1.5 Arguments & Rules of Inference

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1 1.5 Arguments & Rules of Inference Tools for establishing the truth of statements Argument involving a seuence of propositions (premises followed by a conclusion) Premises 1. If you have a current password, then you can log onto the network. 2. You have a current password. Conclusion Therefore You can log onto the network p: You have a current password : You can log onto the network Euivalent representations [( p ) p] p p 1

2 Another example Given the following propositions p The bug is either in module 17 or in module 81 r The bug is a numerical error Module 81 has no numerical error Assuming these statements are true, it is reasonable to conclude: s The bug is in Module 17 If p r then s p r s CSc 28 2

3 An argument An argument in propositional logic is a seuence of propositions All but the final proposition in the list of arguments are called premises The final proposition is called the conclusion Argument Form The Argument Form is valid if every case where all the premises are true, the conclusion is true. The Argument Form is invalid if every case where all premises are true, the conclusion is false in at least one row. CSc 28 3

4 Generalized example of a valid argument Five premises, each represented by a propositional statement Each propositional statement is a function of one or more propositional variables Given all possible combinations of T or F values for each propositional variable, at least one row exists where each Premise is true and the truth value of the Conclusion is also true There are no rows where the truth value for each premise is true and the truth value of the conclusion in false PREMISES CONCLUSION p 1 p 2 p 3 p 4 p 5 c T T T T T T CSc 28 4

5 Using truth tables Is the following argument valid? p p Premise Premise 1 2 Conclusion p p p T T T T T T F F T F F T T F T F F T F F Only one case where Premises are True and Conclusion is True CSc 28 5

6 Rules of Inference (page 33) (instead of using truth tables) Tautology: [p (p )] Rule: Law of Detachment (modus ponens or mode that conforms) p p CSc 28 6

7 Another example p It is snowing today We will go skiing p p CSc 28 7

8 Modus Ponens (law of detachment) [p (p )] Example I smoke p (p ) If I smoke, then I cough Therefore, I cough p: I smoke : I cough CSc 28 8

9 Rule of Inference (page 33) p [ p ( p )] modus ponens (mode that conforms) Tautologies p [ ( p )] p modus tollens (denying the conseuent) r r [( p ) ( r)] ( p r) Hypothetical Syllogism (it must be true) p [( p ) p] Disjunctive Syllogism 9

10 Rule of Inference (page 33) p [ p ( p )] modus ponens (mode that conforms) Tautologies p [ ( p )] p modus tollens (denying the conseuent) r r [( p ) ( r)] ( p r) Hypothetical Syllogism (it must be true) p [( p ) p] Disjunctive Syllogism 10

11 Rule of Inference (page 33) p [ p ( p )] modus ponens (mode that conforms) Tautologies p [ ( p )] p modus tollens (denying the conseuent) r r [( p ) ( r)] ( p r) Hypothetical Syllogism (it must be true) p [( p ) p] Disjunctive Syllogism 11

12 Rule of Inference (page 33) p [ p ( p )] modus ponens (mode that conforms) Tautologies p [ ( p )] p modus tollens (denying the conseuent) r r [( p ) ( r)] ( p r) Hypothetical Syllogism (it must be true) p [( p ) p] Disjunctive Syllogism 12

13 Modus Tollens (denying the conseuent) [ (p )] p p ( p ) p T T F T T T F T F T T F F F F T F F T T T F F T F T F T Whenever the premises are true, the conclusion is true (p ) p Valid Example p: I smoke : I cough I do not cough If I smoke then I cough Therefore, I do not smoke CSc 28 13

14 Rule of Inference (page 33) p p ( p ) Addition Tautologies p ( p ) p Simplification p r [( p) ( )] ( p ) Conjunction r r [( p ) ( p r)] ( r) Resolution 14

15 Example Which rule of inference is the basis for this argument? p It is below freezing now. Therefore, it is either below freezing or raining now. It is below freezing now It is raining now p p Addition Rule p ( p ) CSc 28 15

16 Example (page 33) Which rule of inference is the basis for this argument? It is below freezing and raining now. Therefore, it is below freezing now. p: It is below freezing now : It is raining now p p Simplification Rule ( p ) p CSc 28 16

