Rules of Inference. Arguments and Validity
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1 Arguments and Validity A formal argument in propositional logic is a sequence of propositions, starting with a premise or set of premises, and ending in a conclusion. We say that an argument is valid if and only if the conclusion is true when all premises are true. 61
2 Arguments and Validity Example: 1. If I work, then I get paid. 2. If I get paid, then I pay the bills. 3. Therefore, if I work, then I pay the bills. 62
3 Argument Form An argument form is a sequence of compound propositions involving propositional variables. An argument form is valid if no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true. Example: Argument 1. If I work, then I get paid. 2. If I get paid, then I pay the bills. 3. Therefore, if I work, then I pay the bills. Argument Form 1. p q 2. q r 3. p r 63
4 Modus Ponens (Law of Detachment) Example 1. If it snows today, then we will go skiing. 2. It is snowing today. 3. Therefore, we will go skiing. Argument Form p q p q 64
5 Modus Tollens Example 1. If it snows today, then we will go skiing. 2. We will not go skiing. 3. Therefore, it is not snowing today. Argument Form p q q p 65
6
7 Disjunctive syllogism Example 1. It is sunny, or it is cloudy 2. It is not sunny. 3. Therefore, it is cloudy. Argument Form p q p q 67
8 Addition Example 1. It is sunny. 2. Therefore, it is sunny, or it is cloudy. Argument Form p p q 68
9 Simplification Example 1. It is cloudy, and it is raining. 2. Therefore, it is cloudy. Argument Form p q p 69
10 Conjunction Example 1. It is cloudy. 2. It is raining. 3. Therefore, it is cloudy, and it is raining. Argument Form p q p q 70
11 Resolution Example 1. It s a weekday, or the kids are off of school. 2. It s not a weekday, or Jim is working. 3. Therefore, the kids are off of school, or Jim is working. Argument Form p q p r q r 71
12 Rule of Inference Tautology Name p p q q (p (p q)) q Modus ponens q p q p p q q r p r p q p q ( q (p q)) p ((p q) (q r)) (p r) ((p q) p) q Modus tollens Hypothetical syllogism Disjunctive syllogism 72
13 Rule of Inference p p q Tautology p (p q) Name Addition p q p p q p q (p q) p Simplification (p q) (p q) Conjunction p q p r q r ((p q) ( p r) (q r) Resolution 73
14 Examples: of inference used. In each case identify the rule If it is cloudy, then it s raining. It s not raining. Therefore, it s not cloudy. 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 we will have a barbecue tomorrow. 74
15 Examples: of inference used. In each case identify the rule It is below freezing now. Therefore, it is either below freezing or raining now. It is below freezing and raining now. Therefore, it is below freezing now. Jasmine is skiing, or it is not snowing. It is snowing, or Bart is playing hockey. Jasmine is skiing, or Bart is playing hockey. 75
16 Examples: If you do every problem in this book, then you will learn discrete mathematics. You learned discrete mathematics. Therefore, you did every problem in this book. If you do every problem in this book, then you will learn discrete mathematics. You did not do every problem in this book. Therefore, you did not learn discrete mathematics. 76
17 Common Fallacies Fallacy p q q p p q p q Name Affirming the conclusion Denying the hypothesis 77
18 Constructing Arguments Example: For the given set of premises, what conclusion can be drawn? 1. It is not sunny today, and it is colder than yesterday. 2. We will go swimming only if it is sunny. 3. If we do not go swimming, then we will take a canoe trip. 4. If we take a canoe trip, then we will be home by sunset. Conclusion: 78
19 Constructing Arguments Premises 1. It is not sunny today, and it is colder than yesterday. ( p q) 2. We will go swimming only if it is sunny. (r p) 3. If we do not go swimming, then we will take a canoe trip. ( r s) 4. If we take a canoe trip, then we will be home by sunset. (s t) Conclusion We will be home by sunset. (t) Argument Statement 1. p q 2. p 3. r p 4. r 5. r s 6. s 7. s t 8. t Reason Premise Simplification (1) Premise Modus tollens (2 and 3) Premise Modus ponens (4 and 5) Premise Modus ponens (6 and 7) 79
20 Constructing Arguments Premises 1. If you send me an , then I will finish the project. (p q) 2. If you do not send me an , then I will go home. ( p r) 3. If I go home, then I will go to sleep early. (r s) Conclusion If I do not finish the project, then I will go to sleep early. ( q s) Argument Statement 1. p q 2. q p 3. p r 4. q r 5. r s 6. q s Reason Premise Contrapositive (1) Premise Hypothetical syllogism (2 and 3) Premise Hypothetical syllogism (4 and 5) 80
21 and Quantifiers Rule of Inference x P (x) P (c) for an arbitrary c P (c) for an arbitrary c x P (x) x P (x) P (c) for some element c P (c) for some element c x P (x) Name Universal instantiation Universal generalization Existential instantiation Existential generalization
22 and Quantifiers Premises 1. Everyone taking discrete math has taken calculus. ( x(d(x) C(x))) 2. Marla is taking discrete math class. (D(M arla)) Conclusion Marla has taken calculus. (C(M arla)) Argument Statement 1. x(d(x) C(x)) 2. D(Marla) C(Marla) 3. D(Marla) 4. C(Marla) Reason Premise Universal instantiation (1) Premise Modus ponens (2 and 3)
23 and Quantifiers Premises 1. A student in this class has not read the book. ( x(c(x) B(x))) 2. Everyone in this class passed the first exam. ( x(c(x) P (x))) Conclusion Someone who passed the first exam has not read the book. Argument ( x(p (x) B(x))) Statement 1. x(c(x) B(x)) 2. C(a) B(a) 3. C(a) 4. x(c(x) P (x)) 5. C(a) P (a) 6. P (a) 7. B(a) 8. P (a) B(a) 9. x(p (x) B(x)) Reason Premise Existential instantiation (1) Simplification (2) Premise Universal instantiation (4) Modus ponens (3 and 5) Simplification (2) Conjunction (6 and 7) Existential generalization (8)
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