Compositional truth with propositional tautologies and quantifier-free correctness (Q6139242)

This is another in a series of papers by various authors on what has been dubbed by Ali Enayat the Tarski Boundary Problem. By the well-known theorem of Kotlarski, Krajewski and Lachlan, the theory \(\mathsf{CT}^-\), that extends Peano Arithmetic (\(\mathsf{PA}\)) by compositional axioms for a truth predicate, is conservative over \(\mathsf{PA}\). The theory \(\mathsf{CT}\) obtained by adding to \(\mathsf{CT}^-\) the induction schema for the full language with the truth predicate symbol \(T\) is not a not a conservative extension of \(\mathsf{PA}\), and so is \(\mathsf{CT}_0\), which is \(\mathsf{CT}\) with the induction axioms for formulas with \(T\) is restricted to bounded formulas. Many seemingly innocuous extensions of \(\mathsf{CT}^-\) are pushed into the nonconservative side and often those extensions turn out to be equivalent to \(\mathsf{CT}_0\). By a result of \textit{A. Enayat} and \textit{F. Pakhomov} from [Arch. Math. Logic 58, No. 5--6, 753--766 (2019; Zbl 1477.03250)], prominent among them is obtained by adding to \(\mathsf{CT}^-\) the principle of disjunctive correctness \(\mathsf{DC}\). The main result of Wcisło's paper paper adds to the list \(\mathsf{CT}^-\) extended by the propositional soundness principle \(\mathsf{PS}\) that states that any propositional tautology is true (according to the truth predicate). The proof is in two parts. It is first shown that \(\mathsf{CT}^-+\mathsf{PS}\) extended by the quantifier-free correctness principle \(\mathsf{QFC}\) is equivalent to \(\mathsf{CT}_0\), and to prove this Wcisło applies his own technique of disjunctions with stopping conditions. Then, \(\mathsf{QFC}\) is eliminated by showing that \(\mathsf{CT}^-+\mathsf{QFC}\) is conservative over \(\mathsf{PA}\). This second part uses a variant of the Enayat-Visser construction of satisfaction classes introduced in [\textit{A. Enayat} and \textit{A. Visser}, ``New constructions of satisfaction classes'', Log. Epistemol. Unity Sci. 36, 321--335 (2015)]. Despite relaying on the results by Enayat and Pakhomov, and Enayat and Visser, the paper is self-contained and full proofs of the variants of these results are given in two appendices. The paper is very well written and it can serve as a good introduction to the Tarski boundary problem.