Theorem 7.24 gives us a very general, though nonconstructive, criterion for the existence of realizations, and hence for when a modal logic and a justification logic are counterparts, see Definition 7.2.

There are the obvious counterparts, S4 and LP, K and J, and so on, for which realization has a constructive proof. But we also discussed some less common modal logics, with some suggested justification counterparts.

Justification soundness results are in Section 4.3, and completeness with respect to Fitting models was shown in Section 4.5 using a canonical model construction, and this is enough to invoke Theorem 7.24. Indeed, at this point realization for several logics has already been established, and we just need to point it out. We begin by wrapping up our discussion of these logics and then go on to investigate an infinite family of modal logics we call Geach Logics, and a corresponding family of justification logics. It takes considerable work, but we will show we have realization results for all of these logics. This covers many standard cases and also shows that the fam­ily of modal/justification counterparts is infinite. The extent of the realization phenomenon is not actually known.

8.1

More on the topic Theorem 7.24 gives us a very general, though nonconstructive, criterion for the existence of realizations, and hence for when a modal logic and a justification logic are counterparts, see Definition 7.2.: