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Knowledge and Reality A Lecture Six: Paradoxes I This Week Our look at formal epistemology. Our first look at some paradoxes. Our chance to advance the skills you are expected to acquire. Formal Epistemology Basically this is the way of formalising, using logic, probability theory and mathematics, how to make decisions. It’s applied in numerous areas. Economics(microeconomics and macroeconomics) MAD scheme Formal Epistemology Some choices are easy. If you like Pepsi and hate Coke, what should you decide about which to drink? PEPSI! If you want to go to the pub with your mates but your lover will make your life hell because of it, what do you do? Depends how bad a hell! Formal Epistemology We do some really basic maths to show which option is better. Formal Epistemology Some choices are easy. If you like Pepsi and hate Coke, what should you decide about which to drink? PEPSI! If you want to go to the pub with your mates but your lover will make your life hell because of it, what do you do? Depends how bad a hell! Formal Epistemology We do some really basic maths to show which option is better. Fix a ‘utility’ to picking Coke and Pepsi. If you hate Coke then you’ll lose out. Let’s say -10 Utility Points. If you love Pepsi then you’ll gain. Let’s say 10 Utility Points. The choice is clear – one option gives you 10, the other option loses you 10. Nor does it matter how difficult it is to determine these ‘Utility Points’ Formal Epistemology Some choices are easy. If you like Pepsi and hate Coke, what should you decide about which to drink? PEPSI! If you want to go to the pub with your mates but your lover will make your life hell because of it, what do you do? Depends how bad a hell! Formal Epistemology Going to the pub is worth 20 UP. Say that if your missus/bloke hating you isn’t that bad (say you hate them too, or they’re too mild mannered to be really mean) then it’s -5 UP. Say that if your missus/bloke would do terrible things to you then it’s -25 UP. In the former you do it, in the latter you don’t. And other things can be factored in! If you’d feel guilty at doing it, maybe that’s -20 UP and even in the first case you shouldn’t do it. Formal Epistemology Things can get more complicated. Perhaps if your lover came home he/she would discover you gone and make your life hell to the tune of 25 UP. But you reckon there’s only a 1 in 5 chance of them coming home. If you stay in there’s a 100% chance of losing nothing and gaining nothing. If you go to the pub there’s a 100% chance of you gaining 20 UP and a 20% chance of you losing 25. The calculation is simple: your expected utility gain is: 20 Formal Epistemology Things can get more complicated. Perhaps if your lover came home he/she would discover you gone and make your life hell to the tune of 25 UP. But you reckon there’s only a 1 in 5 chance of them coming home. If you stay in there’s a 100% chance of losing nothing and gaining nothing. If you go to the pub there’s a 100% chance of you gaining 20 UP and a 20% chance of you losing 25. The calculation is simple: your expected utility gain is: 20 – (0.2 x 25 ) Formal Epistemology Things can get more complicated. Perhaps if your lover came home he/she would discover you gone and make your life hell to the tune of 25 UP. But you reckon there’s only a 1 in 5 chance of them coming home. If you stay in there’s a 100% chance of losing nothing and gaining nothing. If you go to the pub there’s a 100% chance of you gaining 20 UP and a 20% chance of you losing 25. The calculation is simple: your expected utility gain is: 20 – (0.2 x 25 ) = 20 – 5 Formal Epistemology Things can get more complicated. Perhaps if your lover came home he/she would discover you gone and make your life hell to the tune of 25 UP. But you reckon there’s only a 1 in 5 chance of them coming home. If you stay in there’s a 100% chance of losing nothing and gaining nothing. If you go to the pub there’s a 100% chance of you gaining 20 UP and a 20% chance of you losing 25. The calculation is simple: your expected utility gain is: 20 – (0.2 x 25 ) = 20 – 5 = 15 Formal Epistemology Easy! But weird things can happen. MAD is one example. The Centipede Game Let’s have another example. I want to play a game. The rules are easy! The Centipede Game Two people, each taking a turn. There are 8 coins. The aim of the game is to get the most coins. On your turn you can: (i) Take a coin and then let it be the other player’s turn. (ii) Take two coins, and then the game ends and we tot up who has the most coins. The game ends when we run out of coins. Centipede Game So you have a principle to maximise utility. What