Two Kinds of Difficulties

Abstract

Two distinct kinds of cases, going back to Crabbé and Ekman, show that the Tennant-Prawitz criterion for paradoxicality overgenerates, that is, there are derivations which are intuitively non-paradoxical but which fail to normalize. We argue that a solution to “Ekman’s paradox” consists in restricting the set of admissible reduction procedures to those that do not yield a trivial notion of identity of proofs. We then discuss a different kind of solution, due to von Plato, and recently advocated by Tennant, consisting in reformulating natural deduction elimination rules in general (or parallelized) form. Developing intuitions of Ekman we show that the adoption of general rules has the consequence of hiding redundancies within derivations. Once reductions to get rid of the hidden redundancies are devised, it is clear that the adoption of general elimination rules offers no remedy to the overgeneration of the Prawitz-Tennant analysis. In this way, we indirectly provide further support for our own solution to Ekman’s paradox.