Well, boys and girls, it's time for another of Turner's Logic Lessons! Todays lesson:

Reductio ad absurdum

(from Lat., meaning "reduction to the absurd")

 

 

Reductio ad absurdum is a rhetorical strategy in which one attempts to prove an assertion by assuming its negation, then proceeding through logical stps until you reach an absurdity (i.e., a false proposiion). It's roughly the logic behind "If that's a woman, then I'm a radish". Your assertion is that that is not a woman, which you prove by assuming it is a woman, and concluding that you are a radish, which is almost certainly false. Yes, I know that it's a poor example, since it is not actually the case that radishhood follows from the femininity of the person in question (but that's a whole other can of worms).

 

In propositional logic terms, if A, B, C and D are propositions (e.g. "The sky is blue", "I am the queen", "Black is white", "God exists"), and you are trying to prove A, you might do so using this strategy. First, assume the negation of A (we'll signify that as "!A").

 

    !A

    !A -> B ("not A implies B")

    B -> C ("B implies C")

    C -> D ("C implies D")

    D -> False ("D implies falsehood, i.e. D is false")
 

 

Now, assuming that all the steps taken are valid--the negation of A really does imply B, B really does imply C, etc.--then what you're left with is "!A -> False", or, if A (the proposition you're trying to prove) is false, then you end up with falsehood, which is logically equivalent to "!A is false, so A is true." *

 

 

 

 

So what's this have to do with this group?

Good question. Atheists generally take this tactic when debating atheists. The atheist generally tries to prove the proposition "God does not exist" (depending on the context and the atheist, that may be "God" in general or a specific religion's God). To do so, the atheist will assume "God exists" (!A, where A = "God does not exist"). From this, he will conclude, after some number of steps, a falsehood, and will therefore have proven his assertion "God does not exist".

For example, the following would be a reductio ad absurdum argument against the existence of the Christian God by the Argument from Evil:

   1) The Christian God exists (assumed the negation of "The Christian God does not exist")

   2) If the Christian God exists, then God is omnipotent, omniscient, and totally benevolent (by definition)

   3) Evil exists

   4) God created everything

   5) If God created everything, then God allowed sin into the world

   6) God allowed sin into the world

   7) If God allowed sin into the world, then he is responsible for all the evil in the world

   8) God is responsible for all the evil in the world

   9) Therefore, God is either not omnipotent or not totally benevolent

  10) But (9) is a contradiction of (2)

  11) Therefore, our original premise ("The Christian God exists") must be false, meaning that the Christian God does not exist.

    Q.E.D.

Of course, this is the point where the Christian tries to invalidate the proof by challenging one or more of the logical steps used in reaching the contradiction, thereby trying to invalidate that particular proof.

Occasionally, some disingenuous theist will see an atheist using reductio ad absurdum and point out that the theist is speaking as if assuming that God exists, so obviously he also believes in God, no matter how much he denies it! This, of course, is utterly invalid.

Theists and Reductio ad Absurdum

Mostly, the people in this group who use this tactic tend to be atheists. However, some theistic arguments might be classified as reductio ad absurdum arguments. For example, the Argument from Design:

    1) Assume God does not exist (negation of "God exists")

    2) If God does not exist, then there is no guiding force to order the universe

    3) There is no guiding force to order the universe

    4) If there is no guiding force to order the universe, then the universe arose from chance

    5) The universe arose from chance

    6) The universe is extremely complex

    7) There is such a small probability that such complexity arose from chance, so it's unreasonable to believe it did

    8) The universe did not arise from chance

    9) But (8) is a contradiction to (5)

    10) Therefore, the universe did not arise from chance

    11) Therefore, there is a guiding force that ordered the universe (i.e. a creator god)

    Q.E.D. 

Again, the atheist then may challenge any of the steps taken to reach the contradiction, thereby trying to invalidate that particular proof. A similar argument might be presented to argue for the Argument from First Cause.

Boring logic stuff for those who wish to learn the reasoning behind it

The following is just for those interested in the logical basis for this device. You can safely skip it if you want.

In logic, a proposition is either a single statement ("The sky is blue") (this is known as an atomic proposition) or multiple statements joined by logical connectives. There are four basic logical connectives in propositional logic:

    And         (^)

    Or           (v)

    Not         (¬)

    Implies   (->)

Each proposition has a truth value; that is, it is either true (T) or false (F). Propositions that contain one or more connectives take on truth values depending on the connectives and the truth values of each proposition it comprises. The rules for determining what truth value such a proposition will take on can be presented in a truth table. For instance, the truth table for "And" looks like this:

A	B	A ^ B
T	T	T
T	F	F
F	T	F
F	F	F

 

 

What this table says is that, if two propositions A and B are joined together by And, then:

    If A is true and B is true, then A and B is true

    If A is true and B is false, then A and B is false

    If A is false and B is true, then A and B is false

    If A is false and B is false, then A and B is false

 

 

 The same can be done for all the connectives, and any arbitrarily complex combination of propositions.

 

 

What we are interested in is "Implies". The truth table for Implies is as follows:

 

 

A	B	A -> B
T	T	T
T	F	F
F	T	T
F	F	T

In a reductio ad absurdum proof, as mentioned, we end up with "!P->False". If the proof is valid, then the proposition "!P->False" has the truth value of True. So we need to look at only the lines of the Implies truth table that correspond to A->B being true while B is false:

 

 

A	B	A ^ B
F	F	F

 

 

Only one such row fits those criteria, as you can see. You can see that A, then, must be false. In a reduction ad absurdum proof, A = !P. So if A is false, then !P is false, which is logically equivalent to saying "P is true". Since P is what you were trying to prove all along, you have now proven that P is true.

 

The End