T E ACH ER ’S N OT ES on L ES SO N 19
REDUCTIO AD ABSURDUM
Intermediate Logic, pp. 143–145
STUDENT OBJECTIVES
frostbite in the snow.) Since the assumption
led to a contradiction, it’s obvious that it was
a false assumption, and it’s obvious that what
you were initially trying to prove is true. Snow
is indeed cold.
1. Use Reductio Ad Absurdum as an aid to writing proofs.
2. Complete Exercise 19.
TEACHING INSTRUCTIONS
1. Explain to students that in this lesson they will
be learning another strategy, like the Conditional Proof, for writing proofs without severe
mental anguish. Have any budding Latin
scholars in the class take a stab at translating
“reductio ad absurdum.” Explain that it means
“reducing to absurdity,” and explain that this
is exactly what the reductio ad absurdum rule
allows us to do: reduce something to absurdity.
Tell students that Reductio ad Absurdum is
a special rule which allows us to assume the
negation of a proposition, deduce a self-contradiction, then conclude the proposition.
2. That’s a mouthful, so break it down. Tell students that in a reductio ad absurdum, as in the
Conditional Proof, you assume something that
you haven’t been given and see where it leads.
Say, for example, that you are trying to prove
(to someone in the Bahamas) that snow is cold.
Say you’re not making very much headway. If
you use reductio ad absurdum you can say,
“All right, let’s assume snow isn’t cold. Where
does that lead us?” Soon you will reach a contradiction (i.e., snow isn’t cold, but people get
3. Explain that the reductio is a common method
of proof in mathematics. For example, if you
want to prove that you can’t divide by zero, it’s
easiest to assume that you can and look for a contradiction to follow from that assumption. Write
this reductio mathematical proof on the board:
0 = 0
Reflexive property of equality
0 * 1 = 0 * 2 Multiplication property of zero
1 = 2
Division by zero
Obviously 1 does not equal 2, so division by
zero doesn’t work.
4. Explain that the reductio can be used in almost
any argument or proof, and that it is a particularly common method in Christian apologetics.
See if students can’t think of arguments they
have heard for the Christian faith that have
used the reductio. Tell them to keep their ears
open for anytime someone says, “All right, let’s
assume that what you’re saying is true.” For
example, “All right, let’s assume that there’s no
divine Lawgiver. But it that’s the case, there’s no
Law that any of us have to obey. How come we
call some things evil and some things good?”
5. Tell students that you are going to try to prove
the rule of Addition without using the rule of
Addition. Have them help you set up the argument using P and Q. It should look like this:
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inter mediate logic
Ordinary Proof:
1) P ⊃ Q
2) ~P ⊃ Q
/∴Q
3) ~Q ⊃ ~P
4) ~Q ⊃ Q
5) ~~Q ∨ Q
6) Q ∨ Q
7) Q
Reductio:
1) P ⊃ Q
2) ~P ⊃ Q
/∴Q
3) ~QR.A.A.
4) ~P
1, 3 M.T.
5) ~~P
2, 3 M.T.
6) ~P • ~~P 4, 5 Conj.
7) Q
3-6 R.A.
Q.E.D
1 Trans
3, 2 H.S.
4 Impl.
5 D.N.
6 Taut.
Q.E.D
1. P
/∴P∨Q
Have students look at it for a moment. They
should see that this argument is impossible to
prove using only the standard rules of inference and replacement, because only the rule of
Addition allows us to introduce new variables.
We’re stuck, unless we can use a reductio ad
absurdum. Finish out the rest of the proof,
using the reductio:
1) P
/∴P∨Q
2) ~(P ∨ Q)
R.A.A. (Assume the negation of a proposition)
3) ~P • ~Q
2 De M.
4) ~P
3 Simp.
5) P • ~P
1, 4 Conj.
(Deduce a self-contradiction)
6) P ∨ Q
2–5 R.A.
(Conclude the original proposition)
Q.E.D.
Make sure students see how you assumed the negation of the conclusion, justified it with R.A.A.
(Reductio ad Absurdum Assumption), and then
followed the regular rules until you obtained a
contradiction of the form p • ~p. Explain that the
contradiction in step five shows that the original
assumption in step two was wrong, and that you
can therefore conclude its opposite (the conclusion) in step six. The justification in step six is
labeled 2–5 for all the steps inside the reductio.
6. Explain that as in the last lesson you are now
going to prove the same argument in two
different ways, one using the normal rules of
inference and replacement, the other using
reductio ad absurdum. The initial argument
should look like this:
1) P ⊃ Q
2) ~P ⊃ Q
/∴Q
Do the two proofs one at a time side-by-side on
the board, having students walk you through
them, or, if you like, you can have half the
class work through it one way and half the
class work through it the other way and see
who finishes faster. (If you have them work on
their own, they may need lots of hints from
you.) The completed proofs should look like
the table at the top of the page.
7. Point out to students that although the two
proofs have exactly the same number of steps,
the reductio proof went (you assume) much
faster and was much easier to develop. Also
point out, if they haven’t noticed yet, that the
justification for step 3 in the reductio has no
line number before R.A.A., because we are
making the assumption out of thin air.
8. Explain that the same considerations that
applied to the Conditional Proof apply to Reductio Ad Absurdum. First of all, the reductio
need not be the entire proof, but can be part of
a larger proof; it is perfectly fine to assume the
negation of something (anything) other than
the conclusion, as long as you go on to deduce a
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