62,240 research outputs found
Analysis of Conditional Expressions
What David Lewis proved in 1976 was stronger than he realized. Not only can no system of logic can have a conditional connective with non-trivial conditional probability, but also no probability space can have even a single non-trivial conditional event. However, Lewis' definition of conditional connective is flawed, and does not apply to his original target, the Stalnaker/Thomasson C2 logic. Lewis assumed a property which Stalnaker's system does not have - McGee's export-import law. Modal models of Stalnaker's C2 exist for every first-order model. Stalnaker's corner connectives, when interpreted as Lycan-style quantified conditionals, do have nontrivial conditional probability. Interpreting propositions as indicator functions instead of sets of possible worlds, the modern Kolmogorov theory of conditional expectation opens new possibilities for simultaneously modeling both objective and subjective probability as the expectation of truth. I use the new interpretation to defend Lycan's theory of conditionals against an objection from Dorothy Edgington
Quantum logic and weak values
In this study, we study weak values from a quantum-logical viewpoint. In
addition, we examine the validity of the counterfactual statements of Hardy's
paradox, which are based on weak values, and we show that these statements have
not been validated. It is also shown that strange weak values may only appear
if they are not (conditional) probabilities.
PACS numbers: 03.65.Ta, 03.65.Ud, 03.65.CaComment: 1 figur
Three-slit experiments and quantum nonlocality
An interesting link between two very different physical aspects of quantum
mechanics is revealed; these are the absence of third-order interference and
Tsirelson's bound for the nonlocal correlations. Considering multiple-slit
experiments - not only the traditional configuration with two slits, but also
configurations with three and more slits - Sorkin detected that third-order
(and higher-order) interference is not possible in quantum mechanics. The EPR
experiments show that quantum mechanics involves nonlocal correlations which
are demonstrated in a violation of the Bell or CHSH inequality, but are still
limited by a bound discovered by Tsirelson. It now turns out that Tsirelson's
bound holds in a broad class of probabilistic theories provided that they rule
out third-order interference. A major characteristic of this class is the
existence of a reasonable calculus of conditional probability or, phrased more
physically, of a reasonable model for the quantum measurement process.Comment: 9 pages, no figur
Quantum Probability Theory
The mathematics of classical probability theory was subsumed into classical
measure theory by Kolmogorov in 1933. Quantum theory as nonclassical
probability theory was incorporated into the beginnings of noncommutative
measure theory by von Neumann in the early thirties, as well. To precisely this
end, von Neumann initiated the study of what are now called von Neumann
algebras and, with Murray, made a first classification of such algebras into
three types. The nonrelativistic quantum theory of systems with finitely many
degrees of freedom deals exclusively with type I algebras. However, for the
description of further quantum systems, the other types of von Neumann algebras
are indispensable. The paper reviews quantum probability theory in terms of
general von Neumann algebras, stressing the similarity of the conceptual
structure of classical and noncommutative probability theories and emphasizing
the correspondence between the classical and quantum concepts, though also
indicating the nonclassical nature of quantum probabilistic predictions. In
addition, differences between the probability theories in the type I, II and
III settings are explained. A brief description is given of quantum systems for
which probability theory based on type I algebras is known to be insufficient.
These illustrate the physical significance of the previously mentioned
differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for
clarity and accuracy made. This is the version to appear in Studies in the
History and Philosophy of Modern Physic
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