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vc wrote:
> Keith H Duggar wrote:
> > vc wrote:
> > > 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.
> >
> > I assume by "and" you mean conjunction? Also do you
> > realize that the "*" you wrote above is /not/ a
> > connective? (which you claimed before it now seems).
>
> You are confused. In any standard exposition of PT, the
> probability of two independent events is the product of
> the probabilities of those two events. In math, the
> product is designated by '*', not by 'and'.
You are confused. Connectives have /nothing/ to do with real multiplication (probabilities are real numbers and * is real multiplication). Connectives are used to build compound /statements/ they are not operators on real numbers. Do you understand? Let me try to put it more simply. The "*" you wrote above is the real operator "multiplication". It is NOT a connective. You are confusing statements with numbers (as you later confuse statements with events).
> >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.
> To counter your possible answer that '1' is a truth value
> rather than the probability, recall my original question
> "Please 'reduce' the above and tell when A and B is true,
> i.e. when P(A and B) = 1."
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?
> > > > 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.
> > > > 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.)
>
> I do not want to assign truth values because your original
> statement was: "Fourth, these connectives (same as logic
> remember) ARE truth functional in PT." Apparently, you've
> made a statement that the connectives are truth functional
> with respect to probability assignments.
No I didn't. My statement is very clear and says nothing of the kind. "when you apply the connectives to truth-valued statements you get truth-valued statements ..."
> Did you you make your assertion with respect to
> probability assignments? yes or no ?
It's clear that the assertion is with respect to truth-value assignments which is a special case ie specialization ie subset of probability assignments. In other words, every truth-value assignment is a probability assignment. NOT every probability assignment is a truth-value assignment. This is very simple do you understand?
> If yes, do you still insist that P(A and B) depends just
> on P(A) and P(B)? If you made your statement with
> respect to truth value assignments, you did not say
> anything relevant to PT.
I said exactly what I meant to say: PT is a /generlization/! Look do you understand that every integer value is also a real value (no quibbling about representations in set theory please)? That every truth-value (only two 0 and 1) is a probability value? That every truth-value assignment is a probability assignment?
> So what is the probability of the event ('the randomly
> chosen point is in the subinterval A' and 'the randomly
> chosen point is in the subinterval B') assuming you know
> the the 'constituent' probabilities and relying on your
> assertion that "probability depends only on the
> constituent probabilities". ?
> Going in the opposite direction what would be the
> probability of 'A or B' ? Is it two? If it's not two,
> what is it?
> So what would be the probability valuation table for let's
> say implication ?
> How about a) proving your statement that "probability
> depends only on the constituent probabilities" using my
> trival challenge; b) explaining what kind of events A and
> B are if P(A) = 1, P(B) = 1, and P(A and B) = 1 as well as
> what would be the probability of 'A or B' assuming your
> probability assignments of one.
> When you've dealt successfully with (a) and (b), we can
> gradute to the book discussion.
Why are you obsessed with "events" when we are talking about logic and it's generalization which applies to /statements/? Why are you asking me to spoon-feed you truth tables and basic probability theory? This is not the proper forum to teach you such fundamentals. If you /still/ require more lessons we will have to move to another group.
Let's recap.
You confused real multiplication with a connective.
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).
You refused to answer (twice now) a simple question (gamma) designed simply to gauge whether you and I have the same understanding of the word /generalization/ and thus whether we are even speaking the same language.
You demanded a conjunction truth table for independent statements. I gave it to you. Your only response was to dismiss it as "cute".
In another post I even gave a /proof/ for the general (ie no independence assumption) reduction of conjunction in the logical limit. Let's see if you will respond or can even understand the proof.
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.
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".
You comment on a book that you do not even seem to have comprehended.
And finally you decided to be a condescending jerk with your "graduate" comment (among others). (Hence my change of tone in this post). I've done enough graduating in my life, probably more than you.
Your arguments have been demolished and your responses are starting to seem like VI commentary. Go educate yourself. Start by comprehending the book you claim to have read.