Mathematico-Deductive Theory (Clark L. Hull – 1943)

The contributions of three more individuals remain to be discussed in this section—the mathematico-deductive theory of Clark L. Hull (1884-1952), the contiguous conditioning theory of Edwin R. Guthrie (1886-1959), and the stimulus sampling theory of William K. Estes (b. 1919). All three are significant to the study of behavioral learning theory, but have virtually no direct influence on today’s practice of instructional design. With the exception of a few inline citations they are completely absent from the sampling of college undergraduate educational psychology books reviewed as part of this study (e.g. Eggen & Kauchak, 1999; O’Donnell et al., 2007; Ormrod, 2003; Sternberg & Williams, 2010; Woolfolk, 1998; 2010). It is difficult to say conclusively why these theories have not influenced instructional design in the same way that the theories of Thorndike, Pavlov, Watson and Skinner have. It may be that practitioners were turned off by the mathematical basis of the theories of Hull and Estes, and Guthrie’s theory may not have seemed a very good fit to the designing of goal-based curriculum, inasmuch as Guthrie was not concerned with the success or failure of achievements but rather with what he called ‘movements.’ Though not as influential as the foregoing theories, each provides an important perspective on learning, and provides valuable data to inform the identification of principles of learning for the present study. Each will be discussed briefly.

development of Hull’s theory can be divided into roughly three periods. From 1929 to 1943 the theory was published in several miniature formalizations in which portions of the projected theory were elaborated (see, for example Hull, 1942).  The second period, from 1943 to 1949, began with the publication of Principles of Behavior (Hull, 1943), which contained perhaps the most influential formulation of the theory. This formulation was radically revised by a research memorandum distributed in 1945 (see Koch, 1954, p. 1). During the third period the postulates were further revised and were published in 1951 in Essentials of Behavior (Hull, 1951). Koch (1954) gives a very detailed analysis of Hull’s theory in each of these three periods that is comparable to, and in agreement with, the conclusions made in Hull’s own posthumous publication of A Behavior System (1952).[1]

Hull’s theory was expressed in the form of postulates based on the following methodology:

The typical procedure in science is to adopt a postulate tentatively, deduce one or more of its logical implications concerning observable phenomena, and then check the validity of the deductions by observation. If the deduction is in genuine disagreement with observation, the postulate must be either abandoned or so modified that it implies no such conflicting statement. If, however, the deductions and the observations agree, the postulate gains in dependability. By successive agreements under a very wide variety of conditions it may attain a high degree of justified credibility, but never absolute certainty. (Hull, 1943, p. 15)

The postulates, as presented by Hull (1943) in Principles of Behavior, are summarized below.[2]

1. Postulate 1 – Sensory input (the afferent neural impulse) and the stimulus trace (afferent impulse decay):

When a stimulus energy (S) impinges on a suitable receptor organ, an afferent neural impulse (s) is generated and is propagated along connected fibrous branches of nerve cells in the general direction of the effector organs, via the brain. During the continued action of the stimulus energy (S), this afferent impulse (s), after a short latency, rises quickly to a maximum of intensity, following which it gradually falls to a relatively low value as a simple decay function of the maximum. After termination of the action of the stimulus energy (S) on the receptor, the afferent impulse (s) continues its activity in the central nervous tissue for some seconds, gradually diminishing to zero as a simple decay function of its value at the time the stimulus energy (S) ceases to act. (p. 47)

2. Postulate 2 – Interaction of afferent neural impulses:

All afferent neural impulses (s) active in the nervous system at any given instant, interact with each other in such a way as to change each into something partially different (s’) in a manner which varies with every concurrent associated afferent impulse or combination of such impulses. Other things equal, the magnitude of the interaction effect of one afferent impulse on a second is an increasing monotonic function of the magnitude of the first. (p. 47)

