Valid and Invalid Arguments - Foundations of Mathematics - Lecture Slides, Slides of Computer Programming

Download Valid and Invalid Arguments - Foundations of Mathematics - Lecture Slides and more Slides Computer Programming in PDF only on Docsity! 1 Valid and Invalid arguments Docsity.com 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. Docsity.com 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. Docsity.com 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. Docsity.com 7 Example of valid argument form p and q; if p then q; q. premises conclusion Critical row ∴ p q p and q if p then q q T T T T T T F F F T F F F F Docsity.com 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. ∴ ∴ ∴ → → → ∴ Docsity.com 11 Valid Argument Forms • Proof by division into cases: p or q p r q r r • Rule of contradiction: ~p c p ∴ → → → ∴ Docsity.com 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? Docsity.com