Modal Interpretation Research Articles

Ambiguous Discourse in a Compositional Context. An Operational Perspective

The processing of sequences of (English) sentences is analyzed compositionally through transitions that merge sentences, rather than decomposing them. Transitions that are in a precise sense inertial are related to disjunctive and non-deterministic approaches to ambiguity. Modal interpretations are investigated, inducing various equivalences on sequences.

現象は説明しきれているか

In this paper, I examine van Fraassen's original version of modal interpretations, which have the increasing significance in the foundational research concerning elementary quantum mechanics. I argue that although van Fraassen's modal interpretation has a salient advantage over the standard Dirac-von Neumann interpretation with the projection postulate, it is confronted with two kinds of interpretive difficulties stemmed from one and the same fact of experience, repeatability of the first kind measurement.

Self-Conditional Probabilities and Probabilistic Interpretations of Belief Functions

We present an interpretation of belief functions within a pure probabilistic framework, namely as normalized self-conditional expected probabilities, and study their mathematical properties. Interpretations of belief functions appeal to partial knowledge. The self-conditional interpretation does this within the traditional probabilistic framework by considering surplus belief in an event emerging from a future observation, conditional on the event occurring. Dempster's original interpretation, in contrast, involves partial knowledge of a belief state. The modal interpretation, currently gaining popularity, models the probability of a proposition being believed (or proved, or known). The versatility of the belief function formalism is demonstrated by the fact that it accommodates very different intuitions.

USE OF GENERALIZED MASS IN THE INTERPRETATION OF DYNAMIC RESPONSE OF BENDING–TORSION COUPLED BEAMS

The interpretation of mode shapes and dynamic response of bending-torsion coupled beams is assessed by using the concept of generalized mass. In the first part of this investigation, the free vibratory motion of bending–torsion coupled beams is studied in detail. The conventional method of interpreting the normal modes of vibration consisting of bending displacements and torsional rotations is shown to be inadequate and replaced by an alternative method which is focussed on the constituent parts of the generalized mass arising from bending and torsional displacements. Basically, the generalized mass in a particular mode is identified and examined in terms of bending, torsion and bending–torsion coupling effects. It is demonstrated that the contribution of individual components in the expression of the generalized mass of a normal mode is a much better indicator in characterizing a coupled mode. It is also shown that the usually adopted criteria of plotting bending displacement and torsional rotations to describe a coupled mode can be deceptive and misleading. In the second part of the investigation, attention is focussed on the dynamic response characteristics of bending–torsion coupled beams when subjected to random bending or torsional loads. A normal mode approach is used to establish the total response. The input random excitation is assumed to be stationary and ergodic so that with the linearity assumption, the output spectrum of the response is obtained by using the frequency response function. The contribution of each normal mode to the overall response is isolated. Particular emphasis is placed on bending-induced torsional response and torsion-induced bending response. A number of case studies involving different types of bending–torsion coupled beams with Cantilever end conditions are presented. The limitations of existing methods of modal interpretation are highlighted, and an insight into the mode selection for response analysis is provided.

A philosopher's understanding of quantum mechanics: possibilities and impossibilities of a modal interpretation

1. Introduction 2. Quantum mechanics 3. Modal interpretations Part I. Formalism: 4. The different versions 5. The full property ascription 6. Joint property ascriptions 7. Discontinuities, instabilities and other bad behaviour 8. Transition probabilities 9. Dynamical autonomy and locality Part II. Physics: 10. The measurement problem 11. The Born rule Part III. Philosophy: 12. Properties, states, measurement outcomes and effective states 13. Holism versus reductionism 14. Possibilities and impossibilities 15. Conclusions.

The modal interpretation of algebraic quantum field theory

We show that Dieks' recent proposal for extending the modal interpretation to algebraic quantum field theory fails to yield a well-defined prescription for which observables in a local spacetime region possess definite values. On the other hand, we demonstrate that there is a well-defined and unique extension of the modal interpretation to the local algebras of quantum field theory (which may, however, face a serious difficulty in connection with ergodic field states).

Quantities in quantum mechanics

The problem of the failure of value definiteness (VD) for the idea of quantity in quantum mechanics is stated, and what VD is and how it fails is explained. An account of quantity, called BP, is outlined and used as a basis for discussing the problem. Several proposals are canvassed in view of, respectively, Forrest's indeterminate particle speculation, the "standard" interpretation of quantum mechanics and Bub's modal interpretation.

