Hi Brian,

On 13 Sep 2012, at 22:04, Brian Tenneson wrote:

Bruno,

You use B as a predicate symbol for "belief" I think.

I use for the modal unspecified box, in some context (in place of the more common "[]"). Then I use it mainly for the box corresponding to Gödel's beweisbar (provability) arithmetical predicate (definable with the symbols E, A, &, ->, ~, s, 0 and parentheses. Thanks to the fact that Bp -> p is not a theorem, it can plays the role of believability for the ideally correct machines.





What are some properties of B and is there a predicate for knowing/ being aware of that might lead to a definition for self-awareness?

Yes, B and its variants:
B_1 p == Bp & p
B_2 p = Bp & Dt
B_3 p = Bp & Dt & t,
and others.




btw, what is a machine and what types of machines are there?

With comp we bet that we are, at some level, digital machine. The theory is one studied by logicians (Post, Church, Turing, etc.).




Is there a generic description for a structure (in the math logic sense) to have a belief or to be aware; something like
A |= "I am the structure A"
?

Yes, by using the Dx = xx method, you can define a machine having its integral 3p plan available. But the 1p-self, given by Bp & p, does not admit any name. It is the difference between "I have two legs" and "I have a pain in a leg, even if a phantom one". G* proves them equivalent (for correct machines), but G cannot identify them, and they obeys different logic (G and S4Grz).




Finally, on a different note, if there is a structure for which all structures can be 1-1 injected into it, does that in itself imply a sort of ultimate structure perhaps what Max Tegmark views as the level IV multiverse?

A 1-1 map is too cheap for that, and the set structure is a too much structural flattening. Comp used the simulation, notion, at a non specifiable level substitution.

Bruno



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



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