Feser said in his publication, "Scholastic Metaphysics - A Contemporary Introduction" that if we reject the PSR, we're undermining any possibility of rational inquiry. Also in Pruss' paper, "PSR and Probability", he says that a localized version of the PSR or rejection of the PSR entails that we can't rationally estimate the probability of any event.
I'll concede that at best it rejects knowledge of cause and effect, which is the basis for scientific reasoning. However I don't see how just because the PSR is false, our thoughts are true or false.
One could also invoke Schopenhauer's fourfold roots of the PSR, without it entailing theism at all.