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Brief Reminder of Constructive Type Theory

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Book cover Linking Game-Theoretical Approaches with Constructive Type Theory

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Abstract

Within Per Martin-Löf’s Constructive Type Theory (CTT for short) the logical constants are interpreted through the Curry-Howard correspondence between propositions and sets. A proposition is interpreted as a set whose elements represent the proofs of the proposition. It is also possible to view a set as a problem description in a way similar to Kolmogorov’s explanation of the intuitionistic propositional calculus.

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Notes

  1. 1.

    Cf. Nordström et al. (1990) and Granström (2011).

  2. 2.

    For a thorough discussion on this issue, see Sundholm (1997, 2001).

  3. 3.

    For a thorough discussion see Granström (2011, pp. 54–76).

  4. 4.

    Martin-Löf used the sign “\(\in \)” in order to indicate that something, say a, is of some type, say B. He even suggests to understand it as a the copula ‘is’. Nordström et al. (1990) also uses this notation while other authors such as Ranta (1994) use the colon. Granström (2011) distinguishes the colon from the epsilon, where the first applies to non-canonical elements and the latter to canonical ones. We will use the colon.

  5. 5.

    Cf. Granström (2011, pp. 111–112).

  6. 6.

    Cf. Ranta (1991, Sect. 3).

References

  • Granström, J. 2011. Treatise on Intuitionistic Type Theory. Dordrecht: Springer.

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  • Martin-Löf, P. 1984. Intuitionistic Type Theory. Naples: Bibliopolis. Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980.

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  • Nordström, B., K. Petersson, and J.M. Smith. 1990. Programming in Martin-Löf’s Type Theory: An Introduction. Oxford: Oxford University Press.

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  • Ranta, A. 1991. Constructing possible worlds. Theoria 57(1–2): 77–99.

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  • Ranta, A. 1994. Type-Theoretical Grammar. Oxford: Clarendon Press.

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  • Sundholm, G. 1997. Implicit epistemic aspects of constructive logic. Journal of Logic, Language and Information 6(2): 191–212.

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  • Sundholm, G. 2001. A plea for logical atavism. In The Logica Yearbook 2000, ed. O. Majer, 151–162. Prague: Filosofia.

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Authors and Affiliations

  1. Instituto de Filosofía; CDHACS, Universidad de Valparaíso, Valparaíso, Chile

    Nicolas Clerbout

  2. UMR-CNRS 8163: STL, University of Lille III, Villeneuve d’Ascq, France

    Shahid Rahman

Authors
  1. Nicolas Clerbout
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  2. Shahid Rahman
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Correspondence to Nicolas Clerbout .

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Clerbout, N., Rahman, S. (2015). Brief Reminder of Constructive Type Theory. In: Linking Game-Theoretical Approaches with Constructive Type Theory. SpringerBriefs in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-19063-1_1

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