[I write:]
#>>> Imagine that 1000000 Alterian dollars turn up in your bank account every
#>>> month. Suppose further that this money is being paid to you by (a) your 
#>>> big-hearted Alterian benefactor, or (b) a bug in an Alterian ATM.  Let's
#>>> suppose that this is a true dichotomy, so P(a)+P(b)=1. Trouble is, Alterius 
#>>> is in a different universe, so that no observations of Alterius are possible
#>>> (except for the banks - you couldn't possibly afford it ;-)
#>>> 
#>>> Now let's examine the case for (a).  There is no evidence whatsoever that
#>>> there is any such thing as a big-hearted Alterian benefactor. However,
#>>> P(exists(b-h A b)) + P(not(exists(b-h A b)) = 1. On the grounds that 
#>>> lack_of_evidence_for is evidence_against when we have a partition like
#>>> that, we dismiss hypothesis (a).
#>>>
#>>> Turning, therefore, to (b), we also find no evidence to support that
#>>> hypothesis.  On the same grounds as before, we dismiss hypothesis (b).
#>>>
#>>> The problem with this is that we have dismissed *all* of the possible
#>>> hypotheses, and even though we know by construction that the money 
#>>> arrives every month, we have proven that it can't, because we
#>>> have dismissed all of its potential causes.
#
#>> That's an *extremely* poor argument, and here's why.
#>>
#>> Premise 1: "...this money is being paid to you by [either] (a) your big-
#>>             hearted Alterian benefactor, or (b) a bug in an Alterian ATM".
#>> 
#>> Thus each monthly appearance of the bucks, should it happen, is an
#>> observation on Alterius, and by construction, is evidence for the
#>> existence of [either the benefactor or the bug in the ATM].
#>> 
#>> Premise 2: no observations on Alterius are possible.
#
#>     #> (except for the banks - you couldn't possibly afford it ;-)
#>
#> You forgot to include this.  My premise is actually:
#>
#> Premise 2:  The cardinality of the set of possible observations on Alterius
#>             is one.
#
#>> This is clearly contradictory to the first.
#
#> Not if you state it properly.
#
#>> Trouble is, on the basis of premise 2, you say that there can be no evidence
#>> of [either the benefactor or the bug], but the first premise leads to the
#>> conclusion that the appearance of the bucks, should it happen, is evidence
#>> for the existence of [either the benefactor or the bug].
#>> 
#>> Voila, a screaming contradiction.
#
#[with my highlights - SC]
#> But in a strawman argument. There is only evidence for OneOf(Benefactor,Bug).
#> No observation to distinguish Benefactor from Bug is possible. That is
#> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#> not evidence for Bug, and neither is it evidence for Benefactor. Nor
#> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#> is true to say that this hypothetical universe appears exactly as
#> if there were no Benefactor/Bug (two statements, both would be false).
#
#This is still contradictory. It reduces to
#
#  (1): Alterian dosh arriving in my account is due to [benefactor or bug].
#
#  (2): this is not evidence for [benefactor], neither is it evidence for
#       [bug] (meaning that it doesn't lend more weight to one than to the
#       other)
#
#  (3): therefore no evidence can exist for [benefactor] and no evidence
#       can exist for [bug].
#
#But (3) relies on a shift in meaning from (2). When you say (paraphrased)
#in (2) that this is not "evidence for [benefactor]", for example, what you
#mean is that it's no *more* evidence for [benefactor] than it is for [bug].

Yes, that's what I mean.

#In (3), however, you've shifted the meaning of "evidence for [benefactor]"
#so that it now means `absolute' "evidence for [benefactor]" rather than
#`relative' "evidence for [benefactor]" w.r.t. [bug].

Not really, I meant evidence that would tend to one over the other.  I
think this is just a communications problem.   What I am trying to say,
in my clumsy way, is that while I buy your theory as far as it relates
to theism making predictions (prayer, your 'Rapture' example), I don't
buy your use of Occam's razor in all cases where A=0.

In my example, one couldn't dismiss
[benefactor] or [bug] on the grounds of simplicity - one of these is necessary
to explain the dosh.  I brought up the 'one-by-one dismissal' process to
show that it would be wrong to do so.  From what you're saying in this
post, it seems you agree, and we're talking at cross-purposes.

