Admissible Rules: From Characterizations to Applications

The admissible rules of a logic (understood as a structural consequence relation) may be described as rules that can be added to the logic without producing any new theorems, or, equivalently, as rules such that any substitution making the premises into theorems, also makes the conclusion into a theorem. However, this equivalence collapses once multiple-conclusion or other, more exotic, admissible rules are considered. The first aim of this paper is to explain how such distinctions can be explained and characterized. The second aim is to explore how these rules can be useful in determining properties of classes of algebras.

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