Lecture 9

Question 1

What is a paradox?

Answer 1

• You have good, rational reasons to choose A and B as true
• But A & B cannot be true at the same time
• Statements that refer to themselves have odd properties: are normally meant to refer to things that are true/false
• e.g liars paradox: for the sentence to be true it has to be false, but if the sentence is false makes it true = no resolution to this paradox
• Wanting to resolve a paradox
• Limitations to do mathematical logical things, as there are also statements that refer to themselves

Question 2

What is the resolution of the St Petersburg Paradox?

Answer 2

• Money has diminishing marginal utility so expected value is not infinite
• If max payout the bank can cover is all the money in the known universe, the expected value of the situation is less than £100 = £100.00 unrealistic

Question 3

What are some complications of the St Petersburg Paradox?

Answer 3

• Diminishing marginal utility of money does not guarantee that the expected value will be finite
• Paradox assumes an implausible premise does not mean conclusion does not follow from premises: paradox might follow logically from premises even if they are implausible

Question 4

What is the Monty Hall Paradox?

Answer 4

• There are 3 doors, two with a goat and one with a car, you pick a door, and monty opens one of the other doors to show a goat
• Options are to stick with choice or switch: switching gives you a 66% chance of winning, and only 33% if you stay
• Has a definite resolution: feels like its 50/50 but wrong = one of the assumptions you make is wrong

Question 5

What are Newcomb’s Paradox?

Answer 5

• Opaque box and transparent box and devil situation
• Premise is not implausible due to studies in neuroscience where brain knows what person will do before person is consciously aware
• Sure thing principle: regardless of the state of the opaque box at the time = taking two boxes seems to get you more money
• Principle of Maximising EU: taking one box has the highest EU
• These two decision principles seem to apply but predict opposite behaviours
• No resolution: as you are predicting both people: self and predictor

Question 6

What is the Allais Paradox?

Answer 6

• Picking amounts of money with different probabilities = but people choose contradicting choices
• If maximising expected value = should pick B and D
• If maximising expected utility = should predict A and then C
• MEU may not be quite right, but works practically mostly
• If Da does not agree with intuitions = worry

Question 7

What is Ellsberg Paradox?

Answer 7

• Pick ball from opaque sack with 30 red balls and 60 black or yellow balls
• When choosing between pulling a red or black = people pick red to eliminate outcome uncertainty
• When choosing between R&Y or B&Y, most people choose B&Y = precise probability of occurring
• Amount you win does not matter on yellow as values are equivalent BUT runs into Sure-thing principle
• Ellsberg Paradox leads to ambiguity aversion
• Inconsistent with MEU as usually formulated
• Normative is still determined

Question 8

What is the confirmation Paradox?

Answer 8

• Starts with formal logic: if A implies B is true, Not B is not A is also true
• Matches intuition e.g if all swans are white, all not white things are not swans
• In science e.g seeing many white swans supports hypothesis that all swans are white, but logically equivalent to not all white things are not swans
• Logically, you can say a black shoe support the hypothesis ‘all black things are not swans’, which is technically the same as the first premise

Question 9

What are the limits of rationality?

Answer 9

Definition of rational behaviour has changed before

Question 10

What is the GRUE paradox?

Answer 10

• AN object is grue if it is green and observed before OR if it is blue and not seen before
• Inductive reasoning justifies that all emeralds are grue and green, but these lead to contradictory future outcomes as they cannot be blue and green after a certain time
• Emphasises choice of language in everyday reasoning

Question 11

What are the differences in concepts?

Answer 11

• Not all concepts are equally good
• Good ones that can actually predict the future = projectable = correctly project past into accurate predictions about future
• Bad ones cannot be used to predict future = non-projectable = do not correctly project past into accurate predictions of the future

Question 12

What is the Old Problem of Induction?

Answer 12

• How to justify the belief that the future will be like the past without being circular
• Admits a practical solution

Question 13

What is the New Problem of Induction?

Answer 13

• Empirical consistency of a scientific theory with past data does not guarantee accurate projectability into the future
• There is no consensus on how a projectability detector should work