Class 8 Team Competitions: Tower of Hanoi. 5 rings on 4 posts. 4 rings on 3 posts. Count Down – working backwards heuristic Goal is to take the last item. 16 items withdrawn 1 – 3 at a time.. 4 million dollar bills as last 4 items – start with 18 items, 1 – 4 at a time. Hobbits and Orcs. 3 Hobbits, 3 Orcs, 3 in the boat. 4 hobbits, 3 Orcs, 2 in the boat.
Class 8 Formal Logic • Do people use formal logic? • Important Terms: • Deductive Reasoning: Problems to which one can apply formal logic and derive an objectively correct solution • Premise: statement of fact taken to be true for the purposes of a logical problem • Conclusion: A statement of fact derived by logical processes. A conclusion is true or false within a problem based on its logical relation to the premises. Whether the conclusion is true in the real world depends on the truth or falseness of the premises
Class 8 Formal Logic • Deductive Reasoning Continued: • Studied in two possible formats: • Conditional Statements; A logical form composed of three statements including an if - then • First premise states, “If condition p is met, then q follows.” • Second premise states whether p or q is true. • Third is a conclusion about p or q as seen in Fig. 10.3 • Syllogisms: Logical form composed of three statements of fact: two premises & conclusion – no if -thn • Inductive Reasoning: Reasoning that allows one to say that a conclusion is more or less likely to be true but does not allow one to say that a conclusion must be true • Conditional Statements • If snow is black, it makes a good hiding place for coal. • Snow is black. • Snow makes a good hiding place for coal. • Syllogism: • All computers are annoyances. • A PC is a computer. • Therefore, a PC is an annoyance. • Inductive Reasoning: • If I cook cabbage, then the house smells funny. • The house smells funny. • I cooked cabbage.
Class 8 Formal Logic - Continued • Brain Structures and Inductive vs. Deductive Reasoning • Goel, et al. (1997) suggest deductive reasoning associated with activation in left interior frontal gyrus and inductive reasoning with broader areas of left frontal lobe and much greater activity in the superior frontal gyrus as seen in Figure. • Osherson, et al. (1998) disagree and found deductive and inductive reasoning associated with supplementary motor area, bilateral cerebellum, right caudate and left thalamus. Probability task alone with cingulate gyrus and right midfrontal gyrus. Deductive task with second visual cortex • More research is need to clarify the discrepancies in these two sets of studies.
Solution: You should turn over the “A” and the “3”. Only 15% of college students get it right. They miss turning over the three. Class 8 Human Success and Failure in Reasoning: Conditional Statements • Philosophers (Aristotle) and psychologists (e.g. Piaget) have assumed that humans are rational and will make correct deductions. However, people may not reason well. • Wason Card problem (Wason, 1968, 1969) The figure shows four cards. Each card has a letter on one side and a digit on the other side. You are to verify whether the following rule is true: If there is a vowel on one side, there is an even number on the other side. You must verify this rule by turning over the minimum number of cards
Class 8 Human Success and Failure in Reasoning - Continued • Concreteness or Familiarity – As shown by Griggs and Cox (1982) – more familiar and/or versions of the Wason problem are more easily solved (72% correct) The cards in front of you have information about four people sitting at a table. On one side of a card is a person’s age and on the other side of the card is what the person is drinking. Here is a rule: If a person is drinking beer, then the person must be over 19 years of age. Select the cards or cards that you definitely need to turn over to determine whether they are violating the rule. Case Based Reasoning: A theory that we reason about problems by remembering similar problems and how they were solved
Class 8 Human Success and Failure in Reasoning - Continued • Pragmatic Reasoning Schemas (Cheng and Holyoak, 1985): Sets of rules defined in relation to goals that can be used to evaluate situations such as permissions or obligations. A key aspect of pragmatic reasoning schemas is that they encourage conclusions that are practical in the real world, as opposed to formal logic, which can lead to conclusions that are technically correct but not useful • Lead to inferences that are practical in solving problems. Logical rules may lead valid inferences that are not much help • Schemas exist for: • Permissions • Obligations • Causations • Prior knowledge activates the appropriate schema to be applied to a problem • Abstract and unfamiliar problems couched in terms of one of these schema are solved more easily.
Class 8 Human Success and Failure in Reasoning - Continued • Evolutionary Perspective –as before : • Humans evolved as social animals • Social networks require us to help or punish other community members • Rules based on this principle are easier to understand • Example – determining who is a cheater but depends on observer’s social perspective as seen in Fig. 10.7 • Cheng and Holyoak (1989) point out subjects solve precaution and permission rules well and this is not part of the social exchange structure • Evolutionary psychologists propose two modules to handle this: • Catching Cheaters • Dealing with precautions • Critics regard this as post-hoc and therefore suspect
Class 8 Human Success and Failure in Reasoning: Syllogisms • Syllogism: A logical form composed of three statements of fact: two premises and a conclusion • Performance is usually bad on these sorts of reasoning tasks due to several factors: • Conversion Errors: An error in dealing with a syllogism in which a person reverses one of the premises. For example the premise reads “All As are Bs” and the participant believes that it is also true that “All Bs are As.” • Conversational implicature: The tendency for people to treat the language of logic as though it has the same meaning as everyday language • Atmosphere : A situation in which two premises of a syllogism are both either positive or negative or use the same quantifier. People are biased to accept as valid a conclusion that maintains the atmosphere • Prior Bias: Real-world knowledge that can influence people’s evaluation of a syllogism. They are more likely to accept as true a syllogism with a conclusion that they know is true and to reject a syllogism with a conclusion that they know is false • No one error or factor is known to be the conclusive factor in most cases of error.
Class 8 Solve “Back To School” Solving Logic Problems • I have handed out to class: • 2 descriptions of how to solve problems. • Five Little Monkeys • Band Camp • We Love Fortunes • All the Groom’s Men • Solutions to the above • Back to school • A Snack at Matt’s Shack • Solutions to 7 & 8. • I worked 7 & 8 in class. Solve “A Snack at Matt’s Shack.” Form teams. Instruct them to come back knowing how to work this level of problem.