# Re: Programming is the Engineering Discipline of the Science that is Mathematics

Date: 9 Jun 2006 06:35:03 -0700

Message-ID: <1149860103.249428.122630_at_f6g2000cwb.googlegroups.com>

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?
**>
*

Irrelevant.

> 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.
*

No.

*> Just as when you apply the gamma function to real numbers
*

> you get real numbers.

Forget the gamma, it's truly irrelevant.

*>
*

> 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