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What is the probability that Mr. X looks like Mr. Y? Clearly the prior
(Mr. X and Mr. Y are identified but there is no additional information
concerning them) probability is virtually zero. This means that for
the argument:

Premise: Mr. X and Mr. Y are identical twins.
Conclusion: Mr. X looks like Mr. Y.

the combination "false premise, true conclusion" is virtually
impossible.

Consider an oversimplified version of the Carnot theorem:

Premise: Heat is an indestructible substance (cannot be converted into
work in the heat engine).
Conclusion (prototype of the second law of thermodynamics): The heat
engine X working between the temperatures T1 and T2 is just as
efficient as the heat engine Y working between the same temperatures.

The premise turned out to be false in the end. Is the prior
probability of the conclusion virtually zero? Is the combination
"false premise, true conclusion" virtually impossible?

If Carnot's conclusion (the prototype of the second law of
thermodynamics) cannot be true, as the above analogy suggests, then
Clausius 1850 argument abandoning Carnot's false premise and deducing
the conclusion from another premise, true this time, must be invalid.
Indeed, Clausius' auxiliary hypothesis given below in capitals: "THE
ONLY CHANGE" and "WITHOUT ANY EXPENDITURE OF FORCE OR ANY OTHER
CHANGE" - is false:

http://web.lemoyne.edu/~giunta/Clausius.html
"Ueber die bewegende Kraft der Warme" 1850 Rudolf Clausius: "Carnot
assumed, as has already been mentioned, that the equivalent of the
work done by heat is found in the mere transfer of heat from a hotter
to a colder body, while the quantity of heat remains undiminished. The
latter part of this assumption--namely, that the quantity of heat
remains undiminished--contradicts our former principle, and must
therefore be rejected... (...) It is this maximum of work which must
be compared with the heat transferred. When this is done it appears
that there is in fact ground for asserting, with Carnot, that it
depends only on the quantity of the heat transferred and on the
temperatures t and tau of the two bodies A and B, but not on the
nature of the substance by means of which the work is done. (...) If
we now suppose that there are two substances of which the one can
produce more work than the other by the transfer of a given amount of
heat, or, what comes to the same thing, needs to transfer less heat
from A to B to produce a given quantity of work, we may use these two
substances alternately by producing work with one of them in the above
process. At the end of the operations both bodies are in their
original condition; further, the work produced will have exactly
counterbalanced the work done, and therefore, by our former principle,
the quantity of heat can have neither increased nor diminished. THE
ONLY CHANGE will occur in the distribution of the heat, since more
heat will be transferred from B to A than from A to B, and so on the
whole heat will be transferred from B to A. By repeating these two
processes alternately it would be possible, WITHOUT ANY EXPENDITURE OF
FORCE OR ANY OTHER CHANGE, to transfer as much heat as we please from
a cold to a hot body, and this is not in accord with the other
relations of heat, since it always shows a tendency to equalize
temperature differences and therefore to pass from hotter to colder
bodies."

In fact, the two-substances process considered by Clausius presupposes
the constant action of an OPERATOR; this operator constantly and
unavoidably undergoes CHANGES, changes that are absent when heat
spontaneously "shows a tendency to equalize temperature differences
and therefore to pass from hotter to colder bodies" (Clausius' new
premise). In othe words, the fact that, spontaneously, heat always
flows from hot to cold (Clausius' new premise) by no means implies
that the operator-driven two-substances process considered by Clausius
is unable to transfer heat from cold to hot. Clausius' argument is
simply invalid.

Pentcho Valev
pvalev@yahoo.com