The obvious objection to everything so far is: if contradiction is fine, if a thing can both be and not be itself, then what stops a person from claiming anything at all? Couldn't you just say the sky is blue and not blue and call it profound? Hegel takes this objection seriously enough to build an entire method around answering it, and the answer is one of the more elegant pieces of argument in the whole system.

Hegel distinguishes between ordinary propositions and speculative propositions. Ordinary propositions are the sentences that formal logic handles perfectly well: "a cat is a mammal," "all bachelors are unmarried." They are not contradictory on their face, and nobody needs Hegel's help to understand them. Speculative propositions look nonsensical by comparison — the most famous is "pure being is pure nothing" — because they assert an identity between two things that formal logic treats as opposites.

The crucial move is that Hegel never simply announces a speculative proposition. He arrives at one, by taking an ordinary proposition with complete seriousness and following its logic wherever that logic actually goes. The opening of the Science of Logic is the clearest demonstration. Start with the ordinary, entirely reasonable claim that pure being — being with no further qualification, no color, no shape, no properties — is completely indeterminate. Now ask what distinguishes it from pure nothing, which is also completely indeterminate. If both have exactly zero distinguishing features, there is no way to tell them apart. The two ideas collapse into each other not because Hegel wanted them to, but because the ordinary proposition, followed honestly, drags the reader there. Nobody chose the contradiction. The logic produced it.

This is why Hegel refuses two easier paths that were both available to him. He could have simply rejected the principle of non-contradiction, the rule, going back to Aristotle's Metaphysics, that a thing cannot both be and not be in the same respect at the same time. He does not reject it. Instead he shows that if you take the principle with total rigor, it generates the very contradictions it was supposed to forbid. The principle refutes itself when followed seriously enough.

Nor does Hegel do what Gottlob Frege later did and simply build a new, cleaner logical system to replace the old one. That move would miss the point of the whole exercise, because any logical system, pushed hard enough, will generate contradiction; inventing a fresh system only relocates the problem rather than removing it, and worse, if you build contradiction into the system as a normal, expected feature from the outset, it stops disturbing anyone and loses the power that made it worth noticing in the first place.

There is a companion challenge aimed at an even more basic law: the Law of Identity, formulated by Gottfried Wilhelm Leibniz as the simple claim that A is A, that each thing is purely and only itself. Hegel's objection is very direct: the moment you try to state the law, you have already broken it. Writing "A is A" requires writing A twice, and the second A is doing different work than the first in the sentence, one is the subject, the other its predicate. You have introduced difference into the very sentence meant to deny that difference matters. As Hegel puts it in the Science of Logic: "Such talk of identity contradicts itself." A thing gains its identity not by standing alone but through its relationship to what it is not; strip away every relation to otherness and you are left with nothing distinctive at all, no boundary separating the thing from everything around it. Otherness, on this account, isn't an intruder threatening identity from outside. It is what makes identity possible in the first place.