LOGAN'S LAWS of LEARNING---Preface

The user is urged to read the preface to the Quad-L Database and
to consult the Glossary-and-Index for an orientation to the way 
the information in the database has been organized into these
"laws."  

The development of a systematic analysis of the "laws of learning"
is very incomplete at the end of the twentiety century.  Partly, I
believe, this is because the Philosophy of Science for Psychology 
has not fully captured some of the fundamental issues we must yet
resolve.  One of these concerns interdependence:

INTERDEPENDENCE POINT

     For three years, my colleagues and I interacted intensively in
search of commonalities and differences among our behavior sciences 
(physiological psychology, experimental psychology, sociology, 
economics, and anthropology/linguistics).  During this time, several
"points" arose with sufficient frequency that we gave them names;
thereafter, one had only to call attention to the relevance of a
point to the topic under discussion and it was unnecessary to review
its implications in detail.

     For example, "continuum point" arises when one appears to treat 
a continuum as if it were a dichotomy.  Although it may be convenient
to establish an arbitrary cut-off and divide instances as being above
or below that value, one must always keep in mind that the underlying
process is continuous in nature.  People who refer to statistical
significance in absolute terms need to be reminded of continuum point.

     Another example is "measurement point."  Most behavioral measures
are relative, arranging instances on an ordinal scale.  When one
treats with ordinal numbers as if they were a cardinal, or even a
ratio scale, the results may be misleading.  People who refer to
statistical interaction between differences at different points along
an ordinal scale need to be reminded of measurement point.

     The purpose of this essay is to develop the concept of "inter-
dependence point."  I believe that this point has very wide relevance
to any analysis of the behavior of organisms but it is rarely made
explicit.  I hope to illustrate some ways in which interdependence 
point enriches or possibly modifies one's analysis.


Historical Background

     When psychologists began to follow the lead of Watson and define
Psychology as a natural science, they chose Newtonian Physics as their
model.  Correlations were useful for making predictions, but an
experimental analysis was required to establish causality.  In the
prototypical experiment, all known relevant factors were held constant
save one, the "independent variable," and if there were systematic
changes in behavior, the "dependent variable," associated with changes
in the independent variable, then unidirectional causality could be
legitimately inferred.

     In physics, for example, metal balls could be rolled down inclined
planes and the measured speed, the dependent variable, was found to 
depend on the angle of incline, the independent variable.  It was easy
to infer that Speed=f(angle) because the flatter the angle, the greater
the resistance to the force of gravity.  In psychology, rats could be
run down straight-alley mazes and the measured speed was found to 
depend on the amount of reinforcement available in the goal box.  It
was easy to infer that Speed=f(Reinforcement) because the larger the
reward, the greater the animal's motivation.

     The relationship of instrumental response speed to the amount of
reinforcement has been observed in enough contexts to qualify as one
of the basic "Laws of Learning."  And, following Hull (1952), this 
relationship could be rationalized in a formal theory of behavior by 
the following equation:

          Speed(IR) = f(sEr) = f(sHr x D x (K x J) - Ir)         (1)

in which IR refers to the instrumental running response, sEr is the
total excitatory potential to make that response, sHr is habit
strength dependent on the number of times the response has been
practiced, D is drive motivation based on time of deprivation of the
relevant commodity, K (based on the amount of reinforcement) and
J (based on the delay of reinforcement) combine to determine the
incentive motivation, and Ir is reactive inhibition based on a
fatigue notion.

     The critical feature of the above formulation is that it is
unidirectional.  The rat's speed of running is a dependent variable
that is determined by the values of the several independent
variables.  In principle, quantitative values could be estimated
for the relevant functions and response speed precisely predicted.

Interdependence:  Correlated Reinforcement

     In order to test the basic premise of the historical approach,
I ran some rats in a straight alley maze under conditions of
negatively correlated reinforcement.  Under those conditions, the
amount of reinforcement depends on the rat's speed; the faster it
runs, the smaller the reinforcer.  Now, if speed is truly determined 
by the amount of reinforcement, and the conditions vary the amount 
of reinforcement with speed, then there is only one possible outcome.
The rat must run at that speed that results in the amount of 
reinforcement that, in turn, determines that same speed.  

