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Thinking about Contradictions pp 75–104Cite as

Imaginary Logic

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

Abstract

This chapter discusses in depth Vasil’ev’s imaginary logic. Vasil’ev criticizes the uniqueness of logic and the absoluteness of logical principles, taking into consideration the conceptions of Gerardus Heymans, Carl Göring, Benno Erdmann, Edmund Husserl and John S. Mill. The key point of his criticism is the assumption of another world, different from ours, and of beings with a different intellectual structure from our own. He then proposes a novel concept of negation, which is not based on the incompatibility between predicates and is not a deduction as it is in our world. In an imaginary world, in which negations are immediate and perceptible, the law of contradiction does not hold. In this imaginary world another logic is valid, imaginary logic, which accepts a third form of judgment near affirmation and negation, namely the indifferent judgment, which asserts that both P and non-P apply to the same object simultaneously. In this new logic, the law of excluded middle does not hold, but the law of excluded fourth does. After an exposition of the different kinds of judgments (individual, universal, and particular), Vasil’ev shows how it is possible to conduct inferences containing indifferent judgments. The chapter closes with three arguments: the analogy between imaginary logic and non-Euclidean geometry, some alternative interpretations of imaginary logic (e.g., a logic that, distinguishing between absolute and relative negation, accepts degrees of falsehood), and the notion of metalogic, that is, a minimal logic which is shared by both Aristotelian logic and imaginary logic.

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Notes

  1. 1.

    Vasil’ev (1912: 208 = 1989: 55 [2003: 128]).

  2. 2.

    Vasil’ev (1912–1913a: 57, 58 = 1989: 99 [1993: 332, 333]).

  3. 3.

    Aristotle , Metaph. γ 4, 1005b11–12, 17–18.

  4. 4.

    Vasil’ev (1912: 208 = 1989: 54 [2003: 127]).

  5. 5.

    Cf. Vasil’ev (1912: 208–209 = 1989: 55 [2003: 128]).

  6. 6.

    Cf. Heymans (1905 2: 64): “[…] man glaubte, die logischen Gesetze entweder als Idealgesetze, welche unabhängig von allem faktischen Denken für sich gelten, oder als Normalgesetze, welche nur aussagen wie das faktische Denken verlaufen soll, auf keinen Fall aber als Realgesetze, welche den Verlauf des faktischen Denkens beschreiben, ansehen zu müssen. Zur Begründung dieser Ansichten pflegt man sich teils auf den allgemeinen Charakter, teils auf den besonderen Inhalt der logischen im Vergleiche mit den psychologischen Gesetzen zu berufen.”

  7. 7.

    Cf. Heymans (1905 2: 62): “[Die psychischen Prozesse lassen] sich sämtlich auf zwei fundamentale, nicht weiter reduzierbare und keine Ausnahme erleidende psychologische Gesetze zurückführen [...]. Diese Gesetze sind: erstens das Gesetz des Widerspruchs (principium contradictionis), […] zweitens das Gesetz des ausgeschlossenen Dritten (principium exclusi tertii).” “Die Gesetze des Widerspruchs und des ausgeschlossenen Dritten haben wir als die Grundgesetze des Denkens kennen gelernt, in genau demselben Sinne, in welchem etwa die Gesetze der Trägheit und des Kräfteparallelogramms die Grundgesetze der Mechanik sind. Die tatsächlich gegebene Organisation des menschlichen Denkens findet in denselben ihren allgemeinsten und erschöpfenden Ausdruck: wir können eben das menschliche Denken definieren als ein Denken nach den Gesetzen des Widerspruchs und des ausgeschlossenen Drittens; so wie wir die mechanische Bewegung definieren können als eine Bewegung nach den Gesetzen der Trägheit und des Kräfteparallelogramms” (ibid., p. 64).

  8. 8.

    Cf. Heymans (1905 2: 6–7, 70). On the argument see also Picardi (1994: 37–39).

  9. 9.

    Cf. Göring (1874–1875: i, 309–310): “die Logik [hat] Gesetze aufgestellt, welche indirekt die Erkenntniss fördern, indem sie den Irrthum ausschliessen: den Satz der Identität, des Widerspruchs und des ausgeschlossenen Dritten.” See also ibid., p. 314.

  10. 10.

