By Fearn T., Brown P.J., Besbeas P.

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4. Suppose now that someone communicates to us the following sentence: α : (Earthquake ∨ Burglary) =⇒ Alarm. 2: Possible relationships between a knowledge base |= ¬α and sentence α . By accepting α, we are considering some of these eight worlds as impossible. In particular, any world that does not satisfy the sentence α is ruled out. Therefore, our state of belief can now be characterized by the set of worlds, Mods(α) = {ω1 , ω3 , ω5 , ω 7 , ω8 }. 4, which rules out any world outside Mods(α). Suppose now that we also learn β : Earthquake =⇒ Burglary, for which Mods(β) = {ω1 , ω2 , ω5 , ω6 , ω 7 , ω8 }.

In particular, any world that does not satisfy the sentence α is ruled out. Therefore, our state of belief can now be characterized by the set of worlds, Mods(α) = {ω1 , ω3 , ω5 , ω 7 , ω8 }. 4, which rules out any world outside Mods(α). Suppose now that we also learn β : Earthquake =⇒ Burglary, for which Mods(β) = {ω1 , ω2 , ω5 , ω6 , ω 7 , ω8 }. Our state of belief is now characterized by the following worlds: Mods(α ∧ β) = Mods(α) ∩ Mods(β) = {ω1 , ω5 , ω 7 , ω8 }. Hence, learning the new information β had the effect of ruling out world ω3 in addition to those worlds ruled out by α.

68 Suppose that we check the first sensor and it is reading normal. 768 ↑ 1 It is possible however for one state of belief to assign a zero probability to the event α ∧ β even though α and β are not mutually exclusive on a logical basis. P1: KPB main CUUS486/Darwiche 36 ISBN: 978-0-521-88438-9 January 30, 2009 17:30 PROBABILITY CALCULUS Hence, our beliefs in these sensor readings are initially dependent. 80 Therefore, even though the sensor readings were initially dependent they become independent once we know the state of temperature.