I have a student that repeatedly writes of “contesting against former arguments”. Is this correct? I know it is normally “contest an argument”, but I’m not sure if the other use is valid also.
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You don't contest against something; you contest that thing.
So in this case your student is incorrect.
According to Collins, both usages (with DO or prepositional phrase) are permitted:
However, the allowable types of DOs and PPs are not spelt out here (can one contest against an argument or only against an opponent?)