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Uses for Truth Tables Determine the truth conditions for any compound statementDetermine the truth conditions for any compound statement Determine whether a statement is a tautology, a contradiction or neither,Determine whether a statement is a tautology, a contradiction or neither, Determine whether two formulae are equivalent.Determine whether two formulae are equivalent. Determine whether an argument is valid or not.Determine whether an argument is valid or not.

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Logical Implication n One statement logically implies another if, but only if, whenever the first is true, the second is true as well n If a statement, S 1, implies S 2 then the conditional (S 1 S 2 ) will be a tautology. Implication is the validity of the conditional. n Truth tables allow one to test for logical implication

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Validity and Logical Implication n An argument is valid if, but only if, its premises logically imply its conclusion.

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Deductive Validity n A characteristic of arguments in which the truth of the premises guarantees the truth of the conclusion. n A characteristic of arguments in which the premises logically imply the conclusion.

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Truth Table Tests for Validity n n Construct a column for each premise in the argument n n Construct a column for the conclusion n n Examine each row of the truth table. Is there a row in which all the premises are true and the conclusion is false. If so, the argument is non- valid. If not, then the argument is valid.

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Truth-Table Test for Validity for R S, R S Modus Ponens (MP) R S R T T S 4 3 T2 T1 S S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T T T T T T T VALID NO

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Truth-Table Test for Validity for R S, S R Fallacy of Affirming the Consequent R S S T T S 4 3 T2 T1 RR Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T YES NON-VALID T T T T T T

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Truth-Table Test for Validity for R S, ~S ~R Modus Tollens (MT) R S ~S T T S 4 3 T2 T1 ~R Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T T T T T T VALID NO

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Truth-Table Test for Validity for R S, ~R ~S Evil Twin of Modus Toll ens R S ~R~R T T S 4 3 T2 T1 ~S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T T T T T T NON-VALID YES

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Truth-Table Test for Validity for R S, ~R S Disjunctive Syllogism (DS) R S ~R T T S 4 3 T2 T1 S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T T T T T T VALID NO

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Truth-Table Test for Validity for R S, R ~S R SR T T S 4 3 T2 T1 ~S~S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? YES NON-VALID T T T T T T T

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Key Ideas n Grouping and Meaning n Paraphrasing Inward n Truth Functional Operators n Truth Tables Using Truth TablesUsing Truth Tables Testing for tautologies and contradictionsTesting for tautologies and contradictions Testing for equivalenceTesting for equivalence n Testing for validity

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