# How can we define in set theory, for each n, a partial truth predicate $Tr_n$ for $\Sigma_n$ ($\Pi_n$) formulae wich is itself $\Sigma_n$ ($\Pi_n$)?

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:

1. $$\forall x \forall y\ [(\ Tr\ \ulcorner \dot x = \dot y \urcorner \leftrightarrow x = y) \ \land \ (\ Tr\ \ulcorner \dot x \in\dot y \urcorner \leftrightarrow x \in y)] ;$$
2. $$\forall \ulcorner \sigma \urcorner\ [ Tr \ulcorner \lnot \sigma \urcorner \leftrightarrow \lnot Tr\ulcorner\sigma\urcorner \ ] ;$$
3. $$\forall \ulcorner \sigma \urcorner \forall \ulcorner \tau \urcorner\ [ Tr \ulcorner \sigma \land \tau \urcorner \leftrightarrow Tr\ulcorner \sigma \urcorner \land Tr\ulcorner\tau\urcorner\ ] ;$$
4. $$\forall \ulcorner \phi (u) \urcorner\ [ Tr \ulcorner \forall u \phi(u) \urcorner \leftrightarrow \forall x Tr \ulcorner \phi(\dot x) \urcorner \ ].$$

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?

Assume KP + "$$\omega$$ exists". Then for each transitive set $$X$$ there is a $$\Sigma_0$$ satisfaction relation, which is a set, i.e. the set $$S_0(X)=\{(\varphi,\vec{x})\bigm|\varphi\text{ is a (formal) }\Sigma_0\text{-formula and }\vec{x}\in X^{<\omega}\text{ and }X\models\varphi(\vec{x})\}.$$ (For $$n<\omega$$, say an $$n$$-partial-$$\Sigma_0$$ satisfaction relation is as above, but just has to work for the first $$n$$ formulas $$\varphi$$ (but still all $$\vec{x}\in X^{<\omega}$$). By induction, for all $$n<\omega$$ there is an $$n$$-partial-$$\Sigma_0$$ satisfaction relation. The statement "$$n<\omega$$ and $$S$$ is an $$n$$-partial-$$\Sigma_0$$ satisfaction relation for $$X$$" is $$\Sigma_0$$ (in free variables $$(n,X,S)$$). So by KP we can collect a set containing one for each $$n$$, and then separate them out and take their union, which is $$S_0(X)$$ as desired.)

Also, for every set $$X$$ the transitive closure of $$X$$ exists (by a simpler argument than the previous one).

The functions $$X\mapsto\mathrm{trancl}(X)$$ and $$X\mapsto S_0(X)$$ are, moreover, $$\Sigma_1$$-definable (in fact the graphs of these functions are $$\Sigma_0$$-definable).

It is now routine to deduce the existence of the truth predicates you asked about. For $$n=1$$: Define $$\mathrm{Sat}_{\Sigma_1}(\varphi,\vec{x})$$ to say "$$\varphi$$ is a (formal) $$\Sigma_1$$ formula of the form $$\varphi(\vec{z})\iff\exists\vec{y}\psi(\vec{y},\vec{z}),$$ in free variables $$\vec{z}$$, where $$\psi$$ is a (formal) $$\Sigma_0$$ formula and $$\mathrm{lh}(\vec{z})=\mathrm{lh}(\vec{x})$$, and there are $$\vec{w}$$ and a transitive set $$X$$ with $$\vec{x},\vec{w}\in X$$ and there is $$S=S_0(X)$$, and $$(\varphi,\vec{w}\frown\vec{x})\in S$$".

Then for all meta $$\Sigma_1$$ formulas $$\varphi$$, we get $$\mathrm{Sat}_{\Sigma_1}(\mathrm{G}(\varphi),\vec{x})$$ iff $$\varphi(\vec{x})$$, where $$G(\varphi)$$ is the code of $$\varphi$$ as a formal formula.

This easily leads to the formula for $$\mathrm{Sat}_{\Pi_1}$$ also.

Then $$\mathrm{Sat}_{\Sigma_2}(\varphi,\vec{x})$$ says "$$\varphi$$ is (formal) $$\Sigma_2$$ formula of the form $$\exists\vec{y}\psi(\vec{y},\vec{z})$$, where $$\psi$$ is $$\Pi_1$$ and $$\vec{z}$$ are free with $$\mathrm{lh}(\vec{z})=\mathrm{lh}(\vec{x})$$, and there is $$\vec{w}$$ such that $$\mathrm{Sat}_{\Pi_1}(\psi,\vec{w}\frown\vec{x})$$".