Learning, Logic, and Topology in a Common Framework

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

russell: the Journal of Bertrand Russell Studies n.s. 36 (winter 2016–17): 163–90 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 c:\users\arlene\documents\rj\type3602\red\rj 3602 134 red.docx 2017-01-09 4:03 PM oeviews ANALYSIS, MATHEMATICS, AND LOGIC IN RUSSELL’S EARLY PHILOSOPHY James Levine Philosophy / Trinity College Dublin 2, Ireland jlevine@tcd.ie Jolen Galaugher. Foreword by Michael Beaney. Russell’s Philosophy of Logical Analysis: 1897–1905. (History of Analytic Philosophy series.) Basingstoke and NewYork: Palgrave Macmillan, 2013. Pp. xii, 218. isbn 13: 978-1-137-302069 . £60; us$110 (hb). n this book Jolen Galaugher discusses a number of key turning-points in Russell’s early philosophy: his break from Idealism in 1898–99; his acceptance of logicism in the period following the International Congress of Philosophy in Paris in August 1900 at which he was impressed by Peano and his students; and his arrival at his theory of descriptions in 1905 in the context of his attempts to grapple with various versions of what Russell called “the Contradiction”. Her overall purpose is to examine the interplay between the view of the analysis of propositions that Russell accepts following his break with Idealism and his developing views within the philosophy of mathematics. The main merits of the book are the way in which it highlights unresolved issues in our understanding of Russell’s early development and brings to bear relevant evidence in attempting to address those issues. However, as Galaugher acknowledges, the book’s “arguments are complicated in places” (p. 2), and it is not an easy read. Like some other books in this series, it appears to be based on a recently completed phd dissertation, and, like some dissertations, it combines a serious engagement with recent scholarship with a manner of presentation that presumes a shared understanding with the reader of a number of central issues and positions, and so does not clearly articulate and defend her understanding of those issues and positions to the uninitiated or sceptical reader. Hence, while the book will be useful to scholars already familiar with the sorts of issues it addresses, it would be hard to recommend it to someone coming to this material for the first time. The book consists of five chapters. In the first, Galaugher outlines some f= 164 Reviews c:\users\arlene\documents\rj\type3602\red\rj 3602 134 red.docx 2017-01-09 4:03 PM central elements of Russell’s position during his Idealist period and some of the steps involved in his dismantling that position. In the second, she argues more specifically, that in rejecting Idealism, Russell was not, as it is sometimes presented, simply following G. E. Moore—that his engagement with Leibniz’s views led Russell to go beyond Moore in certain ways and to adopt his characteristic view that relations are “external”. In the third, she discusses aspects of Russell’s logicism and its development in the period following the Paris Congress. In the fourth, she focuses on Russell’s logicist definition of the cardinal number and the notion of “class” that underpins it, arguing that in the face of the Contradiction, and his changing view of class and propositional function, it becomes questionable whether Russell and Frege should be said to endorse the same definition of cardinal number. In the fifth, she examines aspects of Russell’s coming to reject the theory of denoting concepts in favour of the theory he presents in “On Denoting”, arguing that by doing so Russell reaffirms his commitment to his post-Idealist view of a “decompositional” view of analysis as opposed to Frege’s function/argument style of analysis. Each of the aspects of Russell’s views that Galaugher examines concerns issues on which Russell often changed his mind, sometimes within a quite short period of time. While much of the material relevant to understanding these changes of mind has now been published in the Collected Papers of Bertrand Russell, Russell scholars have not fully assimilated it. Galaugher not only calls attention to a number of features of Russell’s early development that are not yet fully understood...

We study the learnability of classes of computable structures under models that allow finitely many mind changes. Extending classical notions of explanatory learning from informant and from text, we introduce new paradigms where the information source and the convergence requirements are modified. In particular, we define Δ 2 0 -learning, where each atomic fact may be presented with finitely many errors before stabilizing, and c.e.- and d.c.e.-learning , where information is restricted to positive atomic facts that may either never be retracted (c.e.) or be retracted at most once (d.c.e.). We provide syntactic characterizations for these notions: Δ 2 0 -learning coincides with definability by finite existential sentences, and c.e.-learning coincides with TxtEx-learning. For d.c.e.-learning we give partial characterization in restricted languages. Furthermore, for n -learning from informant, where learners are allowed at most n mind changes, we establish a complete syntactic characterization in terms of specific infinitary formulas of bounded depth.

Logical Consequence in Modal Logic: Alternative Semantic Systems for Normal Modal Logics

South-South learning exchange (SSLE) refers to an interactive learning process where peers exchange knowledge and experience to work towards a beneficial change. Despite organizations having recently increased the opportunity to run SSLEs, the SSLE support mechanisms and processes are not well documented in the scientific literature. This study explored experts' perspectives on SSLEs, strengths, weaknesses and mechanisms leading to sustainable outcomes. We conducted a qualitative study using semi-structured interviews on experiences of participants and organizers of SSLEs. Data were collected between 1st September 2021 to 26th November 2021. All data were digitally recorded, transcribed verbatim, and analysed. In the analysis, we adopted an inductive approach derived from thematic analysis. Sixteen experts, who have participated in or facilitated one or more SSLE, were interviewed. The experts' accounts demonstrated an appreciation of participants' empowerment, positive peer-to-peer "mind change" and convincing and powerful hands-on learning of this approach as strengths in the implementation of the SSLE. Being resource heavy, participant and donor reluctance and absence of a validated methodology emerged as main weaknesses of the South-South learning approach, which could impair the effectiveness of this scheme. The strengths of SSLEs are anchored in the theories of experiential and social learning, highlighting SSLE's potential to create an environment that enhances knowledge exchange. the study highlights the challenges SSLE initiatives face. In particular, these include limited commitment and funds, limited evidence of impact, disparate approaches, and the absence of standardized guidelines and evaluation practices.