Consistent histories and relativistic invariance in the modal interpretation of quantum mechanics

Modal interpretations of quantum mechanics assign definite properties to physical systems and specify single-time joint probabilities of these properties. We show that a natural extension, applying to properties at several times, can be given if a decoherence condition is satisfied. This extension defines `histories' of modal properties. We suggest a modification of the modal interpretation, that offers prospects of a more general applicability of the histories concept. Finally, we sketch a proposal to apply the procedure for finding histories and a many-times probability distribution to the context of algebraic quantum field theory. We show that this leads to results that are relativistically invariant.

Delocalized Properties in the Modal Interpretation of a Continuous Model of Decoherence

I investigate the character of the definite properties defined by the Basic Rule in the Vermaas and Dieks' (1995) version of the modal interpretation of quantum mechanics, specifically for the case of the continuous model of decoherence by Joos and Zeh (1985). While this model suggests that the characteristic length that might be associated with the localisation of an individual system is the coherence length of the state (which converges rapidly to the thermal de Broglie wavelength), I show in an exactly soluble case that the definite properties that are possessed with overwhelming probability in this modal interpretation are delocalized over the entire spread of the state.

Evidence theory in multivalued models of modal logic

ABSTRACT A modal logic interpretation of Dempster-Shafer theory is developed in the framework of multivalued models of modal logic, i.e. models in which in any possible world an arbitrary number (possibly zero) of atomic propositions can be true. Several approaches to conditioning in multivalued models of modal logic are presented.

Quasi-Truth In Quasi-Set Theory

Throughout the last two decades, Newton da Costa and his collaborators have developed some frameworks to help the interpretation of science. Two of them are particularly noteworthy: partial structures and quasi-truth (that provide a way of accommodating the openness and partiality of scientific activity), and quasi-set theory (that allows one to take seriously the idea, put forward by several physicists, that we can't meaningfully apply the notion of identity to quantum particles). In this paper I explore the interconnection between these two frameworks. After reviewing the extant formulations of quasi-truth and quasi-set theory, I suggest a way of combining them, advancing a formulation of quasi-truth in quasi-set theory. In this way, a good sense can be made of the idea that quantum mechanics, if not true, is at least quasi-true. I then explore an application of this combined framework, arguing that it provides a conceptual setting appropriate to overcome two (philosophical) difficulties in van Fraassen's modal interpretation of quantum mechanics.

EVIDENCE MEASURES INDUCED BY KRIPKE'S ACCESSIBILITY RELATIONS

Modal logic interpretations of plausibility and belief measures are developed based on the observation that the accessibility relation in a model of modal logic, regarded as a multivalued mapping, induces a plausibility measure and a belief measure on the set of possible worlds.

Two No-Go Theorems for Modal Interpretations of Quantum Mechanics

Modal interpretations take quantum mechanics as a theory which assigns at all times definite values to magnitudes of quantum systems. In the case of single systems, modal interpretations manage to do so without falling prey to the Kochen and Specker no-go theorem, because they assign values only to a limited set of magnitudes. In this paper I present two further no-go theorems which prove that two modal interpretations become nevertheless problematic when applied to more than one system. The first theorem proves that the modal interpretation proposed by Kochen and by Dieks cannot correlate the values simultaneously assigned to three systems. The second and new theorem proves that the atomic modal interpretation proposed by Bacciagaluppi and Dickson and by Dieks cannot correlate the values simultaneously and sequentially assigned to two systems if one assumes that these correlations are uniquely related to the dynamics of the state of the systems.

Dempster–Shafer theory framed in modal logic

In this paper, the modal logic interpretation of plausibility and belief measures on an arbitrary universe of discourse, as proposed by Harmanec et al., is further developed by employing notions from set-valued analysis. In a model of modal logic, a multivalued mapping is constructed from the accessibility relation and a mapping determined by the value assignment function. This multivalued mapping induces a plausibility measure and a belief measure expressed in terms of conditional probabilities of inverse and superinverse images, or equivalently, in terms of conditional probabilities of truth sets of possibilitations and necessitations. Restricting to a finite universe of discourse, multivalued interpretations of basic probability assignments and of commonality functions are also obtained, in terms of conditional probabilities of pure inverse and subinverse images, or equivalently, in terms of conditional probabilities of truth sets of particular logical expressions involving possibilitations and necessitations.

Dempster's rule of conditioning translated into modal logic

A modal logic interpretation of Dempster's rule of conditioning is developed. It is shown that by restricting a model of modal logic in a non-trivial way, the measures induced by this restricted model are, in fact, the conditional measures given the restricting set, corresponding to the measures induced by the original model.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.

