On the Costs of Classical Logic

This article compares classical and nonclassical axiomatisations of the fixed-point semantics developed by Kripke : 690–716, 1975). Following the line of investigation of Halbach and Nicolai : 227–257, 2018), we do not compare KF and PKF qua theories of truth simpliciter, but rather qua axiomatisations of the Kripkean conception of truth. We strengthen the central results of Halbach and Nicolai and Nicolai : 101–130, 2018), showing that, on the one hand, there is a stronger sense in which some variants of KF and some variants of PKF can be taken to be, truth-theoretically, equivalent. On the other hand, we show that this truth-theoretical equivalence is not preserved by some other variants of KF and PKF, arguing that the PKF variants are more adequate axiomatisations of the fixed-point semantics than the corresponding KF variants.