Strongly finite logics : finite axiomatizability and the problem of supremium

Abstract

This is an extended version of a lecture read at the meeting organized by the Lodź section of the Philosophical Society on January 20, 1979. Extended fragments of this paper will appear in “Reports on Mathematical Logic”. This paper, which in its subject matter goes back to works on strongly finite logics (e.g. [8], [9]), is concerned with the following problems: (1) Let Cn1, Cn2 be two strongly finite logics over the same propositional language. Is the supremum of Cn1 and Cn2 (noted as Cn1 U Cn2) also a strongly finite operation? (2) Is any finite matrix (or more precisely, the content of any finite matrix) axiomatizable by a finite set of standard rules? The first question can be found in [9] (and also in [11]). The second conjecture was formulated by Wolfgang Rautenberg, but investigations into this problem had been carried out earlier in works of many logicians (e.g. the known theorem of Mordchaj Wajsberg [7], see also [5]). Moreover, Stephen Bloom [1] posed a conjecture stronger than (2) that: the consequence determined by a finite matrix (a strongly finite consequence, see [9]) is finitely based, i.e. it is the consequence generated by a finite set of standard rules. This hypothesis was, however, disproved by Andrzej Wronźski [10] (and also by Alasdair Urquhart [6]). In the present paper it is shown that neither (1) nor (2) holds true. The negative answer to (2) can be viewed as a generalization of the result given by Andrzej Wronźski [10] (or by [6]) (Fragment tekstu).