Reductio ad absurdum
from 450 BCE Used by Zeno and the Pythagoreans before it was named
Grant what you mean to refute, follow it faithfully until it produces something impossible, and conclude that it was false all along.
- Active
- from 450 BCE
- Where
- Elea and Athens
- Status
- In use
- Tradition
- Forms of inference
What it is
A reductio assumes the opposite of what it wants to establish and derives a contradiction from it. Since a contradiction cannot be true, the assumption cannot be either.
What makes it distinctive is its generosity. The arguer takes on the opponent's claim entirely, argues for it as hard as they can, and lets it destroy itself. Nothing has to be granted at the start except the claim under attack.
How it works
Assume the claim you mean to refute. Reason from it carefully and without help from anything contentious. Arrive at a contradiction — either a plain one, or a consequence nobody involved will accept. Conclude that the assumption is false.
The discipline is in the second step. Every additional premise smuggled in during the derivation is a place where the opponent can say the contradiction came from your material and not from theirs.
The form of the inference
- Assume P, for the sake of argument
- From P, together with what is already agreed, Q follows
- But Q is impossible, or contradicts something already agreed
- Therefore P is false
The argument
Proposition The meter reading is wrong.
- Suppose the meter reading is correct, and the house used 4,200 litres in the quarter.
- The house was unoccupied for eleven of the thirteen weeks, with the mains tap closed at the street.
- With the tap closed no water can enter the property at all, so consumption in those weeks was zero.
- That leaves 4,200 litres used in two weeks, or 300 litres a day.
- The only outlet running in that fortnight was a single garden tap on a timer set for twenty minutes a day, which at the measured flow rate delivers under 80 litres.
Therefore The reading cannot be correct, since accepting it requires the property to have consumed water it had no way of receiving or discharging.
Nothing was assumed except the utility's own figure. That is what makes it a reductio rather than a counter-claim — the argument is conducted entirely on the other side's ground.
What it is good for
Refuting a position without needing a rival position of your own. It is the method of choice against claims that are hard to attack directly, and it is the backbone of mathematical proof.
Where it fails
It fails when the absurdity is not agreed. One side's impossible consequence is the other side's surprising discovery, and a determined opponent can simply accept the conclusion — which is exactly what happened when the Pythagoreans met irrational numbers and what happens routinely in philosophy.
It also fails through smuggled premises. If anything contentious enters during the derivation, the contradiction may have come from that rather than from the assumption, and the refutation collapses into a disagreement about the imported premise.
Intuitionist mathematicians reject one form of it outright, since deriving a contradiction from *not P* shows that *not not P*, and the step from there to *P* is exactly what they will not grant.
Common misconceptions
That it is the same as showing a claim leads to unwelcome consequences. Unwelcome is not impossible, and an argument that only reaches unwelcome is an appeal to consequences.
Also that it requires bad faith. Done properly it requires the opposite: the assumption must be given its strongest reading, or the contradiction is with a straw man.
Where it came from
Zeno of Elea built his paradoxes as reductios of the claim that motion and plurality are real, and the Pythagorean proof of the irrationality of the square root of two is the earliest surviving mathematical example.
What became of it
Universal in mathematics and philosophy, and contested at the foundations by intuitionists from Brouwer onward.
Lasting influence
It is how a large share of mathematics is proved, how Gödel's incompleteness result is structured, and the shape of the Madhyamaka prasanga method, which arrived at the same technique independently in India.
Why it still matters
Undiminished. Every proof by contradiction anyone runs today is this.
Put another way
Loading a bridge until it fails, to find out what it could never have carried.
Texts
Euclid, Elements
Book IX proposition 20, the infinitude of primes, is the cleanest reductio ever written and takes half a page.
Where to start
Read the proof that there is no largest prime. It is five lines and it will teach the method better than any description of it.
The surprising bit
The proof that root two is irrational was so unwelcome to the Pythagoreans, whose whole doctrine held that all things are number, that legend has the man who published it drowned at sea.
In their own words
Reductio ad absurdum is one of a mathematician's finest weapons
Who held it
Euclid, Nagarjuna
Connections
- Relies on
- Modus tollens
- Influenced
- Prasaṅga, The infinite regress
- Parallel to
- Modus tollens
The verdict
The most powerful destructive tool in argument, and it builds nothing.