper des jugements d’existence, ils sont extra-logiques (Cogito; argument ontologique). Et pourtant la Logique a une valeur objective universelle, puisqu’elle s’applique, en particulier, aux objets réels; ainsi s’explique que la nature obéisse aux lois de la Logique. La Logique ne s’occupe même pas du vrai et du faux, car le vrai et le faux sont des qualités de la pensée, et non des objets: la Logique porte sur les relations formelles des objets, non sur la relation de la pensée à ses objets.”

  79. 79.

    Cf. Renouvier (1876).

  80. 80.

    Cf. Vasil’ev (1911/1989: 127; 1912: 222 = 1989: 68 [2003: 140]; 1912–1913a: 61 = 1989: 102 [1993: 335]).

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Raspa, V. (2017). Non-Aristotelian Logic. In: Thinking about Contradictions. Synthese Library, vol 386. Springer, Cham. https://doi.org/10.1007/978-3-319-66086-8_4

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