Reductio ad absurdum
Reductio ad absurdum (Latin: "reduction to absurdity"; pl.: reductiones ad absurdum), 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 recognized and studied in classical Greek philosophy (the Latin term derives from the Greek "εις άτοπον απαγωγή" 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:
- 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, 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 argues denial of the assertion would be untenable; unpleasant or unworkable for society. The third 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).
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 (Elenchus), now called the Socratic method. 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 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.
"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.
- Nicholas Rescher. "Reductio ad absurdum". The Internet Encyclopedia of Philosophy. Retrieved 21 July 2009.
- 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.