23,502 research outputs found
Ignorance and indifference
The epistemic state of complete ignorance is not a probability distribution. In it, we assign the same, unique, ignorance degree of belief to any contingent outcome and each of its contingent, disjunctive parts. That this is the appropriate way to represent complete ignorance is established by two instruments, each individually strong enough to identify this state. They are the principle of indifference (PI) and the notion that ignorance is invariant under certain redescriptions of the outcome space, here developed into the 'principle of invariance of ignorance' (PII). Both instruments are so innocuous as almost to be platitudes. Yet the literature in probabilistic epistemology has misdiagnosed them as paradoxical or defective since they generate inconsistencies when conjoined with the assumption that an epistemic state must be a probability distribution. To underscore the need to drop this assumption, I express PII in its most defensible form as relating symmetric descriptions and show that paradoxes still arise if we assume the ignorance state to be a probability distribution. Copyright 2008 by the Philosophy of Science Association. All rights reserved
Probabilities and Quantum Reality: Are There Correlata?
Any attempt to introduce probabilities into quantum mechanics faces
difficulties due to the mathematical structure of Hilbert space, as reflected
in Birkhoff and von Neumann's proposal for a quantum logic. The (consistent or
decoherent) histories solution is provided by its single framework rule, an
approach that includes conventional (Copenhagen) quantum theory as a special
case. Mermin's Ithaca interpretation addresses the same problem by defining
probabilities which make no reference to a sample space or event algebra
(``correlations without correlata''). But this leads to severe conceptual
difficulties, which almost inevitably couple quantum theory to unresolved
problems of human consciousness. Using histories allows a sharper quantum
description than is possible with a density matrix, suggesting that the latter
provides an ensemble rather than an irreducible single-system description as
claimed by Mermin. The histories approach satisfies the first five of Mermin's
desiderata for a good interpretation of quantum mechanics, including Einstein
locality, but the Ithaca interpretation seems to have difficulty with the first
(independence of observers) and the third (describing individual systems).Comment: Latex 31 pages, 3 figures in text using PSTrick
Quantum Structure in Cognition, Origins, Developments, Successes and Expectations
We provide an overview of the results we have attained in the last decade on
the identification of quantum structures in cognition and, more specifically,
in the formalization and representation of natural concepts. We firstly discuss
the quantum foundational reasons that led us to investigate the mechanisms of
formation and combination of concepts in human reasoning, starting from the
empirically observed deviations from classical logical and probabilistic
structures. We then develop our quantum-theoretic perspective in Fock space
which allows successful modeling of various sets of cognitive experiments
collected by different scientists, including ourselves. In addition, we
formulate a unified explanatory hypothesis for the presence of quantum
structures in cognitive processes, and discuss our recent discovery of further
quantum aspects in concept combinations, namely, 'entanglement' and
'indistinguishability'. We finally illustrate perspectives for future research.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1412.870
Probabilistic Algorithmic Knowledge
The framework of algorithmic knowledge assumes that agents use deterministic
knowledge algorithms to compute the facts they explicitly know. We extend the
framework to allow for randomized knowledge algorithms. We then characterize
the information provided by a randomized knowledge algorithm when its answers
have some probability of being incorrect. We formalize this information in
terms of evidence; a randomized knowledge algorithm returning ``Yes'' to a
query about a fact \phi provides evidence for \phi being true. Finally, we
discuss the extent to which this evidence can be used as a basis for decisions.Comment: 26 pages. A preliminary version appeared in Proc. 9th Conference on
Theoretical Aspects of Rationality and Knowledge (TARK'03
An Ordinal View of Independence with Application to Plausible Reasoning
An ordinal view of independence is studied in the framework of possibility
theory. We investigate three possible definitions of dependence, of increasing
strength. One of them is the counterpart to the multiplication law in
probability theory, and the two others are based on the notion of conditional
possibility. These two have enough expressive power to support the whole
possibility theory, and a complete axiomatization is provided for the strongest
one. Moreover we show that weak independence is well-suited to the problems of
belief change and plausible reasoning, especially to address the problem of
blocking of property inheritance in exception-tolerant taxonomic reasoning.Comment: Appears in Proceedings of the Tenth Conference on Uncertainty in
Artificial Intelligence (UAI1994
- …