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 $\mathrm {CT}_0$ .

We introduce a tool for analysing models of $\text {CT}^-$ , the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan’s theorem that the arithmetical part of models of $\text {CT}^-$ are recursively saturated. We also use this tool to provide a new proof of theorem from [8] that all models of $\text {CT}^-$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for a set of formulae whose syntactic depth forms a nonstandard cut which cannot be extended to a full truth predicate satisfying $\text {CT}^-$ .