Linguistic truth-valued lattice-valued propositional logic system \(\ell P(X)\) based on linguistic truth-valued lattice implication algebra. (English) Zbl 1198.03031
Summary: In the semantics of natural language, quantification may have received more attention than any other subject, and syllogistic reasoning is one of the main topics in many-valued logic studies on inference. Particularly, lattice-valued logic can be applied to describe and treat incomparability by the incomparable elements in its truth-value set. In this paper, we first focus on some properties of linguistic truth-value lattice implication algebras. Secondly, we introduce some concepts of the linguistic truth-value lattice-valued propositional logic system \(\ell P(X)\) whose truth-value domain is a linguistic truth-value lattice implication algebra. Then we investigate the semantic problems of \(\ell P(X)\). Finally, we further probe the syntax of the linguistic truth-value lattice-valued propositional logic system \(\ell P(X)\), and prove the soundness theorem, the deduction theorem and the consistency theorem.
MSC:
| 03B52 | Fuzzy logic; logic of vagueness |
| 03B65 | Logic of natural languages |
| 03G25 | Other algebras related to logic |
References:
| [2] | Delgado, M.; Verdegay, J. L.; Vila, M. A., Linguistic decision making models, International Journal of Intelligent Systems, 7, 479-492 (1992) · Zbl 0756.90001 |
| [3] | Huynh, Van-Nam; Nakamori, Yoshiteru, A satisfactory-oriented approach to multiexpert decision-making with linguistic assessments, IEEE Transactions on Systems, Man, and Cybernetics - Part B, 35, 2, 184-196 (2005) |
| [4] | Ho, N. C.; Wechler, W., Hedge algebras: on algebraic approach to structure of sets of linguistic truth values, Fuzzy Sets and Systems, 35, 281-293 (1990) · Zbl 0704.03007 |
| [5] | Ho, N. C.; Wechler, W., Extended hedge algebras and their application on fuzzy logic, Fuzzy Sets and Systems, 52, 259-281 (1992) · Zbl 0786.03018 |
| [6] | Hájek, P.; Harmancova, D., A hedge for Godel fuzzy logic, International Journal Uncertainty Fuzziness Knowledge-Based Systems, 8, 495-498 (2000) · Zbl 1113.03318 |
| [7] | Hájek, P., On very true, Fuzzy Sets and Systems, 124, 233-329 (2001) · Zbl 0997.03028 |
| [8] | Herrea, F.; Martinez, L., A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems, 8, 6, 746-752 (2000) |
| [9] | Jun, Young Bae; Xu, Yang; Ma, Jun, Redefined fuzzy implicative filters, Information Sciences, 177, 1422-1429 (2007) · Zbl 1111.03325 |
| [10] | Lai, Jiajun; Xu, Yang, Rules of reasoning based on linguistic truth-valued lattice value propositional logic system \(\ell_{vpl}\), Computer Science, 35, 9, 230-232 (2008), (in Chinese) |
| [12] | Lai, Jiajun; Xu, Yang; Chang, Zhiyan, On FP-filters and FPD-filters of lattice implication algebra, International Journal of Applied Mathematice and Informatics, 26, 3-4, 653-660 (2008) |
| [13] | Lai, Jiajun; Xu, Yang, On WLI-ideal space and the properties of WLI-ideals in lattice implication algebra (LIA), International Journal of Applied Mathematics and Computing, 31, 1, 113-127 (2009) · Zbl 1182.03108 |
| [14] | Lai, Jiajun; Xu, Yang; Ma, Jun; Pan, Xiaodong, Congruence relations induced by weak-filters and FWLI-ideals in lattice implication algebra, International Journal of Modern Mathematics, 2, 1, 135-142 (2007) · Zbl 1139.03318 |
