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 $\text {CT}^-$. (shrink)

The main aim of our paper was to present three formal tools for comparing various axiomatic theories of truth. In Section 2 we aimed at showing that there are indeed many different approaches to defining a set of axioms for the notion of truth. In Section 3 we introduced three different \measures of strength" of axiomatic theories of truth, i.e. three reflexive and transitive relations on the set of axiomatic theories of truth. We have explained the intuition behind each of (...) them. The three relations were called : Fujimoto definability, model-theoretical strength, proof-theoretical strength. Then in the last section we described how they order the truth theories introduced in Section 2. We observed that theories made equivalent by the coarser relation can be strictly ordered by the next one. (shrink)

We introduce a principle of local collection for compositional truth predicates and show that it is arithmetically conservative over the classically compositional theory of truth. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the (...) case of the induction scheme. We analyse various further results concerning end-extensions of models of compositional truth and the collection scheme for the compositional truth predicate. (shrink)

This paper is a follow-up to [4], in which a mistake in [6] was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT−with internal induction for total formulae${$, denoted by PT−in [9]). We show that if to PT−the axiom of internal induction forallarithmetical formulae is added, then this theory is semantically stronger than${\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}\left$. In particular the latter is not relatively truth definable in the former. Last but not (...) least, we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in [9]. The truth theory we define is based on Weak Kleene Logic instead of the Strong one. (shrink)

This paper is an exposition of some recent results concerning various notions of strength and weakness of the concept of truth, both published or not. We try to systematically present these notions and their relationship to the current research on truth. We discuss the concept of the Tarski boundary between weak and strong theories of truth and we give an overview of non-conservativity results for the extensions of the basic compositional truth theory. Additionally, we present a natural strong theory of (...) truth which admits a number of apparently unrelated axiomatisations. Finally, we discuss other possible explications of the notion of ‘strength’ of axiomatic theories of truth. (shrink)