    Cf. Göring (1874–1875: i, 311). About the logical laws, Göring writes: “Man wird jedoch durch Beobachtung des natürlichen Denkens sich bald überzeugen, dass es den Satz der Identität weder kennt noch befolgt, vielmehr sich in Widersprüchen herumtummelt, ohne dadurch zu Zweifeln an der Wahrheit seiner Gedanken veranlasst zu werden. [...] Wir werden demnach den Satz der Identität für ein Normalgesetz der Logik halten müssen” (ibid., p. 310). This holds also for the laws of contradiction and of excluded middle, since “die Befolgung des Satzes der Identität macht natürlich den Satz des Widerspruchs überflüssig, denn beide haben denselben Inhalt, der einmal positiv, das andere Mal negativ ausgedrückt wird,” and “verwandt mit dem Satze des Widerspruchs ist der des ausgeschlossenen Dritten” (ibid., p. 311).

  11. 11.

    Cf. Husserl (1900–1901: i, 62 ff., 149 [2001: i, 47 ff., 97]).

  12. 12.

    Vasil’ev (1912: 210 = 1989: 57 [2003: 130]).

  13. 13.

    Stelzner (2001: 259 fn. 52, 260) notes that Vasil’ev failed to consider other equally important conceptions of the logical laws, such as the formalist and the transcendental conceptions, just as he leaves out normative but non-conventionalist views, which can be traced back, for example, to both Sigwart and Frege .

  14. 14.

    Vasil’ev (1912: 211 = 1989: 58 [2003: 130, 131]).

  15. 15.

    Vasil’ev (1912–1913a: 57 = 1989: 98 [1993: 332]).

  16. 16.

    Cf. Erdmann (1892: 375: ff.; 1907 2: i, 527 ff.), Husserl (1900–1901: i, 136 ff. [2001: i, 90 ff.]).

  17. 17.

    Cf. Vasil’ev (1912: 212, fn. 1 = 1989: 58, fn. 2 [2003: 131, fn. 2]).

  18. 18.

    Vasil’ev (1912–1913a: 54–55 = 1989: 96 [1993: 330]).

  19. 19.

    Cf. Vasil’ev (1912: 245 = 1989: 92 [2003: 161–162]; 1912–1913a: 77 = 1989: 119 [1993: 349]). For a very critical assessment of the thesis claimed here by Vasil’ev see Mikirtumov (2013).

  20. 20.

    Here as later, Vasil’ev employs two different ways for marking the ‘not’: the first is written in Cyrillic characters, the second, in italics, is written with Latin letters.

  21. 21.

    Vasil’ev (1912: 212 = 1989: 59 [2003: 132]). I am responsible for the interpolation of Roman numbers into the text. On the nature of negation in Vasil’ev see Bueno (2017: § 6).

  22. 22.

    Cf. Vasil’ev (1912: 221 = 1989: 67 [2003: 139]): “‘A is not non-A’ is true only because in our world there are predicates that are incompatible with A, and we call these predicates non-A.” Cf. also Vasil’ev (1911/1989: 128; 1912–1913a: 62–63, 68 = 1989: 104, 110 [1993: 336, 341]).

  23. 23.

    Cf. Aristotle, Cat. 12, 14b11–22; Int. 9, 18b37–38; Metaph. Θ 10, 1051b6–9.

  24. 24.

    Vasil’ev (1912: 212 = 1989: 59 [2003: 132]); cf. also Vasil’ev (1912–1913a: 69 = 1989: 111 [1993: 341]).

  25. 25.

    Sigwart (19043: i, § 20, p. 159 [1895: i, 122]); cf. also Vasil’ev (1910: 4 = 1989: 13).

  26. 26.

    Cf. Sigwart (19043: i, § 20, pp. 155 ff. [1895: i, 119 ff.]).

  27. 27.

    Vasil’ev (1912–1913a: 71 = 1989: 113 [1993: 343]).

  28. 28.

    Vasil’ev (1912: 213 = 1989: 59 [2003: 132]).

  29. 29.

    Stelzner (2000: 136; 2001: 268).

  30. 30.

    Vasil’ev (1912: 213–214 = 1989: 60 [2003: 133]).

  31. 31.

    Cf. Vasil’ev (1912: 214–215 = 1989: 61 [2003: 133–134]; 1912–1913a: 125–126 = 1989: 112–113 [1993: 343]). Cf. also Kline (1965: 319–320).

  32. 32.

    Cf. Vasil’ev (1912: 215 = 1989: 61–62, 67 [2003: 134, 139]; 1912–1913a: 69–70 = 1989: 111–112 [1993: 342–343]).

  33. 33.

    Vasil’ev (1912: 215 = 1989: 62 [2003: 135]).

  34. 34.

    Vasil’ev (1912: 216 = 1989: 62 [2003: 135]). The parenthetical additions are my own.

