UPDATE: Realized that this is more about the Law of Noncontradiction more than about the Law of the excluded middle, at least the way I discussed it in the post. Maybe I will talk about the relation of Hegel and Law of the Excluded Middle in some other post. One more proof that I should read the posts before committing.
The Law of noncontradiction says that it can’t be both that a proposition is true, and also that its denial to be true.
Did Hegel deny this law? We can say both yes and no.
I guess, it is best to explain this on example.
Take for example the case of being (“is X”), that is – predicating X to something. Now, it is very normal, that either “is X” is true for something, or “is not X”.
And if we stay on the level of being, accepting both “is X”, and “is not X” would be contradiction.
However, think of the case of change. In the case of change, we have a case where both something is X, and is not X.
But, someone would say, that is different because something important is left outside of those propositions. Namely, in case of change, we have “is X at time t1”, and “is not X at time t2”; so those two are not contradiction.
On that we need to point to the notion of time, and where does it come from. For Hegel (and for me), the time is an abstraction from change (and more here, in my comment about SEP article on change). That is, what we have in world are changing things, or changing states of affairs, and ‘time’ is only one way to talk about the relation between two abstract states of the changing states of affairs. that is, we talk for example, which abstract state of affairs came before which other one, or how many times some change taken as a unit repeated while some (measured) change happened.
What Hegel says, then, is this: If you try to describe change through being, the best you can do is say about something that changes that both it “is X”, and that it “isn’t X”. But, this is contradiction, and that doesn’t show that change is impossible, but that simply change can’t be described through being (and non-being). As pointed you can’t use “at time t” to make the distinction between two predicates and avoid the contradiction, because the phenomenon of time is grounded (abstracted from) phenomenon of change.
So, we can point to the how the Law of Noncontradiction isn’t true for Hegel, because when relating richer notion (like that of change), to a poorer one (like that of being), and when thinking on level of the richer one, we can say that both “is X”, and “isn’t X” (or both predicates are contained in change as moments, as Hegel would say).
But also, on the level of the phenomenon of being alone, the Law of Noncontradiction, is seen as valid, and producing contradiction, which points that when thinking of change the notion of being (and non-being) is not enough.
What we can say, I guess, is then that, The Law of Noncontradiction is removed in Hegel’s system as some kind of absolute logical axiom, and is changed with somewhat richer dialectical relations among notions.