On EQ-Fuzzy Logics with Delta Connective

Abstract

In this paper, extension of the EQ-logic by the ∆-connective is introduced. The former is a new kind of many-valued logic which based on EQ-algebra of truth values, i.e. the algebra in which fuzzy equality is the fundamental operation and implication is derived from it. First, we extend the EQ-algebra by the ∆ operation and then introduce axioms and inference rules of EQ ∆ -logic. We also prove the deduction theorem formulated using fuzzy equalities.

On pseudo-equality algebras

“…The first version of such an algebra has been introduced by V. Novák ([25]) under the name of EQ-algebra and a new concept of fuzzy type theory has been developed based on EQ-algebras ( [26]). A fuzzy-equality based logic called EQ-logic has also been introduced ( [27]), while the EQ-logics with delta connective were defined and investigated in [13]. According to [26], a non-commutative EQ-algebra is an algebra (E, ∧, ⊙, ∼, 1) of the type (2, 2, 2, 0) such that the following axioms are fulfilled for all x, y, z, u ∈ E: (E 1 ) (E, ∧, 1) is a commutative idempotent monoid w.r.t ≤ (x ≤ y defined as x ∧ y = x), (E 2 ) (E, ⊙, 1) is a monoid such that the operation ⊙ is isotone w.r.t.…”

“…The first version of such an algebra has been introduced by V. Novák ([25]) under the name of EQ-algebra and a new concept of fuzzy type theory has been developed based on EQ-algebras ( [26]). A fuzzy-equality based logic called EQ-logic has also been introduced ( [27]), while the EQ-logics with delta connective were defined and investigated in [13].…”

Single–valued neutrosophic filters in EQ–algebras

“…This basic structure in fuzzy logic is ordering, represented by ∧-semilattice, with maximal element "1". Further materials regarding EQalgebras are available in the literature too [6,7,9,12]. Algebras including EQ-algebras have played an important role in recent years and have had its comprehensive applications in many aspects including dynam-* Corresponding author.…”

Commutative pseudo-equality algebras

“…The first version of such an algebra has been introduced by V. Novák ([29]) under the name of EQ-algebra and a new concept of fuzzy type theory has been developed based on EQ-algebras ( [30]). A fuzzy-equality based logic called EQ-logic has also been introduced ([31]), while the EQ-logics with delta connective were defined and investigated in [16]. According to [30], a non-commutative EQ-algebra is an algebra (E, ∧, ⊙, ∼, 1) of the type (2, 2, 2, 0) such that the following axioms are fulfilled for all x, y, z, u ∈ E: (E 1 ) (E, ∧, 1) is a commutative idempotent monoid w.r.t ≤ (x ≤ y defined as x ∧ y = x), (E 2 ) (E, ⊙, 1) is a monoid such that the operation ⊙ is isotone w.r.t.…”

“…The first version of such an algebra has been introduced by V. Novák ([29]) under the name of EQ-algebra and a new concept of fuzzy type theory has been developed based on EQ-algebras ( [30]). A fuzzy-equality based logic called EQ-logic has also been introduced ( [31]), while the EQ-logics with delta connective were defined and investigated in [16].…”