you do is altered by what you think your opponent is going to do. With MAD it wasn’t so bad, but here it fails miserably. It ends up that when the game starts I should just take two coins. Imagine we played it with 100 coins (you can play it with as many coins as you like!) Centipede Game When it’s my go, and there are two coins left, it’d be irrational of me to take one coin and let it be your go. To make the most money (assume that’s my aim, and I don’t give a toss about you personally) I should take both coins and end the game. I’d end up with £51 and you end up with £49, rather than us both having £50. Sound right? Centipede Game But you’re a rational kinda person too. So when we get to three coins on the table, and it’s your go you know what I’m going to do next turn. And it’ll mean you end with £49. So when there are three coins on the table, you should take two coins and end the game. You end up with £50 and I end up with £49. Centipede Game Now go back to a turn earlier, where there are four coins on the table and it’s my go. I know that, rationally, you will end the game the next turn. So at that stage I should take both coins and end the game. So I’d have £50 and you’d have £48. Centipede Game So now think about it again a turn earlier. There are five coins on the table, and it’s your go. You know that when it gets to four coins I’ll end the game and you’ll only have £48. Better to end the game then, and you end up with £49 (and I end up with £48). Centipede Game Do you see where this is going? We can do it again – when there are six coins on the table and it’s my turn, I know you’ll end the game when there are five, so I better end it when there are six. When there are seven coins on the table and it’s your turn, you know that’s what I’ll do so you’d better end it there and then. And so on and so forth. Centipede Game So we can keep going. Rationally, it seems, as soon as the game starts I should take two coins and end it. And that can’t be right can it! I only end up with £2 and you end up with nothing. So, again, if we follow rational thinking through to it’s rational conclusions, we end up with apparently irrational activity. Principles of Formal Epistemology Where we’ve gone wrong is that we must be misapplying the principles we thought governed rational choice. For it is demonstrably irrational to take two coins to begin with! Let’s spend the rest of the lecture looking at another example: Newcomb’s Paradox. Principles of Formal Epistemology In this paradox the combination of two intuitively true principles lead us astray. The first is the principle of maximum expected utility. You should do what is likely to bring about the maximum amount of utility. We’ve had examples of this already. Formal Epistemology Some choices are easy. If you like Pepsi and hate Coke, what should you decide about which to drink? PEPSI! If you want to go to the pub with your mates but your lover will make your life hell because of it, what do you do? Depends how bad a hell! Maximum Expected Utility Even in more complex cases, it seems good to go with it. If you have to gamble and betting it on Horse A is 80% likely to yield £30 and betting on Horse B is 20% likely to yield £200, what should you do? Horse A expected yield: 0.8 x 30 = £24 Horse B expected yield: 0.2 x 200 = £40 Ceteris paribus bet on Horse B And things might not be equal: maybe the Mafia will break your legs unless you get them £30. In which case the utility isn’t in line with monetary gain. But for demonstration purposes we’ll stick with simple numbers and money. Newcomb’s Paradox Imagine there is a room with two boxes in it. One box is transparent: it has £1,000. Another box is opaque: but you know it either has nothing in it, or £1,000,000. You can take the contents of either one box, or both boxes. Newcomb’s Paradox Here’s the twist. There’s some Derren Brown- esque character present. He has predicted, with 100% accuracy, which box you will pick. If you choose two boxes, there will be nothing in the opaque box. If you choose one box, there will be £1,000,000 in the opaque box. So, how many boxes do you pick? Newcomb’s Paradox If you pick one box, and pick the one you can see into, you will definitely get £1,000. If you pick on box, and pick the one you can’t see into, what are the chances of you getting £1,000,000? Well, if Derren is a perfect predicator, the chance is 100%. So you’ll definitely get £1,000,000. Newcomb’s Paradox If you pick two boxes, what’s the chances of you getting what? As Derren is the perfect predictor, you have 100% chance of having nothing in the second box. So you will definitely get £1000. Newcomb’s Paradox One boxer (transparent): £1000 Two boxer: £1000 Newcomb’s Paradox One boxer (transparent): £1000 Two boxer: £1000 One boxer (opaque): £1,000,000 The Maximum Expected Utility principle says to pick just one box – the one you can’t see into. Sound good? Principles of Formal Epistemology The other principle is the Dominance Principle. If you have some choices, and one of them, choice X, has no downsides that the others don’t have Principles of Formal Epistemology The other principle is the Dominance Principle. If you have some choices, and one of them, choice X, has no downsides that the others don’t have and there’s at least a chance that doing X will be good for you then you should do it. Principles of Formal Epistemology Example: Stay at home, or go to the pub. Imagine that staying at home has no upsides or downsides. Imagine that going to the pub has no downsides (I get free beer, I have no problem with walking to the pub etc.) but a high chance I’ll have fun. I should go to the pub! Imagine there’s a small chance I’ll have fun. I should still go to the pub! Imagine there’s a one in a trillion chance I’ll have fun. I should still go to the pub! At least, if there’s no difference whatsoever between being in the pub and being at home. Principles of Formal Epistemology One choice dominates the others. It is clearly better, because it is as good as the other choices and has the benefit of (possibly, or even definitely) being better. How does this work in the Newcomb Paradox? Principles of Formal Epistemology One choice dominates the others. It is clearly better, because it is as good as the other choices and has the benefit of (possibly, or even definitely) being better. How does this work in the Newcomb Paradox? Newcomb’s Paradox Imagine your mother could see inside both boxes, and could advise you on how many to take. And let’s imagine you trust your mother implicitly. If there were £1,000,000 in the opaque box and £1,000 in the transparent box, how many boxes would she tell you to pick? Both of them! You’d come away with £1,001,000 rather than £1,000,000. Clearly, then, you’d be well advised to take the advice of your mother in that case. Newcomb’s Paradox What if there were only £1,000 in the transparent box and nothing in the opaque box? Well then your mother would still tell you to pick both boxes as you’d walk away with £1,000 rather than nothing (as if you picked one box, you’d presumably pick the opaque, empty, box). So in that case you also should follow her advice and take both boxes! Newcomb’s Paradox Here’s the rub. It doesn’t matter whether your mother is there or not. You know that either way she’d tell you to take both boxes. So you know that either way it’s in your interests to pick both boxes. That choice dominates the other choices. So, no matter what happens – you should pick both boxes. Newcomb’s Paradox Think of it another way: it’s too late for Derren to change the boxes now. Whatever you decide what’s in the boxes is what’s in the boxes. So, screw it, you may as well take as much as you can – and take both boxes. Newcomb’s Paradox So NP is a paradox because one principle says to do one thing, and another principle says to do another. MEU: Pick one box. DP: Pick two boxes. And both principles look to be good, rational principles. Something has to go! Something has to be revised! A principle concerning how choices are made has to be altered! Broader Skills This topic builds into the idea of broadening your skills. This topic really is about your own responses to the material. In the others, I’ve stressed that you could go off and read books on it. Not so easy with this one – the books are hard! Broader Skills So much of philosophy is your own response, and your own tackling of the material. You need to go away and ruminate on these issues. Broader Skills If you do read the material, you’ll see that a lot of it is tricky. Lots of maths and tables and stuff. That can be okay. You need to master the art of learning what to read and what to ignore. Especially in a discipline like philosophy where you can be dipping in and out of many other disciplines (do applied ethics, read politics and economics; do metaphysics, read physics journals etc.) Broader Skills You have to try and get out of an article what you can. The complex bits may be so complex you won’t get anything out of them. Just skip them. You can’t argue against them, so you may as well assume that it works. If the author says it, then you may as well believe it! It’s called ‘accepting for the purpose of argument’ Broader Skills There’s even a part of the article you’ve got for which you’ll need to do this… Have fun! Next Lecture Induction.