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Critical Thinking: A User ’ s Manual

Critical Thinking: A User ’ s Manual. Chapter 8 Evaluating Truth-Functional Arguments. Truth-Functional Arguments. A truth-functional argument is a deductive argument that contains truth-functional claims. Truth-Functional Claims. Simple claim I have a cat. Negation I do not have cat.

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Critical Thinking: A User ’ s Manual

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  1. Critical Thinking:A User’s Manual Chapter 8 Evaluating Truth-Functional Arguments

  2. Truth-Functional Arguments • A truth-functional argument is a deductive argument that contains truth-functional claims.

  3. Truth-Functional Claims Simple claim I have a cat. Negation I do not have cat. Disjunction I have a cat or I have a dog. Conjunction I have a cat and I have a dog. Conditional If I have a cat, then I have a dog.

  4. Operator Symbols not ~ or  and • if, then  therefore 

  5. Symbolizing Truth-Functional Claims I have a cat. C I do not have a cat. ~ C I have a cat or I have a dog. C  D I have a cat and I have a dog. C • D If I have a cat, then I have a dog. C  D

  6. Translating Conditional Claims ___  ___ Antecedent Consequent

  7. Anatomy of Conditional Claims If __________, then __________. Antecedent Consequent

  8. Your turn! What is the antecedent of the following conditional claim? What is the consequent? If it is raining, then there are clouds in the sky.

  9. Anatomy of Conditional Claims __________ only if __________. Antecedent Consequent

  10. Your turn! What is the antecedent of the following conditional claim? What is the consequent? It is raining only if there are clouds in the sky.

  11. Anatomy of Conditional Claims __________ if __________. Consequent Antecedent

  12. Your turn! What is the antecedent of the following conditional claim? What is the consequent? There are clouds in the sky if it is raining.

  13. Translating “Unless” “unless” = “if not” X unless Y = X if not Y X unless Y = ~ Y  X

  14. Using Proper Punctuation I don’t have a cat but I have a dog. ~ C • D It is not the case that I have both a cat and a dog. ~ (C • D) If I have a cat, then either I have a dog or a fish. C  (D  F) Either I have a cat and a dog, or I have a fish. (C • D)  F

  15. Your turn! Translate the following compound claim into symbolic form. It is not the case that we live in Canada and in South America.

  16. DeMorgan’s Law ~ ( X • Y )  ~ X • ~ Y ~ ( X • Y ) = ~ X  ~ Y ~ ( X  Y )  ~ X  ~ Y ~ ( X  Y) = ~ X • ~ Y

  17. Your turn! Apply DeMorgan’s Law to the following claim. ~ (C • S)

  18. EvaluatingTruth-Functional Arguments • Truth-functional arguments may be valid or invalid. • Demonstrate by identifying argument forms. • Demonstrate using Truth Table Method. • Demonstrate using Shortcut Method.

  19. Anatomy of Conditional Claims If __________, then __________. The antecedent is a sufficient condition for the consequent. The consequent is a necessary condition for the antecedent.

  20. Valid Argument Forms X  Y X  Y Modus Ponens • X is a sufficient condition for Y. • X is true. • Thus, Y must be true.

  21. Valid Argument Forms X  Y ~ Y  ~ X Modus Tollens • Y is a necessary condition for X. • Y is false. • Thus, X must be false.

  22. Invalid Argument Forms X  Y Y  X Affirming the Consequent • Y is a necessary condition for X. • Y is true. • Yet, we cannot conclude that X is true; Y is necessary, not sufficient for X.

  23. Invalid Argument Forms X  Y ~ X  ~ Y Denying the Antecedent • X is a sufficient condition for Y. • X is false. • Yet, we cannot conclude that Y is false; X is sufficient, not necessary for Y.

  24. Your turn! Identify the argument form. P  ~ C C ~ P

  25. Your turn! Identify the argument form. ~ P  C ~ P C

  26. Truth-Functional Definitions • A truth-functional definition specifies when a compound claim is true and when it is false. • Used to complete Truth Tables

  27. Simple Claim I have a cat. C T F

  28. Negation I do not have a cat. ~ C T F

  29. Negation I do not have a cat. ~ C F T T F

  30. Conjunction I have a cat and I have a dog. C • D T T T F F T F F

  31. Conjunction I have a cat and I have a dog. C • D T T T T F F F F T F F F The only time a conjunction is true is when both conjuncts are true.

  32. Disjunction I have a cat or I have a dog. C  D T T T F F T F F

  33. Disjunction I have a cat or I have a dog. C  D T T T T T F F T T F F F The only time a disjunction is false is when both disjuncts are false.

  34. Conditional If I have a cat, then I have a dog. C  D T T T F F T F F

  35. Conditional If I have a cat, then I have a dog. C  D T T T T F F F T T F T F The only time a conditional is false is when the antecedent is true and the consequent is false.

  36. Using the Truth Table Method Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Calculate the number of lines in the truth table using the formula, L = 2n.

  37. C  ~ R / C // R

  38. Using the Truth Table Method Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Calculate the number of lines in the truth table using the formula, L = 2n. Step 4: Assign truth-values to each simple claim in the argument.

  39. C RC  ~ R / C // R T T T F F T F F

  40. C RC  ~ R / C // R T T T T T T T F T F T F F T F T F T F F F F F F

  41. Using the Truth Table Method Step 5: Determine the truth-values of each premise and conclusion using the appropriate truth-functional definitions. You may find it helpful to highlight or draw a box around these final values.

  42. C RC  ~ R / C // R T T T F T T T T F T T F T F F T F F T F T F F F T F F F

  43. C RC  ~ R / C // R T T T T F T TT T F T T T F TF F T F F F T FT F F F T T F FF

  44. Using the Truth Table Method Step 5: Determine the truth-values of each premise and conclusion using the appropriate truth-functional definitions. You may find it helpful to highlight or draw a box around these final values. Step 6: Evaluate whether the argument is valid or invalid by looking for any row with all true premises and a false conclusion. If you find such a row, then the argument is invalid. Otherwise, the argument is valid.

  45. C RC  ~ R / C // R T T T T F T TT T F T T T F TF F T F F F T FT F F F T T F FF Both premises are true, and the conclusion is false, so the argument is invalid.

  46. Your turn! How many lines would your truth table have if there were 3 simple claims in the argument?

  47. Using the Shortcut Method Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Write out the goal truth-values for each premise and conclusion, i.e. all true premises and false conclusion.

  48. C  ~ R / C // R T T F

  49. Using the Shortcut Method Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Write out the goal truth-values for each premise and conclusion, i.e. all true premises and false conclusion. Step 4: Assign the truth-values to simple claims for which there is only one possible truth-value assignment that would result in a true premise or false conclusion.

  50. C  ~ R / C // R T F T T F

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