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.
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