Skip to main content
SpringerLink
Log in

Theory of (n) truth degrees of formulas in modal logic and a consistency theorem

  • Published:
Science in China Series F: Information Sciences Aims and scope Submit manuscript

Abstract

The theory of (n) truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the (n) truth degree of a modality-free formula equals the truth degree of the formula in two-valued propositional logic. Variations of (n) truth degrees of formulas w.r.t. n in temporal logic is investigated. Moreover, the theory of (n) similarity degrees among modal formulas is proposed and the (n) modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Rosser J B, Turquette A R. Many-Valued Logics. Amsterdam: North-Holland, 1952

    MATH  Google Scholar 

  2. Pavelka J. On fuzzy logic I, II, III. Zeitschr.f Math. Logik u Grundlagen d Math. 1979, 25: 119–134, 447-464

    Article  MATH  MathSciNet  Google Scholar 

  3. De Glas M. Knowledge representation in a fuzzy setting. Tech Rep 89/48, Universite’ Pairs VI, Laforia, 1989

    Google Scholar 

  4. Ying M S. A logic for approximate reasoning. J Symbolic Logic, 1994, 59: 830–837

    Article  MATH  MathSciNet  Google Scholar 

  5. Wang G J. Theory of Σ - (α-tautologies) in revised Kleene system. Sci China Ser E-Inf Sci, 1998, 41(2): 188–195

    Article  Google Scholar 

  6. Wang G J, Fu L, Song J S. Theory of truth degrees of propositions in two-valued logic. Sci China Ser A-Math, 2002, 45(9): 1106–1116

    MATH  MathSciNet  Google Scholar 

  7. Wu H B. Theory of generalized tautology in Gödel logic system (in Chinese). Fuzzy Syst Math. 2000, 14(4): 53–59

    Google Scholar 

  8. Li J, Li S P, Xia Y F. Theory of truth degrees of propositions in n valued Lukasiwicz logic (in Chinese). Acta Math Sin, 2004, 47(4): 769–780

    MATH  MathSciNet  Google Scholar 

  9. Pei D. On equivalent forms of fuzzy logic systems NM and IMTL. Fuzzy Sets Sys, 2003, 138: 187–195

    Article  MATH  Google Scholar 

  10. Zhang X F, Meng G W, Zhang A Y. Implication operator families and their applications (in Chinese). Chinese J Comput, 2007, 30(3): 448–453

    MathSciNet  Google Scholar 

  11. Li J, Wang G J. Theory of α-truth degrees in n-valued Gödel propositional logic (in Chinese). J Software, 2007, 18(1): 33–39

    Article  Google Scholar 

  12. Li J, Wang G J. Theory of α-truth degrees in n-valued Lukasiwicz propositional logic (in Chinese). Comput Eng Appl, 2006, 42(31): 16–18

    Google Scholar 

  13. Han C, Wang G J, Xu W. A 16-classification partition of formula set F(S) in revised Kleene System \( \bar W \). Soft Comput, 2006, 10(2): 115–118

    Article  MATH  Google Scholar 

  14. Li J, Wang G J. Theory of truth degrees of propositions in the logic system L n *. Sci China Ser F-Inf Sci, 2006, 49(4): 471–483

    Article  MATH  Google Scholar 

  15. Wang G J, Qian G S, Dang C Y. Theory unified approximate reasoning in L * propositional calculus system and K * predicate calculus system. Sci China Ser F-Inf Sci, 2005, 48(1): 1–14

    Article  Google Scholar 

  16. Wang G J, Wang W. Logical metric spaces (in Chinese). Acta Math Sin, 2001, 44(1): 159–168

    MATH  Google Scholar 

  17. Han B H, Wang G J. The conditional truth degree of formulas in classical propositional logic (in Chinese). Fuzzy Sys Math, 2007, 21(4): 9–15