3. Postulate 3 – Innate behavior tendencies:

Organisms at birth possess receptor effector connections (SUR) which, under combined stimulation (S) and drive (D), have the potentiality of evoking a hierarchy of responses that either individually or in combination are more likely to terminate the need than would be a random selection from the reaction potentials resulting from other stimulus and drive combinations. (p. 66)

4. Postulate 4 – Habit strength[3] as a function of the temporal relation of the conditioned stimulus to the reaction:

Whenever an effector activity (r > R) and a receptor activity (S > s) occur in close temporal contiguity (sCr), and this sCr is closely associated with the diminution of a need (G) or with a stimulus which has been closely and consistently associated with the diminution of a need (Gdot), there will result an increment to a tendency (SHR) for that afferent impulse on later occasions to evoke that reaction. The increments from successive reinforcements summate in a manner which yields a combined habit strength (SHR) which is a simple positive growth function of the number of reinforcements (N). The upper limit (m) of this curve of learning is the product of (1) a positive growth function of the magnitude of need reduction which is involved in primary, or which is associated with secondary, reinforcement; (2) a negative function of the delay (t) in reinforcement; and (3) (a) a negative growth function of the degree of asynchronism (t’) of Sdot and R when both are of brief duration, or (b), in case the action of S’ is prolonged so as to overlap the beginning of R, a negative growth function of the duration (t”) of the continuous action of Sdot on the receptor when R begins. (p. 178)

5. Postulate 5 – Primary stimulus equivalence and stimulus generalization:[4]

The effective habit strength SHbarR is jointly (1) a negative growth function of the strength of the habit at the point of reinforcement (Sdot) and (2) of the magnitude of the difference (d) on the continuum of that stimulus between the afferent impulses of sdot and s in units of discrimination thresholds (j.n.d.’s); where d represents a qualitative difference, the slope of the gradient of the negative growth function is steeper than where it represents a quantitative differences. (p. 199)

6. Postulate 6 – Primary motivation:[5][6]

Associated with every drive (D) is a characteristic drive stimulus (SD) whose intensity is an increasing monotonic function of the drive in question. (p. 253)

7. Postulate 7 – Reaction potential:[7]

Any effective habit strength (SHbarR) is sensitized into reaction potentiality (SER) by all primary drives active within an organism at a given time, the magnitude of this potentiality being a product obtained by multiplying an increasing function of SHR by an increasing function of D. (p. 253)

8. Postulate 8 – Innate inhibition from primary negative drive:

Whenever a reaction (R) is evoked in an organism there is created as a result a primary negative drive (D); (a) this has an innate capacity (IR) to inhibit the reaction potentiality (SER) to that response; (b) the amount of net inhibition (IdotR) generated by a sequence of reaction evocations is a simple linear increasing function of the number of evocations (n); and (c) it is a positively accelerated increasing function of the work (W) involved in the execution of the response; (d) reactive inhibition (IR) spontaneously dissipates as a simple negative growth function of time (t”’). (p. 300)

9. Postulate 9 – Conditioned inhibition— the learned response of not responding:

Stimuli (S) closely associated with the cessation of a response (R) (a) become conditioned to the inhibition (IR) associated with the evocation of that response, thereby generating conditioned inhibition; (b) conditioned inhibitions (SIR) summate physiologically with reactive inhibition (IR) against the reaction potentiality to a given response as positive habit tendencies summate with each other. (p. 300)

10. Postulate 10 – Inhibitory potentiality varies from instant to instance:

Associated with every reaction potential (SER) there exists an inhibitory potentiality (SOR) which oscillates in amount from instant to instant according to the normal “law” of chance. The amount of this inhibitory potentiality associated with the several habits of a given organism at a particular instant is uncorrelated,  and the amount of diminution in SEbarR at the time available. (p. 319)

11. Postulate 11 – Momentary effective reaction must exceed reaction threshold:

The momentary effective reaction potential (SEbardotR) must exceed the reaction threshold (SLR) before a stimulus (S) will evoke a given reaction (R). (p. 344)

