Lecture 1.4: Rules of Inference CS 250, Discrete Structures, Fall 2015 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

Lecture 1.4 - Rules of Inference Course Admin Slides from previous lectures all posted Expect HW1 to be coming in around coming Monday Questions? 6/12/2018 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Outline Rules of Inference 6/12/2018 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Proofs – How do we know? The following statements are true: If I am Mila, then I am a great swimmer. I am Mila. What do we know to be true? I am a great swimmer! How do we know it? 6/12/2018 Lecture 1.4 - Rules of Inference

Proofs – How do we know? A theorem is a statement that can be shown to be true. A proof is the means of doing so. Given set of true statements or previously proved theorems Rules of inference Proof 6/12/2018 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference What rules we study Modus Ponens Modus Tollens Addition Simplification Disjunctive Syllogism Hypothetical Syllogism 6/12/2018 Lecture 1.4 - Rules of Inference

Proofs – How do we know? The following statements are true: If I have taken MA 106, then I am allowed to take CS 250 I have taken MA 106 What do we know to be true? I am allowed to take CS 250 What rule of inference can we use to justify it? 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference – Modus Ponens I have taken MA 106. If I have taken MA 106, then I am allowed to take CS 250.  I am allowed to take CS 250. p p  q  q Inference Rule: Modus Ponens Tautology: (p  (p  q))  q 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference – Modus Tollens I am not allowed to take CS 250. If I have taken MA 106 , then I am allowed to take CS 250.  I have not taken MA 106. q p  q  p Inference Rule: Modus Tollens Tautology: (q  (p  q))  p 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference – Addition I am not a great skater.  I am not a great skater or I am tall. p  p  q Inference Rule: Addition Tautology: p  (p  q) 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference – Simplification I am not a great skater and you are sleepy.  you are sleepy. p  q  p Inference Rule: Simplification Tautology: (p  q)  p 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference – Disjunctive Syllogism I am a great eater or I am a great skater. I am not a great skater.  I am a great eater! p  q q  p Inference Rule: Disjunctive Syllogism Tautology: ((p  q)  q)  p 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference – Hypothetical Syllogism If you are an athlete, you are always hungry. If you are always hungry, you have a snickers in your backpack.  If you are an athlete, you have a snickers in your backpack. Inference Rule: Hypothetical Syllogism Tautology: ((p  q)  (q  r))  (p  r) p  q q  r  p  r 6/12/2018 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Examples Addition Amy is a computer science major.  Amy is a math major or a computer science major. If Ernie is a math major then Ernie is geeky. Ernie is not geeky!  Ernie is not a math major. Modus Tollens 6/12/2018 Lecture 1.4 - Rules of Inference

Complex Example: Rules of Inference Here’s what you know: Ellen is a math major or a CS major. If Ellen does not like discrete math, she is not a CS major. If Ellen likes discrete math, she is smart. Ellen is not a math major. Can you conclude Ellen is smart? M  C D  C D  S M 6/12/2018 Lecture 1.4 - Rules of Inference

Complex Example: Rules of Inference 1. M  C Given 2. D  C Given 3. D  S Given 4. M Given 5. C DS (1,4) 6. D MT (2,5) 7. S MP (3,6) Ellen is smart! 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference: Common Fallacies Rules of inference, appropriately applied give valid arguments. Mistakes in applying rules of inference are called fallacies. 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference: Common Fallacies If I am Bonnie Blair, then I skate fast I skate fast!  I am Bonnie Blair Nope ((p  q)  q)  p Not a tautology. If you don’t give me $10, I bite your ear. I bite your ear!  You didn’t give me $10. Nope 6/12/2018 Lecture 1.4 - Rules of Inference

Rules of Inference: Common Fallacies If it rains then it is cloudy. It does not rain.  It is not cloudy Nope ((p  q)  p)  q Not a tautology. If it is a car, then it has 4 wheels. It is not a car.  It doesn’t have 4 wheels. Nope 6/12/2018 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Today’s Reading Rosen 1.6 Please start solving the exercises at the end of each chapter section. They are fun. 6/12/2018 Lecture 1.4 - Rules of Inference