Conditional Proof and
Reductio ad Absurdum
Conditional proof (CP) can be used to prove a conditional statement by making an assumption that is used to derive a consequent which, in turn, justifies a conditional conclusion.

Consider the following argument:

A (B C)

B

__________

A C


Proof:


1. A (B C) Premise
2. B / A C Premise/Conclusion
       3.       A Assumption
       4.       B C 1, 3 MP
       5.       C 2, 4 MP
6. A C 3-5 CP
The second technique we're going to consider is called proof by reductio ad absurdum (RA). The Latin phrase means a reduction to absurdity or contradiction; the technique is to show that if we accept the premises of an argument, but deny the conclusion, we contradict ourselves. This is a good way to establish that an argument is valid.

The reductio ad absurdum technique is to take the negation of the conclusion, add it as an assumption (just as in conditional proof), and then from the premises and that assumption, derive a contradiction--a statement of the form p ~ p.

(A v B) (C D)

~ (D A)

__________

~A


Proof:


1. (A v B) (C D) Premise
2. ~ (D A) / ~ A Premise/Conclusion
       3.       ~ ~ A Assumption
       4.       A 3 DN
       5.        A v B 4 Add
       6.       C D 1, 5 MP
       7.       D 6 Simp
       8.        ~ D v ~ A 2 DM
       9.        ~ D 3, 8 DS
      10.       D ~ D 7, 9 Conj
11. ~ A 3-10 RA


The reductio part of the argument is indented because our reasoning is based on an assumption.

Comprehension Questions

1.  Identify the justification for third step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (Dv~B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
2.  Identify the justification for fourth step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (D v ~ B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
3.  Identify the justification for fifth step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (D v ~ B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
4.  Identify the justification for sixth step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (D v ~ B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
5.  Identify the justification for seventh step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (D v ~ B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
6.  Identify the justification for eighth step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (D v ~ B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
7.  Identify the justification for the ninth step of the following conditional proof.

1. (A > B) > C Premise
2. C > D / A > (D V ~B) Premise/Conclusion
3. A > (B > C)
4. A
5. B > C
6. B > D
7. ~B v D
8. D v ~B
9. A > (D v ~ B)

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
8.  Identify the justification for the third step of the following reductio absurdum proof.

1. (A > B) > H Premise
2. ~A /C Premise/Conclusion
3. ~C
4. ~(A > B)
5. ~~A . ~B
6. A . ~B
7. A
8. A . ~A
9. H

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
9.  Identify the justification for the fourth step of the following reductio absurdum proof.

1. (A > B) > H Premise
2. ~A /C Premise/Conclusion
3. ~C
4. ~(A > B)
5. ~~A . ~B
6. A . ~B
7. A
8. A . ~A
9. H

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
10.  Identify the justification for the fifth step of the following reductio absurdum proof.

1. (A > B) > H Premise
2. ~A /C Premise/Conclusion
3. ~C
4. ~(A > B)
5. ~~A . ~B
6. A . ~B
7. A
8. A . ~A
9. H

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
11.  Identify the justification for the ninth step of the following reductio absurdum proof.

1. (A > B) > H Premise
2. ~A /C Premise/Conclusion
3. ~C
4. ~(A > B)
5. ~~A . ~B
6. A . ~B
7. A
8. A . ~A
9. H

a) 3,4 modus ponens
b) 7 commutation
c) 6 implication
d) 1 exportation
e) 2,5 hypothetical syllogism
f) 4 implication
g) assumption
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