Proof and Paradox

Open Problems in Proof-Theoretic Semantics

“…So the wellbehaviour of a definition in the case of logic, where we do have cut elimination, is due to the fact that it obeys certain principles, which in the general case cannot be expected to hold. This connects the proof theory of clausal definitions with theories of paradoxes, which conceive paradoxes as based on locally correct reasoning (Prawitz [44] (Appendix B), Tennant [68], Schroeder-Heister [62], Tranchini [71]). For the situation that obtains here, Hallnäs [28] proposed the terms 'total' vs. 'partial' in analogy with the terminology used in recursive function theory.…”

Normalizability, cut eliminability and paradox

“…What Yablo's paradox reveals is that we have to contend not only with vicious circles, but also with vicious helices. Tennant (1982) concentrated on logico-semantic paradoxes, but did also examine Russell's Paradox. Prawitz (1965) had shown how certain naïvely formulated introduction and elimination rules in set theory would-despite the fact that they appeared to admit of a reduction procedure-block the proof that all natural deductions can be brought into normal form.…”

“…Reduction sequences that eventually produce a proof in normal form terminate. Tennant (1982) proposed a proof-theoretic criterion, or test, for paradoxicalitythat of non-terminating reduction sequences initated by the 'proofs of ⊥' associated with the paradoxes in question (p. 271). In that paper, the subsequent focus was on looping reduction sequences.…”

Contractions of Noncontractive Consequence Relations

“…For more on noncontractive approaches to paradoxes, see [25,11,20,36,2,33,15]. For more on nontransitive approaches, see [34,35,28,3,32]. Both approaches are susceptible to a cut-elimination style non-triviality proof (see [20,36] for the noncontractive approaches, and [28] for the nontransitive approaches).…”