My return trip from Lubbock to Rochester took almost 36 hours, due to maintenance issues on three separate aircraft. This leads me to wonder whether American Airlines is erring too far in the direction of safety and too little in the direction of getting people where they want to go — perhaps even recklessly so.
Here’s what the back of my envelope shows:
Continue reading ‘Is American Airlines too Reckless?’
If you get accepted to college because you faked being a sports star, pretty much everyone is outraged. I get that.
If you get accepted at college because you are a sports star, almost nobody seems to mind. That’s what I don’t get.
Either way, you’ve climbed the ladder by prevailing in a largely meaningless zero-sum (and hence socially useless) game, thereby signalling a dollop of narcissism together with a few mostly irrelevant talents or advantages. What’s the difference?
This week, I’ll be in Charleston, South Carolina as part of the annual Adam Smith Week celebration at the College of Charleston. You, along with the rest of the public, are invited to attend any or all of my talks:
Thursday, March 7, 1:45PM, “Is the World Overpopulated?”, Wells Fargo Auditorium
Thursday, March 7, 6:00PM, “What Do the Rich Owe to the Poor?”, Wells Fargo Auditorium
Friday, March 8, 12:00, “Why Be an Econ Major?” (discussion with Dr. Doug Walker)
If you attend, be sure to say hello!
Let’s stipulate that:
A. The border wall is stupid.
B. The border wall would cost about $5 billion.
According to Democratic congressional leadership, these reasons suffice to withhold funding for the border wall.
This is a radical new stance for the congressional leadership, which last year rejected the Trump administration’s bid to cut roughly $300 million a year from the budgets of the National Endowments for the Arts and Humanities. Assuming a 3% interest rate, that’s a present value of about $10 billion — enough to fund two border walls. (Take that, you pesky Canadians!).
One could argue that a border wall is not only stupid but a grotesque symbol of xenophobia. One could equally well argue that a National Endowment for the Arts is not only stupid but a grotesque symbol of government overreach and the politicization of everything.
Continue reading ‘A Matter of Perspective’
The President of the United States tweets that his proposed border wall is essentially “the same thing” as a wall built around the Obamas’ house (or presumably anyone else’s house) to keep away intruders.
No, you idiot. There is absolutely no relevant similarity between a wall somebody builds around his own house and a wall that you build between other people’s houses. The effect of a wall around my house, if I had one (and if I controlled the gates), would be to increase my control over who enters my living room. The effect of a border wall would be to decrease my control over who enters my living room.
That doesn’t prove that the border wall is a bad idea. But if the President believes there are good arguments for his pet project, why does he resort to ridiculous analogies that have absolutely zero chance of being taken seriously by anybody on either side of the issue? I’m pretty sure Rex Tillerson had this right.
I hadn’t expected this escalator business (and see also here) to go on so long, but there have been a lot of smart comments, and a lot of smart disagreements, and a lot of smart changing and re-changing of minds, some of it the unavoidable consequence of the fact that we might all be using language a little differently.
So here is the geeky (i.e. precise!) version of what I want to say.
I. Your journey consists of some time on the stairs and some time on the escalator. You rest for a total of one minute, which you can take on the stairs or on the escalator (or split it if you like).
II. Define some constants:
W = your walking speed
V = the escalator speed
L = the distance from your starting point to your destination
Continue reading ‘Escalators (The Geeky Version)’
There were a lot of great comments on my recent post about escalators, but none better than Bennett Haselton’s, which is so good I want to highlight here in a separate post.
I’m going to strip his argument down to make it even simpler, but this is all Bennett’s idea:
A New Puzzle: You’re boarding an escalator precisely at noon. You know that on a normal day, if you walk the entire way, the ride takes exactly ten minutes. But you also know that this is not a normal day, because the escalator is scheduled to be stopped for maintenance beginning at 12:05, and will at that point turn into the equivalent of a stairway. You’re planning to take a one-minute rest from walking at some point along your journey. Should you rest before 12:05, when the escalator is moving, or after 12:05, when the escalator is stopped?
Answer One:Of course you should rest while the escalator is moving, because that way, at least you make some progress while you rest.
Answer One, Reworded: Of course you shouldn’t rest while the escalator is stopped, because then you’ll spend an entire minute not getting anywhere.
