Document Type : Research Paper

**Authors**

University of Ostrava, NSC IT4Innovations, 30. dubna 22, 702 00 Ostrava, Czech Republic

**Abstract**

In this paper we continue development of formal theory of a special class of

fuzzy logics, called EQ-logics. Unlike fuzzy logics being extensions of the

MTL-logic in which the basic connective is implication, the basic connective in

EQ-logics is equivalence. Therefore, a new algebra of truth values called

EQ-algebra was developed. This is a lower semilattice with top element endowed with two binary

operations of fuzzy equality and multiplication. EQ-algebra generalizes

residuated lattices, namely, every residuated lattice is an EQ-algebra but not

vice-versa.

In this paper, we introduce additional connective $logdelta$ in EQ-logics

(analogous to Baaz delta connective in MTL-algebra based fuzzy logics) and

demonstrate that the resulting logic has again reasonable properties including

completeness. Introducing $Delta$ in EQ-logic makes it possible to prove also

generalized deduction theorem which otherwise does not hold in EQ-logics weaker

than MTL-logic.

fuzzy logics, called EQ-logics. Unlike fuzzy logics being extensions of the

MTL-logic in which the basic connective is implication, the basic connective in

EQ-logics is equivalence. Therefore, a new algebra of truth values called

EQ-algebra was developed. This is a lower semilattice with top element endowed with two binary

operations of fuzzy equality and multiplication. EQ-algebra generalizes

residuated lattices, namely, every residuated lattice is an EQ-algebra but not

vice-versa.

In this paper, we introduce additional connective $logdelta$ in EQ-logics

(analogous to Baaz delta connective in MTL-algebra based fuzzy logics) and

demonstrate that the resulting logic has again reasonable properties including

completeness. Introducing $Delta$ in EQ-logic makes it possible to prove also

generalized deduction theorem which otherwise does not hold in EQ-logics weaker

than MTL-logic.

**Keywords**

[1] P. Cintula, P. Hajek, R. Horck, Formal systems of fuzzy logic and their fragments, Annals

of Pure and Applied Logic, 150 (2007), 40{65.

[2] P. Cintula and C. Noguera, A general framework for Mathematical Fuzzy Logic, In: P. Cin-

tula, P. Hajek, C. Noguera, eds., Handbook of Mathematical Fuzzy Logic - volume 1, Studies

in Logic, Mathematical Logic and Foundations, vol. 37. College Publications, Londres 2011,

103-207.

[3] M. Dyba and V. Novak, EQ-logics: Non-commutative fuzzy logics based on fuzzy equality,

Fuzzy Sets and Systems, 172 (2011), 13{32.

[4] M. El-Zekey, Representable good EQ-algebras, Soft Computing, 14 (2009), 1011{1023.

[5] M. El-Zekey, V. Novak and R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems, 178

(2011), 1{23.

[6] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous

t-norms, Fuzzy Sets and Systems, 124 (2001), 271{288.

[7] S. Gottwald, Mathematical fuzzy logics, Bulletin of Symbolic Logic, 14 (2) (2008), 210{239.

[8] S. Gottwald and P. Hajek, Triangular norm-based mathematical fuzzy logics, In: E. Kle-

ment, R. Mesiar (Eds.), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular

Norms, Elsevier, Amsterdam, (2005), 257{299.

[9] D. Gries and F. Schneider, A Logical Approach to Discrete Math, Springer-Verlag, Heidelberg,

1993.

of Pure and Applied Logic, 150 (2007), 40{65.

[2] P. Cintula and C. Noguera, A general framework for Mathematical Fuzzy Logic, In: P. Cin-

tula, P. Hajek, C. Noguera, eds., Handbook of Mathematical Fuzzy Logic - volume 1, Studies

in Logic, Mathematical Logic and Foundations, vol. 37. College Publications, Londres 2011,

103-207.

[3] M. Dyba and V. Novak, EQ-logics: Non-commutative fuzzy logics based on fuzzy equality,

Fuzzy Sets and Systems, 172 (2011), 13{32.

[4] M. El-Zekey, Representable good EQ-algebras, Soft Computing, 14 (2009), 1011{1023.

[5] M. El-Zekey, V. Novak and R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems, 178

(2011), 1{23.

[6] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous

t-norms, Fuzzy Sets and Systems, 124 (2001), 271{288.

[7] S. Gottwald, Mathematical fuzzy logics, Bulletin of Symbolic Logic, 14 (2) (2008), 210{239.

[8] S. Gottwald and P. Hajek, Triangular norm-based mathematical fuzzy logics, In: E. Kle-

ment, R. Mesiar (Eds.), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular

Norms, Elsevier, Amsterdam, (2005), 257{299.

[9] D. Gries and F. Schneider, A Logical Approach to Discrete Math, Springer-Verlag, Heidelberg,

1993.

[10] D. Gries and F. Schneider, Equational propositional logic, Information Processing Letters,

53 (1995), 145{152.

[11] P. Hajek, Metamathematics of Fuzzy Logic, Dordrecht, Kluwer, 1998.

[12] V. Novak, EQ-algebras: primary concepts and properties, In: Proc. Czech-Japan Seminar,

Ninth Meeting. Kitakyushu& Nagasaki, August 18{22, 2006, Graduate School of Information,

Waseda University, (2006), 219{223.

[13] V. Novak, Which logic is the real fuzzy logic?, Fuzzy Sets and Systems, 157 (2006), 635{641.

[14] V. Novak, EQ-algebras in progress, In: O. Castillo, ed., Theoretical Advances and Applica-

tions of Fuzzy Logic and Soft Computing, Springer, Berlin, (2007), 876{884.

[15] V. Novak, EQ-algebra-based fuzzy type theory and its extensions, Logic Journal of the IGPL,

19 (2011), 512{542.

[16] V. Novak and B. de Baets, EQ-algebras, Fuzzy Sets and Systems, 160 (2009), 2956{2978.

[17] G. Tourlakis, Mathematical Logic, New York, J. Wiley & Sons, 2008.

53 (1995), 145{152.

[11] P. Hajek, Metamathematics of Fuzzy Logic, Dordrecht, Kluwer, 1998.

[12] V. Novak, EQ-algebras: primary concepts and properties, In: Proc. Czech-Japan Seminar,

Ninth Meeting. Kitakyushu& Nagasaki, August 18{22, 2006, Graduate School of Information,

Waseda University, (2006), 219{223.

[13] V. Novak, Which logic is the real fuzzy logic?, Fuzzy Sets and Systems, 157 (2006), 635{641.

[14] V. Novak, EQ-algebras in progress, In: O. Castillo, ed., Theoretical Advances and Applica-

tions of Fuzzy Logic and Soft Computing, Springer, Berlin, (2007), 876{884.

[15] V. Novak, EQ-algebra-based fuzzy type theory and its extensions, Logic Journal of the IGPL,

19 (2011), 512{542.

[16] V. Novak and B. de Baets, EQ-algebras, Fuzzy Sets and Systems, 160 (2009), 2956{2978.

[17] G. Tourlakis, Mathematical Logic, New York, J. Wiley & Sons, 2008.

March and April 2015

Pages 41-61