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2 Day 1Intro Day 2Chapter 1 Day 3Chapter 2 Day 4Chapter 3 Day 5Chapter 4 Day 6Chapter 4 Day 7Chapter 4 Day 8EXAM #1 40% of Exam 1 60% of Exam 1 warm-up

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4 an argument is valid if and only if it is impossible for the conclusion to be false while the premises are true an argument is invalid if and only if it is possible for the conclusion to be false while the premises are true

5 an argument is valid if and only if case there is no case in which the premises are true and the conclusion is false an argument is invalid if and only if case there is at least one case in which the premises are true and the conclusion is false

6 case A case is a possible combination of truth-values assigned to the atomic formulas.

7 case 4 case 3 case 2 case 1 SR cases If an argument form has 2 atomic sentences, then there are 4 cases [4 = 2 2 ]. F T F TT T F F

8 cases If an argument form has 3 atomic sentences, then there are 8 cases [8 = 2 3 ]. QRS case 1 TTT case 2 TTF case 3 TFT case 4 TFF case 5 FTT case 6 FTF case 7 FFT case 8 FFF

9 / not R ; not S if R then S /  R ;  S R  S / conclusion ; premise premise

10 R  S ; S; S F T F T S F 4 F 3 T 2 T 1 / R/ R Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? T T F T T F T F T T F F VALID NO

11 Example 2 Evil Twin of Modus Tollens / not S ; not R if R then S /  S ;  R R  S / conclusion ; premise premise

12 / I don't live in Mass ; I don't live in Boston if I live in Boston then I live in Mass FTT / not S ; not R if R then S

13 F T F T S F 4 F 3 T 2 T 1 R  S ; R; R / S/ S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? T YES INVALID T F T T T F F T F T F

14 / S ; R if R then S / S ; R R  S / conclusion ; premise premise

15 R  S ; R F T F T S F 4 F 3 T 2 T 1 / S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? T T F T F F T T F T F T VALID NO

16 / R ; S if R then S / R ; S R  S / conclusion premise Example 4 Evil Twin of Modus Ponens ; premise

17 / I live in Boston ; I live in Mass if I live in Boston then I live in Mass FTT / R ; S if R then S

18 R  S ; S F T F T S F 4 F 3 T 2 T 1 / R Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? T YES INVALID T F T F T F T F F T T

19 / S ; not R R or S / S ;  R R  S / conclusion ; premise premise

20 R  S ; R; R F T F T S F 4 F 3 T 2 T 1 / S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? F T T T T T F F F T F T VALID NO

21 / not S ; R R or S /  S ; R R  S / conclusion ; premise premise Example 6 Evil Twin of MTP

22 R  S ; R; R F T F T S F 4 F 3 T 2 T 1 / S/ S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? F YES INVALID T T T F F T T T F T F

23 / / not ( R and S)  ( R & S ) RR not R

24 / F T F T S )  F F T T ( R& F F T T R  Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid? T T F F T T T F F F F T VALID NO

25 / / not ( R and S)not R  ( R & S ) RR

26 / Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or invalid?INVALID YES F F T T R  T T F F F T F T S )  F F T T ( R& T T T F F F F T

27 logically equivalent Two formulas are logically equivalent if and only if they have the same truth-value no matter what (in every case).

28 not R and not S = not ( R and S) IT IS JUST LIKE MATH! ZOMBIE REASONING not R or not S = not ( R or S) x 2 + y 2 = ( x + y) 2  x +  y =  ( x + y)    

29 not R and not S  not ( R and S) not R or not S  not ( R or S)

30 F F T T R&  & F T F T S F T F T S ) //  F F T T ( R  Are the two formulas logically equivalent? F F F T T T T F T T F F T F T F T F F F Do the formulas match in truth value?NONO NONO

31 F F T T R  F T F T S F T F T S ) //  F F T T ( R  Are the two formulas logically equivalent? F T T T T F F F T T F F T F T F T T T F Do the formulas match in truth value?NONO NONO

32  ( R&S ) //  R   S F TTTTT F FT T TFFFT T TF T FFTTF T FT T FFFTF T TF Are the two formulas logically equivalent?YES Do the formulas match in truth value?YES

33  ( R  S ) //  R&  S F TTTFT F FT F TTFFT F TF F FTTTF F FT T FFFTF T TF Do the formulas match in truth value? Are the two formulas logically equivalent?YES

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