  35. 35.

    Vasil’ev (1912: 217 = 1989: 63 [2003: 136]).

  36. 36.

    Vasil’ev (1912: 217 = 1989: 64 [2003: 136]).

  37. 37.

    Kant (1781 1–17872: A 151 = B 190 [1998: 279).

  38. 38.

    Sigwart (19043: i, § 23, p. 188 [1895: i, 139]).

  39. 39.

    Vasil’ev (1912: 218 = 1989: 64 [2003: 136]); cf. also Vasil’ev (1912–1913a: 64–65 = 1989: 106–107 [1993: 339]).

  40. 40.

    Vasil’ev (1912: 218–219 = 1989: 65–66 [2003: 137–138]).

  41. 41.

    Sigwart (19043: i, § 23, p. 188 [1895: i, 139]).

  42. 42.

    Kant (1781 1–17872: A 151 = B 190 [1998: 279–280]). Cf. also Kant (1800: Ak. ix, 51 [1992: 559]).

  43. 43.

    Cf. Sigwart (19043: i, § 23, pp. 192–194 [1895: i, 142–144]).

  44. 44.

    Sigwart (19043: i, § 23, p. 196 [1895: i, 145]).

  45. 45.

    Cf. Raspa (1999b: 70–71).

  46. 46.

    Cf. Mill (1872 8/1973–1974: ii, vii, § 5, p. 278: “the Principle of Contradiction (that one of the two contradictories must be false) means that an assertion cannot be both true and false.”

  47. 47.

    Cf. Vasil’ev (1910: 39 = 1989: 46).

  48. 48.

    Cf. Vasil’ev (1911/1989: § 6, pp. 127–128; § 9, p. 129).

  49. 49.

    Vasil’ev (1912–1913a: 121 = 1989: 79 [CitationRef CitationID="CR051">1993</CitationRef>: 350]).

  50. 50.

    That Vasil’ev borrows from Sigwart the interpretation of the Kantian formulation like ‘A is not non-A’ is also confirmed by a passage in “Logic and Metalogic,” where, precisely as Sigwart does (19043: i, § 23, p. 192 [1895: i, 142]), Vasil’ev (1912–1913a: 62 = 1989: 104 [CitationRef CitationID="CR051">1993</CitationRef>: 336]) calls (6) the “Kantian-Leibnizian” formulation. Cf. also Vasil’ev (1912: 217, fn. 1 = 1989: 64, fn. 4 [2003: 136, fn. 4]). In the 1989 edition of Vasil’ev’s writings an error occurs: instead of ‘Канто-Лейбницевской’ the editor wrote ‘антилейбницевской,’ which also caused the translators to make an error in their translation (cf. Vasil’ev 2003: 136, fn. 4).

  51. 51.

    Cavaliere (1991: 60) observes that the formulation of the principle of contradiction borrowed from Sigwart “is not adequate for effectively characterizing his [Vasil’ev’s] proposal,” in that “this formula reintroduces exactly the law of non-contradiction of classical logic: not(A and non-A).”

  52. 52.

    Vasil’ev (1912: 218 = 1989: 64 [2003: 137]).

  53. 53.

    Vasil’ev (1912: 219 = 1989: 66 [2003: 138]). Cf. also Vasil’ev (1912–1913a: 65 = 1989: 107 [1993: 338]): “Imaginary logic never contradicts itself, since it is a system devoid of internal contradictions.”

  54. 54.

    But an interesting point of view on this subject is proposed by D’Ottaviano & Gomes (2017: § 3).

  55. 55.

    Vasil’ev (1912: 219–220 = 1989: 66 [2003: 138]).

  56. 56.

    Vasil’ev (1912–1913a: 63 = 1989: 105 [1993: 337]). Vasil’ev writes ‘S is A and is not A simultaneously’; yet since, as I have endeavoured to show above (see Sect. 5.2), negation in imaginary logic is understood to be predicative, the indifferent judgment is to be read as ‘S is A and non-A simultaneously.’ Priest & Routley (1989: 33), Cavaliere (1992–1993: 136), Suchoń (1999: 134) also agree on this.

  57. 57.

    Vasil’ev (1912: 220 = 1989: 67 [2003: 139]).

  58. 58.

    Vasil’ev (1912–1913a: 64 = 1989: 106 [1993: 337]).

  59. 59.

    Vasil’ev (1912: 235 = 1989: 82 [2003: 153]).

  60. 60.

