Class Note for PHIL 110 at UMass(4)
Class Note for PHIL 110 at UMass(4)
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This 14 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 25 views.
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Date Created: 02/06/15
INTRO LOGIC DAY 23 Derivations in PL 2 Overview Exam 1 Sentential Logic Translations Exam 2 Sentential Logic Derivations Exam 3 Predicate Logic Translations Exam 4 Predicate Logic Derivations 6 derivations 15 points 10 free points Exam 5 very similar to Exam 3 Exam 6 very similar to Exam 4 Exams 5 and 6 will be given on Friday Dec 19 130330 400 Mahar Auditorium provided on exams available on course web page textbook keep this in front of you when doing homework mm 5mm mam m an an 4x x 4x I ll 4mm 1 7 i 7 i x x x 99 z m n m 4 Rz x an a 41 man n 5 an an R25 R25 lRQClJ x c 5 m A x x E E x x don t make UP your own rules Sentential Logic Rules DD ID CD D ampD eto ampl amp0 VO e0 NVO eto Predicate Logic Rules Universal Derivation UD day 2 UniversalOut VO day 1 TildeUniversaIOut NVO day 3 ExistentialIn 3 day 1 ExistentialOut 30 day 2 TildeExistentiaIOut 30 day 3 Rules Already Introduced Day 1 VO VvFv H0 Fo EvFv EII o is an OLD name more about this later Rules to be Introduced Today Universal Derivation UD ExistentialOut EIO Example 1 every F is H everyone is F everyone is H 1 VXFX a HX Pr 2 VXFX Pr 3 W VXHX what 3 W Ha amp Hb amp Hc amp ampampampD is a S Ha ultimately involved b S Hb in showmg c S Hc a universal Example 1a VXFXHX Pr VXFX Pr SHOW VXHX 4 Ha DD 5 FaaHa 1 VO Fa 2 VO Ha 56 a0 one down a zillion to go Example 1b VXFX a Hx Pr VXFX Pr SHOW VXHX Hb DD Fb a Hb 1 VO Fb 2 VO Hb 56 a0 two down a zillion to go Example 1c a zillion to go 1 VXFXHX Pr 2 VXFX Pr 3 SHOW VXHX 4 Hc DD 5 Fcch 1 VO 6 Fc 2 VO 7 Ho 56 a0 three down 11 But wait the derivations all look alike 4 s Ha DD 5 Fa Ha 1vo 6 Fa 2vo 7 Ha 56 0 4 Hb DD 5 Fb Hb 1vo 6 Fb 2vo 7 Hb 56 0 4 S Hc DD 5 Fc Hc 1vo 6 Fc 2vo 7 Ho 56 0 12 The UniversalDerivation Strategy All we need to do is do one derivation with one name say a and then argue that all the other derivations will look the same To ensure this we must ensure that the name is general which we can do by making sure the name we select is NEW a name counts as NEW precisely if it occurs nowhere in the derivation unboxed or uncancelled 13 The UniversalDerivation Rule UD SHOW Vvlv UD v is any variable Fv is any official formula W Fn n replaces v n must be a NEW name 0 Le one that is occurs nowhere in the derivation unboxed or uncancelled 14 Comparison with UniversalOut VO UD VvFv F0 T OLD name SHOW VvFv SHOW Fn T NEW name a name counts as OLD precisely if it occurs somewhere in the derivation unboxed and uncancelled a name counts as NEW precisely if it occurs nowhere in the derivation unboxed or uncancelled Example 2 every F is H if everyone is F then everyone is H 1 VXFX a Hx Pr 2 VXFX a VXHX CD 3 VXFX As 4 VXHX UD a new 5 Ha DD 6 Fa a Ha 1 VO a old 7 Fa 3 VO a old 8 Ha 67 a0 16 Example 3 everyFisG everyGisH everyFisH 1 VXFX a GX Pr 2 VXGX a Hx Pr 3 VXFX a Hx UD a new 4 Fa a Ha CD 5 Fa As 6 Ha DD 7 Fa a Ga 1 VO a old 8 Ga a Ha 2 VO a old 9 Ga 57 a0 10 Ha 89 a0 17 Example 4 I everyone R s everyone everyone is R ed by everyone 1 VXVnyy Pr 2 VXVyRyX UD a new 3 VyRya UD b new 4 S Rba DD 5 Vbey 1 VO b old 6 Rba 5 VO a old 18 ExistentialOut 30 any variable 2 y X w any formula Fn EvFv n replaces v any NEW name a b c d 19 Comparison with UniversalOut V0 30 VvFv EvFv H0 Fn OLD name NEW name a name counts as OLD a name counts as NEW precisely if it occurs precisely if it occurs somewhere nowhere unboxed and uncancelled unboxed or uncancelled 20 10 Example 5 every F is unH no F is H 1 VXFX a HX Pr 2 EIXFX amp Hx ND 3 EXFx amp Hx As 4 X DD 5 Fa amp Ha 3 EIO new 6 Fa a Ha 1 VO old 7 Fa 5 ampO 8 Ha 9 Ha 67 so 10 X 89 XI 21 Example 6 some F is not H not every F is H 1 EXFx amp HX Pr 2 VXFX a Hx D 3 VXFX a Hx As 4 X DD 5 Fa amp Ha 1 EIO new 6 Fa a Ha 3 VO old 7 Fa 5 ampO 8 Ha 9 Ha 67 so 10 X 89 XI 22 11 Example 7 every F is G some F is Hsome G is H 1 VXFX a GX Pr 2 EXFx amp Hx Pr 3 S EXGX amp Hx DD 4 Fa amp Ha 2 EIO new 5 Fa 9 Ga 1 VO old 6 Fa 4 ampO 7 Ha 8 Ga 56 90 9 Ga amp Ha 78 ampI 10 EIXGX amp HX 9 EII Example 8 ifanyone is F then everyone is H if someone is F then everyone is H 1 VXFX a VyHy Pr 2 EIXFX a VXHX CD 3 EIXFX As 4 VXHX UD new 5 S Ha DD 6 Fb 3 EIO new 7 Fb a VyHy 1 VO old 8 VyHy 67 90 9 Ha 8 VO old Example 9 if someone is F then everyone is H if anyone is F then everyone is H 1 EIXFX a VXHX Pr 2 VXFX a VyHy UD new 3 Fa a VyHy CD 4 Fa As 5 S VyHy UD new 6 S Hb DD 7 EIXFX 4 HI 8 VXHX 17 0 9 Hb 8 VO old 25 Example 10 a fragment someone R s someone missing premises everyone R s everyone 1 EIXEInyy Pr 2 Pr 3 S VXVnyy UD new 4 VyRay UD new 5 Rab 6 ElyRcy 1 EIO new 7 Rod 6 EIO new 8 26 THE END
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