17 Example Which rule of inference is the basis for this argument? If it rains today, then we will not have a barbecue today. If we do not have a barbecue today, then we will have a barbecue tomorrow. Therefore, if it rains today, then w will have a barbecue tomorrow. p: It is raining today : We will not have a barbecue to day r: We will have barbecue tomorrow r r ( p ) ( r) ( p r) Hypothetical Syllogism 17

18 More helpful Rules (page 26) De Morgan's Laws of Logic ( p ) p ( p ) p Also (page 28) Conditional Proposition Contrapositive of p 18

19 Using Rules of Inference to Build Arguments Example Show the hypotheses: It is not sunny this afternoon and it is colder than yesterday, We will go swimming only if it is sunny, If we do not go swimming, then we will take a canoe trip, and If we take a canoe trip, then we will be home by sunset Lead to the conclusion: We will be home by sunset p It is sunny this afternoon It is colder than yesterday r We will go swimming s We will take a canoe trip t We will be home by sunset r r s p s t t Premises: p, r p, r s, and s t and conclusion: t CSc 28 19

20 Using the rules p Simplification p p p r p r s s t t r s Hypothetical p s t Syllogism r r t r p Contrapositive p r r Ponens p p r p Q.E.D. Which had been demonstrated r t Ponens p r p t CSc 28 20

21 Example Show the hypotheses: If you send me an message, then I will finish writing the program, If you do not send me an message, then I will go to sleep early, and If I go to sleep early, then I will wake up feeling refreshed. Leads to the conclusion: If I do not finish writing the program, then I will wake up feeling refreshed. p: you send me an message : I will finish writing the program r: I will go to sleep early s: I will wake up feeling refreshed p r r s s Premises: p, p r, and r s and conclusion s CSc 28 21

22 Hypothesis: p, p r, and r s and conclusion s 1. Premise 1 2. p Contrapositive of Premise 1 3. r Premise 2 4. r Hypothetical Syllogism using 2 and 3 5. r s Premise 3 6. s Hypothetical syllogism using 4 and 5 r r s s Q.E.D. Which had been demonstrated You could use a truth table to show that whenever each of the four hypotheses is true, the conclusion is true. CSc 28 22

23 Argument Example (page 34) The bug is either in module 17 or module 81 The bug is a numerical error Module 81 has no numerical error The bug is in module 17 p: The bug is in module 17 : The bug is in module 81 r: The bug is a numerical error r r p CSc 28 23

24 Example - continued r r p Modus Ponens r r p p Use Disjunctive syllogism to conclude p p p CSc 28 24

25 Your turn use Inference Rules to determine validity Show that the hypotheses: Randy works hard If Randy works hard, then he is a dull boy If Randy is a dull boy, then he will not get the job Implies that Randy will not get the job CSc 28 25

26 Using Inference Rule Modus Ponens Hypotheses: Implies that Randy works hard If Randy works hard, then he is a dull boy If Randy is a dull boy, then he will not get the job Randy will not get the job Hypotheses: w: Randy works hard d: Randy is a dull boy j: Randy will get the job w w d d j j w w d d d d j j QED CSc 28 26

27 p Fallacy (sort of resembles Rules of Inference) p Premise Premise 1 2 Conclusion p p p T T T T T T F F F T F T T T F F F T F F Two cases where premises are True but the conclusions are not both True! Argument is Invalid! CSc 28 27

28 The fallacy argument Is this argument valid? If you do every problem in the book, then you will learn discrete mathematics. Therefore you learned discrete mathematics p: You did every problem in this book : You learned discrete mathematics if p and, then p Fallacy of affirming the conclusion You could have learned discrete mathematics using some other strategy CSc p

29 p The fallacy of denying the hypothesis The proposition [( p ) p ] p p ( p ) p T T F T F F T F T F T F T F T T F T F T T T F F not OK F F T T T T T T OK When each premise is true and the conclusion can be either True or False, the Argument is ambiguous and therefore Invalid CSc 28 29

30 What is wrong with the logic If p is true, and p is true, is it correct to conclude that is true? could be true for reasons other than p Can you assume that you did not learn discrete mathematics if you did not do every problem in the book? CSc 28 30

31 1.5 Quantifiers (page 36) Rules of Inference for Propositions, but also there are Rules for Inference for uantifiers Ways of expressing mathematical arguments Example Everyone in this discrete mathematics class has taken a course in computer science. Megan is a student in this class, which means Megan has taken a course in computer science. CSc 28 31

