Abstract: There are at least two ways to work reductio proofs. One way is to generate a contradiction and go straight to the conclusion. The other way is to generate the contradiction and then proceed to go through the steps in the proof showing exactly how the conclusion follows from the contradiction. The first way is to treat reductio ad absurdum as a rule; the second way is to treat reductio ad absurdum as a method. The author argues for teaching reductio proofs as a method.

Sometime in the first few days of a beginning Logic class, I ask my students what it is that is so bad about contradicting ourselves. Most people think of contradictions in terms of hypocritical behavior, such as saying one thing but doing another. We can think of many good examples of this, and there is no need to give one here. One goal of mine in bringing the entire matter up with my Logic students is to get them to see what a sentential/propositional contradiction would look like. So I give the following example:

1. It is raining in Berkeley on the southwest corner of Telegraph and Durant avenues and it is not raining in Berkeley on the southwest corner of Telegraph and Durant avenues at this very moment.

Using a translation dictionary, where 'R' = 'It is raining in Berkeley on the southwest corner of Telegraph and Durant avenues at this very moment', I then scale the sentence down to:

2. R is true and R is not true.

Here is the symbolic translation of (2):

3. (R & ~R)

It doesn't take long before everyone in the class is agreeing that (2) is false. It simply can't be raining and not raining at a specific spot at the same time. So far, so good. At some point, I tell them that, within the system of logic we will be studying, anything and everything follows from a contradiction. I tell them that I can't prove it to them right now, but will be able to do so in the weeks to come, when we introduce the right logical tools.

For the most part, we have a tendency to believe our teachers. They are supposed to be experts and we take as true a lot of what they say based on our belief that they wouldn't be assigned to teach courses at a university if they weren’t experts (more or less, anyway) in the subject of the course. So, since I'm teaching the Logic class, I must be (more or less) an expert in Logic. I'm sure we all try to avoid having our students commit the fallacy of authority. So, the students leave the class and, for the next few weeks, believe at least tentatively that everything follows from a contradiction.

Now move ahead some weeks to when we are working on proofs in a system of Natural Deduction. I have my students work in a system like those used in books by such logicians as Mates, Suppes, Copi, Layman, Herrick, Pospesel, and so on. So, they have to learn the standard rules of inference and replacement, such as Modus Ponens, Hypothetical Syllogism, [Disjunctive] Addition, and DeMorgan's Laws. In addition, the students have to learn to work Conditional Proof problems and do proofs using Reductio ad Absurdum. Many logicians call reductio proofs Indirect Proofs.

But what is reductio ad absurdum, anyway? Is it a rule or is it a method? The idea behind it is that you begin with an argument, assume the denial of the argument's conclusion, work for a contradiction, using the inference and replacement rules, and then, since we know that anything follows from a contradiction, the conclusion must, therefore, follow from the contradiction, we subsequently write the conclusion on a line by itself. QED. Here is an example of a simple proof in Natural Deduction using reductio ad absurdum as it seems to me to be used in the great majority of texts:¹

1. (A -> B)
2. (R -> ~B) |- (A -> ~R)
3. ~(A -> ~R)       Assumption
4. ~(~A v ~R)       3, Material Implication
5. (A & R)       4, DeMorgan's
6. A                5, Simplification
7. R                5, Simplification
8. B                1,6, Modus Ponens
9. ~B                2,7, Modus Ponens
10. (B & ~B)       8,9, Conjunction
11. (A -> ~R)       3-10, Reductio ad Absurdum

This manner of proof uses reduction as a rule whereby we say that anytime an explicit contradiction appears on a line of a proof, it is legitimate to simply write the conclusion on the next line.

I object to the proof above for two reasons. First, I object to this mode of derivation of the very last line of the proof. How does any student know how to derive the conclusion '(A -> ~R)' from the contradiction '(B & ~B)'? Knowing that is not knowing how.

Certainly a fair number of the students in the class would figure out how to do this with just a little time thinking about it. It would be like giving them the following problem:

Problem: Prove the following argument valid.

1. (B & ~B) |- (A -> ~R)

Jumping from line 10 to line 11 in the above proof without any intermediate steps, just because you know that anything follows from a contradiction in this system does not give explicit information about how to prove that everything follows from a contradiction in this system.

My second objection to the above proof, and all proofs of this nature, is that an assumption is opened on line 3 and never explicitly closed. It appears as though the closure itself is assumed somewhere between lines 10 and 11. All reduction proofs use assumptions. All assumption scopes must ultimately be closed so that the last line of the proof exists outside the scope of all assumptions made within the proof. In the end, reduction proofs are Conditional Proofs.

I advocate treating reductio ad absurdum as a method rather than as a rule. As such, there is a standard way to do reduction proofs, showing how the conclusion follows from the contradiction, closing the scope of all assumptions, and deriving the conclusion on a line by itself.

While I admit that the actual proof that anything (in this case '(A -> ~R)) follows from a contradiction is longer, the proof itself is instructive because it shows each step in the proof explicitly, including closing, at line 12, the scope of the assumption opened at line 3. Here is that proof:

1. (A -> B)
2. (R -> ~B) |- (A -> ~R)
3. ~(A -> ~R)          Assumption
4. ~(~A v ~R)          3, Material Implication
5. (A & R)             4, DeMorgan's
6. A                      5, Simplification
7. R                      5, Simplification
8. B                      1,6, Modus Ponens
9. ~B                      2,7, Modus Ponens
10. [B v (A -> ~R)]                8, Addition
11. (A -> ~R)             9,10, Disjunctive Syllogism
12. [~(A -> ~R) -> (A -> ~R)]             3-11, Conditional Proof
13. [~~(A -> ~R) v (A -> ~R)]                   12, Material Implication
14. [(A -> ~R) v (A -> ~R)]                   13, Double Negation
15. (A -> ~R)             14, Replication (Tautology)

Note that line 12 in this proof makes it clear that this is a Conditional Proof proof. This seems important, not wanting students to get the idea that assumptions are convenient "anything goes" moves. Closing each scope of each assumption is necessary.

I would argue, then, that, for the purposes of teaching, the rigor of any system of Natural Deduction should eschew the treatment of reductio ad absurdum as a rule and opt for treating it as a method.

Note

1. Examples of texts that use reductio ad absurdum in this way are: Baronett, Stan, Logic, (Upper Saddle River, NJ, Prentice Hall, 2008; Copi, Irving M., et.al, Introduction to Logic, 14th ed. (Upper Saddle River, NJ, Prentice Hall, 2011); Herrick, Paul, Introduction to Logic (Oxford: Oxford University Press, 2013); Howard-Snyder, Frances, et.al., The Power of Logic, 5th ed. (New York: McGraw Hill, 2013); Hurley, Patrick J., A Concise Introduction to Logic, 12th ed. (Stamford, CT, Cengage Learning, 2015); Copi, Irving M., et.al, Introduction to Logic, 14th ed. (Upper Saddle River, NJ, Prentice Hall, 2011).

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Copyright 2015 by Michael F. Goodman. All rights reserved.