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Introductory Logic PHI 120

Presentation: "Truth Tables – Sequents". Introductory Logic PHI 120. This PowerPoint Presentation contains a large number of slides, a good many of which are nearly identical. If you print this Presentation, I recommend six or nine slides per page. Homework. Study Allen/Hand Logic Primer

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Introductory Logic PHI 120

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  1. Presentation: "Truth Tables – Sequents" Introductory LogicPHI 120 This PowerPoint Presentation contains a large number of slides, a good many of which are nearly identical. If you print this Presentation, I recommend six or nine slides per page.

  2. Homework • Study Allen/Hand Logic Primer • Sec. 1.1, p. 1-2: “validity” • Sec. 2.2, p. 43-4, “validity” & “invalidating assignment • Complete Ex. 2.1, p. 42: i-x Turn to page 40 in The Logic Primer also take out TTs handout

  3. Truth Tables Truth Value of Sentences • Section 2.1 • (quick review)

  4. Atomic sentence Simple

  5. Truth Tables See bottom of Truth Tables Handout Complex Sentences

  6. ~Φ • False?

  7. ~Φ • False – if the statement being negated (Φ) is True

  8. Φ & Ψ • False?

  9. Φ & Ψ • False – if one or both conjuncts are False

  10. Φ & Ψ • False – if one or both conjuncts are False

  11. Φ v Ψ • False?

  12. Φ v Ψ • False – only if bothdisjuncts are False

  13. Φ v Ψ • False – only if bothdisjuncts are False

  14. Φ -> Ψ • False?

  15. Φ -> Ψ • False – if antecedent is True and consequent is False

  16. Φ -> Ψ • False – if antecedent is True and consequent is False

  17. Φ <-> Ψ • False?

  18. Φ <-> Ψ • False – if the two conditions have a different truth value

  19. Φ <-> Ψ • False – if the two conditions have a different truth value

  20. Φ v ~Φ Note the binary structure (P & ~Q) v ~(P & ~Q) • Identify the main connective. • How many atomic sentences are in this WFF?

  21. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Determine the number of rows for the WFF or the sequent as a whole

  22. (P & ~Q) v ~(P & ~Q) • Determine the number of rows for the WFF or the sequent as a whole

  23. TT Method in a Nutshell Determine truth-values of: • atomic statements • negations of atomics • inside parentheses • negation of the parentheses • any remaining connectives

  24. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 3 on Handout • Fill in left main column first.

  25. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 3 on Handout • Fill in left main column first.

  26. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 3 on Handout • Fill in left main column first.

  27. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 4 on Handout • Assign truth-values for negation of simple statements

  28. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 4 on Handout • Assign truth-values for negation of simple statements

  29. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 4 on Handout • Assign truth-values for negation of simple statements

  30. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a conjunction (an “&” statement) false? • Step 5 on Handout • Assign truth-values for innermost binary connectives

  31. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a conjunction (an “&” statement) false?

  32. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a conjunction (an “&” statement) false?

  33. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a conjunction (an “&” statement) false?

  34. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a conjunction (an “&” statement) false?

  35. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 5 on Handout • Assign truth-values for innermost binary connectives

  36. (P & ~Q) v ~(P & ~Q)Φ v ~Φ

  37. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 6a on Handout • Assign truth-values for negation of compounds

  38. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 6a on Handout • Assign truth-values for negation of compounds

  39. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a disjunction (a “v” statement) false? • Step 6b on Handout • Assign truth-values for remaining

  40. (P & ~Q) v ~(P & ~Q)Φ v ~Φ • Step 6b on Handout • Assign truth-values for remaining

  41. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a disjunction (a “v” statement) false?

  42. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a disjunction (a “v” statement) false?

  43. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a disjunction (a “v” statement) false?

  44. (P & ~Q) v ~(P & ~Q)Φ v ~Φ When is a disjunction (a “v” statement) false?

  45. (P & ~Q) v ~(P & ~Q)Φ v ~Φ The values under the governing connective are all T’s.

  46. TTs: Sentences p. 47-8: “tautology,” “inconsistency & contingent” Classifying Sentences

  47. Φ v Ψ Look Under the Main Connective • Tautologies • Only Ts under main operator • Necessarily true

  48. Φ v Ψ Look Under the Main Connective • Tautologies • Only Ts under main operator • Necessarily true

  49. ~Φ Look Under the Main Connective • Inconsistencies • Only Fs under main operator • Necessarily false

  50. ~Φ Look Under the Main Connective • Inconsistencies • Only Fs under main operator • Necessarily false

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