Some consequence relations on propositional formulas

This paper is an extended version of the earlier one, which was presented at the ETRAN 2010 conference. In this version, some theorems are proved, and several typical examples of applications of consequence relations are given. Consequence relations on propositional formulas are binary relations on propositional formulas that represent certain types of entailment — formal or semi-formal derivation of conclusion from a certain set of premises. Some of well known examples are classical implication (standard logical entailment), preference relations (i.e. relations that satisfy Reflexivity, Left logical equivalence, Right weakening, And, Or and Cautious monotonicity) rational relations (i.e. preference relations that also satisfy rational monotonicity), consequence relations (prime examples are qualitative possibilities and necessities) etc. More than two decades various consequence relations are used in automated decision making, product control, risk assessment and so on. The aim of this paper is to give a short overview of the most prominent examples of consequence relations.