Equality propositional logic and its extensions

Document Type : Research Paper


School of Mathematics, Northwest University, Xi'an,710127, China


We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $\boldsymbol{\wedge}$ (conjunction) and $\equiv$ (equivalence). Moreover, the $\Rightarrow$ (implication) connective can be derived as $A\Rightarrow B:=(A\boldsymbol{\wedge}B)\equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective $\Delta$(delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.


[1] M. Bazz, Infi nite-valued Godel logics with 0-1-projections and relativizations, In: P. H´ajek(Ed.), G¨odel’96: Logical
Foundations of Mathematics, Comupter Science, and Physics, Lecture notes in logic, 6, Springer-Verlag, Brno, 1996,
[2] R. A. Borzooei, F. Zebardast, M. Aaly Kologani, Some types of fi lters in equality algebras, Categories and General
Algebraic Structures with Applications, 7, Special issue on the Occasion of Banaschewski’s 90th Birthday, (2017),
[3] R. A. Borzooei, M. Zarean, O. Zahiri, Involutive equality algebras, Soft Computing, 22 (2018), 7505-7517.
[4] H. Casta˜neda, Leibniz's syllogistico-propositional calculus, Notre Dame Journal of Formal Logic XVII, 4 (1976),
[5] L. C. Ciungu, On pseudo-equality algebras, Archive for Mathematical Logic, 53 (2014), 561-570.
[6] L. C. Ciungu, Internal states on equality algebras, Soft Computing, 19 (2015), 939-953.
[7] M. Dyba, V. Nov´ak, EQ-logics: Non-commutative fuzzy logics based on fuzzy equality, Fuzzy Sets and Systems, 172
(2011), 13-32.
[8] M. Dyba, V. Nov´ak, EQ-logics with delta connective, Iranian Journal of Fuzzy Systems, 12(2) (2015), 41-61.
[9] M. El-Zekey, Representable good EQ-algebras, Soft Computing, 14 (2010), 1011-1023.
[10] M. El-Zekey, V. Nov´ak, R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems, 178 (2011), 1-23.
[11] F. Esteva, L. Godo, Monoidal t-norm based logic: Towards a logic for left-continuous t-norms, Fuzzy Sets and
Systems, 124 (2001), 271-288.
[12] S. Gottwald, Mathematical fuzzy logics, Bulletin of Symbolic Logic, 14(2) (2008), 210-239.
[13] D. Gries, F. Schneider, A logical approach to discrete math, Springer-Verlag, Heidelberg, 1993.
[14] L. Henkin, A theory of propositional types, Fundamenta of Mathematicae, 46(52) (1963), 582-585.
[15] S. Jenei, Equality algrbras, Studia Logica, 100 (2012), 1201-1209.
[16] V. Nov´ak, B. D. Baets, EQ-algebras, Fuzzy Sets and Systems, 160 (2009), 2956-2978.
[17] J. Yang, X. L. Xin, P. F. He, Uniform topology on EQ-algebras, Open Mathematics, 15 (2017), 354-364.
[18] J. Yang, X. L. Xin, P. F. He, On topological EQ-algebras, Iranian Journal of Fuzzy Systems, DOI:
[19] F. Zebardast, R. A. Borzooei, M. Aaly Kologani, Results on equality algebras, Information Sciences, 381 (2017),