Here’s the thing about Answer One: It’s completely wrong. It doesn’t make a bit of difference whether you rest from 12:00 to 12:01 or from 12:05 to 12:06 or for any other minute in between. If you don’t believe me, try an example: Suppose the escalator travels, oh, say, 20 yards per minute and your walking speed is 10 yards per minute. Then if you rest from 12:00 to 12:01, with the elevator moving, you’ll have traveled 160 yards by 12:07, and will continue to gain ten yards per minute after that. If instead you rest from 12:05 to 12:06 with the escalator stopped, you’ll have traveled exactly the same 160 yards by 12:07, and will continue to gain exactly the same ten yards per minute after that.
The Old Puzzle: You’re going to travel on a 100 yard staircase followed by a 100 yard escalator. You’re planning to take a one minute rest somewhere along the way. Should you take it on the stairs or on the escalator?
Answer One: You should rest on the escalator, because at least that way you make some progress while you rest. Or to put this another way, you shouldn’t rest on the stairs because then you’ll spend an entire minute not getting anywhere.
This time Answer One gives the right conclusion. But the reasoning can’t be right, because it’s the exact same reasoning that we applied to the New Puzzle, whereupon that reasoning led us totally astray.
Bennett’s lovely example illustrates as starkly as possible why we must reject Answer One even though it sometimes yields the right conclusion. The reason is that it also sometimes leads to the wrong conclusion. I’ve been trying to argue in the abstract that the logic of Answer One makes no sense; Bennett has done us the awesome service of pointing to a concrete example where that logic leads you inarguably astray.
It also illustrates my other main point: The real reason to rest on the escalator in the Old Puzzle is that resting on the escalator buys you more time on the escalator. Bennett has removed that advantage by giving you exactly five minutes on the escalator regardless of where you rest. In other words, when you cook up an example (like Bennett’s) in which resting on the escalator doesn’t buy you more time on the escalator, the argument for resting on the escalator vanishes in a puff of smoke.
This, incidentally, is related to a cryptic comment of my own on that earlier post, where I replied to an inquiry from Bob Murphy about my observation in an old Slate column that the fundamental confusion arises from measuring benefits in distance instead of time. (I claim that this is, in a sense that might not be entirely obvious, an equivalent description of the problem with Answer One.) In the Old Puzzle, you’re on the escalator for a fixed distance; in Bennett’s New Puzzle, you’re on the escalator for a fixed time. That illustrates exactly the distinction I had in mind, and if I find the time, I’ll write out the details sometime soon.
There are two kinds of people in this world: The first kind wonders why people stand still on escalators but not on stairs. The second kind wonders what’s wrong with the first kind. After all, if you stand still on the stairs you never get anywhere.
But people of the first kind are not usually dumb. I could give you a long list of top-rate economists and mathematicians who have been stumped by this puzzle. But I could also give you a long list of equally smart people who have been stumped by why anybody thinks it’s a puzzle in the first place. It’s come up again several times recently, because I included it in Can You Outsmart an Economist? and because I talked about it on my podcast with Bob Murphy, which generated a small flurry of email from listeners. So let me try once again to explain what’s going on here.
Let’s divide this into two parts: First, what’s the right way to think about this problem? Second, why is it a problem in the first place?
Continue reading ‘Ups and Downs’
Suppose, for the sake of argument, that the following three things are true:
Now my questions:
From the Twitter feed of my brilliant friend (and Usenet legend) Tim Pierce:
This is a true story. The names have been changed and some personal details have been blurred by request.
In the American heartland, there lives a university professor named Fletcher. Like most university professors, Fletcher has collaborated with colleagues from around the world, including my friend Zenobia, who teaches at a college in New England. Their collaboration is relatively recent and has not yet resulted in any publications, though some of their joint work has been posted on the web.
Recently, Zenobia has started to receive (by snail mail) a number of magazines — the sort you might find in dentists’ office, like People and Working Mother — addressed to Fletcher, but at Zenobia’s home address. Fletcher purports to know nothing of how this came about, and Zenobia believes him. Moreover, a number of Fletcher’s other friends, collaborators, acquaintances and relatives have also begun to receive the same sorts of magazines with Fletcher’s name, but their own addresses, on the mailing labels.
