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.     

Dualism and its Importance for Medicine

Cartesian dualism has been viewed by medical theorists to be oneof the chief causes of a reductionist/mechanistic treatment ofthe patient. Although I aver that Cartesian dualism is one culprit for the misapprehension of the genuine treatment of patients in termsof both mind and body, I argue that interactive dualism whichstresses the interaction of mind and body is essential to treatpatients with dignity and compassion. Thus, adequate medical carethat is humanistic in nature is difficult (if not impossible)to achieve without physicians adhering to a dualistic frameworkin which the body and person is treated during illness.     

What are symbols symbols of? Situated action,mythological bootstrapping and the emergence of the Self

This paper addresses the question of how symbols should be understood in analytical psychology and psychoanalysis. The point of view examined focuses on the recent turn to more cognitive and developmental models in both disciplines and briefly reviews and critiques the evolutionary and cognitive arguments. The paper then presents an argument based on dynamic systems theory in which no pre-existing template or structure for either mind or behaviour is assumed. Within the dynamic systems model the Self is viewed as an emergent phenomenon deriving from the dynamic patterns existing in a complex system that includes the physiological characteristics of the infant, the intentional attributions of the caregiver and the cultural or symbolic resources that constitute the environment. The symbol can then be seen as a discrete, and in important ways an autonomous, element in the dynamic system. Conclusions are drawn for further research into the nature of the symbol with implications for both theory and practice in analytical psychology and psychoanalysis.     

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.     

On Modal μ-Calculus and Non-Well-Founded Set Theory

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal -calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.     

Categorical Abstract Algebraic Logic Metalogical Properties

Metalogical properties that have traditionally been studied in the deductive system context (see, e.g., [21]) and transferred later to the institution context [33], are here formulated in the -institution context. Preservation under deductive equivalence of -institutions is investigated. If a property is known to hold in all algebraic -institutions and is preserved under deductive equivalence, then it follows that it holds in all algebraizable -institutions in the sense of [36].     

Nonparametric IRT: Testing the bi-isotonicity of isotonic probabilistic models (ISOP)

Nonparametric tests for testing the validity of polytomous ISOP-models (unidimensional ordinal probabilistic polytomous IRT-models) are presented. Since the ISOP-model is a very general nonparametric unidimensional rating scale model the test statistics apply to a great multitude of latent trait models. A test for the comonotonicity of item sets of two or more items is suggested. Procedures for testing the comonotonicity of two item sets and for item selection are developed. The tests are based on Goodman-Kruskal's gamma index of ordinal association and are generalizations thereof. It is an essential advantage of polytomous ISOP-models within probabilistic IRT-models that the tests of validity of the model can be performed before and without the model being fitted to the data. The new test statistics have the further advantage that no prior order of items or subjects needs to be known.     

Dynamic analyses of mother-child interactions in functional and dysfunctional dyads: a synergetic approach

Describes the application of a new analytical approach (derived from synergetics, a complex dynamic systems theory) to home observational data of mother–child interactions in average dyads and dyads with children referred for disruptive behavior problems at home and school (n = 11 in each group). Results show that (1) the two groups differed in their daily interactions in predictable ways, and (2) the most frequent patterns of interactions observed in the two groups brought them back repeatedly to behave in similar ways toward each other. The findings are in keeping with a body of literature on mother–child interactions. However, they add to it by providing multivariate, graphical representations of these interactions and by offering a conceptual framework within which to move from an observational to an inferential level of analysis. At that level, the transactional processes that are characteristic of functional and dysfunctional relationships may become apparent.     

What the Liar Taught Achilles

Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.