Chapter 1, Logic and Proofs (3) 1.6. Rules of Inference

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1 CSI 2350, Discrete Structures Chapter 1, Logic and Proofs (3) Young-Rae Cho Associate Professor Department of Computer Science Baylor University 1.6. Rules of Inference Basic Terminology Axiom: a statement (proposition) that is assumed to be true Logical argument: a sequence of propositions that consist of Conclusion: a single final proposition Premises: all the other propositions Logical rule of inference: a method to derive a new proposition from other propositions based on logic (logical equivalence) 1

2 Valid Arguments Definition of Validity An argument is valid iff it never leads from correct premises to an incorrect conclusion An argument is invalid iff it can lead from correct premises to an incorrect conclusion (called fallacy) Examples If you have a current password, then you can log onto the network Logical representation? Logical analysis? If you have access to the network, then you can change your grade Inference Rules Definition of Inference Rules Patterns establishing valid arguments Patterns establishing that if all premises are true, the conclusion must also be true Form of Inference Rules General form? Each valid inference rule corresponds to an implication that is a tautology: 2

3 Modus Ponens Modus Ponens An inference rule p p q q a.k.a. the mode of affirming (p (p q)) q is a tautology. Prove? Examples? Modus Tollens Modus Ponens An inference rule q p q p a.k.a. the mode of denying ( q (p q)) p is a tautology. Prove? Examples? 3

4 More Inference Rule Examples Basic Inference Rules Addition rule Simplification rule Conjunction rule More Inference Rules Hypothetical syllogism Disjunctive syllogism Resolution rule Example? Formal Proofs Process of a formal proof A formal proof of a conclusion C given premises p 1, p 2,, p n consists of a sequence of steps, each of which applies some inference rule to premises and/or previously proven statements to finally yield the conclusion. Example Premises: It is not sunny and it is cold. We will swim only if it is sunny. If we do not swim, then we will canoe. If we canoe, then we will be home by sunset. Conclusion: We will be home by sunset 4

5 Fallacies Fallacy An inference rule or other proof method that is not logically valid A fallacy may yield a false conclusion Fallacy of Affirming the Conclusion p q is true and q is true, so p must be true Example? Fallacy of Denying the Hypothesis p q is true and p is false, so q must be false Example? Inference Rules for Quantifiers Examples? 5

6 More Proof Examples (1) Example 1 All TAs make easy quizzes. Kelechi is a TA. Therefore, Kelechi will make easy quizzes. True or False? Example 2 At least one of the students in this class is intelligent. David is a student in this class. Therefore, David is intelligent. True or False? More Proof Examples (2) Example 3 For all positive integer n, if n is greater than 4, then n 2 is less than 2 n. Therefore, < True or False? 6

7 1.7. Introduction to Proof Basic Terminology Proof: A valid argument that establishes the truth of a statement. Theorem: A statement that has been proven to be true. Premises, Axioms, Hypothesis, Postulates: Assumptions (often unproven) defining the structures about which we are reasoning More Terminology Lemma: A minor theorem that is used as a stepping-stone to prove a major theorem Corollary: A minor theorem that is proved as an easy consequence of a major theorem Conjecture: A statement whose truth value has not been proven. It may be widely believed to be true. Proof Methods Direct Proof Indirect Proof Proof by contraposition Proof by contradiction Proof by a counterexample 7

8 Direct Proof Process of Direct Proof For proving p q, assume p is true and prove q is true. Example Definition An integer n is even iff n=2k for some integer k An integer n is odd iff n=2k+1 for some integer k Theorem: If n is an odd integer, then n 2 is an odd integer. Proof by Contraposition Process of Proof by Contraposition For proving p q, assume q is true and prove p is true. Example Definition An integer n is even iff n=2k for some integer k An integer n is odd iff n=2k+1 for some integer k Theorem: (For all integers n) If 3n+2 is an odd integer, then n is odd. 8

9 Proof by Contradiction Process of Proof by Contradiction For proving p, assume p is true, and prove that this assumption leads to a contradiction. For proving p q, assume p is true and q is false, and prove both p and p are true. Example Definition An integer n is even iff n=2k for some integer k An integer n is odd iff n=2k+1 for some integer k Theorem: (For all integers n) If 3n+2 is an odd integer, then n is odd. Proof by Counterexamples Counterexample A specific value which makes a universal quantification false. Example This statement is false. Every positive integer is the sum of the square of two integers. 9

10 Mistakes in Proofs Arithmetic Mistakes Example: 1=2 is true Logical Mistakes Example: If n 2 is positive, then n is positive. Questions? Lecture Slides are found on the Course Website, web.ecs.baylor.edu/faculty/cho/