On the Consistency of the Δ1 1-CA Fragment of Frege's Grundgesetze

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing ¹ ₁-comprehension schema would already be inconsistent. In the present paper, we show that this is not the case.     

The Power to Feel Fear and The One to Feel Happiness are the Same

The centre of Adorno's Critical Theory is occupied by the theme of happiness. He speaks of the impaired life, of the unjustness of society, of murderous prejudices, of the atrocities in history, of the dissonance in art, of the unhappy consciousness, because something better can only be described from a point of opposition. Happiness cannot be objectified as possession, it always needs to be experienced subjectively, somatically. (With happiness it is like with truth: One does not have it, one is in it.) Happiness cannot be prescribed and ordered; nothing can be done to guarantee happiness. (Happiness goes beyond doing.) Happiness (like fear) has to do with being open to experience which can overwhelm the self. Sexual and aesthetical experience are models for such overwhelming happiness. The sensation of happiness always is very personal, but in this experience the individual leaves its particularity behind. One has to differentiate between goal and object: Happiness may be a goal, but not it itself, only what obstructs it, can be an object of Critical Theory.     

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.     

Coherentist Contraction

A model of coherentist belief contraction is constructed. The outcome of belief contraction is required to be one of the coherent subsets of the original belief set, and a set of plausible properties is proposed for this set of coherent subsets. The contraction operators obtained in this way are shown to coincide with well-known belief base operations. This connection between coherentist and foundationalist approaches to belief change has important implications for the philosophical interpretation of models of belief change.     

Sets and Classes as Many

In this paper the view is developed that classes should not be understood as individuals, but, rather, as classes as many of individuals. To correlate classes with individuals labelling and colabelling functions are introduced and sets identified with a certain subdomain of the classes on which the labelling and colabelling functions are mutually inverse. A minimal axiomatization of the resulting system is formulated and some of its extensions are related to various systems of set theory, including nonwellfounded set theories.     

Postdecision consolidation in a trend prediction task

To be able to learn from experience it is necessary to correctly apprehend experienced feedback and the situation in which it is provided. The results indicate how post-decision consolidation in complex domains may affect learning. The problem may be particularly pertinent in recurrent decision making where considerable risk is involved. The study explores the changes in aspect (signal) importance from pre- to postdiction as a function of outcome information. By postdiction we mean the remembering of an earlier prediction (cf. Hawkins & Hastie, 1990). Subjects were asked to decide on which of four alternative future price developments would follow a historical price trajectory for different commodities, and to rate the importance of each of the chosen alternative's corresponding aspects. The subjects revealed a bias in their support ratings of aspects--seeing support in aspects that traditionally (by themselves and in many contexts) would be seen as neutral or even counter-indicative of the alternative chosen. After an intermission, the subjects were also given information about what was indicated to be the actual development of the market. One group was told that their decisions were correct (irrespective of what the decisions were), another group that they were incorrect but close, a third group that they were incorrect by far, while a fourth group served as a control. Following this information the subjects were again asked to judge the importance of the aspects for their own prior decision on the most likely future development. The results indicated that outcome feed-back had an effect on post decision restructuring of facts. Subjects in the correct condition showed an average consolidation that increased the support, while the wrong conditions lead to negative consolidation (in retrospect indicating that they never found as much support for their decision in the past as they actually did). Thus, in a choice between consolidating their own initial prediction and the price trajectory they would have to live with, the decision makers consolidated the outcome. Therefore, the results of the study were related to the hindsight bias phenomenon (Fischhoff, 1975) and to Kahneman and Miller's (1986) mutability concept.     

“The Frog Ate the Bug and Made his Mouth Sad”: Narrative Competence in Children with Autism

This study compares the narrative abilities of 13 children with autism, 13 children with developmental delays, and 13 typically developing children matched on language ability. Although groups did not differ in their use of causal language or internal state terms, children with autism and children with developmental delays were less likely than typical children to identify the causes of characters' internal states. Rather, they tended simply to label emotions and explain actions. Children with autism and children with developmental delays also relied on a more restricted range of evaluative devices, which both convey point of view and maintain listener involvement. In addition, the narrative abilities of children with autism were linked to performance on measures of theory of mind and an index of conversational competence, whereas this was not the case among children with developmental delays. Findings are discussed in relation to the social, cognitive, and emotional underpinnings and consequences of narrative activity.     

Effects of noncontingent reinforcement on problem behavior and stimulus engagement: the role of satiation, extinction, and alternative reinforcement

This study examined the effects of noncontingent reinforcement (NCR) with and without extinction on problem behavior and stimulus engagement (consumption of reinforcement) of 4 participants. Reductions in problem behavior using NCR have frequently been attributed to both satiation of the reinforcer and extinction. In the current study, aspects of the NCR treatment effects were difficult to explain based solely on either a satiation or an extinction account. Specifically, it was found that stimulus engagement remained high throughout the NCR treatment analysis, and that problem behavior was reduced to near-zero levels during NCR without extinction. The implications of these findings are discussed with respect to the satiation and extinction hypotheses frequently described in the applied literature. Findings from basic studies examining the effects of response-independent schedules are presented, and are used as the basis for a matching theory account of NCR-related effects. It is proposed that reductions in problem behavior observed during NCR interventions may be a function of the availability of alternative sources of reinforcement.     

Equivalence relations and the reinforcement contingency

Where do equivalence relations come from? One possible answer is that they arise directly from the reinforcement contingency. That is to say, a reinforcement contingency produces two types of outcome: (a) 2‐, 3‐, 4‐, 5‐, or n‐term units of analysis that are known, respectively, as operant reinforcement, simple discrimination, conditional discrimination, second‐order conditional discrimination, and so on; and (b) equivalence relations that consist of ordered pairs of all positive elements that participate in the contingency. This conception of the origin of equivalence relations leads to a number of new and verifiable ways of conceptualizing equivalence relations and, more generally, the stimulus control of operant behavior. The theory is also capable of experimental disproof.