The magazine publishers, when queried, have refused to divulge any information about who is paying for these subscriptions.
Zenobia notes that the only possible direct connection between Fletcher’s name and her own home address is that once, on the occasion of the birth of Fletcher’s first child, she used her Amazon account to send him a baby gift. However, there appears to be no similar connection between Fletcher and any of the others who are receiving magazines addressed to him.
This leaves us with two mysteries:
Continue reading ‘Wednesday Mystery’
A commenter in another thread wanted to talk about trolls, so I’m opening up a new post where they’ll be on topic.
The specific trolls I have in mind operate toll-booths (or troll-booths?), both of which you must pass through to get from Hereville to Thereville. The question is whether you, as a traveler, prefer to have both booths controlled by a single troll, or by separate trolls. (This is Problem 12 in Chapter 3 of Can You Outsmart an Economist?.)
(SPOILER WARNING!)
(Note from the proprietor—I am delighted to present this guest post from my correspondent William Carrington, who might or might not have been inspired by the puzzles in Chapter Nine of Can You Outsmart an Economist?. — SL)
Like zombies and Russian spies, there are more economists among us than you might think. This can be dangerous because studies show that economists are more likely than normal people to graze their goats too long on the town commons, to rat out their co-conspirators in jailhouse interrogations, and to show up drunk on their last day at a job. This appears to be both because unethical people are drawn to economics and because economics itself teaches people to be both untrusting and untrustworthy. This feedback loop has led to the creation of famously difficult economists like John Stuart Mill and….well, it’s a long list. Like halitosis and comb-overs, the problem is worse in Washington.
Can you protect yourself against this unseen risk? Sadly, no, as economists often look all too normal and are hard to pick out from the maladjusted crowds that attend us. This is known as the identification problem in economics, and Norway’s Trygve Haavelmo was awarded a Nobel Prize for his work on this issue. Related work by Ken Arrow, also a Nobelist, proved that an infinitesimal group of economists will bollix up the welfare of an arbitrarily large population of otherwise normal people. It’s most disheartening, but I’m here to offer you a failsafe method for identifying economists. You’ll need an old refrigerator.
Continue reading ‘On Backward Induction’
In honor of the forthcoming visit of Glen Weyl to the University of Rochester, I thought I’d offer a post explaining the idea behind one of Glen’s signature policy reforms: quadratic voting.
Suppose we’re going to hold a referendum on, say, whether to build a street light in our neighborhood.
The problem with giving everybody one vote is that (on both sides of the issue) some people care a lot more about that street light than others do. We’d like those who care more to get more votes.
In fact, we’d like to allocate votes proportional to each voter’s willingness to pay to influence the outcome. There are excellent reasons to think that willingness-to-pay is the right measure of “caring”. Those reasons will be evident to readers with some knowledge of welfare economics and opaque to others, but it would take us to far afield for me to get into them here. (For the record, if you’re encountering this measure for the first time, you’re almost surely raising “obvious” objections to which there are non-obvious but excellent rejoinders.) For this discussion, I’m going to take it as given that this is the right way to allocate votes.
Here’s the problem: If I allocate votes based on willingness to pay, people will simply lie. If you’re willing to pay up to $1 to prevent the street light, but know that you can get more votes by exaggerating your passion, that’s what you’ll probably do.
Okay, then. If we want to allocate votes based on willingness to pay, then we have to make people actually put some money on the table and buy their votes, thereby proving that they care. We could, for example, sell votes for $1 each. That way, people who care more will buy more votes and have more influence, as they should.
Unfortunately, that’s not good enough. If you care more about the issue than I do, you might buy more votes than I do — but there’s no reason to think you’ll buy more votes in direct proportion to your willingness to pay. Let’s suppose, for example, that the ability to cast a vote is worth $2 to you and $4 to me. Then I should get twice as many votes as you. But if votes sell for $3, I might buy quite a few, whereas you’ll buy none at all. That’s a lot more than twice as many.
So let’s try again: Instead of selling votes for a fixed dollar amount, we sell them on an increasing scale. You can buy one vote for a dollar, or two votes for four dollars, or three votes for nine dollars — and we’ll even let you buy in tiny fractions, like 1/10 of a vote for a penny. The price you pay is the square of the number of votes you buy. That’s the definition of quadratic voting.
