Abstract
Formulas of n variables of Łukasiewicz’s sentential calculus can be represented, via McNaughton’s theorem, by piecewise linear functions, with integer coefficients, from hypercube [0, [1]]n to [0, [1]], called Mc-Naughton functions. McNaughton functions, which are truncated functions to [0, [1]] of the restriction to [0, [1]]n of single hyperplanes, are called simple McNaughton functions.
In the present work we describe a class of formulas that can be represented by simple McNaughton functions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Aguzzoli, The Complexity of McNaughton Functions of One Variable, Advances in Applied Mathematics, 21 (1998) 58–77.
C. C Chang, Algebraic Analysis of infinite valued logic, Trans. Amer. Math. Soc. 88 (1958), 467–490.
C. C Chang, A new proof of the completeness of the Łukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959), 74–90.
R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning Trends in Logic, Volume 7, Kluwer, Dordrecht, 1999.
A. Di Nola, G. Georgescu, A. Lettieri, Extending Probabilities to States of MV-algebras, Collegium Logicum, Annals of the Kurt Goedel Society, 3–30, 1999.
P. Hajek, Metamathematics of Fuzzy Logic, Trends in Logic, Kluwer, Dordrecht, (1998).
J. Łukasiewicz, A. Tarski, Untersuchungen uber den Aussagenkalkul, Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 23, 30–50, (1930).
R. McNaughton, A theorem about infinite-valued sentential logic, Journal of Symbolic Logic, 16 (1951) 1–13.
D. Mundici, Interpretation of AFC*-algebras in Łukasiewicz sentential calculus, J. Funct. Analysis 65, (1986), 15–63.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Di Nola, A., Lettieri, A. (2003). Formulas of Łukasiewicz’s Logic Represented by Hyperplanes. In: Bilgiç, T., De Baets, B., Kaynak, O. (eds) Fuzzy Sets and Systems — IFSA 2003. IFSA 2003. Lecture Notes in Computer Science, vol 2715. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44967-1_22
Download citation
DOI: https://doi.org/10.1007/3-540-44967-1_22
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40383-8
Online ISBN: 978-3-540-44967-6
eBook Packages: Springer Book Archive