In a very interesting article, Alain Badiou distinguishes three types or senses of negation: classical, intuitionist, and paraconsistent. Classical negation is strong negation, and obeys both the law of the excluded middle LEM) and the law of non-contradiction (LNC). On the other hand, intuitionist negation is weak negation, and obeys LNC but not LEM, allowing for the existence of intermediate values between P and non-P. Finally, the even « weaker » paraconsistent negation obeys LEM, but not LNC, allowing for both P and non-P to be true in the same domain and at the same time.
Much the same ground is covered, in less compressed form in the video of a talk by Badiou on « The Event as Creative Novelty ». Duane Rousselle gives a useful summary of the talk here.
These three senses of negation are useful in understanding François Laruelle’s evolution from classical philosophy to non-philosophy to non-standard philosophy. It is clear that for Laruelle sufficient philosophy is subjected to the laws of classical negation, at least at the manifest level. In contrast, non-philosophy makes use of an intuitionist acception of negation as it evolves in the multiplicity of values between P (philosophy) and non-P (its classical negation). This underlies the thematic that the « non- » of non-philosophy does not negate absolutely, in the strong sense, but extends what it negates, and so the law of the excluded middle is not respected.
Laruelle’s later move to « non-standard » philosophy is rather a move against the law of non-contradiction, in that we have something (NSP, non-standard philosophy) that is both P and non-P.
Laruelle’s blind spot is in his uniqueness hypothesis: the supposition that there is only one non-philosopher (himself), or later there is only one non-standard philosopher (disciples excepted). The underlying assumption is that all « philosophy » is sufficient, employing classical logic. Thus Laruellle gives a classical reading of Gilles Deleuze’s work, whereas it is far more plausible to read Deleuze in terms of intuitionist logic up to 1969 (his meeting with Guattari) and paraconsistent logic after (this is the sense of his collaboration with Guattari, it allows him to be both philosopher and non-philosopher, i.e. non-standard).