Abstract
We introduce the notion of an alphabetic trace of a cut-free intuitionistic prepositional proof and show that it serves to characterize the equality of arrows in cartesian closed categories. We also show that alphabetic traces improve on the notion of the generality of proofs proposed in the literature. The main theorem of the paper yields a new and considerably simpler solution of the coherence problem for cartesian closed categories than those in [11, 14].
Similar content being viewed by others
References
R. Goldblatt, Topoi, Studies in Logic and the Foundations of Mathematics 98, North-Holland, Amsterdam, 1979.
S. C. Kleene, Permutability of inferences in Gentzen's LK and LJ, Memoirs of the American Mathematical Society 10 (1952), pp. 1–26.
J. Lambek, The mathematics of sentence structure, American Math. Monthly 65 (1958), pp. 154–169.
J. Lambek, On the calculus of syntactic types, Amer. Math. Soc. Proc. Symposia Appl. Math. 12 (1961), pp. 166–178.
J. Lambek, Deductive systems and categories I, Mathematical Systems Theory 2 (1968), pp. 76–122.
J. Lambek, Deductive systems and categories II, Lecture Notes in Mathematics 86 (1969), pp. 57–82.
J. Lambek, Multicategories Revisited, Manuscript, 1988.
J. Lambek, On the Unity of Algebra and Logic, Manuscript, 1988.
G. Osius, Logical and set-theoretical tools in elementary topoi, Lecture Notes in Mathematics 445 (1975), pp. 297–346.
M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1969.
M. E. Szabo, A categorical equivalence of proofs, Notre Dame Journal of Formal Logic 15 (1974), pp. 177–191.
M. E. Szabo, A counter-example to coherence in cartesian closed categories, Canadian Mathematical Bulletin 18 (1975), pp. 111–114.
M. E. Szabo, Polycategories, Communications in Algebra 3 (1975), pp. 663–689.
M. E. Szabo, Algebra of Proofs, Studies in Logic and the Foundations of Mathematics 88, North-Holland, Amsterdam, 1978.
M. E. Szabo, The continuous realizability of entailment, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 29 (1983), pp. 219–233.
M. E. Szabo, A cut elimination theorem for stationary logic, Annals of Pure and Applied Logic 33 (1987), pp. 181–193.
Author information
Authors and Affiliations
Additional information
This research was supported in part by N.S.E.R.C. Grant A-8224 and F.C.A.R. Grant EQ-3491. The author is attached to the Centre interuniversitaire en études catégoriques, McGill University, Montreal.
Rights and permissions
About this article
Cite this article
Szabo, M.E. Coherence in cartesian closed categories and the generality of proofs. Stud Logica 48, 285–297 (1989). https://doi.org/10.1007/BF00370826
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00370826