Dean, Walter and Kurokawa, Hidenori (2015) Kreisel's theory of constructions, the Kreisel-Goodman paradox, and the second clause. In: Piecha, Thomas and Schroeder-Heister, Peter, (eds.) Advances in Proof-Theoretic Semantics. Trends in Logic, 43 . Springer International Publishing, pp. 27-63. ISBN 9783319226859

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Abstract

The goal of this paper is to consider the prospects for developing a consistent variant of the Theory of Constructions originally proposed by Georg Kreisel and Nicolas Goodman in light of two developments which have been traditionally associated with the theory—i.e. Kreisel’s second clause interpretation of the intuitionistic connectives, and an antinomy about constructive provability sometimes referred to as the Kreisel-Goodman paradox. After discussing the formulation of the theory itself, we then discuss how it can be used to formalize the BHK interpretation in light of concerns about the impredicativity of intuitionistic implication and Kreisel’s proposed amendments to overcome this. We next reconstruct Goodman’s presentation of a paradox pertaining to a “naive” variant of the theory and discuss the influence this had on its subsequent reception. We conclude by considering various means of responding to this result. Contrary to the received view that the second clause interpretation itself contributes to the paradox, we argue that the inconsistency arises in virtue of an interaction between reflection and internalization principles similar to those employed in Artemov’s Logic of Proofs.

Item Type:	Book Item
Subjects:	B Philosophy. Psychology. Religion > BC Logic
Divisions:	Faculty of Social Sciences > Philosophy
Library of Congress Subject Headings (LCSH):	Paradox
Series Name:	Trends in Logic
Publisher:	Springer International Publishing
ISBN:	9783319226859
Book Title:	Advances in Proof-Theoretic Semantics
Editor:	Piecha, Thomas and Schroeder-Heister, Peter
Official Date:	2015
Dates:	Date Event 2015 Published
Volume:	43
Page Range:	pp. 27-63
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	28 July 2016
Date of first compliant Open Access:	28 July 2016
Open Access Version:	Publisher

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