[lit-ideas] Re: Back to Popper (and further back to Hume)
- From: Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sat, 9 Dec 2006 13:23:52 +0000 (GMT)
Where were we? Robert Paul refers me to a very long and technical article in
the Stanford Encyclopedia that, he says afair, presents at least two versions
of inductive probability. Afair RP does not identify these two versions [when
there are least three (e.g. enumerative induction, Bayesian logicism,
Bayesian subjectivism)]. Nor does RP indicate which of these, if any, he
supports ? or why.
In other words, Robert Paul acts as though the ball is back in my court. But
without indicating where the ball is ? instead I guess I am supposed to find
it in a thicket of technicality. While this is one way to respond, it ?leaves
a lot to be desired? ? it even leaves unclear whether the ball has, in fact,
been returned.
There is another less than satisfactory aspect to responding to an argument
(I know, I know ? RP says there is no argument; but this does not mean he is
right) by referring to a lengthy, technical article on the topic in general.
It gives the other side a target that is too big and unwieldy for useful
discussion ? it?s as if someone suggests to me that its going to rain because
there are dark clouds above and I respond ?I see no argument and so offer
none back, but why don?t you wade through this technical general manual on
weather forecasting??
While I may take time to deal with some of the various matters discussed in
the article (as best I can), might I refer Robert Paul to a much, much
shorter and much, much less technical piece by Popper that can be found in
the ?Introduction 1982? at xxxvii of ?Realism and the Aim of Science?. I am
assuming that whatever his view of my posts on three boxes, Robert Paul will
agree that at least Popper?s piece contains an argument ? indeed one, like
the Stanford article, that invokes Bayes?s theorem.
Perhaps Robert Paul would like to pick apart this short argument, pinpoint
its errors etc. Others also might like to do so and, since they may not have
access to Popper?s book, here is the argument (reproduced without italics and
subscript):-
?It is sometimes objected to my theory that it cannot answer Nelson Goodman?s
paradox.
That this is not so will be seen from the following considerations
which show, by a simple calculation, that the evidence statement e, ?all
emeralds observed before the 1st January 2000 are green? does not make the
hypothesis h, ?all emeralds are green, at least until February 2000? more
probable than the hypothesis ?all emeralds are blue, for ever and ever, with
the exception of those observed before the year 2000, which are green?. This
is not a paradox, to be formulated and dissolved by linguistic
investigations, but it is a demonstrable theorem of the calculus of
probability. The theorem can be formulated as follows:
The calculus of probability is incompatible with the conjecture that
probability is ampliative (and therefore inductive).
The idea that probability is ampliative is widely held. It is the
idea that evidence e ? say, that all swans in Austria are white ? will
somehow increase the probability of a statement that goes beyond e, such as
h2, ?all (or most) swans in regions bordering Austria are white?. In other
words, the idea is that the evidence makes things *beyond* what it actually
asserts at least a little more probable. (This view was strongly defended by
Carnap, for example).
The view that probability is ampliative was suggested especially by
the following theorem (h=hypothesis; e=empirical evidence; b=background
knowledge):
Let p(h,b) ≠ 0. Further, let e be favourable evidence (that
is, e follows from h in the presence of b, so that p(e,b) ≠ 1 and
p(e,hb) = 1. Then p(h,eb) > p(h,b). That is to say, the favourable evidence e
makes h more probable, even though h says more than e. And this holds for
every new e1, e2 ?., which satisfy similar conditions.
It therefore seems that increasing favourable evidence goes on
supporting h; and so it seems that the support is ampliative.
But this is an illusion, as can be shown as follows:
Let h1 and h2 be any two hypotheses supported by e in the presence
of b, so that
p(e,b) ≠ 1 and p(e,h1b) = p(e,h2b) = 1
Let R1,2 (prior) = p(h1b)/p(h2b) be the ratio of the probabilities of h1 and
h2 prior to the evidence e, and let
R1,2 (posterior) = p(h1,eb)/p(h2,eb)
be the ratio of the two probabilities posterior to the evidence e.
Then we have, for any h1,h2 and e that satisfy the above
conditions:
R1,2 (prior) = R1,2 (posterior).
This follows almost immediately from
p(a,bc) = p(ab,c)/p(b,c),
that is, from Bayes?s theorem.
What does
R1,2 (posterior) = R1,2 (prior)
signify? It says that the evidence does not change the ratio of the prior
probabilities, whether we have calculated them or freely assumed them,
provided the two hypotheses can both explain the evidence e. But this means
that if we let
h1 = all swans in some region greater than Austria are white;
h2 = all swans in the world are non-white except those in Austria
which are white;
e = all swans in Austria are white,
then, assuming any prior probability for h1 and h2 you like: their ratio R1,2
(prior) remains unaffected by the evidence. Thus there is no spill-over, no
ampliative support: there is no ampliative probability, neither for swans nor
for emeralds. And this is not absurd, but tautological (and it is unaffected
by translation).?
As I understand it, this argument might be put in less formal terms ?
layman?s terms, if you like ? as follows:- where increasing amounts of
evidence are consistent with two inconsistent hypotheses, that evidence does
not increase their probability relative to each other, and this means it does
not increase either of their probabilities full stop ? not in any inductive
sense. (Both will, however, remain more probable than hypotheses falsified by
the increasing evidence).
Is this wrong?
Donal
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