Adaptive Fregean Set Theory
This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.
KeywordsFregean set theories Adaptive logics Content guidance Paraconsistency
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Part of the ideas and results were presented in conferences. My gratitude goes (i) to the audience of the 5th World Congress on Universal Logic (Istanbul, Turkey, 20–30 June 2015) for questions and suggestions, especially to Graham Priest with whom I also corresponded on the matter afterwards; (ii) to the attentive and helpful audience of an \(\exists \)ntia et Nomin\(\forall \) Workshop (Krakow, Poland, 9–11 September 2015); and (iii) to both referees for pointing out ways to improve the paper.
- 1.Batens, D., A survey of inconsistency-adaptive logics, in D. Batens, C. Mortensen, G. Priest, and J. Van Bendegem, (eds.), Frontiers of Paraconsistent Logic, Research Studies Press, 2000, pp. 49–73.Google Scholar
- 2.Batens, D., A general characterization of adaptive logics, Logique et Analyse 173–175: 45–68, 2001.Google Scholar
- 4.Batens, D., Towards a dialogic interpretation of dynamic proofs, in C. Dégremont, L. Keiff, and H. Rückert, (eds.), Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman, College Publications, London, 2009, pp. 27–51.Google Scholar
- 5.Batens, D., The consistency of Peano Arithmetic. A defeasible perspective, in P. Allo, and B. Van Kerkhove, (eds.), Modestly Radical or Radically Modest. Festschrift for Jean Paul van Bendegem on the Occasion of His 60th Birthday, College Publications, London, 2014, pp. 11–59.Google Scholar
- 6.Batens, D., Tutorial on inconsistency-adaptive logics, in J. Béziau, M. Chakraborty, and S. Dutta, (eds.), New Directions in Paraconsistent Logic, Springer, 2015, pp. 3–38.Google Scholar
- 8.Batens, D., and K. De Clercq, A rich paraconsistent extension of full positive logic, Logique et Analyse 185–188: 227–257, 2004.Google Scholar
- 10.Brady, R., Universal Logic, CSLI Publications, 2006.Google Scholar
- 12.Odintsov, S. P., and S. 0. Speranski, On algorithmic properties of propositional inconsistency-adaptive logics, Logic and Logical Philosophy 21: 209–228, 2012.Google Scholar
- 16.Priest, G., Inconsistent models of arithmetic. Part I: Finite models, Journal of Philosophical Logic 26: 223–235, 1997.Google Scholar
- 17.Priest, G., In Contradiction. A Study of the Transconsistent, Oxford University Press, 2:2006.Google Scholar
- 19.Shapere, D., Logic and the philosophical interpretation of science, in P. Weingartner, (ed.), Alternative Logics. Do sciences need them?, Springer, 2004, pp. 41–54.Google Scholar
- 20.Straßer, C., Adaptive Logic and Defeasible Reasoning. Applications in Argumentation, Normative Reasoning and Default Reasoning, Springer, 2014.Google Scholar
- 21.Van Bendegem, J., Strict, yet rich finitism, in Z. W. Wolkowski, (ed.), First International Symposium on Gödel’s Theorems, World Scientific, 1993, pp. 61–79.Google Scholar
- 22.Van Bendegem, J., Strict finitism as a viable alternative in the foundations of mathematics, Logique et Analyse 145: 23–40, 1994.Google Scholar