The two halves of disjunctive correctness
Cezary Cie\'sli\'nski, Mateusz {\L}e{\l}yk, Bartosz Wcis{\l}o
TL;DR
This paper investigates the logical strength of two parts of Disjunctive Correctness for the compositional truth predicate, showing one part is non-conservative over PA while the other can be added conservatively.
Contribution
It establishes the equivalence of one disjunctive correctness principle to bounded induction and provides a direct proof of nonconservativeness for DC.
Findings
The principle 'every true disjunction has a true disjunct' is non-conservative.
The converse 'any disjunction with a true disjunct is true' can be added conservatively.
Abstract
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic. In this article, we show that the principle "every true disjunction has a true disjunct" is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On the other hand, the converse implication "any disjunction with a true disjunct is true" can be conservatively added to PA. The methods introduced here allow us to give a direct nonconservativeness proof for DC.
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