Truth and (Circular) Inference -- Re: [om-list] Re: Cyc example

[Tom and other Packers]

Om

    I just wanted to mention that some of my motivation for looking at
things geometrically came from Mark.  I think he told me about his
possibility space a few years ago.  This has certainly influenced my
thinking.

tomp

----- Original Message -----
From: Mark Butler <butlerm at middle.net>
To: OM List <OM-List at onemodel.org>
Sent: Monday, October 02, 2000 12:03 AM
Subject: Re: Truth and (Circular) Inference -- Re: [om-list] Re: Cyc example


Hello everybody,

 I would like to make a few comments on Tom's very well written paper.  Tom
has explained some of this to me before and I wholeheartedly agree with what
he has to say.

If you would like to stretch your mind a bit, I believe I have a compatible
perspective that is like a spatial inversion of Tom's theory.  As part of my
pet programming language design, I worked on a representation where all
statements were converted into volumes in an arbitrarily dimensioned
possibility (or phase) space.

For example, you may be interested in studying the motion of a certain body
on
a two dimensional plane. If you start out with no knowledge at all, the body
could be anywhere, hence you could reasonably model your state of knowledge
as
the set of all points (x,y,t) in possibility space.

But then you could introduce a wall or energy barrier at x = 5 that
constrains
the body to have a limited range of coordinates where x < 5.  Now I would
say
that your knowledge of where the body is could be modeled precisely by the
set
of all points in possibility space such that x < 5.

You know *more*, but in possibility space, the set representing what you
know
has a *smaller* volume.  Indeed if you knew exactly where the body was at
all
times your knowledge would correspond to a zero-width / zero-volume path in
possibility space.

Now before I go on, I would like to point out that "a general statement" has
two different nearly contradictory meanings in English. One meaning
corresponds to ambiguity, e.g. "He could be anywhere by now".  The other
meaning corresponds to (near) universal applicability, e.g. "F = ma".

An ambiguous statement corresponds to a large volume in possibility space,
where most statements of universal applicability are statements that greatly
constrain allowed possibilities, which are clearly low volume defining
statements.

As in Tom's theory, a logical "AND" of two propositions corresponds to the
set
intersection of the corresponding sets.  In like manner, a logical "OR"
corresponds to the set union.  "AND" always moves in the direction of
smaller
PS volume / more knowledge. e.g. "I know his name AND his phone number".
"OR"
always moves in the direction of larger PS volume / less knowledge. e.g. "He
is either at work OR he is at school".

Deductive inference is trivial in possibility space, literally moving from a
PS set to a PS superset.  If a have a rule of universal applicability,
processing it for a specific case is converting a low PS volume statement (a
statement constraining possibilities for many objects) into a high PS volume
statement (a statement constraining possibilities only for the object of
discourse).

Here it helps to see possibility space as containing a very large number of
dimensions for every degree of freedom in the system.  To do the famous "All
men are mortal" example requires a possibility space with an independent
mortality and manhood dimension for each possible being.

We always start with the null statement (saying nothing) - the set of all
points in possibility space.  "All men are mortal" (our main premise)
reduces
that set down to a very low volume subset where the value of each man's
mortality dimension is contrained to be "Mortal".

The second statement, "Socrates is a man" is a high PS volume statement
because it only restricts one dimension out of the infinite number in our PS
phase space.

The logical conjunction, or "AND" of the two premises tells us everything we
know by set intersection, producing an even smaller volume in possibility
space corresponding to "Socrates is a man AND all men are mortal"

The conclusion reached by deduction, "Socrates is mortal" is a proper PS
superset of the possibility space corresponding to our net knowledge about
the
system.  Think of a PS superset as circumscribing the boundaries of where
the
truth may lie.  Reality can be any point or path that lies completely inside
the boundaries circumscribed by the PS set of any true statements.

The more you say, the more tightly you draw those boundaries, and you aren't
extremely careful, you will miss the point in possibility space where the
truth lies completely, more generally known as "logical overstatement".

On the other hand, you cannot be contradicted if you say nothing, because
your
silence corresponds to a comprehensive phase space volume containing all
possibilities.

Then of course, denying the existence of reality or absolute truth is
perfectly analogous to declaring that there is no one point in possibility
space corresponding to "The Truth", or the way things really are.  Rather
you
are completely reduced to worrying about whether a set of statements is
consistent with itself - a difficult task nuch of the time, but hardly
inspiring.

In possibility space, any set of statements that correspond to any non-empty
set of points is a consistent statement. Statements that contradict
eachother
wipe the remaining possibility space out of existence.  For example, X
cannot
both be greater than 5 and less than 5 (at all times).  In such a case you
only have two conclusions:  One is that one of your premises is wrong. Two
is
that the object concerned does not exist at all.  Of course people who do
not
believe in reality can hardly be bothered by the question of whether
something
actually exists.

Now of course, standard induction is simply the process of saying that a
pattern of true constraints on similar dimensions in possibility space
implies
that the same constraint applies to all such similar dimensions.  The trick,
of course, is properly defining the set of objects that have similar
dimensions - in our inductive test case (see Tom's message), correctly
defining who is a man.

In possibility space, an inductive reasoning process is one that proceeds
from
a large PS volume statement about a limited number of actors (i.e.
restricting
a limited number of entity dimensions) and jumps to a low PS volume
statement
about all such entities.

Deductive reasoning is always allowed because your conclusion has less
information (more possibilities) than the combination of your premises.  In
that sense deductive reasoning is like allowed thermodynamical processes -
you
are always allowed to proceed from a high information (low entropy) state to
a
low information (high entropy).

Inductive reasoning is questionable because you are speculating, i.e.
injecting information of your own choosing into your perspective on the
state
of the system. If you are an outside observer, you cannot arbitarily add
information / reduce entropy without converting what you used to *know*
about
the allowed possible states of the system to speculation that the system is
in
a smaller group of states.

This of course leads to the scientific method, where scientists test their
informed speculations (theories) empirically.  Despite its incredible
utility,
scientific principle is still only valid by induction, and is subject to
empirical disproof at any time.  What correctly we know of the principles of
science, more accurately stated, is that our scientific theories predict
reality to a specified statistical approximation.  I.e. Valid scientific
theories place narrow probabilistic constraints (think differential
equations)
on the allowed paths of reality through possiblity space.

Now its clear that if you start out in a state of relative ignorance, you
have
no principles of high information content (low entropy / PS volume) to use
deductively.  That leaves you with inductive reasoning tested empirically,
good for only generating statistical approximations to the truth about
things
that can be easily tested.  As Alan Bloom, so eloquently made the case for
in
"Closing of the American Mind", empirically testable truth is so bland and
uninspiring that people will believe any convincing theory that fills the
gap
left behind - nihilism, deconstructionism, radical relativism, Nietschean
non-relativism - and will nearly destroy societies in the process.

- Mark

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