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Chapter 22: Common Propositional Argument Forms

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Introductory Remarks (p. 220) This chapter introduces some of the most commonly used deductive argument forms. The focus is on valid forms, but there are also three common formal fallacies. If you question the validity or invalidity of any of these argument forms, you can construct a truth table (Chapter 21) that shows that it is impossible for all the premises to be true and the conclusion false, if the argument is valid, or that shows that it is possible for all the premises to be true and the conclusion false, if the argument form is invalid.

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Valid Argument Forms (pp ) In each of the following argument forms, the variables p, q, r, and s can be replaced by statements of any degree of complexity. So, p might be the statement “Mike is a student,” or the statement “Mike is not a student,” or “If Mike is a student and all students find these exercises puzzling, then Mike finds these exercises puzzling.” Simplification –p and q. Therefore, p. or p and q. Therefore, q, –p & q / p or p & q / q (The symbol ‘/ ’ is a conclusion indicator.) –Example: John went to the dance and Louise went bowling. So, Louise went bowling.

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Valid Argument Forms (pp ) Conjunction p q Therefore, p and (yet, but, even though, …) q p q / p & q –Example. Juan went to the dance. Louisa rebuilt her car’s engine. So, Juan went to the dance and Louisa rebuilt her car’s engine. Affirming the Antecedent (modus ponens) If p, then q. p Therefore, q. p q p / q –Example: If Isolde sings soprano, then Tristan sings tenor. Isolde sings soprano. So, Tristan sings tenor.

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Valid Argument Forms (pp ) Denying the Consequent (modus tollens) If p, then q. Not q. Therefore, not p. p q ~q / ~p –Example: If Bertrand buys bread, then Lulu loosens lug nuts. Lulu does not loosen lug nuts. So, Bertrand does not buy bread. Hypothetical Syllogism If p, then q. If q, then r. Therefore, if p, then r. p q q r / p r –If Gustav drives a Toyota, then Tomoji drives a Cadillac. If Tomoji drives a Cadillac, then Hank drives a BMW. So, if Gustav drives a Toyota, Hank drives a BMW.

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Valid Argument Forms (pp ) Disjunctive Syllogism –This assumes an inclusive sense of ‘or’. So a statement of the form “p or q” is true if p is true, if q is true, or if both p and q are true. –The inclusive sense of ‘or’ should be assumed except when there are very good grounds for claiming the exclusive sense, that is, either p or q but not both p and q ([p v q] &~(p & q]). Either p or q. orEither p or q. Not p. Therefore, q.Not q. Therefore, p. p v q orp v q ~p / q~q / p –Example: Either Lynn went to the game or she went to the play. She did not go to the play. So, she went to the game.

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Valid Argument Forms (pp ) Constructive Dilemma If p, then q; and if r then s. Either p or r. Therefore, either q or s. (p q) & (r s) p v r / q v s –Example: If Brunhild bakes bread boisterously, then Gayle grows grapes in great groups; and if Sydney sips sodas suspiciously, then Griselda grumbles gratuitously about grimy grizzlies. Either Brunhild bakes bred boisterously or Sydney sips sodas suspiciously. Ergo, Gayle grows grapes in great groups unless [or] Griselda grumbles gratuitously about grimy grizzlies. –This is like affirming the antecedent when you have two conditionals, you know that the antecedent of at least one is true, but you don’t know which antecedent is true.

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Valid Argument Forms (pp ) Destructive Dilemma If p, then q; and if r then s. Either not q or not s. Therefore, either not p or not r. (p q) & (r s) ~q v ~s / ~p v ~r –Example: If Gertrude is gregarious, then Luis likes lollypops; and if Fran fancies furs, then Helen hates hogs. Either Luis does not like lollypops or Helen does not hate hogs. So, either Gertrude is not gregarious or Fran does not fancy furs. –This is like denying the consequent when you have two conditionals, you know that the consequent of at least one is false, but you don’t know which consequent is false.

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Invalid Forms (pp ) Affirming the Consequent If p, then q. q Therefore, p. p q q / p –Example: If Juanita’s wardrobe is wondrously white, then Fred’s frankfurters are frequently fried. Fred’s frankfurters are frequently fried. Thus, Juanita’s wardrobe is wondrously white. –The problem with the example is that it is not obviously invalid. That’s the general problem with common formal fallacies. –We can construct a deductive counterexample to show that the argument is invalid: If President Bush is over 100 years old, then he is older than his children. President Bush is older than his children. Hence, President Bush is over 100 years old. –The problem is that the first premise is true in virtue of the fact that the antecedent is false and the consequent is true. –While affirming the consequent is a formal fallacy, it provides inductive evidence for its conclusion. Affirming the consequent plays a major role in theory confirmation (see Chapter 27).

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Invalid Forms (pp ) Denying the Antecedent If p, then q. Not p. Therefore, not q. p q ~p / ~q –Example: If Rita writes rhymes regularly, then Wanda wanders willingly through the wide wilderness. Rita does not write rhymes regularly. So, Wanda does not wander willingly through the wide wilderness. –Deductive counterexample: If we are living in the first century C.E., then we are currently living in the 21st century. We are not living in the first century C.E. Therefore, we are not living in the 21st century. –Notice that the first premise is true in virtue of the fact that its antecedent is false.

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Invalid Forms (pp ) Improper Exclusive Disjunctive Syllogism –We usually use the inclusive sense of ‘or’. Occasionally we use the exclusive sense, which is either p or q and not both p and q. If you use an exclusive ‘or’, you would do well to be explicit and add the “and not both.” The problem (and therefore the fallacy) arises when both disjuncts of the disjunctive premise are true. Either p or q.or Either p or q. p. Therefore, not q.q. Therefore, not p. p v qorp v q p / ~qq / ~p –Example: Either Prentice Hall publishes books, or Chicago is in Alaska. Prentice Hall publishes books. So, Chicago is not in Alaska. –Deductive counterexample: Either Prentice Hall publishes books or Chicago is in Illinois. Prentice Hall publishes books. Therefore, Chicago is not in Illinois.

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