Abstract

This article deals with algebraic logic. In particular, it discusses the theory of consequence operations and the general concept of logical independency. The advantage of this general view is its gr- eat applicability: The stated properties of consequence operations hold for almost every logical system. The notion of independency is well known and important in logic, philosophy of science and mathematics. Roughly speaking, a set is independent with respect to a consequence operation, if none of its elements is a consequence of the other elements. The property of being an independent set guarantees therefore that none of its elements is superfluous. In particular, I'm going to show fundamental results for every consequence operation, and hence for every logic: no infinite independent set is finite axiomatizable, and every finite axiomatizable set has relative to a finitary consequence operation an independent axiom system. The main result is that in sentential logic every set of formulas has an independend axiom system.