Re: Programming is the Engineering Discipline of the Science that is Mathematics
Date: 9 Jun 2006 06:35:03 -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.
> 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?
> 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 ?
> 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 ?
> 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.
> 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 ?
>Just as gamma
> reduces to factorial for natural arguments. (Again no quibbles
> about offset by 1 etc).
> > > because in addition to the logically valid modus ponens
> > > and modus tollens, it also gives a foundation for
> > > applying the weak syllogisms
> > It does not -- see above.
> It does. Don't take my word for it, educate yourself. I
> suggest starting with:
> "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.
> -- Keith --
Received on Fri Jun 09 2006 - 15:35:03 CEST