From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Thursday, January 1, 2009 7:56:51 AM Subject: Re: [math-fun] recent illogical conflation On Thursday 01 January 2009, Fred lunnon wrote:
A is a subset of (not C) B is a subset of (not C)
Therefore, A is somehow "equivalent" to B.
It's rather "the enemy of my enemy is my friend"-like, which is also commonly used.
Mmm... It's very tempting to be dismissive of such reasoning, but the bald fact is that associative "logic" of this nature is far more effective in many real-world situations than simple boolean calculus; and is very often the only tool available in the absence of complete information.
Boolean logic is (something like) the special case of probability theory where all probabilities are 0 or 1. If for "is" you read things like "is thereby more likely to be", those "illogical" statements become valid, although in some cases only to the same extent as (famously) it's "valid" to consider it evidence for "all ravens are black" when you look at a non-black thing and discover that it isn't a raven. That "although" is the real point, of course. It can be true that evidence E makes proposition P infinitesimally more likely, but that doesn't justify someone who says "aha, E, therefore P". -- g _______________________________________________ Edwin T. Jaynes, based on the work of Richard Cox, developed probability theory as an extension of deductive logic, in which propositions have degrees of plasuibility, rather than being true or false. If it is assumed that plausibilities are quantified by real numbers, then Cox showed that, to within isomorphism, these plausibilities obey the usual rules of probability. Richard T. Cox: "The Algebra of Probable Inference" Edwin T. Jaynes: "Probability Theory, the Logic of Science" http://bayes.wustl.edu/ Jaynes is interested in the applicatiion of probability, mainly in the form of Bayes' Theorem, to inference from uncertain knowledge, as is the case in science. He states that he has absolutely no interest in the niceities and paradoxes of the infinite or in the measure theoretic approach to the theory. Gene