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

Keith H Duggar wrote:
[...]
>
> > >Here is the reduction:
> > >
> > > A : B : A and B
> > > 0 : 0 : 0
> > > 0 : 1 : 0
> > > 1 : 0 : 0
> > > 1 : 1 : 1
> >
> >
> > So your 'reduction' assigns the probability of one to both
> > P(A) and P(B). Cute.
>
> Huh? What is cute about taking the limit? You asked for a
> truth table, remember? If you want more then look at my
> other post that provides a proof of the above reduction for
> the general case ie not just when A and B are independent as
> you gave in your challenge.
>
> > So what kind of events are A and B if the probability of
> > each of them is one?
>
> A and B are NOT EVENTS! They are STATEMENTS! Just like logic
> PT as a generalization of logic deals with /statements/ not
> /events/. So you are talking nonsense at the moment.

Any probability book would tell you that there is a notion of sample space which is defined as a set of all possible outcomes of some experiment (e.g. throwing a dice). Each subset of such space is called an event. However, the terminology does not matter, you can call an event a 'sentence' describing such event, it really changes nothing.

[...]
> LOL. 1 is a truth-value and 1 is a probability. 0.5 is not a
> truth-value and 0.5 is a probability. 0 is a truth-value and
> 0 is a probability. Does this clear it up?

No, it does not. What kind of sentences do you have in mind whose probability is one ? Please provide a meaningful example. What would be the probability of 'A or B' given that probabilities of A and B are one ? I asked the question before but you did not answer. So what is it, P(A or B) assuming P(A)=P(B)=1 ?

>
> > > > > 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.
> >
> > The probability statements are not truth-valued
> > (true/false, 1/0), they have probability valuations.
>
> Probability statements CAN be truth-valued. Hopefully my
> last statement helped you see this.

You earlier claimed(correctly) that a probability statement is real valued. Are changing your mind ? If a statement is truth valued, it's a logical statement. OK, one can say that the probability of one (a real number) is *interpreted* as a truth value (true), but a question arises as to what two (or more statements) with probability one might mean. You've failed so far to provide a clear explanation and and examples of such statements, in particular what the probability of the disjunction of those statements might be.

>

[...]

Irrelevant stuff skipped.

> You seemingly failed to understand that 1 and 0 are BOTH
> truth-values and probabilities (part of the entire point
> that probability theory is a /generalization/ of logic).

If the real number 1 is interpreted as a probability, then see above.

[...]

> You demanded a conjunction truth table for independent
> statements. I gave it to you. Your only response was to
> dismiss it as "cute".

It is 'cute' for the reasons I described above, namely it's unclear what kind of 'sentences' you can imagine, all with probabilities of one. So far, you've failed to provide an answer. It's not such a hard question, is it ?

> You refused to provide an example showing that the usual
> logical connectives in PT are no longer truth-functional in
> PT.

I did provide a trivial example/puzzle that you refused or were unable to explain.

>
> You jumped to irrelevant "random" "event" terminology when
> we are talking about logic and a generalization of logic
> both of which concern themselves with /statements/ not
> "events".

That's the standard PT terminology.

[...]

To sum up, the OP claimed two things:

1. PT is a logic generalization. My response was that PT is not truth functional in the sense the propositional logic is, namely that the compound sentence truth, in logic, is determined by its constituent's truth values whereas in PT (as a purported generalization) the probability of the compound statement is *not* determined by the probabilities of its constituent statements. To which you responded:

"When you apply the connectives to a probability-valued statements you get probability-valued statements whose probability depends only on the constituent probabilities. "
I've provided an example, trivial to anyone who's read an introduction to PT, and asked to compute P(A and B) given P(A) and P(B). There has been no answer yet. Are you unable to answer the question ?

2. In some case, namely when probabilities are 0 and 1, the probabilistic statements 'reduce' to logical statements. I asked to privide two or more statements whose probability would be one and show what the probability of the disjunction of such statements might be. There has been no answer. Are you unable to answer the question ?

Let's start from scratch, forget the talk about valuations, what is and what is not a connective and try answering (1) and (2), especially (1) in the light of your assertion that 'you get probability-valued statements whose probability depends only on the constituent probabilities'. Received on Sat Jun 10 2006 - 03:54:10 CEST