Rules of Inference. Agenda. Rules of Inference Dr Patrick Chan. p r. p q. q r. Rules of Inference for Quantifiers. Hypothetical Syllogism

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1 Discr ete Mathem atic Chater 1: Logic and Proof 1.5 Rules of Inference Dr Patrick Chan School of Comuter Science and Engineering South China University of echnology Recall John is a co. John knows first aid. herefore, all cos know first aid 3 Agenda Rules of Inference Rules of Inference for Quantifiers Rules of Inference Hyothetical Syllogism q r r Disjunctive Syllogism q q 2 4

2 Argument Premise / Hyothesi s Recall Some students work hard to study. Some students fail in examination. So, some work hard students fail in examination. Argument: Valid? 1 2 Given an argument, where 1, 2,, n be the remises n q be the conclusion he argument is valid when ( 1 2 n ) q is a tautology When all remises are true, the co nclusion sho uld be true When not all remises are true, the co nclu sion can be either true or false 7 If it rains, the floor is wet It rains : It rains q: he floor is wet Conclusi on he floor is wet q therefore Ar gument Argu ment orm Argument in roositional logic is a sequence of roositions Premises / Hyothesis: All excet the final roosition Conclusion: he final roosition Argument form reresents the argument by variables 6 Argument Examle: Argument is valid If it rains, the floor is wet It rains q he floor is wet ( () ) q autology Need to check if the co nclusio n is true or not Must be true q () ( ()) q 8 5 q ocus on this case Check if it hae ns q

3 Rules of Inference How to show an argument is valid? ruth able May be tedious when the number of variables is large Rules of Inference irstly establish the validity of some relatively simle argument forms, called rules of inference hese rules of inference can be used as building blocks to construct more comlicated valid argument forms Rules of Inference Addition q Simlification q Conjunction q q 11 Rules of Inference Modus Ponens Affirm by affirming q Modus ollens Deny by denying q Rules of Inference Resolution = q = / r = = q = r = / q r q r Examle I go to swim or I lay tennis I do not go to swim or I lay football herefore, I lay tennis or I lay football

4 Rules of Inference ( ) Modus Ponens (() ()) q Modus ollens (( q) ()) Hyothetical Syllogism (() (q r)) ( r) Dis junctive Sy llogism (( q) ( )) q Addition () q Simlific ation (() (q)) Conjunction (() (q)) ( q) Resolution (( q) ( r)) (q r) 13 Comarison between Inference and Equivalence Inference () Meaning: If, then q does not mean q Either inference or equivalence rules can be used q imlies is used in roof Equivalence ( q) Meaning: is equal to q q mean q Only equivalence rules can be used q can be roved by showing and q is used in roof Equivalence ( ) is a more restrictive relation than Inference ( ) 15 Rules of Equivalence ( ) Recall Id ent ify Laws Do mination L aws Id emot en t L aws Negation L aws Do uble N egat ion Law ( ) Co mmut ative L aws q q q q Asso ciat iv e Laws Distribut ive Laws (q r) ( q) r (q r) ( q) r (q r) ( q) ( r) (q r) ( q) ( r) Ab so r tion L aws ( q) ( q) De Morg an s Laws ( q) q ( q) q 14 Using Rules of Inference Examle 1: Given: It is not sunny this afternoon and it is colder than yes terday. We will go swimming only if it is sunny If we do not go swimming, then we will ta ke a canoe tri If we take a canoe tri, then we will be home by sunset Can these roositions lead to the conclusion "We will be home by sunset? 16

5 Let : It is sunny this afternoon q: It is colder than yesterday r: We go swimming s: We take a canoe tri t: We will be home by sunset It is not sunny this afternoon and it is colder q than yesterday r We will go swimming only if if it is sunny If we do not go swimming, then we will take r s a canoe tri If we take a canoe tri, then we will be s t home by sunset t We will be home by sunset 17 Using Rules of Inference Or, another resentation method: Hyothesis: q r r s s t Conclusion: t ( q) (r ) ( r s) (s t) (r ) ( r s) (s t) r ( r s ) (s t) s (s t) t By Modus Ponens By Modus Ponens By Modus ollens By Simlif ication 19 Using Rules of Inference Ste Reas on Hyothesis: q r r s s t q r r r s s Simlification using (1) Modus tollens using (2) and (3) Modus onens usi ng (4) and (5) Conclusion: t s t t Modus onens usi ng (6) and (7) herefore, the roositions can lead to the conclusion We will be home by sunset 18 Small Exercise Given: If you send me an message, then I will finish writing the rogram If you do not send me an message, then I will go to slee early If I go to slee early, then I will wake u feeling refreshed Can these roositions lead to the conclusion "If I do not finish writing the rogram, then I will wake u feeling refreshed." 20

