Occasional reflections on Life, the World, and Mathematics

Posts tagged ‘logic’

Quoth the raven, Never Trump

Carl Hempel famously crystallised an obstruction to the formalisation of inductive reasoning as the “Raven paradox”: Suppose I am an ornithologist, concerned to prove my world-shaking hypothesis, “All ravens are black”. I could go out into the field with binoculars and observe ravens. Suppose that over the course of the week I see 198 black ravens, 0 white ravens, 0 green ravens, and so on. These are strong data in favour of my hypothesis, and my publication in the Journal of Chromo-ornithology is assured. (And if they turn it down, I’ve heard there are numerous black studies journals…) But it gets cold out in the field, and sometimes damp, so I could reason as follows: “‘All ravens are black’ is equivalent to ‘all non-black objects are not ravens’.” And in my warm and dry study there may be no ravens, but there are many non-black objects. So I catalogue all the pink erasers and yellow textbooks and white mugs, and list them all as evidence for my hypothesis.

The status of this charming story as a paradox depends on the belief that no one would actually make such an inference. Behold, the president of the United States: Last week the special prosecutor for matters related to Russian interference with the 2016 US election released an indictment of 13 Russians. None of them had worked with the Trump campaign. Trump’s response:

In other words, while it is proving too difficult to collect proof of the contention “No anti-American voter fraud was performed by Trump,” he is collecting evidence that “There were actions not performed by Trump that were anti-American voter fraud.”

Kettle logic on Fox News

Most Republican leaders, in their concern to defend the president from accusations of racist over his terming African nations and Haiti “shithole countries” and saying “get them out”, have resorted to one of two strategies:

  1. Fake news. He didn’t say it, and it’s outrageous to suggest that he did.
  2. Harsh but true. He did say it, and it shows how forthright and unconcerned he is with liberal pieties.

Neither is entirely satisfactory. It is natural, then, that a Fox News correspondent, in the spirit of Freud’s “kettle logic“, combines the two:

I think it’s either fake news or if it’s true, this is how the forgotten men and women in America talk at the bar.

The bar is the new locker room. It’s kind of weird, though, when the best defense for the president’s behaviour is, he’s talking in the formal setting of a negotiation with senators the way even average uneducated Americans would only talk in a private setting when somewhat inebriated.

The unexpected autocracy

One of my favourite logic paradoxes (does everyone have favourite logic paradoxes?) goes by the name of The Unexpected Hanging. There are numerous versions, but a standard story is: A man has been condemned to death for some crime. The judge tells him, “Today is Monday. You are to be hanged at noon some day in the next week, but you will not know until the morning of the day of the hanging which day it will be.” The man then reasons, it can’t be Sunday, because if I haven’t been hanged by Saturday noon, I’ll know it must be Sunday, which would contradict the judge’s order. Since it can’t be Sunday, if we get to Friday afternoon, I’ll know it must be Saturday. Again a contradiction. So it can’t be Saturday. Working backward in this way, he is confident that he cannot be hanged at all. But then Thursday dawns, and he is hanged, and he never anticipated it.

I was thinking about this, particularly in the light of this comment by Josh Marshall:

One thing we can say in Donald Trump’s favor, there was no bait and switch. They told us they would do all of this and more.

It’s true, and I’m not surprised. And yet… Trump did say he would ban Muslims. He would build a wall. He would ban abortion. He would revoke the Affordable Care Act. And yet, at the same time, he was saying over and over again, I’m going to be unpredictable. I won’t say what I’m really going to do. More than that, his whole demeanor suggested that you couldn’t believe the specifics of what he was saying. So, in the end, he does exactly what he said he would do, and it actually is somewhat surprising. (more…)

The tortoise and Achilles enforce property rights

I’ve commented before about how Lewis Carroll’s “What the Tortoise said to Achilles” identifies a paradigmatic gap between principles and action, which may be summarised as “Yeah, what are you going to do about it?”
I was recently reading Jerome K. Jerome’s brilliant comic memoir “Three Men in a Boat”, which reads like a lost work of Mark Twain, if Mark Twain had been a Victorian English dandy (as, I suppose, he almost was in his later years). There was this account:

