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vc wrote:
> In the sense of being a system of reasoning, yes, PT can
> be regarded as a propositional logic generalization.
Ahhh ... FINALLY. I congratulate you on finally having the honesty and courage to admit this. If only it hadn't taken so much arduous vociferous ignorant complaining then we could have had a far more interesting discourse.
> My very simple point was that PT is not truth functional
> and therefore cannot qualify as a generalization of
> propositional logic or any other logic possessing
> truth-functionality.
It has been stated and proven here several times that /in the logical limit/ ie when all values are either certain (1) or impossible(0) probability theory IS truth-functional. Nobody here (myself included) has EVER claimed that PT is "truth-functional" in your understanding of the phrase ie beyond this limit. For one because I don't even apply the concept of "truth-functional" beyond the limit, that is to degrees of /belief/ (Cox formulation). Partly because I have never been talking about degrees of /truth/ which you later migrated to when you started whining about probabilistic logics etc. It should have been clear to you when I introduced a different phrase "probability-functional" that we were talking about different things. (Well, /now/ you are talking about different concepts but originally I don't think you know what you were talking about.)
> As I said before, there are profound connections between
> PT and proopsitional calculus, and PT can be regarded,
> metaphorically, as a generalization of purely logical
> reasoning about uncertainty. However, their areas of
> applicability is quite different and the generalization
> talk serves nothing but to increase confusion.
It is more than "metaphorical", that's just more of your "conjure" "spirit" crap. And stop projecting /YOUR/ confusion and ignorance on others.
> Since I have nothing more to add except PT lacking
> truth-functionality and a system of derivation axioms
> (Abadi/Halpern), this wil be my last comment on
> propositional logic vs. PT.
More of your "Internet Mind" fiction and I believe this is a Celko as well (ie a reference that does not claim what you claim it claims). Abadi and Halper argue NO SUCH THING. And "derivation axioms" is senseless. If you still want to make this claim provide a precise reference (page, theorem, etc) or modify your claim of what A/H argue.
> To be precise, modus ponens is a derivation rule (possibly
> single) in the propositional/predicate logic.
At least now you appear to know that modus ponens is a derivation /rule/ and not a "derivation axiom".
> My challenge merely demonstrates that P(A and B) cannot be
> derived from P(A) and P(B) alone.
In the /logical limit/ YES THEY CAN. This has be proven here and you eventually agreed (though bitched about the proof being "wrong" and tried to hide your backtracking) that the /truth/ tables derived using PT are /exactly the same/ as the usual logical truth tables. Here remember by /truth/ I mean certainty (1) and impossibly (0) not some "degrees of truth" you want to swindle in so you can attack probabilistic or fuzzy logic strawmen.
> I do not know why it is so hard to understand and compare
> PB non-truth-functionality to the propositional logic
> truth-functionality -- it's a simple statement of fact.
I do not know why you can't understand what someone means by "generalization" "specialization" "in the limit" etc.
> Also, I'd like to remind that the OP made the claim, in so
> many words, that P(A and B) can be calculated from P(A)
> and P(B) alone but failed to back up his claim.
I claimed that ONLY IN THE LIMIT OF CERTAINTY AND IMPOSSBILITY. Why is this so hard for you to understand?
vc wrote:
> Bob Badour wrote:
> > vc wrote:
> > > Also, I'd like to remind that the OP made the claim,
> > > in so many words, that P(A and B) can be calculated
> > > from P(A) and P(B) alone but failed to back up his
> > > claim.
> >
> > I have followed this thread. I don't recall where he
> > stated that at all. Could you perhaps find the relevant
> > quote?
>
>
It's /your/ limited understanding and ignorance that leads you to believe those statements are equivalent, bubba. Given that I have employed and discussed the product rule several times along with the Cox derivation of same you are blind if you believe that I claimed P(A and B) = f(P(A),P(B)). Hell, one only need look at the functional requirements used to derive the product rule to realize this is not generally the case. HOWEVER, to repeat, it /IS TRUE/ in the limit of certainty and impossibility since, AGAIN, the truth tables are the same in both logic and PT.
> In the sense of being a system of reasoning, yes, PT can
> be regarded as a propositional logic generalization.
Ahhh ... FINALLY. I congratulate you on finally having the honesty and courage to admit this. If only it hadn't taken so much arduous vociferous ignorant complaining then we could have had a far more interesting discourse.
> > If that was your main point, why didn't you simply state
> > it instead of making philosophical arguments regarding
> > truth-functionality and generalization?
>
Only recently have you started to offer anything remotely mathematical or even logical. Most of your postings did nothing but "conjure" vociferous "spiritual" ignorant "mindless" attacks. Two of which you have now, to your credit, retracted. You still need to retract a few more and apologize.
> It can be so defined, yes, but there are at least two
> problems with such definition. (1)P(false) = 0
What is P(false)? "false" is not a statement in ANY logic or probability I have ever seen. P(false) seems like nonsense as much as P(0), P(1), or P(4.2) are. Perhaps you are using "false" to represent a contradiction. Very strange notation. Something like P(A and ~A) perhaps where "false" = "A and ~A". Bizarre. I'd love some references to material that use this "P(false)" notation.
> is usually derived directly from Cox's postulates or from
> sum/production rules without relying on the limit, so any
> departure from the usual derivation should be clearly
> stated along with possible problems such approach may
> have;
Interesting that you claim there is a "usual derivation" for a seemingly nonsense statement like P(false). Anyhow, please present it. I eagerly await your "usual derivation".
> In the sense of being a system of reasoning, yes, PT can
> be regarded as a propositional logic generalization.
Ahhh ... FINALLY. I congratulate you on finally having the honesty and courage to admit this. If only it hadn't taken so much of your arduous vociferous ignorant complaining then we could have had a far more interesting discourse.