Reductio ad Absurdum Arguments

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Reductio ad Absurdum (“reduction to absurdity”)

The popular reductio ad absurdum argument can be employed in two different ways. Both cases lead to a “reduction to absurdity” by deriving an absurdity that nobody should accept as true. There’s the formal employment and the informal employment.

Formal Version of the reductio ad absurdum

The formal version is also known as an indirect proof or a proof by contradiction. In this type of argument, we validly derive a contradiction. A contradiction cannot be true; ergo, whatever led to that contradiction cannot be true.

The Law of Contradiction is a self-evident truth. Aristotle told us: “The most certain of all basic principles is that contradictory propositions are not true simultaneously.”

A Proof by Contradiction is equivalent to the Law of Excluded Middle.
We’ll see why in a minute!

Any assumed proposition that leads to a contradiction must be false. We have a logical absurdity.

In general, imagine we want to prove proposition P is true. In the reductio ad absurdum, we (1) assume that P being not the case is true, (2) we validly derive a contradiction by that assumption, and (3) thereby conclude that P cannot be false, i.e., it must be true.

Notice what’s happening here! We assume not P in a proof. This leads to a contradiction. Thus, we can conclude that P is true. By the Law of Excluded Middle, since “not P” cannot be true, P must be true. (Likewise, if you prove something to be true, it follows that it cannot be false.) That’s why this argumentative technique is solid.

We can think of it in the valid form of modus tollens:
(1) If P, then Q.
(2) Not Q (because Q is absurd or is a contradiction).
(3) Therefore, not P.

Aristotle himself used indirect proofs!
E.g., in “reducing” certain syllogisms to a valid first figure.

Example 1: “Man can engage in purposeful action.”

This is supposed to be a self-evident proposition (in praxeology), known to be true by self-reflection. We can reflect on the fact that we act. An action has an end and there are means deployed to try to accomplish that end.

Even though it’s a self-evident proposition, we can try an indirect proof. If man doesn’t engage in purposeful action, then he cannot engage in the action to argue that man doesn’t engage in purposeful action. However, to argue that one cannot engage in purposeful action requires engaging in it. Thus, we have a contradiction; ergo, man can engage in purposeful action.

Example 1 is related to the method of retorsion.

Philosopher William Vallicella defines it this way: “Proposition p is such that anyone who denies it falls into performative inconsistency; ergo, p is true.”

Somebody might argue this proposition: “there are no assertions.”
The mere arguing of that is, itself, an assertion! Ergo, we have a contradiction.
And, hence, there are assertions.

Another example, which philosopher Edward Feser has used, is of Heraclitus arguing that “change alone is unchanging” or “nothing persists.”

Yet for a proponent of this worldview to even argue that “nothing persists” is unachievable if nothing in fact persists! But since he can do so, something clearly does persist. In arguing to discover the truth of the matter, truly, we have found unchanging aspects of reality. Arguing back-and-forth between each other with premises, inferences, and conclusions presupposes a solid framework to work within.

Example 2: Extreme monism declares all things are of a single substance. But if all things are of one substance, then living and non-living and man and non-man would be the same. Insofar as the living and non-living and man and non-man are the same, we have multiple contradictions (with the simultaneous affirmations and denials of the same exact attributions), and such contradictions go against the first principle of reasoning.

In the so-called Neo-Scholastic manuals, this is one common way to argue against pantheism. We’re using a reductio ad absurdum because (1) we assume that all things are of one substance, (2) we derive a contradiction based on that assumption, and (3) thereby conclude that it is false that all things are of one substance.

John J. Colligan argues against pantheism this way in his very old philosophical book Cosmology. He writes: “That system which contradicts internal and external experience and reason is absurd. But Pantheism contradicts internal and external experience and reason” (p. 5).

Proving the minor premise is vital to make the argument sound. . .

(1) Pantheism contradicts internal experience, e.g., by our experience that our actions and affections belong to us as distinct beings who have moral responsibility over ourselves.
(2) Pantheism contradicts external experience, e.g., by our experience that there are distinct and diverse bodies that are independent from each other.
(3) Pantheism contradicts reason, e.g., by natural theology’s deduction that God exists, but pantheism makes God finite, limited, changing, composite, etc.

Warning! To be sure, part of Colligan’s argument ventures into the informal version of the reductio ad absurdum. However, consider inferences to the effect that get something like this: “I am a totally distinct individual being and I am totally not a distinct individual being, in the same respect and manner.”

Example 3: “God is all powerful. If God is all powerful, then he could have created the world in any logically possible way and the world has no necessity. If the world has no necessity, then we can’t know the way the world is by abstract speculation apart from experience. Therefore, we cannot know the way the world is by abstract speculation apart from experience.”

This is borrowed from Harry Gensler’s Introduction to Logic (p. 160). We can validly derive that conclusion from the premises by an indirect proof.

Let’s assume it’s false that we can’t know the way the world is by abstract speculation apart from experience. By modus ponens, God could have created the world in any logically possible way and the world has no necessity. By modus tollens, with our assumption, the world has necessity. But now we have a contradiction, as we have derived that the world has necessity and the world has no necessity. Ergo, it validly follows that we can’t know the way the world is by abstract speculation apart from experience.

