Abstract
In the literature one can find three quite different proofs of the completeness of the infinite-valued sentential calculus of Lukasiewicz [8]:
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(i).
the syntactical proof of Rose and Rosser [7], using McNaughton’s theorem,
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(ii).
the algebraic proof of Chang [1, 2], using quantifier elimination in the first-order theory of divisible totally ordered abelian groups, and
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(iii).
the recent proof of Cignoli [3], using the representation of free latticeordered abelian groups.
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References
C. C. Chang, Algebraic analysis of many valued logics,Trans. Amer. Math. Soc.88(1958) 467–490.
C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms,Trans. Amer. Math. Soc.93(1959) 74–80.
R. Cignoli, Free lattice-ordered abelian groups and varieties of MV algebras,Proc. Latin American Symp. Logic, Bahia Blanca 1992, Notas de Logica Matematica, Univ. Nacional del Sur, Bahia Blanca, Argentina, 1994.
G. H. Hardy, E. M. Wright,An Introduction to the theory of numbers, Fifth Edition, Oxford University Press, 1979.
D. Mundici, Interpretation of AFC*-algebras in Lukasiewicz sentential logic,J. Functional Analysis 65(1986) 15–63.
D. Mundici, Normal forms in infinite-valued logic: the case of one variable,Lecture Notes in Computer Science626 (1992) 272–277.
A. Rose, J. B. Rosser, Fragments of many-valued statement calculi,Trans. Amer. Math. Soc.87(1958) 1–53.
A. Tarski, J. Lukasiewicz, Investigations into the Sentential Calculus, In:Logic, Semantics, Metamathematics, Oxford University Press, 1956, pp. 38–59. Reprinted by Hackett Publishing Company, 1981.
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Mundici, D., Pasquetto, M. (1995). A proof of the completeness of the infinite-valued calculus of Łukasiewicz with one variable. In: Höhle, U., Klement, E.P. (eds) Non-Classical Logics and their Applications to Fuzzy Subsets. Theory and Decision Library, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0215-5_6
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