Consequence Relations and Admissible Rules

This paper contains a detailed account of the notion of admissibility in the setting of consequence relations. It is proved that the two notions of admissibility used in the literature coincide, and it provides an extension to multi-conclusion consequence relations that is more general than the one usually encountered in the literature on admissibility. The notion of a rule scheme is introduced to cover rules with side conditions, and it is shown that what is generally understood under the extension of a consequence relation by a rule can be extended naturally to rule schemes. It is shown that such extensions correctly capture the intuitive idea of extending a logic by a rule.