Composition as Trans-Scalar Identity

Representational measurement theory is the dominant theory of measurement within the philosophy of science; and the area in which the theory of conjoint measurement was developed. For many years it has been argued the Rasch model is conjoint measurement by several psychometricians. This paper critiques this argument from the perspective of representational measurement theory. It concludes that the Rasch model is not conjoint measurement as the model does not demonstrate the existence of a representation theorem between an empirical relational structure and a numerical relational structure. Psychologists seriously interested in investigating traits for quantitative structure should use the theory of conjoint measurement itself rather than the Rasch model. This is not to say, however, that empirical relationships between conjoint measurement and the Rasch model are precluded. The paper concludes by suggesting some relevant research avenues.

Chapter 13 - Axiomatic Geometry and Applications

The quantitative method is a powerful tool in the natural and social sciences. Depending on the research problem, the use of quantities may give significant methodological gains. Therefore, it is an important task of philosophers to clarify two related questions: in what sense do mathematical objects and structures exist? What justifies the assignment of numbers and numerals to real objects? The former question is discussed in the ontology of mathematics, the latter in the theory of measurement. This paper defends constructive realism in mathematics: numbers and other mathematical entities are human constructions which can be applied to natural and social reality by means of representation theorems. The axioms of such representation theorems are typically idealizations in the sense analyzed by critical scientific realists. If the axioms are truthlike, then quantities can be regarded as realistic idealizations.

This paper reconstructs and evaluates the representation theorem presented by Ramsey in his essay 'Truth and Probability', showing how its proof depends on a novel application of Holder's theory of measurement. I argue that it must be understood as a solution to the problem of measuring partial belief, a solution that in many ways remains unsurpassed. Finally I show that the method it employs may be interpreted in such a way as to avoid a well known objection to it due to Richard Jeffrey.

This article surveys recent investigations of real sequences (d 1,..., d n ) which arise in the theory of measurement from considerations of uniqueness for numerical representations of qualitative relations on finite sets. The sequences we discuss arise from measurement problems which include measurement of subjective probability, extensive measurement, difference measurement, and additive conjoint measurement. The measurement problems lead to sequences with fascinating combinatorial and number-theoretic properties.The unifying mathematical framework under which we analyze uniqueness of measurement in these diverse areas involves the analysis of the sequences (d 1,...,d n)as the solutions of finite systems of linear equations. Different applications are translated into different restrictions on the types of linear equations that are admissible for each area.Two primary concerns of measurement theory are involved in the work being surveyed: (1) axioms for the qualitative relation that are necessary and sufficient, or at least sufficient, for unique represent ability; (2) the structure of sets of unique solutions. The latter concern leads to combinatorial and number-theoretic problems involving characterizations of unique solutions, counts of numbers of unique solutions, and extreme-value questions. Definitive results and presently open problems are described for the areas covered by the basic theory.KeywordsSubjective ProbabilityIndependent EquationRegular SequenceExtensive MeasurementFibonacci SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter introduces the representational theory of measurement as the relevant formal framework for a metaphysics of quantities. After presenting key elements of the representational approach, axioms for different measurement structures are presented and their representation and uniqueness theorems are compared. Particular attention is given to Hölder’s theorem, which in the first instance describes conditions for quantitativeness for additive extensive structures, but which can be generalized to more abstract structures. The last section discusses the relationship between uniqueness, the hierarchy of scales, and the measurement-theoretic notion of meaningfulness. This chapter provides the basis for Chapter 6, which makes use of more abstract results in measurement theory.

Idiomaticity refers to the situation where the meaning of a lexical unit cannot be derived from the usual meanings of its constituents. As a ubiquitous phenomenon in languages, the existence of idioms often causes significant challenges for semantic NLP tasks. While previous research mostly focuses on the idiomatic usage detection of English verb-noun combinations and the semantic analysis of Noun Compounds (NCs), the idiomaticity issues of Chinese NCs have been rarely studied. In this work, we aim at classifying Chinese NCs into four idiomaticity degrees. Each idiomaticity degree refers to a specific paradigm of how the NCs should be interpreted. To address this task, a Relational and Compositional Representation Learning model (RCRL) is proposed, which considers the relational textual patterns and the compositionality levels of Chinese NCs. RCRL learns relational representations of NCs to capture the semantic relations between two nouns within an NC, expressed by textual patterns and their statistical signals in the corpus. It further employs compositional representations to model the compositionality levels of NCs via network embeddings. Both loss functions of idiomaticity degree classification and representation learning are jointly optimized in an integrated neural network. Experiments over two datasets illustrate the effectiveness of RCRL, outperforming state-of-the-art approaches. Three applicational studies are further conducted to show the usefulness of RCRL and the roles of idiomaticity prediction of Chinese NCs in the fields of NLP.

Quantitative psychology under scrutiny: Measurement requires not result-dependent but traceable data generation

Utilitarianism began as a movement for social reform that changed the world. To understand Utilitarianism, we must understand utility – how is it to be measured, and how aggregate utility of a group can be understood. The authors, a cognitive scientist and a philosopher, pursue these questions from Bentham to the present, examining psychophysics, positivism, measurement theory, meaningfulness, neuropsychology, representation theorems, and dynamics of formation of conventions.

We present an overview of the quantum theory of continuous measurements and discuss some of its important applications in quantum optics. Quantum theory of continuous measurements is the appropriate generalization of the conventional formulation of quantum theory, which is adequate to deal with counting experiments where a detector monitors a system continuously over an interval of time and records the times of occurrence of a given type of event, such as the emission or arrival of a particle.

What is memory and what is it for? How does the brain represent our experiences and use stored information to guide future behavior? In this chapter, we explore the contributions of the hippocampus to memory and to the many aspects of behavior it supports. We start with discussion of the view that hippocampal-dependent memory is a fundamentally relational and compositional representation system, supporting the binding of even arbitrary or accidental relations among constituent elements of experience into memory, representing experience via links that maintain the separate representational integrity of those elements rather than fusing them into wholistic or unitized memories, and supporting the subsequent reactivation of these relational representations. The major portion of the chapter discusses the surprisingly long reach of hippocampal influence, due to the relational and flexible nature of its representations. Hippocampal influence extends both across all manner of relations, including spatial, temporal, and associative relations, and across timescales, including not only the classically described role in long-term memory, but also on the timescale of short-term or working memory and even in moment-to-moment processing. Furthermore, the long reach of hippocampal influence is seen in the flexible use of relational representations in service of a rich variety of behavioral repertoires, helping critically in guiding flexible and adaptive choices—uses of hippocampal representations that clearly stretch the classical definition of memory. Finally, we consider the debt owed to clinical studies in providing insights about the nature of relational representations and about the extent of hippocampal influence.

Extensive measurement with incomparability

A short proof of representability of fork algebras

In this paper a strong relationship is demonstrated between fork algebras and quasi-projective relation algebras. With the help of Tarski's classical representation theorem for quasi-projective relation algebras, a short proof is given for the representation theorem of fork algebras. As a by-product, we will discuss the difference between relative and absolute representation theorems.

Retrieving relevant samples across multiple-modalities is a primary topic that receives consistently research interests in multimedia communities, and has benefited various real-world multimedia applications (e.g., text-based image searching). Current models mainly focus on learning a unified visual semantic embedding space to bridge visual contents & text query, targeting at aligning relevant samples from different modalities as neighbors in the embedding space. However, these models did not consider relations between visual components in learning visual representations, resulting in their incapability of distinguishing images with the same visual components but different relations (i.e., Figure 1). To precisely modeling visual contents, we introduce a novel framework that enhanced visual representation with relations between components. Specifically, visual relations are represented by the scene graph extracted from an image, then encoded by the graph convolutional neural networks for learning visual relational features. We combine the relational and compositional representation together for image-text retrieval. Empirical results conducted on the challenging MS-COCO and Flicker 30K datasets demonstrate the effectiveness of our proposed model for cross-modal retrieval task.