I was demonstrating the paradox you entangle yourself in by proposing that "assumptions are necessary" is the only irrefutable statement. Of course, I too hold that "assumptions are necessary" is irrefutable, but not that it is the only irrefutable proposition.
Furthermore, if we are disputing the necessity of assumptions, disputing the pressupositions of logic itself, then your argument that "assumptions are necessary" is circular, begging the question.
If I claim that the axioms of logic are arbitrary and assumptions are unecessary, it does no good for you to claim, "Before you can talk or even think about reality, knowledge, God, the Flying Spaghetti Monster, truth, and falsehood - an assumption must be made." That begs the question, assumes the proposition under question and is an ineffectual counterargument.
Check out paraconsistent logic, where "P & ~P" can be true.Anyways, if you decide to assume A and ~A and then the laws of logic, then you've assumed a set of premises that could lead to any conclusion.
Logic can be applied to itself and regularly is. For example, we make use of metalogic to prove theorems concerning logic, such as decidability, incompleteness, consistency, etc.It's simply a bad thing to do in RL, but it still has no effect on the correctness of A (Assumption is necessary) even in the presence of ~A. Point being: Make good assumptions. Other point being: Logic doesn't matter in this case- it can't be applied to it's most basic foundation.
In fact, Godel's incompleteness theorems imply that no system of logic can be both consistent and complete, so any demonstration to that fact immediately proves that the system is inconsistent. In other words, it doesn't pass its own standards.
Indeed, your entire argument, that assumptions are necessary, is an example of turning the presuppositions of logic upon themselves.