FOM: Fallis on probability proofs
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Wed Sep 23 12:23:00 EDT 1998
Don Fallis wrote
>scientists (unlike mathematicians) have long since lowered their
>standards and commonly accept inductive evidence for scientific
>claims.
What were the scientists' standards before they were thus "lowered"?
Don also wrote
>There are a number of reasons why mathematicians might want to label
>probabilistic proofs as probabilistic. For one thing, this label gives
>readers a quick idea of the structure of the proof (in the same way that
>the labels "proof by induction" or "proof by reductio" do).
How can the label give information as to the *structure* of the proof?
After all, if I were to say of a conventional proof that it was a
"truth-value proof", this would carry no information at all as to its
*structure*. All I would know is that if the premisses of the proof
have truth-value T, then so does its conclusion. That leaves open
every possibility as to its deductive structure (i.e. the patterning
of steps of inference within the proof).
So how come the probabilistic character of a proof carries information
as to its structure (as Don claims it does)? Presumably a
probabilistic proof has true premisses (probability = 1) and a
conclusion of the explicit form "p(S)>=1-2^(-n)" for rather large
n. But then the validity of the proof will transmit the value T from
its premisses to *this* conclusion!---thereby making the
"probabilistic" proof a special case of ordinary truth-transmitting
proof. This being so, there would appear to be no information as to
its possible structure. All we could glean is that the Kolmogorov
axioms for the probability calculus might be among the premisses of
the proof.
An alternative picture of probabilistic proof might be that its
conclusion doesn't explicitly register the probability of S. Instead,
its conclusion is S itself, but the steps within the proof are made in
such a way that, though they do not guarantee truth-transmission, they
nevertheless guarantee that the probability-value of 1 for each of the
premisses does not degrade below 1-2^(-n) (for some suitably large n).
I am not aware of any such "probability logic" being satisfactorily
developed in the literature, to the point where the authors of
so-called probabilistic proofs would be able reliably to claim that
their proofs could be formalized within that logic, in the way that
authors of conventional proofs can and do reliably claim that their
proofs can be formalized within ordinary predicate calculus.
If there *is* such a probability logic, I'd be most interested to know
where to find it, since it seems that *that* would be the system to
inspect to see whether Don's claim about information concerning proof
structure can be justified (i.e. assigned probability > 0.5 ?!).
Neil Tennant
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