Reductio ad absurdum is a mode of
argumentation that seeks to establish a contention by deriving an
absurdity from its denial, thus arguing that a thesis must be
accepted because its rejection would be untenable. It is a style
of reasoning that has been employed throughout the history of
mathematics and philosophy from classical antiquity onwards.
Table of Contents (Clicking on the links below will take you to those parts of this article)
1. Basic Ideas
Use of this Latin terminology traces back to the Greek
expression hê eis to adunaton
apagôgê, reduction to the impossible, found
repeatedly in Aristotle's Prior Analytics. In its most
general construal, reductio ad
absurdum - reductio for short – is a
process of refutation on grounds that absurd - and
patently untenable consequences would ensue from accepting
the item at issue. This takes three principal forms according as
that untenable consequence is:
- a self-contradiction (ad absurdum)
- a falsehood (ad falsum or even ad
impossibile)
- an implausibility or anomaly (ad ridiculum or
ad incommodum)
The first of these is reductio ad absurdum in its strictest
construction and the other two cases involve a rather wider and
looser sense of the term. Some conditionals that instantiate this
latter sort of situation are:
- If that's so, then I'm a monkey's uncle.
- If that is true, then pigs can fly.
- If he did that, then I'm the Shah of Persia.
What we have here are consequences that are absurd in the
sense of being obviously false and indeed even a bit ridiculous.
Despite its departure from what is strictly speaking so
construed - conditionals with
self-contradictory - time to time conclusions – this
sort of thing is also characterized as an attenuated mode of
reductio. But while all three cases fall into the range of
the term as it is commonly used,
logicians and mathematicians generally have the first and
strongest of them in view.
The usual explanations of reductio fail to acknowledge
the full extent of its range of application. For at the very
minimum such a refutation is a process that can be applied
to
individual propositions or theses
groups of propositions or theses (i.e., doctrines or
positions or teachings)
modes of reasoning or argumentation
definitions
instructions and rules of procedure
practices, policies and processes
The task of the present discussion is to explain the
modes of reasoning at issue with reductio and to
illustrate the work range of its applications.
2. The Logic of Strict Propositional Reductio: Indirect Proof
Whitehead and Russell in Principia
Mathematica characterize the principle of "reductio ad
absurdum" as tantamount to the formula (~p  p)
p of propositional logic. But this view
is idiosyncratic. Elsewhere the principle is almost universally
viewed as a mode of argumentation rather than a specific thesis
of propositional logic.
Propositional reductio is based on the following line of
reasoning:
If p  ~p, then ~p
Here represents assertability, be it absolute or conditional
(that is, derivability). Since p q yields p q this principle
can be established as follows:
Suppose (1) p ~p
(2) p ~p from (1)
(3) p (p & ~p) from (2) since p p
(4) ~(p & ~p) ~p from (3) by contraposition
(5) ~(p & ~p) by the Law of
Contradiction
(6) ~p from (4), (5) by modus
ponens
Accordingly, the above-indicated line of reasoning does not
represent a postulated principle but a theorem that issues from
subscription to various axioms and proof rules, as instanced in
the just-presented derivation.
The reasoning involved here provides the basis for what
is called an indirect proof. This is a process of justificating
argumentation that proceeds as follows when the object is to
establish a certain conclusion p:
(1) Assume not-p
(2) Provide argumentation that derives p from this
assumption.
(3) Maintain p on this basis.
Such argumentation is in effect simply an implementation of the
above-stated principle with ~p standing in place of p.
As this line of thought indicates, reductio argumentation
is a special case of demonstrative reasoning. What we deal with
here is an argument of the pattern: From the situation
(to-be-refuted assumption + a conjunction of preestablished
facts) contradiction
one proceeds to conclude the denial of that to-be-refuted
assumption via modus tollens argumentation.
An example my help to clarify matters. Consider
division by zero. If this were possible when x is not 0 and we took x ÷ 0
to constitute some well-defined quantity Q, then we would have
x ÷ 0 = Q so that x = 0 x Q so that since 0 x (anything) = 0 we
would have x = 0, contrary to assumption. The supposition that
x ÷ 0 qualifies as a well-defined quantity is thereby refuted.
3. A Classical Example of Reductio Argumentation
A classic instance of reductio reasoning in Greek
mathematics relates to the discovery by Pythagoras - disclosed
to the chagrin of his associates by Hippasus of Metapontum in
the fifth century BC - of the incommensurability of the diagonal
of a square with its sides. The reasoning at issue runs as
follows:
Let d be the length of the diagonal of a square and s the length of
its sides. Then by the Pythagorean theorem we have it that d² =
2s². Now suppose (by way of a reductio assumption) that d and s
were commensurable in terms of a common unit n, so that d = n
x u and s = m x u, where m and n are whole numbers (integers)
that have no common divisor. (If there were a common divisor,
we could simply shift it into u.) Now we know that
(n x u)² = 2(m x u)²
We then have it that n² = 2m². This means that n must be even,
since only even integers have even squares. So n = 2k. But now
n² = (2k)² = 4k² = 2m², so that 2k² = m². But this means that
m must be even (by the same reasoning as before). And this
means that m and n, both being even, will have common divisors
(namely 2), contrary to the hypothesis that they do not.
