Stardate 20020929.1157

(On Screen): In such exposure as I've had to pomo ideology (as little as possible, because I don't want to get any of it on me, EEeeeww!) I've gotten the impression that part of it includes an idea that logic is a cultural artifact, local and not global. I actually had someone say that to me in a discussion last year in another forum. But his overall expression was less than articulate, with badly constructed sentences and poor grammar, and I conjecture that I was dealing with a student who had an imperfect understanding of what he was being fed. Nonetheless, this idea that logic is not universal keeps springing out at me whenever I encounter postmodernist arguments.

So far as I know, logic is a form of mathematics, and partakes of the characteristics of mathematics. Its scope is universal, and it doesn't cease to work just because you deny it. Yet it is true that there is more than one kind of logic. I know of at least three.

The earliest form of logic was the syllogism, that exercise in pedanticism we encounter in Greek philosophy. It was reductionist, and is easy to understand. But it's also not very flexible, and trying to do any substantial amount of reasoning with it is painful, sort of like trying to program a Turing Machine to do something nontrivial.

In the 19th century, there was a form of logic which was investigated quite heavily which was sort of a way of representing Venn Diagrams in language. This was deliberate; they were looking for an isomorphism. The idea was to frame the problem in the appropriate way, convert all the statements into operations on an appropriate Venn Diagram, work out the result there, and convert the conclusion back again.

It was really intended to be a more generalized form of syllogism, based on three possible relationships between two sets: all A is B, some A is B, no A is B. Syllogism dealt with the same thing, but where syllogism never dealt with more than two sets at once, the new logic (whose formal name I don't know) tried to generalize to permit statements about the relationship between an arbitrary number of sets at once.

The Reverend Charles Dogdson is best known to most of us as the author of "Alice in Wonderland" and "Through the Looking Glass" under the nom de plume "Lewis Carroll". It's not quite as well known that Dogdson was a professor of mathematics at Oxford for 26 years. (The Alice books are sprinkled with jokes which are hilarious to mathematicians and those with a mathematical bent. Alice's conversation with the White Knight is a classic, full of all kinds of sly jokes about logic and mathematics that go right over the heads of laymen.) He was among those who were fascinated with this new form of logic, and he wrote two books about it and began a third.

What he was trying to do was to popularize it, so he wrote "Symbolic Logic" and "The Game of Logic" for kids, to teach them about it and in hopes of making it fun. But some of the advanced example problems he provides shows why it is that this kind of logic never really caught on:

(1) A logician, who eats pork-chops for supper, will probably lose money;
(2) A gambler, whose appetite is not ravenous, will probably lose money;
(3) A man who is depressed, having lost money and being likely to lose more, always rises at 5 a.m.;
(4) A man, who neither gambles nor eats pork-chops for supper, is sure to have a ravenous appetite;
(5) A lively man, who goes to bed before 4 a.m., had better take to cab-driving;
(6) A man with a ravenous appetite, who has not lost money and does not rise at 5 a.m., always eats pork-chops for supper;
(7) A logician, who is in danger of losing money, had better take to cab-driving;
(8) An earnest gambler, who is depressed though he has not lost money, is in no danger of losing any;
(9) A man, who does not gamble, and whose appetite is not ravenous, is always lively;
(10) A lively logician, who is really in earnest, is in no danger of losing money;
(11) A man with a ravenous appetite has no need to take to cab-driving, if he is really in earnest;
(12) A gambler, who is depressed though in no danger of losing money, sits up till 4 a.m.;
(13) A man, who has lost money and does not eat pork-chops for supper, had better take to cab-driving unless he gets up at 5 a.m.;
(14) A gambler, who goes to bed before 4 a.m., need not take to cab-driving, unless he has a ravenous appetite;
(15) A man with a ravenous appetite, who is depressed though in no danger of losing money, is a gambler.

The idea then is to take all this stuff and try to come to some conclusion about the relationship between logicians and pork-chops and cab-driving and sleeping habits and gambling.

The difficulty with it is twofold. First, it requires that a problem be stated in terms which may not be intuitive, and may not even be accessible. The kinds of statements you see above don't represent the way we really observe problems; it's a synthetic construct which was actually created from the solution diagram. It doesn't represent anything remotely like a real-world problem. It also posits correla