Abstract

This paper, which in its subject matter goes back to works on strongly nite logics , is concerned with the following problems: Let Cn1; Cn2 be two strongly nite logics over the same propositional language. Is the supremum of Cn1 and Cn2 also a strongly nite operation? Is any nite matrix axiomatizable by a nite set of standard rules? The rst question can be found in [9] . The second conjec- ture was formulated by Wolfgang Rautenberg, but investigations into this problem had been carried out earlier in works of many logicians . Moreover, Stephen Bloom [1] posed a conjecture stronger than that: the consequence de- termined by a nite matrix is nitely based, i.e. it is the consequence generated by a nite set of standard rules. This hypothesis was, however, disproved by Andrzej Wronski [10] . In the present paper it is shown that neither nor holds true. The negative answer to can be viewed as a generalization of the result given by Andrzej Wronski [10]