Discrete Structures Lecture Rules of Inference

Transcription

1 Term argument valid remise fallacy Definition A seuence of statements that ends with a conclusion. The conclusion, or final statement of the argument, must follow from the truth of the receding statements, or remises, of the argument. A statement of the argument. Incorrect reasoning Valid s in Proositional Logic DEFINITION 1 An argument in roositional logic is a seuence of roositions. All but the final roosition are called remises and the final roosition is called the conclusion. An argument is valid if the truth of all its remises imlies that the conclusion is true. An argument form in roositional logic is a seuence of comound roositions involving roositional variables. An argument form is valid if no matter which articular roositions are substituted for the roositional variables in its remises, the conclusion is true if the remises are all true. Consider remises 1, 2,, nn and conclusion. The conclusion is valid when ( 1 2 nn ) is a tautology. Rules of Inference for Proositional Logic The tautology ( ( )) is the basis of the rule of inference called modus onens, or the law of detachment. (Modus onens is Latin for mode that affirms.) This tautology leads to the following valid argument form. Please recall that denotes therefore. EXAMPLE 1 Suose that the conditional statement If it snows today, then we will go skiing and its hyothesis, It is snowing today, are true. Then, by modus onens, it follows that the conclusion of the conditional statement, We will go skiing, is true. We will go skiing It is snowing today. If it snows today, then we will go skiing. Therefore, we will go skiing 1

2 EXAMPLE 2 Determine whether the argument given here is valid and determine whether its conclusion must be true because of the validity of the argument if 2 > 3 2, then 2 2 > We know that 2 > 3 2. Conseuently, 2 2 = 2 > = 9 4. " 2 > > 9 4 : The argument form is valid because it matches modus onens. However, one of its remises, 2 > 3, is false. Conseuently we cannot 2 conclude that the conclusion, 2 > 9, is true. The conclusion is, in fact false 4 because 2 < 9. 4 EXAMPLE 3 State which rule of inference is the basis of the following argument: It is below freezing now. Therefore, it is either below freezing or raining now. Proositions: It is below freezing now. It is raining now. : The argument form is addition. 2

3 TABLE 1 Rules of Inference Rule of Inference Tautology Name [ ( )] Modus onens [ ( )] Modus tollens [( ) ( rr)] ( rr) Hyothetical rr syllogism rr [( ) ] Disjunctive syllogism ( ) Addition ( ) Simlification [() ()] ( ) Conjunction [( ) ( rr)] ( rr) Resolution rr rr EXAMPLE 4 State which rule of inference is the basis of the following argument: it is below freezing and raining now. Therefore, it is below freezing now. Proositions ( ) It is raining now. It is below freezing now. it is below freezing and raining now. ( ) It is below freezing now. The argument is a simlification. it is below freezing and raining now. 3

4 EXAMPLE 5 State which rule of inference is the basis of the following 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 barbeue tomorrow. Therefore, if it rains today, then we will have a barbeue tomorrow. Proositions r rr It rains today. We will have a barbeue today We will have a barbeue tomorrow. If it rains today then we will not have a barbeue today. If we do not have a barbeue today then we will have a barbeue tomorrow rr If it rains today then we will have a barbeue tomorrow. The argument is a hyothetical syllogism. Using Rules of Inference to Build s. EXAMPLE 6 Show that the hyothesis 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 tri, and If we take a canoe tri, then we will be home by sunset lead to the conclusion We will be home by sunset. Proositions r s t It is sunny this afternoon. It is colder than yesterday. We will go swimming. We will take a canoe tri. We will be home by sunset. rr rr ss ss tt tt 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 tri. If we take a canoe tri, then we will be home by sunset. 4

5 Resolution Resolution: Ste Proosition Reason 1 Hyothesis 2 Simlification (of ste 1) 3 rr Hyothesis 4 rr Modus tollens using stes 2 and 3 [ (rr )] rr 5 rr ss Hyothesis 6 ss Modus onens using stes 4 and 5 [ rr ( rr ss)] ss 7 ss tt Hyothesis 8 tt Modus onens using stes 6 and 7. [ss (ss tt)] tt Comuter rograms have been develoed to automate the task of reasoning and roving theorems. Many of these rograms make use of a rule of inference known as resolution. This rule of inference is based on the tautology ( ) ( rr) ( rr) EXAMPLE 8 Use resolution to show that the hyotheses Jasmine is skiing or it is not snowing and It is snowing or Bart is laying hockey imly that Jasmine is skiing or Bart is laying hockey. Proositions t u v Jasmine is skiing It is snowing Bart is laying hockey tt uu Jasmine is skiing or it is not snowing uu vv It is snowing or Bart is laying hockey tt vv Jasmine is skiing or Bart is laying hockey [(tt uu) ( uu vv)] ( tt vv) 1. Because the oerator is commutative we can exchange the two exressions enclosed in the suare braces []. 2. Because the is commutative we can exchange the two roositions tt and uu. [( uu vv) ( uu tt)] ( tt vv) Note that the argument form is the same as the resolution argument if u is substituted for, v is substituted for, and t is substituted for r. 5

