Ali Enayat had asked whether two halves of Disjunctive Correctness ([Formula: see text]) for the compositional truth predicate are conservative over Peano Arithmetic (PA). In this paper, 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 [Formula: see text]. The methods introduced (...) here allow us to give a direct nonconservativeness proof for [Formula: see text]. (shrink)

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ($\mathrm {CT}_0$). Furthermore, we (...) extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of$\mathrm {CT}_0$. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of$\mathrm {CT}_0$. (shrink)

In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as $$\Delta _0$$ Δ 0 -induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a (...) routine argument that the principle of quantifier-free correctness is itself conservative. (shrink)

We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic ${\textsf {(PA)}}$. More specifically, let Elementary Arithmetic ${\textsf {(EA)}}$ be the fragment $\mathsf {I}\Delta _0 + \mathsf {Exp}$ of ${\textsf {PA}}$, and let ${\textsf {CT}}^-[{\textsf {EA}}]$ be the extension of ${\textsf {EA}}$ by the commonly studied axioms of compositional truth ${\textsf {CT}}^-$. We investigate both local and global properties of the family of first order theories of the form ${\textsf {CT}}^-[{\textsf {EA}}] +\alpha $, where $\alpha (...) $ is a particular way of expressing “ ${\textsf {PA}}$ is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to ${\textsf {PA}}$ and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of ${\textsf {PA}}$. (shrink)