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.     

Wittgenstein and logic

In his Tractatus Logico-Philosophicus Ludwig Wittgenstein (1889–1951) presents the concept of order in terms of a notational iteration that is completely logical but not part of logic. Logic for him is not the foundation of mathematical concepts but rather a purely formal way of reflecting the world that at the minimum adds absolutely no content. Order for him is not based on the concepts of logic but is instead revealed through an ideal notational series. He states that logic is “transcendental”. As such it requires an ideal that his philosophical method eventually forces him to reject. I argue that Wittgenstein’s philosophy is more dialectical than transcendental.     

Propositional Plausible Logic: Introduction and Implementation

Plausible Logic allows defeasible deduction with arbitrary propositions, and yet when sufficiently simplified it is very similar to the Defeasible Logics of Billington and Nute. This paper presents Plausible Logic, explains some of the ideas behind the definitions, applies Plausible Logic to an example, and proves a coherence result which indicates that Plausible Logic is well behaved. We also report the first complete implementation of propositional Plausible Logic. The implementation has a web interface which makes it available to researchers and students everywhere. The implementation is evaluated experimentally, and is shown to be capable of handling tens of thousands of rules and sufficiently many disjunctions for realistic problems.     

Gentzen-Style Axiomatizations for Some Conservative Extensions of Basic Propositional Logic

We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.     

学校氛围对学生数学成绩影响的跨文化比较:基于多水平分析的结果

学校氛围是学校中被成员所体验并对其行为产生影响的、相对持久而稳定的学校环境特征。它以成员的共同行为感知为基础,对学校和学生的发展具有重要作用。以PISA2003测试的数据为基础,采用多水平线性模型,考察了中国香港、日本、美国和土耳其的15岁学生及其所在学校的校长知觉的学校氛围对学生数学成绩的影响。结果显示:(1)学生知觉的师生关系、数学课堂秩序,校长知觉的学生士气、学校消极行为对数学成绩的显著预测作用具有较高的跨文化一致性;(2)数学课堂上的教师支持、教师负面状态、数学教师间的共识能显著预测部分国家/地区的学生数学成绩;(3)学生知觉的对学校的看法、校长知觉的教师士气对四个国家/地区的学生数学成绩均无显著预测作用。     

Mistaking the instance for the rule: A critical analysis of the truth-table evaluation paradigm

Many studies probe for interpretations of < if A then C> by having people evaluate truth-table cases (<A and C>, < A and not-C>, < not-A and C>, < not-A and not-C>) as making the rule true or false, or being irrelevant. We argue that a single case can never prove a general rule to be true, as philosophy of science has taught any researcher. Giving participants the impossible “true” option would therefore bias results away from this response. In Experiment 1 people judged instead whether cases make a rule false, do not make the rule false, or are irrelevant to the rule. The experimental group (N = 44) showed a significant increase in not-false responses compared with true responses of the control group (N = 39). In Experiments 2 and 3 the experimental groups judged whether cases make a rule true, corroborate it (i.e., make the rule more plausible, but neither true nor false), make it false, or are irrelevant. There was a significant reduction of irrelevant responses as compared to the default true/false/irrelevant task for the control groups. Even < A and C> cases were often no longer considered to make an < if A then C> rule true and were correctly judged to corroborate (vs. verify) rules. Results corroborate our conceptual analyses of the unsuitable “true” response option and put into question arguments that hinge on the presumed likelihood by which people consider truth contingencies to make a rule “true”.     

Predicting stress patterns in an unpredictable stress language: The use of non-lexical sources of evidence for stress assignment in Russian

The main goal of this research was to examine how readers of Russian assign stress to disyllabic words. In particular, we tested the claim that the process of stress assignment in Russian can only be accomplished lexically. Eleven potential non-lexical sources of evidence for stress in Russian were examined in regression and factorial studies. In Study 1, onset complexity, coda complexity, the orthography of the first syllable (CVC1), of the second syllable (CVC2), and of the ending of the second syllable (VC2) were found to be probabilistically associated with stress in Russian disyllables. In Studies 2 and 3, it was shown that Russian speakers do use 3 of these cues (CVC1, CVC2, and VC2) when making stress-assignment decisions. These results provide evidence against the idea that the nature of stress in the Russian language is so unpredictable that stress assignment can only be accomplished lexically. These results also suggest that any successful model of stress assignment in Russian needs to contain mechanisms allowing these 3 orthographic cues to play a role in the stress-assignment process.     

First Order Common Knowledge Logics

In this paper we investigate first order common knowledge logics; i.e., modal epistemic logics based on first order logic with common knowledge operators. It is shown that even rather weak fragments of first order common knowledge logics are not recursively axiomatizable. This applies, for example, to fragments which allow to reason about names only; that is to say, fragments the first order part of which is based on constant symbols and the equality symbol only. Then formal properties of "quantifying into" epistemic contexts are investigated. The results are illustrated by means of epistemic representations of Nash Equilibria for finite games with mixed strategies.