1. Introductory Logic PHI 120
Presentation: “Basic E and I Strategies"
Introductory Logic PHI 120

2. Homework I Memorize the primitive rules Except ->I and RAA
Capable of writing the annotation
Cite how many premises make up each rule
Cite what kind of premises make up each rule
Cite what sort of conclusion may be derived
Except ->I and RAA
See The Rules Handout

3. Homework I Memorize the primitive rules See The Rules Handout
Capable of writing the annotation
m, n &I
Cite how many premises make up each rule
two premise rule
Cite what kind of premises make up each rule
each can be any kind of wff (i.e., the 2 conjuncts)
Cite what sort of conclusion may be derived
a conjunction
See The Rules Handout

4. Homework I Memorize the 10 primitive rules See The Rules Handout
Capable of writing the annotation
m, n &I
Cite how many premises make up each rule
two premise rule
Cite what kind of premises make up each rule
each can be any kind of wff (i.e., the 2 conjuncts))
Cite what sort of conclusion may be derived
a conjunction
See The Rules Handout

5. &E, vE, ->E, <->E
Homework I
Memorize these primitive rules
Capable of writing the annotation
m, n &I
Cite how many premises make up each rule
two premise rule
Cite what kind of premises make up each rule
each can be any kind of wff (i.e., the 2 conjuncts)
Cite what sort of conclusion may be derived
a conjunction
&I, vI, <->I
&E, vE, ->E, <->E
These rules should be memorized by now!
See The Rules Handout

6. Homework II
To memorize the rules, you need to practice doing proofs.
To practice proofs, you need to have the rules memorized
Ex : S1 – S10
First: solve problems with *
* answers in back of book
Second: all others
Seek help with your TA or me, if confused
See class web page for
Office Hours

7. The two major kinds of rules
Review
The two major kinds of rules

8. Rules Elimination Rules (4) Introduction Rules (4) Allow you to break
Every elimination rule will have a premise of the kind indicated by that rule
Introduction Rules (4)
Allow you to make
The conclusion of every introduction rule will be the kind of statement made that rule

9. Solving for the Conclusion
Writing the Proof

10. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R

11. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R (1)
A line of a proof contains four elements:
(i) line number (number within parentheses)
(ii) annotation (at the very right)
(iii) sentence derived (next to line number)
(iv) assumption set (number to very left)

12. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R (1) A
A line of a proof contains four elements:
(i) line number (number within parentheses)
(ii) annotation (at the very right)
(iii) sentence derived (next to line number)
(iv) assumption set (number to very left)

13. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R (1) P -> Q A
A line of a proof contains four elements:
(i) line number (number within parentheses)
(ii) annotation (at the very right)
(iii) sentence derived (next to line number)
(iv) assumption set (number to very left)

14. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A
A line of a proof contains four elements:
(i) line number (number within parentheses)
(ii) annotation (at the very right)
(iii) sentence derived (next to line number)
(iv) assumption set (number to very left)

15. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A (2)
A line of a proof contains four elements:
(i) line number (number within parentheses)
(ii) annotation (at the very right)
(iii) sentence derived (next to line number)
(iv) assumption set (number to very left)

16. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A
A line of a proof contains four elements:
(i) line number (number within parentheses)
(ii) annotation (at the very right)
(iii) sentence derived (next to line number)
(iv) assumption set (number to very left)

17. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A (3)

18. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A
The
Basic
Premises

19. How to Read a Problem S5: P -> Q, P -> R, P ⊢ Q & R
1 (1) P -> Q A
2 (2) P -> R A
3 (3) P A
How to Read a Problem

20. Elimination or Introduction?
Writing the Proof
“Q & R” is not embedded in premises!
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A
General Strategy
Elimination or Introduction?
Whenever solving for a conclusion, ask yourself two questions:
1) What is the conclusion?
2) How is this conclusion embedded in the premises?

21. Elimination or Introduction?
Writing the Proof
“Q & R” is not embedded in premises!
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A
General Strategy
Elimination or Introduction?

22. “Q & R” is not embedded in premises!
Writing the Proof
“Q & R” is not embedded in premises!
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4)
Possible premise for ->E
Possible premise for ->E
Antecedent of (1) and (2)

23. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) m, n ->E

24. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) 1, 3 ->E

25. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Q 1, 3 ->E

26. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Q 1, 3 ->E

27. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Q 1, 3 ->E

28. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E

29. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5)

30. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) m, n ->E

31. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) 2, 3 ->E

32. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) R 2, 3 ->E

33. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) R 2, 3 ->E

34. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) R 2, 3 ->E

35. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E

36. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) ??

37. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) m, n &I

38. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) 4, 5 &I

39. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

40. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & 4, 5 &I

41. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

42. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

43. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

44. Writing the Proof
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E 1,2,3 (6) Q & R 4, 5 &I

45. Writing the Proof
End every proof with two questions:
Is this the main conclusion?
Are the assumptions correct?
S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E 1,2,3 (6) Q & R 4, 5 &I

46. Strategy Key Lesson 2 x 2 Questions A. Solving for a Conclusion
1) What is the conclusion?
2) How is this conclusion embedded in the premises?
B. The End of a Proof
Is the final line the main conclusion?
2) Are the assumptions correct?

47. Homework Memorize primitive rules (except ->I and RAA)
Ex : S1 – S10
First: solve problems with *
* answers in back of book
Second: all others
To memorize the rules, you need to practice doing proofs.
To practice proofs, you need to have the rules memorized