     The analysis is conceptually elegant but empirically false.
Rats run slower than predicted by the above analysis, getting an
amount of reinforcement that would normally result in much faster
speeds.  It is instructive to contrast those results with what
would happen with a comparable experiment involving balls rolling
down inclined planes.  If one made the angle of inclination depend
inversely on speed, the balls would not learn to roll slowly down
a steeply inclined plane; instead, over trials the angle would home 
in on the one value that would keep the angle and speed in equilibrium.  
Speed of inanimate objects in physics is not the same as speed of an
animate organism in psychology.

     One response to these findings is exemplified by Hull (1952).
He contended that the conventional approach was appropriate when the
experimenter controlled the amount of reinforcement, but when the
conditions involved correlated reinforcement, the analysis needed to
be made at what he called the "micromolar" level.  At that level,
different speeds are considered to be different responses and speed,
rather than being used as a measure of response strength, becomes
part of what is learned.


The micromolar approach

     I argued that the true principles of behavior do not change
simply because an experimenter changes the conditions of reinforce-
ment.  If speed is part of what is learned in one situation, it
must be part of what is learned in all situations.  Furthermore,
although in the aforementioned study in which speed was found to
be an increasing function of the amount of reward, the amount was
not correlated with speed, another dimension of the reinforcer,
namely its delay was correlated with speed.  The faster the rat
runs, the sooner it gets the reward.  Surely one does not want to
contend that speed is part of what is learned as regards the delay
of reinforcement and not as regards the amount of that same
reinforcer.

     Accordingly, I proposed that one should adopt the micromolar
approach for both correlated and uncorrelated reinforcement.  It
turns out that Hull's original formulation can be modified to fit
this approach:

          sErx = sHrx x D x (sKrx x sJrx) - sFrx                (2)

where the (x) subscripts indicate responses of different speeds.  
The only other change is that Hull's reactive inhibition (Ir) 
has been changed to an anticipatory effort variable (sFr).  The 
analysis then predicts that the organism will tend to choose 
that speed with the largest sEr.

     Descriptively, for the original condition there are only two
factors that vary with speed, sJr and sFr.  Hence, the rat will
tend to run at that speed at which the marginal utility of getting
the reward a bit sooner equals the marginal disutility of the
additional effort required to run a bit faster.  And what the
original findings show is that the marginal utility of delay
depends on the amount of reinforcement; rats will expend more effort
to get a larger reward sooner.  This is captured by the multipli-
cative relation of sKr and sJr.

     For the condition of correlated reinforcement, sKr also varies
with speed.  Hence, the rat will now learn to run at that speed at
which the marginal utility of a somewhat larger reward equals the
marginal disutility of waiting a bit longer to get it.  (Effort may
be involved to some extent, but not as much as with positively
correlated reinforcement where fast speeds are required to get a
large reward.)  The mathematical complication is that the disutility
of delay is getting larger as the potential amount gets larger.  But
the really important idea is that the same analysis can be used for
both the situation where the experimenter controls the amount of
reinforcement and where the organism controls that variable.


     (The term "micromolar" requires some elaboration.  Responses
differ quantitatively and qualitatively (topographically).  I believe
that all properties of the response are learned, that organisms
learn "movements" rather than "acts" at particular speeds and
forces.  But in the typical experimental context, the qualitative
features of the response are not easily distinguished and hence the
analysis is "molar" rather than "molecular".  Thus, if the conditions
require a rat to take five seconds getting to the food cup in order
to get reinforced, no distinction is made amoung the many ways that
the rat might consume that time.  Indeed, it might run very fast up
and down the alley for five seconds before completing the response,
and it would still be considered a "slow" response.)


