Today I Learned

Some of the things I've learned every day since Oct 10, 2016

130: Consequentia Mirabilis

Consequentia Mirabilis, also known as Clavius’ Law (after the German mathematician) is the principal in classical logic

(\neg X \rightarrow X) \rightarrow X.

That is, it’s the principle that if a proposition’s negation implies the proposition, then the proposition must be true.

Its use is similar in form to that reductio ad absurdum. To use reductio ad absurdum, one assumes that a proposition is false, sees that a contradiction follows, and from this concludes that the proposition must be true. To use consequentia mirabilis, by contrast, one assumes that a proposition is false, sees that the proposition itself follows, and from this concludes that the proposition must be true.

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