What is "reductio ad absurdum"

Demonstration \Dem`on*stra"tion\, n. [L. demonstratio: cf. F. d['e]monstration.]

1. The act of demonstrating; an exhibition; proof; especially, proof beyond the possibility of doubt; indubitable evidence, to the senses or reason.

   Those intervening ideas which serve to show the agreement of any two others are called ``proofs;'' and where agreement or disagreement is by this means plainly and clearly perceived, it is called demonstration.
   --Locke.

2. An expression, as of the feelings, by outward signs; a manifestation; a show. See also sense 7 for a more specific related meaning.

   Did your letters pierce the queen to any demonstration of grief?
   --Shak.

   Loyal demonstrations toward the prince.
   --Prescott.

3. (Anat.) The exhibition and explanation of a dissection or other anatomical preparation.

4. (Mil.) a decisive exhibition of force, or a movement indicating an attack.

5. (Logic) The act of proving by the syllogistic process, or the proof itself.

6. (Math.) A course of reasoning showing that a certain result is a necessary consequence of assumed premises; -- these premises being definitions, axioms, and previously established propositions.

7. a public gathering of people to express some sentiment or feelings by explicit means, such as picketing, parading, carrying signs or shouting, usually in favor of or opposed to some action of government or of a business.

8. the act of showing how a certain device, machine or product operates, or how a procedure is performed; -- usually done for the purpose of inducing prospective customers to buy a product; as, a demonstration of the simple operation of a microwave oven.

   Direct demonstration, or Positive demonstration, (Logic & Math.), one in which the correct conclusion is the immediate sequence of reasoning from axiomatic or established premises; -- opposed to

   Indirect demonstration, or Negative demonstration (called also reductio ad absurdum), in which the correct conclusion is an inference from the demonstration that any other hypothesis must be incorrect.

Latin, literally "reduction to the absurd." Absurdum is neuter of absurdus. See reduction + absurd. The tactic is useful and unobjectionable in proofs in geometry.

n. (context mathematics logic English) The method of proving a statement by assuming the statement is false and, with that assumption, arriving at a blatant contradiction.

n. (reduction to the absurd) a disproof by showing that the consequences of the proposition are absurd; or a proff of a proposition by showing that its negation leads to a contradiction [syn: reductio]

Reductio ad absurdum ( Latin: "reduction to absurdity"; pl.: reductiones ad absurdum), also known as argumentum ad absurdum (Latin: "argument to absurdity", pl.: argumenta ad absurdum), is a common form of argument which seeks to demonstrate that a statement is true by showing that a false, untenable, or absurd result follows from its denial, or in turn to demonstrate that a statement is false by showing that a false, untenable, or absurd result follows from its acceptance.

First recognized and studied in classical Greek philosophy (the Latin term derives from the Greek ἐις ἀτοπον ἀπαγωγή or eis atopon apagoge, "reduction to the impossible", for example in Aristotle's Prior Analytics), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as informal debate.

The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms:

• The Earth cannot be flat, otherwise we would find people falling off the edge.
• There is no smallest positive rational number, because if there were, then it could be divided by two to get a smaller one.

The first example above argues that the denial of the assertion would have a ridiculous result; it would go against the evidence of our senses. The second is a mathematical proof by contradiction, arguing that the denial of the premise would result in a logical contradiction (there is a "smallest" number and yet there is a number smaller than it).