Valid Reasoning. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing February, Outline Truth and Validity Valid Reasoning

Transcription

1 Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February, 2018 Alice E. Fischer Reasoning... 1/23

2 1 Truth is not the same as Validity 2 Alice E. Fischer Reasoning... 2/23

3 Truth is not the same as Validity Definitions. True: correct, accurate, right. A statement that is in accordance with fact or reality. Valid: well founded, sound, reasonable, rational. Executed in compliance with the law. A statement can be true, but we don t talk about valid statements. A line of reasoning can be valid but not true. Valid reasoning starts with premises and applies rules of logic properly to arrive at a conclusion. Alice E. Fischer Reasoning... 3/23

4 Models Outline Truth is not the same as Validity A proposition is true or false depending on the truth values assigned to its variables. A model assigns a truth value to each variable. A proposition can be true in some models and false in others. We often focus on a model that reflects the real world. Alice E. Fischer Reasoning... 4/23

5 Truth is not the same as Validity A deduction can be Valid, Statements about a model can be True Valid True A true statement is a proposition that is true in your model. A false statement is a proposition that is not true in your model. A valid argument is an argument that follows the rules of valid deduction. It might have a true conclusion or a false conclusion, depending on the truth or falsehood of the premises. An invalid argument is an argument that does not follow the rules of valid deduction. Alice E. Fischer Reasoning... 5/23

6 Valid Arguments Outline Truth is not the same as Validity Valid but False: If one of your premises is false, you can make a valid argument and end up with a false conclusion. Invalid but True: With any premises, you can make an invalid argument and end up with a conclusion that just happens to be true. Valid and True: But if you start with true premises and use only valid steps of reasoning, the conclusion will be true. Alice E. Fischer Reasoning... 6/23

7 A Good Argument Outline Truth is not the same as Validity It has been said that a proof is an argument that convinces another mathematician. What makes an argument convincing? Each premise is relevant to the issue. All of the premises are true and you believe that they are true. The collection of premises is sufficient; nothing important is missing. Valid steps of reasoning lead from the premises to the conclusion. Alice E. Fischer Reasoning... 7/23

8 Alice E. Fischer Reasoning... 8/23

9 These rules of inference can be derived directly from the meanings of,, and. The triangle of dots ( ) is read therefore. Rule Name Premises Conclusion Generalization P P Q Elimination P Q, Q P Specialization P Q P Conjunction P, Q P Q Transitivity P Q, Q R P R Contradiction rule P c P Division into cases P Q, P R, Q R R Modus Ponens P, P Q Q Modus Tollens P Q, Q P Alice E. Fischer Reasoning... 9/23

10 Generalization and Elimination The rules of inference are used to progress from the premises to the conclusion in a proof. Generalization lets you add any additional clause to a premise that you know is true: Start with P being true. Then P X is true no matter what X is. Do this if P X was the antecedent in some other premise. Elimination If you have two possibilities and you can rule one out, the other must be true. Start with P Q being true. If you know that Q is false, Then you can conclude that P is true. Alice E. Fischer Reasoning... 10/23

11 Conjunction and Specialization Conjunction lets you combine two true statements into a single true statement. Start with P and Q both being true. Then P Q is true because this is the definition of. Specialization is the opposite of conjunction. It lets you remove a clause from a conjunction: Start with P Q being true. Then P is true independently. Alice E. Fischer Reasoning... 11/23

12 Contradiction and Transitivity Transitivity. Start with P Q and Q R. The Q s in the middle match. Then you can conclude that P R Contradiction. This rule is the basis of proof by contradiction. First, assume that P is false. If this lets you reach a contradiction, Then you know your original assumption was wrong, and P is not false So P is true. Alice E. Fischer Reasoning... 12/23

13 Division into Cases Outline Division into Cases lets you deal with uncertainty. You start with P Q, so you know that one of them is true, but you don t know which. One premise is that P R. One premise is that Q R. Since either P or Q is true, at least one of these implications succeeds. So R is true. Alice E. Fischer Reasoning... 13/23

