Toward a More Natural Expression of Quantum Logic with Boolean Fractions

This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, ‘a if b’ or ‘a given b’, ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be “inapplicable” (or “undefined”) in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b=d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the ‘influence-at-a-distance’, quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that “hidden variables” are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. Partial support for this work is gratefully acknowledged from the In-House Independent Research Program and from Code 2737 at the Space & Naval Warfare Systems Center (SSC-SD), San Diego, CA 92152-5001. Presently this work is supported by Data Synthesis, 2919 Luna Avenue, San Diego, CA 92117.     

Proof-Theoretic Functional Completeness for the Hybrid Logics of Everywhere and Elsewhere

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.     

On Amalgamation in Algebras of Logic

We show that not all epimorphisms are surjective in certain classes of infinite dimensional cylindric algebras, Pinter's substitution algebras and Halmos' quasipolyadic algebras with and without equality. It follows that these classes fail to have the strong amalgamation property. This answers a question in [3] and a question of Pigozzi in his landmark paper on amalgamation [9]. The cylindric case was first proved by Judit Madarasz [7]. The proof presented herein is substantially different. By a result of Németi, our result implies that the Beth-definability Theorem fails for certain expansions of first order logic     

Psychology’s Adaptation to Academic Medical Settings

This introduction provides an overview of the papers that follow. Each paper contributes to the central theme of the issue, psychologists’ adaptation to the realities of academic medical settings. The specific approach of each paper to the shared theme of “adaptation” is described in some detail so the busy reader can go directly to the papers of greatest interest.     

Role of Psychologists in Academic Medicine

With an interdisciplinary approach to understanding human psychosocial development and behavior, psychologists in academic medical settings have much to offer in the areas of administration, research, teaching, and service, including in the realm of organizational systems and behavior within medical schools. In these ways, psychologists can play a large role in the fulfillment of the mission of academic medicine and enhance psychology’s stature in the medical school organization. Two case studies are presented that exemplify the varied roles and responsibilities of psychologists in a medical school. To be successful within medical schools, the following generalizable principles are offered: (1) “add value” to the medical school by demonstrating the diverse and relatively unique abilities of psychologists; (2) be flexible and adaptive to the different requirements of medical academic settings; (3) make a concerted effort to work well with medical students, residents, and physicians who may not be well-versed in certain aspects such as rigorous research methodology and statistics; and (4) find a “good match” between one’s strengths, the needs of the medical setting, and the attributes of the M.D. leader. With such an approach, remarkable accomplishments can be made.     

Teaching ethics and technology with <Emphasis Type="Italic">Agora</Emphasis>, an electronic tool

Courses on ethics and technology have become compulsory for many students at the three Dutch technical universities during the past few years. During this time, teachers have faced a number of didactic problems, which are partly due to a growing number of students. In order to deal with these challenges, teachers in ethics at the three technical universities in the Netherlands — in Delft, Eindhoven and Twente — have developed a web-based computer program called Agora (see www.ethicsandtechnology.com). This program enables students to exercise their ethical understanding and skills extensively. The program makes it possible for students to participate actively in moral reflection and reasoning, and to develop the moral competencies that are needed in their later professional practice. The developers of the program have tried to avoid two traps. Firstly, they rejected, from the outset, a cookbook style of dealing with ethical problems that applied ethics is often taken to be and, secondly, they wanted to design a flexible program that respects the student’s as well as the teacher’s creativity, and that tries to engage students in moral reflection. Agora meets these requirements. The program offers possibilities that extend beyond the requirements that are usually accepted for case-exercises in applied ethics, and that have been realised in several other computer models for teaching ethics. In this article, we describe the main considerations in the development of Agora and the features of the resulting program.     

Why is Law a Normative Discipline? on Hans Kelsen’s ‘Normology’

What does it mean to claim of law that it is a normative discipline? Can the answer be so simple that one need merely refer to law’s normative object of study and the conclusions that the legal participant must allegedly draw from this? What, in any case, is a ‘normative discipline’? The essay attempts to address these questions by analysing Hans Kelsen’s ‘normological’ theory of law through his work on sovereignty and especially by focusing on the normative character of Kelsen’s epistemological claims regarding law. A theoretical critique of Kelsen is offered through Edmund Husserl’s phenomenological account of logic as a normative discipline.     

Semantics for Dual and Symmetric Combinatory Calculi

We define dual and symmetric combinatory calculi (inequational and equational ones), and prove their consistency. Then, we introduce algebraic and set theoretical– relational and operational – semantics, and prove soundness and completeness. We analyze the relationship between these logics, and argue that inequational dual logics are the best suited to model computation.