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ANALYTICITY AND SYNTHETICITY IN TYPE THEORY REVISITED

Published online by Cambridge University Press:  27 June 2023

BRUNO BENTZEN*
Affiliation:
SCHOOL OF PHILOSOPHY ZHEJIANG UNIVERSITY HANGZHOU, CHINA

Abstract

I discuss problems with Martin-Löf’s distinction between analytic and synthetic judgments in constructive type theory and propose a revision of his views. I maintain that a judgment is analytic when its correctness follows exclusively from the evaluation of the expressions occurring in it. I argue that Martin-Löf’s claim that all judgments of the forms $a : A$ and $a = b : A$ are analytic is unfounded. As I shall show, when A evaluates to a dependent function type $(x : B) \to C$, all judgments of these forms fail to be analytic and therefore end up as synthetic. Going beyond the scope of Martin-Löf’s original distinction, I also argue that all hypothetical judgments are synthetic and show how the analytic–synthetic distinction reworked here is capable of accommodating judgments of the forms $A \> \mathsf {type}$ and $A = B \> \mathsf {type}$ as well. Finally, I consider and reject an alternative account of analyticity as decidability and assess Martin-Löf’s position on the analytic grounding of synthetic judgments.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Allen, S. (1987). A Non-Type-Theoretic Definition of Martin-Löf’s Types. Ph.D. Thesis, Cornell University.Google Scholar
Angiuli, C., Harper, R., & Wilson, T. (2017). Computational higher-dimensional type theory. POPL ’17: Proceedings of the 44th Annual ACM SIGPLAN Symposium on Principles of Programming Languages. New York: Association for Computing Machinery, pp. 680693. Available from: https://www.cs.cmu.edu/cangiuli/papers/chtt.pdf.Google Scholar
Bentzen, B. (2020). Sense, reference, and computation. Perspectiva Filosofica, 47(2), 179203. Available from: http://philsci-archive.pitt.edu/17408/.Google Scholar
Bentzen, B. (2022). On different ways of being equal. Erkenntnis, 87(4), 18091830. https://doi.org/10.1007/s10670-020-00275-8 CrossRefGoogle Scholar
Bentzen, B. (2023a). Propositions as intentions. Husserl Studies, 39, 143160. https://doi.org/10.1007/s10743-022-09323-3 CrossRefGoogle Scholar
Bentzen, B. (2023b). Brouwer’s intuition of twoity and constructions in separable mathematics. History and Philosophy of Logic, pp. 121. Published online 5 June 2023. https://doi.org/10.1080/01445340.2023.2210908 CrossRefGoogle Scholar
Brouwer, L. E. J. (1905–1907). Student notebooks. Digital transcriptions by John Kuiper. Available from: http://www.cs.ru.nl/F.Wiedijk/brouwer/transcriptie.pdf.Google Scholar
Brouwer, L. E. J. (1907). Over de Grondslagen der Wiskunde. Ph.D. Thesis, Universiteit van Amsterdam. English translation in Brouwer in 1975. Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North-Holland, pp. 13–101.Google Scholar
Constable, R. L., Allen, S. F., Bromley, H. M., Cleaveland, W. R., Cremer, J. F., Harper, R. W., Howe, D. J., Knoblock, T. B., Mendler, N. P., Panangaden, P., Sasaki, J. T. & Smith, S. F. (1985). Implementing Mathematics with the Nuprl Proof Development System. Englewood Cliffs: Prentice-Hall. Available from: https://www.nuprl.org/book/.Google Scholar
Dummett, M. (2021). Sense and reference from a constructivist standpoint. The Bulletin of Symbolic Logic, 27(4), 485500. Transcription of a talk given at Leiden University on September 26, 1992. https://doi.org/10.1017/bsl.2021.60 CrossRefGoogle Scholar
Dummett, M. A. (1978). Frege’s distinction between sense and reference. In Truth and Other Enigmas. Cambridge: Harvard University Press, pp. 116144. Original work published in 1975.Google Scholar
Dybjer, P. (2012). Program testing and the meaning explanations of intuitionistic type theory. In Dybjer, P., Lindström, S., Palmgren, E., and Sundholm, G., editors. Epistemology Versus Ontology. Dordrecht: Springer, pp. 215241. https://doi.org/10.1007/978-94-007-4435-6_11 CrossRefGoogle Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl. w. Koebner.Google Scholar
Harper, R. (2013). Constructive mathematics is not metamathematics. Available from: https://existentialtype.wordpress.com/2013/07/10/constructive-mathematics-is-not-meta-mathematics/ [Online] (accessed 12 August 2022).Google Scholar
Howard, W. A. (1980). The formulae-as-types notion of construction. In Seldin, J. P. and Hindley, J. R., editors. To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism. London: Academic Press, pp. 479490.Google Scholar
