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  1. 1. Second Thoughts, 4 th ed. Wanda Teays McGraw-Hill Higher Ed. © 2010. Wanda Teays. All rights reserved . Chapter Four Handling Claims, Drawing Inferences
  2. 2. Introduction <ul><li>Knowing the tools of logic helps us work within systems already in place. </li></ul><ul><li>It's sort of like x-ray vision: A firm grasp of logic gives us the ability to see how arguments are structured, to organize that reasoning, and to dismantle it so it can be evaluated. </li></ul><ul><li>This is both useful and empowering. </li></ul><ul><li>What we will do in this chapter is go deeper into analysis and critical thinking skills by examining the different types of claims—and acquire some techniques for handling those claims. </li></ul>
  3. 3. Propositions <ul><li>A proposition is an assertion that is true or false. </li></ul><ul><li>FOR EXAMPLE: These are all propositions: “Bananas are not citrus fruits,” “Long Beach is in California,” “Some CSUN students are from the East Coast.” </li></ul><ul><li>Propositions are not normally expressed as questions or exclamations, unless those are rhetorical forms. </li></ul><ul><li>In classical logic, prescriptive or moral claims (like “You ought to eat spinach” or “Assault guns should be illegal”) were not treated as propositions, because it was difficult to assign them a truth-value. This is also the case with aesthetic judgments, such as “Otis Redding is the greatest rhythm & blues singer ever!” </li></ul><ul><li>That does not mean such claims are just a matter of opinion, since they may rest on a body of knowledge and research. </li></ul>
  4. 4. The 3 Kinds of Propositions <ul><li>Tautologies: Propositions that are always true or true by definition. </li></ul><ul><li>FOR EXAMPLE:  Either he’s a member of an alien species or he’s not. </li></ul><ul><li>Contradictions : Propositions that are always false or false by definition. </li></ul><ul><li>FOR EXAMPLE: He never lived on the East Coast, but he was born and raised in New Jersey. </li></ul><ul><li>Contingent Claims : Propositions that are not necessarily true or false; but are true or false according to the context. They are dependent on what is going on in the world to determine the truth-value. This includes claims for which the truth-value is unknown. </li></ul><ul><li>FOR EXAMPLE: The last lottery winner in New Mexico was a Latina. </li></ul>
  5. 5. The Structure of Propositions <ul><li>A simple proposition is one that is at the atomic level—that is, it does not contain any of the logical connectives “and,” “or”, “not, “if…then,” or “if and only if.” A proposition that contains any of the five logical connectives is called a compound proposition. </li></ul><ul><li>EXAMPLES of Simple Propositions: </li></ul><ul><li>Jasper chewed a small hole in the couch. </li></ul><ul><li>Beach cobbler is awfully tasty. </li></ul><ul><li>65% of senior citizens prefer to sleep with two pillows. </li></ul><ul><li>  </li></ul><ul><li>EXAMPLES of Compound Propositions: </li></ul><ul><li>  Jasper chewed off some of the kitchen window and then bit the computer cord. </li></ul><ul><li>Either Jasper pushed his seed dish onto the floor or there was an earthquake. </li></ul><ul><li>If Jasper climbs too high in the tree, then the blue jay may scare him. </li></ul><ul><li>Jasper eats his corn if and only if it is warm </li></ul>
  6. 6. The Five Types of Compound Propositions <ul><li>  </li></ul><ul><li>1 . Conjunctions: Propositions of the form “A and B.” </li></ul><ul><li>  </li></ul><ul><li>2. Disjunctions: Propositions of the form “Either A or B.” </li></ul><ul><li>  </li></ul><ul><li>3. Negations: Propositions of the form “Not A.” </li></ul><ul><li>  </li></ul><ul><li>4. Conditional claims: Propositions of the form </li></ul><ul><ul><li> “ If A then B” (= “A is sufficient for B”) </li></ul></ul><ul><li> or “B only if A” (= “A is necessary for B”) </li></ul><ul><li>  </li></ul><ul><li>5. Biconditional claims ( Equivalence): Propositions of the form “A if and only if B.” </li></ul><ul><li>(= “A is both necessary and sufficient for B”) </li></ul>
