In Reductio ad Absurdum (RAA), we determine that a proposition P is false when it derives a contradiction. If we use this same derived contradiction as the premises to the Principle of Explosion (POE), we now prove this same proposition P is true. How does that not make Mathematical Logic inconsistent?
Reductio Ad Absurdum
(P → (A ∧ ¬A)) ⊢ ¬P
Principle of Explosion
(1) P → (A ∧ ¬A)
(2) (A ∧ ¬A) ⊢ P
(3) ∴ P
The above seems to show that we can correctly derive both P and ¬P thus making mathematical logic inconsistent. I always thought that the principle of explosion is ridiculous.
The answer to this question seems to be that we can't ever derive the kind of contradiction that POE needs from RAA so inconsistency is never derived.
Why is the Principle of Explosion (that starts with a contradiction) allowed when it is common knowledge that inference stops when reaching a contradiction?
It seems to me that a contradiction in inference is essentially the same thing as dividing by zero in computation. This makes it seem quite foolish to proceed onward with inference to explosion. Why is POE not abolished?
In the (Frank Hubeny) example of disjunction elimination there was no actual need for POE
(1) (P v Q)
(3) ∴ P
Why can't we simply get rid of the Principle of Explosion?
The updated rule would be that inference halts at after contradiction in exactly the same way that computation halts when divide by zero occurs. POE is simply not the way that human reasoning really works. In courtroom testimony a contradiction indicates perjury.
"Well, you'll have to get rid of one of the following: Distjunction introduction A ⊢ A V B"
Yes that one seems to be a good one to get rid of.
A="I like ice cream", B="Donald Trump ate an office building".
The answer to this question seems to be that getting rid of POE requires us to get rid of disjunction introduction which seems like an utterly useless appendage.
WILL LOGIC STILL WORK WITHOUT DISJUNCTION INTRODUCTION?