Reductio ad absurdum
Reductio ad absurdum (Latin: "reduction to absurdity"), also known as argumentum ad absurdum (Latin: argument to absurdity), 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 appearing in classical Greek philosophy (the Latin term derives from the Greek ἡ εἰς ἀτοπον ἀπαγωγη or he 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:
- Rocks have weight, otherwise we would see them floating in the air.
- Society must have laws, otherwise there would be chaos.
- There is no smallest positive rational number, because if there were, 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, against the evidence of our senses. The second argues that the denial would have an untenable result: unacceptable, unworkable or unpleasant for society. The third is a mathematical proof by contradiction, arguing that the denial of the assertion would result in a contradiction (there is a smallest rational number and yet there is a rational number smaller than it).
Greek philosophy 
This technique is used throughout Greek philosophy, beginning with Presocratic philosophers. The earliest Greek example of a reductio argument is supposedly in fragments of a satirical poem attributed to Xenophanes of Colophon (c.570 – c.475 BC). Criticizing Homer's attribution of human faults to the Greek gods, he says that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and oxen bodies. The gods can't have both forms, so this is a contradiction. Therefore the attribution of other human characteristics to the gods, such as human faults, is also false.
The earlier dialogs of Plato (424 – 348 BC), relating the debates of his teacher Socrates, raised the use of reductio arguments to a formal dialectical method, now called the Socratic method. Typically Socrates' opponent would make an innocuous claim, then Socrates by a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in a contradictory conclusion, forcing him to abandon his assertion. The technique was also a focus of the work of Aristotle (384 – 322 BC).
The principle of non-contradiction 
Aristotle clarified the connection between contradiction and falsity in his principle of non-contradiction. This states that an assertion cannot be both true and false. Therefore if the contradiction of an assertion (not-P) can be derived logically from the assertion (P) it can be concluded that a false assumption has been used. This technique, called proof by contradiction has formed the basis of reductio ad absurdum arguments in formal fields like logic and mathematics.
The principle of non-contradiction has seemed absolutely undeniable to most philosophers. However a few philosophers such as Heraclitus and Hegel have accepted contradictions. In recent years the discovery of contradictions at the foundations of mathematics, such as Russell's paradox, have led a few philosophers such as Graham Priest to reject the principle of non-contradiction, giving rise to theories such as dialethism and paraconsistent logic which accept that some statements can be both true and false.
Straw Man argument 
An argument similar to reductio ad absurdum often seen in polemical debate is the straw man logical fallacy. A straw man argument attempts to refute a given proposition by showing that a slightly different or inaccurate form of the proposition (the "straw man") is absurd or ridiculous, relying on the audience not to notice that the argument does not actually apply to the original proposition. For example:
- Politician A: "We should not serve schoolchildren sugary desserts with lunch and further worsen the obesity epidemic by doing so."
- Politician B: "What, do you want our children to starve?"
See also 
- Nicholas Rescher. "Reductio ad absurdum". The Internet Encyclopedia of Philosophy. Retrieved 21 July 09.
- Daigle, Robert W. (1991). "The reductio ad absurdum argument prior to Aristotle". Master's Thesis. San Jose State Univ. Retrieved August 22, 2012.
- Bobzian, Suzanne (2006). "Ancient Logic". Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Stanford University. Retrieved August 22, 2012.
- "Reductio ad absurdum". New World Encyclopedia. 2007. Retrieved August 22, 2012.