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From: George Levy <glevy.domain.name.hidden>

Date: Wed, 29 Sep 2004 11:59:42 -0700

Bruno Marchal wrote:

*> Hi George, [out-of-line message]
*

*> perhaps you could try to motivate your "qBp == If q then p".
*

*> I don't see the relation with "if q is 1 then p is known, and and if q
*

*> is 0
*

*> then p is unknown". How do you manage the "known" notion.
*

Imagine a three port device such as an electrically controlled switch.

Let's say that this device has three lines connected to it: an input

connected to p, a control connected to q and an output that we'll call qBp.

If the control sets the switch to OFF (ie. q=0) , the output is not

connected to the input. Therefore for anyone observing the output, the

value of p is unknown, i.e., qBp = x. The electronic value of x can be

any arbitrary value except 0 and 1 which are reserved for the possible

known binary values.

If the control sets the switch to ON (ie. q=1), the output is connected

to the input. Therefore for anyone observing the output, the value of p

is known. It is either 0 or 1 depending on what the input p is.

George

*>
*

*>
*

*>
*

*>
*

*>
*

*> At 11:44 28/09/04 -0700, you wrote:
*

*>
*

*>
*

*>> I am still working to express Lob's formula using the simplest
*

*>> possible electronic circuit. I am trying to use the well known
*

*>> three-state concept in electronic as a vehicle for expressing belief .
*

*>>
*

*>> Let's first define the operator B as a binary operator that uses two
*

*>> arguments and has one result. Thus the expression qBp means that if
*

*>> q is 1 then p is known, and and if q is 0 then p is unknown. i.e: qBp
*

*>> == If q then p.
*

*>>
*

*>> Physically this can be implemented by using three-state electronic
*

*>> technology. According to this technique, an electrical line can be
*

*>> defined by two voltage levels (eg., 1 and 0) and two impedances (eg.,
*

*>> HIGH and LOW). Thus an electrical line can have three states:
*

*>>
*

*>> 1) a LOW impedance ON state with a low voltage symbolized by 0
*

*>> 2) a LOW impedance ON state with a high voltage symbolized by 1
*

*>> 3) a HIGH impedance OFF state for "unknown" and symbolized by x.
*

*>> Physically x could be an arbitrary voltage level other than the ones
*

*>> assigned for 0 and 1. If a high impedance line is in contact with a
*

*>> low impedance line the low impedance line dominates.
*

*>>
*

*>> The truth table for qBp is
*

*>>
*

*>> q p qBp
*

*>> 0 0 x
*

*>> 0 1 x
*

*>> 1 0 0
*

*>> 1 1 1
*

*>>
*

*>>
*

*>> AND and OR can easily be defined in terms of 0, 1 and x for two
*

*>> propositions p and q
*

*>>
*

*>> AND
*

*>> p q pq
*

*>> 0 0 0
*

*>> 0 1 0
*

*>> 0 x 0
*

*>> 1 0 0
*

*>> 1 1 1
*

*>> 1 x x
*

*>> x 0 0
*

*>> x 1 x
*

*>> x x x
*

*>>
*

*>> OR
*

*>> p q p+q
*

*>> 0 0 0
*

*>> 0 1 1
*

*>> 0 x x
*

*>> 1 0 1
*

*>> 1 1 1
*

*>> 1 x 1
*

*>> x 0 x
*

*>> x 1 1
*

*>> x x x
*

*>>
*

*>> For a digital implementation it is necessary to express
*

*>> "implication" in terms of logical operators using AND, OR , NOT
*

*>> operators.
*

*>> In general we can convert implication p -> q to a digitally
*

*>> impementable form: -p + q.
*

*>> Now let's convert Lob's formula in terms of AND, OR and NOT operators.
*

*>> Originally Lob's formula is B(Bp -> p) -> Bp.
*

*>>
*

*>> Since we have defined B as a binary operator we must specify what its
*

*>> inputs are. Let the left input for the first B be b1 and that for the
*

*>> second B be b2.
*

*>> Lob's formula becomes
*

*>> b1B(b2Bp -> p) -> b1Bp
*

*>>
*

*>>
*

*>> Accordingly, Lob's formula is: ~b1B(~(b2Bp)+ p) + b1Bp
*

*>>
*

*>> The truth table is
*

*>>
*

*>> b2 b1 p b1Bp ~(b1Bp)+ p ~b2B(~(b1Bp)+ p)
*

*>> ~b2B(~(b1Bp)+ p) + b1Bp
*

*>>
*

*>>
*

*>> 0 0 0 x x x
*

*>> x
*

*>> 0 0 1 x 1 x
*

*>> x
*

*>> 0 1 0 0 1 x
*

*>> x
*

*>> 0 1 1 1 1 x
*

*>> 1
*

*>> 1 0 0 x x x
*

*>> x
*

*>> 1 0 1 x 1 0
*

*>> x
*

*>> 1 1 0 0 1 0
*

*>> 0
*

*>> 1 1 1 1 1 0
*

*>> 1
*

*>>
*

*>>
*

*>> I am not sure where this is leading but here it is.
*

*>>
*

*>> George
*

*>
*

*>
*

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

*>
*

*>
*

Received on Wed Sep 29 2004 - 15:02:09 PDT

Date: Wed, 29 Sep 2004 11:59:42 -0700

Bruno Marchal wrote:

Imagine a three port device such as an electrically controlled switch.

Let's say that this device has three lines connected to it: an input

connected to p, a control connected to q and an output that we'll call qBp.

If the control sets the switch to OFF (ie. q=0) , the output is not

connected to the input. Therefore for anyone observing the output, the

value of p is unknown, i.e., qBp = x. The electronic value of x can be

any arbitrary value except 0 and 1 which are reserved for the possible

known binary values.

If the control sets the switch to ON (ie. q=1), the output is connected

to the input. Therefore for anyone observing the output, the value of p

is known. It is either 0 or 1 depending on what the input p is.

George

Received on Wed Sep 29 2004 - 15:02:09 PDT

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