6 Fallacies Fallacies are incorrect arguments. Consider the following fallacies. Proosition Descrition [( ) ] Fallacy of affirming the conclusion. [( ) ] Fallacy of denying the hyothesis. EXAMPLE 10 Is the following argument valid? If you do every roblem in this book, then you will learn discrete mathematics. You learned discrete mathematics. Therefore, you did every roblem in the book. Proositions You did every roblem in the book. You learned discrete mathematics. If you did every roblem in the book then you learned discrete mathematics You learned discrete mathematics. Therefore, you did every roblem in the book. This is an examle of an incorrect argument using the fallacy of affirming the conclusion. Indeed, it is ossible for you to learn discrete mathematics in some way other than by doing every roblem in the book. (You may learn discrete mathematics by reading, listening to lectures, doing some, but not all, the roblems in this book, and so on.) EXAMPLE 11 Let and be as in Examle 10. If the conditional statement is true, and is true, is it correct to conclude that is true? In other words, is it correct to assume that you did not learn discrete mathematics if you did not do every roblem in the book, assuming that if you do every roblem in this book, then you will learn discrete mathematics? It is ossible that you learned discrete mathematics even if you did not do every roblem in this book. The incorrect argument is of the form If you did every roblem in the book then you learned discrete mathematics You did not do every roblem in the book. Therefore, you did not learn discrete mathematics.. This is an examle of an incorrect argument using the fallacy of denying the hyothesis. 6

7 Rules of Inference for Quantified Statements TABLE 2 Rules of inference for Quantified Statements. Rule of Inference Name xxxx(xx) PP(cc) Universal instantiation PP(cc) for an arbitrary cc xxxx(xx) Universal generalization xxxx(xx) PP(cc) for some element cc Existential instantiation PP(cc) for some element cc xxxx(xx) Existential generalization Name Universal instantiation Universal generalization Existential instantiation Existential generalization Descrition The rule of inference that is used to conclude that PP(cc) is true, where c is a articular member of the domain, given the remise xxxx(xx). Examle: We can conclude that Lisa, a woman, is wise from the remise All women are wise. The rule of inference that states that xxxx(xx) is true, given the remise that PP(cc) is true for all elements c in the domain. Universal generalization is used when we show that xxxx(xx) is true by taking an arbitrary element c from the domain and showing that PP(cc) is true. The element c that we select must be an arbitrary, and not a secific, element of the domain. That is, when we assert from xxxx(xx) the existence of an element c in the domain, we have no control over c and cannot make any other assumtions about c other than it comes from the domain. Examle: We can conclude that all women are wise if we select Lisa arbitrarily from among all women and find that she is wise. The rule of inference that allows us to conclude that there is an element c in the domain for which PP(cc) is true if we know that xxxx(xx) is true. We cannot select an arbitrary value of c here, but rather it must be a c for which PP(cc) is true. Usually we have no knowledge of what c is, only that it exists. Because it exists, we may give it a name (c) and continue our argument. Examle: From the statement that there exists a woman who is wise we can conclude that there is a articular woman who is wise and by testing women we found a articular woman, Lisa, who is wise. The rule of inference that is used to conclude that xxxx(xx) is true when a articular element c in the domain with PP(cc) true is known. That is, if we know one element c in the domain for which PP(cc) is true, then we know that xxxx(xx) is true. Examle: We can conclude that there exists a woman who is wise from the knowledge that Lisa, a woman, is wise. 7

8 EXAMPLE 12 Show that the remises Everyone in this discrete mathematics class has taken a course in comuter science and Marla is a student in this class imly the conclusion Marla has taken a course in comuter science. Premise DD(xx) CC(xx) xx(dd(xx) CC(xx)) xx is a student in this discrete mathematics class. x has taken a course in comuter science. Everyone in this discrete mathematics class has taken a course in comuter science DD(MMMMMMMMMM) Marla is a student in this class CC(MMMMMMMMMM) Marla has taken a course in comuter science Ste Statement Reason 1 xx(dd(xx) CC(xx)) Premise 2 DD(MMMMMMMMMM) CC(MMMMMMMMMM) Universal instantiation from ste 1 3 DD(MMMMMMMMMM) Premise 4 CC(MMMMMMMMMM) Modus Ponens from stes 2 and 3. 8

9 EXAMPLE 13 Show that the remises A student in this class has not read the book, and Everyone in this class assed the first exam imly the conclusion Someone who assed the first exam has not read the book. Proositional functions CC(xx) BB(xx) PP(xx) Premises xx(cc(xx) BB(xx)) xx(cc(xx) PP(xx)) xx is in this class. x has read the book. x assed the first exam. There exists a student in this class who has not read the book. Every student in this class has assed the first exam. xx(pp(xx) BB(xx)) There exists a student that assed the exam and did not read the book. Ste Statement Reason 1. xx(cc(xx) BB(xx)) Premise 2. CC(aa) BB(aa) Existential instantiation from (1) 3. CC(aa) Simlification from (2) 4. xx(cc(xx) PP(xx)) Premise 5. CC(aa) PP(aa) Existential instantiation from (4) 6. PP(aa) Modus Ponens from (3) and (5) 7. BB(aa) Simlification from (2) 8. PP(aa) BB(aa) Conjunction from (6) and (7) 9. xx(pp(xx) BB(xx)) Existential generalization from (8) Combining Rules of Inference for Proositions and Quantified Statements xx(pp(xx) QQ(xx)) PP(aa), where a is a articular element in the domain QQ(aa) Universal modus onens EXAMPLE 14 Assume that For all ositive integers n, if n is greater than 4, then nn 2 is less than 2 nn is true. Use universal modus onens to show that < PP(nn) nn > 4 QQ(nn) nn 2 < 2 nn The domain of n is all ositive integers. nn(pp(nn) QQ(nn)) For all ositive integers n, if n is greater than 4, then nn 2 is less than 2 nn PP(100) 100 > 4 QQ(100) < Universal modus onens 9