Why the square, as opposed to the cube or the square root or the exponential? There really is something special about the square. To appreciate it, try an example: If a vote is worth, say, $8 to you, you’ll keep buying additional votes as long as you can get them for less than $8 each, and then stop. With quadratic voting, one vote costs you a dollar. You’ll take it! A second vote costs you an extra $3 (bringing the total to $4). You’ll take that too! A third vote costs you an extra $5, a fourth costs you an extra $7, and a fifth costs you an extra $9. So you’ll buy 4 votes and then stop. You can similarly check that if a vote is worth $24 to your cousin Jeter, Jeter will buy twelve votes and then stop. Jeter cares three times as much as you do, and he buys three times as many votes. And with a little calculus, you can check that if Aunt Murgatroyd’s vote is worth four or five or nine or twenty times more to her than your vote is to you, she’ll buy exactly four or five or nine or twenty times as many votes as you do. That’s exactly what we wanted. In that sense, this voting scheme works — and, except for minor variations, it’s the only scheme that works.
Continue reading ‘Quadratic Voting: A Pre-Primer’
I am delighted to announce that Glen Weyl, of Microsoft and Princeton University, will be visiting us at the University of Rochester on Tuesday, November 6 (election day!) and will speak on the topic Radical Markets: Uprooting Capitalism for a Just Society, based on the (extremely) provocative and original book of almost the same title, co-authored by Weyl and Eric Posner. The lecture is free and open to the public. The event will be held in Wegmans Hall, room 1400, and will begin promptly at 6:30PM.
Is it rational to play Megamillions, with the current jackpot of a billion dollars or so?
Yes and no. Here are three questions:
There is no irrational answer to any of these questions. Some people like to play the lottery; some don’t. Some people are safety fanatics; some aren’t. Some, but not all, love huge risks with huge potential payouts.
But there is such a thing as an irrational pattern of answers. I know there are millions who answer yes to question 1, because I see them buying tickets. I’m guessing that almost all of those people would answer no to question 2. And I’m guessing that a fair number of those would answer no to question 3 as well. Perhaps you”re one of that fair number. If so, I declare you irrational.
Ten Megamillions tickets give you about a one in thirty-million chance to win the jackpot. One in thirty-million is also pretty close to the chance you’ll be struck by lightning in the next seven weeks. If you’re willing to buy the lottery tickets but not immunity from the lightning, you’re telling me that winning the jackpot means more to you than staying alive. So you should go for the coin flip in question three. If you didn’t, I declare you irrational.
Well, so what? Some economist called you irrational. Why should you care?
You should care because this is exactly the kind of irrationality that will allow me to take all your money. Keep reading to see why.
But first, let’s acknowledge that I’ve ignored a few things, like the possibility that you’ll win one of the lesser prizes in the lottery. I’m assuming that, compared to the jackpot, those are negligible as reasons to buy a ticket. (This is consistent with the observed fact that a whole lot of people buy tickets only when the jackpot is large.) If you don’t like that assumption, we can avoid it by tweaking the numbers in the questions, in ways that I strongly suspect won’t change most people’s answers.
Now let’s get to the fun part where I take all your money. Suppose you’ve answered yes/no/no to the three questions above. Then I’ll rig up the following experiment: I fill a bag with 30 million white balls, one black ball, and one blue ball. I plan to pull a ball from the bag, but before I do, I’ll make you two offers. You can take them or leave them; it’s entirely up to you.
You’ve already told me that you’d pay $20 for ten Megamillions tickets, giving you a one in thirty-million chance at roughly a billion dollars. So of course you’ll happily go for A. And you’ve already told you that it’s not worth it to you to give up $20 to avoid a one in thirty-million chance of electrocution. So of course you’ll also happily go for B.
So: You (voluntarily) give me $20 and accept my $20. So far we’re even. And, I might add, you’re very glad to be playing. You got two things and you wanted both of them.
