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From: INTERNET:[EMAIL PROTECTED][:]
Sent: Thursday, July 01, 1999 2:06 AM
MH: I think it's time that we accepted that induction is just as rational
as deduction. Whether we do or not, the interesting "problem of induction"
is not whether induction is "true" or "rational", "axiomatic" or
"self-evident", but rather _when_ does it work and how far. There are
massive literatures in statistics, psychology, machine learning, and UAI
(to name but a few) that address this important problem and have generated
many interesting and useful results, theoretical and empirical. At this
point, Hume is a true red herring.
CDC: I think Hume still has something to say. In particular, let me lift
this quote off the back cover of my copy of An Enquiry Concerning Human
Understanding:
DH: All the objects of human reason or enquiry may be naturally divided
into too kinds, to wit, Relations of Ideas, and Matters of Fact. Of the
first kind are the sciences of Geometry, Algebra, and Arithmetic; and in
short, every affirmationwhich is either intuitively or demonstratively
certain. That the square of the hypothenuse is equal to the square of the
two side, is a proposition which expresses a relation between these
figures. That three times five is equal to the half of thirty, expresses a
relation between these numbers. Propositions of this kind are discoverable
by the mere operation of thought, without dependence on what is anywhere
existent in the universe. Though there never were a circle or triangle in
nature, the truths demonstrated by Euclid would for ever retain their
certainty and evidence.
DH: Matters of fact, which are the second objects of human reason, are not
ascertained in the same manner; nor is our evidence of their truth, however
great, of a like nature with the foregoing. The contrary of every matter
of fact is still possible; because it can never imply a contradiction, and
is conceived by the mind with the same facility and distinctness, as if
ever so conformable to reality. That the sun will not rise tomorrow is no
less intelligible a proposition, and implies no more contradiction, than
the affirmation, that it will rise. We should in vain, therefore, attempt
to demonstrate its falsehood. Were it demonstratively false, it would
imply a contradiction, and could never be distinctly conceived by the mind.
CDC: Before I continue, I will make the jump that what Hume refers to as
Relations of Ideas is what I will refer to deduction, while Matters of Fact
refers to induction. As an example of this usage, consider these snippets
from FP Ramsey's essay entitled Truth and Probability:
FPR: Deduction -- the Logic of Consistency. A formal deduction does not
increase our knowledge, but only brings out clearly what we already know in
another form. We are bound to accept it's validity on pain of being
inconsistent with ourselves. The logical relation which justifies the
inference is that the sense or import of the conclusion is contained in
that of the premises.
FPR: But in the case of an inductive argument this does not happen in the
least; it is impossible to represent it as resembing a deductive argument
and merely weaker in degree; it is absurd to say that the sense of the
conclusion is partially contained in that of the premisses. We could
accept the premisses and utterly reject the conclusion without any sort of
inconcistency or contradiction.
FPR: Induction-- the Logic of Truth. Four snips: 1) But I think it would
be a pity, out of deference to authority, to give up trying to say anything
useful about induction. 2) .. human logic or the logic of truth, which
tells men how they should think, is not merely independent of but sometimes
actually incompatible with formal [dedutive] logic. 3) In spite of all
this nearly all philosophical thought about human logic and especially
induction has tried to reduce it in some way to formal [deductive] logic.
4) We all agree that a man who did not make inductions would be
unreasonable: the question is only what this means.
MH: In a practical sense, we cannot put greater trust in deduction,
because of our fallible premises and rules, not to mention our fallible
inferencing. As I understand it, the rationale for induction is just about
as weak and circular as the rationale for deduction:
CDC: Now, perhaps I can make my point. The rational for deduction is
circular, but not weak. Deduction is infallible. It just doesn't say
anything by itself. The rationale for induction, on the other hand, is
weak but not circular. It is a matter of tradition, authority, opinion,
... and if it goes unchallenged and undoubted, religion. As I take
mathematics to be quantitative deduction, I think it is entirely
independent of induction. That is, if it's mathematics, then it isn't
induction.
CDC: The problem with just "accepting" induction, is that it is not so
clear what it is that we are accepting. It is relatively easy to get
agreement that good theories fit the facts and that simpler theories are
better than more complex ones. But these are competing principles. There
is a balance to be struck and we clearly don't all go about striking this
balance in the same way -- which is why we end up with justification
problems. However, while inductive justifcation is important in public
policy, it isn't necessarily a UAI problem. However, if you need to
convince someone else that the robot you built is intelligent...
