notes_3 - Branden Fitelson Philosophy 290 Notes 1 '...

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Unformatted text preview: Branden Fitelson Philosophy 290 Notes 1 ' & Conditionals Seminar: Day 3 Administrative: Ive added a link to Hajeks SEP entry (good background on probability) The lecture notes for my PHIL 148 class may also be useful on this score Oct. 5 : Alan Hajek will present most of the Chapter 5-related material Ill present this week and next week (ch. 4 and intro to Ch. 5). Then, well move on to student presentations. Volunteers for Chapter 6 / 7 (in 2 weeks)? More Background: A Brief (Classical) Probability Calculus Primer Back to Bennett: Chapter 2 Wrap-Up, and then Chapters 3 & 4 Ramsey and Grice (or, more accurately, Ramsey and Bennetts Grice) Jackson on Conventional Implicature, the Ramsey Test, , and The or-to-if inference [one more time] Next: Chapter 4 Lead-up to The Equation, and Lewisian attacks on it UCB Philosophy C 2 & 3 ( ) & 4 ( ) B 09 / 14 / 04 Branden Fitelson Philosophy 290 Notes 2 ' & A Brief (Classical) Probability Calculus Primer 1: Small Boolean Algebras For present purposes, it su ffi ces to think of the probability calculus as a (very simple) quantiative augmentation of classical, Boolean logic (or algebra). In fact, we wont even need to worry about algebras with more than three atomic propositions! Two representations of a 3-atom Boolean algebra: X Y Z States T T T s 1 T T F s 2 T F T s 3 T F F s 4 F T T s 5 F T F s 6 F F T s 7 F F F s 8 X Y Z s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 In a 3-atom algebra, there are 2 3 8 states . Each proposition in an algebra can be expressed as a disjunction of states . From here, probability is easy . UCB Philosophy C 2 & 3 ( ) & 4 ( ) B 09 / 14 / 04 Branden Fitelson Philosophy 290 Notes 3 ' & A Brief (Classical) Probability Calculus Primer 2: Probability Models 1 A probability function Pr pq on a Boolean algebra B is just a function that maps each state s i of B to a real number on the unit interval r , 1 s such that the sum of all the Pr p s i q is equal to one. Example of a 2-atom B plus a Pr: X Y States Pr p s i q T T s 1 1 6 T F s 2 1 4 F T s 3 1 8 F F s 4 11 24 X Y s 1 s 2 s 3 s 4 The area of the box is 1, since Pr( T ) = 1. On the left, we have a stochastic truth-table representation of x B , Pr y , and on the right we have a Venn Diagram representation in which the area of the regions is proportional to the probability of the corresponding propositions. A pair x B , Pr y consisting of a Boolean algebra of propositions B and a probability function Pr over B is called a (classical) probability model ....
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notes_3 - Branden Fitelson Philosophy 290 Notes 1 '...

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