[From Bill Powers (920114.1400)]
Martin Taylor (930114.1430) --
I have reconsidered one of my main arguments concerning the
"Critique of pure reason." In the course of proving myself wrong,
however, I have come across another consideration that may have
some minor importance, depending on how many dozen people before
me have already seen it. First the proof that I was wrong:
Suppose we have n propositions which can be true or false. On the
face of it, there are 2^n possible ways to combine these
propositions to make a compound statement. So 3 propositions
represented by a string of 3 bits can be used to distinguish 8
compound statements or hypotheses.
In forming logical combinations of n propositions, there are 2^n
minterms, each containing all of the n variables either primed
(not) or unprimed. The minterms are ANDed together, each
multiplied by a coefficient of 1 or 0 telling whether to include
that minterm in the logical OR of the minterms to make up the
final function. If we have a standard systematic way of
exhaustively listing successive minterms, we can state the
function simply by listing the coefficients of the minterms. For
n variables, we thus must transmit a string of 2^n minterm
coefficients to isolate one particular compound proposition made
of n independent propositions.
The number of logical functions of n variables is 2^(2^n). The
number of states representable by the string of 2^n minterm
coefficients is 2^(2^n). I therefore stand corrected by myself:
the same number of distinctions is made either way.
HOWEVER:
The mistake I made was to confuse the number of propositions
represented by the bits in a string with the number of logical
functions of that same number of propositions, which is very much
larger and would required a longer string to distinguish any one
of the combinations. It takes a lot more than n bits to
distinguish a unique logical combination of n propositions. To
use just n bits is equivalent to considering only one minterm.
Having realized this mistake, I then realized that there may be
something more to say from a different angle.
I think I understood John Gabriel to say that he was dealing with
binary strings of length 18, in which each bit represented a
separate proposition's truth value. The on-the-surface
implication is that there are 2^18 distinct combinations of
propositions thus represented.
From the above development, however, it might seem that a string
of 18 digits representing truth values of propositions might
actually correspond to a compound proposition of no more than
log2(18) variables, or 4+ variables. The actual number of digits
needed to specify a particular hypothesis made of 4 independent
propositions would be 16, because there would be 2^4 minterms.
The question that comes to mind is this: if one seems to be faced
with the possibility of 18 (well, 16) independent propositions,
could it be that behind the scenes the meanings of these
propositions are related in such a way that they could be reduced
to a combination of 4 propositions: ab'c'd or abcd' or ... some
such? If we could find such related meanings, we could reduce the
apparent complexity of a decision to a much simpler form. A
universe of 256 propositions would be equivalent, transformed, to
a universe of only 8 propositions.
Just to give a rough idea of the rough idea in my head, consider
this example:
x = I got liability insurance for the car
y = I sold the car to Pete
z = The car ran over a valuable cat.
Now consider this list of propositions and the minterm
coefficients for combinations of the above variables:
No problems with owning this car (000)
I am liable for a lot of damages (001)
Pete doesn't feel guilty about buying the car (010) or (110)
Looks like I sold the car just in time (011)
I wish I hadn't bought that insurance (100) or (101)
I feel guilty but I don't owe any damages (101)
Pete is glad he bought the car (110)
Somebody else is liable for damages (111)
And so on. Clearly, there are many derivative propositions that
could be stated for different combinations of truth values for
the three basic propositions -- a lot more than 3 derivative
propositions. The tantalizing possibility is that, given an
extended set of propositions like the second set, would there be
some way to find a basic set of 3 underlying propositions?
This sort of transformation could only be done through the
meanings, not the abstract algebra. John mentioned a Japanese guy
in connection with meanings. Is this something similar?
I still have other reservations on this subject, but will let
them go for now. Sorry about the enthusiastic false alarm.
ยทยทยท
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RE: Feedback is too slow
Who made the claim was presumably not an electronics person.
Too right. Much of this confusion can be traced back to early
cyberneticians, almost none of whom had engineering knowledge of
control systems. Psychologists tended to treat anyone with the
label cyberneticist as a technonerd, so when they decided to
believe something a cyberneticist wrote they invested it with
full authority. That's how a lot of myths about control theory
got into circulation.
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Best,
Bill P.