#(3) is still in contradiction to (1).
#
#Some sums may help. With B = benefactor, b = bug, d = dosh arrives in account:
#
#  (1) implies  P(B+b | d) = 1
#
#Assuming that P(Bb | d) = 0, so it's either the benefactor *or* a bug
#which is responsible if the bucks arrive, but not both, then
#
#   P(B+b | d)   =   P(B | d) + P(b | d)
#
#so
#
#   P(B | d) + P(b | d)  =  1
#
#but (3) implies that
#
#   P(B | d)  =  0  and  P(b | d)  =  0.

No, this isn't what I meant.  P(B | d) = 0.5 and P(b | d) = 0.5, with
necessarily no new observation (we've already seen the dosh) to change
those estimates.  I was trying to say (again, in my clumsy way) that
it would be _wrong_ to assign 0 probability to either of these.  And that's
precisely what use of the Razor does in the case of gods - gods are
one class of hypothesis (there are many others) belonging to a set of
hypotheses _one_of_which_ is necessary to explain something which otherwise 
would _not_ be satisfactorily explained.  It can be thrown out or
retained on grounds of non-rational preference, not of science or statistics.   
Alternatively, one could chuck out or retain the lot, on the grounds
that the answer can't be known, or that the notional probability estimates
are effectively useless, being equal (agnosticism/weak atheism).

#> As they do when the set M is filled by "the universe is caused by x",
#> where x is gods, pink unicorns, nothing, etc.  - and no observation
#> tends to one conclusion over the other.
#
#Exactly the point I was making, I think. So we don't "throw out" any of
#these, contrary to your assertion above that we do.

Some people do, Simon, and they think they are doing excellent science.
My sole point was that they aren't.

#>> Only observations which directly contradict the hypothesis H[i] (i.e. x
#>> where P(x | H[i]) = 0) can cause P(H[i]) to go to zero after a finite
#>> number of observations. Only in this case do we get to throw any of the
#>> hypotheses out.
#
#> Exactly my point, though I may have been unclear.
#
#You said the diametric opposite, which I guess is the source of my confusion.

I was merely trying to illustrate the incorrectness of doing so.

#> What I'm trying to say is that while you are correct to say that absence of
#> evidence can sometimes be evidence of absence, this does not hold true for
#> all, or perhaps any, versions of theism - and it isn't true that those for
#> which it does not hold can be discarded using the razor.
#
#On the contrary, those for which it does not hold are *exactly* those which
#can be discarded using the Razor. See my post on the other branch of this
#thread.

Then you seem to be guilty of the contradiction you accuse me of.  If
the razor holds for gods, then it holds for all like hypotheses.  Which
means that you're assigning P(x | H[i]) =0 for all i, though we've already
established that it's not correct to do so when SUM(P(x|H[i]))=1 over
all i.

#> Simply put, anyone who claims to have a viable proof of the existence or
#> non-existence of gods, whether inductive or no, is at best mistaken, and
#> at worst barking mad.
#
#Luckily I make no such claim, and have specifically said as much on numerous
#occasions. You wouldn't be constructing a strawman here, would you Frank?
#Although that doesn't, of course, rule out my being barking mad in any case
#(I could be barking mad in my spare time, with apologies to Cleese et al).
#
#But I think you miss the point once again. When I say that something is
#"evidence against" an hypothesis, that doesn't imply that observation of
#the said something necessarily *falsifies* the hypothesis, reducing the
#estimate of P(H | data) to zero. If it *reduces* this quantity, it's still
#evidence against H.

No, I got that.  I'm talking about the case when A=0.  You're clearly
correct when A!=0.   And I'm not constructing a strawman (though it's
certainly possible that I've misunderstood what you're saying).  However,
by any standards, a system that says when A=0, gods are highly unlikely,
and when A!=0 gods can be dismissed using the Razor, is a system purporting
to be an inductive proof that gods either don't exist, or are unnecessary
to explain any or all phenomena.  In my experience, systems such as this
(including those which purport to prove that gods exist) always contain
a fallacy upon close examination.  If that's not what you're saying, then
please put me straight.