We introduce a notion of syntactical truth predicate (s.t.p.) for the second order arithmetic PA$^2$. An s.t.p. is a set T of closed formulas such that: (i) T(t = u) if and only if the closed first order terms t and u are convertible, i.e., have the same value in the standard interpretation (ii) T(A $\rightarrow$ B) if and only if (T(A) $\Longrightarrow$ T(B)) (iii) T($\forall$ x A) if and only if (T(A[x $\leftarrow$ t]) for any closed first order term t) (iv) T($\forall$ X A) if and only if (T(A[X $\leftarrow\triangle$]) for any closed set definition $\triangle = \{ x \mid D(x)\}$). S.t.p.'s can be seen as a counterpart to Tarski's notion of (model-theoretical) validity and have main model properties. In particular, their existence is equivalent to the existence of an $\omega$-model of PA$^2$, this fact being provable in PA$^2$ with arithmetical comprehension only.