# Re: the duality between Bp and Bp &p.

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On 20 Apr 2011, at 01:15, Stephen Paul King wrote:```
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```Hi Bruno,

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You mentioned in a previous mail “the duality between Bp and Bp &p”. Could you elaborate on this? Is it a Stone or a Pontryagin duality? (these are different!)
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Not at all. It is more akin to the cartesian mind-body duality. Bp gives the third person discourse given by the machine, and Bp & p gives the first person knower linked to that machine. Bp is third person self-reference, like when you discuss with your doctor about your temperature and body constitution, and Bp & p is the Plotinus inner god, or soul, which the amchine cannot even name.
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Also, are there any restriction on the content of the proposition p?
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p is for any arithmetical proposition (assuming comp)

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p is for any sigma_1 proposition (assuming comp, and assuming that the machine assumes comp too). In that case the hypostases are restricted to the propositions accessed by the Universal Dovetailer.
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```Could a model of a possible world be a p?
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No. p are (roughly speaking) finite syntactical object. A model is an infinite structure. p might be satisfied, or not, by a model (true in that model).
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In the modal context, and in a model, we can use a stone-like duality to "model" a proposition by a set of models. This is useful for the hypostase which have no Kripke semantics. But this is technic. In Bp & p, it is better to think of p as a syntactical arithmetical proposition (it can be a huge one!).
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I ask this because so far you seem to only consider p that are tautologically true (such as 2+2 = 4)
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p can be 2+2=5. In that case, the correct machine will not know p.

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```and thus are trivially independent of observer notions.
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The observer is defined in the third person way by its body/program or relative number (relative to some probable universal environement/ history).
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```What about the contents of Observer Moments? Could they be p?
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They can be accessible "p", like "I am in Washington" after a self- duplication experience, or in the universal dovetailing.
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I suspect that the in the “&p” (and p is true) is where the concreteness, that I have referred to before, lies hidden; for in situations where the truth or untruth of a proposition that involves possible worlds (not just the truth of arithmetic statements) there is a requirement of a concrete realization between the observer of that world and the possible world. This latter idea is explicit in the Everett and Rovelli Interpretations. (Concreteness is the property of being “this and not anything else” that is invariant to point of view.)
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Perhaps. I would say that the concreteness is in both B and p (in Bp & p). Also the duality above, the cartesian one, extends itself into the duality between Bp & Dt and Bp & p & Dt (or Dp, it is equivalent). The concreteness is then made even more concrete by imposing the existence of a model satisfying the proposition (assuming the machine talk first-order logic, Dt is equivalent with the existence of a model by Gödel's completeness theorem). But remember that, unlike Everett and Rovelli, we do not assume QM, nor the existence of a primitive physical world. A concrete realization is just a local and relative very probable (with the comp- measure) universal number.
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Bruno

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

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