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Reductio Ad Absurdum
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The Latin phrase reductio ad absurdum means "reduction to the absurd." It is used to refer to the process of demonstrating that an idea is probably false by first assuming its truth, and then showing how that truth leads to absurd conclusions which cannot possibly be true. The process is ... used in ethical philosophy by assuming the moral validity of some principle, and then showing that acceptance of it would lead to very unethical consequences.
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Abstract: It is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg-Kohn theorem for ensembles of fractionally occupied states for isolated many-electron Coulomb systems with Coulomb-type external potentials by Gross et al. [Phys. Rev. A 37, 2809 (1988)] is self-contradictory since the to-be-refuted assumption (negation) regarding the ensemble one-electron densities and the assumption about the external potentials are logically incompatible to each other due to the Kato electron-nuclear cusp theorem. It is ... proved that the Kato theorem itself provides a satisfactory proof of this theorem.
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It seems that the Reductio ad absurdum method of disproving a proposition is fundamentally flawed. Every mathemathical proposition that rests solely upon it needs to have chapter and verse re-opened. What new gems of mathematics lie discarded in the 'Reductio wasteland'?
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The first of these is reductio ad absurdum in its strictest construction and the other two cases involve a rather wider and looser sense of the term. Some conditionals that instantiate this latter sort of situation are:
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By the way, was Bennett's tactic described above a "reductio-ad-absurdum"? It doesn't seem to be one: rather, he was rejecting prudential justifications of what is, for him, a moral-ontological stance. His way of doing this was to demonstrate (in his lights) that once you admit ANY prudential justifications of a moral crime, you in effect allow ALL prudential justifications: and then it's no longer a moral issue.
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Occasionally, some disingenuous theist will see an atheist using reductio ad absurdum and point out that the theist is speaking as if assuming that God exists, so obviously he ... believes in God, no matter how much he denies it! This, of course, is utterly invalid.
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