    Vasil’ev (1912–1913a: 64–65 = 1989: 106 [1993: 338]); cf. also Vasil’ev (1912: 223, 235, 243 = 1989: 70, 82, 90–91 [2003: 142, 153, 160]).

  61. 61.

    Through a close analysis of the continuous, which leads him to call his own metaphysical concept “synechism” (cf. Peirce [1898]/1992: 261), and of the related phenomenon of vagueness, Peirce arrives at the hypothesis that there is something intermediate, like “a border line between affirmation and negation” (1905: CP 5.450), that is neither the one nor the other. He proposes two perspectives in order to handle such a state of indeterminacy: the first regards the introduction of a third truth-value, or even more, alongside the true and the false (1976: iii/1, 742–750); the second consists in measuring the truth-grade of a proposition (1976: iii/1, 751–754). On a metric conception of truth-values, Peirce speaks already in 1885. Cf. Peirce (1885: CP 3.365; W 5, 166): “According to ordinary logic, a proposition is either true or false, and no further distinction is recognized. This is the descriptive conception, as the geometers say; the metric conception would be that every proposition is more or less false, and that the question is one of amount.” With reference to this cf. Dipert (1981: 571–572 and fn. 4). I have dealt with these arguments in Raspa (1999b: 292–322; 2008b: 198–210).

  62. 62.

    Cf. Łukasiewicz (1920).

  63. 63.

    Vasil’ev (1912: 221 = 1989: 67 [2003: 139]).

  64. 64.

    Vasil’ev (1912: 221 = 1989: 68 [2003: 140]). As a further confirmation of the fact that Vasil’ev is constantly referring to Sigwart ’s Logik, one may consider that, immediately after this, he states that “the law of identity establishes the logical constancy of concepts” (ibid.). Now, according to Sigwart (1904 3: i, § 23, p. 191 [1895: i, 141]), the so-called “constancy of ideas [Constanz der Vorstellungen],” or “the unambiguity of the act of judgment,” “would form the content of a principle of identity” taken as the “positive rendering [positive Kehrseite]” of the principle of contradiction.

  65. 65.

    Cf. Vasil’ev (1912: 222 = 1989: 69 [2003: 141]): “Every real thought is always manifested in a judgment. Therefore, to think a contradiction actually means to conceive a special kind of judgment of contradiction, viz. an indifferent one, alongside with the affirmative and negative ones.” On the conceivability of the contradiction with a special attention to representation see Raspa ( 2015). On the relation between conceivability and imagination related to Vasil’ev see Bueno (2017: § 4).

  66. 66.

    Vasil’ev (1912–1913a: 63 = 1989: 104 [1993: 336]). Another point distinguishing the approaches of Vasil’ev and Łukasiewicz regards the different meanings each assigns to negation, which itself plays a fundamental part in the way the principle of contradiction is intended. In imaginary logic, the relationship between incompatible predicates and negative facts is the inverse of what obtains in traditional logic. In the latter, there are incompatible predicates but not negative facts; in imaginary logic, there are negative facts but not incompatible predicates. In the former, negation has its foundation in the logical relation of incompatibility; in the latter, in perception, because in the imaginary world, negations, like positive facts, are the objects of sensation and perception, and there are negative facts. Łukasiewicz embraces the traditional conception, according to which a negation is true, in so far as the opposite positive state of affairs does not exist; for Vasil’ev, instead, a negation is true, in so far as the corresponding negative fact subsists. For both the truth of negation entails the falsity of affirmation, but for different reasons.

  67. 67.

    Cf. Vasil’ev (1912: 222–223 = 1989: 69–70 [2003: 141]; 1912–1913a: 59–63 = 1989: 101–104 [1993: 334–336]).

  68. 68.

    Cf. Vasil’ev (1912: 223–224 = 1989: 70–71 [2003: 142–143]).

  69. 69.

    Cf. Suchoń (1999: 132), Schumann (2006: 29, fn. 9).

  70. 70.

    Cf. Vasil’ev (1912: 224–225 = 1989: 71–72 [2003: 143–144]). In a footnote, Vasil’ev (1912: 225, fn. 1 = 1989: 72, fn. 8 [2003:144, fn. 8]) refers us back explicitly to the 1910 article.

  71. 71.

    Cf. V. A. Smirnov (1986: 209–210; 1989a: 633 ff.); but cf. also Markin & Zaitsev (2002) (see Sect. 6.4).

  72. 72.

    Vasil’ev (1912: 225 = 1989: 72 [2003: 144]); cf. also Vasil’ev (1912–1913a: 66 = 1989: 107–108 [1993: 339]).

