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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements and Such Todays Music: The Rolling Stones I have posted my solutions to HW #4 (with the shortest proofs I know). HW #5 is due today @ 4pm. Ive posted a handout entitled Working with LMPL Interpretations, which contains model answers for LMPL semantics problems. HW #6 has been posted, and will be due next Thursday @ 4pm. + The final is in class next Thursday. Youll be given 3 hours to do it. Ive posted two important handouts concerning the final exam: The (Complete) Natural Deduction Rules Handout (provided at final). A sample final exam, which has the same structure as the actual final. This sample will be discussed, in detail, in lecture tomorrow. Today: Chapter 6 Natural deduction proofs in LMPL UCB Philosophy Chapter 6 06/23/10 Branden Fitelson Philosophy 12A Notes 2 ' & The Rule of -Introduction Rule of -Introduction : For any sentence , if has been inferred at line j in a proof, then at line k we may infer [ ( ) , labeling the line j I and writing on its left the numbers that occur on the left of j. a 1 ,. . . , a n (j) . . . a 1 ,. . . , a n (k) ( ) j I Where [ ( ) is obtained syntactically from by: Replacing one or more occurrences of in by a single variable . Note: the variable must not already occur in the expression . [This prevents double-binding , e.g. , ( x)( x)(Fx & Gx) .] And, finally, prefixing the quantifier [ ( ) in front of the resulting expression (which may now have both [ s and [ s occurring in it). UCB Philosophy Chapter 6 06/23/10 Branden Fitelson Philosophy 12A Notes 3 ' & $ % The Rule of -Elimination Rule of -Elimination : For any sentence [ ( ) and constant , if [ ( ) has been inferred at a line j, then at line k we may infer , labeling the line j E and writing on its left the numbers that appear on the left of j. a 1 ,. . . , a n (j) ( ) . . . a 1 ,. . . , a n (k) j E Where is obtained syntactically from [ ( ) by: Deleting the quantifier prefix [ ( ) . Replacing every occurrence of in the open sentence by one and the same constant . [This prevents fallacies , e.g. , ( x)(Fx Gx) Fa Gb .] Note: since means everything , there are no restrictions on which individual constant may be used in an application of E. UCB Philosophy Chapter 6 06/23/10 Branden Fitelson Philosophy 12A Notes 4 ' & The Rule of -Introduction: Some Background It is useful to think of a universal claim [ ( ) as a conjunction which asserts that the predicate expression is satisfied by all objects in the domain of discourse ( i.e. , the conjunction [ a & (b & (c & ...)) is true)....
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This note was uploaded on 11/26/2011 for the course PHILOSOPHY 101 taught by Professor Buechner during the Fall '06 term at Rutgers.
- Fall '06