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From: Marchal <marchal.domain.name.hidden>

Date: Wed Nov 17 06:18:16 1999

Russell Standish wrote:

*>My point is that linearity appears to be essential, and that it
*

*>deserves an explanation. I give what might charitably be called a
*

*>speculation that it is related to "causality" of the multiverse (or
*

*>subset of the ensemble (aka plenitude, although I avoid that term in
*

*>the paper). Of course, if anyone else has a better idea, I'd love to
*

*>hear it!
*

Well, it seems that in my chapter 5 of my thesis at

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

I derive ..., well, trying to be modest, I will say that

I derive the shadow of linearity (in fact the skeleton of the

lattice of subspaces of Hilbert spaces) from comp + the theetetic

approach to belief, knowledge, and observation.

I modelize "scientific third person communication by

self-referentially correct universal machine" by the Godelian

modal logic of provability, known as G.

It happens by the "Solovay" miracle that I inherit also of

what I call privately the machines' Guardian Angels, which is an

extension G* of the logic G.

G proves what the machine can prove about its own provability.

G* proves what is true about the machine's provability.

Read Bp as the machine proves p, and read BT as the machine proves

the universaly true tautology.

Now B-T means the machine is inconsistent.

G proves -B-T -> -B-B-T (this is Godel's second incompleteness

theorem).

G* proves -B-T (the guardian angels proves the machine is

consistent), but

of course the machine doesn't prove that.

In general G is closed for B, and G* is closed for -B-.

B is the modal box, and -B- is the modal diamond.

The axioms of G are

B(p->q)->(Bp->Bq) Kripke Axiom

B(Bp->p)->Bp "Lob's theorem"

The rule of inference are

p p->q and p

------- ---

q B p

Known as modus ponens and necessitation respectively.

G* has as axioms all theorems of G, and as ONLY rule of inference

the modus ponens.

Now the theetetic ideas are to define Kp, the knowledge of p,

by p & Bp. It is the true and justified communication proposed by

Thaetetus to Socrate.

We get a logic of knowledge S4Grz which can also be formalised:

The axiome of S4Grz are

B(p->q)->(Bp->Bq) Kripke Axiom

B(B(p->Bp)->p)->p Grzegorczyk formula

the rule are also modus ponens and necessitation.

G* does not change S4Grz. That is S4Grz* = S4Grz. From the point

of view of the subject, truth is identified with proof, like

intuitionnistic philosophy (Brouwer's philosophy of the subject).

Ultimately this is used to tackle the first person point of view.

To get the logic of observation, I do two things, which are linked

to the thought experiment UDA.

Op is "I observe p".

1) I modelize Op by (Bp & -B-p). This is linked

to the fact that I do not take the world I am dead, or I am not

reconstituted, into account for the computation of the probability

on the domain of reconstitution. I impose consistency of the

possible continuation.

2) I modelise the universal dovetailer by the set of sigma1 proposition.

a sigma1 proposition is a verifiable arithmetical proposition. It has

the form "it exists n such that P(n)" with P decidable.

That gives two logics: Z1 and Z1*. Now the "result" (simplified a

little bit) is that Z1* prove B-B-p if and only if QL

proves p, where QL stands for Quantum Logic. This gives an embryo of

a purely arithmetical interpretation of quantum logic, which, overall,

corresponds to the searched logic of the computational continuations,

which by UDA corresponds to the searched logic of physics.

I have not been able to axiomatize completely Z1 and Z1*, and I suspect

that it is not possible. But thanks to Solovay's powerfull theorem I

have been able to program theorem provers which are complete for these

logics.

Once you have a quantum logic, you begin to smell a basic LINEAR

realm. The difference between Z1 and Z1* (guardian angel point of view)

makes it possible to distinguish the physical measure, which is

communicable, from the physical sensation (observable but uncommunicable

truth).

The logic of (Bp & -B-p & p) (reapplication of the theetetic idea) gives

an unexpected notion of true qualia.

I hope this gives you a raw feeling of my way to "derive" physics from

psychology. I guess it is both too concise and too long, but it can

give you hints for my chapter 5.

The problem now is to derive the explicit measure from a suitable

semantics for the Z1(*) logics.

