Clark Hull and his Mathematical Model of Learning
Biography: Clark Hull was born near Akron, New York, but spent most of his boyhood in Michigan. He attended the academy associated with Alma College before enrolling at the University of Wisconsin. In his 20s, he contracted typhoid fever and polio, illnesses that left him with lifelong disabilities including a withered leg. Gifted with mechanical ability, he designed and made his own leg brace to compensate. Later, he designed an automated correlation calculating machine (Hull, 1925) to aid his work on aptitude testing. After holding a series of jobs, he returned to Wisconsin and received a PhD in psychology in 1918. He remained there until 1929, when he moved to Yale’s Institute of Human Relations as a research professor. Before arriving at Yale, his research was eclectic and included books on aptitude testing (Hull, 1928) and hypnosis and suggestibility (Hull, 1933a). He also was interested in concept formation and verbal learning.
Contributions: Hull’s interests turned strictly to rat learning after arriving at Yale. He spent the rest of his career providing an alternative to Tolman’s line of research while providing a synthetic theory that combined aspects of Thorndike’s law of effect and Pavlovian conditioning. Like Tolman, Hull believed that much could be learned about human behavior by running laboratory experiments using the white rat. Although his theory was extremely influential during his lifetime, it is of only historical interest today. Nevertheless, any history of psychology that neglects or omits it is incomplete. Additionally, failure to understand Hull and his influence makes it difficult to see properly how subsequent attempts to understand learning developed.
Hull wished to make psychology as scientific an enterprise as physics. His two models were Newton’s Principia and Euclid’s Elements. From both, he adopted the hypothetico-deductive system and the tight logic of inferred theorems constructed from a minimal set of a priori postulates and definitions. He believed that psychology would advance only when theory and observations were closely linked. Then, those types of investigations would yield “facts of intrinsic importance” while “indicating the truth or falsity of the theoretical system from which the phenomena were originally deduced” (Hull, 1935, p. 493). Later (pp. 512–513) he wrote:
"Scientific theory in its best sense consists of the strict logical deduction from definite postulates of what should be observed under specified conditions. If the deductions are lacking or are logically invalid, there is no theory; if the deductions involve conditions of observation which are impossible of attainment, the theory is metaphysical rather than scientific; and if the deduced phenomenon is not observed when the conditions are fulfilled, the theory is false."
Systematically, he set out to do for psychology what Newton had done for physics and Euclid had done for geometry. He used Thorndike’s law of effect and Pavlov’s analysis of classical conditioning as a starting point. He retained Watson’s S-R model but added intervening variables. He anchored his intervening variables, via operational definitions, to both the S side and the R side of the S-R formulation, something he claimed other theorists failed to do. In two major books (Hull, 1943; Hull, 1952), he specified the details of his system. True to his advice, the later book incorporated theoretical changes forced by the accumulation of new experimental data. His system was dynamic, designed to change in the face of unexpected new data. In explicit contrast to Tolman’s approach, Hull wanted to explain learning through the interaction of stimulus variables and intervening variables only. Purposive behavior had no place in his system. Ultimately, his system failed to explain learning. But, while he was alive, his system inspired a large number of psychologists to pursue his vision of a mechanistic explanation for learning. It is worthwhile to briefly examine some of Hull’s variables and how they interacted.
Hull’s system was complex. Marx and Hillix (1963, p. 247) described its final form as including:
"…a total of eighteen postulates and twelve corollaries was produced. In accordance with the hypothetico-deductive procedure that Hull intended to follow, these primary principles were to be used deductively to predict secondary principles, such as “goal gradient and latent learning”
The basic structure of the system consisted of three types of variables: stimulus, organismic or intervening, and response. The four stimulus variables were measurable. They were the number of reinforced trials, stimulus deprivation level, stimulus intensity, and size of the reinforcer.
Each of these, in turn was connected to a corresponding intervening variable, habit strength sHr, drive D, stimulus intensity dynamism V, and incentive K, respectively. Together, those four variables accounted for acquisition of a learned response, its maintenance, or its decline. Their mathematical relationship was multiplicative; thus should any one of them drop to zero then the product of all of them would be zero as well. Other intervening variables accounted for extinction and spontaneous recovery: reactive inhibition Ir and conditioned inhibition sIr; individual differences, oscillation sOr; and consistency of learned response, threshold sLr.
The mathematical equation of the intervening variables just mentioned equaled yet another one: overall or net reaction potential sĒr. The response variables, too, were measurable. They were response latency str, amplitude A, number of responses until extinction n, and response probability p. These last variables, naturally enough, were the ones measured as the rats interacted with the laboratory apparatus under various experimental conditions. Here are all of the intervening variables and their mathematical relationships in the final version of Hull’s equation:
sĒr = (sHr × D × V × K) − Ir − sIr − sLr +/− sOr
The biggest change between the final version and previous versions was the addition of incentive (K). Hull added incentive because of experiments by Crespi (1942) that demonstrated that rats ran faster when the food reward in the goal box was made larger and slower when it was made smaller.
Hull was an S-R theorist. He believed that learning was strengthened by repetition (through habit strength) and that reinforcement was related to the satisfaction of internal drive states such as hunger and thirst. In other words reinforcement was a biological drive and as it was satisfied it played less of a role in predicting behavior. But, later as it renewed itself as a drive, its role grew again. Think of hunger as an example. Food only acts as a reinforcer when hunger is high. Extinction was accounted for by the rapid accumulation of reactive inhibition following unreinforced trials. Spontaneous recovery occurred because reactive inhibition was only temporary.
Criticism of Hullian formulations led to further patchwork and repair of the theory. After Hull’s death in 1952, interest by others dropped considerably. His most prominent student, Kenneth Spence, carried on Hull’s tradition but very quickly dropped his support for the drive reduction view of reinforcement and was much less concerned about maintaining the formal structure Hull had created.
Eleanor Gibson, too, was one of Hull’s students. She came to him after being rejected as a potential graduate student by Yerkes; he rejected any female student (Rodkey, 2010). Her dissertation was couched in Hullian terminology although her underlying views were more functional. Later in her career she carved out a research career at Cornell University because Smith College, where her husband James Gibson worked, would not allow the hiring of married couples. At Cornell, she and Richard Walk developed the visual cliff apparatus, at first working with rats and eventually with human infants. She eventually was hired as a professor at Cornell and received the National Medal of Science in 1992, an award rarely given to psychologists.