12. Postulate 12 – Probability of striated-muscle reaction evocation:

Other things equal, the probability (p) of striated-muscle reaction evocation is a normal probability (ogival) function of the extent to which the effective reaction potential (SEbarR) exceeds the reaction threshold (SLR). (p. 344)

13. Postulate 13 – The greater the momentary effective reaction potential, the shorter will be the latency between stimulus and response:

Other things equal, the latency (StR) of a stimulus evoking a striated-muscle reaction is a negatively accelerated decreasing monotonic function of the momentary effective reaction potential (SEbardotR), provided the latter exceeds the reaction threshold (SLR). (p. 344)

14. Postulate 14 – Extinction:

Other things equal, the mean number of unreinforced striated-muscle reaction evocations (n) required to produce experimental extinction is a simple linear increasing function of the effective reaction potential (SEbarR) provided the latter at the outset exceeds the reaction threshold (SLR). (p. 344)

15. Postulate 15 –  Amplitude of response:

Other things equal, the amplitude (A) of responses mediated by the autonomic nervous system is a simple linear increasing function of the momentary effective reaction potential (SEbardotR). (p. 344)

16. Postulate 16 –  Incompatible response:

When the reaction potentials (SER) to two or more incompatible reactions (R) occur in an organism at the same time, only the reaction whose momentary effective reaction potential (SEbardotR) is greatest will be evoked. (p. 344)

In later work, Hull also described the following influential ideas:

1. Incentive Motivation. In 1943, Hull assumed that the greater the amount of reward the greater the amount of drive reduction.

From these considerations, coupled with the amount-of-reinforcement hypothesis, it may be inferred that the successful reaction will be more strongly conditioned to the stimulus aggregate arising from a large piece of food than to that from a small one. Therefore, given a normal hunger drive, the organism will execute the correct one of several acts originally evoked by the situation more promptly, more vigorously, more certainly, and more persistently when a large amount of food is stimulating its receptors than when they are stimulated by a small amount. (pp. 131-132)

Since the amount of need reduction presumably varies with the amount of the reinforcing agent consumed by the organism, it follows as a strong probability from the dependence of reinforcement upon the amount of need reduction that the increment of habit strength (SHR) per reinforcement will be an increasing function of the amount of the reinforcing agent employed. (p. 132)

Experiments (such as those by Crespi (1942, 1944) and Zeaman (1949), as cited in Hergenhahn, 1982, p. 140) led Hull to conclude that organisms learn just as rapidly for a small reward as for a large one, but performance, once the behavior is learned, varies according to the size of the reward.

2. Stimulus-Intensity Dynamism. “The greater the intensity of the stimulus, the greater the probability that a learned response will be elicited” (Hergenhahn, 1982, p. 141).

3. Drive Reduction vs. Drive Stimulus Reduction. Hull’s theory was originally termed a drive reduction theory of learning. He revised this terminology to drive stimulus reduction for two reasons. The first is because of the latency between the time a drive-satisfying reward is introduced and the actual reduction of the drive itself. Hull decided that drive reduction was too far removed from the presentation of the reinforcement to explain how learning was taking place. The second reason is that a 1950 study reported by Sheffield and Roby (cited in Hull, 1952, p. 153) found that hungry rats were reinforced by nonnutritive saccharine—a substance that could not possibly have reduced the hunger drive. Hull concluded that the ingestion of the saccharine-sweetened water reduces hunger tension (the drive stimulus) but not the drive itself, and thus served as a reinforcer.

4. Fractional Antedating Goal Reactions. As Pavlov discovered, an organism can develop a conditioned response to stimuli experienced just prior to the behavior that is reinforced. It is this conditioned response to antecedent stimuli that Hull refers to as “anticipatory goal reactions” (1951, p. 24) or “fractional antedating goal reactions” (1952, pp. 124-155). This response is a fraction of the end goal response. This fractional response brings the organism closer to the end goal. As the response is made, the firing of kinesthetic receptors in the organism causes proprioceptive stimuli that simultaneously reinforce the response and stimulate an additional response.

