Take the following general sentence of propositional logic:

S = P → ~P (In words: “If P is true then P is false.”)

This is clearly an archetype of a self-contradictory statement.

The truth of a logical implication S = P → Q can be negated in just one way, namely when you observe a case where P entails ~Q instead of Q, and so directly contradicts S. That is, there is a complementary instance of S, its negation ~S = S' = P → ~Q.

If we replace Q with ~P, we have our original self-contradictory sentence S, and its negation becomes S' = P → ~(~P) = P → P, which is always true. (Technically, it’s called a tautology, which is indeed a special category of truth, viz. something trivially true.)

However, S' is not an axiom unless P itself is an axiom, in which case S' is merely an awkward restatement of P.

So, once again, the negation of a self-contradictory proposition is not necessarily an axiom.

By turning the situation around and starting with the tautology S = P → P where P could be an axiom or a theorem, we can now negate S in the opposite direction as before, and so arrive at the situation where the negation of a theorem creates the self-contradiction S = P → ~P.

If the above still doesn’t satisfy your requirements then I don’t understand properly what you are driving at, in which case perhaps you need to delve into some formal logic and the arcane field of axiomatic formal systems to crystallise your meaning.

'Luthon64