Mat2345 Negation 1.5 Inference Week 2 Modus Ponens Modus - PowerPoint PPT Presentation

Mat2345 Week 2 Chap 1.5, 1.6 Week2 Mat2345 Negation 1.5 Inference Week 2 Modus Ponens Modus Tollens Chap 1.5, 1.6 Rules Fallacies Practice Fall 2013 1.6 Proofs Methods

Student Responsibilities — Week 2 Mat2345 Week 2 Chap 1.5, 1.6 Week2 Negation Reading : Textbook, Sections 1.5 – 1.6 1.5 Inference Modus Assignments : as given in the Homework Assignment list Ponens (handout) — Secs. 1.5 & 1.6 Modus Tollens Rules Attendance : Dryly Encouraged Fallacies Practice 1.6 Proofs Methods

Week 2 Overview Mat2345 Week 2 Chap 1.5, 1.6 Week2 Negation Finish up 1.1–1.4 1.5 Inference Modus Ponens 1.5 Rules of Inference Modus Tollens Rules 1.6 Introduction to Proofs Fallacies Practice 1.6 Proofs Methods

Negating Quantifiers Mat2345 Week 2 Chap 1.5, 1.6 Care must be taken when negating statements with quantifiers. Week2 Negation Negations of Quantified Statements 1.5 Inference Modus Statement Negation Ponens Modus All do Some do not Tollens (Equivalently: Not all do ) Rules Fallacies Some do None do Practice (Equivalently: All do not ) 1.6 Proofs Methods

Practice with Negation Mat2345 Week 2 What is the negation of each statement? Chap 1.5, 1.6 1. Some people wear glasses. Week2 Negation 1.5 Inference 2. Some people do not wear glasses. Modus Ponens Modus Tollens 3. Nobody wears glasses. Rules Fallacies Practice 4. Everybody wears glasses. 1.6 Proofs Methods 5. Not everybody wears glasses.

Some Notes of Interest Mat2345 Week 2 Chap 1.5, 1.6 DeMorgan’s Laws: Week2 ¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q ¬ ( p ∨ q ) ≡ ¬ p ∧ ¬ q Negation 1.5 Inference Modus Ponens p → q is false only when p is true and q is false Modus Tollens Rules p → q ≡ ¬ p ∨ q Fallacies Practice 1.6 Proofs The negation of p → q is p ∧ ¬ q Methods

Which Are Equivalent? Mat2345 Week 2 Chap 1.5, 1.6 Week2 Direct Inverse Converse Contrapositive Negation 1.5 Inference p q ¬ p ¬ q p → q ¬ p → ¬ q q → p ¬ q → ¬ p Modus Ponens T T F F Modus T F F T Tollens Rules F T T F Fallacies F F T T Practice 1.6 Proofs Methods

1.5 Rules of Inference Theorems, Lemmas, & Corollaries Mat2345 Week 2 A theorem is a valid logical assertion which can be proved Chap 1.5, 1.6 using: Week2 other theorems Negation axioms : statements given to be true 1.5 Inference Modus Ponens Rules of Inference : logic rules which allow the deduction of Modus conclusions from premises. Tollens Rules Fallacies A lemma is a pre–theorem or result which is needed to Practice prove a theorem. 1.6 Proofs Methods A corollary is a post–theorem or result which follows directly from a theorem.

Mathematical Proofs Mat2345 Week 2 Proofs in mathematics are valid arguments that establish the Chap 1.5, 1.6 truth of mathematical statements. Week2 Negation Argument : a sequence of statements that ends with a 1.5 Inference conclusion. Modus Ponens Modus Tollens Valid : the conclusion or final statement of the argument Rules must follow from the truth of the preceding statements, or Fallacies premises , of the argument. Practice 1.6 Proofs Methods An argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false.

Mat2345 If it rains, then the squirrels will hide Week 2 It is raining. Chap 1.5, 1.6 ----------------------------------------- Week2 The squirrels are hiding. Negation p = it rains / is raining 1.5 Inference q = the squirrels hide / are hiding Modus Ponens Premise 1 : p → q Premise 2 : p Conclusion : q Modus Tollens Associated Implication : (( p → q ) ∧ p ) → q Rules (( p → q ) ∧ p ) → q Fallacies p q Practice T T 1.6 Proofs T F Methods F T F F Are the squirrels hiding?

Mat2345 If you come home late, then you are grounded. Week 2 You come home late. Chap 1.5, 1.6 --------------------------------------------- You are grounded. Week2 p = Negation q = 1.5 Inference Premise 1 : Modus Ponens Premise 2 : Modus Conclusion : Tollens Rules Associated Implication : Fallacies p q Practice T T 1.6 Proofs T F Methods F T F F Are you grounded?

Modus Ponens — The Law of Detachment Mat2345 Both of the prior examples use a pattern for argument called Week 2 modus ponens , or The Law of Detachment . Chap 1.5, 1.6 Week2 p → q Negation p 1.5 Inference ------ Modus Ponens q Modus Tollens Rules or Fallacies Practice (( p → q ) ∧ p ) → q 1.6 Proofs Methods Notice that all such arguments lead to tautologies , and therefore are valid .

Mat2345 If a knee is skinned, then it will bleed. Week 2 This knee is skinned. Chap 1.5, 1.6 ----------------------------------------- It will bleed. Week2 p = Negation q = 1.5 Inference Premise 1 : Modus Ponens Premise 2 : Modus Tollens Conclusion : Rules Associated Implication : Fallacies p q Practice T T 1.6 Proofs T F Methods F T F F ( Modus Ponens ) – Did the knee bleed?

