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INTRO LOGIC

INTRO LOGIC. DAY 04. Schedule for Unit 1. warm-up. 40% of Exam 1. 60% of Exam 1. CHAPTER 3. VALIDITY IN SENTENTIAL LOGIC. Validity in General. an argument is valid if and only if it is im possible for the conclusion to be false while the premises are true.

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INTRO LOGIC

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  1. INTRO LOGIC DAY 04

  2. Schedule for Unit 1 warm-up 40% of Exam 1 60% of Exam 1

  3. CHAPTER 3 VALIDITY IN SENTENTIAL LOGIC

  4. Validity in General an argument is valid if and only if it is impossiblefor the conclusion to be false while the premises are true an argument is invalid if and only if it is possiblefor the conclusion to be false while the premises are true

  5. Validity in Sentential Logic an argument is valid if and only if there is nocasein which the premises are true and the conclusion is false an argument is invalid if and only if there is at least onecasein which the premises are true and the conclusion is false

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

  7. Example 1 R S case 1 case 2 case 3 case 4 If an argument form has 2 atomic sentences, then there are 4cases [4 = 22]. T T T F F T F F

  8. Example 2 If an argument form has 3 atomic sentences, then there are 8 cases [8 = 23].

  9. Example 1 Modus Tollens premise ; premise / conclusion if Rthen S ;not S /not R R  S ;S /R

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

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

  12. Counterexample if R then S ;not R /not S if I live in Boston then I live in Mass ;I don't live in Boston /I don't live in Mass T T F

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

  14. Example 3 Modus Ponens premise ; premise / conclusion if Rthen S ; R / S R  S ;R /S

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

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

  17. Counterexample if R then S ; S / R if I live in Boston then I live in Mass ;I live in Mass /I live in Boston T T F

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

  19. Example 5 Modus Tollendo Ponens(disjunctive syllogism) premise ; premise / conclusion Ror S ;not R / S R  S ;R /S

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

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

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

  23. Example 7 not R / not ( R and S) R /  ( R & S )

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

  25. Example 8 not ( R and S) / not R  ( R & S ) / R

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

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

  28. Examples 7 and 8 ZOMBIE REASONING not ( R and S) = not R andnot S  not ( R or S) = not R ornot S  IT ISJUSTLIKE MATH! ( x + y)2 = x2+ y2  ( x + y) = x +y 

  29. Truth-Table for 7  ( R & S ) //  R &  S F T T T F T F F T T T F F F T F T F T F F T T F F F T T F F F T F T T F Do the formulas match in truth value? NO Are the two formulas logically equivalent? NO

  30. Truth-Table for 8  ( R  S ) //  R   S F T T T F T F F T F T T F F T T T F F F T T T F T F T T F F F T F T T F Do the formulas match in truth value? NO Are the two formulas logically equivalent? NO

  31. Valid Equivalence – 1 Do the formulas match in truth value? YES Are the two formulas logically equivalent? YES

  32. Valid Equivalence – 2 Do the formulas match in truth value? YES Are the two formulas logically equivalent? YES

  33. THE END

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