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Use a truth table to determine the validity or invalidity of this argument. First, translate into standard form “Martin is not buying a new car, since he said he would buy a new car or take a Hawaiian vacation and I just heard him talking about his trip to Maui.” C or H H _ Not C Now, translate into symbols C v H H _ ~ C 2

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C v H H _ ~ C Now, build a truth table. “C” and “H” each need a column and will serve as references. Next, we need a column for each premise and the conclusion. C H T T F F T F C v H ~C 3

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C v H H _ ~ C Fill in the truth values for the first premise based on the rule of disjunction: A disjunction is false if and only if both disjuncts are false. C H T T F F T F Our truth table now tells us whether or not the argument is valid. What do you think? 4 C v H T T T F ~C F F T T

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C H T T F F T F C v H T F ~C F T HTFTFHTFTF C v H H _ ~ C Is it possible for the premises to be true and the conclusion false ? 5 “Martin is not buying a new car, since he said he would buy a new car or take a Hawaiian vacation and I just heard him talking about his trip to Maui.”

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O → (H & S) ~H v ~S _ ~ O There are no cases where the premises are true and the conclusion false; this is a valid argument. Can you provide an interpretation? O H S T T T T T F T F T T F F F T T F T F F F T F F F ~H ~S F F F T T F T T F F F T T F T T H & S T F T F O → (H&S) T F T ~H v ~S F T F T ~ O F T 6

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What conclusion does the following demonstrate to be true and how do you know it is true? P R. R S. P. /

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How do you know the following are true: P & Q P P & Q Q

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3. R & S. S P. If the above is true, what else is true?

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What should the next line of the argument line be and how do you know? (P & Q) R. S. S ~R. ~R. ~(P & Q) Now what should the next line be? MP. Modus Tollens (MT): If a consequent of a conditional that appears as one premise appears appears as the negation of that consequent in another premise, then the negation of the antecedent should appear as the next line in the argument.

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4. P → Q ~P → S ~Q / ∴ S How do you know S is true if the premises are true?

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7. ~S (P & Q) → R R → S / ∴ ~(P & Q) How do you know ~(P & Q) is true if the premises are true?

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8. P → ~(Q & T) S → (Q & T) P / ∴ ~S

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P → Q R → S P v R / ∴ Q v S How do you know Q v S is true?

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2. P → S P v Q Q → R / ∴ S v R How do you know S v R is true?

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5. (P v Q) → R Q / ∴ R How do you get to R?

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1. R → P Q → R / ∴ Q → P How we get to Q → P?

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9. (P v T) → S R → P R v Q Q → T / ∴ S Talk us to S.

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6. ~P ~(R & S) v Q ~P → ~Q / ∴ ~(R & S) 1, 3, MP ~(R & S)2, 4, Disjunctive Argument (DA): The negative of one disjunct in a disjunction implies the truth of the other disjunct.

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Section 3.5 Symbolic Arguments

Section 3.5 Symbolic Arguments

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