# Reductio ad Absurdum ## Intro Sometimes the fastest way to prove a claim is to imagine the opposite is true and see where it leads. If the opposite ends in something obviously absurd, the claim must be right. That move has a Latin name: *reductio ad absurdum*, "reduction to absurdity." The setup is short. To prove that P is true, assume that P is false. Work out what follows. If what follows is a contradiction, or something nobody would accept, then the assumption that P is false has to be wrong. So P is true. A simple math example. To prove that the square root of 2 is irrational, you assume the opposite, that it can be written as a fraction in lowest terms. Then you work through the algebra and end up showing the numerator and denominator both have to be even, which contradicts your starting assumption that the fraction was in lowest terms. The contradiction proves your assumption was false. So the square root of 2 must be irrational. The argument form runs through philosophy, mathematics, theology, and Christian apologetics. C. S. Lewis used a reductio when he argued that if naturalism is true, then human reasoning is just the noise of chemicals, in which case we cannot trust the reasoning that led us to naturalism. The argument turns the position on itself. There are two strict versions and one looser version. The strict ones derive a formal contradiction, like "Q and not-Q." The looser one derives a consequence the opponent would not actually accept, even if it is not strictly self-contradictory. Both versions show up constantly in live debate. This page lays out the formal structure, the two strict types, and the common ways the move can be misused. ## In full A foundational logical argument form (Latin: *"reduction to absurdity"*) in which one **assumes the opposite of what one intends to prove, derives a contradiction or evidently false consequence from that assumption, and concludes that the assumption must be false**, validating the original thesis. Used throughout philosophy, mathematics, theology, and Christian apologetics as one of the principal tools for defeating opposing positions by showing that they entail consequences their proponents would reject. The general schema: > To prove **P**: > 1. Assume **not-P** (the contradiction of what you want to prove). > 2. Derive from **not-P** (in combination with the opponent's other commitments) some statement **Q** that is either (a) a logical contradiction, (b) demonstrably false, or (c) absurd / unacceptable on the opponent's own terms. > 3. Conclude: **not-P** is false; therefore **P** is true. The valid formal version is **proof by contradiction** (deriving a logical contradiction, e.g., *Q ∧ not-Q*). The looser apologetic / philosophical version is **reductio in the rhetorical sense**, deriving a consequence the opponent would not actually accept, even if the consequence is not strictly self-contradictory. --- ## The two strict types ### 1. Formal proof by contradiction If **not-P** logically entails *(Q ∧ not-Q)*, then **not-P** is necessarily false, and **P** must be true. This is the mathematician's bread-and-butter, Euclid's proof that the primes are infinite is the canonical example: > **Theorem.** There are infinitely many prime numbers. > > **Proof.** Assume the opposite: there are only finitely many primes, *p₁, p₂, …, pₙ*. Consider the number *N = (p₁ × p₂ × … × pₙ) + 1*. Either *N* is itself prime (in which case it is a prime not in the list, contradicting the assumption) or *N* has a prime factor (which cannot be any of *p₁, p₂, …, pₙ* because *N* leaves remainder 1 when divided by each, so the prime factor is again not in the list). Either way we contradict the assumption that *p₁, p₂, …, pₙ* are all the primes. So the assumption is false. There are infinitely many primes. ∎ The contradiction is *logically* compelling. Anyone who accepts the premises must accept the conclusion. ### 2. Reductio to evident falsity If **not-P** entails some claim **Q** that is *not* a logical contradiction but is *evidently false* (or absurd, or unacceptable on the opponent's own terms), then **not-P** is dialectically defeated. This is the rhetorical / philosophical / theological deployment. > **Example.