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A sequent calculus for type assignment

Published online by Cambridge University Press:  12 March 2014

Jonathan P. Seldin
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The sequent calculus formulation (L-formulation) of the theory of functionality without the rules allowing for conversion of subjects of [3, §14E6] fails because the (cut) elimination theorem (ET) fails. This can be most easily seen by the fact that it is easy to prove in the system

and

but not (as is obvious if α is an atomic type [an F-simple])

The error in the “proof” of ET in [14, §3E6], [3, §14E6], and [7, §9C] occurs in Stage 3, where it is implicitly assumed that if [x]X ≡ [x] Y then X ≡ Y. In the above counterexample, we have [x]x ≡ ∣ ≡ [x](∣x) but x ≢ ∣x. Since the formulation of [2, §9F] is not really satisfactory (for reasons stated in [3, §14E]), a new seguent calculus formulation is needed for the case in which the rules for subject conversions are not present. The main part of this paper is devoted to presenting such a formulation and proving it equivalent to the natural deduction formulation (T-formulation). The paper will conclude in §6 with some remarks on the result that every ob (term) with a type (functional character) has a normal form.

The conventions and definitions of [3], especially of §12D and Chapter 14, will be used throughout the paper.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 1 , March 1977 , pp. 11 - 28
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

[1]Andrews, Peter B., Resolution in type theory, this Journal, vol. 36 (1971), pp. 414–432.Google Scholar
[2]Curry, Haskell B. and Feys, R., Combinatory logic, vol I, North-Holland, Amsterdam, 1958; reprinted 1968 and 1974.Google Scholar
[3]Curry, Haskell B., Hindley, J. Roger and Seldin, Jonathan P., Combinatory logic, vol. II, North-Holland, Amsterdam, 1972.Google Scholar
[4]Curry, Haskell B., Foundations of mathematical logic, McGraw-Hill, New York, 1963.Google Scholar
[5]Girard, Jean-Yves, Une extension de l'interprétation de Gödel à l'analyse, et son application à l'élimination des coupures dans l'analyse et la théorie des types, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J. E., Editor), North-Holland, Amsterdam, 1971, pp. 63–92.CrossRefGoogle Scholar
[6]Gödel, Kurt, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280–287.CrossRefGoogle Scholar
[7]Hindley, J. Roger, Lercher, Bruce and Seldin, Jonathan P., Introduction to cominatory logic, London Mathematical Society Lecture Note No. 7, Cambridge University Press, 1972.Google Scholar
[8]Howard, W. A., Assignment of ordinals to terms for primitive recursive functionals of finite type, Intuitionism and proof theory, Proceedings of the Summer Conference at Buffalo, New York, 1968 (Myhill, J., Kino, A., and Vesley, R. E., Editors), North-Holland, Amsterdam, 1970, pp. 443–458.CrossRefGoogle Scholar
[9]Prawitz, Dag, Natural deduction, a proof-theoretical study, Almqvist & Wiksell, Stockholm, 1965.Google Scholar
[10]Sanchis, L. E., Functionals defined by recursion, Notre Dame Journal of Formal Logic, vol. 8 (1967), pp. 161–174.CrossRefGoogle Scholar
[11]Schütte, Kurt, Theorie der Funktionale endlicher Typen, mimeographed lecture notes, Munich, 1968.Google Scholar
[12]Seldin, Jonathan P., A normal form theorem for generalized functionality, (abstract), Notices of the American Mathematical Society, vol. 21 (1974), p. A–23, no. 711–02–20.Google Scholar
[13]Seldin, Jonathan P., A sequent calculus formulation of type assignment with equality rules for the λβ-calculus, in preparation; abstract, this Journal, vol. 41 (1976), p. 284.Google Scholar
[14]Seldin, Jonathan P., Studies in illative combinatory logic, Dissertation, University of Amsterdam, 1968.Google Scholar
[15]Seldin, Jonathan P., A stratification theorem for generalized functionality in combinatory logic, (abstract), this Journal, vol. 37 (1972), pp. 431–432.Google Scholar
[16]Seldin, Jonathan P., The theory of generalized functionality. I, in preparation.Google Scholar
[17]Spector, Clifford, Provably recursive functionals of analysis, a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. V (1962), pp. 1–27.Google Scholar
[18]Stenlund, Sören, Combinators, λ-terms, and proof theory, Riedel, Dordrecht, 1972.CrossRefGoogle Scholar
[19]Tait, W. W., Intensional interpretations of functionals of finite type. I, this Journal, vol. 32 (1967), pp. 198–212.Google Scholar