FOM: Inductive reasoning

[Mark Steiner]

Subject:
            Re: FOM: Inductive reasoning
       Date:
            Thu, 20 Jan 2000 13:01:20 +0200
      From:
            Mark Steiner <marksa at vms.huji.ac.il>
        To:
            Joe Shipman <shipman at savera.com>
 References:
            1





Joe Shipman wrote:
>
> 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


        This is really not my field, but I (like Prof. Black) had in
mind cases
in which the mathematical hypothesis is decidable (the usual case where
mathematicians talk about something "probably" true), but not known to
be true or false, so it is a logical consequence of axioms of
mathematics to which I assign probability 1.  As I understand this, the
only sense you could give to a probability other than 0 or 1 is an
subjective sense, i.e. degree of rational belief.  But, as Prof. Black
again pointed out, even the usual models of subjective probability obey
the usual axioms of probability, and also assign "logical omniscience"
to the agent, so that the agent must assign probability 1 to any
theorem, if the axioms have probability 1 (which they usually do).

        This idealization won't capture the mathematician's sense of
"probably
true" or "probably provable," but I am not aware of some other one.