In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or an apagogical argument, is the form of argument that attempts to establish a claim by showing that following the logic of a proposition or argument would lead to absurdity or contradiction.
This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called proof by contradiction. In formal logic, this technique is captured by an axiom for " reductio ad absurdum", normally given the abbreviation RAA, which is expressible in propositional logic. This axiom is the introduction rule for negation (see negation introduction).
Examples
The "absurd" conclusion of a
reductio ad absurdum argument can take a range of forms:
-
The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find people falling off the edge.
-
There is no smallest positive rational number . If there were, then would also be a rational number, it would be positive, and we would have . This contradicts the hypothetical minimality of among positive rational numbers, so we conclude that there is no such smallest positive rational number.
The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses (empirical evidence).
Greek philosophy
Reductio ad absurdum was used throughout
Greek philosophy. The earliest example of a reductio argument can be found in a satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475
Common Era).
Criticizing
Homer's attribution of human faults to the gods, Xenophanes 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 ox 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.
Greek mathematicians proved fundamental propositions using reductio ad absurdum. Euclid of Alexandria (mid-4th – mid-3rd centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples.
The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method (elenchus), also called the Socratic method.
The technique was also a focus of the work of Aristotle (384–322 BCE), particularly in his Prior Analytics where he referred to it as demonstration to the impossible (, 62b).
Another example of this technique is found in the sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.
Buddhist philosophy
Much of
Madhyamaka Buddhist philosophy centers on showing how various
Essentialism ideas have absurd conclusions through
reductio ad absurdum arguments (known as
prasaṅga, "consequence" in Sanskrit). In the Mūlamadhyamakakārikā, Nāgārjuna's
reductio ad absurdum arguments are used to show that any theory of substance or essence was unsustainable and therefore, phenomena (
dharmas) such as change, causality, and sense perception were empty (
sunya) of any essential existence. Nāgārjuna's main goal is often seen by scholars as refuting the essentialism of certain Buddhist
Abhidharma schools (mainly
Vaibhasika) which posited theories of
svabhava (essential nature) and also the Hindu
Nyaya and
Vaisheshika schools which posited a theory of ontological substances (
dravyatas).
[Wasler, Joseph. Nagarjuna in Context. New York: Columibia University Press. 2005, pgs. 225-263.]
Example from Nāgārjuna's Mūlamadhyamakakārikā
In 13:5, Nagarjuna wishes to demonstrate consequences of the presumption that things essentially, or inherently, exist, pointing out that if a "young man" exists in himself then it follows he cannot grow old (because he would no longer be a "young man"). As we attempt to separate the man from his properties (youth), we find that everything is subject to momentary change, and are left with nothing beyond the merely arbitrary convention that such entities as "young man" depend upon.
13:5
- A thing itself does not change.
- Something different does not change.
- Because a young man does not grow old.
- And because an old man does not grow old either.
Modern philosophy
Contemporary philosophers have also utilized appeals to the
reductio ad absurdum argument within their respective scholarly works. Included among them are:
The Philosophical Review, Duke University Press on behalf of Philosophical Review Jul., 1977, Vol. 86, No. 3 (Jul., 1977), pp. 418-421 The Actor and the Spectator by Lewis Beck, book reviewed by Stephen Griffith on JSTOR.org
-
Robert L. Holmes - in his criticism of deterrence theory based upon the deontological prima facie presumption against killing innocent life. ( On War and Morality, 1989)
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
(not-
Q) cannot both be true. Therefore, if a proposition and its negation can both be derived logically from a premise, it can be concluded that the premise is false. This technique, known as indirect proof or proof by contradiction,
has formed the basis of reductio ad absurdum arguments in formal fields such as
logic and mathematics.
See also
Sources
-
Pasti, Mary. Reductio Ad Absurdum: An Exercise in the Study of Population Change. United States, Cornell University, Jan., 1977.
-
Daigle, Robert W.. The Reductio Ad Absurdum Argument Prior to Aristotle. N.p., San Jose State University, 1991.
External links
