I threw intuition and self-evidence in together simply out of laziness. They both fall to the criticism I set out, that if someone says that they do not see modus ponens as intuitive or self-evident, then there is no recourse left. However on the - what will we call it - status function account? - there is a recourse: you haven't used modus ponens the way we do. You've done it wrong. Notice that this is the same answer we might give a child who adds seven and three and gets eleven.
It might be argued that this still relies on an authority, and that may be so, but it is at least a more
distributed authority.
I'll add that intuition is fine in other situations - when judging a personality, or picking a path, or what you will. Notice that it is idiosyncratic even there: if challenged, can you justify picking this individual or that path? And here we have to share criteria; or perhaps conveniently our intuitions coincide.
The piece of autobiography displays laudable self awareness. I might be inclined to call your interpretation an insight rather than an intuition.
Non of this should be taken to detract from the import of intuition, nor to the respect in which it ought be held.
There's an approach to logic - and rationality - that supposes there are foundational propositions from which it is built, that justify everything that follows. Logic is seen here as a hierarchy; the epitome being axiomatic constructions. This fell into disfavour in the eighties, replaced by natural deduction, sequent calculus and such. Gentzen-style. I was taught, and indeed have taught, both approaches. Justification in the newer approach is contextual, dialogical, and structurally horizontal, focusing on rules of inference rather than axioms. Logic became more dynamic — a tool for reasoning, not a blueprint for metaphysical truth.
This lead to a picture of logic not as a hierarchy so much as a network, and to a pictures of justification not in terms of foundations but in terms of coherence.
This directly parallels the differences being played out here. At the risk of taking us back to the topic of the tread, we have those who see a need to find some absolute immovable foundation for what is real, and those who see what is real as grounded in context and action, in what we are doing.
Why does no one agree with it? — Count Timothy von Icarus
A trivial logic might begin with p^~p, from which everything follows. Formally, it's complete and consistent, but utterly unable to help us in deciding what is the case and what isn't. Go ahead and agree with it, if you like. It won't get you far. In such a trivial logic, everything is the case. That's why we don't use them, and why (almost?) no one agrees with them. But it seems we agree that they are useless. pv~p is much more interesting.
This is an example of how the choice of logics might be made. Pick one that does the job you want done, or that will extend and enhance the conversation.