14 The Way of the Bridge and The Way of Removal Modus Ponens From P and P R we conclude R. This is our most basic and most important rule of deduction. You can demonstrate its soundness using a truth table. Modus Tollens From P R and R, we can conclude P. This is like a statement of the contrapositive of Modus Ponens. You can demonstrate its soundness using a truth table. Alice E. Fischer Reasoning... 14/23

15 Preparing for class: What conclusion(s) can I draw here? Premises (facts): 1 My cat is not needy after he is fed. 2 My cat is a pest if and only if he feels needy. 3 Right now, my cat is sitting on my keyboard. 4 Pesty cats climb and sit on keyboards. 5 With a cat on the keys, a computer is not useable. 6 I can get ready for class if and only if I use my computer. So... what can I prove? Alice E. Fischer Reasoning... 15/23

16 Define the Symbols Outline N F P C D K U R Use these symbols: My cat is needy. My cat has been fed. My cat is a pest right now. Pesty cats climb or sit on keyboards. A cat is climbing on my desk. A cat is on my keyboard. My computer is useable. I can get ready for class. Premises: 1 F N 2 P N 3 K 4 P D K 5 K U 6 U R Alice E. Fischer Reasoning... 16/23

17 Theorem 1: I have a Problem I can t get ready for class. Proof: 3. K Write down the relevant premises. 5. K U 6. U R 7. R U Definition of. 8. U Modus Ponens, premises 3 and R Modus Tollens, steps 7 and 8. Alice E. Fischer Reasoning... 17/23

18 Theorem 2: My Cat has a Problem My cat is hungry Proof: 1. F N Write down the relevant premises. 2. P N 3. K 4. P D K 5. D K Premise 3 and generalization rule. 6. P Premise 4 and step N Premise 2 and step F Premise 1, step 7, Modus Tollens. Alice E. Fischer Reasoning... 18/23

19 Theorem 3: I can solve this problem! If I feed my cat, he won t be a pest. Proof: 0. F Suppose I feed the cat. 1. F N 2. N P 3. N Assumption 0, premise N P Contrapositive of premise 2, definition of. 5. P Steps 8, 9, modus ponens. So if I want to get any work done, I should get up and feed the cat. Alice E. Fischer Reasoning... 19/23

20 To buy or not to buy... Is this valid reasoning? Symbolize it and prove it. Valentine s Day 1 If I can pay for these roses, I will take them to my sweetheart. 2 If I have enough cash or good credit, I can pay. 3 I do not have enough cash in my pocket to pay for the flowers. 4 I have a Discover card in my pocket that the florist will accept. 5 A credit card is good if the expiration date is in the future, the credit balance is low, and the store accepts that kind of card. 6 My Discover card expires next year. 7 My Discover card has a low balance right now. I can get those flowers for my sweetheart. Alice E. Fischer Reasoning... 20/23

21 Define the Symbols Outline P F C A X L G Use these symbols: I can pay for the flowers I take flowers to my sweetheart. I have enough cash. The florist accepts my Discover. The credit card has expired. The balance on my card is low. My credit is good. Premises: 1 P F 2 (C G) P 3 C 4 A 5 ( X L A) G 6 X 7 L Alice E. Fischer Reasoning... 21/23

22 Can I buy the flowers? Outline Proof: 1. P F First, we write down the premises. 2. (C G) P 3. C 4. A 5. ( X L A) G 6. X 7. L 8. X L A conjunction, steps 4, 6, 7 9. G modus ponens, steps 5, C G generalization, step P modus ponens, steps 2, 10 F modus ponens, steps 1, 11 Alice E. Fischer Reasoning... 22/23

23 Summary Outline Premises can be true or false. Reasoning can be valid or invalid. Valid reasoning leads to true conclusions if it starts with true premises. Valid reasoning can lead to a false conclusion when a premise is false. We presented several fundamental rules of inference: Generalization / Elimination Specialization / Conjunction Contradiction / Transitivity / Division into cases Modus Ponens Modus Tollens Alice E. Fischer Reasoning... 23/23