Very True Operators

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]).