Monthly Archive for July, 2018

Monday Solution

Published
by
Steve Landsburg
on July 30, 2018
in Logic, Math and Puzzles
. 14 Comments

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’

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Friday Puzzle

Published
by
Steve Landsburg
on July 27, 2018
in Puzzles
. 10 Comments

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.

Click here to comment or read others’ comments.

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Thursday Solution

Published
by
Steve Landsburg
on July 26, 2018
in Math and Puzzles
. 38 Comments

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:

  • I can prove P
  • It’s tautological that “P implies the moon is made of green cheese”
  • Therefore I can conclude by modus ponens that the moon is made of green cheese.

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’

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Down on the Farms

Published
by
Steve Landsburg
on July 25, 2018
in Current Events, Policy and Politics
. 30 Comments

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’

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Wednesday Puzzle

Published
by
Steve Landsburg
on July 25, 2018
in Puzzles
. 21 Comments

Here’s what you should know about me: 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.)

Here’s what else you should know: Last night, in the course of an imaginary chat with an imaginary Bob Murphy, I found myself admitting out loud that “I cannot prove that there is no God.”

First Puzzle: Can I in fact prove that there is no God?

Second Puzzle: Can I prove that there is a God?

Third Puzzle: Based on the information given, can you determine whether there is a God?

I’ll answer tomorrow, or in a few days depending on how the comments play out.

Click here to comment or read others’ comments.

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The Elephant in the Brain

Published
by
Steve Landsburg
on July 17, 2018
in Books
. 14 Comments

elephantI haven’t been blogging much lately, but John Lanchester’s boneheaded review of Robin Hanson and Kevin Simler’s The Elephant in the Brain seemed to demand a response. But I see that Robin himself has saved me the trouble with his devastating refutation here.

This does remind me that I never fulfilled my intention to review the The Elephant myself. This fell through the cracks because I’d accumulated substantial notes while reading it, lost my notes, decided to search for them before reviewing, and then sort of fell temporarily out of the blogosphere for a while. But it is a genuinely terrific book; my lost notes contain a long list of minor quibbles, but the bottom line is that I learned a lot (about others and about myself) and had a lot of fun along the way. There was never a moment when I wanted to put this book down.

The book, for those who have somehow managed not to hear about it, is about the hidden motives for human behavior — the motives we hide from each other, and the motives we hide from ourselves. The table of contents promises to enlighten us about how hidden motives drive our body language, laughter, conversation, and our choices in consumption, art, charity, education, medicine, religion and politics — and the book delivers on those promises with a heady mix of sparkling logic and striking evidence. There are plenty of fun facts you’ll be repeating at cocktail parties, and plenty of deep insights that will make you smarter forever.

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Letter from an Infantryman

Published
by
Steve Landsburg
on July 8, 2018
in Outrage, Personal and Tragedy
. 11 Comments

armydad

I found this among my father’s papers. He wrote it as a 20-year old infantryman who had been in combat for about six months.

I am struck by the eloquence, and doubly struck that he managed to be eloquent in the medium of pen-and-ink, with no copy/paste/delete and not even any crossouts:

Monday, Jan. 8 (1945)

Dear Mother and Dad:

Well, the new year has arrived and with it, sadly enough, have come no great changes. The war is still being fought, I and millions of other boys are still several thousands of miles away from home and our loved ones, and it almost seems as if there will never be an end to this useless, heart-breaking, killing war.

Whether a man is German, American, or French, he looks just the same when he is wounded, dying or dead. The battlefield bullet is a great leveler; it can make the biggest man very small or the weakest man a hero, but in this war most of the heroes are dead.

We who are actively engaged in defeating the enemy would not hesitate to lay down our arms and surrender if we thought that the people who make the peace will fail to make it permanent. The mere thought that our comrades may have died for nothing, that we may have a brief pause from this war so that we can raise sons to fight another war would cause us many sleepless nights. The last thing one dying soldier said to me was that he was dying on the battlefield so that his son would not.

I may sound very bitter and full of resentment and frankly I am. This war should have been averted in 1918 and the ensuing years, but instead of preventing war, the American people actually encouraged it by ignoring everything that was going on around them. For the sake of all the men who have gone through this hell, we must not let this happen again. We must not have allowed so many of our boys to have died in vain.

I can’t possibly express the resentment these boys feel when they hear about these “Victory in Europe Celebrations”, and when they hear about the lotteries that are held to determine the date of the European victory. Here their own sons are being killed, maimed and crippled for life, and they trouble themselves with such trivial tripe. What is the matter with the American public? Is it entirely aloof to this war?

Perhaps I don’t sound like a twenty-year-old kid anymore, but I’ve seen things that I shall never forget, ghastly things that I shudder to think about. I think that a just punishment for any of these “Victory in Europe Celebration” planners would be to pick up a soldier’s boot on a battlefield and find the foot still in it, or sweat out just one artillery barrage. If they could just realize what is going on they would spend all their spare time praying for the safety of their boys and thanking God that America has been spared everything but an army.

Aside from being a little angry, I’m feeling fine. I’ve received several of your packages and everything is swell. I know that God has been answering your prayers, and he will continue to watch over me.

Love, Norman

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End of an Era

Published
by
Steve Landsburg
on July 8, 2018
in Personal
. 20 Comments

dad2
Norman Landsburg (1924-2018), a survivor of the worst ravages of the Great Depression, a survivor of the trenches in France, where he landed in the wake of the Normandy invasion (and, according to what I think I’ve just learned while going through his papers, was awarded two bronze stars that he never once mentioned to his wife of 68 years or any of his three children), who transcended a series of hard knocks that would have led many to despair and struggled every day, often against mighty odds, to make a better life for his family, succumbed tonight to complications from Alzheimer’s disease.

His bullheadedness was his greatest vice and his greatest virtue. I owe him a couple of good slaps upside the head (not that I ever got one from him, but he deserves them anyway) and eternal gratitude for the way he eased my setting forth and filled my world with possibilities. Neither debt will ever be paid.

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