## On the Emergence of Reasons in Inductive Logic (2001)

Citations: | 2 - 2 self |

### BibTeX

@MISC{Paris01onthe,

author = {J. B. Paris and M. Wafy},

title = {On the Emergence of Reasons in Inductive Logic },

year = {2001}

}

### OpenURL

### Abstract

We apply methods of abduction derived from propositional probabilistic reasoning to predicate probabilistic reasoning, in particular inductive logic, by treating finite predicate knowledge bases as potentially infinite propositional knowledge bases. It is shown that for a range of predicate knowledge bases (such as those typically associated with inductive reasoning) and several key propositional inference processes (in particular the Maximum Entropy Inference Process) this procedure is well defined, and furthermore yields an explanation for the validity of the induction in terms of `reasons'.

### Citations

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(Show Context)
Citation Context ...rn to considering solutions Bel to S 1 i=1 K(a i ) based on principles of common sense. Common sense principles were introduced explicitly in [8] (several of these had appeared earlier, especially in =-=[12-=-], a paper drawing similar conclusions albeit from rather stronger initial assumptions) as constraints on the process of assigning beliefs from (nite, linear) probabilistic knowledge bases. In this pa... |

75 |
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(Show Context)
Citation Context ...= 0; on my assigning subjective probability function Bel. Thus in this note we are identifying knowledge with a satisable set of linear constraints on a probability function Bel where, as usual (see [6]), a function Bel : SL ! [0; 1] is a probability function if it satises that for all ; 2 SL; (P 1) If j= then Bel() = 1; (P 2) If j= :( ^ ) then Bel( _ ) = Bel() +Bel(): [It is easy to ch... |

52 |
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(Show Context)
Citation Context ... their strengths. 2 The Maximum Entropy solution We now turn to considering solutions Bel to S 1 i=1 K(a i ) based on principles of common sense. Common sense principles were introduced explicitly in =-=[8]-=- (several of these had appeared earlier, especially in [12], a paper drawing similar conclusions albeit from rather stronger initial assumptions) as constraints on the process of assigning beliefs fro... |

31 |
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(Show Context)
Citation Context ...and so one might argue, should be normative for intelligent agents like ourselves. Such a conclusion for induction would stand squarely opposed to the conventional Carnapian approach (see for example =-=[5]-=-, [1], [2]) based on considerations of symmetry etc. which yield continuous de Finetti measures. 6 Acknowledgements We would like to thank Graham Little for his considerable help in the proof of Theor... |

24 | Common sense and maximum entropy
- Paris
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(Show Context)
Citation Context ...ains a subject of research. The emergence of `reasons' in this fashion is particularly intriguing in the case of the maximum entropy inference process which, we have previously argued, for example in =-=[7]-=-, corresponds to the idealization of common sense, and so one might argue, should be normative for intelligent agents like ourselves. Such a conclusion for induction would stand squarely opposed to th... |

11 | In defence of the maximum entropy inference process
- Paris, Vencovská
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(Show Context)
Citation Context ...ns albeit from rather stronger initial assumptions) as constraints on the process of assigning beliefs from (nite, linear) probabilistic knowledge bases. In this paper (subsequently improved in [6], [=-=10-=-] and [7]) it was shown that the Maximum Entropy Inference Process, ME, is the only inference process which satises all these common sense principles. To expand on this result and its context, in [8] ... |

10 |
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(Show Context)
Citation Context ...(a i ); where K(a 1 ) consists of a (satisable)snite set of linear constraints (over the reals) c 1 Bel( 1 ) + c 2 Bel( 2 ) + ::: + c mBel(m ) = d; on a subjective probability function Bel : SL ! [0; 1], with 1 ; 2 ; :::; m sentences from SL (1) and K(a i ) is the result of replacing a 1 everywhere in K(a 1 ) by a i . So, for example, with the above interpretation of Q 1 ; P (a 1 ); etc. my kn... |

9 | A method of updating that justifies minimum cross-entropy - Paris - 1992 |

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5 |
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(Show Context)
Citation Context ... might argue, should be normative for intelligent agents like ourselves. Such a conclusion for induction would stand squarely opposed to the conventional Carnapian approach (see for example [5], [1], =-=[2]-=-) based on considerations of symmetry etc. which yield continuous de Finetti measures. 6 Acknowledgements We would like to thank Graham Little for his considerable help in the proof of Theorem 3.2. We... |

5 |
A method for updating that justi®es minimum cross entropy
- Paris, Vencovska
(Show Context)
Citation Context .... This has already been proved in a number of cases although conrming it in full generality remains a topic for future investigation. Turning now to CM 1 , its motivation (which wassrst explained in [=-=9-=-], see also [6]) is rather dierent from that of MD or ME. Brie y, given a set of constraints K as above (on a probability function Bel on sentences of the language with propositional variables p 1 ; p... |

3 |
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(Show Context)
Citation Context ...n the case where K(a 1 ) is a complete set of reasons as in (1.1) the limit in (2.1) exists and equals the canonical solution. [For a proof of this result and the other main results in this paper see =-=[13-=-] or [11].] Indeed, in this case the situation is particularly simple because each sequence ME( n [ j=1 K(a j ))( r ^ i=1 P i (a n i )); is eventually constant. More interesting is the case of a gene... |

2 |
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(Show Context)
Citation Context ...ere is another way of saying that the limit solutions for ME, MD, CM 1 correspond to canonical solutions of complete sets of reasons. According to the celebrated theorem of de Finetti, [3], (see also =-=[-=-4]), if B is an exchangeable probability function on the sentences of the language with propositional variables P (a 1 ); P (a 2 ); :::, that is B( r ^ i=1 P i (a n i )) depends only on r and P i , ... |

1 |
Su signi soggetivo della probabilita', Fund
- Finetti
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(Show Context)
Citation Context ...nt shortly). There is another way of saying that the limit solutions for ME, MD, CM 1 correspond to canonical solutions of complete sets of reasons. According to the celebrated theorem of de Finetti, =-=[3-=-], (see also [4]), if B is an exchangeable probability function on the sentences of the language with propositional variables P (a 1 ); P (a 2 ); :::, that is B( r ^ i=1 P i (a n i )) depends only on... |

1 |
Some limit theorems for
- Paris, Vencovská
(Show Context)
Citation Context ...se where K(a 1 ) is a complete set of reasons as in (1.1) the limit in (2.1) exists and equals the canonical solution. [For a proof of this result and the other main results in this paper see [13] or =-=[1-=-1].] Indeed, in this case the situation is particularly simple because each sequence ME( n [ j=1 K(a j ))( r ^ i=1 P i (a n i )); is eventually constant. More interesting is the case of a general (ni... |