It's interesting to take a look at Hegel, who makes the valiant attempt to turn Western logic into ontology in what must be one of the most difficult works of the entire philosophical tradition.
Here's a tit-bit from Hegel on contradiction:
"Es bewegt sich etwas nur, nicht indem es in diesem Jetzt hier ist und in einem anderen Jetzt dort, sondern indem es in einem und demselben Jetzt hier und nicht hier, indem es in diesem Hier zugleich ist und nicht ist. Man muß den alten Dialektikern die Widersprüche zugeben, die sie in der Bewegung aufzeigen, aber daraus folgt nicht, daß darum die Bewegung nicht ist, sondern vielmehr, daß die Bewegung der daseiende Widerspruch selbst ist." (Logik II Werke Bd. 6, Suhrkamp, Anm. 3 zu "3. Der Widerspruch" S.76)
"Something moves not in that in this now it is here and in another now there, but only in that in one and the same now it is here and not here, in that in this here it is and is not simultaneously. One has to concede to the old dialecticians [e.g. Zenon ME] the contradictions which they point out in movement, but it does not follow from this that therefore movement is not [non-existent], but rather, that movement is the existing contradiction itself."
Notice that the term "simultaneously" is temporal, just as the term, _hama_, in Aristotle's formulation of the principle of non-contradiction in Met. Book Gamma, meaning 'simultaneously', also is temporal. Contra Hegel, it is not "one and the same now", but the simultaneity of presence and absence at the same time, i.e. in the same ecstatically stretched, time-clearing, that enables something in motion to be here and also not here.
Movement is the "existing contradiction", and this contradiction can only exist because the temporal dimensions of presence and absence presence together, i.e. 'simultaneously'. Hegel does not get this far, but Heidegger does, at least implicitly. The 3D 'time-clearing', as I call it, is capacious, capacious enough also for contradictions to 'exist', i.e. stand out ec-statically in the existential time-clearing. It is the tacit presupposition of Aristotle's ontology of movement itself: _dynamis - energeia - entelecheia_: potential - energetic movement - actual, perfected presence. The "existing contradiction" of movement is only possible as such in its identity with the temporally 3D, triple vision of human being (Dasein) itself.
The 'contradiction' of the simultaneity of absencing and presencing is what enables movement at all. The logical principle of non-contradiction excludes such simultaneity -- all the worse for logic. And such simultaneity is nothing other than the unity of presence and absence in the time-clearing.
I also think that with the so-called 'scientific discovery' of quantum indeterminacy around 1925 (Heisenberg and Schrödinger), which was not 'experimental' but the blind consequence of the mathematical theories employed, the end of science was reached, i.e. its final, teleological, com-pletion, consummation, and simultaneously its demise. Its nemesis was time itself which ultimately repudiates mathematization.
With the publication of Sein und Zeit in 1927, the alternative was already given, i.e. the keys to another age but, of course, science did not give up and has not taken up Sein und Zeit one whit. On the contrary, it has continued for a further century without let-up on the road of effectivity, remaining blind and without any resolution of the creakiness of its foundations, of its unsolved problems, starting with that of so-called quantum gravity, an open sore festering now for well nigh a century.
Modern science is wounded, and a wounded beast is all the more savage and dangerous. It hides its wound and will remain in self-denial for as long as possible. Through its agents, the scientists, it will preserve its privileges, its honours and glowing reputation, at all costs, repudiating any other thought that comes along, dismissing it as fanciful, as 'non-verifiable', as 'useless'.
And who notices today that this is our situation? Not the media, not the scientists, physical or social, of course, and not the scholars, including the academic philosophers, even those with an hermeneutical, phenomenological bent.
Thinking thinks very slowly, waiting patiently for the attentive recipients of its messages.