Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension—in the sense of Feferman—is as strong as the schematic extension of KF, thus matching the strength of predicative analysis.

关于经典与非经典真理论的证明理论强度的问题最近受到了一些关注。一个特别方便的案例研究涉及不动点语义的经典和非经典公理化。众所周知，四值或三值逻辑中的非经典公理化明显弱于它们的经典公理化。在本文中，我们考虑为一级蕴含添加一个合适的条件——Hannes Leitgeb 最近在标签HYPE下研究的逻辑。我们特别表明，通过制定PKF over HYPE理论，人们获得了一种关于定点模型的合理理论，同时在理论上与经典对应物相提并论肯夫。此外，我们确定它的图式扩展——在 Feferman 的意义上——与KF的图式扩展一样强，从而与预测分析的强度相匹配。