Invariance principles underlie many key theories in modern science. They provide the explanatory and predictive framework necessary for the rigorous study of natural phenomena ranging from the structure of crystals, to magnetism, to relativistic mechanics. Vigo (2008, 2009)introduced a new general notion and principle of invariance from which two parameter-free (ratio and exponential) models were derived to account for human conceptual behavior. Here we introduce a new parameterized exponential “law” based on the same invariance principle. The law accurately predicts the (...) subjective degree of difficulty that humans experience when learning different types of concepts. In addition, it precisely fits the data from a large-scale experiment which examined a total of 84 category structures across 10 category families (R-Squared =.97, p < .0001; r= .98, p < .0001). Moreover, it overcomes seven key challenges that had, hitherto, been grave obstacles for theories of concept learning. (shrink)

Two theories of propositional deductive reasoning are considered: the mental models of P. N. Johnson-Laird et al and the mental logic of M. D. Braine. The model theory is said to account for practically all of the known phenomena of deductive propositional reasoning, offer a general theory of conditionals, account for the most important aspects of Braine's theory, and predict new phenomena that rule theories cannot explain. It is argued that the model theory is flawed in a way that is (...) difficult to overcome, conditionals are seriously misrepresented, the algorithms proposed to implement it either allow invalid inferences or are psychologically useless, Braine's theory accounts for all of the new phenomena worth considering, and the model theory can predict Braine's results only at the cost of self-refutation. It is concluded that the mental model theory of propositional reasoning offers no reason to reject the program of mental logic. (shrink)

An alternative account of human concept learning based on an invariance measure of the categorical stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or size. To do this we introduce a mathematical framework based on the notion of a Boolean differential operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the category in respect (...) to its dimensions. Using this framework, we propose that the structural complexity of a Boolean category is indirectly proportional to its degree of categorical invariance and directly proportional to its cardinality or size. Consequently, complexity and invariance notions are formally unified to account for concept learning difficulty. Beyond developing the above unifying mathematical framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications. Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the degree of learning difficulty of a large class of categories (in particular, the 41 category types studied by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature, 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g., cognitively tractable). (shrink)

What is the relationship between the degree of learning difficulty of a Boolean concept (i.e., a category defined by logical rules expressed in terms of Boolean operators) and the complexity of its logical description? Feldman [(2000). Minimization of Boolean complexity in human concept learning. Nature, 407(October), 630–633] investigated this question experimentally by defining the complexity of a Boolean formula (that logically describes a concept) as the length of the shortest formula logically equivalent to it. Using this measure as the independent (...) variable in his experiment, he concludes that in general, the subjective difficulty of learning a Boolean concept is well predicted by Boolean complexity. Moreover, he claims that one of the landmark results and benchmarks in the human concept learning literature, the Shepard, Hovland, and Jenkins learning difficulty ordering, is precisely predicted by this hypothesis. However, in what follows, we introduce a heuristic procedure for reducing Boolean formulae, based in part on the well-established minimization technique from Boolean algebra known as the Quine–McCluskey (QM) method, which when applied to the SHJ Boolean concept types reveals that some of their complexity values are notably different from the approximate values obtained by Feldman. Furthermore, using the complexity values for these simpler expressions fails to predict the correct empirical difficulty ordering of the SHJ concept types. Motivated by these findings, this note includes a brief tutorial on the QM method and concludes with a brief discussion on some of the challenges facing the complexity hypothesis. (shrink)