There is a crazy idea that since Peano arithmetic has not been proven to be consistent (and cannot be proven so, by Goedel's Theorem), if you ever find a counterexample you can use nondisjunctions like these to immediately effectively prove anything. This may have dubious effects on objective reality. "I have one apple and one apple. Since I can prove that 1+1=5, I now have 5 apples." And boom! You got 5 apples.
There is a crazy idea that since Peano arithmetic has not been proven to be consistent (and cannot be proven so, by Goedel's Theorem), if you ever find a counterexample you can use nondisjunctions like these to immediately effectively prove anything. This may have dubious effects on objective reality. "I have one apple and one apple. Since I can prove that 1+1=5, I now have 5 apples." And boom! You got 5 apples.