## Constructing a Logic of Plausible Inference: a Guide To Cox's Theorem (2003)

Venue: | International Journal of Approximate Reasoning |

Citations: | 11 - 0 self |

### BibTeX

@ARTICLE{Horn03constructinga,

author = {Kevin S. Van Horn},

title = {Constructing a Logic of Plausible Inference: a Guide To Cox's Theorem},

journal = {International Journal of Approximate Reasoning},

year = {2003},

volume = {34},

pages = {3--24}

}

### Years of Citing Articles

### OpenURL

### Abstract

Cox's Theorem provides a theoretical basis for using probability theory as a general logic of plausible inference. The theorem states that any system for plausible reasoning that satisfies certain qualitative requirements intended to ensure consistency with classical deductive logic and correspondence with commonsense reasoning is isomorphic to probability theory. However, the requirements used to obtain this result have been the subject of much debate. We review Cox's Theorem, discussing its requirements, the intuition and reasoning behind these, and the most important objections, and finish with an abbreviated proof of the theorem.

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Citation Context ...e fundamental objection to representing one's degree of certainty or belief in a proposition with a single value. In particular, two popular approaches to plausible inference|belief function theories =-=[14,15]-=- and possibility theory [10]|are twodimensionalstheories in which one's certainty in a proposition is represented by a pair of numbers. Such theories unavoidably lack universal comparability. With reg... |

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Citation Context ...e fundamental objection to representing one's degree of certainty or belief in a proposition with a single value. In particular, two popular approaches to plausible inference|belief function theories =-=[14,15]-=- and possibility theory [10]|are twodimensionalstheories in which one's certainty in a proposition is represented by a pair of numbers. Such theories unavoidably lack universal comparability. With reg... |

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Citation Context ...d to show that those requirements force our system of plausible reasoning to be isomorphic to probability theory, giving us the Bayesian approach to plausible reasoning. Our proof borrows from Jaynes =-=[3] and-=- Paris [4]. We begin by deriving some properties of F . The simple fact that \^" is associative turns out to have great ramications. It ensures that F is associative, which in turn limits the pos... |

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Citation Context ...ics deal with the latter, and hence have aims distinct from ours. Failure to distinguish these distinct concepts has in the 2 past led to unnecessary controversy [11]. On this issue, Dubois and Prade =-=[12-=-] write: \. . . [name omitted] fails to understand the important distinction between two totally dierent problems. . . These are the handling of gradual (thus non-Boolean) properties whose satisfactio... |

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Citation Context ...ny choices available, as the issue is not suciently resolved to allow widespread agreement on the proper solution. On the other hand, the rapidly growing use of Bayesian methods by working scientists =-=[19]-=- suggests that the issue need not be a serious stumbling block in practice. Another motivation for two-dimensional theories has been the perception that proper application of Bayesian methods requires... |

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Citation Context .... Given our requirements w is continuous and strictly increasing, so the two approaches are equivalent, but the distinction becomes important if we relax the strictness requirement on F of Section 10 =-=[-=-23]. 9 such that F as T, there exists some c 2 P 0 such that asb. For every y 1 ; y 2 ; y 3 2 P 0 there exists some consistent X with a basis of at least three atomic propositions|call them A 1 , A ... |

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Citation Context ...sing arbitrary values for the various (A i j X), then using the above equalities. R4 is not without controversy. Paris [4] highlights its crucial importance in his proof of Cox's Theorem, and Halpern =-=[24]-=- shows that omitting R4 allows one to construct an explicit counterexample to Cox's Theorem. Halpern goes further to argue that R4 is unreasonable forsnite domains (those with only asnite set of atomi... |

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Citation Context ...values issnite only if, in addition to restricting ourselves to a single,snite problem domain, we also restrict ourselves to asnite set of possible states of information for that problem domain. Snow =-=[28-=-] argues against such a restriction and for innite gradations of plausibility within even a single,snite domain: \It often happens that sentences of interest include some that describe events for whic... |

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Citation Context ...jodin have investigated an alternative form of Cox's Theorem that does not rely on universality, but instead uses notions of non-informative renability and information independence forsnite domains [2=-=9,30-=-]. Other dierences include replacing R3 with a requirement that (A_B j X), where A and B are exclusive, be a strictly increasing function of (A j X) and (B j X). One trivial consequence of adding R4 i... |

1 |
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Citation Context ...jodin have investigated an alternative form of Cox's Theorem that does not rely on universality, but instead uses notions of non-informative renability and information independence forsnite domains [2=-=9,30-=-]. Other dierences include replacing R3 with a requirement that (A_B j X), where A and B are exclusive, be a strictly increasing function of (A j X) and (B j X). One trivial consequence of adding R4 i... |

1 |
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Citation Context ...r strictly decreasing in both arguments on (F; T] 2 . It is possible to slightly weaken our requirements on F without losing our main result (that our system must be isomorphic to probability theory) =-=[31]-=-. Unfortunately, we cannot get by with requirements nearly as weak as those on S 0 . In particular, the requirement that F be strictly increasing is essential; if we relax this to a requirement that F... |