I often heard that if you can prove from some axiomatic system some proposition "P", and you can prove proposition "not P", then it means that you can prove any proposition in that axiomatic system. But why is it so ? Is it possible to prove this somehow using proof theory or mathematical logic ?
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2See Ex falso (or Principle of explosion): in many logic proof systems we have that from $P \land \lnot P$ we can derive $Q$, for $Q$ whatever. – Mauro ALLEGRANZA Mar 02 '18 at 13:17
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1The rules is grounded in the definition of Logical consequence. See the post: Prove tautological consequence from a set of premises for details: a formula $Q$ whatever is logical consequence of an inconsistent set of premises. – Mauro ALLEGRANZA Mar 02 '18 at 13:18
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Prove proposition A by contradiction. Assume (not A). Show (P and not P), which is a contradiction. Therefore, not (not A) which is equivalent to A. (i.e. by propositional logic [(P and not P) --> A] is a tautology for all formulas A and P ). – Ned Mar 02 '18 at 13:53
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See Ex falso (or Principle of explosion): in many proof systems we have that from $P∧¬P$ we can derive $Q$, for $Q$ whatever.
This rule can be a "basic" rule of the system (like the ($\bot$E) rule of Natural Deduction) or it can be derived from other basic rules (see Wiki's proof).
The rules is grounded in the definition of Logical consequence: a formula $Q$ whatever is logical consequence of an inconsistent set of premises.
But see Paraconsistent Logic for an overview of systems rejecting Ex falso.
Mauro ALLEGRANZA
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3The OP might be interested to know that there are some proof systems where proofs of $P$ and of $\neg P$ don't automatically lead to proofs of everything, but such "paraconsistent" proof systems do not match the usual notions of truth and are (I think it's fair to say) of far less interest than traditional systems that include the "ex falso quodlibet" rule. – Andreas Blass Mar 02 '18 at 14:06