Is is an improvement of Weber's and Fechner's Laws
on
sensations and stimuli.
According to the neutrosophic theory, between
an
<idea> (= spiritual) and an <object> (= material)
there are infinitely many states.
Then , how can we mix an <idea> with an <object> and
obtain something in between: s% spiritual and m%
material (s+m = 100)?
[kind of chemical alloy].
Or, as Boethius, a founder of scholasticism, urged to
"join faith to reason" in order to reconcile the
Christian judgement with the rational judgement.
For example <mind> and <body> co-exist.
Gustav
Theodor Fechner, who inaugurated the experimental
psychology, obsessed with this problem, advanced the
theory that every object is both mental and physical
(psychophysics).
Fechner's Law, S = k log R, with S the sensation, R
the stimulus, and k a constant,
which is derived from Weber's Law, Delta R / R = k,
with Delta R the increment of stimulus just
detectable,
should be improved, because the function log R is
indefinitely increasing as R ---> infinity, to
ln R
S(R) = k -------, for R in [Rm, RM],
ln RM
and
S(R) = 0, for R in [0, Rm) union with (RM, infinite),
where k is a positive constant depending on three
parameters:
individual being, type of sensation, and the kind of
stimulus, and Rm, RM represent the minimum and maximum
stimulus magnitude respectively perceptible by the
subject,
the second one bringing about the death of sensation.
Fechner's "functional relation", as well as later
psychologists' power law R = kS^n, with n depending on
the kind of stimulus, were upper unbounded, while the
beings are surely limited in perception.
S: [0, infinite) ---> {0} union with [Sm, SM], with Sm,
SM
the minimum and maximum perceptible sensation
respectively; of course here subjectivity plays its role,
but we consider the average beings' perception extrema.
Of course Rm > 1, S(Rm) = Sm, and S(RM) = SM = k.
Ln, increasing faster, replaces log because the
sensation is more rapidly increasing at the beginning,
and later going on much slower.
At R = RM, S attaints its maximum, beyond whom it
becomes flat again, falling to zero.
The beings have a low and high threshold respectively,
a range where they may feel a sensation.
Graph of Fechner's Law Improvement:
/\
S |
|
----------------------.------
SM |
.
|
.
| .
| .
| .
| .
--------.----------------------
Sm |
|
|......______________........>
0 Rm
RM R
For example in acoustics: a sound is
not heard
at the beginning and, if it constantly keeps enlarging
its intensity, at a given moment we hear it, and for a
while its loudness increases in our ears, until the
number of decibels - getting bigger than
our possibility of hearing - breaks our eardrums...
We would not hear anything anymore, our sensation
died...
Now, if at a given moment t0 the stimulus R
remains constant equal to R0 (between the conscious
limits of the being, for a long period of time t), and
the sensation S(R0) = c, then we get the following
formulas:
In the case when the stimulus in not physically or
physiologically damaging the individual being:
Sdec(t) = c log1/e(t+1/e) = -c ln(t+1/e),
for 0 <= t <= exp(-Sm/c)-1/e, and 0 otherwise;
which is a decreasing function;
In the case when the stimulus is hurting the
individual being:
Sinc(t) = c ln(t+e), for 0 <= t <= exp(SM/c)-e, and 0
otherwise;
which is an increasing function until the sensation
reaches its upper bound;
where c, as a constant, depends on individual being,
type of sensation, and kind of stimulus.
Examples:
a) If a prisoner feels a constant smell in his closed
room for days and days, isolated from the exterior,
and he doesn't go outside to change the environment,
he starts to feel it less and less and after a
critical moment he becomes inured to the smell
and do not feel it anymore -
thus the sensation disappears under the low
perceptible limit.
b) If a water drop licks constantly, at the same
interval of time, with the same intensity, on the head
of a prisoner tied to a pillar, the prisoner after a
while will feel the water drop heavier and heavier,
will mentally get ill and out of his mind,
and will even physically die - therefore again
disappears the sensation, but above the high
limit. See how one can kill someone with a... water
drop!
c) If one permanently plays the same song for days and
days to a person enclosed in a room without any other
noise from outside, that person will be driven crazy,
even psychologically die, and the sensation will
disappear.
Weber's Law can be improved to Delta R / ln
R =
k, with R defined on [Rm, RM], where k is a constant
depending on individual being, type of sensation, and
kind of stimulus, due to the fact that the relative
threshold Delta R increases slower with respect to R.
Reference:
Smarandache, Florentin, "Neutrosophy. / Neutrosophic
Probability, Set, and Logic", American Research
Press,
Rehoboth, 1989.