"According to both Plato and Aristotle,[2] Heraclitus was said to have denied the law of non-contradiction. This is quite likely[3] if, as Plato pointed out, the law of non-contradiction does not hold for changing things in the world. If a philosophy of Becoming is not possible without change, then (the potential of) what is to become must already exist in the present object. In "We step and do not step into the same rivers; we are and we are not", both Heraclitus's and Plato's object simultaneously must, in some sense, be both what it now is and have the potential (dynamis) of what it might become."
https://en.wikipedia.org/wiki/Law_of_noncontradiction

In my number theoretical research I've traced Gödel's incompleteness theorems back to Law of Identity, while showing that consistent number theory is possible without it. Maybe I'll write something about it some time.