Thomas Osborne in an earlier post (Oct. 25, 2013) regarding Garrigou-Lagrange and Africanus Spir raises interesting questions regarding the relation of Garrigou-Lagrange and Leibnitz. This calls to mind Gilson's imputations about GL's thought--the slur of a Wolff-like rationalism in GL kindred with the thought of Leibnitz. These suggestions of Gilson appear to be founded in political animosity (of the kind depicted by Shook in his biography of Gilson), not in any genuine correspondence of doctrine between GL and Leibnitz. GL expressly argues--with Thomas--against the view that there is a "best of all possible worlds"; against the view that all truths are analytic in the sense that they would be (as they are for Leibnitz) genuine entailments of the divine essence such that all created reality in its particularity could be deductively drawn from the vision of God. GL expressly says that sufficient reason is "analytic" only in the sense that it may be defended by an indirect demonstration or reductio ad absurdum. What can one say? There is only the word "analytic" between them, no common teaching. Moreover, Thomas does say, e.g., in Book II, Chapter 15 of the Summa contra gentiles, that “that which has no cause is something first and immediate; hence it is necessary that it be by reason of itself and in consequence of what it is” (“Quod causam non habet, primum et immediatum est; unde necesse est ut sit per se et secundum quod ipsum.”) This is an analogical not univocal principle of sufficient reason. One recalls that Humani Generis also refers to sufficient reason, clearly not designating by that language the doctrine of Leibnitz.
The argument against Hume is actually quite interesting. It is that every real relation requires a foundation in the real, because a non-existent foundation cannot sustain a real relation. This is per se nota. So, for a thing to be related to existence, there must be some real foundation for that relation either in the thing or outside the thing. Without such a foundation, the thing is not related since relation as such requires foundation. Thus when Hume says that a thing can come into being--and thus have a relation to existence--but that nonetheless it is neither its own reason for being nor is there a reason for being outside itself, he is saying that there is a thing that has a real relation to existence and does not have a real relation to existence. This is because for Hume nothing about it relates it to existence since the foundation of the relation is neither in it nor is it outside it, which is simply to say that there is no foundation whatsoever for such a relation. GL expressly deals with the claim that this is simply a petitio principii in explaining it as an indirect argument. His argument against Hume amounts to saying that nonbeing is not the foundation for a relation to existence--which is evident--and that since real foundation is required for the claim for real relation, this means that a thing cannot intelligibly be affirmed to have a real relation to existence if it is neither its own reason for being nor is there a reason for being extrinsic to it.
A similar but in some ways simpler way to respond to Hume is simply to observe Hume's failure to distinguish real and conceptual possibility. That something may be conceived or even only imagined does not suffice to indicate that it is really possible: the latter requires real evidence. What is not in the premises cannot be in the conclusion. So, if we are to conclude to real possibility, we must do so from real evidence. Now, when it is said something could come from nothing without any cause, one wishes then to know what the real evidence is for this proposition. But it is impossible that there be real evidence for this proposition: because one could never really distinguish between a thing having no real cause for being, and its having one of which one were merely contingently ignorant. Since there could never then be real evidence that something coming from nothing without a cause is really possible, the proposition that it is really possible for something to come from nothing without a cause is one that there could never be a reason to hold: a point analogous to GL's observation that where there is no foundation for a claim of real relation to existence such a claim is an absurd impossibility. It is, as it were, a bad cognitive check, more pathology than proposition. But this, too, is an indirect argument or demonstration.