Interdependence Point

     Generalizing on the preceding analyses, I now venture what may
be called "interdependence point:"

          a.  The inference that S is the unidirectional cause 
     of R from the empirical relationship, R=f(S) is inappropriate 
     when R and S are interdependent.

          b.  When the effect of a variable when controlled by 
     the environment differs from its effect when controlled by 
     the organism, a single analysis based on interdependence 
     is appropriate.

          c.  In such an analysis, quantitative as well as 
     qualitative dimensions of the response are treated as 
     part of what is learned.


    In calling interdependence point, one must be cautious about
several similar concepts.  Indeed, for clarification, it might be
better to use the term, "interdetermination" point of "intercontrol"
point.  In a statistical interaction between two independent
variables, the effect of each depends on the value of the other.
Hence, it would be legitimate to say that variables that interact
are, in some sense, interdependent.  For example, as noted above,
the effect of delay of reinforcement depends on the amount of
reinforcement, but the actual delay is not determined (controlled)
by the amount.  The present point does not apply to interactions.

     There are also other kinds of interrelationships for which the
term interdependence might be appropriate but to which the present
point does not apply.  For example, a symbiotic relationship in
which one organism depends on another for some service while
reciprocating with some other service could be called interdependent.
As another example, interdisciplinary relatinships such as that 
between psychology and economics involve interdependence in the
sense that economic affairs are dependent on the behavior of
organisms in the economy, and some behavior depends on the economic
system that prevails.  To untangle all possible types of inter-
relationships is beyond my analytic skill, but interdependence
point is restricted to those in which each variable determines
and is determined by the other variable.


              The Generality of Interdependence Point

     I believe that interdependence point is of very widespread 
generality in the behavior sciences generally and in psychology
in particular.  I say this in part because the very terms,
stimulus and response, are themselves interdependent.  To
determine whether an event is a stimulus, it must be shown
capable of determining the occurrence of some response, and to
determine that an action is a response, it must be shown that
it is determined by some stimulus.  (An effort to condition
hair growth to a magnetic field would fail on both counts.)  
But there are several contexts that illustrate this generality 
more specifically.

The tickle dilemma.

     Among the many perplexing childhood observations is the fact
that one simply cannot tickle oneself.  Of course, some people
are not ticklish at all, but even the most ticklish of people
cannot do it to themselves.  Here then is a instance where the
effect of a stimulus differs when controlled by the person from
when it is controlled by someone else.

     This dilemma affords the opportunity to introduce another
point, namely "question point."  This point is that, when one is
baffled by an apparently unanswerable question, sometimes the
answer can be inferred from the inverse question.  One familiar
example is statistical inference; one cannot readily test whether
an obtained difference is general, but one can test the inverse
(null) hypothesis, and to the extent that one can reject that
hypothesis, one can infer the generality of the result.

     In the case of the tickle dilemma, rather than asking why
you cannot tickle yourself, perhaps you should ask why someone
else can tickle you.  That question is at least amenable to the
hypothesis that "tickle" is not an intrinsic property of certain
tactual stimuli.  Conceivably, being ticklish was acquired in 
early infancy as a result of appropriate tactual stimuli being 
associated with a playful, giggling interaction.  This would
account for individual differences in ticklishness and it would
imply that ticklishness may be subject to extinction.  

     Of course, I do not know whether this is indeed the solution
to the tickle dilemma but I included it to emphasize another
difference between Newtonian Physics and Psychology.  It has long
been recognized that the reflexive S-R is actually S-O-R and that
the effect of a putative stimulus depends on how it is perceived 
by the organism.  This "perception point" is an important adjunct 
to interdependence point.

     For example, the original "law of learning" relating 
instrumental performance to amount of reinforcement is actually 
restricted to naive organisms.  An intermediate amount of 
reinforcement looks large to an organism accustomed to smaller 
einforcements but small to one accustomed to larger reinforcements.  
More generally, behavioral laws are rarely independent of path.  
Although the speed of a ball rolling down an inclined plane is 
independent of its past history of rolling down other inclines, 
he speed of a rat running down an alley maze depends importantly 
of its past history with that and/or other reinforcers.