"Рэймонд Смаллиан. Две философские сценки (англ.)" - читать интересную книгу автора

trust the machine, shouldn't I trust the machine, should I, shouldn't I,
should I, shouldn't I." (He decided to use the word "should" in your
empirical sense.) He got nowhere! So he then decided to "formalize" the
whole argument. He reviewed his study of symbolic logic, took the axioms of
first-order logic, and added as nonlogical axioms certain relevant facts
about the machine. Of course the resulting system was inconsistent--he
formally proved that he should trust the machine if and only if he
shouldn't, and hence that he both should and should not trust the machine.
Now, as you may know, in a system based on classical logic (which is the
logic he used), if one can prove so much as a single contradictory
proposition, then one can prove any proposition, hence the whole system
breaks down. So he decided to use a logic weaker than classical logic--a
logic close to what is known as "minimal logic"--in which the proof of one
contradiction does not necessarily entail the proof of every proposition.
However, this system turned out too weak to decide the question of whether
or not he should trust the machine. Then he had the following bright idea.
Why not use classical logic in his system even though the resulting system
is inconsistent? Is an inconsistent system necessarily useless? Not at all!
Even though given any proposition, there exists a proof that it is true and
another proof that it is false, it may be the case that for any such pair of
proofs, one of them is simply more psychologically convincing than the
other, so simply pick the proof you actually believe! Theoretically the idea
turned out very well--the actual system he obtained really did have the
property that given any such pair of proofs, one of them was always
psychologically far more convincing than the other. Better yet, given any
pair of contradictory propositions, all proofs of one were more convincing
than any proof of the other. Indeed, anyone except the epistemologist could
have used the system to decide whether the machine could be trusted. But
with the epistemologist, what happened was this: He obtained one proof that
he should trust the machine and another proof that he should not. Which
proof was more convincing to him, which proof did he really "believe"? The
only way he could find out was to consult the machine! But he realized that
this would be begging the question, since his consulting the machine would
be a tacit admission that he did in fact trust the machine. So he still
remained in a quandary.
FRANK: So how did he get out of it?
DOCTOR: Well, here is where fate kindly interceded. Due to his absolute
absorption in the theory of this problem, which consumed about his every
waking hour, he became for the first time in his life experimentally
negligent. As a result, quite unknown to him, a few minor units of his
machine blew out! Then, for the first time, the machine started giving
contradictory information--not merely subtle paradoxes, but blatant
contradictions. In particular, the machine one day claimed that the
epistemologist believed a certain proposition and a few days later claimed
he did not believe that proposition. And to add insult to injury, the
machine claimed that he had not changed his belief in the last few days.
This was enough to simply make him totally distrust the machine. Now he is
fit as a fiddle.
FRANK: This is certainly the most amazing thing I have ever heard! I
guess the machine was really dangerous and unreliable all along.