** *"Suppose the materialist is right that all human thought is purely the result of physical particles obeying physical laws. Then the materialist's own thought, including the thought 'materialism is true', is itself just particles obeying laws, not a rational conclusion from evidence. The materialist's argument for materialism is self-undermining: if it is sound, we have no reason to trust the reasoning that produced it. So either materialism is false, or we have no rational warrant for believing it. Either way, the materialist position is dialectically defeated."* (See [Argument from Reason](/codex/argument-from-reason/) for the formal version.) The conclusion is not strictly *self-contradictory*; it is *self-defeating*, the opponent's position, if true, would undercut the rational basis on which they hold the position. --- ## Reductio in classical and contemporary apologetics Reductio is the workhorse argumentative move in much Christian apologetics, especially the **presuppositional** school ([Cornelius Van Til](/codex/cornelius-van-til/), [Greg Bahnsen](/codex/greg-bahnsen/), John Frame). The presuppositional move: > *"For the sake of argument, suppose the Christian worldview is false. On atheism / naturalism / materialism, the very preconditions of intelligibility (logic, meaning, moral knowledge, the reliability of perception, the uniformity of nature, the existence of universal abstract objects like numbers) cannot be accounted for. So if atheism were true, no argument would work, including the argument for atheism. The atheist's argument presupposes the very framework atheism cannot supply. Therefore atheism is self-defeating; the Christian framework is necessary even for the atheist's denial to be intelligible."* This is the [Transcendental Argument for God](/codex/transcendental-argument-for-god/) in its reductio shape (TAG). See [Stealing from God Argument](/codex/stealing-from-god-argument/) and [Presuppositionalism](/codex/presuppositionalism/) for the developed Christian deployments. **Other apologetic reductio deployments:** - **The [Argument from Reason](/codex/argument-from-reason/)** (C. S. Lewis, *Miracles* ch. 3-4; Victor Reppert, *C. S. Lewis's Dangerous Idea*): naturalism cannot ground the reliability of reason; therefore reasoning *from* naturalism *to* naturalism is self-undermining. - **The [Moral Arguments](/codex/moral-arguments/) from objective values**: if objective moral values exist, theism is the best explanation; the materialist who denies objective values denies what they implicitly affirm in moral discourse (the wrongness of the Holocaust, child abuse, etc.). - **The [Atheism is a Belief](/codex/argument-from-science-as-faith-based-guillen/) defeater**: if atheism is "merely the lack of belief," then newborns are atheists, rocks are atheists, the term has been emptied of meaning, a reductio of the lacktheist redefinition. - **The Problem-of-Evil counter-reductio**: if evil exists as a *real* moral category (necessary for the atheist's argument to have force), then objective moral standards exist; objective moral standards require a moral law-giver; theism is presupposed by the very objection, turning the Problem of Evil into evidence *for* God's existence. (See [Problem of Evil](/codex/problem-of-evil/).) --- ## Christ's use of reductio in the Gospels [Jesus](/codex/jesus/) repeatedly used reductio in dispute with His opponents: ### [Mark 12:35-37](/codex/mark-12-35-37/), The Messiah's Lordship Jesus to the scribes: *"How say the scribes that Christ is the Son of [David](/codex/david/)? For David himself said by the Holy Ghost, The Lord said to my Lord, Sit thou on my right hand, till I make thine enemies thy footstool. David therefore himself calleth him Lord; and whence is he then his son?"* ([Mark 12:35-37](/codex/mark-12-35-37/)). The reductio: if the Messiah is *merely* David's son (i.e., a human descendant of equal or lesser stature), then David would not call him *Lord*; but David does call him Lord in [Psalm 110](/codex/psalms-110/); therefore the Messiah is more than David's biological descendant. The scribes' restricted Messianic doctrine is reduced to incoherence with their own Scripture. ### [Matthew 22:23-33](/codex/matthew-22-23-33/), The Resurrection against the Sadducees The Sadducees pose a reductio against resurrection (the seven-brother case where a woman is widowed seven times; "whose wife will she be in the resurrection?"). Jesus counter-reductios: *"the children of this world marry, and are given in marriage: but they which shall be accounted worthy to obtain that world, and the resurrection from the dead, neither marry, nor are given in marriage."