lecture_3_2x2

# lecture_3_2x2 - Branden Fitelson Philosophy 148 Lecture 1...

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Branden Fitelson Philosophy 148 Lecture 1 ' & \$ % Philosophy 148 — Announcements & Such Administrative Stuﬀ Branden’s Thursday oﬃce hours will be 2:30–3:30 this week. Raul’s oﬃce hours will be 10–12 Wed., and by appointment. Section times have been determined. Sections will meet Tuesday, 10–11 and Wednesday, 9–10. You should have received an email assigning you to a section. Otherwise, please see Raul about this. Section locations will be announced soon. Meanwhile, 301 Moses. Last Time: Finite Boolean Algebras & Some Overview Stuﬀ Today’s Agenda Review of Key Facts About Finite Propositional Boolean Algebras Some Additional “Big Picture” Stuﬀ (on Logic & Epistemology) An Algebraic Approach to Probability Calculus Next: An Axiomatic Approach to Probability Calculus UCB Philosophy Logical Background & Algebraic Probability 01/29/08 Branden Fitelson Philosophy 148 Lecture 2 ' & \$ % Overview of Finite Propositional Boolean Algebras I Consider a logical language L containing n atomic sentences. These may be sentence letters ( X , Y , Z , etc. ), or they may be atomic sentences of monadic or relational predicate calculus ( Fa , Gb , Rab , Hcd , etc. ). The Boolean Algebra B L set-up by such a language will be such that: B L will have 2 n states (corresponding to the state descriptions of L ) B L will contain 2 2 n propositions , in total. * This is because each proposition p in B L is equivalent to a disjunction of state descriptions. Thus, each subset of the set of state descriptions of L corresponds to a proposition of B L . * Note: there are 2 2 n subsets of a set of size 2 n . · The empty set of state descriptions corresponds to “the empty disjunction”, which corresponds to the logical falsehood : . · Singelton sets of state descriptions correspond to “disjunctions with one member”. [All other subsets are “normal” disjunctions.] UCB Philosophy Logical Background & Algebraic Probability 01/29/08 Branden Fitelson Philosophy 148 Lecture 3 ' & \$ % Overview of Finite Propositional Boolean Algebras II Example. Let L have three atomic sentences: X , Y , and Z . Then, B L is: 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 YZ s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 Examples of reduction to disjunctions of state descriptions of L : X & X ’ is equivalent to the empty disjunction: . X & ( Y & Z) ’ is equivalent to the singleton disjunction: s 3 . X (Y ’ is equivalent to: s 1 s 2 s 3 s 8 . In general: p ±² W { s i | s i ² p } . And, if { s i | s i ² p } = ∅ , then p ±²⊥ . UCB Philosophy Logical Background & Algebraic Probability 01/29/08 Branden Fitelson Philosophy 148 Lecture 4 ' & \$ % Inductive Logic — Basic Motivation and Ideas Intuitively, not all “logically good” arguments are deductively valid.

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lecture_3_2x2 - Branden Fitelson Philosophy 148 Lecture 1...

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