Abstract
We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, in the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning.
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Frege, G. (1893). Grundgesetze der Arithmetik. I. Band. reprinted 1962. Hildesheim: Georg Olms Verlagsbuchhandlung.
Frege, G. (1903). Grundgesetze der Arithmetik. II. Band. reprinted 1962. Hildesheim: Georg Olms Verlagsbuchhandlung.
Frege, G. (2013). Basic Laws of Arithmetic. Derived using concept-script. Volumes I & II. Oxford: Oxford University Press.
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Stockholm: Almqvist & Wiksell.
Schroeder-Heister, P. (2016). Open Problems in Proof-Theoretic Semantics. In: Advances in Proof-Theoretic Semantics. Ed. by T. Piecha and P. Schroeder-Heister. Berlin: Springer, 253–83.
Tennant, N. (1978). Natural Logic. Edinburgh University Press.
Tennant, N. (2010). Inferential Semantics. In: The Force of Argument: Essays in Honor of Timothy Smiley. Ed. by J. Lear and A. Oliver. Routledge: London, 223–57.
Tennant, N. (2017). Core Logic. Oxford: Oxford University Press.
Tennant, N. (2018). A logical theory of truthmakers and falsitymakers. In: Handbook on Truth. Ed. by M. Glanzberg. Oxford: Oxford University Press, 355–393.
Tennant, N. (2020). Does Choice Really Imply Excluded Middle? Part I: Regimentation of the Goodman–Myhill result, and its immediate reception. Philosophia Mathematica 28, 139–171.
Tennant, N. (2021). Does Choice Really Imply Excluded Middle? Part II: Historical, philosophical and foundational reflections on the Goodman–Myhill result. Philosophia Mathematica 29.
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65, 261–81.
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Tennant, N. (2024). Frege’s Class Theory and the Logic of Sets. In: Piecha, T., Wehmeier, K.F. (eds) Peter Schroeder-Heister on Proof-Theoretic Semantics. Outstanding Contributions to Logic, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-031-50981-0_3
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