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notes_17_2x2 - Branden Fitelson ' Philosophy 12A Notes 1...

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Branden Fitelson Philosophy 12A Notes 1 ' $ % TV On the Radio : Dear Science Administrative Stuff HW #3 will be returned Wed. Resubs due Friday (as usual). + When you turn in resubmissions, make sure that you staple them to your original homework submission . – Take-Home Mid-Term will be posted on Friday. Today: Chapter 3, Finalé — Brief Final Words on LSL Semantics An actual LSAT problem. [LSAT logic problems are LSL problems!] Some famous logic puzzles (again, these are just LSL problems). Then: Chapter 4 — Natural Deduction Proofs for LSL Validity ( ± ) vs Proof ( ` ) — some introductory remarks. Our natural deduction proofs system for LSL. Soon — MacLogic — a useful computer program for proofs. + Natural deductions are the most challenging topic of the course. UCB Philosophy Chapter 3 Finalé, Chapter 4 Intro 10/06/08 Branden Fitelson Philosophy 12A Notes 2 ' $ % Expressive Completeness: Rewind, and More Extra-Credit Q . How can we define in terms of | ? A . If you naïvely apply the schemes I described last time, then you get a 187 symbol monster : [ p q ± , A | A , where A is given by the following 93 symbol expression: (((p | (q | q)) | (p | (q | q))) | ((p | (q | q)) | (p | (q | q)))) | (((q | (p | p)) | (q | (p | p))) | ((q | (p | p)) | (q | (p | p)))) There are simpler definitions of using | . E.g. , this 43 symbol answer: [ p q ± , ((p | (q | q)) | (q | (p | p))) | ((p | (q | q)) | (q | (p | p))) I offered extra-credit for a shorter solution. One of the students — while still in class! — came up with the following 19-symbol solution: [ p q ± , ((p | p) | (q | q)) | (p | q) I had a hunch this might be the shortest possible solution. And, in fact, it is ! I wrote a computer program to check all the shorter candidates (there are actually only several hundred possibilities one has to check). More E.C. Find the shortest possible definitions of (1) [ p q ± , (2) [ p q ± , and (3) [ p q ± in terms of p , q , and the NAND operator | . UCB Philosophy Chapter 3 Finalé, Chapter 4 Intro 10/06/08 Branden Fitelson Philosophy 12A Notes 3 ' $ % LSL and LSAT Logic Problems: A Sample Question A university library budget committee must reduce exactly five of eight areas of expenditure--G, L, M, N, P, R, S, and W--in accordance with the following conditions: If both G and S are reduced, W is also reduced. If N is reduced, neither R nor S is reduced. If P is reduced, L is not reduced. Of the three areas L, M, and R, exactly two are reduced. Which one of the following could be a complete and accurate list of the areas of expenditure reduced by the committee? (A) G, L, M, N, W
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notes_17_2x2 - Branden Fitelson ' Philosophy 12A Notes 1...

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