Understanding Truth Values of WFFs in Logic

1. Homework 3

2. True or False? True or false? For each of the following WFFs, determine whether it is true or false on the given evaluation. Evaluation: P = T, Q = T, R = T 1. ~(P ↔ Q) 2. ~((P v (Q → R)) & ~P)

3. True or False? True or false? For each of the following WFFs, determine whether it is true or false on the given evaluation. Evaluation: P = T, Q = T, R = T 1. ~(P ↔ Q) 2. ~((P v (Q → R)) & ~P)

4. Write Down the Evaluation

5. Copy the Evaluations beneath the Sentence Letters

6. Consult Biconditional Truth-Table

7. Consult Biconditional Truth-Table

8. Copy Down the Answer

9. Consult the Negation Truth-Table

10. Consult the Negation Truth-Table

11. Copy Down the Answer

12. The Answer is “False”

13. True or False? True or false? For each of the following WFFs, determine whether it is true or false on the given evaluation. Evaluation: P = T, Q = T, R = T 1. ~(P ↔ Q) 2. ~((P v (Q → R)) & ~P)

14. From My Inbox Dear Michael, Are there too many or too little parentheses in this formula: ~((P v (Q → R)) & ~P) Thanks, --Concerned Student

15. Nope! ~((P v (Q → R)) & ~P)

16. Set #1 ~((P v (Q → R)) & ~P)

17. Set #2 ~((P v(Q → R)) & ~P)

18. Set #3 ~((P v(Q → R))& ~P)

19. Definition of WFF • All sentence letters are WFFs. • If φ is a WFF, then ~φ is a WFF. • If φ and ψ are WFFs, then (φ & ψ), (φ v ψ), (φ → ψ), (φ ↔ ψ) are also WFFs. • Nothing else is a WFF.

20. Demonstration Using the definition we can show that certain sequences of symbols are WFFs. For example ~((P v (Q → R)) & ~P) is a WFF.

21. All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R is a WFF

22. All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R is a WFF

23. ~((P v (Q → R)) & ~P)

24. By (iii) if φ and ψ are WFFs, then (φ → ψ) is a WFF.

25. ~((P v (Q → R)) & ~P)

26. ~((P v (Q→R)) & ~P)

27. All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R is a WFF

28. ~((P v (Q→R)) & ~P)

29. ~((P v (Q→R)) & ~P)

30. By (iii) if φ and ψ are WFFs, then (φ v ψ) is a WFF.

31. ~((P v (Q→R)) & ~P)

32. ~((Pv(Q→R)) & ~P)

33. All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R is a WFF

34. ~((Pv(Q→R)) & ~P)

35. ~((Pv(Q→R)) & ~P)

36. By (ii) if φ is a WFF, then ~φ is a WFF.

37. ~((Pv(Q→R)) & ~P)

38. ~((Pv(Q→R)) & ~P)

39. By (iii) if φ and ψ are WFFs, then (φ & ψ) is a WFF.

40. ~((Pv(Q→R)) & ~P)

41. ~((Pv(Q→R))&~P)

42. By (ii) if φ is a WFF, then ~φ is a WFF.

43. ~((Pv(Q→R))&~P)

44. ~((Pv(Q→R))&~P)

45. Scope Every occurrence of a connective in a WFF has a scope. The scope of that occurrence is the smallest WFF that contains it. For example The scope of “&” in “(~(~P&Q)→P)” is (~P&Q) • (~(~P & Q) → P) is not a WFF. • (~(~P & Q) → P)is not a WFF. • (~(~P & Q) → P) is a WFF, but is bigger than (~P&Q)

46. Occurrences Notice that the same symbol can occur different times in the same formula, and that its different occurrences can have different scopes. • ~((~P & Q) & (R ↔ Q)) • ~((~P & Q) & (R ↔ Q)) • ~((~P & Q) & (R ↔ Q)) • ~((~P & Q) & (R ↔ Q))

47. Scope of → ~((P v(Q → R))& ~P)

48. Scope of v ~((P v(Q → R))& ~P)

49. Scope of & ~((P v(Q → R))& ~P)

50. Scope of Main Connective ~((P v(Q → R))& ~P)