dorsal/arxiv
View SchemaReality and Probability: Introducing a New Type of Probability Calculus
| Authors | Diederik Aerts |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205165 |
| URL | https://arxiv.org/abs/quant-ph/0205165 |
| DOI | 10.1142/9789812778024_0009 |
Abstract
We consider a conception of reality that is the following: An object is 'real' if we know that if we would try to test whether this object is present, this test would give us the answer 'yes' with certainty. If we consider a conception of reality where probability plays a fundamental role it can be shown that standard probability theory is not well suited to substitute 'certainty' by means of 'probability equal to 1'. The analysis of this problem leads us to propose a new type of probability theory that is a generalization of standard probability theory. This new type of probability is a function to the set of all subsets of the interval [0, 1] instead of to the interval [0, 1] itself, and hence its evaluation happens by means of a subset instead of a number. This subset corresponds to the different limits of sequences of relative frequency that can arise when an intrinsic lack of knowledge about the context and how it influences the state of the physical entity under study in the process of experimentation is taken into account. The new probability theory makes it possible to define probability on the whole set of experiments within the Geneva-Brussels approach to quantum mechanics, which was not possible with standard probability theory. We introduce the structure of a 'state experiment probability system' and derive the state property system as a special case of this structure. The category SEP of state experiment probability systems and their morphisms is linked with the category SP of state property systems and their morphisms
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"abstract": "We consider a conception of reality that is the following: An object is\n\u0027real\u0027 if we know that if we would try to test whether this object is present,\nthis test would give us the answer \u0027yes\u0027 with certainty. If we consider a\nconception of reality where probability plays a fundamental role it can be\nshown that standard probability theory is not well suited to substitute\n\u0027certainty\u0027 by means of \u0027probability equal to 1\u0027. The analysis of this problem\nleads us to propose a new type of probability theory that is a generalization\nof standard probability theory. This new type of probability is a function to\nthe set of all subsets of the interval [0, 1] instead of to the interval [0, 1]\nitself, and hence its evaluation happens by means of a subset instead of a\nnumber. This subset corresponds to the different limits of sequences of\nrelative frequency that can arise when an intrinsic lack of knowledge about the\ncontext and how it influences the state of the physical entity under study in\nthe process of experimentation is taken into account. The new probability\ntheory makes it possible to define probability on the whole set of experiments\nwithin the Geneva-Brussels approach to quantum mechanics, which was not\npossible with standard probability theory. We introduce the structure of a\n\u0027state experiment probability system\u0027 and derive the state property system as a\nspecial case of this structure. The category SEP of state experiment\nprobability systems and their morphisms is linked with the category SP of state\nproperty systems and their morphisms",
"arxiv_id": "quant-ph/0205165",
"authors": [
"Diederik Aerts"
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"categories": [
"quant-ph"
],
"doi": "10.1142/9789812778024_0009",
"title": "Reality and Probability: Introducing a New Type of Probability Calculus",
"url": "https://arxiv.org/abs/quant-ph/0205165"
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