> >Suppose you have a million holes in the ground, and you want to see
> >if at least one of them contains a marble. You examine 200,000
> >holes at random and find no marble. This is good evidence that, if
> >there are any marbles at all, their number is quite small.
>
> Assuming that the marbles were themselves distributed randomly, which you
> did not specify. Otherwise there could be 500,000 of them, but placed in
> specific arrangements meant to defeat a random search. As such, your lack
> of evidence at this point really tells you nothing about how many there are.
No assumptions need be made about the distribution of the marbles.
The fact that the search is _random_, means that there is _no_
distribution strategy which can defeat it. It would be impossible to
arrange the marbles in such a way as to defeat the search because
there is no way of predicting, ahead of time, which holes will be
searched.
Suppose there were in fact 100 marbles. The chance of not finding one
after 200,000 searches, can be calculated. It is:
999,900 999,899 999,898 799,901
---------.---------.---------. ... .---------
1,000,000 999,999 999,998 800,001
which is a _very_ small number --- we're talking around 1/100,000,000.
In fact, further calculations would show that if no marble were found
after 200,000 searches, we could have 99% statistical confidence that
there were fewer than 20 marbles.
> >It is not however enough evidence to conclude that there are no
> >marbles at all. If however you examined 950,000 of the holes and
> >still there were no marbles found, it would be strong evidence that
> >are no marbles.
>
> Not really. Your hypothesis is that there is at least one marble present,
> but nothing is said about the probability of there being more than one.
"at least one marble present" is mathematical speak for "one marble,
or two marbles, or three marbles, or four marbles, ..." --- in other
words, it is an expression often used when one wants to talk about the
probability of there being "one or more than one" marbles present. I
hope this clarifies things --- a technical point in any case.
> That makes the null hypothesis that there are no marbles. As such,
> to verify or refute either hypothesis you would have to examine every single
> hole; even 950,000 holes would be insufficient to verify or refute either
> hypothesis. Even 999,999 holes would be insufficient because the last hole
> still has a 50/50 chance of containing a marble.
It is _possible_, after searching 999,999 holes without success, that
the last hole contained the marble, but _very_ _very_ improbable ---
certainly not 50/50! A 999,999 hole search would not give you
_absolute_ certainty of no marbles, but it would give you extremely
high _statistical_confidence_ of no marbles.
> Despite appearances, however, this is not a case where a lack of
> evidence verifies one hypothesis and refutes another, because in
> this kind of study the lack of marbles is not a lack of evidence;
> you would expect to find some holes empty in any event.
Well this "marbles scenario" is analogous to the "missing fossil
scenario", though the latter is obviously not so neat. There are a
finite number of potential fossil locations on earth (though we may
not know where they all are). Theoretically we could break this up
into small regions --- our "holes". Finding a pre-Cambrian fossil in
a region is equivalent to finding a marble. Clearly there are lots of
complications, and probability calculations are much more difficult
than in the marble scenario, but the essential idea is still there.
My point is that just as "finding many empty holes" is admissible
evidence as to the number of marbles, so too is "finding many fossil
beds with no pre-Cambrian fossils" evidence as to the number of
pre-Cambrian fossils.
> >I don't believe the "lack of pre-Cambrain fossils argument" correctly
> >matches the "appeal from ignorance" logical falacy. If no one had
> >bothered to look for fossilized Cambrian ancestors, and then somebody
> >claimed that because there were no such specimins they hadn't evolved,
> >_then_ this would be an appeal from ignorance falacy. But the fact
> >that people have been looking without success, is admissible evidence.
>
> The fallacy doesn't depend upon whether you look for the evidence or not; it
> only depends upon whether you use a lack of evidence to try to prove or
> disprove a concept. The fallacy doesn't say, claiming a concept is false
> because no one has tried to prove it true; it says, claiming a concept is
> false because it hasn't been proven true. The implication is that people
> could have been trying to prove it true but had not yet succeeded.
You are correct in suggesting that _trying_to_prove,_without_success_
does not invalidate a claim --- *providing* the very process of
_trying_to_prove_ does not in itself yield evidence in favour of the
counter claim.
I am saying that, a large enough search for pre-Cambrian fossils which
results in none being found, can be taken as positive evidence in
favour of the proposition that no pre-Cambrian fossils exist. And
thus, this kind of argument cannot be classified as an "appeal from
ignorance" falacy.
Mark.
_/~~~~~~~~\___/~~~~~~\____________________________________________________
____/~~\_____/~~\__/~~\__________________________Mark_Phillips____________
____/~~\_____/~~\________________________________mark@ist.flinders.edu.au_
____/~~\HE___/~~\__/~~\APTAIN_____________________________________________
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"They told me I was gullible ... and I believed them!"