Fractional antedating goal reactions were Hull’s answer to how maze learning occurred, suggesting that chains are established not only through instrumental conditioning, as proposed by Skinner, but also through classical conditioning—i.e., in Hull’s view, a combination of the two. The rat, having been rewarded in the end goal box of the maze, begins to associate, in anticipatory fashion, stimuli that it experienced just prior to entering the goal box with its own kinesthetic response that moved it into the goal box, and into contact with the reward. The reward reinforces the actions made in response to stimuli just prior to entering the goal box. The stimuli just prior to entering the goal box reinforce actions made just prior to encountering those stimuli.  Similarly, each set of stimuli encountered on the way to the goal box becomes associated with the prior response that brought the rat into that state and each serves as a cue for action. This chain continues all the way back to the start of the maze so that the stimulus of the start box is linked through one fractional antedating goal response after another to the goal box.

The sixteen postulates and four additional ideas stated above leave out the very lengthy details typical in Hull’s writing. As an example, Mathematico-deductive theory of rote learning (Hull, 1940), is filled with over three hundred pages of mathematical equations and proofs of theorems derived from the basic postulates. The ideas are conveyed primarily through symbols and symbolic relations more than the narrative prose common to the presentation of most learning theories, and familiar to most readers. Doubtless this resulted in the theory falling outside the grasp of, or beyond the limits of reasonable utility for, most educational practitioners.

In addition to the complexity of the theory, the fact that it had some fairly radical changes in a relatively short period of time, and the falling out of favor of ‘systems’ theories by the mid 1900s may also account for its lack of adoption in practical pedagogical application.


[1] Hull’s intention was to write a three volume series to cover “in an elementary manner the range of ordinary mammalian behavior” (Hull, 1952, p. vii). The first volume is Principles of Behavior (Hull, 1943), updates and revisions to which were published in Essentials of Behavior (Hull, 1951).   The second volume is A Behavior System (1952), the manuscript for which was written by Hull, but the proofing and print of which occurred after he passed away. The third volume was never written.

[2] For convenience of reference I have labeled them with titles which briefly summarize the core idea of the postulate, for example, “Sensory input and the stimulus trace,” Interaction of afferent neural impulses,” Innate behavior tendencies,” etc. However, the reader will please note that these titles are assigned by me, not Hull.

[3] Habit strength, one of Hull’s most important concepts, refers to the strength of the association between a stimulus and a response. As the number of pairings between the two goes up, the habit strength goes up. The mathematical formula that describes this relationship is as follows, where N is the number of pairings between stimulus and response:

SHR = 1 – 10-0.0305 N

[4] c.f. Thorndike’s theory of identical elements (Thorndike, 1914a, p. 268) and the transfer of training (p. 276).

[5] Hull (1943) explained that “significant empirical evidence…has led to the tentative conclusion that all primary drives produce their effects by the action of various chemicals in the blood” (p. 251):

Drive substances, such as the various endocrine secretions, are conceived either to be released into the blood by certain kinds of strong stimulation or as themselves initiating stimulation of resident receptors through their evocation of action by selected portions of the body, e.g., the intestinal tract and the genitalia. In both cases the energy effecting this receptor activation is called the drive stimulus (SD). (p. 252)

[6] Hergenhahn and Olsen (1982, p. 131-148) explained this postulate as follows:

Biological deficiency in the organism produces a drive state. Each drive state has specific stimuli associated with it (e.g. hunger pains, dry mouth, etc…). The existence of specific drive stimuli make it possible to teach an animal to behave in one way under one drive and another way under another.

[7] Hull’s formula for reaction potential:

Reaction Potential = SER = SHR x D

Operant Conditioning (Burrhus Fredric Skinner – 1938)Contiguous Conditioning (Edwin R. Guthrie – 1930) >

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