On the degree of complexity of sentential logics. A couple of examples

The first part of the paper is a reminder of fundamental results connected with the adequacy problem for sentential logics with respect to matrix semantics. One of the main notions associated with the problem, namely that of the degree of complexity of a sentential logic, is elucidated by a couple of examples in the second part of the paper. E.g., it is shown that the minimal logic of Johansson and some of its extensions have degree of complexity 2. This is the first example of an exact estimation of the degree of natural complex logics, i.e. logics whose deducibility relation cannot be represented by a single matrix. The remaining examples of complex logics are more artificial, having been constructed for the purpose of checking some theoretical possibilities.The paper was presented to the Polish Philosophical Society, Wrocaw Branch, at its meeting on March 27^th, 1980.The authors wish to thank both the referees of Studia Logica for their helpful and very insightful remarks. Following their criticism, we have been able to improve the style and structure of our presentation. In particular, we are indebted to the referees for pointing out a gap in the original proof of Theorem 2, and we have incorporated into the revised text a corrected proof of step (2.1) which one of them was kind enough to supply in detail.