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Valid and Invalid arguments

Valid and Invalid arguments. 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 . Examples of Arguments. It is raining or it is snowing;

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Valid and Invalid arguments

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  1. Valid and Invalid arguments

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

  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.

  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.

  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.

  6. Checking the validity of an argument form • Construct truth table for the premises and the conclusion; • Find the rows in which all the premises are true (critical rows); • 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.

  7. Example of valid argument form p and q; if p then q; q. premisesconclusion Critical row

  8. Example of invalid argument form p or q; if p then q; p. premisesconclusion Critical row Critical row

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

  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: pq; q r; pr.

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

  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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