At the very beginning of the paper Classes and truths in set theory (2012, APAL) by Kentaro Fujimoto, you can find this proposition:
For each n ≥ 1, we can define in KPω [Kripke-Platek set theory with infinity] a partial truth predicate $Tr_n$ for $\Sigma_n$-formulae ($\Pi_n$-formulae) which in itself is $\Sigma_n$ ($\Pi_n$); we mean here by a partial truth predicate a predicate which satisfies the Tarskian truth axioms restricted to bounded complexity, where the Tarskian truth axioms are the following:
- $\forall x \forall y\ [(\ Tr\ \ulcorner \dot x = \dot y \urcorner \leftrightarrow x = y) \ \land \ (\ Tr\ \ulcorner \dot x \in\dot y \urcorner \leftrightarrow x \in y)] ;$
- $\forall \ulcorner \sigma \urcorner\ [ Tr \ulcorner \lnot \sigma \urcorner \leftrightarrow \lnot Tr\ulcorner\sigma\urcorner \ ] ;$
- $\forall \ulcorner \sigma \urcorner \forall \ulcorner \tau \urcorner\ [ Tr \ulcorner \sigma \land \tau \urcorner \leftrightarrow Tr\ulcorner \sigma \urcorner \land Tr\ulcorner\tau\urcorner\ ] ;$
- $\forall \ulcorner \phi (u) \urcorner\ [ Tr \ulcorner \forall u \phi(u) \urcorner \leftrightarrow \forall x Tr \ulcorner \phi(\dot x) \urcorner \ ]. $
Here, $\sigma$ and $\tau$ range over (codes of) sentences, and $\phi$ ranges over (codes of) formulae. We are working with a formalized (in set theory) language $\mathscr L_\in^\infty$ $\supseteq \mathscr L_\in$, in wich we have a constant symbol $\dot x$ for each x $\in \pmb V$ (the standard set's universe).
The strategy for the "coding" of the syntax of $\mathscr L_\in^\infty$ is the same strategy that can be found, for example, in Devlin's book "Constructibility".
Well, in this context, how can we find the desidered partial truth predicates?