Husserl, E. (1970). Logical Investigations (I/II). London: Routledge and Kegan Paul. Translated by J. N. Findlay. Vol. I. First published in 1900, vol. II in 1901.Google Scholar
Kant, I. (1781). Kritik der reinen Vernunft. Revised Second Edition published by Johann Friedrich Hartknoch in 1787. English translation in Paul Guyer and Allen Wood in 2010. The Cambridge Companion to Kant’s Critique of Pure Reason. Cambridge: Cambridge University Press.Google Scholar
Klev, A. (2019). Eta-rules in Martin-Löf type theory. The Bulletin of Symbolic Logic, 25(3), 333359. https://doi.org/10.1017/bsl.2019.21 CrossRefGoogle Scholar
Kreisel, G. (1962). Foundations of intuitionistic logic. In Logic, Methodology, and the Philosophy of Science. Stanford: Stanford University Press, pp. 198210.Google Scholar
Martin-Löf, P. (1975). An intuitionistic theory of types: Predicative part. In Rose, H. E. and Shepherdson, J. C., editors. Logic Colloquium ’73: Proceedings of the Logic Colloquium, Bristol. Amsterdam: North-Holland, pp. 73118. https://doi.org/10.1016/S0049-237X(08)71945-1 CrossRefGoogle Scholar
Martin-Löf, P. (1982). Constructive mathematics and computer programming. In Logic, Methodology and Philosophy of Science, VI (Hannover, 1979). Studies in Logic and the Foundations of Mathematics, Vol. 104. Amsterdam: North-Holland, pp. 153175. Available from: https://doi.org/10.1016/S0049-237X(09)70189-2 Google Scholar
Martin-Löf, P. (1985). On the meanings of the logical constants and the justifications of the logical laws. In Atti degli incontri di logica matematica. Scuola di Specializzazione in Logica Matematica, Vol. 2. Siena, Italy: Universitá di Siena, pp. 203281. Available from: https://ncatlab.org/nlab/files/MartinLofOnTheMeaning96.pdf. Reprinted in Nordic Journal of Philosophical Logic, 1(1), 11–60.Google Scholar
Martin-Löf, P. (1987). Truth of a proposition, evidence of a judgement, validity of a proof. Synthese, 73(3), 407420. https://doi.org/10.1007/BF00484985 CrossRefGoogle Scholar
Martin-Löf, P. (1993). Philosophical aspects of intuitionistic type theory. Transcriptions of lectures given at Leiden University from 23 September to 16 December. Available from: https://pml.flu.cas.cz/uploads/PML-LeidenLectures93.pdf.Google Scholar
Martin-Löf, P. (1994). Analytic and synthetic judgments in type theory. In Parrini, P., editor. Kant and Contemporary Epistemology. The University of Western Ontario Series in Philosophy of Science, Vol. 54. Dordrecht: Kluwer Academic Publishers, pp. 8799.CrossRefGoogle Scholar
Martin-Löf, P. (2014). Truth of empirical propositions. Transcriptions of lectures given at Leiden University on February 4, 2014. Available from: https://pml.flu.cas.cz/uploads/PML-Leiden04Feb14.pdf.Google Scholar
Martin-Löf, P. (2021). The sense/reference distinction in constructive semantics. The Bulletin of Symbolic Logic, 27(4), 501513. Transcription of a talk given at Leiden University on 25 August, 2001. https://doi.org/10.1017/bsl.2021.61 CrossRefGoogle Scholar
Sundholm, G. (1994). Ontologic versus epistemologic: Some strands in the development of logic, 1837–1957. In Logic and Philosophy of Science in Uppsala: Papers from the 9th International Congress of Logic, Methodology and Philosophy of Science. Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 373384. Available from: https://doi.org/10.1007/978-94-015-8311-4_25 CrossRefGoogle Scholar
Sundholm, G. (2004). Antirealism and the roles of truth. In Niiniluoto, I., Sintonen, M., & Woleński, J., editors. Handbook of Epistemology. Dordrecht: Kluwer Academic Publishers, pp. 437466. https://doi.org/10.1007/978-1-4020-1986-9_11 CrossRefGoogle Scholar
Sundholm, G. (2013). Containment and variation; two strands in the development of analyticity from Aristotle to Martin-Löf. In van der Schaar, M., editor. Judgement and the Epistemic Foundation of Logic. Dordrecht: Springer, pp. 2335. https://doi.org/10.1007/978-94-007-5137-8_3 CrossRefGoogle Scholar
Sundholm, G. (1990). Sätze der logik: An alternative conception. In Rudolf, H. and Brandl, J., editors. Wittgenstein—Towards a Re-Evaluation, Proceedings of the 14th International Wittgenstein Symposium. Kirchberg Am Wechsel, 13–20 August 1989. Vienna: Verlag Hölder-Pichler-Tempsky, pp. 5961.Google Scholar
Troelstra, A. S., & van Dalen, D. (1988). Constructivism in mathematics. Vol. I. Studies in Logic and the Foundations of Mathematics, Vol. 121. Amsterdam: North-Holland.Google Scholar
Wang, H. (1990). Reflections on Kurt Gödel. Cambridge: MIT Press.Google Scholar