  73. 73.

    Vasil’ev (1912: 226 = 1989: 73 [2003: 145]).

  74. 74.

    Cf. Vasil’ev (1912: 226–227 = 1989: 73–74 [2003: 145–146]; 1912–1913a: 66 = 1989: 108 [1993: 339]).

  75. 75.

    Where “in” marks the universal indifferent judgment and “inp” the particular indifferent judgment.

  76. 76.

    Cf. Vasil’ev (1912: 228–229 = 1989: 75–76 [2003: 147–148]). In “Logic and Metalogic” Vasil’ev (1912–1913a: 67 = 1989: 109 [1993: 340]) states that “[t]he second figure of syllogism in imaginary logic is impossible.”

  77. 77.

    Vasil’ev (1912: 231 = 1989: 78 [2003: 150]).

  78. 78.

    Cf. Vasil’ev (1912: 229–230 = 1989: 76–77 [2003: 148–149]).

  79. 79.

    Cf. Mangione & Bozzi (1993: 17).

  80. 80.

    Cf. Łukasiewicz (1910a/1987: 8): “only then shall we know if such a principle [of contradiction] is really supreme, the true corner stone of all of our logic, or if it is possible to transform it or even to reject it, in order to create a non-Aristotelian system of logic, just as the system of non-Euclidean geometry was founded by means of the transformation of the axiom of the parallels”; Peirce (quoted in Carus 1910a: 45): “Before I took up the general study of relatives, I made some investigation into the consequences of supposing the laws of logic to be different from what they are. It was a sort of non-Aristotelian logic, in the sense in which we speak of non-Euclidean geometry.” See also Sect. 2.3, 4.3.

  81. 81.

    Vasil’ev (1912: 208 = 1989: 54 [2003: 128]).

  82. 82.

    Cf. Lobachevsky (1835).

  83. 83.

    Cf. Vasil’ev (1912: 232–233 = 1989: 79–80 [2003: 151]).

  84. 84.

    Lobachevsky (1835–1838/1883–1886: i, 301).

  85. 85.

    Vasil’ev (1912: 233 = 1989: 81 [2003: 152]).

  86. 86.

    Cf. Riemann (1867).

  87. 87.

    Vasil’ev (1912: 234 = 1989: 81 [2003: 152]).

  88. 88.

    Vasil’ev (1912: 235 = 1989: 82 [2003: 153]).

  89. 89.

    Vasil’ev (1912: 238 = 1989: 85 [2003: 156]).

  90. 90.

    Cf. Bolzano (1837: i, 415–417 [2014: i, 298–299]).

  91. 91.

    Bolzano (1837: i, 419 [2014: i, 301]).

  92. 92.

    Vasil’ev (1912: 241 = 1989: 88 [2003: 158]).

  93. 93.

    Stelzner (2014) proposes another type of application of imaginary logic to our real world. It is to consider such logic from an epistemic point of view. He proposes, then, to interpret imaginary worlds as epistemic worlds. According to Stelzner (2014: 57), “[i]n our real world, epistemic situations (or epistemic worlds) are given, whereas epistemic subjects have contradictory epistemic attitudes or perform contradictory epistemic or linguistic acts. […] The existence of epistemic contradictions is entirely compatible with the soundness of the law of contradiction in our world.” Therefore, Stelzner assumes as his starting point one of the results of Łukasiewicz’s investigations concerning the principle of contradiction (see Sect. 4.3). Cf. also Stelzner (2017).

  94. 94.

    Vasil’ev (1911/1989: 130 [184]).

  95. 95.

    Cf. Bolyai (1832 [1987]). D’Ottaviano & Gomes (2015: 274) suggest that Vasil’ev anticipates “aspects of what is today known as universal logic.” Cf. also D’Ottaviano & Gomes (2017: § 4).

  96. 96.

    Cf. Vasil’ev (1912: 242–243 = 1989: 89–90 [2003: 159–160]; 1912–1913a: 72–74, 77 = 1989: 115–116, 119 [1993: 344–345, 348]).

  97. 97.

    Except in a context where the notion of metalogic has not yet been introduced (cf. Vasil’ev 1912: 221 = 1989: 68 [2003: 140]).

  98. 98.

    Cf. Vasil’ev (1912–1913a: 74–77 = 1989: 116–119 [1993: 346–349]).

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Raspa, V. (2017). Imaginary Logic. In: Thinking about Contradictions. Synthese Library, vol 386. Springer, Cham. https://doi.org/10.1007/978-3-319-66086-8_5

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