  This paper is a follow-up to "Models of PT${}^-$ with internal induction for
total formulae." We give a strenghtening of the main result on the semantical
non-conservativity of the theory of PT${}^-$ with internal induction for total
formulae (PT${}^- +$ INT(tot)). We show that if to PT${}^-$ the axiom of
internal induction for all arithmetical formulae is added (PT${}^-$), then this
theory is semantically stronger than PT${}^- +$ INT(tot). In particular the
latter is not relatively truth definable (in the sense of Fujimoto) 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 "The expressive power of
truth."
