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Boolean Algebra
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
The two-element Boolean algebra β0 = {0, 1} only expresses two extreme or opposite relationships such as “negative” and “positive,” “no” and “yes,” “off” and “on,” and “false” and “true.” Thus, in order to express degrees of relationships, we introduce a new algebra known as an incline algebra to expand β0 to a closed unit interval [0, 1]. For a detailed account of incline algebra, see Cao et al. [16].
A variety of algebras closely related to subordination algebras
Published in Journal of Applied Non-Classical Logics, 2022
We will now prove that the categories and are isomorphic. The idea is to obtain the binary relation of the objects in out of the ternary relation of the objects in . This idea has also been used in Bezhanishvili, Bezhanishvili, Sourabh, et al. (2017) to obtain the duality between and by applying the generalised Jónsson–Tarski duality to the map obtained from a subordination ≼ of a Boolean algebra seen as a map from the product Boolean algebra to the two element Boolean algebra.
Metainferential duality
Published in Journal of Applied Non-Classical Logics, 2020
Bruno Da Ré, Federico Pailos, Damian Szmuc, Paula Teijeiro
Naturally, inference schemata are -valid if and only if all their inference tokens are valid. In addition, when some logic is induced by a single matrix , we may interchangeably refer to as . This being said, we can identify Classical Logic (, for short) with the (matrix) logic induced by the logical matrix , where is the usual two-element Boolean algebra counting with elements and to represent truth and falsity, respectively. With respect to Classical Logic, then, a system will be said to be (inferentially) subclassical if and only if some inference schema that is valid in Classical Logic does not hold in it.
Two proofs of the algebraic completeness theorem for multilattice logic
Published in Journal of Applied Non-Classical Logics, 2019
Oleg Grigoriev, Yaroslav Petrukhin
Let be the two element Boolean algebra (in what follows BA) with 1 (top) and 0 (bottom) as its only members. We define Boolean valuation δ as a mapping from the set to as follows: for any , iff and for any , iff (where j is supposed to satisfy ). Then we have the following lemma connecting this basic structure of Boolean algebra and ultralogical multilattices via extensions of the valuations. We will write to express the validity of a formula ϕ in .