1. Predicate Logic

2. TRUTH-TABLE REMINDERS

3. The problem people had the most trouble with was 1e: construct a truth-table for: (P & (~Q & R)) Many of you only had four rows. The very first principle of truth-tables is to write out ALL your variables to the left, and ALL the possible combinations of truth values under them.

4. This is OK PQR(P&(~Q&R)) TTT TTF TFT TFF FTT FTF FFT FFF

5. This is also OK PQR(P&(~Q&R)) TTT FTT TFT FFT TTF FTF TFF FFF

6. This is also OK PQR(P&(~Q&R)) FFF FFT FTF FTT TFF TFT TTF TTT

7. The important point is that each possible combination of truth-values for P, Q, and R shows up in the rows. Notice that if I want to find, for example, P = T, Q = F, and R = T, I can.

8. PQR(P&(~Q&R)) TTT TTF TFT TFF FTT FTF FFT FFF

9. PQR(P&(~Q&R)) TTT FTT TFT FFT TTF FTF TFF FFF

10. PQR(P&(~Q&R)) FFF FFT FTF FTT TFF TFT TTF TTT

11. For a WFF containing only one variable, like “~(P&~~~P),” you only need two rows on the truth-table, because there are only two possibilities: P is true or P is false.

12. Only Two Rows P~(P&~~~P) TTTFFTFT FTFFTFTF

13. For a WFF with 2 variables, you’re going to need 4 rows, to represent the four possibilities: P = T, Q = T P = T, Q = F P = F, Q = T P = F, Q = F

14. For a WFF with 3 variables (for example, P, Q, and R) you need 8 rows, because there are two possibilities for P (T or F), 2 possibilities for Q (T or F) and two possibilities for R (T or F), for a total of 8 possibilities: 2 x 2 x 2 = 8

15. But not just any 8 lines will do. You have to have 8 lines that represent all 8 unique possibilities.

16. Bad Truth-Table Construction PQR(P&(~Q&R)) TTT TFT FTF FFF TTT TFT FTF FFF

17. Only Represents 4 Different Possibilities PQR(P&(~Q&R)) TTT TFT FTF FFF TTT TFT FTF FFF

18. Standard Form This is why logicians always use a standard way of writing truth-tables. Suppose I had to write a table for 4 variables (16 rows). Here’s what I’d do: P rows: TTTTTTTTFFFFFFFF Q rows: TTTTFFFFTTTTFFFF R rows: TTFFTTFFTTFFTTFF S rows: TFTFTFTFTFTFTFTF

19. HOMEWORK #2

20. (P → Q) ├ ((P & R) → Q) 11. (P → Q)A 22. (P & R)A 23. P2 &E 1,24. Q1,3 →E 15. ((P & R) → Q)2,4 →I

21. Proving Conditionals Remember our strategy: if you want to prove a conditional: 1.Assume the antecedent. 2.Prove the consequent. 3.Use →I

22. ~~P ├ P 11. ~~PA 22. ~PA (for ~E) 1,23. (~P & ~~P)1,2 &I 14. P2,3 ~E

23. Hard Problems If you run into a problem that looks difficult, often your best option is to use ~E or ~I. 1.Figure out what you want to prove. 2.Assume the opposite. 3.Prove a contradiction. 4.Use ~E or ~I.

24. (P → (Q → R)) ├ ((P & Q) → R) 11. (P → (Q → R))A 22. (P & Q)A (for →I) 23. P2 &E 24. Q2 &E 1,25. (Q → R)1,3 →E 1,26. R4,5 →E 17. ((P & Q) → R)2,6 →I

25. Assume the Antecedent for →I It’s very important here to continue to follow our strategy for proving conditionals. In line #2 I assumed the antecedent. This is because the rule for →I requires that I assume the antecedent. It says that if I assume φ and derive ψ, then I can write down (φ → ψ) on any future line (depending on whatever ψ depended on except φ).

26. Bad Strategy 11. (P → (Q → R))A 22. PA 33. QA 1,24. (Q → R)1,2 →E 1,2,35. R3,4 →E ?6. ????

27. Correct Step, Bad Result 11. (P → (Q → R))A 22. PA 33. QA 1,24. (Q → R)1,2 →E 1,2,35. R3,4 →E 1,36. (P → R)2,5 →I

28. Correct Step… 11. (P → (Q → R))A 22. PA 33. QA 1,24. (Q → R)1,2 →E 1,2,35. R3,4 →E 2,36. (P & Q)2,3 &I

29. Incorrect Step! 11. (P → (Q → R))A 22. PA 33. QA 1,24. (Q → R)1,2 →E 1,2,35. R3,4 →E 2,36. (P & Q)2,3 &I 17. ((P & Q) → R)6,5 →I

