I mainly agree. I think this is true not only for "sanity", but also for "happiness", "cleverness", and a lot of *positive* predicate.
A clever person will not say "I am clever" (or will look stupid)
An happy person will rarely say "I am happy" (or will look or be unhappy)


(Unless special context, and nuances).

Those adjectives are solution of the equation

                                 Bx -> ~x.

Communicating x entails the contrary of x.

So it is interesting that something as sober and crisp as Peano Arithmetic or Zermelo-Fraenkel set theory, or any sound machines, already obey that equation for any x of the shape D#. (Dt, DBf, DDt, DDDt, DDDDDDDBBBBBBf, ...), and can prove it once enough rich. Lobian machine cannot prove any sentence beginning by a diamond. Any possibility (consistency) need some sort of act of faith.

Very generally, inconsistency, that is Bf, can be used to modelise

Death (cf Bf can only be true in the cul-de-sac world of the Kripke multiverse)
Lies     (to communicate the false, 1)
Error    (to communicate the false, 2)
Dream  (to communicate the false to oneself)
Madness (as you say)

Dt -> DBf can then be read respectively: (reading the D by "I can").

If I am alive I can die
If I am honest I can lie
If I am correct I can fail
If I am awake I can dream
If I am sane I can be mad

Bruno


Le 30-déc.-05, à 05:33, George Levy a écrit :

Bruno Marchal wrote:

Godel's result, known as Godel's second incompleteness theorem, is that no consistent machine can prove its own consistency:

            IF M is consistent then M cannot prove its consistency


Bruno,

After I read your email, we had a gathering of family and friends, and my head being full of the subject of this post. I wanted to test the idea of Godel's second incompleteness theorem on the average people just to see how they would respond. I found the right place in the discussion to insert the paraphrase:

If I am sane, it is impossible to know for sure that I am sane.

This povoked some hilarity, especially with my kids (young adults) who probably view me as some kind of nutty professor. While this statement is mathematically true, it was not considered serious by the people I was talking with. I guess that the average human has no doubt about his own sanity.(But my kids had some doubts about mine) One way to prove that you are crazy is to assert that you are sane. This means that the average human is crazy! :-)

George




http://iridia.ulb.ac.be/~marchal/


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