The abstract variable-binding calculus

Theabstract variable binding calculus (VB-calculus) provides a formal frame-work encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the standard formalism of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course here and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculus by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that _T s=t iff _D s=t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifierexists uncountably many and a fragment of Riemann integration.The work of the first author was supported in part by National Science Foundation Grant #DMS 8805870.     

Artificial grammar learning by 1-year-olds leads to specific and abstract knowledge.

Four experiments used the head-turn preference procedure to assess whether infants could extract and remember information from auditory strings produced by a miniature artificial grammar. In all four experiments, infants generalized to new structure by discriminating new grammatical strings from ungrammatical ones after less than 2 min exposure to the grammar. Infants acquired specific information about the grammar as demonstrated by the ability to discriminate new grammatical strings from those with illegal endpoints (Experiment 1). Infants also discriminated new grammatical strings from those with string-internal pairwise violations (Experiments 2 and 3). Infants in Experiment 4 abstracted beyond specific word order as demonstrated by the ability to discriminate new strings produced by their training grammar from strings produced by another grammar despite a change in vocabulary between training and test. We discuss the implications of these findings for the study of language acquisition.     

Mental comparisons involving abstract attributes

Adult subjects in two experiments were presented pairs of stimuli that differed in varying degree on an abstract semantic attribute, and were required to choose the one with the higher value on the given dimension. Subjects in Experiment 1 chose the more pleasant member of a pair of pictures, concrete nouns, or abstract nouns. Those in Experiment 2, presented a pair of pictures or concrete nouns, chose the one whose referent had the higher monetary value. Theoretical interest centered on the effects of semantic distance, stimulus mode, and individual differences in imagery and verbal ability on choice time. In both experiments, response times (1) decreased with increases in semantic distance, (2) were faster for pictures than words (and for concrete than abstract words in Experiment 1), and (3) were faster for high- than for lowimagery participants. The results are completely consistent with a dual-coding (image vs. verbal) interpretation: Pleasantness and value, though conceptually abstract, are attributes of things rather than words, and they are accordingly represented in and processed by a system specialized for dealing with nonverbal information.     

Handedness shapes children's abstract concepts

Can children’s handedness influence how they represent abstract concepts like kindness and intelligence? Here we show that from an early age, right‐handers associate rightward space more strongly with positive ideas and leftward space with negative ideas, but the opposite is true for left‐handers. In one experiment, children indicated where on a diagram a preferred toy and a dispreferred toy should go. Right‐handers tended to assign the preferred toy to a box on the right and the dispreferred toy to a box on the left. Left‐handers showed the opposite pattern. In a second experiment, children judged which of two cartoon animals looked smarter (or dumber) or nicer (or meaner). Right‐handers attributed more positive qualities to animals on the right, but left‐handers to animals on the left. These contrasting associations between space and valence cannot be explained by exposure to language or cultural conventions, which consistently link right with good. Rather, right‐ and left‐handers implicitly associated positive valence more strongly with the side of space on which they can act more fluently with their dominant hands. Results support the body‐specificity hypothesis ( Casasanto, 2009 ), showing that children with different kinds of bodies think differently in corresponding ways.     

Collaboration facilitates abstract category learning

We examined the effects of collaboration (dyads vs. individuals) and category structure (coherent vs. incoherent) on learning and transfer. Working in dyads or individually, participants classified examples from either an abstract coherent category, the features of which are not fixed but relate in a meaningful way, or an incoherent category, the features of which do not relate meaningfully. All participants were then tested individually. We hypothesized that dyads would benefit more from classifying the coherent category structure because past work has shown that collaboration is more beneficial for tasks that build on shared prior knowledge and provide opportunities for explanation and abstraction. Results showed that dyads improved more than individuals during the classification task regardless of category coherence, but learning in a dyad improved inference-test performance only for participants who learned coherent categories. Although participants in the coherent categories performed better on a transfer test, there was no effect of collaboration.     

The abstract representations in speech processing

Speech processing by human listeners derives meaning from acoustic input via intermediate steps involving abstract representations of what has been heard. Recent results from several lines of research are here brought together to shed light on the nature and role of these representations. In spoken-word recognition, representations of phonological form and of conceptual content are dissociable. This follows from the independence of patterns of priming for a word's form and its meaning. The nature of the phonological-form representations is determined not only by acoustic-phonetic input but also by other sources of information, including metalinguistic knowledge. This follows from evidence that listeners can store two forms as different without showing any evidence of being able to detect the difference in question when they listen to speech. The lexical representations are in turn separate from prelexical representations, which are also abstract in nature. This follows from evidence that perceptual learning about speaker-specific phoneme realization, induced on the basis of a few words, generalizes across the whole lexicon to inform the recognition of all words containing the same phoneme. The efficiency of human speech processing has its basis in the rapid execution of operations over abstract representations.     

The learnability of abstract syntactic principles

Children acquiring language infer the correct form of syntactic constructions for which they appear to have little or no direct evidence, avoiding simple but incorrect generalizations that would be consistent with the data they receive. These generalizations must be guided by some inductive bias – some abstract knowledge – that leads them to prefer the correct hypotheses even in the absence of directly supporting evidence. What form do these inductive constraints take? It is often argued or assumed that they reflect innately specified knowledge of language. A classic example of such an argument moves from the phenomenon of auxiliary fronting in English interrogatives to the conclusion that children must innately know that syntactic rules are defined over hierarchical phrase structures rather than linear sequences of words (e.g., [Chomsky, 1965] , [Chomsky, 1971] , [Chomsky, 1980] and [Crain and Nakayama, 1987] ). Here we use a Bayesian framework for grammar induction to address a version of this argument and show that, given typical child-directed speech and certain innate domain-general capacities, an ideal learner could recognize the hierarchical phrase structure of language without having this knowledge innately specified as part of the language faculty. We discuss the implications of this analysis for accounts of human language acquisition.     

