Counting distinctions: on the conceptual foundations of Shannon’s information theory

Categorical logic has shown that modern logic is essentially the logic of subsets (or “subobjects”). In “subset logic,” predicates are modeled as subsets of a universe and a predicate applies to an individual if the individual is in the subset. Partitions are dual to subsets so there is a dual logic of partitions where a “distinction” [an ordered pair of distinct elements (u, u′) from the universe U] is dual to an “element”. A predicate modeled by a partition π on U would apply to a distinction if the pair of elements was distinguished by the partition π, i.e., if u and u′ were in different blocks of π. Subset logic leads to finite probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered |U|² pairs from the finite universe. That yields a notion of “logical entropy” for partitions and a “logical information theory.” The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon’s theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon’s theory based on the logical notion of “distinctions.” This paper is dedicated to the memory of Gian-Carlo Rota—mathematician, philosopher, mentor, and friend.     

A plea for logical objects

An account of validity that makes what is invalid conditional on how many individuals there are is what I call a conditional account of validity. Here I defend conditional accounts against a criticism derived from Etchemendy’s well-known criticism of the model-theoretic analysis of validity. The criticism is essentially that knowledge of the size of the universe is non-logical and so by making knowledge of the extension of validity depend on knowledge of how many individuals there are, conditional accounts fail to reflect that the former knowledge is basic, i.e., independent of knowledge derived from other sciences. Appealing to Russell’s pre-Principia logic, I defend conditional accounts against this criticism by sketching a rationale for thinking that there are infinitely many logical objects.     

The development of category learning strategies: What makes the difference?

Category learning can be achieved by identifying common features among category members, distinctive features among non-members, or both. These processes are psychologically and computationally distinct, and may have implications for the acquisition of categories at different hierarchical levels. The present study examines an account of children’s difficulty in acquiring categories at the subordinate level grounded on these distinct comparison processes. Adults and children performed category learning tasks in which they were exposed either to pairs of objects from the same novel category or pairs of objects from different categories. The objects were designed so that for each category learning task, two features determined category membership whereas two other features were task irrelevant. In the learning stage participants compared pairs of objects noted to be either from the same category or from different categories. Object pairs were chosen so that the objective amount of information provided to the participants was identical in the two learning conditions. We found that when presented only with object pairs noted to be from the same category, young children (6 ? YO ? 9.5) learned the novel categories just as well as older children (10 ? YO ? 14) and adults. However, when presented only with object pairs known to be from different categories, unlike older children and adults, young children failed to learn the novel categories. We discuss cognitive and computational factors that may give rise to this comparison bias, as well as its expected outcomes.     

Using logical relevance for question answering

We propose a novel method of determining the appropriateness of an answer to a question through a proof of logical relevance rather than a logical proof of truth. We define logical relevance as the idea that answers should not be considered as absolutely true or false in relation to a question, but should be considered true more flexibly in a sliding scale of aptness. This enables us to reason rigorously about the appropriateness of an answer even in cases where the sources we are getting answers from are incomplete or inconsistent or contain errors. We show how logical relevance can be implemented through the use of measured simplification, a form of constraint relaxation, in order to seek a logical proof than an answer is in fact an answer to a particular question. We then give an example of such an implementation providing a set of specific rules for this purpose.     

国有大中型企业职工满意感研究

本文在对山东省国有大中型企业职工进行大规模抽样问卷凋查的基础上．分析了满意感调查表的信度和效度。并探讨了运用该调查表所得出的～些重要结论。对所获数据的分析表明：(1)满意感量表具有良好的再测信度和结构效度;(2)国有大中型企业职工总体满意感和五个方面的具体满意感不太平衡;(3)国有大中型企业职工满意感受到职工个人的性别、年龄、受教育程度、婚姻状况、工作性质、进厂方式,以及职工所属企业所在地区、企业规模、效益、转轨情况等因素的影响。     

Intra‐sentential context effects on the interpretation of logical metonymy⋆

Verbs such as enjoy in the student enjoyed the book exhibit logical metonymy: enjoy is interpreted as enjoy reading. Theoreticalwork [Computational Linguistics 17 (4) (1991) 409; The Generative Lexicon, MIT Press, Cambridge, MA, 1995] predicts that this interpretation can be influenced by intra‐sentential context, e.g., by the subject of enjoy. In this article, we test this prediction using a completion experiment and find that the interpretation of a metonymic verb is influenced by the semantic role of its subject. We present a Bayesian model that accounts for the interpretation of logical metonymy and achieves a good fit on our experimental data. We show that the parameters of the model can be estimated from completion data or from corpus data.     

Overshadowing by fixed- and variable-duration stimuli

Two experiments investigated the effect of the temporal distribution form of a stimulus on its ability to produce an overshadowing effect. The overshadowing stimuli were either of the same duration on every trial, or of a variable duration drawn from an exponential distribution with the same mean duration as that of the fixed stimulus. Both experiments provided evidence that a variable-duration stimulus was less effective than a fixed-duration cue at overshadowing conditioning to a target conditioned stimulus (CS); moreover, this effect was independent of whether the overshadowed CS was fixed or variable. The findings presented here are consistent with the idea that the strength of the association between CS and unconditioned stimulus (US) is, in part, determined by the temporal distribution form of the CS. These results are discussed in terms of time-accumulation and trial-based theories of conditioning and timing.     

Deferred Interpretations: Why Starting Dickens is Taxing but Reading Dickens Isn't

Comprehenders often need to go beyond conventional word senses to obtain an appropriate interpretation of an expression. We report an experiment examining the processing of standard metonymies (The gentleman read Dickens) and logical metonymies (The gentleman began Dickens), contrasting both to the processing of control expressions with a conventional interpretation (The gentleman met Dickens). Eye movement measures during reading indicated that standard (producer-for-product) metonymies were not more costly to interpret than conventional expressions, but logical metonymies were more costly to interpret than both standard metonymies and conventional expressions. These results indicate that constructing alternative senses is sometimes taxing and that not all types of deferred interpretations are processed in the same way. The results suggest that a critical factor in determining the attendant cost of constructing alternative senses is whether compositional operations must generate unexpressed semantic structure to realize an extended sense of an expression.