As the title of this thesis suggests, it should be a contribution to the field of fuzzy logic. In order to make it clear, we firstly explain what we understand under the term “fuzzy logic”. The word “fuzzy” in a relation with mathematics was likely used for the first time by LA Zadeh in his paper [38] on fuzzy sets. He came with an idea to introduce a new kind of a set (so-called fuzzy set) to which its elements belong with a certain degree. In the classical setting an element either belongs to a set or not. If A is a classical set then the formula x∈ A is either absolutely true or absolutely false. In the case of a fuzzy set A, an element x can attain more than two degrees of its membership. Thus the formula x∈ A may be only partially satisfied. In order to define fuzzy sets in a proper way, we need a logical calculus which is able to cope with partially true statements. Such logical calculus is called fuzzy logic. Similarly as in the classical setting fuzzy logic can be propositional or predicate. We will focus here only on propositional fuzzy logics. This thesis is devoted to the research direction started by Hájek. He introduced one of the most successful fuzzy logics, so-called Hájek’s Basic Logic (BL for short). In his monograph [17] one can find a lot of interesting results about this calculus (like completeness theorem) and also connections to other already known calculi (like Lukasiewicz logic or Gödel logic). The logic BL has an algebraic type of semantics, so-called BL-algebras. They play an analogous role for BL as Boolean algebras for the classical logic. BL-algebras form in fact a subvariety of residuated lattices1. A motivation example of a BL-algebra is the real unit interval endowed with a continuous t-norm2 interpreting a conjunction and the corresponding residuum interpreting an implication. Such BL-algebra is often