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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such Peter Tosh : Rastafari Is Administrative Stuff HW #3 due today , usual drill (truth-table methods for validity). I have posted two handouts: (1) solutions to problems from lecture on logical truth, equivalence, etc., and (2) three examples of the short truth-table method for validity (to be discussed today). + Make sure you study my handouts. They tend to be useful. Today: Chapter 3, Final Final Remarks on LSL Semantics Expressive Completeness: re-cap + some additional remarks. Rewind: Some Chapter 2 problems revisited in light of Chapter 3. An actual LSAT problem (easy, given our knowledge of Chapter 3). Next: Chapter 4 Natural Deduction Proofs for LSL Natural deductions are the most challenging topic of the course. UCB Philosophy Chapter 3 Final, Chapter 4 Intro 10/03/08 Branden Fitelson Philosophy 12A Notes 2 ' & Expressive Completeness: Recap Fact . The set of 4 connectives h , & , , i is expressively complete. [ p q , [ (p q) & (q p) Fact . The set of 3 connectives h , & , i is expressively complete. [ p q , [ p q Fact . The pairs h , & i and h , i are both expressively complete. [ p q , [ ( p & q) The h , i strategy is similar [ [ p & q , [ ( p q) ]. Consider the binary connective | such that [ p | q [ (p & q) . Fact . | alone is expressively complete! How to express h , & i using | : [ p , [ p | p , and [ p & q , [ (p | q) | (p | q) I called | NAND in a previous lecture. NOR is also expressively complete. UCB Philosophy Chapter 3 Final, Chapter 4 Intro 10/03/08 Branden Fitelson Philosophy 12A Notes 3 ' & $ % Expressive Completeness: Additional Remarks and Questions Q . How can we define in terms of | ? A . If you navely 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))) Can anyone give an even simpler definition of using | ? Extra-Credit! How could you show that the pair h , i is expressively complete? Fact . No subset of h , & , , , i that does not contain negation is expressively complete. [This is a 140A question, beyond our scope.] Let denote the truth-function ( i.e. , the trivial function that always returns ). How could you show that h , i is expressively complete?...
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This note was uploaded on 09/23/2009 for the course PHIL 12A taught by Professor Fitelson during the Spring '08 term at University of California, Berkeley.
- Spring '08