Proofs, what they prove and their representations: remarks in connection with identity of proofs and the normalisation thesis
Article Sidebar
Main Article Content
Abstract
A topic of de Castro Alves (2019) stands in need of re-visitation, namely: possible ways of specifying restrictions on the notion of proof and some other related ones that are relevant in connection with the discussion on identity of proofs. This effort is dedicated to start a compensation for relevant shortcomings of some ideas proposed in that work. More concretely: by taking some very generic traits of proofs as a departure point, we proceed to the identification of possible outset conditions upon the investigation of identity of proofs (instead of proposing a taxonomy of criteria of identity of proofs, as in de Castro Alves (2019)). We will describe and briefly comment on two kinds of such conditions: one given in terms of how the identity of proofs is conditioned by the identity of what is proved, and other in terms of how equivalence relations between proof (re)presentations are conditioned by, on the one hand, how many distinct (collections/kinds of) proofs can be (re)presented by them, and, on the other, how many distinct (re)presentations a collection of proofs may have. To exemplify the meaningfulness of these considerations, they will be used here as basis for some critical remarks on the normalisation thesis on identity of proofs.
Article Details
Copyright Notice
The author of the article or book reviews submitted and approved for publication authorizes the editors to reproduce it and publish it in the journal O que nos faz pensar, with the terms “reproduction” and “publication” being understood in accordance with the definitions of the Creative Commons Attribution-NonCommercial 4.0 International license. The article or book reviews may be accessed both via the World Wide Web – Internet (WWW – Internet), and in printed form, its being permitted, free of charge, to consult and reproduce the text for the personal use of whoever consults it. This authorization of publication has no time limit, with the editors of the journal O que nos faz pensar being responsible for maintaining the identification of the author of the article.
References
DE CASTRO ALVES, T. A questão da identidade de provas, a tese da normalização e alguns de seus pressupostos semânticos básicos. Perspectiva Filosófica 47.2, pp. 425-452, 2020.
DOŠEN, K. Identity of proofs based on normalization and generality, The Bulletin of Symbolic Logic, vol. 9, pp. 477-50, 2003.
DOŠEN, K. and PETRIĆ, Z. The maximality of the typed lambda calculus and of cartesian closed categories, Publ. Inst. Math. (N.S.) 68(82), pp. 1-19, 2000.
DOŠEN, K. and PETRIĆ, Z. The maximality of cartesian categories, Math. Logic Quart. 47, pp. 137-144, 2001.
KREISEL, G. A survey of proof theory II, in: J.E. Fenstad ed., Proceedings of the Second Scandinavian Logic Symposium. Amsterdam: North-Holland,1971, pp. 109-170.
KREISEL, G. Matehmatical Logic, in: T.L. Saaty ed., Lectures on Modern Mathematics, v. 3. J. Wiley & Sons, 1965, pp. 95–195.
NOVAES, C.D. The Dialogical Roots of Deduction - Historical, Cognitive, and Philosophical
Perspectives on Reasoning. Cambridge: Cambridge University Press, 2020.
PRAWITZ, D. Natural Deduction. A Proof-Theoretical Study. Stockholm: Almqvist & Wiksell, 1965.
PRAWITZ, D. Ideas and results in proof theory, in: J.E. Fenstad ed., Proceedings of the Second Scandinavian Logic Symposium. Amsterdam: North-Holland,, 1971, pp. 235-307.
WIDEBÄCK, F. Identity of Proofs. Doctoral dissertation, University of Stockholm. Stockholm: Almqvist & Wiksell, 2001.
WITTGENSTEIN, L. The Blue and Brown Books. Oxford: Basil Blackwell, 1958.
WITTGENSTEIN, L. Philosophische Untersuchungen. Kritisch-genetische Edition. Herausgegeben von Joachim Schulte. Frankfurt: Wissenschaftliche Buchgesellschaft, 2001.