Friday, 25 October 2013

Triple equivocation: when three wrongs make a right

You've presumably noticed that some people talk obvious drivel. They're not usually much trouble: obvious drivel is easy to refute (or just ignore). What's harder to deal with is people who talk plausible-sounding rubbish: arguments that you know must be wrong somehow, but you just can't see how. For instance:

Philosophers have various names for the different types of fallacious reasoning; they're useful to know about because they can help you spot where an argument has gone wrong. One of the subtlest types of fallacy is called equivocation; this involves using the same word in two different senses, to construct something that looks like good reasoning but is in fact nonsense. Here's a classic example:
  1. A hot dog is better than nothing.
  2. Nothing is better than a nice, juicy steak.
  3. Therefore, a hot dog is better than a nice, juicy steak
What's gone wrong here? The word nothing has been used in two different ways. In (1), it means nothing at all, whereas in (2), it means no possible food. So although each premise is (arguably) correct, and the argument looks valid, the conclusion is false.

The nicest example I have seen is the one William Lane Craig uses to explain equivocation:
  1. Socrates is Greek.
  2. Greek is a language.
  3. Therefore, Socrates is a language.
The word Greek has been used in two different senses here: in the first premise, it's a nationality, and in the second it's a natural language. So, again, two true premises but a seemingly obviously false conclusion.

Now that all seems plain enough. But here's the shock: Socrates, contrary to all expectation, is, in fact, a language. I stumbled across a web site the other day that describes Socrates as "[a] programming language embedded in PLT Scheme that supports advanced separation of concerns using predicate dispatching".

This puts the example in a completely new light. We thought we had a false conclusion because of equivocation on the word Greek; it now turns out that Socrates means two different things here (a philosopher and a programming language); and language is also being used in two different ways (a natural language and a programming language).

So we have two true premises; a true conclusion; and a triple equivocation, on Greek, Socrates and language.

Sometimes, it seems, three wrongs do make a right. Who'd have thought it?

Sunday, 23 June 2013

The Eggy Mattress Puzzle

Excellent little puzzle from Puzzle Man Dave.

You're standing outside a 120-storey building, which has a mattress on the ground outside it. You have two identical eggs; and what you want to know (for no doubt excellent, but irrelevant, reasons) is the highest floor at which you can drop an egg out of the window and have it land on the mattress without breaking. You don't mind getting your mattress a bit eggy in the process, but you need a way of guaranteeing to find the highest floor.

Obviously even if you just had one egg then you could still find out the answer: you'd drop it out of the first floor, and if it didn't break, then you'd go for the second floor, and then the third, until it smashed. That would be the only procedure that would guarantee to discover the answer: if you started by dropping it out of floor five, for instance, and it broke, you'd have no idea whether it would have broken from floor four, and you'd be out of eggs. It works, but in the worst case it takes 120 drops.

But if you have two eggs, you can do better than that. You can guarantee to find out the answer in many fewer than 120 drops.

What's the optimal method, the one that guarantees to find the highest floor at which an egg won't break, but minimizes the number of drops required in the worst case?