Now the drawing. Most of the time I’ll draw a white ball, and nothing happens. But occasionally I’ll draw a blue or a black. When I do, I’ll tell you (perfectly honestly, because I’m honestly exploiting your irrationality, not trying to trick you) that I’ve drawn a non-white ball. There’s a fifty-fifty chance it’s black, in which case you will die, and a fifty-fifty chance it’s blue, in which case you win the billion. You are effectively facing the very coin flip that you told me in Question 3 that you prefer to avoid. Of course, then, you’ll gladly pay me a few bucks to call everything off. Now I’m ahead.
We might have to play this game several million times before I draw a non-white ball and take your money, but as long as you’re committed to your stated preferences, you should be willing to play several million times — or to let me set my computer to playing a virtual version for us several million times per hour. In a few hours, I’ll win some money from you. In another few hours, I’ll win more. And so on till you’re broke.
When an economist calls you irrational, it almost always means that if you follow through on your stated preferences, a sufficiently clever opponent can take all your money, leaving you smiling along the way. It’s worth being alert to such things.
If you thought this was fun, you should take the ten-question Irrationality Test in Chapter Six of Can You Outsmart an Economist?. Come back to the blog and let me know how you did.
This Monday (October 15), I’ll be in at Webber International University in Babson Park, Florida, talking on the subject “What Do the Rich Owe to the Poor?”. The talk is at 7pm in the Yentes Conference Center. Please join us if you’re in the area!
Somewhere on my shelves, there is a math book with a page very like the following:
I know this because I remember seeing it (or at least I think I do), but I can’t quite remember which book it’s in.
Fortunately, I didn’t try to steal this joke for Can You Outsmart an Economist?, because it turns out there’s an actual erratum in the main text. It comes in Chapter 16 where a series of problems leads the reader to discover the basics of option pricing. The arithmetic in those problems is all correct, except for one thing: In order to keep the math easy, I assumed an interest rate of 50%. But with an interest rate that high (and given the other assumptions in the chapter), nobody would be mucking around with buying options in the first place; we’d all just be putting our money in the bank and getting rich in a hurry.
So what I should have done is assumed an interest rate of 20%. If you’ve got a copy of the book, you should pencil in the following changes:
My new book is now on sale! Readers of this blog will recognize some but not nearly all of these 100+ puzzles (146, actually, by my count). If you’ve enjoyed my puzzle posts, you’ll probably enjoy these extended discussions of some past puzzles, and the many more that are entirely new. Most of these puzzles are designed to teach important lessons about economics, broadly defined to encompass all purposeful human behavior. All of them are also designed to be fun.
Once you’ve had a look, please don’t hesitate to share your opinions right here on the blog — or better yet (especially if your opinions are positive!) don’t hesitate to share them on Amazon or on Goodreads.
Or, if you’d prefer to taste the milk before you buy the cow, here is the introduction, absolutely free of charge.
You can read a few advance reviews here. And remember, the more copies you buy, the sooner I’ll write the sequel.
You’re a policymaker in a country where people buy widgets that are produced both at home and abroad. You can set (separate) excise tax rates on domestic production and imports. (The tax on imports is, of course, what we usually call a tariff.) What tax rates should you set?
The Economics 101 answer makes two assumptions:
1. You care only about the economic welfare of your citizens (and not at all about foreigners).
2. You can’t affect foreign prices (i.e. your country is a negligible portion of the world market for widgets). The fancy way to say this is that the supply of imports is perfectly elastic.
From these assumptions, it follows that both tax rates should be zero. In fact, we can relax assumption 1) and allow you to care as much as you want about the welfare of foreigners; the conclusion doesn’t change.
But suppose we relax these assumptions in a different way:
1A. You care about both the economic welfare of your citizens and (separately) about the tax revenue earned by your government. (I continue to assume, however, that you don’t care about foreigners.)
2A. The foreign supply curve might not be perfectly elastic. Contrary to the Economics 101 assumption, this gives you some market power that you might want to exploit. (I continue to assume, though, that you take the foreign supply curve as given. In particular, this means that your policies do not affect foreign tax rates, so I am assuming away things like retaliatory tariffs.)
Now what’s your best policy? I can’t answer that because you have two competing goals (economic welfare and tax revenue) and I don’t know how much weight you put on one versus the other. But surely if I can show you that Policy A delivers on both goals better than Policy B, you’ll want to reject Policy B. The existence of Policy A leads me to call Policy B inefficient, and surely you’ll want to reject any inefficient policy.
