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

From: Keith H Duggar <duggar_at_alum.mit.edu>
Date: 8 Jun 2006 19:26:59 -0700
Message-ID: <1149820019.451370.87850@f6g2000cwb.googlegroups.com>


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?

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).

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.) 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. 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

Received on Thu Jun 08 2006 - 21:26:59 CDT

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