Right Reasoning, pt. V, Reductio Ad Absurdum

We come now to a favorite of mine, the...

V. Reductio ad absurdum

A. Formally stated:

1. In order to prove P,
2. Presuppose, for argument’s sake, its contrary, not-P.
3. Argue that, in presupposing not-P, we’d have to conclude Q.
4. Demonstrate the absurdity and/or inviability of Q.
5. Conclude, P must be true.

B. 1. Illustrated from a pro-abortion argument:

i. In order to prove that, (P) women do not have the right to destroy the life within them,

ii. presuppose the opposite that, (not-P) women do have the right to destroy the life within them, since “women have a right over their own bodies” and that that “tissue” within them “is their own body.”

iii. However, if we presuppose this (not-P), then we must conclude that (Q) a. the aborted “tissue” (i.e., the child) has identical DNA to the woman, and b. that approximately half of women who have abortions also have penises (because approximately half of those aborted are boys).

iv. But both of these conclusions (Q) are utterly absurd.

v. Therefore, (P) it is not the case that women have the right to destroy the life within them.

B. 2. Illustrated from Scripture (Matt 12:24—26):

i. (P-assumed: I [Jesus] do not cast out demons by Beelzebul).

ii. Assume though that, (not-P) Satan casts out Satan

iii. If (not-P) Satan casts out Satan, then (Q) his kingdom is divided, ruined and cannot continue.

iv. But, since demonic activity is continuing, clearly (Q) is absurd.

v. Therefore, (P) I do not cast out demons by the power of Beelzebul.

B. 3. Again, Illustrated from Scripture (Matt 22:41ff):

i. (P-assumed: I [Jesus] am the Christ, more than merely David’s son)

ii. Assume (not-P), that I am merely David’s physical descendent (the Pharisees’ answer, see v. 42).

iii. If (not-P), then (Q) Jesus could not be David’s “Lord” (especially in first century near Eastern culture!!).

iv. But since David, writing “in the Spirit,” in Psalm 110:1, calls him “Lord,” (Q) is absurd.

v. Therefore, (P) I am the Christ, more than merely David’s son.

(NOTE: “And no one was able to answer him a word, nor from that day did anyone dare to ask him any more question” v. 46.)

Douglas Groothuis comments on this passage in this way:

“Jesus’ point was not to deny the Christ’s ancestral lineage to David, since Jesus himself is called ‘the son of David’ in the Gospels (Matthew 1:1), and since Jesus accepts the title without rebuke (Matthew 20:30—31). Rather, Jesus is denying that the Christ is merely the son of David; he is also Lord, and was so at the time of David. By using this reductio ad absurdum argument, Jesus attempts to expand his audience’s understanding of who the Christ is and that he himself is the Christ” (On Jesus, 34).

C. Fallacy factor:

Because the reductio is an indirect or negative argument, it is not as susceptible to particular fallacies. In essence, however, its just an expanded modus tollens. And since the modus tollens is subject to both informal fallacies and the formal fallacy of denying the antecedent, review those pitfalls there.

Right Reasoning, pt. II, Modus Tollens

To continue our series in valid forms of argument...

II. Modus Tollens (Latin, lit. “The mode of taking”):

A. Formally stated:

1. If P is the case, the Q is also the case;
2. Not-Q,
3. Therefore, not-P

B. 1. Illustrated from ethics:

(Here I cite Gregory Koukl and Francis Beckwith’s Relativism: Feet Firmly Planted in Mid-Air, 1999, 65.)

“In like fashion [to B. F. Skinner’s conclusions in Beyond Freedom and Dignity], relativists must remove the words praise and blame from their vocabularies. But if the notions of praise and blame are valid, then relativism must be false.”

The line of reasoning is modus tollens.

i. If (P) moral relativism is the case, then (Q) the words praise and blame are unintelligible.

ii. But, (not-Q) the words praise and blame are valid moral categories (even presupposed by relativists!).

iii. Therefore, (not-P) relativism is not the case.

B. 2. Illustrated from Scripture (1 John 2:19, speaking of the antichrist heretics):

“They went out from us, but they were not of us; for if they had been of us, they would have continued with us. But they went out, that it might become plain that they all are not of us.”

Again, modus tollens.

i. If (P) they had been of us, then (Q) they would have continued with us.

ii. But (not-Q) they did not continue with us.

iii. Therefore, (not-P) they were not of us.

B. 3. Illustrated from Sir Arthur Conan Doyle’s Sherlock Holmes, “The Adventure of Silver Blaze” (since Sherlock Holmes is Beaner new favorite movie!):

“A dog was kept in the stalls, and yet, though someone had been in and fetched out a horse, the dog had not barked. Obviously the visitor was someone whom the dog knew well...”

i. If (P) the horse thief was a stranger, then (Q) the dog would have barked, since the dog always barks at strangers.

ii. But (not-Q) the dog never barked.

iii. Therefore, (not-P) the horse thief was not a stranger.

C. The fallacy factor:

As with the modus ponens, discussed in pt. I of the Right Reasoning series, the modus tollens is susceptible to a formal logical fallacy, namely the fallacy of 'denying the antecedent.' (Remember, in the case of an “If...then” conditional statement, the “If” clause is the antecedent, and the “then” clause is the consequent.)

Obviously, the fallacious attempt at a modus tollens would formally follow like this:

1. If P, then Q;
2. Not-P,
3. Therefore, not-Q.

However, the conclusion, though perhaps true, doesn’t follow from the premises. This too is an invalid line of inference.

Consider this.

1. If (P) the doctrine of the perseverance of the saints (i.e., that the elect will persevere in their faith to the end) is true, then (Q) no Christian would denounce the faith.

2. However, (not-Q) some once-professed Christians do denounce the faith.

3. Therefore, (not-P) the doctrine of the perseverance of the saints is not true.

It is amazing how many Arminians believe this to be a reasonable argument!

D. This last argument is valid, yet unsound: (see section D of Right Reasoning, pt I).