* ([Luke 20:34-35](/codex/luke-20-34-35/)). The Sadducee assumption, that resurrection life is just continuation of earthly marriage, is the absurd assumption; the *opponent's* objection is reduced by exposing its hidden premise. Jesus then deploys His own positive reductio: *"as touching the dead, that they rise: have ye not read in the book of [Moses](/codex/moses/), how in the bush God spake unto him, saying, I am the God of Abraham, and the God of Isaac, and the God of Jacob? He is not the God of the dead, but the God of the living: ye therefore do greatly err"* ([Mark 12:26-27](/codex/mark-12-26-27/)). The reductio: if God said *"I am"* the God of Abraham/Isaac/Jacob (present-tense, in [Exodus 3](/codex/exodus-3/)), and God is the God of the living not the dead, then Abraham/Isaac/Jacob must be living, therefore there is resurrection. The Sadducees' denial of resurrection contradicts their own canonical text. ### John 8, The "before Abraham was, I am" climax Jesus disputes with religious opponents who challenge His authority: *"Your father Abraham rejoiced to see my day: and he saw it, and was glad."* They retort: *"Thou art not yet fifty years old, and hast thou seen Abraham?"* Jesus' answer is the most explosive reductio in the Gospels: *"Verily, verily, I say unto you, Before Abraham was, I am"* ([John 8:58](/codex/john-8-58/)). The opponents' reductio (you're too young to have known Abraham) is collapsed by Jesus' positive claim of pre-existence and divine self-naming (the *ego eimi*, the I AM of [Exodus 3:14](/codex/exodus-3-14/)). The reductio runs back upon the questioners; *"then took they up stones to cast at him"* (8:59), they understood exactly what reductio Jesus had performed. --- ## Common errors and limitations ### Error 1, Reductio of a position the opponent does not actually hold The reductio works only if the opponent actually holds the position you are reducing. If the version of materialism you reduce is a straw man, the actual materialist can simply say *"that's not my view."* The discipline is to steel-man the opponent's actual position before reducing it. (See *Steelmanning* if hub exists, or *charitable interpretation* generally.) ### Error 2, Treating dialectical defeat as logical disproof A successful reductio in the dialectical sense (the opponent's position entails consequences they would reject) is not the same as a logical disproof. A stubborn opponent can *accept* the absurd consequence (the *moral realist* atheist accepts moral realism without theistic grounding; the *eliminative materialist* accepts that there is no genuine thought). The reductio shows the *cost* of the position; it does not necessarily compel the opponent to abandon it. The honest deployment of reductio acknowledges where the opponent might bite the bullet. ### Error 3, Constructive vs non-constructive proof In mathematics, proof by contradiction sometimes establishes that **P** is true without giving any *example* or *construction* of how **P** is realized. Some mathematicians (the intuitionists, Brouwer school) reject non-constructive reductio for this reason. In theology and apologetics, however, the reductio of *not-P* combined with positive evidence for *P* is the standard pattern, and the non-constructive worry does not typically arise. ### Error 4, Reductio without context A reductio that derives an absurd consequence from premises *the opponent does not share* fails. The premises must be the *opponent's* premises. The presuppositional reductio works because it uses premises the atheist actually holds (the reliability of reason, the existence of meaning, etc.) and shows those premises require Christian foundations. --- ## Apologetic deployment **Use reductio when:** - The opponent advances a position whose consequences they would reject. - You want to defeat a position rather than positively establish your own. - The opponent is making a self-undermining claim (the materialist arguing against the rationality of theism; the relativist asserting the absoluteness of relativism; the nihilist deriving a moral imperative from nihilism). **Do not use reductio when:** - The opponent's actual position is unclear (steel-man first). - The opponent is genuinely willing to accept the "absurd" consequence (bite the bullet). - The conversation is not dialectical but pastoral / wounded (reductio in a wounded conversation is cruel; see [Listening Tools](/codex/listening-tools/) #5 for the pastoral alternative). **Live-cite kit:** - *"The materialist's argument for materialism is itself the product of particles obeying laws, not a rational inference. So if materialism is true, no reason to trust the reasoning that produced it."