Normalisation and Subformula Property for a System of Classical Logic with Tarski's Rule, and a Correction

TL;DR

This paper formalizes a classical logic system with general rules, introduces normalization procedures, and proves the subformula property, while addressing and correcting previous errors in the theoretical framework.

Contribution

It develops a normalization procedure for a classical logic system with Tarski's rule, ensuring deductions are in normal form and satisfy the subformula property, with a correction to prior inaccuracies.

Findings

Deductions can be transformed into normal form

Normal form deductions satisfy the subformula property

Correction of previous errors does not affect main results

Abstract

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of `maximal formula', `segment' and `maximal segment' suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski's Rule is treated as a general introduction rule for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic. The Correction added to the end of the paper corrects an error: Theorem 2 is mistaken, and so is a corollary drawn from it as well as a corollary that was concluded by the same mistake. Luckily this does not affect the main result of the paper.