So which pairs of tax rates are efficient?
Continue reading ‘Efficient Tariffs’
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I keep hearing that the matter of Brett Kavanaugh is “just like Clarence Thomas and Anita Hill all over again”.
Seriously?
Point the First: Clarence Thomas stood accused of boorishness. Brett Kavanaugh stands accused of violent attempted rape. If all the accusations against Thomas were true, he deserved an elbow to the ribs. If the accusations against Kavanaugh are true, he should probably be in jail.
To suggest that there is even a rough equivalence here is sheer madness.
Continue reading ‘Kavanaugh v. Thomas’
First: Neither you nor I know who’s telling the truth, who’s honestly misremembering, who’s dissembling, and who’s doing some combination of all three. But if we want to think about what should happen next in the real world, it pays to think first about what should happen next in a hypothetical world where we can somehow be sure that Professor Ford’s account is 100% accurate. In that world, we need to ask this question: Should a night of bad behavior at age 17 be punished by a career derailment at age 53 (assuming there’s been no punishment in the interim)?
The answer, of course, depends on the benefits and costs of that punishment.
First the benefits: Punishments are beneficial when they deter other bad behavior. So we should embrace this punishment only insofar as we believe that some future 17 year old boy will be deterred from committing sexual assualt by the prospect of a career derailment 35 years down the line. How big is that deterrent effect? Not only do I have no idea; I also have no idea how to start forming an idea. That is, I can’t think of any good empirical strategy for measuring such a deterrent effect. We have good data on the deterrent effect of imprisonment, of capital punishment, and of fines — but not, as far as I know, on the deterrent effect of long-delayed career consequences. (I’ll be very glad if any reader can prove me wrong about this.) I do, however, have a guess. My guess is that for boys of the social milieu that Kavanaugh came from, this effect could loom pretty large. True, no 17 year old really expects to be nominated to the Supreme Court some day, but plenty of 17 year olds at prep schools have lavish dreams of future success, and seem to care quite a bit about preserving that future. If my guess is right, the benefits of killing this nomination might be pretty big.
Continue reading ‘Quick Thoughts on Kavanaugh’
Having recently dealt with some of the same customer service issues at both Vanguard and Fidelity, I can make these recommendations with confidence:
1) Keep your money at Vanguard.
2) If you’ve made the unfortunate mistake of keeping some of your money at Fidelity and you ever need customer service, and if your customer service representative turns out to be named Cameron Marcil, hang up immediately and try again.
Two decades after hiring Paul Krugman, the New York Times has doubled down by hiring the venomous Sarah Jeong, whose old tweets include the following rhetorical question:
Are white people genetically disposed to burn faster in the sun, thus logically being only fit to live underground like groveling goblins?
According to Jeong’s supporters, the tweet needs to be read in context — it was, you see, intended as a parody of Andrew Sullivan’s audacious piece in New York magazine, advocating research — or at least opposing the suppression of research — into racial differences in IQ.
I’m all for parody. I’m all for taking other people’s logic (and my own!), pushing it to its limits, seeing where it leads, and thereby calling attention to its weaknesses. And I am outraged when authors engaged in this enterprise are taken out of context. If I say “X”, and if “Y” is both analogous to X and clearly outrageous, then Sarah Jeong or anyone else ought to be able to tweet “Y” by way of making fun of me, without having to face down a gang of yahoos accusing her of believing “Y”.
But that’s not what this is about. Because — and here is the crux of the matter — the analogue to
is
which is not at all the same thing as
The problem here is not that Sarah Jeong believes white people are fit only to live underground like groveling goblins. (I feel pretty confident, in fact, that she believes no such thing.) The problem here is that she is attempting to refute Andrew Sullivan’s logic by writing down an analogy (so far so good) and then, having done so, tacking on the phrase “being only fit to live underground like groveling goblins”, which in no way reflects anything Andrew Sullivan said, and which Sarah Jeong pulled out of her ass.
Continue reading ‘In the Matter of Sarah Jeong’
The answer to Friday’s puzzle is YES. If I am a logic machine who only states what I can prove, and if I say “If I can prove there is no God, then there is no God”, it does follow that I can prove there is no God.
