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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 Truth-Value Assignment A truth-value assignment specifies a unique truth-value (either T or F) for

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

  2. Truth-Value Assignment A truth-value assignment specifies a unique truth-value (either T or F) for each atomic formula. 1

  3. Consider the formula ( A → ( A ∨ B )). 2

  4. Consider the formula ( A → ( A ∨ B )). The atomic subformulas are A and B 2

  5. Consider the formula ( A → ( A ∨ B )). The atomic subformulas are A and B There are 4 truth-value assignments for this formula: 1. A is T, B is T 2. A is T, B is F 3. A is F, B is T 4. A is F, B is F 2

  6. How many truth value assignments are there for a single atomic proposition A ? 3

  7. How many truth value assignments are there for a single atomic proposition A ? 2 3

  8. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 3

  9. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 3

  10. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 How many truth value assignments are there for three atomic propositions A , B , and C ? 3

  11. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 How many truth value assignments are there for three atomic propositions A , B , and C ? 8 3

  12. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 How many truth value assignments are there for three atomic propositions A , B , and C ? 8 How many truth value assignments are there for four atomic propositions A , B , C and D ? 3

  13. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 How many truth value assignments are there for three atomic propositions A , B , and C ? 8 How many truth value assignments are there for four atomic propositions A , B , C and D ? 16 3

  14. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 How many truth value assignments are there for three atomic propositions A , B , and C ? 8 How many truth value assignments are there for four atomic propositions A , B , C and D ? 16 How many truth value assignments are there for n atomic propositions A 1 , A 2 , . . . , A n ? 3

  15. How many truth value assignments are there for a single atomic proposition A ? 2 How many truth value assignments are there for two atomic propositions A and B ? 4 How many truth value assignments are there for three atomic propositions A , B , and C ? 8 How many truth value assignments are there for four atomic propositions A , B , C and D ? 16 How many truth value assignments are there for n atomic propositions A 1 , A 2 , . . . , A n ? 2 n 3

  16. Truth Assignments Given a truth assignment for all the atomic propositions in ϕ , how do we determine the truth value of ϕ ? 4

  17. Conjunction Eric had steak and wine. ( S ∧ W ) S W T T T F F T F F 5

  18. Conjunction Eric had steak and wine. ( S ∧ W ) S W T T T F F T F F 5

  19. Conjunction Eric had steak and wine. ( S ∧ W ) S W ( S ∧ W ) T T T T F F F T F F F F 5

  20. Conjunction Eric had steak and wine. ( S ∧ W ) S W ( S ∧ W ) T T T T F F F T F F F F 5

  21. Conjunction Eric had steak and wine. ( S ∧ W ) S W ( S ∧ W ) T T T T F F F T F F F F 5

  22. Conjunction Eric had steak and wine. ( S ∧ W ) S W ( S ∧ W ) T T T T F F F T F F F F 5

  23. Conjunction Eric had steak and wine. ( S ∧ W ) S W ( S ∧ W ) T T T T F F F T F F F F 5

  24. Conjunction Eric had steak and wine. ( S ∧ W ) S W ( S ∧ W ) T T T T F F F T F F F F 5

  25. Truth-Table for Conjunction ( ϕ ∧ ψ ) ϕ ψ T T T T F F F T F F F F 6

  26. Disjunction Eric had steak or wine. ( S ∨ W ) S W ( S ∨ W ) T T T T F T F T T F F F 7

  27. Disjunction Eric had steak or wine. ( S ∨ W ) S W ( S ∨ W ) T T T T F T F T T F F F 7

  28. Disjunction Eric had steak or wine. ( S ∨ W ) S W ( S ∨ W ) T T T T F T F T T F F F 7

  29. Disjunction Eric had steak or wine. ( S ∨ W ) S W ( S ∨ W ) T T T T F T F T T F F F 7

  30. Disjunction Eric had steak or wine. ( S ∨ W ) S W ( S ∨ W ) T T T T F T F T T F F F 7

  31. Truth-Table for Disjunction ( ϕ ∨ ψ ) ϕ ψ T T T T F T F T T F F F 8

  32. Negation Eric didn’t have steak. ¬ S S ¬ S T F F T 9

  33. Negation Eric didn’t have steak. ¬ S S ¬ S T F F T 9

  34. Negation Eric didn’t have steak. ¬ S S ¬ S T F F T 9

  35. Truth-Table for Negation ϕ ¬ ϕ T F F T 10

  36. P ¬ ( Q ∨ R ) P ∧ ¬ ( Q ∨ R ) T T T P ∧ ¬ ( Q ∨ R ) T T F F F T F F F F ¬ ( Q ∨ R ) T P T Q ∨ R ¬ ( Q ∨ R ) T F F T Q ∨ R F Q R Q ∨ R T T T T F T F T T Q F R F F F F 11