Modus Tollens — Example Mat2345 Week 2 If Frank sells his quota, he’ll get a bonus. Frank doesn’t get a bonus. Chap 1.5, 1.6 ------------------------------------- Week2 Frank didn’t sell his quota. Negation p = q = 1.5 Inference Modus Premise 1 : p → q Premise 2 : ∼ q Conclusion : ∼ p Ponens Modus Thus, the argument converts to: (( p → q ) ∧ ∼ q ) → ∼ p Tollens Rules (( p → q ) ∧ ∼ q ) → ∼ p p q Fallacies T T Practice 1.6 Proofs T F Methods F T F F Did Frank sell his quota or not?

Modus Tollens Mat2345 Week 2 An argument of the form: Chap 1.5, 1.6 p → q Week2 Negation ∼ q 1.5 Inference ------ Modus ∼ p Ponens Modus Tollens or Rules Fallacies (( p → q ) ∧ ∼ q ) → ∼ p Practice 1.6 Proofs Methods is called Modus Tollens , and represents a valid argument.

Mat2345 If the bananas are ripe, I’ll make banana bread. Week 2 I don’t make banana bread. Chap 1.5, 1.6 ------------------------------------- The bananas weren’t ripe. Week2 p = Negation q = 1.5 Inference Modus Premise 1 : p → q Premise 2 : ∼ q Conclusion : ∼ p Ponens Modus Thus, the argument converts to: (( p → q ) ∧ ∼ q ) → ∼ p Tollens Rules (( p → q ) ∧ ∼ q ) → ∼ p p q Fallacies T T Practice T F 1.6 Proofs Methods F T F F Were the bananas ripe or not?

Other Famous Rules of Inference Mat2345 p Week 2 Addition ∴ p ∨ q Chap 1.5, 1.6 p ∧ q Week2 Simplification ∴ p Negation p → q 1.5 Inference q → r Modus ∴ p → r Hypothetical syllogism Ponens p ∨ q Modus Tollens ¬ p Rules ∴ q Disjunctive syllogism Fallacies p Practice q 1.6 Proofs Conjunction ∴ p ∧ q Methods ( p → q ) ∧ ( r → s ) p ∨ r Constructive dilemma ∴ q ∨ s

Rules of Inference for Quantifiers Mat2345 ∀ xP ( x ) Week 2 ∴ P ( c ) Universal Instantiation (UI) Chap 1.5, 1.6 P ( c ) (for arbitrary c ) Week2 ∴ ∀ xP ( x ) Universal Generalization (UG) Negation 1.5 Inference P ( c ) (for some c ) Modus ∴ ∃ xP ( x ) Existential Generalization Ponens Modus ∃ xP ( x ) Tollens ∴ P ( c ) (for some c ) Existential Instantiation Rules Fallacies Practice In Universal Generalization, x must be arbitrary. 1.6 Proofs In Universal Instantiation, c need not be arbitrary but often Methods is assumed to be. In Existential Instantiation, c must be an element of the universe which makes P ( x ) true.

Proof Example Mat2345 Every human experiences challenges. Week 2 Kim Smith is a human. Chap 1.5, 1.6 ------------------------------------- Kim Smith experiences challenges. Week2 Negation H(x) = x is a human 1.5 Inference C(x) = x experiences challenges Modus k = Kim Smith, a member of the universe Ponens Modus Predicate 1 : ∀ x [ H ( x ) → C ( x )] Predicate 2 : H ( k ) Conclusion : C ( k ) Tollens Rules The proof: Fallacies (1) ∀ x [ H ( x ) → C ( x )] Hypothesis (1) Practice (2) H ( k ) → C ( k ) step (1) and UI 1.6 Proofs (3) H ( k ) Hypothesis 2 Methods (4) C(k) steps 2 & 3, and Modus Ponens Q.E.D.

Fallacies Mat2345 Fallacies are incorrect inferences. Week 2 Chap 1.5, 1.6 An argument of the form: Week2 Negation p → q 1.5 Inference ∼ p Modus ------ Ponens ∼ q Modus Tollens Rules or Fallacies Practice (( p → q ) ∧ ∼ p ) → ∼ q 1.6 Proofs Methods is called the Fallacy of the Inverse or Fallacy of Denying the Antecedent , and represents an invalid argument.

Fallacy of the Inverse — Example Mat2345 If it rains, I’ll get wet. Week 2 It doesn’t rain. Chap 1.5, 1.6 ------------------------------------- I don’t get wet. Week2 p = Negation q = 1.5 Inference Premise 1 : p → q Premise 2 : ∼ p Conclusion : ∼ q Modus Ponens Modus Thus, the argument converts to: (( p → q ) ∧ ∼ p ) → ∼ q Tollens Rules (( p → q ) ∧ ∼ p ) → ∼ q p q Fallacies T T Practice T F 1.6 Proofs Methods F T F F Did I get wet?

Did the Butler Do It? Mat2345 Week 2 If the butler is nervous, he did it. Chap 1.5, 1.6 The butler is really mellow. (i.e., not nervous) Therefore, the butler didn’t do it. Week2 Negation Translate into symbols: 1.5 Inference Modus Ponens Modus Tollens Rules Fallacies Practice 1.6 Proofs Methods