SPECIALTY GRAND CHALLENGE article Front. Chem., 26 March 2013Sec. Analytical Chemistry https://doi.org/10.3389/fchem.2013.00005

The relationship between language and other aspects of conceptual development is one of the central issues in child language acquisition. One view holds that language is a special capacity, separate from other areas of cognition and learning. Another maintains that language is part of a larger, more general cognitive system, and is crucially dependent on other cognitive domains. Recent research has turned to blind children and their acquisition of language as a way of evaluating whether and how language development relies on the non-linguistic context. Vision and the Emergence of Meaning addresses this complex problem through a detailed empirical analysis of early language development in a group of blind, partially sighted and fully sighted children who took part in a pioneering longitudinal investigation at the University of Southern California. By exploring the strategies which blind children bring to selected aspects of the language learning task, Anne Dunlea not only identifies some important differences between blind and sighted children, but also offers new insights on semantic and pragmatic development in general. Further, the study demonstrates the role of conceptual information in language learning and, at a more fundamental level, reveals a convergence of early language and conceptual development.

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.

In his classic 1936 essay “On the Concept of Logical Consequence”, Alfred Tarski used the notion of satisfaction to give a semantic characterization of the logical properties. Tarski is generally credited with introducing the model-theoretic characterization of the logical properties familiar to us today. However, in his book, The Concept of Logical Consequence, Etchemendy argues that Tarski's account is inadequate for quite a number of reasons, and is actually incompatible with the standard model-theoretic account. Many of his criticisms are meant to apply to the model-theoretic account as well.

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels.

Can Tarski's definition of truth be taken to be an analysis of truth – i.e., a method of defining formal truth predicates that are capable of playing the role of truth in all theoretical contexts in which that notion is needed? It is argued that although Tarski's truth predicates can play many of the roles demanded of truth, they cannot play the role of truth in theories of meaning and interpretation. Crucial to the argument is the observation that there is a conceptual connection between our ordinary notions of truth and meaning, in virtue of which statements of the truth conditions of sentences provide some information about their meanings. Since there is no corresponding connection between Tarski's notion of truth and the notion of meaning, statements of the Tarski‐truth conditions of sentences provide no information about meaning. The chapter closes with a detailed defense of Tarski's model‐theoretic analysis of logical truth and logical consequence against John Etchemendy's objections to it.

Computability theoretic learning theory (machine inductive inference) typically involves learning programs for languages or functions from a stream of complete data about them and, importantly, allows mind changes as to conjectured programs. This theory takes into account algorithmicity but typically does not take into account feasibility of computational resources. This paper provides some example results and problems for three ways this theory can be constrained by computational feasibility. Considered are: the learner has memory limitations, the learned programs are desired to be optimal, and there are feasibility constraints on learning each output program as well as other constraints to minimize postponement tricks.

The relation between Software Engineering and General Systems Theory is explored. They are found to be dual meta-disciplines which need each other. It is discovered that there is a simple connection between the background variables of Klir's GSPS and Software Design Methodological Viewpoints. This connection motivates a look al the ontological assumptions underlying the two meta-disciplines which underlies the epistemological framework of GSPS. That framework is used to attempt to understand the nature of the different manifestations of software. This leads to an in depth look at the essence of software which in turn leads to the extension of the epistemological lattice of GSPS. The article shows the complex ways in which Software Engineering Methods and General Systems Theory intertwine al the ontological and epistemological levels which causes us to understand both differently than we would have understood them separately. The point of the article is to show how these two meta-disciplines need and intimately involve each other at a fundamental level and in fact co-arise from each other in spite of the fact that one is extremely general and the other extremely concrete. The object is to establish a basis for software engineers and general systems theoriests to learn from each other based on a common framework for understanding Software Systems

<title>ABSTRACT</title> <p>The United States military stands to greatly benefit from perpetual advances in vehicle-borne 360-degree Situational Awareness (SA) systems. However, in recent years, a gap has emerged that hinders development of vehicle-borne 360 SA. At a fundamental level, military ground vehicle designers require unambiguous requirements to build effective 360-degree SA systems; and, critical decision-makers must define requirements that offer substantial operational value. To ensure that 360-degree SA systems effectively address Warfighter requirements, the military ground vehicle research and development communities must better understand vehicle-borne 360 SA evaluation parameters and their relevance to current military operations. This paper will therefore describe a set of evaluation parameters across five broad categories that are vital to effective 360-degree SA: namely, vehicle-mounted visual sensors, data transmission systems, in-vehicle displays, intelligent cuing technologies, and human factors issues. This paper clearly explains the links between these parameters and current military operations; and, it argues that such parameters are critical to uniting stakeholders under a common framework to ensure that 360-degree SA systems provide Warfighters with the means to make sound decisions in combat.</p>

This chapter is an examination of the evidence of the effects of the continental Psychologismusstreit on Moore’s introduction to mental science and moral science via his teachers. Moore’s teachers were influential mental scientists in their own right, but their critical approach to continental idealism, particularly in their discussions of Kant, is an important element in an account of Moore’s early philosophical development. This chapter examines the influences on Moore’s thought via surviving archival material, including Moore’s lecture notes and essays. The focus is on giving an account of the way in which an act/object distinction—or an act/content/object distinction—had begun to surface explicitly in the mental science and philosophical literature of the late nineteenth century, along with a deepening awareness of a gap between the logical and metaphysical properties of mental states and their purely psychological ones that began to take hold across a variety of disciplines on the continent and in Britain.