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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: 9 Jun 2006 21:24:35 -0700
Message-ID: <1149913475.098135.291900@j55g2000cwa.googlegroups.com>


vc wrote:
> Keith H Duggar wrote:
> > 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).

This confirms that you have not comprehended Jaynes' book or any other material relevant to probability theory as a generalization of logic. Had you read and comprehended that book then you would know that notions of sample space and events are not needed to formulate probability theory. You would have known that there are at least two formulations Kolmogorov's (the one you refer to) and the arguably more general Cox formulation. And, finally, you would have known that anyone referring to probability theory as generalized logic is without question referring to the Cox formulation.

> 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.

Viewing PT from a Cox formulation versus Kolmogorov changes A LOT. Since you are ignorant of the Cox formulation you simply do not understand what changes.

> > 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.

Please tell me you are joking? You do not understand that 1 is both a truth-value and a probability?

> > 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.

Finally you see that a probability of 1 can at least be *interpreted* as a truth value. Of course, who knows what you had in mind when adding the word *interpreted*. And no, I haven't "changed my mind". I have consistently maintained that a probability of 1 *is* a truth-value. Remember, this is the Cox formulation. If you were more knowledgeable you would have immediately known this when reading my original post:

  "And interestingly, one view of logic is as a   specialization of conditional probability theory. One that   deals only with certainty (1) and impossibility (0) rather   than a range of probability." -- KHD

Here "one view" = "Cox formulation".

And by the way, no "question arises as to what ... probability one might mean" in Cox. You really should learn more about this formulation. It's quite interesting although getting a copy of Cox's paper may be somewhat difficult for some given it's age.

> > 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.

No you didn't! Stop snipping relevant context. Here is the context restored:

> > You refused to provide an example showing that the usual
> > logical connectives in PT are no longer truth-functional
> > in PT. The very definition of truth-functional of course
> > requiring truth-values which you "do not want to assign".
> > Seemingly because you wrongly believe truth-values are not
> > (also) probabilities.

Your example was not a example of non-truth-functional connectives because you applied them to non-truth-values!

> > 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.

No, that is standard Kolmogorov terminology. Not standard in the "one view of probability theory" ie Cox formulation that is clearly what is being described. This is the nature of VI.

You are clearly ignorant of any other formulation of probability theory (even though you implied you had read a book steeped in the Cox formulation) besides Kolmogorov. Being thus ignorant you decided to vociferously attack a remark made by someone that knew about alternatives.

> 1. PT is a logic generalization.

No, I the OP claimed quote:

  "interestingly, one view of logic is as a specialization   of conditional probability theory. One that deals only   with certainty (1) and impossibility (0) rather than a   range of probability." -- KHD

In other words to paraphrase myself and Cox:

  "interestingly, one view of probability theory, the Cox   formulation, is as a generalization of logic. One that   deals with a range of probability corresponding to a   degree of rational belief bounded in the extremes by   certainty (1) and impossibility (0)."

> 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

No, your response was quote

  "PT cannot be 'a generalization of logic' because PT   'connectives' (+/*) are not truth functional." -- vc

Which, I tried to explain to you is wrong for two reasons. First, (+/*) are NOT connectives in PT they are the real operators addition and multiplication. Second, PT uses the SAME connectives as logic. The connectives haven't changed they are still truth-functional as well as probabilityfunctional.

> 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."

No, you have completely screwed the context. That statement of mine was a response to your

  "PT cannot be 'a generalization of logic' because PT   'connectives' (+/*) are not truth functional." -- vc

You then responded TO ME with the simple "No". I have never responded to your claim above as you seem to recognize below.

> 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 ?

Rest assured there is an answer and I am able to answer it. I choose not to answer for two reasons. First, you are acting like a VI and I'm not here to teach you basic theory, in this case the Cox formulation. Second, responding to your challenge is not needed to refute your nonsense claim that "PT 'connectives' (+/*) are not truth functional." As I have done many times now.

["what does P(A) = 1 mean" nagging]

> > 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 ?

> What kind of sentences do you have in mind whose
> probability is one?

> Please provide a meaningful example.

What are you talking about? It's not hard. It's trivial and irrelevant! For proving probability theorems or that probability is a generalization of logic it does not matter what A, B, C etc stand for! "My first name is Keith" "My last name is Duggar" "My first name is Keith or my last name is Duggar". Does that satisfy you? I hope so because it is TOTALLY irrelevant (and somewhat VI honestly).

[disjunction nagging]

> 2. In some case, namely when probabilities are 0 and 1,
> the probabilistic statements 'reduce' to logical
> statements. I asked to provide 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 ?

> 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 ?

> 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.

Dude, did you bother to look at my conjunction proof in another post? Given that what makes you think I cannot prove the same (ie reduction to logic) for disjunction? Why can't you attempt it yourself? Why can't you comprehend Jaynes' book? Why can't you admit that before this thread you were ignorant of Cox probability? Why can't you admit that this means you have been acting like a VI so far?

One last time I will provide one of these basic proofs. And I will provide more than you ask for, that is below is a proof for the complete reduction to logic in the limit of certainty (1) and impossibility (0).

P(A) = 1
P(A or B) = P(~(~A and ~B))
P(A or B) = 1 - P(~A and ~B)
P(A or B) = 1 - P(~B|~A)P(~A)
P(A or B) = 1 - P(~B|~A)(1-P(A))
P(A or B) = 1 - P(~B|~A)(0)
P(A or B) = 1 - 0
P(A or B) = 1

P(B) = 1
(same as above just swap A and B)

P(A) = 0
P(B) = 0
P(~A) = 1
P(~B) = 1
P(~A and ~B) = 1 [conjunction proved in previous post]
P(A or B) = P(~(~A and ~B))
P(A or B) = 1 - P(~A and ~B)
P(A or B) = 1 - 1
P(A or B) = 0

thus

A : B : A or B
0 : 0 : 0
0 : 1 : 1
1 : 0 : 1
1 : 1 : 1

Again back to your "what does the sentence mean" nagging, it does not matter what A and B mean. Just as their meaning (ie their interpretation under some model) does not matter when proving logical theorems.

> 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'.

No, I don't want to start over. I've said and /proven/ all that was needed. I think it's time for you to admit that

  "PT cannot be 'a generalization of logic' because PT   'connectives' (+/*) are not truth functional."

is non-sense, that you were entirely ignorant of the Cox formulation of probability theory, that this formulation is a generalization of logic, and that you did not comprehend (or even read) Jaynes' book even though you commented on it.

If you cannot admit any of these, then it is time for you to provide a reference to a respected source that echos your claims that

  "PT cannot be 'a generalization of logic' because PT   'connectives' (+/*) are not truth functional."

and that PT "does not" provide a foundation for applying the weak syllogisms. Just as I provided Jaynes' book and Cox's work/papers that more thoroughly demonstrate the contrary.

Received on Fri Jun 09 2006 - 23:24:35 CDT

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