Cut elimination for the second order propositional logic with Hilbert's ε-symbol, extensionality, and full comprehension

Abstract

The cut-elimination technique is well developed for classical higher order systems, but, since the normal form (not normalization) theorem was established by Mints[1, 2, 3, 4, 5], not much further progress has been achieved for the logical systems with an ε-symbol.

We give a cut elimination procedure for the second order propositional logic with Hilbert's ε-symbol, extensionality and full comprehension.

We prove that selective elimination of the ε-symbol and quantifiers is possible, so that a cut formula becomes essentially quantifier- and epsilon-free, after which cuts can be eliminated by the standard Gentzen procedure.