Hello, I would like to know if this already has been researched.


There has been lot of research done, where logics are limited. They are often limited in the axioms or inference rules, which makes them weaker.


However, I am interested if someone has researched logics that are limited in arithmetic hierarchy. I am interested in a system that has only sentences of $\Pi^0\_2$.


Has someone worked that out?


Lucas



logics restricted in arithmetic hierarchy 
$\Pi\_2$ statements can be modeled in the form of a "question and answer." Specifically, the statement $(\forall a \in A)(\exists b \in B)\phi(a,b)$ can be thought of as follows: $A$ is a set of questions, $B$ is a set of answers, and $\phi(a,b)$ determines whether $b$ is a correct answer to question $a$. It turns out that this scenario lends itself to interpreting Girard's Linear Logic. This is described in detail by Andreas Blass in [Questions and Answers — A Category Arising in Linear Logic, Complexity Theory, and Set Theory](http://www.math.lsa.umich.edu/~ablass/qa.pdf); in fact, Andreas Blass has [several papers on the subject](http://www.math.lsa.umich.edu/~ablass/ll.html).


