Abstract
This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A → (A →B)) → (A →B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Bachman,Transfinite Zahlen, Springer-Verlag, Berlin 1955.
F. B. Fitch,Symbolic Logic, The Ronald Press, New York 1952.
V. N. Grisin,A nonstandard Logic, and its application to set theory,Studies in Formalized Languages and Nonclassical Logics, Nauka, Moscow 1974, pp. 135–171. (In Russian)
D. Prawitz,Natural Deduction, Almqvist and Wiksell, Stockholm 1965.
P. Suppes,Introduction to Logic, Van Nostrand, New York 1957.
R. B. White,A note on natural deduction in many-valued logic,Notre Dame Journal of Formal Logic 15 (1974), pp. 167–168.
R. B. White,A demonstrably consistent type-free extension of the logic BCK,Mathematica Japonica 32 (1987), pp. 149–169.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
White, R.B. A consistent theory of attributes in a logic without contraction. Stud Logica 52, 113–142 (1993). https://doi.org/10.1007/BF01053067
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01053067