An assertion or claim is any statement that is arguable and provable.
To justify a claim is to prove that it is true.
A counterclaim is a claim that refutes another claim. In the context of your own writing, a counterclaim is an argument that goes against your assertion.
To rebut or refute an argument means to argue that it is false.
To qualify a claim is to argue that it is partially true (one part might be true and another false; or it might be true in some circumstances but not others)
Here is how to address a counterclaim in your writing. Depending on how thoroughly you want to address the counterclaim, you could both introduce and rebut it in a single paragraph or you can introduce the counterclaim in one paragraph and rebut it in the next.
Explain the counterclaim. Make sure you include both the claim and the logic that supports it.
Begin your rebuttal with a transition that shows contrast.
Rebut the counterclaim as thoroughly as you can, being sure to incorporate your own evidence when necessary. Here are some potential rebuttals you can make against the counterclaim:
Their evidence is wrong/outdated/unreliable. (Introduce better evidence.)
They are misinterpreting their evidence. (Analyze the evidence they present differently.)
Their logic is faulty. (Identify gaps in their reasoning.)