Reductio ad absurdum
In logic, reductio ad absurdum (Latin for "reduction to absurdity"; or argumentum ad absurdum, "argument to absurdity") is a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible. Traced back to classical Greek philosophy in Aristotle's Prior Analytics (Greek: ἡ Εις άτοπον απαγωγή, translit. hê eis atopon apagôgê, lit. 'reduction to the impossible'), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.
The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:
- 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 argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof by contradiction which argues 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 B.C.E.). Criticizing Homer's attribution of human faults to the gods, he states 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 cannot 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 B.C.E.), 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 B.C.E.).
Greek mathematicians proved fundamental propositions utilizing reductio ad absurdum. Euclid of Alexandria (c. mid 3rd - 4th centuries B.C.E.) and Archimedes of Syracuse (c. 287 – c. 212 B.C.E.) are two very early examples.
Principle of non-contradiction
Aristotle clarified the connection between contradiction and falsity in his principle of non-contradiction, which states that a proposition cannot be both true and false. That is, a proposition and its negation cannot both be true. Therefore if a proposition and its negation (not-Q) can both be derived logically from a premise, it can be concluded that the premise is false. This technique, called proof by contradiction has formed the basis of reductio ad absurdum arguments in formal fields like logic and mathematics.
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 deny that the principle of explosion holds for all sentences in logic, which amounts to denying that all contradictions entail everything, which 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
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.
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