Reductio ad absurdum

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).

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 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 dialogues of Plato (424 – 348 BC), relating the debates of his teacher Socrates, raised the use of reductio arguments to a formal dialectical method (Elenchus), now called the Socratic method which is taught in law schools. Typically Socrates' opponent would make an innocuous assertion, 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 an absurd or 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.

The principle of explosion and paraconsistent logic

A curious logical consequence of the principle of non-contradiction is that a contradiction implies any statement; if a contradiction is accepted, any proposition (or its negation) can be proved from it. This is known as the principle of explosion (Latin: ex falso quodlibet, "from a falsehood, anything [follows]", or ex contradictione sequitur quodlibet, "from a contradiction, anything follows"), or the principle of Pseudo-Scotus.

$\forall Q: (P \and {\sim}P) \rightarrow Q\;$

"for all Q, P and not-P implies Q"

The discovery of contradictions at the foundations of mathematics at the beginning of the 20th century, such as Russell's paradox, threatened the entire structure of mathematics due to the principle of explosion. This has led a few philosophers such as Newton da Costa, Walter Carnielli and Graham Priest to reject the principle of non-contradiction, giving rise to theories such as paraconsistent logic and its particular form, dialethism, which accepts that there exist statements that are both true and false.

Paraconsistent logics usually deny that the principle of explosion holds for all sentences in logic, which amounts to denying that a contradiction entails everything (what is called “deductive explosion”). The Logics of Formal Inconsistency (LFIs) are a family of paraconsistent logics where the notions of contradiction and consistency are not coincident; although the validity of the principle of explosion is not accepted for all sentences, it is accepted for consistent sentences. Most paraconsistent logics, as the LFIs, also reject the principle of non-contradiction.

Straw man argument

Main article: Straw man

A fallacious 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") has an absurd, unpleasant, or ridiculous consequence, relying on the audience failing to notice that the argument does not actually apply to the original proposition. For example, in a 1977 appeal of a U.S. bank robbery conviction, a prosecuting attorney said in his closing argument

I submit to you that if you can't take this evidence and find these defendants guilty on this evidence then we might as well open all the banks and say, "Come on and get the money, boys", because we'll never be able to convict them.

The prosecutor was using this "straw man" to attempt to alarm the appellate judges; the chance that any precedent set by this one particular case would literally make it impossible to convict any bank robbers was undoubtedly remote.

See also

• Argument from fallacy
• Contraposition
• List of Latin phrases
• Mathematical proof

External links

• Reductio ad absurdum entry in the Internet Encyclopedia of Philosophy