The logic of tied implications, part 2: Syntax

Morsi, Nehad N.; Lotfallah, Wafik Boulos; El-Zekey, Moataz

doi:10.1016/j.fss.2006.03.004

Cited by 9 publications

(4 citation statements)

References 11 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We devote the second part of this work [69] to syntax. There, we construct a predicate calculus for prelinear tied adjointness algebras.…”

Section: Resultsmentioning

confidence: 99%

“…This way, duality works to establish some new inferences from their duals, without new proofs. This duality principle will be made precise, within syntax, in Part 2 [69]. As this particular duality is based upon reversing the order of L and keeping the order of P, its formulation would have been impossible if L had to be indiscernible from P.…”

Section: K(k(a Y) Y) A(a H (X Y)) H (H (X Y) Z)mentioning

confidence: 99%

“…Note that in a BL-algebra, both lattice operations are defined in terms of T and I. Whereas in our system, as in MTL-algebras, each of the join and meet operations will be obtainable, only, in terms of the other one plus the other connectives, see Lemma 6.6 and Part 2 [69].…”

Section: Prelinear Tied Adjointness Algebrasmentioning

confidence: 99%

See 2 more Smart Citations

The logic of tied implications, part 1: Properties, applications and representation

Morsi

Lotfallah

El-Zekey³

2006

Fuzzy Sets and Systems

Self Cite

View full text Add to dashboard Cite

A conjunction T ties an implication operator A if the identity A(a, A(b, z)) = A(T (a, b), z) holds [We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contains, besides residuated implications, several other implications employed in fuzzy logic. Nevertheless, we show that the algebraic inequalities of residuated algebras remain true for our tied implications, but in forms that distribute roles over the five connectives of the algebra. We apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules. We prove its equivalence to a scheme with one compound rule, when both schemata are interpreted by the compositional rule of inference, and all connectives are taken from one tied adjointness algebra. Then we quote another application of this rich theory, a notion of many-valued rough sets, which exhibit the basic mathematical behaviour of the rough sets of Pawlak. A comparator H is said to be prelinear if it satisfies H (y, z) ∨ H (z, y) = 1 for all y, z (Hájek). We introduce prelinear tied adjointness algebras, in which two comparators are prelinear. We provide a representation of those algebras, as subdirect products of tied adjointness chains, on the lines of Hájek's representation of BL-algebras. But our representations are more economical, because we employ minimal prime filters (on residuated lattices) only; rather than all prime filters.

show abstract

“…We devote the second part of this work [69] to syntax. There, we construct a predicate calculus for prelinear tied adjointness algebras.…”

Section: Resultsmentioning

confidence: 99%

Section: K(k(a Y) Y) A(a H (X Y)) H (H (X Y) Z)mentioning

confidence: 99%

See 1 more Smart Citation

The logic of tied implications, part 1: Properties, applications and representation

Morsi

Lotfallah

El-Zekey³

2006

Fuzzy Sets and Systems

Self Cite

View full text Add to dashboard Cite

show abstract

“…The relationship between the operations ↑ and ↓ is in fact an instance of Morsi's duality[55] combined with the inversion duality (i.e., the duality between R and R −1 ).…”

mentioning

confidence: 99%