Given a class $C$ of arithmetical sentences, an arithmetical theory $T$ is said to be $C$-sound if all the theorems of $T$ which are in $C$ are true. For instance, $T$ is $\Sigma_1$-sound if all the $\Sigma_1$ theorems of $T$ are true.

Now, for some classes $C$, like the class of $\Sigma_1$ sentences, the statement "$T$ is $C$-sound" is expressible in the language of arithmetic. For other classes, like the class of all arithmetical sentences, $C$-soundness (which is just soundness) can't be definable in the language of arithmetic. But there are some classes, like the class of $\Pi_1$ sentences, which have a rarer property: not only is $C$-soundness definable in the language of arithmetic, it can be defined within $C$ itself.

However, I'm looking for an even rarer property:

does there exist a class $C$ for which $C$-soundness is definable within $C$,

andwhich is also closed under negation?

Or, failing that:

does there exist a class $C$ for which "$T$ is

not$C$-sound" is expressible within $C$?