Reductio ad Absurdum

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.

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:

1. a self-contradiction (ad absurdum)
2. a falsehood (ad falsum or even ad impossibile)
3. 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 (that is, 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 (that is,
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. References and Further Reading

* 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.