ファン・フラーセンの量子論解釈について

This paper deals with van Fraassen's 'no-collapse' interpretation, or 'modal interpretation'. In this interpretation he avoids 'collapse' by supposing that quantum mechanical states, unlike classical states, specify possibilities rather than actualities. But my argument will show that van Fraassen's interpretation is confronted with some difficulties concerning values of observables.

Dynamics for Modal Interpretations

An outstanding problem in so-called modal interpretations of quantum mechanics has been the specification of a dynamics for the properties introduced in such interpretations. We develop a general framework (in the context of the theory of stochastic processes) for specifying a dynamics for interpretations in this class, focusing on the modal interpretation by Vermaas and Dieks. This framework admits many empirically equivalent dynamics. We give some examples, and discuss some of the properties of one of them. This approach is applicable to a wider class of theories, in particular, those using (discrete) strict effective—as in decoherence theory—superselection rules.

Van Fraassen and Ruetsche on Preparation and Measurement

Ruetsche (1996) has argued that van Fraassen's (1991) Copenhagen Variant of the Modal Interpretation (CVMI) gives unsatisfactory accounts of measurement and of state preparation. I defend the CVMI against Ruetsche's first argument by using decoherence to show that the CVMI does not need to account for the measurement scenario which Ruetsche poses. I then show, however, that there is a problem concerning preparation, and the problem is more serious than the one Ruetsche focuses on. The CVMI makes no substantive predictions for the everyday processes we take to be measurements.

Maximal Beable Subalgebras of Quantum Mechanical Observables

The centerpiece of Jeffrey Bub's book Interpreting the Quantum World is a theorem (Bub and Clifton 1996) which correlates each member of a large class of no-collapse interpretations with some 'privileged observable'. In particular, the Bub-Clifton theorem determines the unique maximal sublattice L(R,e) of propositions such that (a) elements of L(R,e) can be simultaneously determinate in state e, (b) L(R,e) contains the spectral projections of the privileged observable R, and (c) L(R,e) is picked out by R and e alone. In this paper, we explore the issue of maximal determinate sets of observables using the tools provided by the algebraic approach to quantum theory; and we call the resulting algebras of determinate observables, "maximal *beable* subalgebras". The capstone of our exploration is a generalized version of Bub and Clifton's theorem that applies to arbitrary (i.e., both mixed and pure) quantum states, to Hilbert spaces of arbitrary (i.e., both finite and infinite) dimension, and to arbitrary observables (including those with a continuous spectrum). Moreover, in the special case covered by the original Bub-Clifton theorem, our theorem reproduces their result under strictly weaker assumptions. This added level of generality permits us to treat several topics that were beyond the reach of the original Bub-Clifton result. In particular: (a) We show explicitly that a (non-dynamical) version of the Bohm theory can be obtained by granting privileged status to the position observable. (b) We show that Clifton's (1995) characterization of the Kochen-Dieks modal interpretation is a corollary of our theorem in the special case when the density operator is taken as the privileged observable. (c) We show that the 'uniqueness' demonstrated by Bub and Clifton is only guaranteed in certain special cases -- viz., when the quantum state is pure, or if the privileged observable is compatible with the density operator. We also use our results to articulate a solid mathematical foundation for certain tenets of the orthodox Copenhagen interpretation of quantum theory. For example, the uncertainty principle asserts that there are strict limits on the precision with which we can know, simultaneously, the position and momentum of a quantum-mechanical particle. However, the Copenhagen interpretation of this fact is not simply that a precision momentum measurement necessarily and uncontrollably disturbs the value of position, and vice-versa; but that position and momentum can never in reality be thought of as simultaneously determinate. We provide warrant for this stronger 'indeterminacy principle' by showing that there is no quantum state that assigns a sharp value to both position and momentum; and, a fortiori, that it is mathematically impossible to construct a beable algebra that contains both the position observable and the momentum observable. We also prove a generalized version of the Bub-Clifton theorem that applies to "singular" states (i.e., states that arise from non-countably-additive probability measures, such as Dirac delta functions). This result allows us to provide a mathematically rigorous reconstruction of Bohr's response to the original EPR argument -- which makes use of a singular state. In particular, we show that if the position of the first particle is privileged (e.g., as Bohr would do in a position measuring context), the position of the second particle acquires a definite value by virtue of lying in the corresponding maximal beable subalgebra. But then (by the indeterminacy principle) the momentum of the second particle is not a beable; and EPR's argument for the simultaneous reality of both position and momentum is undercut.