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Reasoning for Humans: Clear Thinking in an Uncertain World PHIL 171 Eric Pacuit Department of Philosophy University of Maryland pacuit.org Recap: Truth Tables ( ) ( ) T T T T T T T F F T F T F

  1. Reasoning for Humans: Clear Thinking in an Uncertain World PHIL 171 Eric Pacuit Department of Philosophy University of Maryland pacuit.org

  2. Recap: Truth Tables ( ϕ ∧ ψ ) ( ϕ ∨ ψ ) ϕ ψ ϕ ψ T T T T T T T F F T F T F T F F T T F F F F F F ϕ ψ ( ϕ → ψ ) ϕ ψ ( ϕ ↔ ψ ) T T T T T T T F F T F F F T T F T F F F T F F T ϕ ¬ ϕ T F F T 1

  3. Construct a truth table for ϕ Is there a row in which ϕ is false? yes no Is there a row in which ϕ ϕ is a tautology is true? yes no ϕ is a contradiction ϕ is contingent 2

  4. Construct a truth table with columns for ϕ and ψ ϕ and ψ are Is there a row in which ϕ and ψ tautologically no have different truth values? equivalent yes Is there a row in which ϕ ϕ and ψ are no and ψ have the same contradictory truth values? yes ϕ and ψ are Is there a row in which ϕ no mutually exclusive and ψ are both true? yes ϕ and ψ are satisfiable 3

  5. Classifying Arguments 4

  6. Is it possible that the formulas ( A ∧ B ) and ( A ∨ C ) can both be true at the same time? 5

  7. Is it possible that the formulas ( A ∧ B ) and ( A ∨ C ) can both be true at the same time? ( A ∧ B ) ( A ∨ C ) A B C T T T T T T T F T T T F T F T T F F F T F T T F T F T F F F F F T F T F F F F F 5

  8. Is it possible that the formulas ( A ∧ B ) and ( A ∨ C ) can both be true at the same time? A B C ( A ∧ B ) ( A ∨ C ) T T T T T T T F T T T F T F T T F F F T F T T F T F T F F F F F T F T F F F F F 6

  9. Is it possible that the formulas ( A ∧ B ) and ( A ∨ C ) can both be true at the same time? A B C ( A ∧ B ) ( A ∨ C ) T T T T T T T F T T T F T F T T F F F T F T T F T F T F F F F F T F T F F F F F 6

  10. Is it possible that the formulas ( A ∧ B ) and ( A ∨ C ) can both be true at the same time? Yes...There are two truth assignments that make both formulas true. ( A ∧ B ) ( A ∨ C ) A B C T T T T T T T F T T T F T F T T F F F T F T T F T F T F F F F F T F T F F F F F 6

  11. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? 7

  12. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? A B C ¬ A ( ¬ A ∧ B ) ( A ∨ C ) ¬ C T T T T T T T T T F T T T T T F T F T T T T F F F T T T F T T F T T T F T F F T F T F F T F T T T F F F F T F T 8

  13. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? A B C ¬ A ( ¬ A ∧ B ) ( A ∨ C ) ¬ C T T T F T T T T T F F T T T T F T F T T T T F F F T T T F T T T T T T F T F T T F T F F T T T T T F F F T T F T 8

  14. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? A B C ¬ A ( ¬ A ∧ B ) ( A ∨ C ) ¬ C T T T F F T T T T F F F T T T F T F F T T T F F F F T T F T T T T T T F T F T T F T F F T T F T T F F F T F F T 8

  15. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? A B C ¬ A ( ¬ A ∧ B ) ( A ∨ C ) ¬ C T T T F F T T T T F F F T T T F T F F T T T F F F F T T F T T T T T T F T F T T F T F F T T F T T F F F T F F T 8

  16. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? A B C ¬ A ( ¬ A ∧ B ) ( A ∨ C ) ¬ C T T T F F T F T T F F F T T T F T F F T F T F F F F T T F T T T T T F F T F T T F T F F T T F T F F F F T F F T 8

