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Very True Operators

Chapter

Abstract

Inspired by the considerations of Zadeh (Synthesis, 30:407–428, 1975, [252]), Hajek in (Fuzzy Sets and Systems, 124:329–333, 2001, [105]) formalized the fuzzy truth-value very true. He enriched the language of the basic fuzzy logic BL by adding a new unary connective vt and introduced the propositional logic \(BL_{vt}\). The completeness \(BL_{vt}\) was proved in Liu and Wang (On v-filters of commutative residuated lattices with weak vt-operators, 2009, [168]) by using the so-called \(BL_{vt}\)-algebra, an algebraic counterpart of \(BL_{vt}\). In 2006, Vychodil (Fuzzy sets and systems, 157:2074–2090, 2006, [237]) proposed an axiomatization of unary connectives like slightly true and more or less true and introduced \(BL_{vt, st}\)-logic which extends \(BL_{vt}\)-logic by adding a new unary connective “slightly true” denoted by “st.” Noting that bounded commutative \(R\ell \)-monoids are algebraic structures which generalize, e.g., both BL-algebras and Heyting algebras (an algebraic counterpart of the intuitionistic propositional logic), Rachunek and Salounova taken bounded commutative \(R\ell \)-monoids with a vt-operator as an algebraic semantics of a more general logic than Hajeks fuzzy logic and studied algebraic properties of \(R\ell _{vt}\)-monoids in Rachunek (Soft Comput, 15:327–334, 2011, [196]).

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsMVGR College of EngineeringVizianagaramIndia

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