The (Non-)classicality of (Non-)classical Mathematics
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
I would like to thank Michèle Friend and Mihir K. Chakraborty for putting this special issue together, and for their help to produce a better paper than the originally submitted. Special thanks are deserved to an anonymous referee for their comments, corrections and suggestions, as well as to Charlie Donahue and the audience at the IV Conference of the Latin American Association for Analytic Philosophy for discussion, and to Elisángela Ramírez-Cámara for discussion and her invaluable help in producing a readable English version. This work was written under the support of the PAPIIT projects IA401015 “Tras las consecuencias. Una visión universalista de la lógica (I)” and IA401117 “Aspectos filosóficos de las lógicas contraclásicas”.
- 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