  17. Is it possible that the formulas ( ¬ A ∧ B ) and ( A ∨ C ) and ¬ C can all be true at the same time? No...there is no row in which all these formulas are true. ( ¬ A ∧ B ) ( A ∨ C ) A B C ¬ A ¬ C T T T F F T F T T F F F T T T F T F F T F T F F F F T T F T T T T T F F T F T F F T F F T T F T F F F F T F F T 8

  18. Valid Argument : 9

  19. Valid Argument : An argument is valid provided that there is no truth value assignment that makes all the premises true and the conclusion false. 9

  20. Valid Argument : An argument is valid provided that there is no truth value assignment that makes all the premises true and the conclusion false. (So, any truth-value assignment that makes all the premises true also makes the conclusion true). Invalid Argument : 9

  21. Valid Argument : An argument is valid provided that there is no truth value assignment that makes all the premises true and the conclusion false. (So, any truth-value assignment that makes all the premises true also makes the conclusion true). Invalid Argument : An argument is invalid just in case it is not valid, i.e., if there is some truth-value assignment that makes the premises true and the conclusion false. Counterexample : A truth-value assignment that makes the premises of an argument true and its conclusion false is called a counterexample to the argument. 9

  22. Valid Argument : An argument is valid provided that there is no truth value assignment that makes all the premises true and the conclusion false. (So, any truth-value assignment that makes all the premises true also makes the conclusion true). Invalid Argument : An argument is invalid just in case it is not valid, i.e., if there is some truth-value assignment that makes the premises true and the conclusion false. Counterexample : A truth-value assignment that makes the premises of an argument true and its conclusion false is called a counterexample to the argument. So, an argument if valid if there are no counterexamples. 9

  23. A → B A ∴ B Is this argument valid? 10

  24. A → B A ∴ B Is this argument valid? Yes. 10

  25. A → B A ∴ B Is this argument valid? Yes. Why? 10

  26. Modus Ponens A → B A ∴ B A B A → B T T T T F F F T T F F T 11

  27. Modus Ponens A → B A ∴ B A B A → B T T T T F F F T T F F T 11

  28. Modus Ponens A → B A B ∴ A B A → B T T T T F F F T T F F T 11

  29. Modus Ponens A → B A ∴ B Modus Ponens is valid because there is no truth-value assignment that makes the premises true ( A , A → B ) and the conclusion ( B ) false. 11

  30. Construct a truth table with columns for ϕ 1 , ϕ 2 , . . . , ϕ n , and ψ . Is there a row in which ϕ 1 , ϕ 2 , . . . , ϕ n are all true and ψ is false ? yes no The argument is invalid The argument is (there is a counterexample) valid 12

  31. A → B B ∴ A Is this argument valid? 13

  32. A → B B ∴ A Is this argument valid? No. 13

  33. A → B B ∴ A Is this argument valid? No.Why? 13

  34. Affirming the Consequent A → B B ∴ A A B A → B T T T T F F F T T F F T 14

  35. Affirming the Consequent A → B B ∴ A A B A → B T T T T F F F T T F F T 14

  36. Affirming the Consequent A → B B ∴ A A B A → B T T T T F F F T T F F T 14

  37. Affirming the Consequent A → B B ∴ A Affirming the Consequent is not valid because there is a truth-value assigment that makes the premises true and the conclusion false. Namely, the truth-value function that sets A to F and B to T. 14

  38. Disjunctive Syllogism A ∨ B ¬ A ∴ B A B ¬ A A ∨ B T T F T T F F T F T T T F F T F 15

  39. Disjunctive Syllogism A ∨ B ¬ A ∴ B A B ¬ A A ∨ B T T F T T F F T F T T T F F T F 15

  40. Disjunctive Syllogism A ∨ B ¬ A ∴ B A B ¬ A A ∨ B T T F T T F F T F T T T F F T F 15

  41. Disjunctive Syllogism A ∨ B ¬ A ∴ B Disjunctive Syllogism is valid because there is no truth-value assignment that make the premises true ( ¬ A and A ∨ B ) and the conclusion ( B ) false. 15

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