FOM: Inductive reasoning

[Joe Shipman]

Steiner, replying to Black:

>>Obviously, to get an idealization of such nondeductive reasoning, we
have (a) to distinguish between objective and subjective probability;
(b) lay down that the set of probability 1 propositions is not closed
under mathematical deduction. <<

I don't understand the necessity for either of these considerations.

For mathematical propositions that are independent of, say, ZFC, there
are different possible mathematical universes in which the propositions
could be true or false.  The identification of a probability not equal
to 0 or 1 with the class of universes in which GCH is true need not be
"subjective".  Only if you somehow know you are referring to a
particular mathematical universe is such a label appropriate, because
there is an objective "fact of the matter" even though you don't know
what it is.

Let x be the trillionth bit of Chaitin's number Omega, an
algortihmically random sequence of 0s and 1s.  We will never know what x
is; is the estimate P(x=1)=0.5 subjective or objective?  If you believe
in the set of integers as a completed whole, you can define the truth
set for arithmetic and it is a perfectly objective question whether x=0
or x=1.  Most of us would accept this and say that 0.5 is a SUBjective
probability.  But what about GCH?  It is much harder to say that there
is an objective truth value, even though in any particular model GCH is
either true or false.  An intermediate case is CH -- if we take modern
theoretical physics seriously (I personally don't take it this
seriously), then our ontology includes real numbers and higher-type
objects as well, and CH may have an objective truth value (relative to
OUR universe, meaning the physical universe we live in).

Second, I don't see why you require the set of probability 1
propositions not be closed under mathematical deduction.  Do you simply
mean to say that some propositions need not have probability 1 or 0?

If you use a Boolean algebra of "truth values", you can construct a
universe in which all axioms of ZFC have "probability" 1, all logical
deductions preserve the property of having probability 1, and CH does
not have probability 1 (Scott and Solovay "Boolean-valued-model"
adaptation of Cohen's independence proof for CH).  You can get
propositions with "truth values" strictly between 0 and 1 while having
mathematical deduction respect "truth value" and not lead you from
propositions to other propositions with "smaller" truth values.  The
only obstacle to turning this into a proper generalization of logical
validity which allows for weaker positive conclusions than "logical
truth" is that the Boolean algebra is not simply an algebra of
probabilities.

Technical question here for logicians: in the Boolean-valued model
approach to independence proofs, is there ever a way to "project" the
algebra of "truth values" into the probability space [0,1] so that
(some) statements with intermediate truth values can be assigned a
probability strictly between 0 and 1in a consistent way?  Is there a
Boolean-valued model in which the axioms of ZFC have value 1 but CH has
a value strictly between 0 and 1?  (I already know there are BVM's in
which it has value 0, showing it is independent, and BVM's in which it
has value 1, showing it is consistent).

-- Joe Shipman