Abstract
The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.
Similar content being viewed by others
References
Chang, C. C., 1958, 'Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88, 467-490.
Chang, C. C., 1959, 'A new proof of the completeness of the Łukasiewicz axioms', Trans. Amer. Math. Soc. 93, 74-80.
Cignoli, R., 1993, 'Free lattice-ordered abelian groups and varieties of MV-algebras', In: “Proceedings of the IXth Latin American Symposium on Mathematical Logic, I”, Notas de Lógica Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina 38, 113-118.
Mangani, P., 1973, 'On certain algebras related to many-valued logics', (Italian), Bollettino Unione Matematica Italiana 8, 68-78.
Mundici, D., 1986, 'Interpretation of AF C*-algebras in Łukasiewicz sentential calculus', J. Functional Analysis 65, 15-63.
Mundici, D., 1993, 'Ulam game, Łukasiewicz logic, and AF C*-algebras', Fundamenta Informaticae 18, 151-161.
Panti, G., 1995, 'A geometric proof of the completeness of the calculus of Łukasiewicz', J. Symbolic Logic, 60, 563-578.
Rose, A., and J. B. Rosser, 1958, 'Fragments of many-valued statement calculi', Trans. Amer. Math. Soc. 87, 1-53.
Tarski, A., J. Łukasiewicz, 1956, 'Investigations into the Sentential Calculus', In: ”Logic, Semantics, Metamathematics”, Oxford University Press, 38-59, reprinted by Hackett Publishing Company, Indianapolis, 1983.
Ulam, S. M., 1976, Adventures of a Mathematician, Scribner's, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cignoli, R., Mundici, D. An Elementary Proof of Chang's Completeness Theorem for the Infinite-valued Calculus of Lukasiewicz. Studia Logica 58, 79–97 (1997). https://doi.org/10.1023/A:1004991931741
Issue Date:
DOI: https://doi.org/10.1023/A:1004991931741