Thank you for writing. You noted your dissatisfaction with Jaynes'
explanation of Cox's position on complementary negation so far as Jaynes
understood it, and discussed your own ideas about negation.
Cox's position can be better understood by considering the question
of whether probability is a complete description of belief, and what would
be needed to show that if it were true.
Your introspection convinces you that the lung cancer example
evokes at least two feelings:
[1] An appreciation that some evidence bears on one and not the
other of a pair of alternatives, and
[2] The simultaneous awareness of the complementary logical
relationship between the alternatives, having nothing much
to do with how or why the evidence bears
As you suggest, your own TBM can handle both feelings, although [2] is
relegated to the status of mere pragmatism, while [1] is fully credal.
We know from the work of our mutual friend Marianne Belis that both
feelings can also be represented even-handedly, and she can always recover
a Bayesian trajectory of belief if she wishes. [1] need not be ignored
nor [2] sanitized in order to attain a consistent and tractable description
of the hypothetical belief-state in this example.
Both of you employ more than one modeling dimension to depict your
conceptions of belief. Unaided probability, of course, attempts the job
with just one dimension.
Perhaps beliefs do have some very simple structure. But if not,
then any terse one-dimensional description of them is apt to be incomplete,
just as shadows lose something of the solid objects which cast them.
Whether beliefs really do have great simplicity is an empirical
question, and so beyond what can be established affirmatively by prioristic
argument, Cox's chosen form. Cox taught the incompleteness of specifically
scalar models from homely empirical considerations, and refrained from
strenuous completeness claims for any probabilistic model.
That Cox would grasp the limitations of his form of argument is
plausible. As a physicist active in the early 20th century, Cox surely
noticed the spectacular contemporary revelation of Newton's incompleteness.
Newton had his reasonable assumptions, deductive development,
innovative mathematics, applications to a variety of phenomena which thus
came to be organized into a coherent whole, and just a few loose ends
in odd circumstances hardly worth discussing by practical folk. Cox would
also have noticed that Newton's incomplete physics has enduring value
because it describes something important, well enough and simply.
Cox seems to have taken both parts of the Newtonian example to
heart. He shows that probability offers a reasonable description of some
crucial aspects of belief. Whether other descriptions might reveal more
he leaves for modes of investigation better suited to the task. The final
paragraphs of his 1978 paper make for rewarding reading on that point.
Amicalement, comme toujours -
Paul