6 Let : you send me an me ssage q: I will finish writing the rogram r: I will go to slee early s: I will wake u feeling refreshed If you send me an message, then I will finish writing the rogram If you do not send me an message, r then I will go to slee early If I go to slee early, then I will wake u r s feeling refreshed If I do not finish writing the rogram, q s then I will wake u feeling refreshed 21 Small Exercise Or, another resentation method: Hyothesis: r r s Conclusion: q s () ( r) (r s) ( q ) ( r) (r s) ( q r) (r s) ( q s) By Hy othetical Sy llogism Cont raosit iv e By Hy othetical Sy llogism 23 Small Exercise Ste Reas on Hyothesis: r r s Conclusion: q s q r q r r s q s Contraositive of (1) Hyothetical Syllogism using (2) and (3 ) Hyothetical Syllogism using (4) and (5 ) herefore, the roositions can lead to the conclusion If I do not finish writing the rogram, then I will wake u feeling refreshed 22 Using Rules of Inference allacies Are the following arguments correct? Examle 1 Hyo thesis If you success, you w ork hard You w ork hard Con clusion You success Examle 2 (allacy of denying the hyothesis) Hyo thesis If you success, you w ork hard You do not success Con clusion You do not w ork hard (allacy of affirmi ng the conclusion) q q 24

7 Rules of Inference for Quantifiers Universal Instantiation Existential Instantiation x P(x) P(a) x P(x) P(c) for some element c where ais a articular member of t he domain Universal Generalization Existential Generalization P(b) for an arbitrary b P(d) for some element d x P(x) x P(x) Be noted that b that we select m ust be an arbitrary, and not a secific 25 Let DC(x): x studies in discrete mathematics CS(x): x studies in co muter science Domain of x: student Everyone in this discrete x (DC(x) CS(x)) mathematics class has taken a course in comuter science DC(Marla) Marla is a student in this class CS(Marla) Marla has taken a course in comuter science 27 Rules of Inference for Quantifiers Examle 1 Given Everyone in this discrete mathematics class has taken a course in comuter science Marla is a student in this class hese remises im ly the conclusion "Marla has taken a course in comuter science" 26 Rules of Inference for Quantifiers : x (DC(x) CS(x)) DC(Marla) Conclusion : CS(Marla) Ste Reas on x (DC(x) CS(x)) DC(Ma rla) CS(Marla ) Universal Instantiatio n from (1) 3. DC(Ma rla) 4. CS(Ma rla) Modus onens usi ng (2) and (3) herefore, the roositions can lead to the conclusion Marla has taken a course in comuter science 28

8 Using Rules of Inference for Quantifiers Or, another resentation method: : x (DC(x) CS(x)) DC(Marla) Conclusion : CS(Marla) x (DC(x) CS(x)) DC(Marla) By Univ ersal Instantiation (DC(Marla) CS(Marla)) DC(Marla) CS(Marla) By Modus onens 29 Let C(x): x in this class RB(x): x reads the book PE(x): x asses the first exam Domain of x: any erson A student in this class has not x (C(x) RB(x)) read the book Everyone in this class assed x (C(x) PE(x)) the first exam Someone who assed the first x (PE(x) RB(x)) exam has not read the book We cannot define the doma in as stude nt in this class since the co nclusio n mea ns a nyo ne 31 Small Exercise Given A student in this class has not read the book Everyone in this class assed the first exam hese remises imly the conclusion "Someone who assed the first exam has not read the book" 30 Small Exercise Ste : Conclusion : x (C(x) RB(x)) x (C(x) PE(x)) x (C(x) RB(x)) C(a) RB(a) C(a) x (C(x) PE(x)) C(a) PE(a) PE(a ) RB(a) PE(a ) RB(a) x (PE(x) RB(x)) x (PE(x) RB(x)) Reas on Existential Instantiation from (1) Simlification from (2) Universal Instantiatio n from (4) Modus onens from (3) and (5) Simlification from (2) Conjunction fro m (6) and (7) Existential Genera lization from (8) herefore, the roositions can lead to the conclusion Someone who assed the first exam has not read the book 32

9 Small Exercise Or, another resentation method: ( x (C(x) RB(x))) ( x (C(x) PE(x))) C(a) RB(a) ( x (C(x) PE(x))) By Existential Instantiation C(a) RB(a) (C(a) PE(a)) By Univ ersal Instantiation PE(a) RB(a) By Modus onens x (PE(x) RB(x)) By Existential Generalization : Conclusion : x (C(x) RB(x)) x (C(x) PE(x)) x (PE(x) RB(x)) 33 Combining Rules of Inference Examle: Given or all ositive integers n, if n is greater than 4, then n 2 is less than 2 n is true. Show that < Combining Rules of Inference he rules of inference of Proositions and Quantified Statements can be combined Universal Modus Ponens Universal Modus Ponens x (P(x) Q(x)) P(a), where a is a articular element in the domain Q(a) x (P(x) Q(x)) Q(a), where a is a articular element in the domain P(a) ( x (P(x) Q(x))) (P(a)) By Un iv ersal Instantiation (P(a) Q(a)) (P(a)) Q(a) By Mo dus Po nen s ( x (P(x) Q(x))) ( Q(a)) By Un iv ersal Instantiation (P(a) Q(a)) ( Q(a)) P(a) By Mo du s o llen s 34 Combining Rules of Inference Examle: or all ositive integers n, if n is greater than 4, then n 2 is less than 2 n n (P(n) Q(n)) P(n): n > 4 Q(n): n 2 < 2 n P(100) (since 100 > 4) Q(100) (100 2 < ) By Universal Modus Ponens 36

10 Summary What we have learnt in revious lectures? Proosition Oerator Predic ates Quantifier ruth able Rules of Equivalence Rules of Inference Show if an argument is valid his is called the formal roof very clear and recise extremely long and hard to follow 37