Re: Programming is the Engineering Discipline of the Science that is Mathematics
Date: 9 Jun 2006 12:27:33 -0700
> Keith H Duggar wrote:
> > vc wrote:
> > > Keith H Duggar wrote:
> > > > Probability theory as a generalization of logic is
> > > > useful
> > >
> > > PT cannot be 'a generalization of logic' because PT
> > > 'connectives' (+/*) are not truth functional.
> > First, if you don't believe that PT can be seen as a
> > generalization of logic, then I have a simple
> > question. In limit of all probabilities being either 0
> > or 1, what does PT reduce to?
> Let P(A and B ) = P(A)*P(B) where P stands for
> probabilities of respective events. Please 'reduce' the
> above and tell when A and B is true, i.e. when P(A and B)
> = 1.
I assume by "and" you mean conjunction? Also do you realize
that the "*" you wrote above is /not/ a connective? (which
you claimed before it now seems). Here is the reduction:
A : B : A and B
0 : 0 : 0
0 : 1 : 0
1 : 0 : 0
1 : 1 : 1
> > Second, do you understand what "generalization" means?
> > Would you claim that the gamma function is /not/ a
> > generalization of the factorial because it is not
> > limited to naturals?
It's relevant to making sure we agree on what generalization means. So perhaps you can answer the question next time around? Is gamma a generalization of factorial?
> > Third, +/* are not the connectives of PT. PT uses the
> > same connectives as logic: conjunction, disjunction, and
> > negation (whatever symbol you decide to give them).
> Cool, so what are the truth tables for those connectives
> in PT, or alternatively what are the derivation rules?
For truth-valued statements they are exactly the same as those for logic! I wrote the one for conjunction above, I'm sure you know the others.
> > Fourth, these connectives (same as logic remember) ARE truth
> > functional in PT. That is when you apply the connectives to
> > truth-valued statements you get truth-valued statements
> > whose truth depends only on the constituent truth-values.
> > (If you don't agree to this then provide a counter-example.)
> Consider the real interval [0..1] with two subintervals A
> and B whose respective lengths are say 1/3 and 1/8.
> Assuming the uniform distribution, a randomly chosen point
> would have the probability P(A) = 1/3 to be in subinterval
> A and the probability P(B)=1/8 to be in subinterval B.
> What is the probability P(A and B), the probability of the
> randomly chosen point being both in A and B ?
What part of "apply the connectives to truth-valued statements" did you miss? Furthermore, I have no idea what /statements/ A and B are supposed to be. In PT just as in logic the connectives apply to /statements/ not lengths, points, subintervals, etc. Regardless, you assigned probability rather than truth values. Try again with truth-valued statements.
> > Just as when you apply the gamma function to natural
> > numbers you get a natural numbers (no zero quibbles
> > please).
> > When you apply the connectives to a probability-valued
> > statements you get probability-valued statements whose
> > probability depends only on the constituent
> > probabilities.
> > Just as when you apply the gamma function to real
> > numbers you get real numbers.
> Forget the gamma, it's truly irrelevant.
Once I know we agree on what "generalization" is.
> > This is why PT is a /generalization/ of logic. It
> > reduces to logic when applied to truth-valued
> > statements.
> So how about reducing the example I've given above ?
I did, and it was warm and tart and trivial on my tongue.
> > "Probability Theory: The Logic of Science" - ET Jaynes
> That's unfortunate that Jaynes included a chapter on
> similarities between logic and probabilistic reasoning in
> his otherwise interesting book. If he did not, there
> would have been much fewer confused readers.
LOL. Amusing dismissal. I guess the title of his point is also confusing? Have you actually read the book or did you just look at a TOC after I mentioned it? Cause, if you have read it then you must realize that almost everything I have said here and the responses to your trivial challenges are all explained with much greater care in the that book and other sources. So can you point to flaws in his reasoning then? Since you are adamant that PT is not a generalization of logic perhaps you can point me to one of the surely numerous resources demonstrating this? Received on Fri Jun 09 2006 - 21:27:33 CEST