The symmetric implicational methods for fuzzy reasoning characterizes the solution B* (A*) of the formula (A → 1B) → 2 (A* → 1B*) for the fuzzy modus ponens (fuzzy modus tollens), where →1 and →2 are two different implications. In this study, we provide a predicate formal representation of the solution for the symmetric implicational methods based on the LΠ formal logic system, including detailed logic proofs. We bring the symmetric implicational methods within a logical framework and provide a sound logic foundation for the symmetric implicational methods of fuzzy reasoning.
CintulaP., The LΠ and propositional and predicate logics, Fuzzy Sets and Systems124 (2001), 289–302.
2.
CintulaP., Advances in the LΠ and logics, Arch Math Logic42 (2003), 449–468.
3.
DybaM. and NovákV., EQ-logics: Non-commutative fuzzy logics based on fuzzy equality, Fuzzy Sets and Systems172 (2011), 13–32.
4.
EstevaF. and GodoL., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems124 (2001), 271–288.
5.
EstevaF., GodoL., MontagnaF., The LΠ and logics: two complete fuzzy systems joining Łukasiewicz and product logics, Arch Math Logic40 (2001), 39–67.
6.
GeorgescuG. and PopescuA., Non-commutative fuzzy structures and pairs of weak negations, Fuzzy Sets and Systems143(1) (2004), 129–155.
7.
HájekP., Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998.
8.
LuoM. and ZhouK., Logical foundation of the quintuple implication inference methods, Int J Approx Reason101 (2018), 1–9.
9.
NovákV., PerfilievaI. and GottwaldS., Fuzzy relations equations via basic predicate fuzzy logic, in: D. Ruan, P. dHondt, E.E. Kerre (Eds.), Computational Intelligent Systems for Applied Research, Proceedings on 5th International FLINS Conference, Gent, Sept. 16-18, World Scientific, Singapore, 2002, pp. 113–120.
10.
PeiD.W., On the strict logic foundation of fuzzy reasoning, Soft Computing8 (2004), 539–545.
11.
PeiD.W., Formalization of implication based fuzzy reasoning method, Int J Approx Reason53 (2012), 837–846.
12.
PeiD.W., On the strict logic foundation of fuzzy reasoning, Soft Comput8 (2004), 539–545.
13.
PeiD.W., On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Systems138 (2003), 187–195.
14.
TangY. and YangX., Symmetric implicational method of fuzzy reasoning, Int J Approx Reason54(8) (2013), 1034–1048.
15.
TangY. and PedryczW., On the a(u, v)-symmetric implicational method for R-and (S, N)-implications, Int J Approx Reason92 (2018), 212–231.
16.
WangG.J., The full implication triple I method of fuzzy reasoning, Sci China Ser E29 (1999), 43–53.
17.
WangD.G., MengY.P. and LiH.X., A fuzzy similarity inference method for fuzzy reasoning, Computers & Mathematics with Applications56(10) (2008), 2445–2454.
18.
ZadehL.A., Outline of new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man and Cybernetics3 (1973), 28–33.
19.
ZadehL.A., The concept of a linguistic variable and its application to approximate reasoning, Part 1-3, Information Sciences8 (1975), 199–249, 301–357; 9 (1975) 43–80.
20.
ZhouB.K., XuG.Q. and LiS.J., The quintuple implication principle of fuzzy reasoning, Inf Sci297 (2015), 202–215.
