Logic and Reasoning

1. Logic and Reasoning

2. Objective • Spot valid and invalid reasoning. • Be able to construct a validreasoning. • Make appropriate predictions based on acceptable premises. • Logically draw conclusionsfrom experimental result. Statement VS Reasoning Statement – True or False Reasoning – Valid or Invalid

3. Logic and Reasoning • Logically draw conclusions / Construct a valid reasoning. • Spot valid and invalid reasoning. Statement VS Reasoning Statement – True or False Reasoning – Valid or Invalid Premises: What we assume to be true / given. Conclusion: What we derive from the premises. Premise Premise (something assumed to be true) If you study hard, you will get A. You study hard. Reasoning Reasoning You will get A Conclusion Conclusion (something derived from the premises) “False conclusion may comes from invalid reasoning orfalse premises”. Conclusion/Premise: True/False (T/F) Reasoning: Valid/Invalid (V/I) Only all true premises and valid reasoningcanguarantee true conclusion. In math term, Premise is called Axiom, Conclusion is called Theorem, Lemma, Reasoning is called Proof. In experimental science, Empirical scientists tell us whether statements are true. Logicians tell us whether reasoning is valid. Premise Statement: True/False Conclusion

4. Reasoning Truth VS Validity Conclusion Premises • They are not the same. Truth for statements. Validity for argument/reasoning. Premises: Dogs have eight legs. [If x is a dog, then x has eight legs.] Spooky is a dog. Conclusion: Spooky has eight legs. valid The argument is valid. However, the conclusion is false. For further clarification, see lecture note.

5. Note Reasoning Conclusion Premises • Valid reasoning does not guarantee a true conclusion. • Invalid reasoning does not guarantee a false conclusion. • A false conclusion does not guarantee invalidity. • True premises and a true conclusion together do not guarantee validity. No valid argument can have true premise and false conclusion.

6. Some Important Equivalent … from checking the truth table … 1. Double Negation 2. Commutative Law 3. Associative Law 4. Distributive Law 5. 6. Contra-positive 7. 8. 9. De Morgan’s Law 10. 11 11.

7. Are these arguments/reasoning valid or invalid? Argument 1 Argument 2 valid? valid? Argument 3 Argument 4 valid? valid?

8. Argument 1: Are these arguments/reasoning valid or invalid? Premises: If it rains, then the garden is wet. The garden is wet. invalid Conclusion: It rains. Activity: Class Discussion Ex) Premises: If x = 2p, then sin x = 0. sin x = 0. Conclusion: Therefore, x = 2p. Invalid e.g., x = p Showing one counter-exampleis enough for confirming invalid reasoning.

9. Argument 2: Are these arguments/reasoning valid or invalid? Premises: If it rains, the garden is wet. It rains. valid (next page) Conclusion: The garden is wet. Activity: Class Discussion Ex) Premises: If x = 2p, then sin x = 0. x = 2p. Conclusion: Therefore, sin x = 0. Valid? Showing one true examples is not enough for confirming invalid reasoning. You need to show that all possible casesare true.

10. How to investigate validity of the reasoning (argument) Valid? Truth Table No valid argument can has true premise andfalse conclusion. Logic Derivation T T T T T T T F F F T T F F T F T T F F Try to find Counter-Example, then show the Contradiction p Contradiction T F T x q T T T

11. at least one case that is invalid is valid • Assume that there is one case that noone case that Proof of Valid Reasoning by Contradiction Method No valid argument can have true premise and false conclusion. [Using Contra-positive Equivalence] Proof by Contradiction Method • Then show that this is not possible – there is no such case - by (finding) contradiction.

12. Valid Reasoning (Argument) • A reasoning (an argument) is said to be valid if and only if, by virtue of logic, • the truth of the premise P guarantees the truth of the conclusion Q, • if P is true, Q is necessarily/always true, • is a tautology. • In this case, we write • A reasoning that is not valid is said to be invalid.

13. Argument 3: Are these arguments/reasoning valid or invalid? Premises: If it rains, the garden is wet. It does not rain. invalid Conclusion: The garden is not wet. Activity: Class Discussion Premises: If x = 2p, then sin x = 0. x 2p. Conclusion: Therefore, sin x 0. Invalid e.g., x = p, sin x = 0

14. Argument 4: Are these arguments/reasoning valid or invalid? Premises: If it rains, the garden is wet. The garden is not wet. valid (next page) Conclusion: It does not rain. Activity: Class Discussion Premises: If x = 2p, then sin x = 0. sin x 0. Conclusion: Therefore, x 2p. Valid?

