25/03 - Lucas Rosenblatt (Colloquium Logicae)
Capturing Naive Validity in the Strict-Tolerant Approach
Rejecting the structural rule of Cut has been recently proposed as a strategy to avoid both the usual semantic paradoxes and the so-called Validity Paradox. In this paper we consider if a theory that rejects Cut is capable of accurately representing its own notion of validity. We claim that the standard rules governing a naive validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a serious problem: an internalized version of Cut can be proved. We also evaluate a number of possible ways of escaping this difficulty.