Some Closure Properties of Finite Definitions

There is no known syntactic characterization of the class of finite definitions in terms of a set of basic definitions and a set of basic operators under which the class is closed. Furthermore, it is known that the basic propositional operators do not preserve finiteness. In this paper I survey these problems and explore operators that do preserve finiteness. I also show that every definition that uses only unary predicate symbols and equality is bound to be finite.     

Effects of ethanol on reinforced variations and repetitions by rats under a multiple schedule.

Response sequences emitted by five Long-Evans rats were reinforced under a two-component multiple schedule. In the REPEAT component, food pellets were contingent upon completion of a left-left-right-right (LLRR) sequence on two levers. In the VARY component, pellets were contingent upon variable sequences (i.e., a sequence was reinforced only if it differed from each of the previous five sequences). The rats learned to emit LLRR sequences in the REPEAT component and variable sequences in VARY. Intraperitoneal injections of ethanol (1.25, 1.75, and 2.25 g/kg) significantly increased sequence variability in REPEAT, thereby lowering reinforcement probability, but had little effect on sequence variability in the VARY component. These results extend previous findings that alcohol impairs the performance of reinforced repetitions but not of reinforced variations in response sequences.     

Auditory expectations for newly acquired structures

Our study investigated whether newly acquired auditory structure knowledge allows listeners to develop perceptual expectations for future events. For that aim, we introduced a new experimental approach that combines implicit learning and priming paradigms. Participants were first exposed to structured tone sequences without being told about the underlying artificial grammar. They then made speeded judgements on a perceptual feature of target tones in new sequences (i.e., in-tune/out-of-tune judgements). The target tones respected or violated the structure of the artificial grammar and were thus supposed to be expected or unexpected. In this priming task, grammatical tones were processed faster and more accurately than ungrammatical ones. This processing advantage was observed for an experimental group performing a memory task during the exposure phase, but was not observed for a control group, which was lacking the exposure phase (Experiment 1). It persisted when participants realized an in-tune/out-of-tune detection task during exposure (Experiment 2). This finding suggests that the acquisition of new structure knowledge not only influences grammaticality judgements on entire sequences (as previously shown in implicit learning research), but allows developing perceptual expectations that influence single event processing. It further promotes the priming paradigm as an implicit access to acquired artificial structure knowledge.     

The influence of temporal regularities on the implicit learning of pitch structures

Implicit learning is the acquisition of complex information without the intention to learn. The aim of this study was to investigate the influence of temporal regularities on the implicit learning of an artificial pitch grammar. According to the dynamic attending theory (DAT) external regularities can entrain internal oscillators that guide attention over time, inducing temporal expectations that influence perception of future events. In the present study, the presentation of the artificial pitch grammar in the exposure phase was temporally either regular or irregular for one of two participant groups. Based on the DAT, it was hypothesized that the regular temporal presentation would favour implicit learning of tone structures in comparison to the irregular temporal presentation. Results demonstrated learning of the artificial grammar for the group with the regular exposure phase and partial learning for the group with the irregular exposure phase. These findings suggest that the regular presentation helps listeners to develop perceptual expectations about the temporal occurrence of future tones and thus facilitates the learning of the artificial pitch grammar.     

Scientific Discovery on Positive Data via Belief Revision

A model of inductive inquiry is defined within a first-order context. Intuitively, the model pictures inquiry as a game between Nature and a scientist. To begin the game, a nonlogical vocabulary is agreed upon by the two players along with a partition of a class of structures for that vocabulary. Next, Nature secretly chooses one structure (the real world) from some cell of the partition. She then presents the scientist with a sequence of atomic facts about the chosen structure. With each new datum the scientist announces a guess about the cell to which the chosen structure belongs. To succeed in his inquiry, the scientist's successive conjectures must be correct all but finitely often, that is, the conjectures must converge in the limit to the correct cell. A special kind of scientist selects his hypotheses on the basis of a belief revision operator. We show that reliance on belief revision allows scientists to solve a wide class of problems.     

Evaluations of Outcome Sequences

We extend the literature on intertemporal choice by investigating how possession of knowledge related to the present value of future outcomes (PV knowledge) affects the extent to which individuals weight certain attributes when evaluating outcome sequences. While PV-knowledgeable individuals can ascribe value to attributes according to their PV relevance (or irrelevance), unknowledgeable individuals cannot do so. Such knowledge, therefore, likely interacts with outcome-sequence attributes to affect the extent to which individuals exhibit impatience when evaluating outcome sequences. The main experimental findings indicate that higher PV knowledge increases the extent to which individuals value impatience (as opposed to improvement). However, these findings also reinforce a need to distinguish among impatience, improvement, and PV because some higher PV knowledge participants willingly sacrifice PV while exhibiting impatience (while others do so in order to gain improvement). Overall, PV considerations appear central, but not determinative, in higher PV-knowledgeable individuals' evaluations of outcome sequences.     

Conditional Probability and Defeasible Inference

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. The research of both authors was supported in part by a grant from NSF, and, for Parikh, also by support from the research foundation of CUNY.     

The mental representation of ordinal sequences is spatially organized

In the domain of numbers the existence of spatial components in the representation of numerical magnitude has been convincingly demonstrated by an association between number magnitude and response preference with faster left- than right-hand responses for small numbers and faster right- than left-hand responses for large numbers (Dehaene, S., Bossini, S., & Giraux, P. (1993) The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371-396). Because numbers convey not only real or integer meaning but also ordinal meaning, the question of whether non-numerical ordinal information is spatially coded naturally follows. While previous research failed to show an association between ordinal position and spatial response preference, we present two experiments involving months (Experiment 1) and letters (Experiment 2) in which spatial coding is demonstrated. Furthermore, the response-side effect was obtained with two different stimulus-response mappings. The association occurred both when ordinal information was relevant and when it was irrelevant to the task, showing that the spatial component of the ordinal representation can be automatically activated.     

Choice in a variable environment: effects of unequal reinforcer distributions

Six pigeons were trained in a procedure in which sessions included seven unsignaled components, each offering two pecking keys, and each providing a potentially different reinforcer ratio between the two keys. Across conditions, various combinations of reinforcer ratios and reinforcer-magnitude ratios were used to create unequal reinforcer distributions between the two alternatives when averaged across a session. The results extended previous research using the same basic procedure that had included only reinforcer distributions symmetrical around 1:1. Data analyses suggested that the variables controlling choice operated at a number of levels: First, individual reinforcers had local effects on choice; second, sequences of successive reinforcers obtained at the same alternative (continuations) had cumulative effects; and, third, when these sequences themselves occurred with greater frequency, their effects further cumulated. A reinforcer obtained at the other alternative following a sequence of continuations (a discontinuation) had a large effect and apparently reset choice to levels approximating the sessional reinforcer ratio.     

Selective Revision

We introduce a constructive model of selective belief revision in which it is possible to accept only a part of the input information. A selective revision operator ο is defined by the equality K ο α = K * f(α), where * is an AGM revision operator and f a function, typically with the property ⊢ α → f(α). Axiomatic characterizations are provided for three variants of selective revision. This revised version was published online in June 2006 with corrections to the Cover Date.