We stopped under the willows by Kempton Park, and lunched. It is a pretty little spot there: a pleasant grass plateau, running along by the water’s edge, and overhung by willows. We had just commenced the third course—the bread and jam—when a gentleman in shirt-sleeves and a short pipe came along, and wanted to know if we knew that we were trespassing. We said we hadn’t given the matter sufficient consideration as yet to enable us to arrive at a definite conclusion on that point, but that, if he assured us on his word as a gentleman that we were trespassing, we would, without further hesitation, believe it.

He gave us the required assurance, and we thanked him, but he still hung about, and seemed to be dissatisfied, so we asked him if there was anything further that we could do for him; and Harris, who is of a chummy disposition, offered him a bit of bread and jam.

I fancy he must have belonged to some society sworn to abstain from bread and jam; for he declined it quite gruffly, as if he were vexed at being tempted with it, and he added that it was his duty to turn us off.

Harris said that if it was a duty it ought to be done, and asked the man what was his idea with regard to the best means for accomplishing it. Harris is what you would call a well-made man of about number one size, and looks hard and bony, and the man measured him up and down, and said he would go and consult his master, and then come back and chuck us both into the river.

In other words, you can accept the principles of property rights as far as the most devoted libertarian would wish to push them, but they still get you nowhere without a plan of action to enforce them.

Association by guilt

Guilt by association — you’re friends with a terrible person so you must also be one — is generally recognised as a pernicious logical fallacy. But what should we call this comment by Israeli Foreign Ministry spokesman Paul Hirschson, explaining why Norwegian trauma surgeon Mads Gilbert has been banned from returning to Gaza after he made critical remarks about the Israeli military activities this past summer? Dr Gilbert, he opined, is

not on the side of decency and peace and he’s got a horrible track record. I wouldn’t be surprised if his acquaintances are among the worst people in the world.

In other words, he’s a terrible person, so I’m sure his friends are too. Is this association by guilt?

Logical crisis in Parliament

One of my favourite logical paradoxes (What, doesn’t everyone have a favourite logical paradox?) is the Unexpected Hanging. I first read about it from Martin Gardner, but you can read a summary here. A recent article in the Oxford Times, about the career plans of a local MP, raised a corresponding problem in the political realm:

Conservative Tony Baldry MP has announced he will stand down at the next General Election.

In a statement Sir Tony said: “One of the consequences of now having five year fixed term Parliaments is that if I succeed in being re-elected at the forthcoming general election, given my age, most people will assume that Parliament will be my last.

“I think this creates a danger that I may be unable to be as effective as I would wish to be; and that the constituency will be distracted from more important issues by the need to choose my successor.”

We might summarise his argument as beginning with three axioms:

  1. It is irresponsible for a politician to stand for parliament if the strong expectation is that this would be his or her last term.
  2. There is a strong expectation that someone will not stand for parliament past age 70.
  3. Terms are now fixed at five years.

Consider, now, the position of a responsible politician, whom we will call Tony65, who who will be, shall we say, 65 years old at the next election. He could stand for parliament now, but he will be 70 at the next election (axiom 3), and so the strong expectation is that he will not stand at the next election (axiom 2). Hence, by axiom 1, and the fact that he is responsible, he should not stand in this election.

But now consider Tony’s younger colleague; call him Tony60, who will be 60 years old at the next election. If Tony60 is elected, he will be 65 at the following election. As a responsible politician, like Tony65, he will stand down at that age. Thus this would be his last term, so by axiom 1 he will not stand at this election either.

By the same reasoning, Tony55 will not stand, since we know that he won’t stand again when he is 60. And so, we can keep backing it up to Tony20, who also can’t stand, because he won’t stand again when he is 25. (18 is the minimum age for election to the House of Commons.) Thus, we have proved:

Theorem: No responsible politician can stand for election to the House of Commons.

This theorem does not apply, for obvious reasons, to the House of Lords, so there will still be space for responsible (if unelected) government.

 

 

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