FOM: More on pobabilistic proof

Neil Tennant neilt at
Thu Sep 24 08:53:53 EDT 1998

Don Falls wrote

>There is a common presumption (among many philosophers and
>mathematicians) that deductive evidence is epistemically superior to
>inductive evidence. ...  Scientists commonly appeal to inductive
>evidence for the truth of scientific claims.  As a result, their
>acceptance of probabilistic proofs is not very surprising.

I take it that we are talking about *contemporary* scientists here,
and about *contemporary* scientific methodology (so that Aristotle and
even Bacon would be virtually irrelevant to this discussion). Don's
claim that scientists "commonly appeal to inductive evidence for the
truth of scientific claims" states the obvious. There is nothing else
to which they CAN appeal!  If E is adduced as scientific evidence for
theoretical claim T, it is NEVER the case that E deductively implies
T. If E did deductively imply T, then we would complain that adducing
E as evidence for T was simply begging the question. 

For example, let T be the claim "Tadpoles grow into frogs"---the sort
of claim that even Aristotle might have made. Here's a claim E that
deductively implies T: "Tadpoles grow into frogs and snakes lay
eggs". If someone were to offer E as "scientific *evidence*" for T,
one would (quite rightly) think he or she was cognitively deficient. 

A second example: let T be the claim "hearts pump blood" (the great
discovery of the 'other Harvey'). Here's a claim E that deductively
implies T: "hearts pump blood and livers purify it". Again, if someone
were to offer E as "scientific *evidence*" for T, one would think
(quite rightly) that he or she was cognitively deficient.

(By "cognitively deficient" here, I mean deficient in both logical and
scientific understanding.)

Don makes out that scientists have a lower epistemic threshold for
their empirical hypotheses (than mathematicians do for their
mathematical claims), and thinks that this might help explain why
scientists would accept probabilistic proofs in mathematics more
easily than mathematicians would. 

On the contrary, there is good reason for any scientist to insist on
*mathematical* standards of proof (i.e. deductive proof) for
*mathematical* propositions. This is so that, in the event of
experimental falsification of a scientific prediction from a testable
scientific hypothesis, the blame for the failure can be laid at the
door of the *empirical* component of the overall set of premisses,
rather than at the door of the *mathematical* component. If all the
mathematics used in making the prediction has been properly proved
according to the deductive standards of mathematicians, then it can be
"insulated off" from any doubt, and the theoretical failure can be
located in the empirical component. 

(I know it is fashionable for Quineans to insist that *any* claim is
in principle revisable. But when was the last time any scientists were
heard emerging from the lab saying "Hey, guys, we've just got to give
up the commutativity of addition on the integers, in the light of
these observations"? Or "Hey, guys, yesterday's meter readings showed
that the power set axiom of ZFC is looking really shaky?".)

Yet Don makes out that *only*

>Mathematicians, however, try not to appeal to inductive evidence for
>the truth of mathematical claims 

and asks

>Is mathematicians' distaste for
>inductive evidence due to the epistemic inferiority of inductive
>evidence or is it due simply to an aesthetic preference on the part of

It's the former. Both scientists' and mathematicians' distaste for
inductive evidence for *mathematical* claims is due to its epistemic
inferiority to proper deductive (not: 'probabilistic') PROOF. 

When Don went on to write

>... from the fact that some labels (such as "truth value proof") do
>not convey information about the structure of a proof, I don't think that
>you can infer that no labels convey such information.

he was not grasping the point of my simple analogy. I had written

"...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)."

Then I posed the crucial question that Don failed to answer:

"So how come the probabilistic character of a proof carries information
as to its structure (as Don claims it does)?" 

To help the reader in thinking about possible answers to this
question, I outlined two possibilities as to the kind of thing that a
probabilistic proof might be.

The first possibility was that a probabilistic proof has true
premisses (probability = 1) and a conclusion of the explicit form
"p(S)>=1-2^(-n)" for rather large n.

The second possibility was that a probabilistic proof has a conclusion
that 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 argued that the first possibility would be of no help. For "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."

Don did not challenge this claim. This is important, since (in
my own view) it is the *first* possibility that is actually the case with
so-called 'probabilistic' proofs. Hence the label 'probabilistic'
carries no information whatsoever about the structure of the proof.

Instead, Don embraced the second possibility, writing:

>Your alternative picture, however, seems to capture the meaning of the term
>"probabilistic proof" (at least as it has been used on FOM).

But in doing so he failed to address the following comments I had made:

"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."

So I need to repeat my closing request for references to the literature:

"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 ?!)."  

Don, can you tell us what formal system of probabilistic proof you
have in mind? What are the well-formed formulae of the language? What
are its rules of inference? How is "probabilistic soundness" proved?
How is it that the mere clue that a given proof is in *this* system
carries information as to its structure? These are dark and mysterious
matters to me, and a little light shed by you would be most welcome.

Neil Tennant

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