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.
two or more propositions are asserted that cannot both possibly be true.
When the principle of explosion: (A ∧ ¬A) ⊢ B ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]') https://en.wikipedia.org/wiki/Principle_of_explosion
This diverges from the model of the syllogism because it fails to maintain a semantic link between the terms. Because it diverges from the syllogism it is easy to see that it allows semantic nonsense to be construed as correct reasoning.
Major premise: All humans are mortal.:
Minor premise: All Greeks are humans.:
Conclusion: All Greeks are mortal.:
Each of the three distinct terms represents a category. From the example above, humans, mortal, and Greeks: mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term;
Thus maintaining a semantic link between the premises and the conclusion.
The major premise defines a universal property M of set H.
The minor premise defines a G subset of H.
H has the property of M:
∀h ∈ H ( M(h) )
G is a subset of H:
G ⊆ H ∴ G has the property of M