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
3 Church Citation1956, 25fn67; Peirce 1931-58, vol. 3, para. 365, hereafter cited as CP 3.365.
25 Like Peirce, I will express rules as conditionals and cases as antecedents of such conditionals.
49 CP 5.448; emphasis in original.
67 Dummett Citation1959, 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 Citation1976, 141.
83 Manuscript R339; reproduced in Fisch and Turquette Citation1966, 75.
86 Manuscript R637, 30; quoted in Bellucci Citation2017, 354.
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