Basics: Logic, aka "It's illogical to call Mr. Spock logical"

This is another great basics topic, and it's also one of my pet peeves. In general, I'm a big science fiction fan, and I grew up in a house where every saturday at 6pm, we all gathered in front of the TV to watch Star Trek. But one thing which Star Trek contributed to our vocabulary, for which I will never forgive Gene Rodenberry, is "Logic". As in, Mr. Spock saying "But that would not be logical.".

The reason that this bugs me so much is because it's taught a huge number of people that "logical" means the same thing as "reasonable". Almost every time I hear anyone say that something is logical, they don't mean that it's logical - in fact, they mean something almost exactly opposite - that it seems correct based on intuition and common sense.

If you're being strict about the definition, then saying that something is logical by itself is an almost meaningless statement. Because what it means for some statement to be "logical" is really that that statement is inferable from a set of axioms in some formal reasoning system. If you don't know what formal system, and you don't know what axioms, then the statement that something is logical is absolutely meaningless. And even if you do know what system and what axioms you're talking about, the things that people often call "logical" are not things that are actually inferable from the axioms.

Logic, in the sense that we generally talk about it, isn't really one thing. Logic is a name for the general family of formal proof systems with inference rules. There are many logics, and a statement that is a valid inference (is logical) in one system may not be valid in another. To give you a very simple example, most people are familiar with the fact that in logic, if you have a statement "A", then either the statement "A or not A" must be true. In the most common simple logic, called propositional logic, that's a tautology - that is, a statement which is always true by definition. But in another common and useful logic - intuitionistic logic - "A or not A" is not necessarily true. You cannot infer anything about whether it's true or false without proving whether A is true or false.

More....

So when someone says, a la Mr. Spock, that something is logical, the correct thing to do is to whack them in the head with a logic textbook for saying something nonsensical.

reposted from: http://scienceblogs.com/goodmath/2007/01/basics_logic_aka_its_illogical_1.php
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