Accordingly, since that initial commensurability assumption
engendered a contradiction, we have no alternative but to reject it.
The incommensurability thesis is accordingly established.
As indicated above, this sort of proof of a thesis by reductio
argumentation that derives a contradiction from its negation is
characterized as an indirect proof in mathematics. (On the
historical background see T. L. Heath, A History of Greek
Mathematics [Oxford, Clarendon Press, 1921].)
The use of such reductio argumentation was common in
Greek mathematics and was also used by philosophers in
antiquity and beyond. Aristotle employed it in the Prior
Analytics to demonstrate the so-called imperfect syllogisms when
it had already been used in dialectical contexts by Plato (see
Republic I, 338C-343A; Parmenides 128d). Immanuel Kant's
entire discussion of the antinomies in his Critique of Pure
Reason was based on reductio argumentation.
The mathematical school of so-called intuitionism
has taken a definite line regarding the limitation of reductio
argumentation for the purposes of existence proofs. The only
valid way to establish existence, so they maintain, is by
providing a concrete instance or example: general-principle
argumentation is not acceptable here. This means, in specific,
that one cannot establish ( x)Fx by deducing an absurdity from
( x)~Fx. Accordingly, intuitionists would not let us infer the
existence of invertebrate ancestors of homo sapiens from the
patent absurdity of the supposition that humans are vertebrates
all the way back. They would maintain that in such cases where
we are totally in the dark as to the individuals involved we are
not in a position to maintain their existence.
4. Self-Annihilation: Processes that Engender Contradiction
Not only can a self-inconsistent statement (and thereby a
self-refuting, self-annihilating one) but also a self-inconsistent
process or practice or principle of procedure can be "reduced to
absurdity." For any such modus operandi answers to some
instruction (or combination thereof), and such instruction can
also prove to be self-contradictory. Examples of this would be:
Never say never.
Keep the old warehouse intact until the new one is
constructed. And build the new warehouse from the
materials salvaged by demolishing the old.
More loosely, there are also instructions that do not
automatically result in logically absurd (self-contradictory)
conclusions, but which open the door to such absurdity in
certain conditions and circumstances. Along these lines, a
practical rule of procedure or modus operandi would be reduced
to absurdity when it can be shown that its actual adoption and
implementation would result in an anomaly.
Consider an illustration of this sort of situation. A man
dies leaving an estate consisting of his town house, his bank
account of $30,000, his share in the family business, and several
pieces of costume jewelry he inherited from his mother. His
will specifies that his sister is to have any three of the valuables
in his estate and that his daughter is to inherent the rest. The
sister selects the house, a bracelet, and a necklace. The executor
refuses to make this distribution and the sister takes him to
court. No doubt the judge will rule something like "Finding for
the plaintiff would lead ad absurdum. She could just as well
have also opted not just for the house but also for the bank
account and the business, thereby effectively disinheriting the
daughter, which was clearly not the testator's wish." Here we
have a juridical reductio ad absurdum of sorts. Actually
implementing this rule in all eligible cases - its generalized
utilization across the board - would yield an unacceptable and
untoward result so that the rule could self-destruct in its actual
unrestricted implementation. (This sort of reasoning is common
in legal contexts. Many such cases are discussed in David
Daube Roman Law [Edinburgh: Edinburgh University Press,
1969], pp. 176-94.)
Immanuel Kant taught that interpersonal practices
cannot represent morally appropriate modes of procedure if they
do not correspond to verbally generalizable rules in this way.
Such practices as stealing (i.e., taking someone else's
possessions without due authorization) or lying (i.e. telling
falsehoods where it suits your convenience) are rules
inappropriate, so Kant maintains, exactly because the
corresponding maxims, if generalized across the board, would
be utterly anomalous (leading to the annihilation of property-
ownership and verbal communication respectively. Since the
rule-conforming practices thus reduce to absurdity upon their
general implementation, such practices are adjudged morally
unacceptable. For Kant, generalizability is the acid test of the
acceptability of practices in the realm of interpersonal dealings.
5. Doctrinal Annihilation: Sets of Statements that Are Collectively Inconsistent
Even as individual statements can prove to be
self-contradictions, so a plurality of statements (a "doctrine" let us
call it) can prove to be collectively inconsistent. And so in this
context reductio reasoning can also come into operation. For
example, consider the following schematic theses:
A B
B C
C D
Not-D
In this context, the supposition that A can be refuted by a
reductio ad absurdum. For if A were conjoined to these
premisses, we will arrive at both D and not-D which is patently
absurd. Hence it is untenable (false) in the context of this
family of givens.
