Tutorial on Admissible Rules in Gudauri

Most theorems have more than one proof and most theories have more than one axiomatization. Certain proofs or axiomatizations are preferable to others because they are shorter or more transparent or for some other reason. Our aim is to describe or study the possible proofs of a theorem or the possible axiomatizations of a theory. As the former is a special instance of the latter, by considering a theory consisting of one theorem, it suffices to consider theories. To describe the possible axiomatizations of a theory we first have to specify what we mean by a theory and what counts as an axiomatization of it. We assume that theories are given by consequence relations, and consider an arbitrary consequence relation to be an axiomatization of the theory if it has the same theorems as the consequence relation of the theory. In [1] Avron argues convincingly that in general a logic is more than its set of theorems, meaning that there exist logics which have the same set of theorems but which nevertheless do not seem to be equal. For example, because the proofs of certain theorems differ with the logic. Then the question what counts as an axiomatization of a certain theory becomes more complex in that one wishes to axiomatize certain other characteristics of the theory, such as certain inference steps, rather than just its theorems. In this paper, however, we restrict ourselves to the set of theorems as that part of a theory that an axiomatization has to capture. And as we will see, already in this case the variety of possible axiomatizations of a theory can be quite complicated and is in many cases not yet well-understood. Thus our main aim is a description of the consequence relations that have the same theorems as a given consequence relation. As it turns out, admissible rules are the central notion here, where a rule is admissible in a theory if it can be added to a theory but no new theorems can be proved in the extension. Clearly, such extensions are axiomatizations of the original theory, which is why admissible rules are so important in this setting.