Class Note for PHIL 110 at UMass(21)
Class Note for PHIL 110 at UMass(21)
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This 8 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 15 views.
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Date Created: 02/06/15
INTRO LOGIC DAY 17 Translations in PL 3 Existential Quantifier someone is happy there is someone who is happy there is some x x is happy EIX Hx REVIEW of DAY 1 and DAY 2 ExistentialNegative Quantifier someone is unhappy there is someone who is not happy there is some x x is not happy EIX Hx D Universal Quantifier everyone is happy no matter who you are you are happy no matter who x is x is happy Vx Hx UniversalNegative Quantifier eve ryo ne is unhappy no matter who you are you are not happy no matter who x is x is not happy Vx Hx D NegativeExistentialQuantifier 3 no one is happy there is no one who is happy there is no x x is happy 3x Hx 7 D NegativeUniversal Quantifier not everyone is happy not no matter who you are you are happy not no matter who x is x is happy Vx Hx Quantifier Specification some IE QuantifierSpecification every I I some Freshman is Happy I I every Freshman is Happy I there is someone who is F and who is H no matter who you are IF you are F THEN you are H there is some X X is F and X is H no matter who X is IF X is F THEN X is H EIXFX amp HX VX FX a HX D D Quantifier Specification no I no Freshman is Happy I there is no one who is F and who is H thereisnox XisF and XisH EXFX amp HX forday3 10 12 Multiple Quantification l sentences with more than one quantifier GENERAL STRATEGY Count the number of quantifiers in original sentence Determine the overall structure ofthe sentence Work on constituents separately Substitute constituents back into overall formula Example 2 every CAT is a PET but not every PET is a CAT every C is P but not every P is C VXCX PX ampN VxPx4gtCx VXCX PX VxPx4gtCx 13 15 Example 1 everyone is FRIENDLY but not everyone is HAPPY everyone is F but not everyone is H VXFX amp VXHX VXFX VXHX Example 3 if everyone is FRIENDLY then everyone is HAPPY W everyoneis F then everyone is H 14 VXFX E VxHx VXFX VxHx 16 Example 4 if every STUDENT is FRIENDLY then every STUDENT is HAPPY every 8 is F then every Sis H VXSX Fx VXSX Hx VXSX Fx VXSX Hx 17 Any versus Every Basic Principle both any and every are universal quantifiers BUT they are usually not interchangeable Sometimes they are interchangeable any one can Dance every one can Dance VxDx if can Dance then any one can Di VxDx if can Dance then every one can 19 Usually they are not interchangeable is every one here 739 is any one here Jay doesn t respect Jay doesn t respect every one 397 quot any one if every one can fix your if any one can fix your car then I can 397quot car then I can no one respects 1 no one respects every one any one 18 20 Difference between every and any 2 the scope of every is narrow the scope of any is wide 21 Example NotAny Jay doesn t respect anyone does Jay respect a Rja no NRja does Jay respect b ij no Nij does Jay respect c ch no Nch etc no matter who you are Jay does not respect you no matter who x is Jay does not respect x VX ij any V has wide scope not has narrow scope 23 Example NotEvery Jay doesn t respect everyone not Jay respects everyone Vijx not has wide scope every V has narrow scope 22 NotAny None NotSome D Jay respects no one there is no one whom Jay respects there is no X Jay respects X EX ij Recall Ev Vv Elx ij Vx ij Jay respects no one Jay doesn t respect anyone 24 Example FEVERY if everyone fails then satan wins if everyonefails then satanwins VXFX a Ws VXFX Ws evew has narrow scope ifthen has wide scope 25 How do we SHOW such a formula 1 SHOW Vxe a Ws CD 2 Vx Fx As 3 SHOW Ws D 26 Example FANY if anyone fails then satan wins if a fails then satan wins if b fails then satan wins if c fails then satan wins etc if anyonefails then satanwins 27 In Other Words no matterwho you are if you fail then satan Wins no matter who X is if X fails then satan Wins Vx Fx Ws any has wide scope ifthen has narrow scope 28 How do we SHOW such a formula 2 1 SHOW VxFx aWs UD 2 SHOW Fa a Ws CD 3 Fa As 4 SHOW Ws UD Universal Derivation later 29 Special Note Sometimes but not always if any if some Vx Fx q Elex gt d provided d has no free occurrence of x 30
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