The Origin of Probability and Entropy

Knuth, Kevin H.

doi:10.1063/1.3039020

Cited by 8 publications

(24 citation statements)

References 32 publications

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“…By introducing probability as a bi-valuation defined on a lattice of statements Knuth quantifies the degree to which one statement implies another [3][4][5][6][7][8][9][10][11]. This generalization from logical implication to degrees of implication not only mirrors Cox's notion of plausibility as a degree of belief, but includes it.…”

Section: Introductionmentioning

confidence: 94%

“…This most concise question is called the central issue, as it is the question which, when answered, answers all real questions that lie above it in the lattice of real questions [4][5][6][7][8][9][10]. Since a question is the set of statements that can be given as an answer to that question, each question represents a set of answers.…”

Section: What Is a Question?mentioning

confidence: 99%

“…Thus, in order to go from deduction to induction, we need to generalize the algebra of deduction to a calculus of induction, by way of a bi-valuation on the lattice of propositions. In what follows, we derive the constraints on a bi-valuation measure, called probability, that quantifies the degree to which one proposition implies another [3][4][5][6][7][8][9][10][11].…”

Section: Valuations On the Specific Lattice Of Propositionsmentioning

confidence: 99%

“…Logical deduction is straightforward in this framework since every proposition in the lattice implies (i.e., is included by) all the proposition above it with certainty-for example, x implies x, x ∨ y, x ∨ y ∨ z, etc. The lattice of propositions is in this sense an algebra of deduction [3][4][5][6][7][8][9][10][11].…”

Section: Valuations On the Specific Lattice Of Propositionsmentioning

confidence: 99%

“…Ideal questions are mathematical abstractions since they do not entertain all the possible mutually exclusive statements (34) as answers, with the exception of the ideal question ABC that is answered by every statement. Real questions are constructed by taking the set unions ∪ of the ideal questions such that all the possible mutually exclusive statements (34) are entertained as answers [4][5][6][7][8][9][10].…”

Section: What Is a Question?mentioning

confidence: 99%

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Inquiry Calculus and the Issue of Negative Higher Order Informations

Erp

Linger

Gelder

2017

Entropy

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Abstract:In this paper, we will give the derivation of an inquiry calculus, or, equivalently, a Bayesian information theory. From simple ordering follow lattices, or, equivalently, algebras. Lattices admit a quantification, or, equivalently, algebras may be extended to calculi. The general rules of quantification are the sum and chain rules. Probability theory follows from a quantification on the specific lattice of statements that has an upper context. Inquiry calculus follows from a quantification on the specific lattice of questions that has a lower context. There will be given here a relevance measure and a product rule for relevances, which, taken together with the sum rule of relevances, will allow us to perform inquiry analyses in an algorithmic manner.

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Section: Introductionmentioning

confidence: 94%

Section: What Is a Question?mentioning

confidence: 99%

Section: Valuations On the Specific Lattice Of Propositionsmentioning

confidence: 99%

Section: Valuations On the Specific Lattice Of Propositionsmentioning

confidence: 99%

Section: What Is a Question?mentioning

confidence: 99%

See 3 more Smart Citations

Inquiry Calculus and the Issue of Negative Higher Order Informations

Erp

Linger

Gelder

2017

Entropy

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On the perturbation of self-organized urban street networks

Benoit

Jabari

2019

Appl Netw Sci

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The complexity of urban street networks is well recognised to reside in their information networks. An information network maps roads to nodes and maps junctions to edges. Information networks of self-organized cities typically have a scale-free degree distribution. A recent fluctuating mesoscopic model links their scale-freeness to the preservation of their average amount of surprisal. Here the surprisal measures the astonishment and indecision of city-dwellers. The Metropolis algorithm may in theory allow to generate fluctuating information networks, with the scaling exponent acting as inverse temperature, in thermallike equilibria. In this paper, we show how the Metropolis algorithm can apply to urban street networks along a case study. Our case study, Old Ahmedabad, sustains Metropolis equilibria and offers promising thermodynamic-like results. Our work opens doors to a statistical physics framework for understanding how self-organized urban street networks develop.

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Measuring on Lattices

Knuth¹

2009

AIP Conference Proceedings

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Previous derivations of the sum and product rules of probability theory relied on the algebraic properties of Boolean logic. Here they are derived within a more general framework based on lattice theory. The result is a new foundation of probability theory that encompasses and generalizes both the Cox and Kolmogorov formulations. In this picture probability is a bi-valuation defined on a lattice of statements that quantifies the degree to which one statement implies another. The sum rule is a constraint equation that ensures that valuations are assigned so as to not violate associativity of the lattice join and meet. The product rule is much more interesting in that there are actually two product rules: one is a constraint equation arises from associativity of the direct products of lattices, and the other a constraint equation derived from associativity of changes of context. The generality of this formalism enables one to derive the traditionally assumed condition of additivity in measure theory, as well introduce a general notion of product. To illustrate the generic utility of this novel lattice-theoretic foundation of measure, the sum and product rules are applied to number theory. Further application of these concepts to understand the foundation of quantum mechanics is described in a joint paper in this proceedings.Comment: 13 pages, 7 figures, Presented at the 29th International Workshop on Bayesian and Maximum Entropy Methods in Science and Engineering: MaxEnt 200

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The Origin of Probability and Entropy

Cited by 8 publications

References 32 publications

Inquiry Calculus and the Issue of Negative Higher Order Informations

Inquiry Calculus and the Issue of Negative Higher Order Informations

On the perturbation of self-organized urban street networks

Measuring on Lattices

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