Once again, it was our commenter Leo who got this first (in Friday’s comment thread, graciously rot-13’d).
As with Thursday’s solution to Wednesday’s puzzle, there are two key relevant background facts:
A) An inconsistent system can prove anything.
B) A sufficiently complex consistent system cannot prove its own consistency. (This is Godel’s second incompleteness theorem.)
Here’s the logic:
1) I’ve asserted that “if I can prove there is no God, then there is no God”. We know that I assert only things that I can prove. Therefore I can prove this assertion.
2) That means I can also prove the equivalent assertion that “if there is a God, I cannot prove otherwise”.
3) Therefore, if I take my axioms and add the axiom “There is a God”, then I can prove that there is something I cannot prove.
4) Therefore, if I take my axioms and add the axiom “There is a God”, then I can prove that my axiom system is consistent (by Background Fact A.)
5) Therefore, if I take my axioms and add the axiom “There is a God”, my axiom system is inconsistent. (Because only an inconsistent system can prove its own consistency — that is, Background Fact B.)
6) Therefore the statement “There is a God” must contradict my axiom system.
7) This can happen only if my axiom system is able to prove that “There is no God”.
So yes, I can prove there is no God.
Continue reading ‘Monday Solution’
A followup to Wednesday’s puzzle:
The assumptions are the same as on Wednesday:
I am basically a logic machine. There are certain axioms that I believe, and I never say anything out loud unless it can be deduced from those axioms via the rules of logic. (Fortunately, I can talk about many things, because my axioms include everything from the usual axioms for arithmetic to a rich set of beliefs about ontology, ethics, psychology, and everything else I care about.)
Now I’ve found myself in a whole new imaginary conversation with the same old imaginary Bob Murphy. This time I found myself saying out loud that “If I can prove there is no God, then surely there is no God.”
The Puzzle: Can I in fact prove there is no God?
Solution forthcoming on Monday.
Here are the answers to yesterday’s puzzle. The first correct solution came from our commenter Leo (comment #18 on yesterday’s post).
The assumptions of the problem were: Everything I say out loud can be deduced from my axioms. My axioms include the ordinary axioms for arithmetic, among other things. And I recently said out loud that “I cannot prove that God does not exist”.
The questions were: Can I prove there is no God? Can I prove there is a God? And is there enough information her to determine whether there actually is a God?
The answers are yes, yes and no: Yes, I can prove there is no God. Yes, I can also prove there is a God. And no, you can’t use any of this to determine whether there is a God.
To explain, I’ll use the phrase “logical system” to refer to a system of axioms sufficiently strong to talk about basic arithmetic (and perhaps a whole lot of other things), together with the usual logical rules of inference. It’s given in the problem statement that I am a logical system.
Here are two background facts about logical systems:
A. An inconsistent logical system can prove anything at all. That’s because it’s tautological that if P is self-contradictory, then any statement of the form “P implies Q” is valid. If I’m inconsistent, that means I can prove at least one statement (call it P) that’s self-contradictory. Then if I want to prove, say, that the moon is made of green cheese, I note that:
B. No consistent logical system can prove its own consistency. This is Godel’s celebrated Second Incompleteness Theorem.
Now here’s the argument:
Continue reading ‘Thursday Solution’
Suppose there’s a guy in your neighborhood who routinely harasses strangers on the street, calling them ugly names, maybe threatening them with violence, but always stopping short of anything that’s actually illegal.
You consider this bad behavior, so you work to pass some laws that will discourage it. Maybe you criminalize the behavior; maybe you tax it.
The new laws turn out to be somewhat effective. The guy tones it down. He still harasses people, but only half as much.
Question: Do we owe this guy something? Should the taxpayers cut him a check so he won’t feel so bad about having to rein himself in?
I’m going to go out on a limb and guess that most of you will answer “no”.
Here’s why I ask:
The President of the United States believes that under current circumstances, much international trade is a bad thing and ought to be discouraged. Unfortunately, there’s a bunch of farmers out there who have been behaving very badly (i.e. trading with foreigners) and the law hasn’t done much to stop them. So the President has expanded the scope of the law to punish this bad behavior via tariffs. And then he’s turned right around and announced a plan to compensate the bad guys.
Continue reading ‘Down on the Farms’
