A sequent calculus for type assignment
- 12 March 1977
- journal article
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- Vol. 42 (1), 11-28
- https://doi.org/10.2307/2272314
Abstract
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.Keywords
This publication has 8 references indexed in Scilit:
- Combinators, λ-Terms and Proof TheoryPublished by Springer Science and Business Media LLC ,1972
- Simultane Rekursionen in der Theorie der Funktionale endlicher TypenArchive for Mathematical Logic, 1971
- Une Extension De ĽInterpretation De Gödel a ĽAnalyse, Et Son Application a ĽElimination Des Coupures Dans ĽAnalyse Et La Theorie Des TypesPublished by Elsevier BV ,1971
- Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite TypePublished by Elsevier BV ,1970
- Functionals defined by recursion.Notre Dame Journal of Formal Logic, 1967
- Foundations of Mathematical LogicThe Mathematical Gazette, 1964
- Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematicsProceedings of Symposia in Pure Mathematics, 1962
- ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTESDialectica, 1958