On 8/21/2012 2:28 PM, Bruno Marchal wrote:


On 21 Aug 2012, at 12:12, Roger Clough wrote:

Hi Bruno and Stephen,
This is the bicameral mind again. Right brain must accept left brain decisions for human safety. Ought must rule over is (or else we'd all be nazis, Hume, for the safety of humanity) Passion must rule over reason (or else we'd all be nazis, Hume, for the safety of humanity)
Acceptace of proof dominates proof (common sense psychology)
Thus you can objectively, mathematically prove that 2+2=4, but you still have to subjectively accept that psychologically.
Woman always gets the last word.

No problem here. That fits nicely with the Bp versus Bp & p duality, which is just the difference between "rational belief" and "rational knowledge" (true rational belief).

It took time to realize that when we define the rational belief by formal proof, which makes sense in the ideal correct machine case, although knowledge and belief have the same content (the same arithmetical p are believed), still, they obey to different logics. This is a consequence of incompleteness. Rational beliefs obey to a modal logic known as G (or GL, Prl, K4W, etc.) and true rational belief obeys to a logic of knowledge (S4), indeed known as S4Grz.

G is

[](p -> q) -> ([]p -> []q)
[]p -> [][]p
[]([]p -> p) -> []p

with the rules A, A->B  /  B and A / []A

S4Grz is

[](p -> q) -> ([]p -> []q)
[]p -> [][]p
[]([](p -> []p) -> p) -> p

with the rules A, A->B  /  B and A / []A

Bruno

Dear Bruno,

It might help us immensely if you could tell us how to read these symbolic representations. Not all of us speak that language! There are English words for all of these symbols!

--
Onward!

Stephen

"Nature, to be commanded, must be obeyed."
~ Francis Bacon

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