Entailment with Near Surety of Scaled Assertions of High Conditional Probability

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given A B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl"s System-Z ⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL.     

Likelihood inference on the underlying structure of IRT models

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.We thank Dr. Gorman and Dr. Rogers of the Educational Testing Service for providing the data analyzed in Section 4. We also thank three reviewers for comments and suggestions.This revised article was published online in August 2005 with the PDF paginated correctly.     

Transitive responding in hooded crows requires linearly ordered stimuli

Eight crows were taught to discriminate overlapping pairs of visual stimuli (A+ B-, B+ C-, C+ D-, and D+ E-). For 4 birds, the stimuli were colored cards with a circle of the same color on the reverse side whose diameter decreased from A to E (ordered feedback group). These circles were made available for comparison to potentially help the crows order the stimuli along a physical dimension. For the other 4 birds, the circles corresponding to the colored cards had the same diameter (constant feedback group). In later testing, a novel choice pair (BD) was presented. Reinforcement history involving stimuli B and D was controlled so that the reinforcement/nonreinforcement ratios for the latter would be greater than for the former. If, during the BD test, the crows chose between stimuli according to these reinforcement/nonreinforcement ratios, then they should prefer D; if they chose according to the diameter of the feedback stimuli, then they should prefer B. In the ordered feedback group, the crows strongly preferred B over D; in the constant feedback group, the crows' choice did not differ significantly from chance. These results, plus simulations using associative models, suggest that the orderability of the postchoice feedback stimuli is important for crows' transitive responding.     

Experience and inference: how far will science carry us?

This paper begins with a view of the remarkable understanding of infant and child development that has evolved from research and observation. The limitations of this contribution from science to the multi-dimensional context-based individuality of each human in his or her intersubjective realm are then considered. For a contemporary view we must recognize the influence of the variability of experiences and the inferences drawn from them. Inferences involve symbolization and culturally derived archetypes as illustrated in a clinical example.     

The influence of hierarchy on probability judgment

Consider the task of predicting which soccer team will win the next World Cup. The bookmakers may judge Brazil to be the team most likely to win, but also judge it most likely that a European rather than a Latin American team will win. This is an example of a non-aligned hierarchy structure: the most probable event at the subordinate level (Brazil wins) appears to be inconsistent with the most probable event at the superordinate level (a European team wins). In this paper we exploit such structures to investigate how people make predictions based on uncertain hierarchical knowledge. We distinguish between aligned and non-aligned environments, and conjecture that people assume alignment. Participants were exposed to a non-aligned training set in which the most probable superordinate category predicted one outcome, whereas the most probable subordinate category predicted a different outcome. In the test phase participants allowed their initial probability judgments about category membership to shift their final ratings of the probability of the outcome, even though all judgments were made on the basis of the same statistical data. In effect people were primed to focus on the most likely path in an inference tree, and neglect alternative paths. These results highlight the importance of the level at which statistical data are represented, and suggest that when faced with hierarchical inference problems people adopt a simplifying heuristic that assumes alignment.     

Transitive choices by a simple, fully connected, backpropagation neural network: implications for the comparative study of transitive inference.

In search of the minimal requirements for transitive reasoning, a simple neural network was trained and tested on the non-verbal version of the conventional "five-term-series task" – a paradigm used with human adults, children and a variety of non-human species. The transitive performance of the network was analogous in several aspects to that reported for children and animals. The three effects usually associated with transitive choices i.e. "symbolic distance", "lexical marking" and "end-anchor", were also clearly shown by the neural network. In a second experiment, where the training conditions were manipulated, the network failed to match the behavioural pattern reported for human adults in the test following an ordered presentation of the premises. However, it mimicked young children's performance when tested with a novel comparison term. Although we do not intend to suggest a new model of transitive inference, we conclude, in line with other authors, that a simple error-correcting rule can generate transitive behaviour similar to the choice pattern of children and animals in the binary form of the five-term-series task without requiring high-order logical or paralogical abilities. The analysis of the training history and of the final internal structure of the network reveals the associative strategy employed. However, our results indicate that the scope of the associative strategy used by the network might be limited. The extent to which the conventional five-term-series task, in absence of appropriate manipulations of training and testing conditions, is suitable to detect cognitive differences across species is also discussed on the basis of our results. Accepted after revision: 29 May 2001 Electronic Publication     

When Hyperpropositions Meet ...

With each proposition P we associate a set of proposition (a hyperproposition) which determines the order in which one may retreat from accepting P, if one cannot fully hold on to P. We first describe the structure of hyperpropositions. Then we describe two operations on propositions, subtraction and merge, which can be modelled in terms of hyperpropositions. Subtraction is an operation that takes away part of the content of a proposition. Merge is an operation that determines the maximal consistent content of two propositions considered jointly. The merge operation gives rise to an inference relation which is, in a certain sense, optimally paraconsistent.     

Deductivism Within Pragma-Dialectics

The present paper elaborates a deductivist account of natural language argu-ment in the context of pragma-dialectics. It reviews earlier debates, criticizes some standard misconceptions in the literature, and argues that the identification and analysis of deductive argument schemes can be the basis of a compelling theory of argumentative discourse.     

Training requirements and retention characteristics of serial list organization by macaque monkeys

This work evaluated the prospect that organizational accounts of the retention of list information by monkeys might be an artifact of familiarity with conditional relationships. Seven sophisticated macaques were trained on four five-item lists. Each acquisition selectively excluded one of the internal conditional pairs of the typical four-problem sequence (AB,BC,CD,DE) that defines a five-item serially ordered list. Then, all possible novel pairings and the trained pairs appeared together in a test. After this, the previously omitted pair was trained and animals were retested. On all tasks, initial tests revealed little organization and much intersubject variability of characteristic choice strategies, but subsequent inclusion of all four conditional pairs always yielded organized serial choice. On both the four-problem tests and in a later retention, errors were directly related to interitem distance between the objects paired on test trials. These results helped to specify the conditions required for demonstration of non-human primate analogs of transitivity, and showed that even sophisticated monkeys organize information in retention only if they know all interitem relationships. Received: 7 October 1998 / Accepted after revision: 10 October 1999     

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.We thank David Spiegelhalter for suggesting applying the Gibbs sampler to structural equation models to the first author at a 1994 workshop in Wiesbaden. We thank Ulf Böckenholt, Chris Meek, Marijtje van Duijn, Clark Glymour, Ivo Molenaar, Steve Klepper, Thomas Richardson, Teddy Seidenfeld, and Tom Snijders for helpful discussions, mathematical advice, and critiques of earlier drafts of this paper.