I am aware of some study into the logic of provability. It is generally taken to be intermediate in strength between S4 and S5 modal logics. Is there corresponding study into something like the logic of satisfiability or (semantic) validity?
The only mention of the topic I have been able to find is in this article by John Burgess, where he calls it "validity logic" (and says it is an S5 modal logic) and doesn't seem to cite any other work on the topic.
Is this subject more widely studied (perhaps under a different name)?