    MathSciNet  Google Scholar 

  18. Wang G J, Song J S. Graded method in propositional logic (in Chinese). Acta Electr Sin, 2006, 34(2): 252–257

    Google Scholar 

  19. Li B J, Wang G J. Theory of truth degrees of formulas in Lukasiwicz n-valued propositional logic and limit theorem. Sci China Ser F-Inf Sci, 2005, 48(6): 727–776

    Article  Google Scholar 

  20. Wang G J. Comparison of deduction theorems in diverse logic systems. New Math Nat Comput, 2005, 1(1): 65–77

    Article  MathSciNet  Google Scholar 

  21. Zhou X N, Wang G J. Consistency degrees of theories in some systems of propositional fuzzy logic. Fuzzy Sets Syst, 2005, 152(3): 321–331

    Article  MATH  Google Scholar 

  22. Wang G J, Zhang W X. Consistency degrees of finite theories in Lukasiwicz propositional fuzzy logic. Fuzzy Sets Syst, 2005, 149: 275–284

    Article  MATH  Google Scholar 

  23. Zhou H J, Wang G J. A new theory consistency index based on deduction theorems in several logic system. Fuzzy Sets Syst, 2006, 157: 427–443

    Article  MATH  Google Scholar 

  24. Zhou H J, Wang G J. Generalized consistency degrees of theories w.r.t. formulas in several standard complete logic systems. Fuzzy Sets Syst, 2006, 157: 2058–2073

    Article  MATH  Google Scholar 

  25. Zhou H J, Wang G J, Zhou W. Consistency degrees of theories and methods of graded reasoning in n-valued R 0-logic (NM-Logic). Int J Approx Reason, 2006, 43: 117–132

    Article  MATH  Google Scholar 

  26. Zhou H J, Wang G J. Characterizations of maximal consistent theories in the formal deductive system L * (NM-Logic) and Cantor space. Fuzzy Sets Syst, 2007, 158: 2951–2604

    Google Scholar 

  27. Wang G J, Leung Y. Integrated semantic and logic metric spaces. Fuzzy Sets Syst, 2003, 136: 71–91

    Article  MATH  MathSciNet  Google Scholar 

  28. Wang G J, Hui X J. Randomization of classical inference patterns and its application. Sci China Ser F-Inf Sci, 2007, 50(6): 867–877

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang G J, Hui X J. Generalization of the fundamental theorem of probability logic (in Chinese). Acta Electr Sin, 2007, 35(7): 1333–1340

    MATH  Google Scholar 

  30. Wang G J. Quantified logic (I). Chinese J Engine Math (in Chinese), 2006, 23(2): 191–215

    MATH  Google Scholar 

  31. Wang G J. Introduction to Mathematical Logic and Resolution Principle (in Chinese). 2nd ed. Beijing: Science Press, 2007

    Google Scholar 

  32. Wang G J, Qin X Y, Zhou X N. An intrinsic fuzzy set on the universe of discourse of predicate formulas. Fuzzy Sets Syst, 2006, 157: 3145–3158

    Article  MATH  MathSciNet  Google Scholar 

  33. Blackburn P, Rijke M, Venema Y. Modal Logic. Cambridge: Cambridge University Press, 2001

    MATH  Google Scholar 

  34. Wang G J. Non-classical Propositional Logic and Approximate Reasoning (in Chinese). 2nd ed. Beijing: Science Press, 2008

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Shaanxi Normal University, Xi’an, 710062, China

    GuoJun Wang & QiaoLin Duan

  2. Research Center for Science, Xi’an Jiaotong University, Xi’an, 710049, China

    GuoJun Wang

Authors
  1. GuoJun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. QiaoLin Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to GuoJun Wang.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 10331010 and 10771129), and the Foundation of 211 Construction of Shaanxi Normal University

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, G., Duan, Q. Theory of (n) truth degrees of formulas in modal logic and a consistency theorem. Sci. China Ser. F-Inf. Sci. 52, 70–83 (2009). https://doi.org/10.1007/s11432-009-0008-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-009-0008-x

Keywords

Search

Navigation