30. ~(P v Q) ├ ~P 11. ~(P v Q)A 22. PA (for ~I) 23. (P v Q)2 vI 1,24. ((P v Q) & ~(P v Q))1,3 &I 15. ~P2,4 ~I

31. (P ↔ ~P) ├ Q 11. (P ↔ ~P)A 22. PA (for ~I) 13. ((P → ~P) & (~P → P))1 ↔E 14. (P → ~P)3 &E 1,25. ~P2,4 →E 1,26. (P & ~P)2,5 &I 17. ~P2,6 ~I

32. (P ↔ ~P) ├ Q 17. ~PA 18. (~P → P)3 &E 19. P7,8 →E 110. (P v Q)9 vI 111. Q7,10 vE

33. Dependencies Even though I had P and ~P at lines 2 and 5 in my proof and I could have proved Q (my goal), I did not. Why? Because both P and ~P depended on line 2.

34. (P ↔ ~P) ├ Q 11. (P ↔ ~P)A 22. PA (for ~I) 13. ((P → ~P) & (~P → P))1 ↔E 14. (P → ~P)3 &E 1,25. ~P2,4 →E

35. (P ↔ ~P) ├ Q 11. (P ↔ ~P)A 22. PA (for ~I) 13. ((P → ~P) & (~P → P))1 ↔E 14. (P → ~P)3 &E 1,25. ~P2,4 →E 26. (P v Q)2 vI 1,27. Q2,6 ~I

36. Dependencies 1,27. Q2,6 ~I What does this line mean? Another way of writing it is: (P ↔ ~P), P ├ Q This is not what we want to show!

37. Instead, what I do is prove ~P again– this time depending only on 1. Then I prove P again, this time depending only on 1. Then I’m free to prove Q, depending only on 1, and my proof is complete.

38. Prove ~P Again 11. (P ↔ ~P)A 22. PA (for ~I) 13. ((P → ~P) & (~P → P))1 ↔E 14. (P → ~P)3 &E 1,25. ~P2,4 →E 1,26. (P & ~P)2,5 &I 17. ~P2,6 ~I

39. Prove P Again 17. ~PA 18. (~P → P)3 &E 19. P7,8 →E

40. Then Prove Q 17. ~PA 18. (~P → P)3 &E 19. P7,8 →E 110. (P v Q)9 vI 111. Q7,10 vE

41. Very Hard Bonus Question! The very hard bonus question asked you to prove: ├ (((P → Q) → P) → P) Here’s some strategy. We want to prove a conditional, so we should assume its antecedent, ((P → Q) → P). Now we want to derive its consequent, P. One way to do this is if we had (P → Q). Since that’s a conditional, we assume its antecedent, P and try to prove Q.

42. ├ (((P → Q) → P) → P) 11. ((P → Q) → P)A (for →I) 22. PA (for →I) 33. ~PA (for ~E) 24. (P v Q)2 vI 2,35. Q3,4 vE 36. (P → Q)2,5 →I 1,37. P1,6 →E

43. ├ (((P → Q) → P) → P) 1,38. (P & ~P)3,7 &I 19. P3,8 ~E 10. (((P → Q) → P) → P) 1,9 →I

44. Here P shows up in the proof 3 times. Only on the third time am I done with the proof, because only then can I conclude (((P → Q) → P) → P) depending on no assumptions at all.

45. ├ (((P → Q) → P) → P) 11. ((P → Q) → P)A (for →I) 22. PA (for →I) 23. (((P → Q) → P) → P)1,2 →I

46. ├ (((P → Q) → P) → P) 11. ((P → Q) → P)A (for →I) 22. PA (for →I) 33. ~PA (for ~E) 24. (P v Q)2 vI 2,35. Q3,4 vE 36. (P → Q)2,5 →I 1,37. P1,6 →E 38. (((P → Q) → P) → P)1,7 →I

47. ├ (((P → Q) → P) → P) 1,38. (P & ~P)3,7 &I 19. P3,8 ~E -----10. (((P → Q) → P) → P) 1,9 →I

48. PREDICATE LOGIC

49. Remember that the goal of logic is to develop formal tests for validity. This is done by finding deductively valid argument forms. In SL, we have two tests to determine valid argument forms: the truth-table test, and derivations. Any argument that has a form that is valid according to the truth-table test is valid, and any argument that has a form that can be proven is valid.

