It is difficult to justify assuming p in the antecedent if we know a priori that it is false in the consequent. I think in general that it is not sufficient to shoot down another argument if one's own presuppositions don't work either. This is just the Golden Rule. One paradigm I have found helpful is the notion from mathematical logic of a conservative extension. This is a first-order formalization of a procedure in model theory found in Church, Tarski, and Kleene, among others.
Quine is probably good here too.
If such consistency is provable then the new axiom may safely be added. If not, the new axiom is rejected. T' is called a conservative extension of the initial theory T. This line of thinking may also tie in to the Carnap-Popper debates about falsifiability, and Thomas Kuhn's Structure of Scientific Revolutions. At the risk of sounding Clarkian, this is in some ways how I think about presuppositions. I think the Decalogue and Christology are central to our faith.