  7. 7. NECESSARY & SUFFICIENT CONDITIONS <ul><li>A is SUFFICIENT for B. </li></ul><ul><li>If A occurs, B will occur as well. </li></ul><ul><li>“ If A then B” </li></ul><ul><li>Being hungry is sufficient for Betty Jean to eat a donut. </li></ul><ul><li> If she is hungry, then Betty Jean will eat a donut </li></ul><ul><li>A is NECESSARY for B. </li></ul><ul><li>If A does not occur, B will not occur either. </li></ul><ul><li>“ B only if A” </li></ul><ul><li>“ If not A then not B” </li></ul><ul><li>“ If B then A” </li></ul><ul><li>Having jelly inside is necessary for the donut to be tasty. </li></ul><ul><li> If there’s no jelly inside, then the donut is not tasty. </li></ul><ul><li> If the donut is tasty, then there must be jelly inside. </li></ul>
  8. 8. “Only if” <ul><li>EXAMPLES of “ Only if ” Claims </li></ul><ul><li>Jasper will go in his cage only if he’s forced. </li></ul><ul><li> If he’s not forced, Jasper will not go in his cage. </li></ul><ul><li> If the Jasper went in his cage, then he was forced. </li></ul><ul><li>Watering the pear tree is necessary for it to bear fruit. </li></ul><ul><li> If we don’t water the pear tree, then it won’t bear fruit. </li></ul><ul><li> If it bears fruit, then we watered the pear tree. </li></ul><ul><li>  </li></ul><ul><li>“ P only if Q.” P happens only if Q does. </li></ul><ul><li> If Q happens, then P must have happened. </li></ul><ul><li> Q is necessary for P. </li></ul><ul><li> “ If not Q then not P.” </li></ul><ul><li> “ If P then Q.” P  Q </li></ul><ul><li>Remember: Any time the “only if” is written as “P only if Q” that can automatically be rewritten as </li></ul><ul><li> “ If P then Q”—so the “then” is in the location of the “only if”. </li></ul>
  9. 9. Structure of a Conditional Claim <ul><li>If that is my pet bird, Jasper , then he will enjoy some corn for dinner. </li></ul><ul><li> </li></ul><ul><li>Antecedent Consequent </li></ul><ul><li>Biconditional Claim : A if and only if B. </li></ul><ul><li>FOR EXAMPLE: </li></ul><ul><li>Jason will go to Salem if and only if Anthea comes. </li></ul><ul><li>If Jason goes to Salem, then Anthea comes along and if Anthea goes to Salem, Jason is going too. </li></ul>
  10. 10. Forms of the Categorical Propositions <ul><li>All P is Q. This is called an “A” claim. </li></ul><ul><li>“ All birds are hawks.” </li></ul><ul><li>No P is Q. This is called an “E” claim. </li></ul><ul><li>“ No hawk is a chicken.” </li></ul><ul><li>Some P is Q. This is called an “I” claim. </li></ul><ul><li> “ Some hawks are very large birds.” </li></ul><ul><li>Some P is not Q. This is called an “O” claim. </li></ul><ul><li>‘ Some hawks are not birds found in Texas.” </li></ul><ul><li>  </li></ul>
  11. 11. Quantity & Quality of Propositions <ul><li>Quantity —Universal or Particular </li></ul><ul><li>  </li></ul><ul><li>Universal </li></ul><ul><li>“ All P is Q.” All flies are insects. </li></ul><ul><li>“ No P is Q”No fly is a spider. </li></ul><ul><li>Particular </li></ul><ul><li>“ Some P is Q” </li></ul><ul><li>Some ducks are snail-eaters. </li></ul><ul><li>“ Some P is not Q” </li></ul><ul><li>Some raccoons are not pesky animals. </li></ul><ul><li>  </li></ul><ul><li>A and E  Universal claims </li></ul><ul><li>I and O  Particular claims </li></ul><ul><li>Quality —Positive or Negative </li></ul><ul><li>Positive: </li></ul><ul><li>“ All P is Q” </li></ul><ul><li>All chocolate is a delicious treat. </li></ul><ul><li>“ Some P is Q” </li></ul><ul><li>Some delicious treats are donuts. </li></ul><ul><li>Negative: </li></ul><ul><li>“ No P is Q” </li></ul><ul><li>No donut is a healthy snack. </li></ul><ul><li>“ Some P is not Q” </li></ul><ul><li>Some healthy snacks are not vegetables. </li></ul><ul><li>A and I  Positive claims </li></ul><ul><li>E and O  Negative claims </li></ul>
  12. 12. The Five Logical Connectives Connective NEGATION Symbol ~ Name NOT Expression ~ A AND & Conjunction A & B OR ∨ Disjunction A V B IF/THEN Or ONLY IF -> Conditional A-> B IF and ONLY IF P  Q Biconditional or Equivalence A  B
  13. 13. Only P is Q <ul><li>Forms of “Only P Is Q” (~P  ~Q)  (Q  P) </li></ul><ul><li> If it’s not P then, it’s not Q.  If it’s a Q, then it’s a P. </li></ul><ul><li>All Q is P </li></ul><ul><li>  </li></ul><ul><li>EXAMPLES: </li></ul><ul><li>Only Americans eat hamburgers. </li></ul><ul><li> If they are not Americans, they won't eat hamburgers </li></ul><ul><li> If they eat hamburgers, they are Americans. </li></ul><ul><li> All people who eat hamburgers are Americans. </li></ul><ul><li> ~A  ~H ( Form 1 )  H  A ( Form 2 ) </li></ul>