| [15] | Lai, Jiajun; Xu, Yang, Logical properties of lattice filters of lattice implication algebra, Journal of Southwest Jiaotong University, 15, 4, 353-356 (2007) · Zbl 1150.03344 |
| [16] | Lai, Jiajun; Xu, Yang, Inequality of lattice implication algebra, Journal of Jiangnan University, 6, 3, 366-370 (2007), (in Chinese) |
| [18] | Lianzhen, Liu; Kaitai, Li, Fuzzy implicative and Boolean filters of Ro algebras, Information Sciences, 171, 61-71 (2005) · Zbl 1066.03062 |
| [19] | Ma, J.; Chen, S.; Xu, Y., Fuzzy logic form the viewpoint of machine intelligence, Fuzzy Sets and Systems, 157, 5, 628-634 (2006) · Zbl 1092.68096 |
| [20] | Ma, Jun; Li, Wenjiang; Ruan, Da; Xu, Yang, Filter-based resolution principle for lattice-valued propositional logic LP \((X)\), Information Sciences, 177, 1046-1062 (2007) · Zbl 1114.03019 |
| [22] | Pan, Xiaodong; Xu, Yang, Lattice implication ordered semigroups, Information Sciences, 178, 403-413 (2008) · Zbl 1133.06303 |
| [23] | Pei, D. W., Unified full implication algorithms of fuzzy reasoning, Information Sciences, 178, 520-530 (2008) · Zbl 1125.68122 |
| [25] | Qin, K. Y.; Xu, Y., Lattice-valued propositional logic (II), Journal of Southwest Jiaotong University, 2, 1, 22-27 (1994) · Zbl 0824.03009 |
| [26] | Qin, K. Y.; Xu, Y.; Song, Z. M., Some kinds of approximate reasoning based on system LP \((X)\), Fuzzy Systems and Mathematics, 12, 2, 55-60 (1998), (in Chinese) · Zbl 1333.03055 |
| [28] | Takeuti, G.; Titani, S., Globalization of intuitionitic set theory, Annals of Pure and Applied Logic, 33, 195-211 (1987) · Zbl 0633.03050 |
| [29] | Vychodil, Vilem, Truth-depressing hedges and BL-logic, Fuzzy Sets and Systems, 157, 2074-2090 (2006) · Zbl 1114.03023 |
| [30] | Wang, Guojun; Zhou, Hongjun, Quantitative logic, Information Sciences, 179, 3, 226-247 (2009) · Zbl 1167.03020 |
| [34] | Xu, Y.; Liu, J.; Ruan, D.; Lee, T. T., On the consistency of rule bases based on lattice-valued first-order logic LF \((X)\), International Journal of Intelligent Systems, 21, 399-424 (2006) · Zbl 1156.68581 |
| [35] | Xu, Y.; Ruan, D.; Qin, K. Y.; Liu, J., Lattice-Valued Logic - An Alternative Approach to Treat Fuzziness Incomparability. Lattice-Valued Logic - An Alternative Approach to Treat Fuzziness Incomparability, Studies in Fuzziness and Soft Computing, vol. 132 (2003), Springer: Springer Berlin · Zbl 1048.03003 |
| [36] | Xu, Y.; Qin, K. Y., Lattice-valued propositional logic(I), Journal of Southwest JiaoTong University, 1, 2, 123-128 (1993) · Zbl 0807.03020 |
| [37] | Xu, Y., Lattice implication algebras, Journal of Southwest Jiaotong University, 28, 1, 20-27 (1993), (in Chinese) · Zbl 0784.03035 |
| [39] | Zadeh, Lotfi A., Is there a need for fuzzy logic?, Information Sciences, 178, 2751-2779 (2008) · Zbl 1148.68047 |
| [40] | Zadeh, Lotfi A., Toward a generalized theory of uncertainty (GTU) - an outline, Information Sciences, 172, 1-40 (2008) · Zbl 1074.94021 |
| [41] | Zadeh, Lotfi A., Fuzzy logic and approximate reasoning, Sythese, 30, 407-428 (1975) · Zbl 0319.02016 |
| [42] | Zadeh, Lotfi A., A theory of approximate reasoning, (Yager, R. R.; Ovchinnikov, S.; Tong, R. M.; Nguyen, H. T., Fuzzy Sets and Applications: Selected papers by L.A. Zadeh (1987), Wiley: Wiley New York), 367-411 · Zbl 0671.01031 |
| [43] | Part III, Information Sciences, 9, 43-80 (1975) · Zbl 0404.68075 |
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