Abstract
This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 〈0, 1〉 of reals. These are special cases of a residuated lattice 〈L, ∨, ∧, ⊗, →, 1, 0〉. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Gödel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.
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References
J. F. Baldwin, Fuzzy logic and fuzzy reasoning, International Journal of Man-Machine Studies 11 (1979), pp. 465–480.
C. C. Chang, H. J. Keisler, Continuous Model Theory, Princeton University Press, Princeton 1966.
C. C. Chang, H. J. Keisler, Model Theory, North-Holland, Amsterdam 1973.
D. Dubois, H. Prade, Fuzzy Sets and Systems. Theory and Applications, Academic Press, New York, 1980.
B. R. Gainess, Foundations of fuzzy reasoning, International Journal of Man-Machine Studies 8 (1976), pp. 623–668.
E. Giles, A formal system for fuzzy reasoning. Fussy Sets and Systems 2 (1979), pp. 233–257.
J. A. Gougen, The logic of inexact concepts, Synthese 19 (1968–69), pp. 325–373.
J. A. Goguen, Concept representation in natural and artificial languages: axioms, extensions and applications for fuzzy sets, International Journal of Man-Machine Studies 6 (1974), pp. 513–561.
S. Gottwald, Set theory for fuzzy sets of higher level, Fussy Sets and Systems 2 (1979), pp. 125–151.
M. Katz, Controlled-Error Theories of Proximity and Dominance, in: H.J. Skala, S. Termini, E. Trillas (eds.), Aspects of Vagueness, D. Reidel, Dordecht 1984.
L. T. Kóczy, M. Hajnal, A new attempt to axiomatize fuzzy algebra with an application example, Probl. Control and Inf. Theory 6 (1977), pp. 47–66.
V. Novák, Fuzzy Sets and their Applications, SNTL, Prague 1985 — in Czech.
J. Pavelka, On fuzzy logic, Zeitschrift fur mathematik Logik und Grundlagen der Mathematic 25 (1979), pp. 45–52; 119–134; 447–464.
H. Rasiowa, R. Sikorski, The Mathematics of Meta-mathematics, PWN, Warszawa 1963.
J. R. Shoenfield, Mathematical Logic, Addison-Wesley, New York 1967.
L. A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning, Inf. Sci. 8 (1975), 199–257; pp. 301–357; 9 (1975), pp. 43–80.
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Novák, V. First-order fuzzy logic. Stud Logica 46, 87–109 (1987). https://doi.org/10.1007/BF00396907
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DOI: https://doi.org/10.1007/BF00396907