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ANTIREALISM AND CONSTRUCTIVISM: BROUWER’S WEAK COUNTEREXAMPLES

Published online by Cambridge University Press:  03 October 2012

CHARLES McCARTY
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Abstract

Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to Test, the recognition condition, and the BHK interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the BHK, then every relevant instance of the classical principle Test is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is seen, in this instance, to fail.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 1 , March 2013 , pp. 147 - 159
Copyright
Copyright © Association for Symbolic Logic 2012

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References

BIBLIOGRAPHY

Brouwer, L. E. J. (1908). De onbetrouwbaarheid der logische principes. Tijdschrift voor Wijsbegeerte, 2, 152–158. English translation in Brouwer (1975, pp. 107–111).Google Scholar
Brouwer, L. E. J. (1975). Collected Works, Vol. 1. Amsterdam: North-Holland Publishing Co.Google Scholar
Dummett, M. A. E. (1975a). The justification of deduction. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press 1978, pp. 290–318. Originally published in The Proceedings of the British Academy, Vol. 59. pp. 201–231.Google Scholar
Dummett, M. A. E. (1975b). The philosophical basis of intuitionistic logic. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press 1978, pp. 215–247.Google Scholar
Dummett, M. A. E. (1977). Elements of Intuitionism. Oxford, UK: Clarendon Press. Second Edition: 2000.Google Scholar
Goodman, N. D. (1968). Intuitionistic arithmetic as a theory of constructions. PhD Dissertation. Palo Alto, CA: Stanford University, p. v+111.Google Scholar
Heyting, A. (1930). Die formalen Regeln der intuitionistischen Logik I, II, III. Sitzungsberichte der Preussischen Akademie von Wissenschaften, Physikalisch-mathematische Klasse, 9, pp. 42–56, 57–71, 158–169.Google Scholar
Koss, M. (2009). Re-examining Brouwer’s earliest challenge to classical logic. Paper presented to the Fall 2009 Meeting of the Indiana Philosophical Association. 6 December 2009, p. 24.Google Scholar
Kreisel, G. (1962). Foundations of intuitionistic logic. In Nagel, E., Suppes, P., and Tarski, A., editors. Logic, Methodology, and Philosophy of Science I. Palo Alto, CA: Stanford University Press, pp. 198–210.Google Scholar
McCarty, C. (2008). The new intuitionism. In van Atten, M., Boldini, P., Bourdeau, M., and Heinzmann, G., editors. One Hundred Years of Intuitionism (1907–2007). Boston, MA: Birkhuser, pp. 37–49.CrossRefGoogle Scholar
McCarty, C. (2011). Proofs and constructions. In Sommaruga, G., editor. Foundational Theories of Classical and Constructive Mathematics. Berlin: Springer-Verlag, pp. 209–225.Google Scholar
Scott, D. S. (1975). Data types as lattices. In Müller, G. H., editor. ISILC Logic Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel, 1974, Vol. 499 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, pp. 579–651.Google Scholar
Troelstra, A. S., & van Dalen, D. (1988). Constructivism in Mathematics, Vol. 1. Amsterdam: North-Holland, p. xx+342+XIV.Google Scholar