Formal Operations and Rationality

1. Formal Operations and Rationality

2. Formal Operations • Using the real vs. the possible • Inductive vs. deductive reasoning • Inductive: Specific to general, generalizing from cases we have seen to infer information about cases we haven't. • Deductive: General to specific, involves deciding what must be true given the rules of logic and some starting set of facts(premises).

3. Perry (1968) • Thinking across Undergraduates: • Early stage: authorities are the keepers of knowledge (thinking tightly coupled with logic) • Later stage: shift toward relativism– equal experts disagree… often leads to skepticism, if equally right no point in developing further knowledge • Last state: Recognize that there is no absolute truth but viewpoints are supported with evidence that aids in your assessment of quality. Problem seeking begins.

4. Inductive Reasoning Problem

5. Deductive Reasoning Problem In a small hospital, there was only 1 room left, which had space for 2 patients. The nurses were instructed to put patients in the remaining room according to these rules: 1. A rubella patient cannot be put with a tuberculosis patient, unless they are both female. 2. If a rubella patient is male, then he can be put with a female emphysema patient. 3. A cancer patient cannot be put with an emphysema patient if one of them is male and the other female. 4. A tuberculosis patient cannot be put with an emphysema patient if they are the same sex. Which of the following patients could be put in the room if a female emphysema patient was already accepted into the room? a) female rubella patient b) female tuberculosis patient c) male cancer patient d) male tuberculosis patient

6. Reasoning • Conditional Reasoning: compares the status of two things, objects, or ideas (e.g., if, then statements) • Syllogistic Reasoning: Specific three statement logical form (e.g., two premises and a conclusion)

7. Conditional Reasoning • Conditional Reasoning: A reasoning problem comparing the status of two things, objects, or ideas. A conclusion is said to be conditional upon the “truth” of the relationship of these premises. • Goal: Determine if the relation between two events is true

8. Conditional Reasoning Cont. Premise: If this is a glasses case, then this is a container. This is a glasses case. Conclusion: Therefore, this is a container.

9. Common Errors in Conditional Reasoning 1. Making only one model of the antecedent and consequence. Not every logical possibility is examined. If she meets her friend, then she will go to the play. She did not meet her friend. Therefore, she did not go to the play INVALID For example: She meets her mom

10. Common Errors in Conditional Reasoning 2. Trying to confirm a hypothesis, rather than trying to disprove it. Looking for positive, but not negative evidence. Selection task. RULE: If a card has a vowel on one side, then it has an even number on the other side. CARDS: E J 6 7

11. Common Errors in Conditional Reasoning 2. Trying to confirm a hypothesis, rather than trying to disprove it. Looking for positive, but not negative evidence. Selection task. RULE: If a card has a vowel on one side, then it has an even number on the other side. CARDS: E J 6 7 ANSWER: E (affirm the antecedent) and 7 (deny the consequent).

12. Syllogisms • Syllogism: A three statement logical form; the first two parts state the premise, which are taken to be true, the third part states a conclusion based on those premises. • Major premise All M are P • Minor premise All S are M • Conclusion All S are P

13. The Propositional Calculus • 1. Affirming the antecedent (Valid) Modus Ponens If this is an apple, then this is a fruit. This is an apple. Therefore, this is a fruit. • 2. Affirming the consequent (Invalid) If this is an apple, then this is a fruit. This is a fruit. Therefore, this is an apple.

14. The Propositional Calculus • 3. Denying the antecedent (Invalid) If this is an apple, then this is a fruit. This is not an apple. Therefore, this is not a fruit. • 4. Denying the consequent (Valid) Modus Tollens If this is an apple, then this is a fruit. This is not a fruit. Therefore, this is not an apple.

15. Example Syllogisms • All scientists are great thinkers. • Einstein was a scientist. • Therefore, Einstein was a great thinker. • All politicians are great thinkers. • Bush is a politician. • Therefore, Bush is a great thinker.

16. Example Syllogisms • All polar bear are animals • Some animals are white • Therefore, some polar bears are white. • All A are B • Some B are C • Therefore, some A are C

17. Example Syllogisms • All polar bears are animals • Some animals have venomous poison • Therefore, some polar bears are poisonous • Confirmation Bias: The tendency for people to search for evidence that confirms their decisions far more than is logical.