Interpolation and definability in abstract logics

A semantical definition of abstract logics is given. It is shown that the Craig interpolation property implies the Beth definability property, and that the Souslin-Kleene interpolation property implies the weak Beth definability property. An example is given, showing that Beth does not imply Souslin-Kleene.     

The acquisition of abstract letter codes

A modification of Posner's letter-matching paradigm proposed by Beller (1971) was employed to study the development of abstract visual and name codes for letters. Second-, fourth-, and sixth-grade students (ages 8, 10, and 12 years, respectively) were presented in advance one of a pair of to-be-matched letters. Case agreement (upper- and lowercase) between the prime letter and the letter pair was manipulated to differentiate facilitation based on concrete features of the stimulus letter from that based on the letter's abstract visual code. For physical matches second graders showed greater facilitation from same-case primes than from different-case primes, whereas sixth graders benefited equally from both types of prime. For same-name and different conditions, no qualitative developmental differences were revealed as all subjects benefited from letter primes irrespective of case agreement. The acquisition and utilization of abstract visual codes for letters are a later acquisition than name codes, which were clearly operative at the lowest grade level in the present experiment.     

Searle's abstract argument against strong AI

Discussion of Searle's case against strong AI has usually focused upon his Chinese Room thought-experiment. In this paper, however, I expound and then try to refute what I call his abstract argument against strong AI, an argument which turns upon quite general considerations concerning programs, syntax, and semantics, and which seems not to depend on intuitions about the Chinese Room. I claim that this argument fails, since it assumes one particular account of what a program is. I suggest an alternative account which, however, cannot play a role in a Searle-type argument, and argue that Searle gives no good reason for favoring his account, which allows the abstract argument to work, over the alternative, which doesn't. This response to Searle's abstract argument also, incidentally, enables the Robot Reply to the Chinese Room to defend itself against objections Searle makes to it.     

The abstract representations in speech processing

Speech processing by human listeners derives meaning from acoustic input via intermediate steps involving abstract representations of what has been heard. Recent results from several lines of research are here brought together to shed light on the nature and role of these representations. In spoken-word recognition, representations of phonological form and of conceptual content are dissociable. This follows from the independence of patterns of priming for a word's form and its meaning. The nature of the phonological-form representations is determined not only by acoustic-phonetic input but also by other sources of information, including metalinguistic knowledge. This follows from evidence that listeners can store two forms as different without showing any evidence of being able to detect the difference in question when they listen to speech. The lexical representations are in turn separate from prelexical representations, which are also abstract in nature. This follows from evidence that perceptual learning about speaker-specific phoneme realization, induced on the basis of a few words, generalizes across the whole lexicon to inform the recognition of all words containing the same phoneme. The efficiency of human speech processing has its basis in the rapid execution of operations over abstract representations.     

In defense of abstract conceptual representations

An extensive program of research in the past 2 decades has focused on the role of modal sensory, motor, and affective brain systems in storing and retrieving concept knowledge. This focus has led in some circles to an underestimation of the need for more abstract, supramodal conceptual representations in semantic cognition. Evidence for supramodal processing comes from neuroimaging work documenting a large, well-defined cortical network that responds to meaningful stimuli regardless of modal content. The nodes in this network correspond to high-level “convergence zones” that receive broadly crossmodal input and presumably process crossmodal conjunctions. It is proposed that highly conjunctive representations are needed for several critical functions, including capturing conceptual similarity structure, enabling thematic associative relationships independent of conceptual similarity, and providing efficient “chunking” of concept representations for a range of higher order tasks that require concepts to be configured as situations. These hypothesized functions account for a wide range of neuroimaging results showing modulation of the supramodal convergence zone network by associative strength, lexicality, familiarity, imageability, frequency, and semantic compositionality. The evidence supports a hierarchical model of knowledge representation in which modal systems provide a mechanism for concept acquisition and serve to ground individual concepts in external reality, whereas broadly conjunctive, supramodal representations play an equally important role in concept association and situation knowledge.     

An abstract setting for Henkin proofs

A general result is proved about the existence of maximally consistent theories satisfying prescribed closure conditions. The principle is then used to give streamlined proofs of completeness and omitting-types theorems, in which inductive Henkin-style constructions are replaced by a demonstration that a certain theory respects a certain class of inference rules.     

An abstract dynamic access control architecture

This paper presents an abstract architecture capturing the most essential characteristics of security architectures for the kind of access control required in multi-agent systems. A security architecture is usually component based; it is dynamically extensible, reflective, and should be able to support a variety of policy strategies and enforcement mechanisms. Therefore, in designing and implementing such an architecture, it is important to accurately describe its structure, components, and their interactions; and we should also provide support for verifying properties that the architecture must satisfy. The abstract Dynamic Access Control Architecture, or DACA, we propose provides the flexibility to express several complex characteristics abstractly, ultimately resulting in a more structured foundation for implementing security architectures for access control.     

Learning abstract relations from using categories

When people learn categories, the importance of the features and relations in the category representation reflects both their diagnosticity for classification and their relevance to the use of the category. In earlier work in which the influence of category use on the representation has been shown, only cases in which the features and relations were simple, observable, and very specific were examined. Learners may begin to understand the underlying similarities of category members by using the categories. In the four experiments presented here, learners applied a simple category-specific formula to category members. The test results showed that the learners had incorporated relations among features from this use, including cases in which the relations were abstract. This learning occurred even though the relations were actually not predictive of category membership but just perceived to be so as a function of the use.