50. Write Down ALL Possibilities PQ TT TF FT FF

51. Write Down Truth-Table for Premises PQ~(P v Q) TTF TFF FTF FFT

52. Write Down Truth-Table for Conclusion PQ~(P v Q)~P TTFF TFFF FTFT FFTT

53. Is the Conclusion True When the Premises are ALL True? PQ~(P v Q)~P TTF*F TF F FT T FFTT

54. Soundness and Completeness In fact, SL is sound and complete, meaning: [sound] if you can prove φ ├ ψ, then the truth- table test will show φ ╞ ψ [complete] if the truth-table test shows φ ╞ ψ, then you can prove φ ├ ψ.

55. Valid Arguments that Don’t Pass But, there are still valid arguments that don’t pass the truth-table test for validity and aren’t provable in SL. (This is why failing the truth-table test does not show that an argument is invalid). For example: Premise: Michael is human. Conclusion: Therefore, someone is human.

56. SL Translations In SL, we’d translate “Michael is human” as a single sentence letter, for example: M. How would we translate “Someone is human”? It’s not a conjunction, or a disjunction, or a conditional, or a biconditional. So we have to translate it as a single sentence letter, for example: S.

57. Invalid Argument Form MSMS TTTT TFTF FTF*T FF F

58. SL Not Expressive Enough The problem here is that our logic is not expressive enough. All simple English sentences get translated as sentence letters in SL. Therefore, no argument involving only simple sentences has a valid SL form, even though many such arguments are valid: Everyone will die (at some point). Therefore, Michael will die (at some point).

59. Therefore, we have a more expressive logic, predicate logic PL, which represents the parts of simple sentences. In PL, a sentence like “Michael is human” will have its parts “Michael” and “is human” translated separately. In particular, PL has a specific grammatical category for singular terms.

60. Singular Terms A singular term is an expression that names or identifies a particular individual, like a person or a city. English singular terms include:

61. Singular Terms Proper names: ‘Michael,’ ‘Jenny,’ ‘Hong Kong,’ etc. ‘the’ + description: ‘the tallest man in the world,’ ‘the country with the second largest economy,’ ‘the third Wednesday of March,’ etc. ‘that’ ‘this’ or ‘that’ + description, ‘this’ + description: ‘this pencil,’ ‘that table,’ etc.

62. NOT Singular Terms ‘Every happy person’ ‘No one in Hong Kong’ ‘A bird with red feathers’ ‘Beautiful dresses’ None of these expressions name a particular individual or thing.

63. Translating Singular Terms To translate singular terms into PL, we will use lowercase Roman letters: a, b, c, d, e, f, g, etc. So we might translate “Michael” as “m.” If we’re just doing logic (and not translating) usually we choose a, b, and c to be our singular terms.

64. Variables In addition, PL contains a special grammatical category called variables. Variables are a lot like singular terms, but they do not name or represent anything in particular. Variables: x, y, z (and if need be w, v, and u) We do not use these letters to translate singular terms!

65. Variables Replace Singular Terms Consider how variables are used in arithmetic: 5 + 7 = 12 5 + x = 12 y + x = 12 y + x = z “5,” “7,” and “12” all name particular numbers. “x,” “y,” and “z” do not. But they go in the same places singular terms go.

66. Variables Replace Singular Terms In logic, things are very similar, except variables can replace more than just singular terms for numbers: Michael gave that book to Sam Michael gave x to Sam y gave x to Sam y gave x to z

67. Open Sentences If you take an English sentence, remove one or more singular terms and replace them with variables, the result is an open sentence. Michael gave x to Sam x gave y to z This past winter, x went home to visit x’s grandmother. John went to the party but z stayed home.

68. Predicates Predicate logic also contains expressions that translate predicates. In traditional grammar, a sentence like “Michael is human” has “Michael” as its subject and “is human” as its predicate. In logic, we simply identify predicates as open sentences. So “x is human” is a predicate, but “is human” is not.

69. Translating Predicates Predicates in PL are translated as capital Roman letters: A, B, C, D, E, F, G, etc. We have a preference for the letters F, G, and H when there is no other reason to choose.

70. Our Fragment Predicate logic is a lot harder than sentential logic, so to make it easier, the system in the reading (PL) only has “monadic” or “one-place” predicates– predicates containing only one place for a singular term.

71. Only 1-Place Predicates OK: x is human Michael gave z to Sam y gave that book to Sam Michael gave that book to y NOT OK: x gave that book to y x admires z

72. Sample Translations EnglishPL Michael is human.Hm Japan is an island.Ij That dog ate my homework.Ad The Chief Executive likes beans.Bc x likes beans.Bx It is raining in Hong Kong.Rh

73. More Complicated WFFs Just as in SL, we can combine PL WFFs with truth-functional connectives: (Hm & Bc) ((Rh ↔ Ij) v (Ad → ~Bc))