32 P (x) Definition Statement involving variable x D A Set (the domain of discourse of P) P Example P (n) Propositional function with respect to D If for each x D, P (x) is a proposition n an odd integer D Z + For each n Z +, P (n) is true or false but not both True if n is odd, False if n is even CSc 28 32

33 Example ( universal uantifier) Everyone in this discrete mathematics class has taken a course in computer science. Megan is a student in this class, which means Megan has taken a course in computer science. Student x D(x): x is in the Discrete math class C(x): x has taken a course in Computer Science means " for every" Premises: x( D( x)) C( x)) D( Megan) Conclusion: C( Megan) CSc 28 33

34 Example P (d) Domain is the set { d 1,, d n ) Determine whether x P (x) is true or false for i = 1 to n if ( P (d i ) ) then false else true endif CSc 28 34

35 Definition ( existential uantifier ) x D x P (x) P (x) is true if true for at least one x in D P (x) is false if false for every x in D CSc 28 35

36 Example x 2 x P 2 x R x x 2x 2 2x 5x 2 0 ( 2x1)( x 2) 0 x 1, x 2 P x 2 x 2 1 is true for x , CSc 28 36

37 Generalized De Morgan s Laws of Logic Each pair of propositions in (a) and (b) have he same truth values: ( a) ( x P( x) ); x P( x) ( b) ( x P( x) ); x P( x) (page 44) CSc 28 37

38 x R 1 P( x) 2 1 x 1 1 P( x) is false, when 2 1 is true x is true for eve ry real number x 1 1 Therefore, 2 1 is false for every real number x 1 1 Conseuently, x 2 1 must be false x 1 xp( x) is false because xp( x) is true xp( x) must be euivalent to ( xp( x) ) Example (page 44) CSc 28 38

39 P (x) x loves U2 x P (x) Using De Morgan s domain is the set of rock fans DeMorgan (x P (x)) euivalent to x P (x) There exists at least one rock fan who does not like U2 CSc 28 39

40 Rules of Inference for Quantified Statements xp( x) Pc () Universal instantiation Pc () xp( x) Universal generalization xp( x) Pc () Existential instantiation Pc () xp( x) Existential generalization CSc 28 40

41 Rules of Inference for Quantified Statements xp( x) Pc () Universal instantiation Pc () xp( x) Universal generalization xp( x) Pc () Existential instantiation Pc () xp( x) Existential generalization CSc 28 41

42 Rules of Inference for Quantified Statements xp( x) Pc () Universal instantiation Pc () xp( x) Universal generalization xp( x) Pc () Existential instantiation Pc () xp( x) Existential generalization CSc 28 42

43 Rules of Inference for Quantified Statements xp( x) Pc () Universal instantiation (if c D) Pc () xp( x) Universal generalization (for every c D) xp( x) Pc () Existential instantiation (for some c D) Pc () xp( x) Existential generalization (for some c D) CSc 28 43

44 Table (page 48) CSc 28 44

45 Combining Rules of Inference Rules including both Propositions and Quantified Statements x( P( x) Q( x)) P( a), where a is a particular element in the domain Qa ( ) CSc 28 45

46 A famous argument All men are mortal. Socrates is a man. therefore, Socrates is mortal Translate using rules of inference: For all humans Humans that are male are mortal Implies that Socrates is a male, therefore Socrates is mortal h: Humans P(h): Human is male M(h): Mortal Modus Ponens h( P( h) M ( h)) P(Socrates) M (Socrates) CSc 28 46

47 Example 14 (page 71) x( P( x) Q( x)) P( a), where a is a particular element in the domain Qa ( ) For all positive integers, x P(x): x > 4, Q(x): x 2 < 2 x Show that < x( P( x) Q( x)) P(100) Q(100) CSc 28 47

48 Argument from Lewis Carroll Premises: All lions are fierce and Some lions do not drink coffee Conclusion: Some fierce creatures do not drink coffee. Domain creatures: P(x): x is a lion Q(x): x is fierce R(x): x drinks coffee x [P(x) Q(x)] x [P(x) R(x)] x [Q(x) R(x)] CSc 28 48

49 Your turn For each argument, determine whether the argument is correct or incorrect and explain why. a. All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic. b. Every computer science major takes discrete math. Natasha is taking discrete math. Therefore, Natasha is a computer science major. c. All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit. d. Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day. CSc 28 49

50 Examples CSc 28 50

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