Argument that leads to a logical absurdity
Reductio ad absurdum
, painting by
John Pettie
exhibited at the
Royal Academy
in 1884
In
logic
,
reductio ad absurdum
(
Latin
for "reduction to absurdity"), also known as
argumentum ad absurdum
(
Latin
for "argument to absurdity") or
apagogical arguments
, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
[1]
[2]
[3]
[4]
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. Formally, the proof 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
) and it is sometimes named to make this connection clear. It is a consequence of the related mathematical proof technique called
proof by contradiction
.
Examples
[
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]
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, 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
because such a number would have to not be divisible by two, which is impossible.
The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses.
[5]
The second example is a mathematical
proof by contradiction
(also known as an indirect proof
[6]
), 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).
[7]
Greek philosophy
[
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]
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
BCE
).
[8]
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.
[9]
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.
[10]
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
.
[11]
Typically, Socrates' opponent would make what would seem to be an innocuous assertion. In response, Socrates, via 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 and adopt a position of
aporia
.
[6]
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 (
Greek
:
? ε?? τ? ?δ?νατον ?π?δειξι?
,
lit.
"demonstration to the impossible", 62b).
[4]
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
[
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]
Much of
Madhyamaka
Buddhist philosophy
centers on showing how various
essentialist
ideas have absurd conclusions through
reductio ad absurdum
arguments (known as
prasa?ga
, "consequence" in Sanskrit). In the
M?lamadhyamakak?rik?
,
N?g?rjuna
'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. N?g?rjuna'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
Ny?ya
and
Vai?e?ika
schools which posited a theory of ontological substances (
dravyatas
).
[13]
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
[
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]
- 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.
Principle of non-contradiction
[
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]
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.
[15]
[16]
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
,
[6]
has formed the basis of
reductio ad absurdum
arguments in formal fields such as
logic
and mathematics.
See also
[
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]
Sources
[
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]
- Hyde, Dominic; Raffman, Diana (2018).
"Sorites Paradox"
. In
Zalta, Edward N.
(ed.).
Stanford Encyclopedia of Philosophy
(Summer 2018 ed.).
- Garfield, Jay L. (1995),
The Fundamental Wisdom of the Middle Way
, Oxford: Oxford University Press
- 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.
References
[
edit
]
- ^
"Reductio ad absurdum | logic"
.
Encyclopedia Britannica
. Retrieved
2019-11-27
.
- ^
"Definition of REDUCTIO AD ABSURDUM"
.
www.merriam-webster.com
. Retrieved
2019-11-27
.
- ^
"reductio ad absurdum"
,
Collins English Dictionary ? Complete and Unabridged
(12th ed.), 2014 [1991]
, retrieved
October 29,
2016
- ^
a
b
Nicholas Rescher.
"Reductio ad absurdum"
.
The Internet Encyclopedia of Philosophy
. Retrieved
21 July
2009
.
- ^
DeLancey, Craig (2017-03-27),
"8. Reductio ad Absurdum"
,
A Concise Introduction to Logic
, Open SUNY Textbooks
, retrieved
2021-08-31
- ^
a
b
c
Nordquist, Richard.
"Reductio Ad Absurdum in Argument"
.
ThoughtCo
. Retrieved
2019-11-27
.
- ^
Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan (30 March 2012).
The Power of Logic
(5th ed.). McGraw-Hill Higher Education.
ISBN
978-0078038198
.
- ^
Daigle, Robert W. (1991).
"The reductio ad absurdum argument prior to Aristotle"
.
Master's Thesis
. San Jose State Univ
. Retrieved
August 22,
2012
.
- ^
"Reductio ad Absurdum - Definition & Examples"
.
Literary Devices
. 2014-05-18
. Retrieved
2021-08-31
.
- ^
Joyce, David (1996).
"Euclid's Elements: Book I"
.
Euclid's Elements
. Department of Mathematics and Computer Science, Clark University
. Retrieved
December 23,
2017
.
- ^
Bobzien, Susanne (2006).
"Ancient Logic"
.
Stanford Encyclopedia of Philosophy
. The Metaphysics Research Lab, Stanford University
. Retrieved
August 22,
2012
.
- ^
Wasler, Joseph.
Nagarjuna in Context.
New York: Columibia University Press. 2005, pgs. 225-263.
- ^
Ziembi?ski, Zygmunt (2013).
Practical Logic
. Springer. p. 95.
ISBN
978-9401756044
.
- ^
Ferguson, Thomas Macaulay; Priest, Graham (2016).
A Dictionary of Logic
. Oxford University Press. p. 146.
ISBN
978-0192511553
.
External links
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]