  37. P ¬ ( Q ∨ R ) P ∧ ¬ ( Q ∨ R ) T T T P ∧ ¬ ( Q ∨ R ) T T F F F T F F F F ¬ ( Q ∨ R ) T P T Q ∨ R ¬ ( Q ∨ R ) T F F T Q ∨ R F Q R Q ∨ R T T T T F T F T T Q F R F F F F 11

  38. P ¬ ( Q ∨ R ) P ∧ ¬ ( Q ∨ R ) T T T P ∧ ¬ ( Q ∨ R ) T T F F F T F F F F ¬ ( Q ∨ R ) T P T Q ∨ R ¬ ( Q ∨ R ) T F F T Q ∨ R F Q R Q ∨ R T T T T F T F T T Q F R F F F F 11

  39. P ¬ ( Q ∨ R ) P ∧ ¬ ( Q ∨ R ) T T T P ∧ ¬ ( Q ∨ R ) T T F F F T F F F F ¬ ( Q ∨ R ) T P T Q ∨ R ¬ ( Q ∨ R ) T F F T Q ∨ R F Q R Q ∨ R T T T T F T F T T Q F R F F F F 11

  40. P ¬ ( Q ∨ R ) P ∧ ¬ ( Q ∨ R ) T T T P ∧ ¬ ( Q ∨ R ) T T F F F T F F F F ¬ ( Q ∨ R ) T P T Q ∨ R ¬ ( Q ∨ R ) T F F T Q ∨ R F Q R Q ∨ R T T T T F T F T T Q F R F F F F 11

  41. P ¬ ( Q ∨ R ) P ∧ ¬ ( Q ∨ R ) T T T P ∧ ¬ ( Q ∨ R ) T T F F F T F F F F ¬ ( Q ∨ R ) T P T Q ∨ R ¬ ( Q ∨ R ) T F F T Q ∨ R F Q R Q ∨ R T T T T F T F T T Q F R F F F F 11

  42. 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 ¬ ϕ T F F T 12

  43. Find truth tables for the formulas • P ∧ Q • ¬ ( P ∧ Q ) • ¬ P ∨ ¬ Q • ¬ P ∧ ¬ Q 13

  44. P Q ( P ∧ Q ) ¬ ( P ∧ Q ) ( ¬ P ∨ ¬ Q ) ( ¬ P ∧ ¬ Q ) T T T F F F T F F T T F F T F T T F F F F T T T ( P ∧ Q ) and ¬ ( P ∧ Q ) are contradictory : they always have opposite truth values 14

  45. Material Conditional If Eric had steak, then he had wine. ( S → W ) S W ( S → W ) T T T T F F F T T F F T 15

  46. Material Conditional If Eric had steak, then he had wine. ( S → W ) S W ( S → W ) T T T T F F F T T F F T 15

  47. Material Conditional If Eric had steak, then he had wine. ( S → W ) S W ( S → W ) T T T T F F F T T F F T 15

  48. Material Conditional If Eric had steak, then he had wine. ( S → W ) S W ( S → W ) T T T T F F F T T F F T 15

  49. Material Conditional If Eric had steak, then he had wine. ( S → W ) S W ( S → W ) T T T T F F F T T F F T 15

  50. Truth-Table for the Conditional ( ϕ → ψ ) ϕ ψ T T T T F F F T T F F T 16

  51. 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 T T T T F F F T T F F T ϕ ¬ ϕ T F F T 17

  52. A truth table for a formula ϕ is a table, where each row is a truth assignment for the atomic propositions in ϕ and there is a column for ϕ (and possible subformulas of ϕ ) list the truth values of ϕ for each truth assignment. 18

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