Van-Hung Le
ABSTRACT
This paper proposes an axiomatization of mathematical fuzzy logic with many dual hedges. We extend an axiomatization of mathematical fuzzy logic with one truth-stressing and one truth-depressing hedges as an expansion of a core fuzzy logic with new unary connectives by Esteva et al. for many truth-stressing and truth-depressing hedges, in which each hedge can have its own dual one. Our motivation is that, in the real world, we usually use many hedges at the same time, e.g., very, highly, rather, and slightly, to express different levels of emphasis, and each hedge seems to have a dual one, e.g., slightly (true) and rather (true) can be seen as a dual hedge of very (true) and highly (true), respectively. The proposed logic not only covers a large class of hedge functions but also has all completeness properties as the underlying core fuzzy logic w.r.t. the class of their chains as well as a number of special subclasses of their chains, including standard completeness.
- L. Behounek and P. Cintula. Fuzzy logics as the logics of chains. Fuzzy Sets and Systems, 157(5): 604--610, 2006. Google Scholar
Digital Library
- P. Cintula, P. Hájek, and C. Noguera, editors. Handbook of Mathematical Fuzzy Logic. Studies in Logic, Mathematical Logic and Foundations. College Publications, 2011.Google Scholar
- P. Cintula and C. Noguera. Implicational (semilinear) logics i: a new hierarchy. Arch. Math. Log., 49(4): 417--446, 2010.Google ScholarCross Ref
- F. Esteva and L. Godo. Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124(3): 271--288, 2001.Google ScholarCross Ref
- F. Esteva, L. Godo, and C. Noguera. A logical approach to fuzzy truth hedges. Inf. Sci., 232: 366--385, 2013. Google ScholarDigital Library
- P. Hájek. Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, The Netherlands, 1998.Google ScholarCross Ref
- P. Hájek. On very true. Fuzzy Sets and Systems, 124(3): 329--333, 2001.Google ScholarCross Ref
- P. Hájek and D. Harmancová. A hedge for gödel fuzzy logic. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8(4): 495--498, 2000. Google ScholarDigital Library
- C.-H. Nguyen and V.-N. Huynh. An algebraic approach to linguistic hedges in zadeh's fuzzy logic. Fuzzy Sets and Systems, 129(2): 229--254, 2002. Google ScholarDigital Library
- C. H. Nguyen and W. Wechler. Hedge algebras: An algebraic approach to structure of sets of linguistic truth values. Fuzzy Sets and Systems, 35: 281--293, 1990. Google ScholarDigital Library
- C. H. Nguyen and W. Wechler. Extended hedge algebras and their application to fuzzy logic. Fuzzy Sets and Systems, 52: 259--281, 1992. Google ScholarDigital Library
- H. Rasiowa. An Algebraic Approach to Non-classical Logic. North-Holland, Amsterdam, 1974.Google Scholar
- V. Vychodil. Truth-depressing hedges and bl-logic. Fuzzy Sets and Systems, 157(15): 2074--2090, 2006.Google ScholarCross Ref
- W. J. Blok and D. L. Pigozzi. Algebraizable Logics. Memoirs of the American Mathematical Society. American Mathematical Society, 1989.Google Scholar
- L. A. Zadeh. A fuzzy-set-theoretic interpretation of linguistic hedges. Journal of Cybernetics, 2(3): 4--34, 1972.Google ScholarCross Ref
Recommendations
A logical approach to fuzzy truth hedges
The starting point of this paper are the works of Hajek and Vychodil on the axiomatization of truth-stressing and-depressing hedges as expansions of Hajek's BL logic by new unary connectives. They showed that their logics are chain-complete, but ...
Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics
In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with ...
Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics
In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with ...
Comments