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 x^k + y^k = z^k, 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.