*>> I ask you a question:
*

*>>
*

*>> Do you agree with the following statement:
*

*>>
*

*>> ``(Understanding why we must exclude the white rabbit)
*

*>>
*

*>> is equivalent with
*

*>>
*

*>> (Understanding why me must justify the laws of physics)" ?
*

*>>
*

*>
*

*>Not equivalent per se, but exclusion of the white rabbit appears to be
*

*>necessary in order to justify the laws of physics.
*

That is indeed the main point. I agree.

Bruno

Received on Wed Nov 17 1999 - 06:18:16 PST

Date: Wed Nov 17 06:18:16 1999

Russell Standish wrote:

Well, it seems that in my chapter 5 of my thesis at

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

I derive ..., well, trying to be modest, I will say that

I derive the shadow of linearity (in fact the skeleton of the

lattice of subspaces of Hilbert spaces) from comp + the theetetic

approach to belief, knowledge, and observation.

I modelize "scientific third person communication by

self-referentially correct universal machine" by the Godelian

modal logic of provability, known as G.

It happens by the "Solovay" miracle that I inherit also of

what I call privately the machines' Guardian Angels, which is an

extension G* of the logic G.

G proves what the machine can prove about its own provability.

G* proves what is true about the machine's provability.

Read Bp as the machine proves p, and read BT as the machine proves

the universaly true tautology.

Now B-T means the machine is inconsistent.

G proves -B-T -> -B-B-T (this is Godel's second incompleteness

theorem).

G* proves -B-T (the guardian angels proves the machine is

consistent), but

of course the machine doesn't prove that.

In general G is closed for B, and G* is closed for -B-.

B is the modal box, and -B- is the modal diamond.

The axioms of G are

B(p->q)->(Bp->Bq) Kripke Axiom

B(Bp->p)->Bp "Lob's theorem"

The rule of inference are

p p->q and p

------- ---

q B p

Known as modus ponens and necessitation respectively.

G* has as axioms all theorems of G, and as ONLY rule of inference

the modus ponens.

Now the theetetic ideas are to define Kp, the knowledge of p,

by p & Bp. It is the true and justified communication proposed by

Thaetetus to Socrate.

We get a logic of knowledge S4Grz which can also be formalised:

The axiome of S4Grz are

B(p->q)->(Bp->Bq) Kripke Axiom

B(B(p->Bp)->p)->p Grzegorczyk formula

the rule are also modus ponens and necessitation.

G* does not change S4Grz. That is S4Grz* = S4Grz. From the point

of view of the subject, truth is identified with proof, like

intuitionnistic philosophy (Brouwer's philosophy of the subject).

Ultimately this is used to tackle the first person point of view.

To get the logic of observation, I do two things, which are linked

to the thought experiment UDA.

Op is "I observe p".

1) I modelize Op by (Bp & -B-p). This is linked

to the fact that I do not take the world I am dead, or I am not

reconstituted, into account for the computation of the probability

on the domain of reconstitution. I impose consistency of the

possible continuation.

2) I modelise the universal dovetailer by the set of sigma1 proposition.

a sigma1 proposition is a verifiable arithmetical proposition. It has

the form "it exists n such that P(n)" with P decidable.

That gives two logics: Z1 and Z1*. Now the "result" (simplified a

little bit) is that Z1* prove B-B-p if and only if QL

proves p, where QL stands for Quantum Logic. This gives an embryo of

a purely arithmetical interpretation of quantum logic, which, overall,

corresponds to the searched logic of the computational continuations,

which by UDA corresponds to the searched logic of physics.

I have not been able to axiomatize completely Z1 and Z1*, and I suspect

that it is not possible. But thanks to Solovay's powerfull theorem I

have been able to program theorem provers which are complete for these

logics.

Once you have a quantum logic, you begin to smell a basic LINEAR

realm. The difference between Z1 and Z1* (guardian angel point of view)

makes it possible to distinguish the physical measure, which is

communicable, from the physical sensation (observable but uncommunicable

truth).

The logic of (Bp & -B-p & p) (reapplication of the theetetic idea) gives

an unexpected notion of true qualia.

I hope this gives you a raw feeling of my way to "derive" physics from

psychology. I guess it is both too concise and too long, but it can

give you hints for my chapter 5.

The problem now is to derive the explicit measure from a suitable

semantics for the Z1(*) logics.

That is indeed the main point. I agree.

Bruno

Received on Wed Nov 17 1999 - 06:18:16 PST

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