Circle graphs and monadic second-order logic

This article is part of a project consisting in expressing, whenever possible, graph properties and graph transformations in monadic second-order logic or in its extensions using modulo p cardinality set predicates or auxiliary linear orders. A circle graph is the intersection graph of a set of chords of a circle. Such a set is called a chord diagram. It can also be described by a word with two occurrences of each letter, called a double occurrence word. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation by a chord diagram, and this diagram can be defined by monadic second-order formulas with the even cardinality set predicate. By using the (canonical) split decomposition of a circle graph, we define in monadic second-order logic with auxiliary linear orders all its chord diagrams. This construction uses the fact that the canonical split decomposition of a graph can be constructed in monadic second-order logic with help of an arbitrary linear order. We prove that the order of first occurrences of the letters in a double occurrence word w that represents a connected circle graph determines this word in a unique way. The word w can be defined by a monadic second-order formula from the word of first occurrences of letters. We also prove that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width.     

Second-Order Science and Policy

In March 2016, an interdisciplinary group met for two days and two evenings to explore the implications for policy making of second-order science. The event was sponsored by SITRA, the Finnish Parliament's Innovation Fund. Their interest arose from their concern that the well-established ways, including evidence-based approaches, of policy and decision making used in government were increasingly falling short of the complexity, uncertainty, and urgency of needed decision making. There was no assumption that second-order science or second-order cybernetics would reveal any practical possibilities at this early stage of enquiry. On the other hand, some members of the group are practioners in both policy and in facilitating change in sectors of society. Thus, the intellectual concepts were strongly grounded in experience. This is an account of the deliberations of that group and some reflections on what came out of the various shared contributions and ensuing dialogues. The overall conclusion of the event is that there definitely are possibilities that are worthy of further research and exploration.     

Learning to fear a second-order stimulus following vicarious learning

Vicarious fear learning refers to the acquisition of fear via observation of the fearful responses of others. The present study aims to extend current knowledge by exploring whether second-order vicarious fear learning can be demonstrated in children. That is, whether vicariously learnt fear responses for one stimulus can be elicited in a second stimulus associated with that initial stimulus. Results demonstrated that children's (5–11 years) fear responses for marsupials and caterpillars increased when they were seen with fearful faces compared to no faces. Additionally, the results indicated a second-order effect in which fear-related learning occurred for other animals seen together with the fear-paired animal, even though the animals were never observed with fearful faces themselves. Overall, the findings indicate that for children in this age group vicariously learnt fear-related responses for one stimulus can subsequently be observed for a second stimulus without it being experienced in a fear-related vicarious learning event. These findings may help to explain why some individuals do not recall involvement of a traumatic learning episode in the development of their fear of a specific stimulus.     

The monadic second-order logic of graphs XV: On a conjecture by D. Seese

A conjecture by D. Seese states that if a set of graphs has a decidable monadic second-order theory, then it is the image of a set of trees under a transformation of relational structures defined by monadic second-order formulas, or equivalently, has bounded clique-width. We prove that this conjecture is true if and only if it is true for the particular cases of bipartite undirected graphs, of directed graphs, of partial orders and of comparability graphs. We also prove that it is true for line graphs, for interval graphs and for partial orders of dimension 2. Our treatment of certain countably infinite graphs uses a representation of countable linear orders by binary trees that can be constructed by monadic second-order formulas. By using a counting argument, we show the intrinsic limits of the methods used here to handle this conjecture.     

Hintikka’s Logical Revolution

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.     

幼儿对所有权权利的理解

对物品的所有权意味着所有者对所有物具有多重权利, 如触碰、使用、更改、追踪和转移等。研究发现, 3岁幼儿就能理解, 所有者对自己的物品具有触碰权和使用权, 而他人不具有。但是, 他们理解所有物的更改权、追踪权和转移权, 却相对滞后。这提示对不同所有权权利表征的发展可能是分化的。此外, 3岁幼儿还理解所有者具有赋予他人使用所有物的权利, 还会积极维护这种权利, 并对阻止权利实施的行为表示抗议, 说明他们也能理解二级所有权权利。为什么幼儿对不同所有权权利的表征会出现分化, 其背后的机制需要未来研究的探索。此外, 某些公共物品(如公共汽车)本身存在着所有权权利相分离的情况, 幼儿是如何表征的, 也值得我们进行研究。不同文化对所有权权利的侧重不同, 提示我们有必要对所有权权利认知的发展进行跨文化检验。     

What Is Philosophy of Music Education and Do We Really Need It?

The article deals with the problem of the disciplinary identification of thephilosophy of music education. It explores alternative approaches to thephilosophy of music education and its relation to musical pedagogy. On thebasis of this analysis an account of the philosophy of music education as aphilosophical discipline is suggested and its specific function identified.     

智能机器研究的早期历史回顾-阿兰·图林的理论贡献

本文回顾了提出著名"理想计算机"理论的英国数学家阿兰·图林在机器智能研究领域的重要贡献,探索了二战前他对智能计算机研究的开端及一些思想来源,分析了他1947年在会议上发表的关于"智能机器"的报告以及1950年关于"机器智能可能性"的著名论文,并探讨了图林智能计算机思想与维纳等人控制论思想的相互影响。     

Relationship between Chinese children's imaginary companions and their understanding of second-order false beliefs and emotions

We investigated relationship between Chinese children's imaginary companions (ICs) and their understanding of second-order false beliefs and emotions in 180 children, aged 5–6 years old. We examined the potential differences in second-order false belief understanding and emotion understanding between children with and without ICs, children with egalitarian IC relationships and hierarchical IC relationships, as well as children with invisible friends and personified objects. The results revealed that children with ICs had better second-order false belief understanding and emotion understanding than children without ICs. Compared with children with hierarchical relationships, children with egalitarian relationships had better second-order false belief understanding. However, children with invisible friends and personified objects did not differ on their understanding of second-order false beliefs and emotions. The results suggest that compared with IC types, IC status and child-IC relationship qualities may be more relevant to children's theory of mind. It will be interesting for the future researchers to investigate the underlying mechanism of the differences between children with egalitarian IC relationships and hierarchical IC relationships.