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assignment_1 - Philosophy 148 Assignment#1 This assignment...

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Philosophy 148 — Assignment #1 02/14/08 This assignment is due Thursday, 2/28/08. Answer all questions. If you work in a group, list your group members at the top of your submitted work. 1 Problem #1 In this class, we take a probability function Pr ( ) to be a real-valued function on a (essentially, sentential) language L containing finitely many atomic sentences satisfying the following three ( Kolmogorov ) axioms, for all sentences p, q ∈ L : 1. Pr (p) 0. 2. If p > , then Pr (p) = 1. 3. If p & q , then Pr (p q) = Pr (p) + Pr (q) . In Ch. 6 of his book Choice and Chance: An Introduction to Inductive Logic (posted on website), Skyrms adopts the following six probability “rules” : ( i ) If p > , then Pr (p) = 1. ( ii ) If p , then Pr (p) = 0. ( iii ) If p q , then Pr (p) = Pr (q) . ( iv ) If p & q , then Pr (p q) = Pr (p) + Pr (q) . ( v ) Pr ( p) = 1 - Pr (p) . ( vi ) Pr (p q) = Pr (p) + Pr (q) - Pr (p & q) . Here are the two problems you must solve: ( a ) Prove Skyrms’s six rules from our axioms. Specifically, prove all of his six rules from our axioms (2) and (3) alone. You may not use any results proved in class until you’ve proved them (yourself) as “lemmas” for the purpose of this problem. You may prove Skyrms’s rules in any order, and once you have proved a result you may use it in subsequent proofs.

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