The Logic of Quantum Measurements in terms of Conditional Events

This paper shows that the non-Boolean logic of quantum measurementsis more naturally represented by a relatively new 4-operationsystem of Boolean fractions—conditional events—thanby the standard representation using Hilbert Space. After therequirements of quantum mechanics and the properties of conditionalevent algebra are introduced, the quantum concepts of orthogonality,completeness, simultaneous verifiability, logical operations,and deductions are expressed in terms of conditional eventsthereby demonstrating the adequacy and efficacy of this formulation.Since conditional event algebra is nearly Boolean and consistsmerely of ordered pairs of standard events or propositions,quantum events and the so-called "superpositions" of statesneed not be mysterious, and are here fully explicated. Conditionalevent algebra nicely explains these non-standard "superpositions"of quantum states as conjunctions or disjunctions of conditionalevents, Boolean fractions, but does not address the so-called"entanglement phenomena" of quantum mechanics, which remainphysically mysterious. Nevertheless, separating the latter phenomenafrom superposition issues adds clarity to the interpretationof quantum entanglement, the phenomenon of influence propagatedat faster than light speeds. With such treacherous possibilitiespresent in all quantum situations, an observer has every reasonto be completely explicit about the environmental–instrumentalconfiguration, the conditions present when attempting quantummeasurements. Conditional event algebra allows such explicationwithout the physical and algebraic remoteness of Hilbert space.     

Intuitive physics and cognitive algebra: A review

IntroductionIntuitive physics explores how people without a formal instruction in physics intuitively understand physical phenomena. After a general overview of the topics of current research in intuitive physics and a discussion of current debates, this paper provides an introduction to Information Integration Theory (IIT).ObjectiveBy means of examples, it is shown how IIT can be used to directly compare the algebraic structure of physical laws and the algebraic structure of cognitive representations of these laws.MethodThe review considers about 40 years of research on the application of IIT in the field of intuitive physics. Occasionally, reference is also made to intuitive physics studies outside this theoretical framework.ResultsThe reviewed studies highlight four main factors that affect the degree of consistency between physical laws and cognitive algebraic laws: the participants’ age, their familiarity with the event under study, the type of task, and possible learning processes.ConclusionThe last part of the article discusses the implications of the results of the reviewed studies for the two main current hypotheses on the nature of intuitive physics, namely, that intuitive physics may be based on sub-optimal heuristics or may be based on the internalization of physical laws.     

Products of modal logics and tensor products of modal algebras

One of natural combinations of Kripke complete modal logics is the product, an operation that has been extensively investigated over the last 15 years. In this paper we consider its analogue for arbitrary modal logics: to this end, we use product-like constructions on general frames and modal algebras. This operation was first introduced by Y. Hasimoto in 2000; however, his paper remained unnoticed until recently. In the present paper we quote some important Hasimoto's results, and reconstruct the product operation in an algebraic setting: the Boolean part of the resulting modal algebra is exactly the tensor product of original algebras (regarded as Boolean rings). Also, we propose a filtration technique for Kripke models based on tensor products and obtain some decidability results.     

SIMONE WEIL'S SPIRITUAL CRITIQUE OF MODERN SCIENCE: AN HISTORICAL‐CRITICAL ASSESSMENT

Simone Weil is widely recognized today as one of the profound religious thinkers of the twentieth century. Yet while her interpretation of natural science is critical to Weil's overall understanding of religious faith, her writings on science have received little attention compared with her more overtly theological writings. The present essay, which builds on Vance Morgan's Weaving the World: Simone Weil on Science, Necessity, and Love (2005), critically examines Weil's interpretation of the history of science. Weil believed that mathematical science, for the ancient Pythagoreans a mystical expression of the love of God, had in the modern period degenerated into a kind of reification of method that confuses the means of representing nature with nature itself. Beginning with classical (Newtonian) science's representation of nature as a machine, and even more so with the subsequent assimilation of symbolic algebra as the principal language of mathematical physics, modern science according to Weil trades genuine insight into the order of the world for symbolic manipulation yielding mere predictive success and technological domination of nature. I show that Weil's expressed desire to revive a Pythagorean scientific approach, inspired by the “mysterious complicity” in nature between brute necessity and love, must be recast in view of the intrinsically symbolic character of modern mathematical science. I argue further that a genuinely mystical attitude toward nature is nascent within symbolic mathematical science itself.     

Congruences on Dynamic Algebras

The lattice Cong of all dynamic congruences on a given dynamic algebra is presented. Whenever is separable with zero we define dynamic ideal on , given rise to the lattice Ide. The notions of kernel of a dynamic congruence andthe congruence generated by a dynamic ideal are introduced todescribe a Galois connection between Cong and Ide. We study conditions under which a dynamic congruence is determined byits kernel.     

代数应用题项目生成的结构分析方法

自动化项目生成是近年来兴起的测量领域, 是一种以项目认知加工理论为基础的原则性项目设计(principled item design)模式。其中, 如何在项目认知模型基础上, 通过任务结构分析的方式系统全面的鉴别和提取任务特征是一个关键环节。基于已有文献中代数应用题的命题分析法、网络语言分析法、关系-函数分析法、任务分析地图等四种结构分析方法, 研究探索了能够服务于自动化项目生成的代数应用题任务结构分析方法。该分析表明, 前三种方法分别对应于个体解题过程需要形成的三种中介表征, 即问题陈述背后的命题表征、事件时空关系的情境模型、以及变量间数量关系的问题模型, 第四种方法从过程角度分析了问题解决的认知需求。然而, 要实现项目生成的特征提取需求, 尚需对现有四种方法所揭示问题特征的心理现实性、特征提取的系统性和完备性、任务领域的适用范围、以及不同方法的整合等问题开展进一步研究。     

Maximal Subalgebras of MVn-algebras. A Proof of a Conjecture of A. Monteiro

For each integer n ≥ 2, MV_n denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MV_n are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MV_n are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV₃, the mentioned characterization of maximal subalgebras of A can be given in terms of prime filters of the underlying lattice of A, in the form that was conjectured by A. Monteiro. Mathematics Subject Classification (2000): 06D30, 06D35, 03G20, 03B50, 08A30. Presented by Daniele Mundici