Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

Exercises
for
Section 5: Rules of Inference

4. Tautological Implications and Tautological Equivalences 5. Rules of Inference 6. Arguments and Proofs Main Logic Page "Real World" Page
Answers to Odd-Numbered Exercises

In each of the following exercises, supply the missing statement or reason, as the case may be. (To make life simpler, we shall allow you to write ~(~p) as just p whenever it occurs. This saves an extra step in practice.)

Statement Reason Statement Reason
1. 1. p~q Premise 2. 1. ~pq Premise
2. p Premise 2. ~p Premise
3. - - - - 1,2 Modus Ponens 3. - - - - 1,2 Modus Ponens
3. 1. (~pq) ~(qr) Premise 4. 1. (~pq)(q~r) Premise
2. ~pq Premise 2. ~pq Premise
3. - - - - 1,2 Modus Ponens 3. - - - - 1,2 Modus Ponens
5. 1. (~pq)~(qr) Premise 6. 1. (~pq)(q~r) Premise
2. qr Premise 2.~(q~r) Premise
3. - - - - 1,2 Modus Tollens 3. - - - - 1,2 Modus Tollens
7. 1. ~(~pq) Premise 8. ~(p~q) Premise
2. - - - - De Morgan 2. - - - - De Morgan
9. 1. (pr)~q Premise 10. 1. (~pq)(q~r) Premise
2. ~qr Premise 2. (q~r)s Premise
3. - - - - 1,2 Transitive Law 3. - - - - 1,2 Transitive Law
11. 1. (pr)~q Premise 12. 1. (~pq)(q~r) Premise
2. ~qr Premise 2. (q~r)s Premise
3. ~r Premise 3. ~s Premise
4. - - - - 1,2 Transitive Law 4. - - - - 1,2 Transitive Law
5. - - - - 3,4 Modus Tollens 5. - - - - 3,4 Modus Tollens
13. 1. (pq)r Premise 14. 1. ~(pq)s Premise
2. ~r Premise 2. ~s Premise
3. - - - - 3. - - - - 1,2 Disjunctive Syllogism
15. 1. p(rq) Premise 16. 1. (pq)r Premise
2. ~r Premise 2. ~r Premise
3. - - - - 2, Addition of ~q 3. - - - - 1,2 Modus Tollens
4. - - - - 3, De Morgan 4. - - - - 3, De Morgan
5. - - - - 1,4 Modus Tollens 5. - - - - Simplification
17. 1. (pq)r Premise 18. 1. pr Premise
2. q Premise 2. p Premise
3. p Premise 3. s Premise
4. - - - - 3,2 Rule C 4. - - - - 1,2 Modus Ponens
5. - - - - 1,4 Modus Ponens 5. - - - - 3,4 Rule C
19. 1. ~(~pq)p~q - - - - 20. 1. [(~pq)~q]p - - - -
21. 1. p~(qr) 22. 1. (st)(q~r) Premise
2. qr Premise 2. (st) Premise
3. ~p - - - - 3. q~r - - - -
23. 1. ~p(rs) 24. 1. ~pq Premise
2. p(rs) 2. ~p~q - - - -
25. 1. ~[p~(qr)] Premise 26. 1. (~p~q)p Premise
2. ~[~p~(qr)] - - - - 2. ~(~p~q)p - - - -
3. p(qr) - - - - 3. (pq)p - - - -
27. 1.(pq)(rs) Premise 28. 1. (pq)~r Premise
2. p Premise 2. ~p~q Premise
3. pq - - - - 3. ~(pq) - - - -
4. rs - - - - 4. ~r - - - -
5. r - - - - 5. ~rs - - - -
29. 1. p~q Premise 30. 1. (pq)(r~s) Premise
2. ~q~r Premise 2. ~rs Premise
3. (r~p)t Premise 3. ~(r~s) - - - -
4. p~r - - - - 4. ~(pq) - - - -
5. r~p - - - - 5. ~p~q - - - -
6. t - - - - 6. ~p - - - -
31. 1. p~p Premise 32. 1. ~p Premise
2. p - - - - 2. p Premise
3. ~p - - - - 3. ~p~p - - - -
4. ~pq - - - - 4. p~p - - - -
5. pq - - - - 5. pp - - - -
6. q - - - - 6. ~pp - - - -
33. 1. p~(~p) - - - - 34. 1. ~pp Premise
2. pp - - - - 2. t Premise
3. ~pp - - - - 3. t~p - - - -
4. (~pp)~q - - - - 4. ~pt - - - -
5. ~p(p~q) - - - - 5. (~pp)(~pt) - - - -
6. ~p(~qp) - - - - 6. ~p(pt) - - - -
7. p(~qp) - - - -
8. p(qp) - - - -

Convert each of the following into a symbolic proof, and supply the justifications for each step.

Communication and Reasoning Exercises

41. Complete the following sentence. The Modus Tollens rule of inference says that, if bothandappear on a list of statements known to be true, then we can add.

42. Complete the following sentence. The Modus Ponens rule of inference says that, if bothandappear on a list of statements known to be true, then we can add.

43. Modify Example 6 to produce a proof that uses every type of inference rule we have discussed. (Try replacing q by b and referring to Example 5.)

44. Explain why the following is not a reasonable candidate for a new rule of inference:

A
AB

4. Tautological Implications and Tautological Equivalences 5. Rules of Inference 6. Arguments and Proofs Main Logic Page "Real World" Page
Answers to Odd-Numbered Exercises

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble