Rules of inference Section 1.5 Monday, December 02, 2019 MSU/CSE 260 Fall 2009 1 © 2006 by A-H. Esfahanian. All Rights Reserved.

Review: Logical Implications Monday, December 02, 2019 Review: Logical Implications ? #1 #2 #1 #2 Answer? T Yes F No The above tables summarizes the main truth table for #1 and #2, which may consists of many atomic propositions. Every proof technique must directly or indirectly examine all the rows of the main truth table to come to an affirmative answer. MSU/CSE 260 Fall 2009 2 © 2006 by A-H. Esfahanian. All Rights Reserved.

Monday, December 02, 2019 Terminology Axiom or Postulate: An underlying assumption; often used to begin a logical argument with. Rules of inference: Rules explaining how conclusions are drawn from axioms/postulates. Proof: A sequence of propositions that forms a valid argument. Fallacy: Incorrect reasoning (invalid argument) MSU/CSE 260 Fall 2009 3 © 2006 by A-H. Esfahanian. All Rights Reserved.

Terminology… Theorem: A proposition that can be shown to be true. Monday, December 02, 2019 Terminology… Theorem: A proposition that can be shown to be true. Lemma: A simple theorem used in the proof of other theorems. Corollary: A fact that can be immediately deduced from a Theorem/Lemma. Conjecture: A proposition whose correctness is unknown. MSU/CSE 260 Fall 2009 4 © 2006 by A-H. Esfahanian. All Rights Reserved.

Monday, December 02, 2019 Rules of inference Rules of inference are used to draw conclusions from hypotheses. These are the logical implication questions for which the answer is YES Consider the question: Does [p  (p  q)] logically imply q The answer is YES as [p  (p  q)]  q is a tautology It is the basis of the rule of inference called modus ponens, which can be represented by the symbolic form p p  q  q which means that whenever p is true and p  q is true we can conclude that q is true. In other words [p  (p  q)] logically implies q The name "modus ponens" comes from the Latin word "modus" meaning method, and the Latin word "ponens" meaning affirming. MSU/CSE 260 Fall 2009 5 © 2006 by A-H. Esfahanian. All Rights Reserved.

Example Consider the argument: You have a CSE account if you are taking CSE 260. You are taking CSE 260. Therefore, you have a CSE account. This argument is an instance of modus ponens: p: You are taking CSE 260. q: You have a CSE account. Then the argument has the form: p  q p  q Thus, the argument is valid. MSU/CSE 260 Fall 2009

A Few Tautologies p → (p  q) (p  q) → p p  q → (p  q) Monday, December 02, 2019 A Few Tautologies p → (p  q) (p  q) → p p  q → (p  q) [p  (p → q)] → q [¬ q  (p → q)] → ¬ p [(p → q)  (q → r)] → (p → r) [(p  q)  ¬p] → q Each of these tautologies can be formalized as a rule of inference for use in justifying claims in a valid argument. MSU/CSE 260 Fall 2009 7 © 2006 by A-H. Esfahanian. All Rights Reserved.

Rules of Inference p  p  q p logically implies (p  q) p → (p  q) Monday, December 02, 2019 Rules of Inference p  p  q p logically implies (p  q) p → (p  q) is a tautology Addition p  q  p (p  q) logically implies p (p  q) → p Simplification q  p  q p  q logically implies (p  q) (p  q) → (p  q) Conjunction p  q  q [p  (p → q)] logically implies q [p  (p → q)] → q Modus ponens MSU/CSE 260 Fall 2009 © 2006 by A-H. Esfahanian. All Rights Reserved.

Rules of Inference p  p  q p logically implies (p  q) p → (p  q) Monday, December 02, 2019 Rules of Inference p  p  q p logically implies (p  q) p → (p  q) is a tautology Addition p  q  p (p  q) logically implies p (p  q) → p Simplification q  p  q p  q logically implies (p  q) (p  q) → (p  q) Conjunction p  q  q [p  (p → q)] logically implies q [p  (p → q)] → q Modus ponens MSU/CSE 260 Fall 2009 © 2006 by A-H. Esfahanian. All Rights Reserved. 9

Rules of Inference p  p  q p logically implies (p  q) p → (p  q) Monday, December 02, 2019 Rules of Inference p  p  q p logically implies (p  q) p → (p  q) is a tautology Addition p  q  p (p  q) logically implies p (p  q) → p Simplification q  p  q p  q logically implies (p  q) (p  q) → (p  q) Conjunction p  q  q [p  (p → q)] logically implies q [p  (p → q)] → q Modus ponens MSU/CSE 260 Fall 2009 © 2006 by A-H. Esfahanian. All Rights Reserved. 10

Rules of Inference p  p  q p logically implies (p  q) p → (p  q) Monday, December 02, 2019 Rules of Inference p  p  q p logically implies (p  q) p → (p  q) is a tautology Addition p  q  p (p  q) logically implies p (p  q) → p Simplification q  p  q p  q logically implies (p  q) (p  q) → (p  q) Conjunction p  q  q [p  (p → q)] logically implies q [p  (p → q)] → q Modus ponens MSU/CSE 260 Fall 2009 © 2006 by A-H. Esfahanian. All Rights Reserved. 11

Rules of Inference Resolution ¬ q p  q  ¬ p Monday, December 02, 2019 Rules of Inference ¬ q p  q  ¬ p [¬ q  (p → q)] logically implies ¬ p [¬ q  (p → q)] → ¬ p is a tautology Modus tollens p → q q → r  p → r [(p→q)  (q→r)] logically implies (p→r) [(p→q)  (q→r)] → (p→r) Hypothetical syllogism p  q ¬p  q [(p  q)  ¬p] logically implies q [(p  q)  ¬p] → q Disjunctive syllogism ¬ p  r  q  r [(p  q)  (¬p  r)] logically implies q  r [(p  q)  (¬p  r)] → q  r Resolution MSU/CSE 260 Fall 2009 12 © 2006 by A-H. Esfahanian. All Rights Reserved.

Rules of Inference Resolution ¬ q p  q  ¬ p Monday, December 02, 2019 Rules of Inference ¬ q p  q  ¬ p [¬ q  (p → q)] logically implies ¬ p [¬ q  (p → q)] → ¬ p is a tautology Modus tollens p → q q → r  p → r [(p→q)  (q→r)] logically implies (p→r) [(p→q)  (q→r)] → (p→r) Hypothetical syllogism p  q ¬p  q [(p  q)  ¬p] logically implies q [(p  q)  ¬p] → q Disjunctive syllogism ¬ p  r  q  r [(p  q)  (¬p  r)] logically implies q  r [(p  q)  (¬p  r)] → q  r Resolution MSU/CSE 260 Fall 2009 13 © 2006 by A-H. Esfahanian. All Rights Reserved. 13

Rules of Inference Resolution ¬ q p  q  ¬ p Monday, December 02, 2019 Rules of Inference ¬ q p  q  ¬ p [¬ q  (p → q)] logically implies ¬ p [¬ q  (p → q)] → ¬ p is a tautology Modus tollens p → q q → r  p → r [(p→q)  (q→r)] logically implies (p→r) [(p→q)  (q→r)] → (p→r) Hypothetical syllogism p  q ¬p  q [(p  q)  ¬p] logically implies q [(p  q)  ¬p] → q Disjunctive syllogism ¬ p  r  q  r [(p  q)  (¬p  r)] logically implies q  r [(p  q)  (¬p  r)] → q  r Resolution MSU/CSE 260 Fall 2009 14 © 2006 by A-H. Esfahanian. All Rights Reserved. 14

Rules of Inference Resolution ¬ q p  q  ¬ p Monday, December 02, 2019 Rules of Inference ¬ q p  q  ¬ p [¬ q  (p → q)] logically implies ¬ p [¬ q  (p → q)] → ¬ p is a tautology Modus tollens p → q q → r  p → r [(p→q)  (q→r)] logically implies (p→r) [(p→q)  (q→r)] → (p→r) Hypothetical syllogism p  q ¬p  q [(p  q)  ¬p] logically implies q [(p  q)  ¬p] → q Disjunctive syllogism ¬ p  r  q  r [(p  q)  (¬p  r)] logically implies q  r [(p  q)  (¬p  r)] → q  r Resolution MSU/CSE 260 Fall 2009 15 © 2006 by A-H. Esfahanian. All Rights Reserved. 15

Example Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer. Therefore, if the program crashed, someone input a text value for an integer. If valid, what rule of inference is used? If not, how do you know it is invalid? p → q p: The program crashed. q → r q: An exception was raised.  p → r r: Someone input a text value for an integer. Hypothetical Syllogism – its valid! MSU/CSE 260 Fall 2009

Example Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer value. Therefore, the program did not crash, If valid, what rule of inference is used? If not, how do you know its invalid? p → q p: The program crashed. q → r q: An exception was raised.   p r: Someone input a text value for an integer. [(p → q)  (q → r ) →  p] is not a tautology; therefore, the argument is not valid. MSU/CSE 260 Fall 2009

Example Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer value. No one input a text value for an integer. Therefore, the program did not crash, If valid, what rule of inference is used? If not, how do you know its invalid? p → q p: The program crashed. q → r q: An exception was raised.  r r: Someone input a text value for an   p integer. MSU/CSE 260 Fall 2009

Example Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer value. No one input a text value for an integer. Therefore, the program did not crash, If valid, what rule of inference is used? If not, how do you know its invalid? [ ( (p → q)  (q → r )   r ) →  p ] is a tautology; therefore, the argument is valid. We will shortly develop methods of proof so we don’t have to always convert an inference into a formula and demonstrate that the formula is a tautology; but first … . MSU/CSE 260 Fall 2009

Rules of Inference for Quantifications Monday, December 02, 2019 Rules of Inference for Quantifications Rule of Inference Name Comments x P(x)  P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain P(c)  x P(x) Universal generalization (UG) for an arbitrary c, not a particular one x P(x) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)  x P(x) Existential generalization (EG) Finding one c such that P(c) MSU/CSE 260 Fall 2009 20 © 2006 by A-H. Esfahanian. All Rights Reserved.

Rules of Inference for Quantifications Monday, December 02, 2019 Rules of Inference for Quantifications Rule of Inference Name Comments x P(x)  P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain P(c)  x P(x) Universal generalization (UG) for an arbitrary c, not a particular one x P(x) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)  x P(x) Existential generalization (EG) Finding one c such that P(c) MSU/CSE 260 Fall 2009 21 © 2006 by A-H. Esfahanian. All Rights Reserved. 21

Rules of Inference for Quantifications Monday, December 02, 2019 Rules of Inference for Quantifications Rule of Inference Name Comments x P(x)  P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain P(c)  x P(x) Universal generalization (UG) for an arbitrary c, not a particular one x P(x) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)  x P(x) Existential generalization (EG) Finding one c such that P(c) MSU/CSE 260 Fall 2009 22 © 2006 by A-H. Esfahanian. All Rights Reserved. 22

Rules of Inference for Quantifications Monday, December 02, 2019 Rules of Inference for Quantifications Rule of Inference Name Comments x P(x)  P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain P(c)  x P(x) Universal generalization (UG) for an arbitrary c, not a particular one x P(x) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)  x P(x) Existential generalization (EG) Finding one c such that P(c) MSU/CSE 260 Fall 2009 23 © 2006 by A-H. Esfahanian. All Rights Reserved. 23

Example: Socrates is mortal All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Define M(x): “x is mortal.” Define the universe to be all men. Then the argument being made is: x M(x)  M(Socrates) which is an example of universal specification MSU/CSE 260 Fall 2009

Example: Show that there is no “largest” integer: That is, show x y P(x, y) where P(x, y) denotes the predicate “y > x” Let x be an arbitrary integer. Then P(x, x+1 ) is true by laws of arithmetic ( x+1 > x ). It follows that y P(x, y) from existential generalization. In turn, it follows that x y P(x, y) from universal generalization. MSU/CSE 260 Fall 2009