The (Non-)classicality of (Non-)classical Mathematics
- First Online:
Graham Priest has recently argued that the distinctive trait of classical mathematics is that the conditional of its underlying logic—that is, classical logic—is extensional. In this article, I aim to present an alternate explanation of the specificity of classical mathematics (and actually for L-mathematics, for a significative amount of instances of 'L').
I examine Priest's argument for his claim and show its shortcomings. Then I deploy a model-theoretic presentation of logics that allows comparing them, and the mathematics based on them, more fine-grainedly.
Such a model-theoretic presentation of logics suggests that the specific character of classical logic consists in the structure that it confers to its truth values and in the structure of the evaluation indices of its formulas, and that this trait is useful to explain the specific character of the logics and the mathematics based on them.
The extensionality of the conditional in classical logic is a by-product of other structural features of a logic, which are more likely to be what gives a kind of mathematics based on it its specific character.
KeywordsConditional Extensionality Classical mathematics Non-classical mathematics LP K3 Satisfiability conditions
- Beall, J. C. (2010). On truth, abnormal worlds, and necessity. In M. Peliš & V. Punčochář (Eds.), Logica Yearbook 2009 (pp. 17–33). London: College Publications.Google Scholar
- Bell, J. L. (1998). A primer of infinitesimal analysis. Cambridge: Cambridge University Press.Google Scholar
- Meyer, R. (1976). Relevant arithmetic. Bulletin of the Section of Logic, 5(4), 133–135.Google Scholar
- Priest, G. (2008). An introduction to non-classical logic. From if to is (2nd ed). Cambridge: Cambridge University Press.Google Scholar
- Priest, G. (forthcoming). What is the specificity of classical mathematics?Google Scholar