5 ms·
> a single inconsistency in a formal logical system means you can prove anything. if both A and !A are true, then you can prove anything you wish via contradict
by ArtB 12y ago
> a single inconsistency in a formal logical system means you can prove anything. if both A and !A are true, then you can prove anything you wish via contradiction.
I believe this has been solved in intuitionistic logic (http://en.wikipedia.org/wiki/Intuitionistic_logic http://en.wikipedia.org/wiki/Intuitionistic_logic). If you don't except the law of the excluded middle and require constructive proofs you avoid many of these shenanigans.
- MereInterest 12y agoA question. Wouldn't this line of reasoning still work? It is entirely a constructive proof, as opposed to the GP's proof by contradiction. http://en.wikipedia.org/wiki/Principle_of_explosion#The_proof-theoretic_argument http://en.wikipedia.org/wiki/Principle_of_explosion#The_proo...
- quchen 12y agoAlthough intuitionistic logic does not have the axiom ------ (LEM) A ∨ ¬A that is, the law of the exluded middle (LEM) can not be derived "from nothing" for all propositions, it is still an inconsistency if you can prove A ∧ ¬A to be true, since ------ (Assumption) A ∧ ¬A ------ (Assumption) ------ (∧E2) A ∧ ¬A ¬A ------ (∧E1) ------ (Definition of ¬) A A ⇒ ⊥ ---------------------------- (⇒E) ⊥ --------------- (⇒I, discharge assumption) A ∧ ¬A ⇒ ⊥ --------------- (Definition of ¬) ¬(A ∧ ¬A) In other words, assuming A ∧ ¬A holds you can prove the false proposition ⊥ (from which you can prove anything). (Note that the De Morgan's law ¬(A ∧ ¬A) ⇒ A ∨ ¬A does not hold without the LEM, so the above proof cannot be simplified in terms of it!)