10.6.07

the universal tautology conjecture

to begin with, i bring something interesting to the table. this concerns formal logic, one of the most simplistically beautiful things to learn in my field, and one of the most rewarding; i shall be happy to explain the basics of predicate, propositional and fuzzy logic to any member of the set of homo sapiens capable of looking at a negation and asking in all honesty, "What the fuck is this line thing for?", so please ask if it's necessary. logic is a tool everyone should have access to.

there are some statements which are always agreed to be true; these are tautologies, and there are an infinite number of them. for all P, (P \/ ¬P) - "either P is true, or not P is true" - is one; i'm not talking outside the realms of formal calculae here, so "There is too a God" and "Lepht is a narrow-minded bastard" are not.

by the same token, a contradiction is a statement which is always agreed to be false. for all P, (P /\ ¬P) is one of the simplest ones - that is, "both P and not P are true".

there's a problem with this, though. logic comes in different systems; the statements i've showed you are tautological or contradictory under propositional logic. if i quantified them, they would be so under predicate logic, too. but there are many logical systems.

the one that chucks the proverbial spanner in the mail sorting machine and giggles like a heathen in church as it all explodes in a fountain of shredded paper and jiffy-bag shrapnel, so to speak, is fuzzy logic. fuzzy's a new system, and it's all fucked up: the fundamental difference between it and the older systems is that truth, in the old ones, is a binary value.

this deserves further inspection. under a system with binary truth values, truth is one of two things: 0 or 1, T or F. any value P can be either true, or false; it can't be neither, and it can't be both.

fuzzy logic kicks that in the balls: under this new system, truth is a double. so P can have any truth value from 0 to 1: it can have one of 0.009, or 0.56, or 0.987. the biggest ramification of this is, then, that P can belong to the groups "true" and "false" at the same time: that's why it's called "fuzzy": it blurs the once-concrete lines of 0 and 1 into a strange continuum of half-truths and grey areas that can't be used in the same precise way.

enter the propositional tautology. analyse it again, and unfortunately it loses its status: under fuzzy logic, P and ¬P can both have a non-zero truth value, meaning superficially that neither one of them is false.

fuck.

this, friends, is my problem. it's my opinion that for any given tautology or contradiction, Q, a system of formal logic can be devised for which Q does not hold; that's my conjecture, the "universal tautology problem". i haven't tried it exhaustively, of course, but i'd like to write an algorithm that could; it would be non-halting, though, until it found a tautology or contradiction it couldn't invalidate, and so it probably wouldn't finish during my lifetime (in fact my research hypothesis for the experiment involvong such an algorithm would be that the machine eventually stops, the null being that it's a non-halter).

there are many more famous problems in logic (at some point i'll chew through Bertrand Russell's paradox of naive set theory, purely because it's so stunningly obvious post hoc and so easy to miss when you first encounter sets). it's the ability to dissect these kinds of problem that make homo sapiens sapiens so awesomely intelligent.

and i'm not talking about myself. let the first game of Correct Lepht begin: discuss, my friends, discuss!

Lepht

2 comments:

Anonymous said...

"Both P and not P are not untrue." A statement that under fuzzy logic is still always true.

Amirite?

Envy.

Ian said...

@anon: Well, as it happens, there's a logical calculus with exactly four truth values; true, false, both true and false, and neither true nor false. Under this system, "Both P and ~P are not untrue" is not a tautology. Boom.