# Re: Use of Three-State Electronic Level to Express Belief

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

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