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A common type of reductio ad absurdum is proof by contradiction (also called indirect proof), where a proposition is proved true by proving that it is impossible for it to be false. That is to say, if A being false implies that B must also be false and it is known that B is true, then A cannot be false and therefore A is true.
Two simple examples of reductio ad absurdum are:
Proposition: "Raising taxation rates always results in increased tax revenue."
Proposition: "Lowering taxation rates always results in increased tax revenue."
These can both be disproved using reductio ad absurdum as follows:
"If taxes were raised to 100% of income, individuals would not work, and companies would not operate, resulting in zero income, and thus zero tax. That is less than current tax income, thus the proposition is false."
"If taxes were lowered to 0%, no taxes at all would be collected. Zero will always be less revenue than even the lowest non-zero tax rate would produce, thus the proposition is false."
An example concerns the following statement, attributed to physicist Niels Bohr: "The opposite of every great idea is another great idea." Carl Sagan used a reductio ad absurdum argument to counter this claim. If this statement is true, he argued, then it would certainly qualify as a great idea - it would automatically lead to a corresponding great idea for every great idea already in existence. But if the statement itself is a great idea, its opposite ("It is not true that the opposite of every great idea is another great idea", provided "opposite" is a synonym of "negation" in Bohr's aphorism) must also be a great idea. The original statement is disproven because it leads to an absurd conclusion: that an idea can be great regardless of whether it is true or false.
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