I've noticed that many engineers like to draw pictures of abstract
concepts in order to understand them better. I'm not sure if it can
be attributed to the engineering education, or whether people that
like to think visually are just more like to take engineering. In any
case, the TGrid is a simple 2D graph of strength of belief plotted
against comparative evidence for any given proposition.
The strength of belief, S, reflects how strongly you believe the
proposition, P, is true or false. A +1 on this (the vertical) axis
corresponds to ultimate certainty that the proposition is true. A
-1 means that you are ultimately certain the belief is false. In
between these two extremes, a 0 on this axis means you can't say
one way or the other, either because of lack of information or
fuzziness in terms. An S-value, x, assigned to any proposition P
is equivalent to the negative value of x assigned to the logical
opposite of P:
S(P) -> -S(~P)
For example, if P is "God exists" and is assigned an S-value of
0.8, then that is equivalent to "God does not exist" having an
S-value of -0.8.
Another way to think about strength of belief is in terms of gambling.
How much are you willing to bet that the proposition is true (or false)?
[3] This is a bit easier to understand with respect to lack of information,
e.g. "I believe this coin flip will come up heads" would be given an S
value of 0.5 (assuming a fair coin, a fair toss, a rational speaker,
etc., etc.). But a 0.5 S-value can also reflect an ill-defined concept
in the sense that there is a 50% chance that the audience will have
a different definition of the terms of the proposition such that the
truth value will be reversed. Almost always it is some combination of
fuzzy terminology and uncertainty.
Now for the horizontal axis: the comparative evidence, E, corresponds
to how much evidence exists is support of the proposition compared to
how much evidence supports the opposite proposition. A +1 means that
there is overwhelming evidence for, and none against. A -1 means that
there is overwhelming evidence against and nothing for. A 0 could mean
there is equal amounts for and against, or there is no evidence either
way. [4] In a very real way, the E-value for any proposition reflects the
S-value of the sum of all evidence related to P.
There are two more features of the TGrid I'd like to point out before
trying to use it. There are two triangular shaded regions, one in the
upper left of the graph and another opposite the first in the lower
right. These regions are labelled "blind faith" and correspond to all
the points on the TGrid where an S-value deviates from the S-line by
more than 1. And what is the S-line? That represents the skeptical
position: that the strength of belief in a proposition should be
proportional to the evidence in support of that proposition. [5]
Now, given the TGrid, I can show how John's definition of faith differs
from mine, and why we both think the dictionary supports our own views.
I think John would agree that the more evidence there is supporting
a given proposition, the less faith is used. But he has also said that he
thinks that every belief contains an element of faith. Based on this, I
think John's version of faith can be modelled by the length of the line
from the point of belief, S(E), to the top of the grid, S=+1. The length
of the line is given by 1-S(E). In the TGrid image there is a point labelled
"b" in the upper right quadrant. Let's say that the E-value of point B
is about 0.4 and the S-value is slightly higher, 0.45. The length of the
faith line in this case is 0.55 (1-0.45), which could be interpreted as
a large component of faith, though a long way from blind faith. If, on
the other hand, the proposition is believed to be false given the same
amount of evidence, the length of the faith line would be measured from
the bottom edge (S = -1) giving a faith value of 1.45. This is clearly in
the region of blind faith.
My version of faith is measured in a similar fashion but with a different
base line. I think faith corresponds to how much the strength of belief
deviates from the skeptical position, i.e. how far the point lies from the
S-line: |S(E)-E|. In the case of point b on the grid, the faith component
is given by |0.45-0.4| = 0.05. There is still a faith component in that
belief, but far less than the one given John's version. The key difference
is that given my definition it is possible to reduce faith to 0 by staying
on the S-line, even if attaining an E-value of +1 is practically impossible.[6]
Now let's see how these two versions of faith fit with the dictionary
definition:
> From: John ''Storm of Drones'' Williamsby <prefect@tricon.net>
> Subject: Re: virus: Religion, Zen, post-structuralism, and the failure of , logic
> Date: Monday, June 23, 1997 8:07 PM
>
>My Webster's Encyclopedic Dictionary from 1992 defines "faith" as "complete
>exceptence of a truth that cannot be demonstrated or proved by the process
>of logical thought."
I take this to mean that faith describes the situation where the S-value is
+1 (complete acceptance of truth) and the E-value is 0 (cannot be
demonstrated). Using John's faith metric, this describes the line that lies
along the y-axis from the origin to the top, a length of 1.0 (i.e. complete
faith, though not blind faith which is impossible in the absence of evidence
one way or the other). Now using my version of faith, deviation from the
S-line, the answer is also 1.0 (complete faith). The dictionary is consistent
with both of our versions!
Enough for now. Does the framework make sense?
[1] Even though the diagram uses quantitative measures, they are assigned
subjectively.
[2] Hopefully the image will show up below in the hypermail archives.
[3] I'm not just talking money bets here. We often gamble our reputations,
careers, relationships and lives on our beliefs. Every time you step off a
curb you are betting your life that your senses are not deceiving you about
oncoming traffic, so the strength of belief value for that proposition would
be very high (0.9999 or more).
[4] An example of a proposition that I would give an E-value of 0.0 is
"there is intelligent life elsewhere in the universe". This is partly due to
fuzziness of definitions for "intelligent" and "life" but mostly due to
lack of information. I would also assign an E-value of 0.0 to "there
is snaffly broja elsewhere in the universe" because I have no idea
what that means.
[5] Sorry about any confusion between S-values and the S-line; I
wasn't thinking ahead when I drew the diagram.
[6] I think attaining an E-value of +1 is, in fact, practically impossible
for any empirical propositions. But it may be possible for special cases
like logical tautologies (if A then A) and mathematical theorems.
-- David McFadzean david@lucifer.com Memetic Engineer http://www.lucifer.com/~david/ Church of Virus http://www.lucifer.com/virus/