In his book “After Finitude”, he explains that the principle of facticity (which he also refers to as “the principle of unreason”) stands in contrast to Leibniz’s “Principle of Sufficient Reason”, which states that anything that happens does so for a definite reason.
From pg. 33 of After Finitude:
“But we also begin to understand how this proof [the ontological proof of God] is intrinsically tied to the culmination of a principle first formulated by Leibniz, although already at work in Descartes, viz., the principle of sufficient reason, according to which for every thing, every fact, and every occurence, there must be a reason why it is thus and so rather than otherwise.
For not only does such a principle require that there be a possible explanation for every worldly fact; it also requires that thought account for the unconditioned totality of beings, as well as for their being thus and so. Consequently, although thought may well be able to account for the facts of the world by invoking this or that global law - nevertheless, it must also, according to the principle of reason, account for why these laws are thus and not otherwise, and therefore account for why the world is thus and not otherwise. And even were such a ‘reason for the world’ to be furnished, it would yet be necessary to account for this reason, and so on ad infinitum.
If thought is to avoid an infinite regress while submitting to the principle of reason, it is incumbent upon it to uncover a reason that would prove capable of accounting for everything, including itself - a reason no conditioned by any other reason, and which only the ontological argument is capable of uncovering, since the latter secures the existence of an X through the determination of this X alone, rather than through the determination of some entity other than X - X must be because it is perfect, and hence causa sui, or sole cause of itself.
If every variant of dogmatic metaphysics is characterized by the thesis that *at least one entity* is absolutely necessary (the thesis of real necessity) it becomes clear how metaphysics culminates in the thesis according to which *every* entity is absolutely necessary (the principle of sufficient reason). Conversely, to reject dogmatic metaphysics means to reject all real necessity, and a fortiori to reject the principle of sufficient reason, as well as the ontological argument, which is the keystone that allows the system of real necessity to close in upon itself. Such a refusal enjoins one us to maintain that there is no legitimate demonstration that a determinate entity should exist unconditionally.”
As to the principle of non-contradiction:
Pg. 60:
“We are no longer upholding a variant of the principle of sufficient reason, according to which there is a necessary reason why everything is the way it is rather than otherwise, but rather the absolute truth of a *principle of unreason*. There is no reason for anything to be or to remain the way it is; everything must, without reason, be able not to be and/or be other than it is.
What we have here is a principle, and even, we could say, an anhypothetical principle; not in the sense in which Plato used this term to describe the Idea of the Good, but rather in the Aristotelian sense. By ‘anhypothetical principle’, Aristotle meant a fundamental proposition that could not be deduced from any other, but which could be proved by argument. This proof, which could be called ‘indirect’ or ‘refutational’, proceeds not by deducing the principle from some other proposition - in which case it would no longer count as a principle - but by pointing out the inevitable inconsistency into which anyone contesting the truth of the principle is bound to fall. One establishes the principle without deducing it, by demonstrating that anyone who contests it can do so only by presupposing it to be true, thereby refuting him or herself. Aristotle sees in non-contradiction precisely such a principle, one that is established ‘refutationally’ rather than deductively, because any coherent challenge to it already presupposes its acceptance. Yet there is an essential difference between the principle of unreason and the principle of non-contradiction; viz. what Aristotle demonstrates ‘refutationally’ is that no one can *think* a contradiction, but he has not thereby demonstrated that contradiction is absolutely impossible. Thus the strong correlationist could contrast the facticity of this principle to its absolutization - she would acknowledge that she cannot think contradiction, but she would refuse to acknowledge that this proves its absolute impossibility. For she will insist that nothing proves that what is possible in-itself might not differ toto caelo from what is thinkable for us. Consequently the principle of non-contradiction is anhypothetical with regard to what is thinkable, but not with regard to what is possible.”
Continuing on pg. 77:
“It could be objected that we have conflated contradiction and inconsistency. In formal logic, an ‘inconsistent system’ is a formal system all of whose well-formed statements are true. If this formal system comprises the operator of negation, we say that an axiomatic is inconsistent if *every* contradiction which can be formulated within it is true. By way of contrast, a formal system is said to be non-contradictory when (being equipped with the operator of negation) it does not allow *any* contradiction to be true. Accordingly, it is perfectly possible for a logical system to *be* contradictory without thereby being inconsistent - all that is required is that it give rise to *some* contradictory statements which are true, without permitting *every* contradiction to be true. This is the case with ‘paraconsistent’ logics, in which some but not all contradictions are true. Clearly then, for contemporary logicians, it is not non-contradiction that provides the criterion for what is thinkable, but rather inconsistency. What every logic - as well as every logos more generally - wants to avoid is a discourse so trivial that it renders every well-formulated statement, as well as its negation, equally valid. But contradiction is logically thinkable so long as it remains ‘confined’ within limits such that it does not entail the truth of every contradiction.”
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