Given the truth of finitely many formulae Fa, Fb, Fc,… in a language
with names a, b, c,… for infinitely many different individuals; it
would be nice to infer that (for all x)Fx has non-zero probability.
Hume and Popper say one can’t, and I believe them.
Nevertheless(!) can one define a language L like, but not necessarily
identical to, that of ordinary first order logic (though it must have
names for infinitely many individuals and some wff that are "singular"
and some that are "universal"); such that a probability P may be defined
on a sigma-algebra of sets of wff of L with this property:
For some universal wff U, there is a finite set S of singular wff, st
S does not logically imply U,
but P(U given S) = P({U} intersect S)/P(S) <> 0
?
I feel that I have not worded that very clearly. My apologies if I
haven’t. What I’m trying to ask (of course) is "Can finitely-many
singular statements ‘give’ a universal statement a non-zero
probability?"
PP
In article <38BD9ED1.6B8A5…@cwcom.net>,
Peter Percival <peter.perci…@cwcom.net> wrote:
- Hide quoted text — Show quoted text -
> Given the truth of finitely many formulae Fa, Fb, Fc,… in a language
> with names a, b, c,… for infinitely many different individuals; it
> would be nice to infer that (for all x)Fx has non-zero probability.
> Hume and Popper say one can’t, and I believe them.
> Nevertheless(!) can one define a language L like, but not necessarily
> identical to, that of ordinary first order logic (though it must have
> names for infinitely many individuals and some wff that are "singular"
> and some that are "universal"); such that a probability P may be
defined
> on a sigma-algebra of sets of wff of L with this property:
> For some universal wff U, there is a finite set S of singular wff, st
> S does not logically imply U,
> but P(U given S) = P({U} intersect S)/P(S) <> 0
> ?
> I feel that I have not worded that very clearly. My apologies if I
> haven’t. What I’m trying to ask (of course) is "Can finitely-many
> singular statements ‘give’ a universal statement a non-zero
> probability?"
> PP
Yes, of course, given the context. The snowflake example comes to
mind. It depends on the way the propositions are worded. You might want
to read "Scientific Reasoning" by Howson and Urbach for a defense of
Bayesianism. They get heavily into the probability. I think Popper is
wrong (even though he is real big in my book) about meaning not being
important in the philosophy of science. I think the Raven’s paradox
supercedes the problem of induction. Of course, this thing didn’t come
along ’till way after Hume’s death (another one in my hall of fame.)
Give Quine a try. He’s great also. Him and Popper disagree with what you
are driving at above, Keep up the good work, PP.
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