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1 Valid and Invalid arguments

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2 Definition of Argument Sequence of statements: Statement 1; Statement 2; Therefore, Statement 3. Statements 1 and 2 are called premises. Statement 3 is called conclusion.

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3 Examples of Arguments It is raining or it is snowing; It is not snowing; Therefore, it is raining. If x=2 then x<5; x<5; x is an even integer; Therefore, x=2.

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4 Argument Form If the premises and the conclusion are statement forms instead of statements, then the resulting form is called argument form. Ex: If p then q; p; q.

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5 Validity of Argument Form Argument form is valid means that for any substitution of statement variables, if the premises are true, then the conclusion is also true. The example of previous slide is a valid argument form.

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6 Checking the validity of an argument form 1)Construct truth table for the premises and the conclusion; 2)Find the rows in which all the premises are true (critical rows); 3)a. If in each critical row the conclusion is true then the argument form is valid; b. If there is a row in which conclusion is false then the argument form is invalid.

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7 Example of valid argument form p and q; if p then q; q. premises conclusion Critical row pqp and qif p then qq TTTTT TFF FTF FFF

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8 Example of invalid argument form p or q; if p then q; p. premises conclusion Critical row pqp or qif p then qp TTTTT TFTF FTTTF FFF

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9 Valid Argument Forms Modus ponens: If p then q; p; q. Modus tollens: If p then q; ~q; ~p.

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10 Valid Argument Forms Disjunctive addition: p; p or q. Conjunctive simplification: p and q; p. Disjunctive Syllogism: p or q; ~q; p. Hypothetical Syllogism: p q; q r; p r.

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11 Valid Argument Forms Proof by division into cases: p or q p r q r r Rule of contradiction: ~p c p

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12 A more complex deduction Knights always tell the truth, and knaves always lie. U says: None of us is a knight. V says: At least three of us are knights. W says: At most three of us are knights. X says: Exactly five of us are knights. Y says: Exactly two of us are knights. Z says: Exactly one of us is a knight. Which are knights and which are knaves?

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