Pretty neat, right? That’s the power of logic!

Example 4: Prove that the square root of 2 is irrational.

Mathematicians often make use of the reductio ad absurdum in proofs. How can we prove that √2 is not rational, that is, it’s not a number that can be written as a fraction?

We’ll prove this indirectly! (No interest in math? Skip ahead!)

So, assume that √2 is not irrational, i.e., it can be written as a fraction a/b, where a and b have no common factors (meaning it’s irreducible). Based on this assumption, we can algebraically get 2 = (a^2)/(b^2). This yields 2b^2 = a^2, which means a^2 is even.

By definition, an even integer is of the form 2k. We’ll here take it as a fact that if a^2 is even, then a must be even (which can be separately proved). Hence, it follows, a = 2k. This yields 2b^2 = (2k)^2 = 4k^2. With algebra, we can then get b^2 = 2k^2, which means b^2 is even and so b is even.

Now, look closely to spot the contradiction! a = 2k because it’s even and b = 2r because it’s even. But then, we have a common factor of 2. This contradicts supposedly having no common factors. That is, we’re led to the derivation that 2 does not divide both a and b and 2 does divide both a and b. (Put differently, a/b is irreducible and is not irreducible.)

Ergo, √2 is not rational and so must be irrational.

It takes a lot of practice to get good at mathematical proofs. If you’re new to them and want to learn to write them, a first good step is to get comfortable reading them.

How to Prove it: A Structured Approach by Daniel Velleman is one of the best textbooks to learn mathematical proofs. It’s commonly assigned to math majors in college. And, unlike most textbooks, is not insanely expensive to buy a new copy.

Indirect proofs have a long history.

For example, Euclid (325 – 265 BC) used the reduction ad absurdum to prove that there must be infinitely many prime numbers. His famous work The Elements not only contains geometry, it has some number theory, too.

Informal Version of the reductio ad absurdum

While the formal version of the reductio ad absurdum leads to an outright contradiction, the informal version leads to a falsehood or implausibility.

Let’s imagine we’re examining a worldview. To determine its truth, we first look at its premises. We might then take those premises to validly derive conclusions. Perhaps one of those conclusions is “obviously” false.

Insofar as that “obviously” false conclusion was validly derived, at least one premise of that worldview must be false. We, therefore, would have good reason to reject that worldview, or at least one premise of it.

This is very handy in debate! Your opponent supports position X. You show that (1) X logically has the consequence Y, (2) that Y is absurd, and thus (3) X should be rejected.

Socrates (469-399 BC) is especially famous, in Plato’s dialogues, of making informal reductio ad absurdum arguments. Socrates would logically demonstrate, through a series of back-and-forth questioning, the “absurdity” of someone’s position. This would constitute Socrates’ “elenchus,” i.e., his refutation of the opponent.

Example 1:

Note that there are two main issues in Plato’s Republic. First, what is justice? Second, how should the ideal city (a.k.a. Kallipolis – the beautiful city) be organized? Socrates seeks an answer. He first comes to Cephalus.

Cephalus, a rich and old man, tells Socrates that justice is only about following customary rules or paying your debts.

Socrates replies: Does this always lead to justice or right results? What if a murderer or madman wants his knife back? You borrowed his knife. If you follow Cephalus, then justice demands you give it back to him. But you shouldn’t give it back! Therefore, since the view of Cephalus leads to an absurd conclusion, we should reject that view.

Example 2:

Economist Frédéric Bastiat (1801 – 1850) was a master at the reductio ad absurdum. He wrote a marvelous satire called “The Candlemakers’ Petition.” Candles have an “unfair” competition with the Sun. Why not snuff out the Sun? Imagine the economic boost that candlemakers would receive!

“What we pray for is that it may please you to pass a law ordering the shutting up of all windows, skylights, dormer-windows, outside and inside shutters, curtains, blinds, bull’s-eyes; in a word, of all openings, holes, chinks, clefts, and fissures, by or through which the light of the sun has been in use to enter houses, to the prejudice of the meritorious manufactures with which we flatter ourselves that we have accommodated our country — a country that, in gratitude, ought not to abandon us now to a strife so unequal.”

It’s sometimes argued that protective tariffs are needed (1) due to “unfair” competition that offers low costs goods and (2) to economically boost specific industries. Bastiat’s argument shows absurd consequences, if we logically follow these arguments consistently.

Example 3: This philosophical worldview, if true, leads to radical skepticism. However, because radical skepticism is absurd, this philosophical worldview should be rejected.

Example 4: This moral philosophy, if true, leads to denying the immorality of torturing innocent people. However, because it’s absurd to deny the immorality, this moral philosophy should be rejected.

Difficulties are immediately raised here!

The formal reductio ad absurdum leads to a straight-out contradiction. Such a reductio ad absurdum is a definitive proof.

The informal reductio ad absurdum leads to an absurdity, not to a straight-out contradiction. What does it mean to be “absurd”? And how do we reply to somebody who takes what we find “absurd” not to be “absurd”? Or are there occasions when the so-called “absurd” consequences of a set of premises should be accepted as true?

We can only take things case-by-case. That’s often the nature with informal logic. Nonetheless, we should argue, if necessary, why we think something is “absurd.” That is, we shouldn’t “beg the question.”