Reasons given in proof or rebuttal are called...

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Reasons given in proof or rebuttal are called arguments. An argument is a connected set of statements where the premises provide support for a conclusion. In proof, you lay out premises that justify the conclusion; in rebuttal, you offer counterpremises or reasons aimed at challenging the opposing view, and that whole interaction is still an argument. Evidence refers to the data or facts used within an argument to back up the premises, hypotheses are testable conjectures, and theories are broad, well-supported explanations. The term that best captures the act of presenting those reasons to justify or oppose a claim is arguments.