A Pragmatic-Semiotic Defence of Bivalence

Abstract

Since Peirce defined the first operators for three-valued logic, it is usually assumed that he rejected the principle of bivalence. However, I argue that, because bivalence is a principle, the strategy used by Peirce to defend logical principles can be used to defend bivalence. Construing logic as the study of substitutions of equivalent representations, Peirce showed that some patterns of substitution get realized in the very act of questioning them. While I recognize that we can devise non-classical notations, I argue that, when we make claims about those notations, we inevitably get saddled with bivalent commitments. I present several simple inferences to show this. The argument that results from those examples is ‘pragmatic’, because the inevitability of the principle is revealed in use (not mention); and it is ‘semiotic’, because this revelation happens in the use of signs.

Acknowledgements

I want to thank Emily Beattie, Francesco Bellucci, Jeffrey Brian Downard, David Gilbert, Diana Heney, Travis LaCroix, Robert Lane, Ahti-Veikko Pietarinen, Chris Stephens, audience members at the meetings of the Charles S. Peirce Society and the Canadian Philosophical Association, students in my Advanced Formal Logic course, as well as anonymous reviewers for this journal.

Notes

1 Brady 2000.

2 Peirce 2019; Peirce 2020; and more volumes forthcoming.

3 Church 1956, 25fn67; Peirce 1931-58, vol. 3, para. 365, hereafter cited as CP 3.365.

4 Anellis 2012.

5 Fisch and Turquette 1966, 73–5.

6 Lane 1998, 1–2.

7 Quine 1986, 84.

8 Priest 1995b, 5.

9 Priest 1995b.

10 Van Benthem et al. 2006.

11 Haack 1996.

12 Suszko 1975.

13 Priest 2007.

14 CP 5.250.

15 CP 5.253; see Champagne 2009.

16 Bellucci 2017, 16.

17 Bellucci 2017.

18 Pietarinen and Bellucci 2016.

19 Bellucci 2017, 42.

20 Hausman 2002.

21 Champagne and Pietarinen 2020.

22 Bellucci 2016, 255; Bellucci 2017, 25.

23 CP 2.623.

24 Margolis 1998.

25 Like Peirce, I will express rules as conditionals and cases as antecedents of such conditionals.

26 CP 2.466.

27 Bellucci 2017, 26.

28 Carroll 1895, 279; see Zhang 2017, 2.

29 Dutilh Novaes 2015.

30 Champagne 2016.

31 Legg 2012, 13; emphasis in original.

32 Peirce 1976, 4, 175; emphasis in original.

33 Peirce 1976, 4, 175–6; emphasis in original.

34 Peirce 1976, 4, 47–8.

35 Pietarinen 2006, 77–102; Hintikka 2007.

36 Champagne 2015c.

37 Hull and Atkins 2017; Legg and Franklin 2017.

38 Hitchcock 2003, 70–1.

39 Braver 2012, 169.

40 Braver 2012.

41 Peirce 1976, 4, 48.

42 Thomson 2011, 24.

43 Cellucci 2009.

44 Gentzen 1969, 78.

45 Dennett 1991.

46 Benacerraf 1973.

47 CP 5.253.

48 Sayward 1989.

49 CP 5.448; emphasis in original.

50 Priest 1995a.

51 Lane 1998, 51.

52 CP 2.647.

53 Stjernfelt 2014.

54 CP 5.447.

55 Rumfitt 2015, 3.

56 CP 1.172.

57 Vargas and Moore 2021.

58 Peirce 1976, 3, 851; emphasis in original. Quoted in Lane 1999, 290.

59 Lane 1998, 208.

60 Fisch and Turquette 1966, 79.

61 Roberts 1973, 64–86.

62 Ma and Pietarinen 2018, 3642.

63 Peirce 2020, 448; emphasis added.

64 Hintikka 1962.

65 Pietarinen 2019.

66 Peirce 2020, 450.

67 Dummett 1959, 149–50. Note that semantic paradoxes do not have to pose the level of worry often attributed to them. See, for instance, the Peirce-inspired argument developed by Prior 1976, 141.

68 Legg 2014.

69 Raatikainen 2004, 133.

70 Howat 2013, 453–5.

71 Misak 2004, 157; Lane 2018, 181.

72 CP 5.265.

73 Everett forthcoming.

74 Pietarinen 2020, 135–6; emphasis in original.

75 CP 5.265.

76 Dummett 1978, 291–2; Haack 1976.

77 Cf. the modus morons of Haack 1996, 186–7.

78 CP 2.204.

79 CP 1.623.

80 Howat 2014, 482; see Champagne 2015a.

81 Brandom 1994.

82 Bellucci 2017, 43.

83 Manuscript R339; reproduced in Fisch and Turquette 1966, 75.

84 CP 5.402.

85 Béziau 2012, 251.

86 Manuscript R637, 30; quoted in Bellucci 2017, 354.

87 Bellucci 2017, 354.

88 CP 5.253.

89 CP 5.397; see Champagne 2014.

90 Champagne 2015b.

91 Short 2007, 288.

92 Thomson 2011, 23.

93 Howat 2014.