When someone is "caught out in a contradiction" in this
way their position self-destructs in a reduction to absurdity. An
example is provided by the exchange between Socrates and his
accusers who had charged him with godlessness. In elaborating
this accusation, these opponents also accused Socrates of
believing in inspired beings (daimonia). But here inspiration is
divine inspiration such a daimonism is supposed to be a being
inspired by a god. And at this point Socrates has a ready-made
defense: how can someone disbelieve in gods when he is
acknowledged to believe in god-inspired beings. His accusers
here become enmeshed in self-contradiction. And their position
accordingly runs out into absurdity. (Compare Aristotle,
Rhetorica 1398a12 [II xxiii 8].)
6. Absurd Definitions and Specifications
Even as instructions can issue in absurdity, so can
definitions and explanations. As for example:
A zor is a round square that is colored green.
Again consider the following pair:
A bird is a vertebrate animal that flies.
An ostrich is a species of flightless bird.
Definitions or specifications that are in principle unsatisfiable
are for this very reason absurd.
7. Per Impossible Reasoning
Per impossible reasoning also proceeds from a patently
impossible premiss. It is closely related to, albeit distinctly
different from reductio ad absurdum argumentation. Here we
have to deal with literally impossible suppositions that are not
just dramatically but necessarily false thanks to their logical
conflict with some clearly necessary truths, be the necessity at
issue logical or conceptual or mathematical or physical. In
particular, such an utterly impossible supposition may negate:
a matter of (logico-conceptual) necessity ("There are
infinitely many prime numbers").
a law of nature ("Water freezes at low temperatures").
Suppositions of this sort commonly give rise to per
impossibile counterfactuals such as:
- If (per impossible) water did not freeze, then ice would
not exist.
- If, per impossible, pigs could fly, then the sky would
sometimes be full of porkers.
- If you were transported through space faster than the
speed of light, then you would return from a journey
younger than at the outset.
- Even if there were no primes less than 1,000,000,000, the
number of primes would be infinite.
- If (per impossible) there were only finitely many prime
numbers, then there would be a largest prime number.
A somewhat more interesting mathematical example is as
follows: If, per impossible, there were a counterexample to
Fermat's Last Theorem, there would be infinitely many
counterexamples, because if xk + yk = zk, then (nx)k + (ny)k =
(nz)k, for any k.
With such per impossible counterfactuals we envision
what is acknowledged as an impossible and thus necessarily
false antecedent, doing so not in order to refute it as absurd (as
in reductio ad absurdum reasoning), but in order to do the best
one can to indicate its "natural" consequences.
Again, consider such counterfactuals as:
- If (per impossible) 9 were divisible by 4 without a
remainder, then it would be an even number.
- If (per impossible) Napoleon were still alive today, he
would be amazed at the state of international politics in
Europe.
A virtually equivalent formulation of the very point at issue with
these two contentions is:
Any number divisible by 4 without remainders is even.
By the standards of Napoleonic France the present state
of international politics in Europe is amazing.
However, the designation per impossible indicates that it
is the conditional itself that concerns us. Our concern is with
the character of that consequence relationship rather than with
the antecedent or consequent per so. In this regard the situation
is quite different from reductio argumentation by which we seek
to establish the untenability of the antecedent. To all intents and
purposes, then, counterfactuals can serve distinctly factual
purpose.
And so, often what looks to be a per impossible
conditional actually is not. Thus consider
- If I were you, I would accept his offer.
Clearly the antecedent/premiss "I = you" is absurd. But even
the slightest heed of what is communicatively occurring here
shows that what is at issue is not this just-stated impossibility
but a counterfactual of the format:
- If I were in your place (that is, if I were circumstanced in
the condition in which you now find yourself), then I
would consult the doctor.
Only by being perversely literalistic could the absurdity of that
antecedent be of any concern to us.
One final point. The contrast between reductio and per
impossible reasoning conveys an interesting lesson. In both
cases alike we begin with a situation of exactly the same basic
format, namely a conflict of contradiction between an
assumption of supposition and various facts that we already
know. The difference lies entirely in pragmatic considerations,
in what we are trying to accomplish. In the one (reductio) case
we seek to refute and rebut that assumptions so as to establish
its negation, and in the other (per impossible) case we are trying
to establish an implication - to validate a conditional. The
difference at bottom thus lies not in the nature of the inference
at issue, but only in what we are trying to achieve by its means.
The difference accordingly is not so much theoretical as
functional - it is a pragmatic difference in objectives.
8. Sources
David Daube, Roman Law (Edinburgh: Edinburgh University Press, 1969), pp. 176-94.
M. Dorolle, "La valeur des conclusion par l'absurde," Révue philosophique, vol. 86 (1918), pp. 309-13.
T. L. Heath, A History of Greek Mathematics, vol. 2 (Oxford: Clarendon Press, 1921), pp. 488-96.
A. Heyting, Intuitionism: An Introduction (Amsterdam, North-Holland Pub. Co., 1956).
William and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1962), pp. 7-10.
J. M. Lee, "The Form of a reductio ad absurdum," Notre Dame Journal of Formal Logic, vol. 14 (1973), pp. 381-86.
Gilbert Ryle, "Philosophical Arguments," Colloquium Papers, vol. 2 (Bristol: University of Bristol, 1992), pp. 194-211.
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