15. How to investigate validity of the reasoning (argument) Valid? Try to find Counter-Example, then show the Contradiction Contradiction p T F T q x F T F T Truth Table No valid argument can has true premise andfalse conclusion. Logic Derivation F T T T F T F T T F T F F F T T F T T F T T T F

16. How to investigate validity of the reasoning (argument) Contra-positive Equivalent Argument2 (already proofed) valid? valid valid

17. Rule of Inference Modus Ponendo Ponens Modus Tollendo Tollens valid valid Logical Fallacies • Fallacy of The Converse • Fallacy of The Converse

18. Some Important Implications 1. Modus Ponens 2. Modus Tollens 3. Simplification 4. Addition 5. Modus Tollendo Ponens 6. Hypothetical Syllogism 7. Biconditional-Conditional 8. Conditional- Biconditional 9. Constructive dilemma

19. Logically Draw Conclusions Premises: She does not like A and she likes B. She does not like B or she likes U. If she likes U, then U are happy. Conclusions: She likes who? and Who are happy? Activity: Class Discussion

20. A = She likes A. B = She likes B. U = She likes U. H = U are happy. Premises: She does not like A and she likes B. She does not like B or she likes U. If she likes U, then U are happy. Conclusions: ?

21. ……… (1) Premises are assumes to be true. ……… (2) ……… (3) She doesn’t like A. From (1) with Simplification ……… (4) She likes B. ……… (5) From (2) and (5) with Modus Tollendo Ponens She likes U. ……… (6) From (3) and (6) with Modus Ponens U are happy. ……… (7)

22. Logically Draw Conclusions Premises: If it rains or it is humid, then I wear blue shirt. If it is cold, then I do not wear blue shirt. It rains. Conclusions: What is the weather condition? What color of the shirt I wear? Activity: Class Discussion

23. R = It rains. H = It is humid. B = I wear blue shirt. C = It is cold. Premises: If it rains or it is humid, then I wear blue shirt. If it is cold, then I do not wear blue shirt. It rains. Conclusions: ? Activity: Class Discussion

24. ……… (1) Premises are assumed to be true. ……… (2) It rains ……… (3) ……… (4) From (3) with addition I wear blue shirt. ……… (5) From (1),(4) with Modus Ponens It is not cold. From (2),(5) with Modus Tollens ……… (6) However, we can’t determine the truth value of H. (we don’t know whether it is humid or not.

25. Logically Draw Conclusions Premises:If I am bored, then I go to a movie. If I am not bored, then I go to a library. If I do not go to a movie, then I do not go to a library. Conclusions: Where do I go? Activity: Class Discussion

26. B = I am bored. M = I go to a movie. L = I go to a library. Premises: If I am bored, then I go to a movie. If I am not bored, then I go to a library. If I do not go to a movie, then I do not go to a library. Conclusions: ? Activity: Class Discussion

27. ……… (1) Premises are assumed to be true. ……… (2) ……… (3) ……… (4) From (3) with Contrapositive From (2),(4) with Hypothetical Syllogism ……… (5) By Tertium non datur (Principle of Excluded Middle) ……… (6) From (1),(5),(6) with Constructive dilemma ……… (7) I goes to a movie.

28. Necessary and Sufficient Condition Example) The one who graduates, must pass this course. P: Graduate, Q: the one who passes this course. x What is the relation between P and Q? p x q P is necessary or sufficient condition of Q ? P is sufficient condition of Q. Q is necessary or sufficient condition of P ? Q is necessary condition of P. 49

29. Example of statement usually used in conversation “จะเป็น p ได้ ต้องเป็น q เท่านั้น” Example) P only if Q แต่การที่เป็น q ไม่ได้แปลว่าจะเป็น p โดยอัตโนมัติ p q การที่ไม่ใช่ q นั้น แสดงว่าไม่ใช่ p What is necessary / sufficient condition of what ? P is sufficient condition of Q. Q is necessary condition of P.

30. Example of statement usually used in conversation Example) P if and only if Q q P if Q p P only if Q q p

31. Some Logic: Necessary and Sufficient Conditions (Deductive Reasoning) Implication (Conditional Statement): p  q Note: There is also “inductive reasoning.” p p p q q q q whenever p 52

32. Conditional Statements: If p, then q: PV = mRT (If/Under-the-condition-of/) For a fixed gas and mass of the gas, (if/under-the-condition-of/) and for a fixed temperature: If volume increases, then pressure decreases. If pressure does not decrease, then volume does not increase. Pressure decreases or volume does not increase. For a fixed gas and mass of the gas, and for a fixed pressure: If temperature decreases, then volume decreases. If volume does not decreases, then temperature doest not decreases. Temperature does not decreases, or volume decreases.

33. Real Life Example Objective: by experiment varying P Design Exp: ….. Doing Exp: ….. Result: By the way, the experimental result should be ….? Basic Knowledge Of Mech Material Premises lab conclusion Predicted Result

34. Predicted Result Basic Knowledge Of Mech Material Prediction of Expected Result should have a linear relationship. If … some assumptions … If … some assumptions … V M

35. theory theory Discussion: lab Cause of Error? lab maybe possible inaccurate E Not likely possible Not likely possible maybe possible inaccurate I maybe possible Not likely possible inaccurate L maybe possible Not likely possible inaccurate Position C maybe possible inaccurate load P maybe possible Err in mass of each dish Err in support dish

36. Rule of Inference Modus Tollendo Tollens Modus Ponendo Ponens valid valid - Investigate the validity of argument (reasoning). - Make a theoretical predictions. // Logically draw conclusions. - Hypothesis Testing 60