A Note on Abstract Consequence Structures

Abstract

Tarski’s pioneer work on abstract logic conceived consequence structures as a pair (X, Cn) where X is a non empty set (infinite and denumerable) and Cn is a function on the power set of X, satisfying some postulates. Based on these axioms, Tarski proved a series of important results. A detailed analysis of such proofs shows that several of these results do not depend on the relation of inclusion between sets but only on structural properties of this relation, which may be seen as an ordered structure. Even the notion of finiteness, which is employed in the postulates may be replaced by an ordered substructure satisfying some constraints. Therefore, Tarski’s structure could be represented in a still more abstract setting where reference is made only to the ordering relation on the domain of the structure. In our work we construct this abstract consequence structure and show that it keeps some results of Tarski’s original construction.