## To Properly Reflect Physicists’ Reasoning about Randomness, We Also Need a Maxitive (Possibility

Venue: | Measure”, Proc. FUZZIEEE’2005 |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Finkelstein_toproperly,

author = {Andrei M. Finkelstein and Olga Kosheleva and Tanja Magoc and Erik Madrid and Scott A. Starks and Julio Urenda},

title = {To Properly Reflect Physicists’ Reasoning about Randomness, We Also Need a Maxitive (Possibility},

booktitle = {Measure”, Proc. FUZZIEEE’2005},

year = {},

pages = {84--108}

}

### OpenURL

### Abstract

According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cup will start lifting up. In contrast, physicists believe that events with extremely small probability cannot occur. In this paper, we show that to get a consistent formalization of this belief, we need, in addition to the original probability measure, to also consider a maxitive (possibility) measure. We also show that the resulting advanced and somewhat difficult-to-described definition can be actually viewed as a particular case of something very natural: the general notion of boundedness. c ○ 2007 World Academic Press, UK. All rights reserved. 1

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Citation Context ...ery real number p0 > 0, there exists a set T (p0) for which ∀E ∈ A (m(E) > p0 ↔ E ∩ T (p0) = ∅). (1) To describe our main result, we need to recall the definition of a maxitive (possibility) measure =-=[8, 22, 24]-=-: Definition 2. A mapping m from sets to real numbers (and possibly a value +∞) is called a maxitive (possibility) measure if for every family of sets Eα for which m(Eα) and m(∪Eα) are defined, we hav... |

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Citation Context ...ery real number p0 > 0, there exists a set T (p0) for which ∀E ∈ A (m(E) > p0 ↔ E ∩ T (p0) = ∅). (1) To describe our main result, we need to recall the definition of a maxitive (possibility) measure =-=[8, 22, 24]-=-: Definition 2. A mapping m from sets to real numbers (and possibly a value +∞) is called a maxitive (possibility) measure if for every family of sets Eα for which m(Eα) and m(∪Eα) are defined, we hav... |

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Citation Context ...n-monotonic character of expert reasoning into consideration, then the lottery paradox stops being a paradox, it becomes simply one of the non-monotonic features of expert reasoning; see, e.g., Poole =-=[20, 21]-=- (see also [15]). Specifically, if we use formalisms like default logic that have been designed to capture commonsense reasoning, we can explain the above paradox. From the pragmatic viewpoint, this a... |

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Citation Context ...enience, all the proofs are placed in a special Appendix. Comment about the result and about the proof. This result is in perfect accordance with a recent paper by D. Dubois, H. Fargier, and H. Prade =-=[7]-=- in which the authors prove that the only uncertainty theory coherent with the notion of accepted belief is possibility theory. Moreover, even our proof is similar to the proofs from [7, 8]. In princi... |

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Citation Context ...er of expert reasoning into consideration, then the lottery paradox stops being a paradox, it becomes simply one of the non-monotonic features of expert reasoning; see, e.g., Poole [20, 21] (see also =-=[15]-=-). Specifically, if we use formalisms like default logic that have been designed to capture commonsense reasoning, we can explain the above paradox. From the pragmatic viewpoint, this approach is very... |

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Citation Context ...physics, explicitly stated that such inverse processes “may be regarded as impossible, even though from the viewpoint of probability theory that outcome is only extremely improbable, not impossible.” =-=[3]-=-. Example 3. If we toss a fair coin 100 times in a row, and get heads all the time, then a person who is knowledgeable in probability would say that it is possible – since the probability is still pos... |

1 | et al: To Properly Reflect Physicists’ Reasoning about Randomness [8 - Finkelstein - 1988 |

1 |
Use of Maxitive (Possibility
- Finkelstein, Kosheleva, et al.
(Show Context)
Citation Context ...alness) in the sense of the above definition. Thus, the seemingly complex notion of randomness is, in fact, very mathematically natural. Comment. Some of the results from this paper first appeared in =-=[9, 10, 14]-=- 2 Formalizing Physicists’ Reasoning about Typicalness: Formulation of the Problem 2.1 Physicists Assume That Initial Conditions and Values of Parameters Are Not Abnormal To a mathematician, the main ... |

1 |
On a general notion of boundedness
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(Show Context)
Citation Context ...alness) in the sense of the above definition. Thus, the seemingly complex notion of randomness is, in fact, very mathematically natural. Comment. Some of the results from this paper first appeared in =-=[9, 10, 14]-=- 2 Formalizing Physicists’ Reasoning about Typicalness: Formulation of the Problem 2.1 Physicists Assume That Initial Conditions and Values of Parameters Are Not Abnormal To a mathematician, the main ... |