Some unifying fixed point principles

This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of metamathematics, recursion theory or combinatory logic is presupposed, although this paper deals with quite general abstractions of standard results in those three areas. Our purpose is to show how some apparently diverse results in these areas can be derived from a common construction. In Section 1 we consider five classical fixed point arguments (or rather, generalizations of them) which we present as problems that the reader might enjoy trying to solve. Solutions are given at the end of the section. In Section 2 we show how all these solutions can be obtained as special cases of a single fixed point theorem. In Section 3 we consider another generalization of the five fixed point results of Section 1 and show that this is of the same strength as that of Section 2. In Section 4 we show some curious strengthenings of results of Section 3 which we believe to be of some interest on their own accounts.     

Some new double induction and superinduction principles

Some new double analogues of induction and transfinite recursion are given which yields a relatively simple proof of a result of Robert Cowen, [2] which in turn is a strengthening of an earlier result of Smullyan [1], which in turn gives a unified approach to Zorn's Lemma, the transfinite recursion theorem and certain results about ordinal numbers.     

On the adequacy of current empirical evaluations of formal models of categorization

Categorization is one of the fundamental building blocks of cognition, and the study of categorization is notable for the extent to which formal modeling has been a central and influential component of research. However, the field has seen a proliferation of noncomplementary models with little consensus on the relative adequacy of these accounts. Progress in assessing the relative adequacy of formal categorization models has, to date, been limited because (a) formal model comparisons are narrow in the number of models and phenomena considered and (b) models do not often clearly define their explanatory scope. Progress is further hampered by the practice of fitting models with arbitrarily variable parameters to each data set independently. Reviewing examples of good practice in the literature, we conclude that model comparisons are most fruitful when relative adequacy is assessed by comparing well-defined models on the basis of the number and proportion of irreversible, ordinal, penetrable successes (principles of minimal flexibility, breadth, good-enough precision, maximal simplicity, and psychological focus).     

Reasoning with conditionals: a test of formal models of four theories

The four dominant theories of reasoning from conditionals are translated into formal models: The theory of mental models (Johnson-Laird, P. N., & Byrne, R. M. J. (2002). Conditionals: a theory of meaning, pragmatics, and inference. Psychological Review, 109, 646-678), the suppositional theory (Evans, J. S. B. T., & Over, D. E. (2004). If. Oxford: Oxford University Press), a dual-process variant of the model theory (Verschueren, N., Schaeken, W., & d'Ydewalle, G. (2005). A dual-process specification of causal conditional reasoning. Thinking &Reasoning, 11, 278-293), and the probabilistic theory (Oaksford, M., Chater, N., & Larkin, J. (2000). Probabilities and polarity biases in conditional inference. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26, 883-899). The first three theories are formalized as multinomial models. The models are applied to the frequencies of patterns of acceptance or rejection across the four basic inferences modus ponens, acceptance of the consequent, denial of the antecedent, and modus tollens. Model fits are assessed for two large data sets, one representing reasoning with abstract, basic conditionals, the other reflecting reasoning with pseudo-realistic causal and non-causal conditionals. The best account of the data was provided by a modified version of the mental-model theory, augmented by directionality, and by the dual-process model.     

Six principles for biologically based computational models of cortical cognition

This review describes and motivates six principles for computational cognitive neuroscience models: biological realism, distributed representations, inhibitory competition, bidirectional activation propagation, error-driven task learning, and Hebbian model learning. Although these principles are supported by a number of cognitive, computational and biological motivations, the prototypical neural-network model (a feedforward back-propagation network) incorporates only two of them, and no widely used model incorporates all of them. It is argued here that these principles should be integrated into a coherent overall framework, and some potential synergies and conflicts in doing so are discussed.     

Some topological properties of paraconsistent models

In this work, we investigate the relationship between paraconsistent semantics and some well-known topological spaces such as connected and continuous spaces. We also discuss homotopies as truth preserving operations in paraconsistent topological models.     

Some deterministic models of concept identification

It is shown that deterministic models can compete effectively with stochastic models in summarizing concept identification behavior. Three groups of deterministic models are examined. Examination of individual learners' trial by trial behavior in a concept experiment shows: (1) One person exhibited behavior consistent with a Hypothesis Permutation (HP) model despite being a nonlearner who showed no evidence of improvement over a period of 24 trials. However, when all 50 persons studied in each of two treatment groups were examined, only 22 members of one group and 10 of the other showed no inconsistencies with deterministic local consistency assumptions. (2) Certain deterministic computer programs could find at least one satisfactory order for predicting all responses by 18 of the 22 consistent solvers and 6 of the 10 consistent solvers, respectively, in the two groups just mentioned. For these 24 persons, then, a less restrictive deterministic model is adequate than for the others. (3) Those 38 original members of the first treatment group who met a stringent learning criterion were compared with respect to predictions generated by stochastic and mathematized deterministic models. One deterministic model (RSS-U 9-state) is in some respects the best of the models examined, but this success is a partial reflection of estimating eight parameters from the data.     

Some evidence on additive learning models

Deterministic and probabilistic additive learning models forsignal detection/recognition replace the fixed criterion of classical detection models with one which shifts from trial to trial in the light of the preceding trial events. Data from a sinusoid in noise detection task without feedback and an auditory amplitude recognition task with feedback are used to test these models with respect to their predictions about asymptotic response frequency, and, where possible, by likelihood ratio tests. These and some previous experiments show that whether or not feedback is given subjects do not universally probability match, overmatch, undermatch, or keep response probability constant over discriminability, so that none of the testable special models can fit more than a proportion of subjects. The likelihood ratio tests confirm this conclusion for the special deterministic models. The six-parameter general deterministic model does nonsignificantly better than an ad hoc six-parameter response runs model in fitting the recognition data and significantly better than the five-parameter memory recognition model of Tanner, Rauk, andAtkinson (1970). Monte Carlo methods are used to confirm the applicability of asymptotic response frequency results to practically feasible sample sizes.     

Some multi-attribute models in examination assessment

Examination assessment can be likened to multi-attribute models in value theory by treating candidates as alternatives that have to be compared and examination components as the different attributes on which they are assessed. Then the mark combination process is equivalent to a value function, the main difference being that in this context the examiners' judgement is modelled rather than their preferences. This paper consists of five parts. In the first, a formal representation of the problem is given. In the next three sections, three additive representations are studied, each involving a different set of properties for the marks and the combination process. Finally, there is a short discussion and comparison of the three representations.     

Some local models for correlation experiments

This paper constructs two classes of models for the quantum correlation experiments used to test the Bell-type inequalities, synchronization models and prism models. Both classes employ deterministic hidden variables, satisfy the causal requirements of physical locality, and yield precisely the quantum mechanical statistics. In the synchronization models, the joint probabilities, for each emission, do not factor in the manner of stochastic independence, showing that such factorizability is not required for locality. In the prism models the observables are not random variables over a common space; hence these models throw into question the entire random variables idiom of the literature. Both classes of models appear to be testable.Work on this paper was supported, in part, by National Science Foundation Grant SES 79-25917.     

Some results on random utility models

This paper extends earlier work on the properties of Thurstone choice models to the general random utility model (RUM), in which the components of the random utility vector are not necessarily independent and identically distributed, even apart from location shifts. The condition for two RUMs to be equivalent is established, and relationships between the choice axiom, decomposition, and other conditions are explored. In the special case of the Thurstone model, the conditions considered are all equivalent to the double exponential distribution.