PaL Diagrams: A Linear Diagram-Based Visual Language

Linear diagrams have recently been shown to be more effective than Euler diagrams when used for set-based reasoning. However, unlike the growing corpus of knowledge about formal aspects of Euler and Venn diagrams, there has been no formalisation of linear diagrams. To fill this knowledge gap, we present and formalise Point and Line (PaL) diagrams, an extension of simple linear diagrams containing points, thus providing a formal foundation for an effective visual language.We prove that PaL diagrams are exactly as expressive as monadic first-order logic with equality, gaining, as a corollary, an equivalence with the Euler diagram extension called spider diagrams. The method of proof provides translations between PaL diagrams and sentences of monadic first-order logic

Fuzzy Natural Logic in IFSA-EUSFLAT 2021

The present book contains five papers accepted and published in the Special Issue, “Fuzzy Natural Logic in IFSA-EUSFLAT 2021”, of the journal Mathematics (MDPI). These papers are extended versions of the contributions presented in the conference “The 19th World Congress of the International Fuzzy Systems Association and the 12th Conference of the European Society for Fuzzy Logic and Technology jointly with the AGOP, IJCRS, and FQAS conferences”, which took place in Bratislava (Slovakia) from September 19 to September 24, 2021. Fuzzy Natural Logic (FNL) is a system of mathematical fuzzy logic theories that enables us to model natural language terms and rules while accounting for their inherent vagueness and allows us to reason and argue using the tools developed in them. FNL includes, among others, the theory of evaluative linguistic expressions (e.g., small, very large, etc.), the theory of fuzzy and intermediate quantifiers (e.g., most, few, many, etc.), and the theory of fuzzy/linguistic IF–THEN rules and logical inference. The papers in this Special Issue use the various aspects and concepts of FNL mentioned above and apply them to a wide range of problems both theoretically and practically oriented. This book will be of interest for researchers working in the areas of fuzzy logic, applied linguistics, generalized quantifiers, and their applications

Students' empathy and classroom climate as predictors of attitudes towards immigrants: a case study in three EU countries

The development of positive attitudes towards immigrants among students can be addressed at the individual level through their empathic abilities and at the school or classroom level, where the classroom climate plays an important role.In the present study, we have taken a closer look into the relationship between attitudes towards immigrants, two components of empathy (perspective taking, empathic concern), quality of student-teacher relations (both positiveand negative) and inclusive classroom climate(presented as perceived intercultural sensitivity of teachers) in a sample of 814 8th-grade students in three EU countries (Slovenia, Croatia, and Sweden). The findings showonlyempathy (especially perspective taking) was associated with better attitudes towards immigrantsin all three countries. Additionally, in two out of three countries, the importance of the relationship with teachers and inclusiveclassroomclimate was important as well. The results are discussed in the light of guidelines for school practice

Sixth Annual Catalogue of Iowa State Normal School, 1881-82

Table of Contents: Etching of the Iowa State Normal School Grounds ... 1Board of Directors ... 3Faculty ... 4Catalogue of Students 1882-83 ... 5Summary ... 15Calendar for 1882-83 ... 16Synopsis of the Course of Instruction ... 17The Science and Art of Teaching - Course of Study ... 19Text Books ... 50Expenses ... 51Summary of Information ... 52Alumni ... 56Table Showing Courses of Study in Years and Terms ... 58https://scholarworks.uni.edu/uni_catalogs/1115/thumbnail.jp

Students' Production and Processing of Mathematical Explanations.

Discussion as a mechanism for learning has been emphasized in both curriculum standards and psychological theories. However, US students get few opportunities to explain their mathematical thinking during classroom instruction. This project is aimed at answering three related questions about the role of discussion in elementary student learning. 1) Are there fewer student explanations in the US than in higher achieving East Asian locales? What predicts the prevalence of student explanations and do those predictors vary across countries? A machine-learning system was developed and validated for identifying explanations using transcripts of 232 mathematics classes in Japan, Hong Kong, and United States. Results suggest that Japan and Hong Kong lessons feature more student explanations than US lessons do. In all countries, lessons with a higher proportion of student talk showed more explanation; in Hong Kong and Japan (but not in the US), teachers’ requests for procedures and reasoning, as well as their language modeling of contradicting opinions predicted increased student explanations. One reason for this difference may be that teachers in the East Asian settings were more stringent in what they accepted as an adequate explanation. 2) Do US students differ from their Chinese peers in the quality of their mathematical explanations? Chinese and US elementary students were interviewed about mathematical equivalence. Results indicated US students underperformed their Chinese peers in the accuracy and mathematical richness of students’ explanations. 3) Do students process peer explanations differently than those of adults? US elementary students watched matched mathematical explanations from children vs. adults. The ones who watched peer-produced explanations were more likely to recognize the insufficiency of explanations and provide elaborated reasoning. Moreover, difference in processing moderated the learning gain from pre- to post-tests of their understanding of mathematical equivalence. Overall, the current study showed that although US students are providing and hearing both fewer and lower quality explanations, they may still benefit from hearing peers’ less fluent or flawed explanations. Teachers play a crucial role in directing student explanations, but this role goes beyond the questions they ask to the way in which they socialize students about what is an acceptable explanation.PhDEducation & PsychologyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108964/1/xypan_1.pd

