Elementary Logic

1. Elementary Logic PHIL 105-302 Intersession 2013 MTWHF 10:00 – 12:00 ASA0118C Steven A. Miller Day 4

2. Formalizing review Symbolization chart: It is not the case = ~ And = & Or = v If … then = → If and only if = ↔ Therefore = ∴

3. Logical semantics Our interpretations are concerned with statements’ truthand falsity. Principle of bivalence: Every statement is either true or false (and not both).

4. Logical semantics Negation semantics “The Cubs are the best team” is true, then … what’s false? “It is not the case that the Cubs are the best team.”

5. Logical semantics Negation semantics Likewise, if: “The Cubs are the best team” is false, then … what’s true? “It is not the case that the Cubs are the best team.”

6. Logical semantics Negation semantics (truth table) P ~P T F F T

7. Logical semantics Conjunction semantics “My name is Steven and my name is Miller.” is true when “My name is Steven Miller.”

8. Logical semantics Conjunction semantics “My name is Steven and my name is Miller.” is false when “My name is not Steven or Miller, or both.”

9. Logical semantics Conjunction semantics (truth table) P Q P & Q T TT T F F F T F F FF

10. Logical semantics Disjunction semantics “My name is Steven or my name is Miller.” is true when “My name is Steven or Miller, or both.”

11. Logical semantics Disjunction semantics “…or both”: “Soup or salad?”

12. Logical semantics Disjunction semantics Inclusive disjunction: this, or that, or both Exclusive disjunction: this, or that, but not both

13. Logical semantics Disjunction semantics For our purposes, unless stated otherwise, all disjunctions are inclusive: “or” means: this, or that, or both

14. Logical semantics Disjunction semantics (truth table) P Q P v Q T TT T F T F T T F FF

15. Logical semantics Disjunction semantics Exclusive disjunction symbolization: (P v Q) & ~(P & Q)

16. Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T TTT T F T F F T T F F FFF

17. Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T TT F T T F T TF F T TTF F FF T F

18. Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T TT F FT T F TTTF F T TTTF F FFFTF

19. Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T TTFFT T F TTTF F T TTTF F FFFTF

20. Logical semantics Material conditional semantics Follows the rules of deductive validity (in fact, every argument is an if-then statement). Is false only when antecedent (premises) is true and consequent (conclusion) is false.

21. Logical semantics Material conditional semantics This can be counter-intuitive, see: If there are fewer than three people in the room, then Paris is the capital of Egypt.

22. Logical semantics Material conditional semantics If there are fewer than three people in the room, then Paris is the capital of Egypt. Antecedent = false Consequent = false

23. Logical semantics Material conditional semantics (truth table) P Q P → Q T TT T F F F T T F FT

24. Logical semantics Biconditional semantics Biconditional is conjunction of two material conditionals with the antecedent and consequent reversed: P ↔ Q = (P → Q) & (Q → P)

25. Logical semantics Biconditional semantics (truth table) P Q (P → Q) & (Q → P) T TTT T F F T F T TF F F T T

26. Logical semantics Biconditional semantics (truth table) P Q (P → Q) & (Q → P) T TTTT T F FFT F T T F F F FTTT

27. Logical semantics Biconditional semantics (truth table) P Q (P ↔ Q) T TT T F F F T F F F T

28. Seventh Inning Stretch (“…Buy Me Some Peanuts …”)

29. Logical semantics Combining truth tables Always work from the operator that affects the least of the formula to that which affects the most of it. ~[(P & ~Q) v (Z ↔ Q)]

30. Logical semantics Combining truth tables P Q ~~ (P & Q) T TT T F F F T F F FF

31. Logical semantics Combining truth tables P Q ~~ (P & Q) T T F T T F T F F T TF F F T F

32. Logical semantics Combining truth tables P Q ~~ (P & Q) T TTFT T F FTF F T F TF F FFTF

33. Logical semantics Combining truth tables P Q ~~ (P & Q) T TTFT T F FTF F T FTF F FFTF

35. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T TTTTT T F T F F T F T F T T F F FFFFF

36. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T T TT T F FT F F T F T TF T T F F F T FFFF

37. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T F T TT T F FT F F F T F T TF T T T F F F T FFFFF

38. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T TFT F TTTT T F FT F FF T T F T TF T TTT F F FTFFFFFF

39. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T TFT F T F TTT T F FT F FFF T T F T TF T T F TTF F FTFFF T FFF

40. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T TFT F TT F TTT T F FT F F T F FTT F T TF T T F F TTF F FTFFF T T FFF

41. Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T TFTFTTFTTT T F FTFFTFFTT F T TFTTFFTTF F FTFFFTTFFF

43. Three kinds of formulas Tautologies – true in all cases P P v ~P T T F F F T

44. Three kinds of formulas Tautologies – true in all cases P P v ~P T TTF F FT T

45. Three kinds of formulas Tautologies – true in all cases P P v ~P T TTF F FTT

46. Three kinds of formulas Contradictory (or truth-functionally inconsistent) – false in all cases P P & ~P T T F F F T

47. Three kinds of formulas Contradictory (or truth-functionally inconsistent) – false in all cases P P & ~P T T F F F FFT

48. Three kinds of formulas Contradictory (or truth-functionally inconsistent) – false in all cases P P & ~P T TFF F FFT

49. Three kinds of formulas Contingent – can be both true and false Z R Z & R T TT T F F F T F F FF