*, the [Argument from Reason](/codex/argument-from-reason/) deployment - *"If atheism is just the lack of belief, newborns are atheists. The redefinition empties the term."*, the [Atheism is a Belief](/codex/argument-from-science-as-faith-based-guillen/) deployment - *"If your moral outrage at the Crusades is justified, you're using a moral standard atheism can't ground."*, the [Moral Arguments](/codex/moral-arguments/) / [Hypocrisy](/codex/hypocrisy/) deployment - Jesus' [Mark 12:35-37](/codex/mark-12-35-37/), [Matthew 22](/codex/matthew-22/), John 8, the scriptural precedent for skilled reductio in apologetic discourse --- ## See also - [Logic](/codex/logic/), parent category - [Law of Non-Contradiction](/codex/law-of-non-contradiction/), the foundation of all reductio - [Argument from Reason](/codex/argument-from-reason/), the foundational reductio against naturalism - [Transcendental Argument for God](/codex/transcendental-argument-for-god/), the developed presuppositional reductio - [Stealing from God Argument](/codex/stealing-from-god-argument/), the popular-deployment form of the transcendental reductio - [Atheism is a Belief](/codex/argument-from-science-as-faith-based-guillen/), reductio of the lacktheist redefinition - [Moral Arguments](/codex/moral-arguments/), the moral reductio - [Presuppositionalism](/codex/presuppositionalism/), the apologetic school built largely on reductio - [Cumulative Case for Christian Theism](/codex/cumulative-case-for-christian-theism/), the meta-apologetic that incorporates multiple reductios alongside positive arguments - [Quick Objection Responses](/codex/quick-objection-responses/), many 30-second responses are reductio-shaped - [Apologetics](/codex/apologetics/)
## Common questions this page answers **Q: What is reductio ad absurdum?** *Reductio ad absurdum* (Latin for "reduction to absurdity") is the argument form that assumes the opposite of what you intend to prove, derives a contradiction or evidently false consequence from that assumption, and concludes the assumption must be false. To prove P, assume not-P, show it leads to absurdity, therefore P. Used throughout philosophy, mathematics, theology, and Christian apologetics. **Q: What is the difference between strict and loose reductio?** A *strict* reductio derives a formal contradiction (Q and not-Q). A *loose* reductio derives a consequence the opponent would not actually accept, even if it is not strictly self-contradictory. Both versions show up constantly in live debate. The strict form is mathematically rigorous; the looser form is rhetorically powerful in argument about live worldviews. **Q: How is reductio used in apologetics?** [C.S. Lewis](/codex/c-s-lewis/)'s [Argument from Reason](/codex/argument-from-reason/) is a reductio: if naturalism is true, then human reasoning is just the noise of chemicals, in which case we cannot trust the reasoning that led us to naturalism. The argument turns the position on itself. [Frank Turek](/codex/frank-turek/)'s *Stealing from God* uses the CRIMES acronym (Causality, Reason, Information, Morality, Evil, Science) to show that the atheist's tools for arguing against God all presuppose what only theism can ground. See [Transcendental Argument for God](/codex/transcendental-argument-for-god/). **Q: Is reductio different from modus tollens?** They are closely related but not identical. Modus tollens is the inference rule: if P then Q; not Q; therefore not P. Reductio uses modus tollens within a broader argument structure: assume P, derive Q (which is absurd or contradictory), use modus tollens to conclude not-P. Reductio is the *strategy*; modus tollens is one of the *steps* within it. See [Modus Ponens](/codex/modus-ponens/) for the companion rule. **Q: When does a reductio go wrong?** Three common failures. *Wrong target*: deriving an absurdity from a stronger or different premise than the opponent actually holds (this is often a [straw man](/codex/straw-man/)). *Question-begging consequence*: deriving "consequences" only the apologist finds absurd; the opponent does not feel the pressure. *Hidden assumption*: the contradiction comes from a smuggled-in premise the opponent rejects. Run the reductio cleanly on the opponent's actual claims and an absurdity they would reject.