A logic is a pair (P,Q) where P is a set of formulas of a fixed propositional language and Q is a set of rules. A formula is deducible from X in the logic (P, Q) if it is deducible from XP via Q. A matrix is strongly adequate to (P, Q) if for any , X, is deducible from X iff for every valuation in , is designated whenever all the formulas in X are. It is proved in the (...) present paper that if Q = {modus ponens, adjunction } and P {E, R, E +, R +, E I, R I } then there exists a matrix strongly adequate to (P, Q). (shrink)

This abstract is a contribution to the paper by R. Wojcicki [2]. We use the terminology introduced there, . In this note we show that 1 0 For each n 2 there exists a logic whose degree of uniformity and degree of complexity are both equal to n. 2 0 There is a logic with the degree of uniformity equal to 2 @0 but with the degree of complexity equal to @0.

This paper is a continuation of [1] and so all the concepts and notations introduced in [1] will be used freely. We shall consider here the lattice of all standard consequence operations in the implicationless language intermediate between the intuitionistic and the classical consequence operation. The problem of characterizing this lattice was suggested by Prof. Wolfgang Rautenberg. We shall show that the lattice in question is denumerable and it is isomorphic to a chain of type ∗.