Задачи по практической логике

В монографии доказывается, что созданное автором представление логики естественного языка посредством линейно-табличных диаграмм наиболее эффективно для решения первоочередных задач логики и для обучения их решению. Дана представительная подборка образцов задач и их решений для такого доказательства. В оптимизированной для обучения форме изложены основные положения авторской концепции практической логик

The background and use of the term 'idea' by Malebranche, Locke and Leibniz

The general distinction between uses of the term "idea" which we draw is between occurrences in the mind and dispositions for them as opposed to concepts. Locke uses "idea" in the first way, Malebranche uses it in the second. Leibniz allows that the mind is infinite and that dispositions in the body correspond to dispositions in the mind; thus he is able to maintain that idea are both concepts and dispositions in the mind. We explain concepts in terms of conventional rules, for the most part linguistic and especially mathematical. We call a system of conventional rules an objective structure and, as those who took ideas to be concepts held that they are concepts of divine science, we treat God as the unique objective structure. The question in seventeenth century theories of ideas is how that body of knowledge comprising ideas and their relations is applicable to thing. In the first four chapters, we consider concepts and the Cartesian programme to reduce the description of everything but that which applies concepts to mathematical descriptions. Descartes, Malebranche, and Leibniz held that the lack of simplicity and exactness in human knowledge arises from the correspondence between microscopic activities in the body and mental occurrences. With occurrences in the body explained mechanically, it was held, the world can be described with maximum simplicity and exactness. Extended things are law-obeying configurations to which concepts are applied; thinking things are rule-following things by virtue of applying these concepts. But the parts played by convention and behaviour are left out of their accounts and, omitting these, the world cannot be shown to be anything more than a diagram, perhaps portrayed only in the mind of the investigator. In the antepenultimate chapter, we discuss two related views which led the rationalists to maintain that all rational beings naturally follow a unique objective structure: their position on the correspondence between the activity of the body and occurrences in the mind (illustrated in their theories of vision) and the view that divine science is the standard for all scientific formulations. In the penultimate chapter, we present evidence that rationalist accounts of cognition were in fact modelled on rule-governed activity, Plato's theory of knowledge and Ideas is compared with rationalist accounts and is found to have less relevance to rule-governed activity, Kant, we admit, saw the relevance of rules, but no more than the rationalists. In the ninth chapter, we discuss Malebranche's vision in God (which most clearly presents ideas as concepts), its relation to Descartes' and Leibniz's positions and its dependent on occasionalism. In the fifth chapter, we argue against Chomsky's innatist position and, more generally, claims in the behavioural and social sciences to explain human knowledge in terms of internalized components and covert activities. It is also maintained that Chomsky's innatism bears little resemblance to that of seventeenth century rationalism. We discuss in the sixth through the eighth chapters the Scholastic back-ground to the use of the term "idea" and theories of ideas. In the sixth chapter, the pervasive influence of Suarez is established, as is the prevalence of nominalism in the seventeenth century and its connection with Gaszendism and eventually Locke. Suarez combined aspects of Thomism and nominalism, Thomism was concerned with so-called spiritual objects of knowledge, which roughly act as standards and are the contribution of the knower to what is known; rationalism's account of knowledge maintained these aspects of Thomism, nominalism, on the other hand, presented what we shall call a causal or genetic account of knowledge (according to which our knowledge arises from causal relations and operations of the intellect) and was concerned with so-called material objects got from sensation (while allowing for spiritual operations). The distinction between spiritual and material objects and faculties is introduced in the sixth chapter. In the seventh chapter, we discuss the bridge between these facilities, the intellectus agens, which served as an objective structure in Thomist accounts. In the eight chapter, we discuss uses of “spiritual”, “idea” and “mind”, beginning with Scholastic uses, but concentrating on the differences between Descartes and Gassendi. Locke’s causal account is discussed in the final chapter. We emphasise his divergence from Cartesianism, such as his view on the narrow compass of the understanding, his treatment of mathematical ideas as signs and his reliance on mental dispositions. Locke’s position suffers from the omission of concepts