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The expressive unary truth functions of $$n$$-valued logic. (English) Zbl 1098.03032
To every (normal) consequence relation there corresponds a notion of closure such that the set of closed sets forms a complete lattice. If the consequence relation is finitary, this lattice has a minimal closed basis consisting of the irreducibles [see R. Wójcicki, Theory of logical caculi. Basic theory of consequence operations. Dordrecht: Kluwer Academic Publishers (1988; Zbl 0682.03001)]. If all these irreducibles are maximally consistent, the lattice is said to be expressive. Some truth functions force irreducibles to be maximally consistent and hence are called expressive. In “The expressive truth conditions of two-valued logic” [Notre Dame J. Formal Logic 43, 221–230 (2002; Zbl 1050.03008)] the author has identified all expressive truth functions of two-valued logic. This paper identifies the expressive truth functions of $$n$$-valued logic, by characterizing the unary ones.
##### MSC:
 03B50 Many-valued logic 03B22 Abstract deductive systems
##### Keywords:
many-valued logics; abstract deductive systems
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##### References:
  Beall, J. C., and B. C. van Fraassen, Possibilities and Paradox , Oxford University Press, Oxford, 2003.  Gabbay, D. M., and H. Wansing, editors, What Is Negation? , vol. 13 of Applied Logic Series , Kluwer Academic Publishers, Dordrecht, 1999. · Zbl 0957.00012  Martin, N. M., and S. Pollard, Closure Spaces and Logic , vol. 369 of Mathematics and its Applications , Kluwer Academic Publishers Group, Dordrecht, 1996. · Zbl 0855.54001  Pollard, S., ”The expressive truth conditions of two-valued logic”, Notre Dame Journal of Formal Logic , vol. 43 (2002), pp. 221–30. · Zbl 1050.03008  Pollard, S., and N. M. Martin, ”Contractions of closure systems”, Notre Dame Journal of Formal Logic , vol. 35 (1994), pp. 108–15. · Zbl 0804.06004  Pollard, S., and N. M. Martin, ”Closed bases and closure logic”, The Monist , vol. 79 (1996), pp. 117–27.  Rasiowa, H., An Algebraic Approach to Nonclassical Logics , vol. 78 of Studies in Logic and the Foundations of Mathematics , North-Holland Publishing Co., Amsterdam, 1974. · Zbl 0299.02069  van Fraassen, B. C., Formal Semantics and Logic , Macmillan, New York, 1971. · Zbl 0253.02002  Weaver, G., ”Compactness theorems for finitely-many-valued sentential logics”, Studia Logica , vol. 37 (1978), pp. 413–16. · Zbl 0415.03018  Wójcicki, R., Theory of Logical Calculi: Basic Theory of Consequence Operations , vol. 199 of Synthese Library , Kluwer Academic Publishers Group, Dordrecht, 1988. · Zbl 0682.03001  Woodruff, P. W., ”On compactness in many-valued logic. I”, Notre Dame Journal of Formal Logic , vol. 14 (1973), pp. 405–7. · Zbl 0245.02022
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