Capturing naive validity in the Cut-free approach
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Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the 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 for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty.
KeywordsValidity Paradoxes Strict-tolerant logic Substructural logics Cut
We are very grateful to Elia Zardini, Dave Ripley and two anonymous referees for extremely helpful comments on previous versions of this paper. Some of this material was presented at conferences in Campinas (CLE) and Buenos Aires (Buenos Aires Logic Group). We also owe thanks to the members of these audiences for their valuable feedback.
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