  14. 14. The Only P is Q <ul><li>Forms of “The only P is Q” (~Q  ~P)  ( P  Q) </li></ul><ul><li> If it’s not Q then, it’s not P. </li></ul><ul><li> If it’s a P, then it’s a Q. </li></ul><ul><li> All P is Q </li></ul><ul><li>  </li></ul><ul><li>EXAMPLE: </li></ul><ul><li>The only outer gear the Count owns is a cape. </li></ul><ul><li> If it’s not a cape, it’s not outer gear the Count owns. </li></ul><ul><li> If it’s outer gear the Count owns, then it’s a cape. </li></ul><ul><li> All the outer gear the Count owns are capes. </li></ul>
  15. 15. Unless <ul><li>Forms of “P unless Q” (~ Q  ~ P)  (P Q) </li></ul><ul><li>P unless Q If not Q then P </li></ul><ul><li>Either P or Q </li></ul><ul><li>EXAMPLES: </li></ul><ul><li>We will go on a picnic unless it rains. </li></ul><ul><li> If it does not rain, we will go on a picnic. </li></ul><ul><li> Either we went on a picnic or it rained. </li></ul><ul><li> ~R  P ( Form 1 ) </li></ul><ul><li> R v P ( Form 2) </li></ul><ul><li>Unless Joe stops the car, he’s going to hit the moose. </li></ul><ul><li> If Joe does not stop the car, he’s going to hit the moose. </li></ul><ul><li> Either Joe stops the car or he’s going to hit the moose. </li></ul><ul><li> ~M  J ( Form 1 ) </li></ul><ul><li> M v J ( Form 2 ) </li></ul><ul><li>  </li></ul>
  16. 16. Rules of Replacement: DeMorgan’s Laws <ul><li>DeMorgan’s Law #1: Not both ~ (P & Q)  ~P  ~Q </li></ul><ul><li>Not both P and Q —-negates either P or Q. </li></ul><ul><li>It is not true that both P and Q.  Either not P or not Q.   </li></ul><ul><li>EXAMPLE </li></ul><ul><li>Not both Kung Pao beef and lasagna are Chinese food. </li></ul><ul><li>Either Kung Pao beef is not Chinese food or lasagna is not Chinese food. </li></ul><ul><li>DeMorgan’s Law #2 Neither/nor ~ (P  Q)  (~P & ~Q) </li></ul><ul><li>Neither P nor Q —-negates BOTH P and Q. </li></ul><ul><li>It is not true that either P or Q.  “ Not P and not Q. </li></ul><ul><li>EXAMPLE </li></ul><ul><li>Neither frogs nor salamanders are in the water.  Frogs are not in the water and salamanders are not in the water. </li></ul>
  17. 17. Transposition & Material Implication <ul><li>Form of Transposition (P  Q)  (~Q  ~P) </li></ul><ul><li>If P then Q  If not Q then not P. ( Hint : flip & switch). </li></ul><ul><li>FOREXAMPLE </li></ul><ul><li>If Lisa doesn't hurry, then she will be late to school. </li></ul><ul><li> If she was not late to school on time, then Lisa hurried. </li></ul><ul><li>Form of Material Implication (P  Q)  (~P  Q) </li></ul><ul><li>If P then Q  Either not P or Q </li></ul><ul><li>Note: You can go from the conditional “if/then” </li></ul><ul><li>to the disjunction “either/or” and vice versa. </li></ul><ul><li>  </li></ul><ul><li>FOR EXAMPLE If Marty brings potato salad, Luke will fry chicken. </li></ul><ul><li> Either Marty will not bring potato salad or Luke will fry chicken. </li></ul>
  18. 18. Exportation <ul><li>Form of Exportation [(A & B)  C]  [A  (B  C)] </li></ul><ul><li>  If A and B, then C </li></ul><ul><li> If A then, if B then C. </li></ul><ul><li>Note: Do you see how the second conjunct in the antecedent was shipped back to the consequent? </li></ul><ul><li>EXAMPLES </li></ul><ul><li>If he spills paint and doesn't wipe it up, then there will be a mess. </li></ul><ul><li> If he spills paint, then, if he doesn't wipe it up, there will be a mess. </li></ul><ul><li>If the flashlight’s on, then, if we aim it in the cave, the bats fly out. </li></ul><ul><ul><li> If the flashlight’s on and we aim it in the cave, the bats fly out. </li></ul></ul><ul><li>  </li></ul>
  19. 19. Equivalence <ul><li>Equivalence . This is also known as a biconditional” or “if and only if” “ claim. </li></ul><ul><li>Forms of Biconditionals </li></ul><ul><li>“ P if and only Q” can be written in two ways: </li></ul><ul><li> If P then Q, and, if Q then P. </li></ul><ul><li> If P then Q, and, if not Q then not P. </li></ul><ul><li>Translation of form 1: (P  Q) & (Q  P) </li></ul><ul><li>Translation of form 2: (P  Q) & (~Q  ~ P) </li></ul><ul><li>  </li></ul><ul><li>EXAMPLE </li></ul><ul><li>Fish can swim if and only if they are in the water. </li></ul><ul><li> If fish are in the water then they can swim, and if fish can swim then they are in the water. </li></ul><ul><li> If fish are in the water then they can swim, and if fish are not in the water then they cannot swim. </li></ul><ul><li>Translation of form 1: (F  W) & (W  F) </li></ul><ul><li>Translation of form 2: (F  W) & (~ F  ~ W) </li></ul><ul><li>  </li></ul>