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2 EXAM #3 25 translations from English into Predicate Logic 4 points each Only final formula is graded. Do intermediate work on scratch paper.

3 if anyone is F, then everyone is H16i‘any’ if everyone is F, then everyone is H36hmultiple quantification every A and B is C16g disjunctive combinations every F S is HAPPY16econjunctive combinations only SENIORS are HAPPY16f‘only’ every SENIOR is HAPPY26cspecified quantifiers everyone is HAPPY26bsimple quantifiers JAY and KAY are HAPPY26aun-quantified formulas example# HW Chapter 6

4 every F RESPECTS some G27d2 quantifiers, 3 predicates no STUDENT RESPECTS everyone27c2 quantifiers, 2 predicates everyone RESPECTS someone27b2 quantifiers, 1 predicate KAY RESPECTS every STUDENT21 quantifier, 2 predicates everyone RESPECTS KAY27a1 quantifier, 1 predicate Example# HW Chapter 7 every one who RESPECTS every F RESPECTS every G 27ecomplex predicates

5   x Rxj no-one respects Jay  x Rxk every one respects Kay  x Rjx Jay respects some one   x ( Fx & Rxj ) no F respects Jay  x ( Fx  Rxk ) every F respects Kay  x ( Fx & Rjx ) Jay respects some F

6  x   y Ryx there is someone whom no-one respects  x   y Rxy there is someone who respects no-one  x  y Ryx there is someone whom everyone R’s  x  y Rxy everyone respects someone (or other)   x  y Ryx no-one is respected by everyone   x  y Rxy no-one respects everyone  x  y Rxy someone respects someone  x  y Rxy everyone respects everyone

7  x  y ( Gy & Rxy ) there is someone who R’s no G  x ( Fx &  yRxy ) there is some F who R’s everyone   x  y ( Gy  Rxy ) no one respects every G  x ( Fx   yRxy ) every F respects someone (or other)  x  y ( Gy & Rxy ) some one respects some G  x ( Fx   yRxy ) every F respects everyone

8  x ( Fx &  y(Gy & Rxy) ) there is some F who R’s no G  x ( Fx &  y(Gy  Rxy) ) there is some F who R’s every G   x ( Fx &  y(Gy  Rxy) ) no F respects every G  x ( Fx   y(Gy & Rxy) ) every F respects some G (or other)  x ( Fx &  y(Gy & Rxy) ) some F respects some G  x ( Fx   y(Gy  Rxy) ) every F respects every G

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10 Simple Predicates PxPx x is a Politician no further analysis required CxCx x is a Citizen Complex Predicates no Citizen Respects x Jay Respects x x Respects everyone must be further analyzed xx x Respects him/herself

11   y ( Cy & Ryx ) no Citizen Respects x  y ( Cy  Ryx ) every Citizen Respects x  y ( Py  Rxy ) x Respects every Pol  y ( Py & Rxy ) x Respects at least one Pol  y Ryx someone Respects x  y Rxy x Respects everyone Rxx x Respects him/herself Rjx Jay Respects x RxkRxk x Respects Kay

12  y ( Py & Rxy ) x R’s at least one P xx Cx xx every citizen respects at least one politician  x { Cx   y ( Py & Rxy ) } respects at least one politicianeverycitizen iseveryC   {} then  you if you are C you are 

13 xx no politician is respected by every citizen  y ( Cy  Ryx ) every C Respects x x is R’ed by every C } &Px xx   x { Px &  y ( Cy  Ryx ) } respected by every citizen nopolitician noP is  there is no one who is Pand who is  {

14 every one who respects every pol is a moron  y ( Py  Rxy ) x R’s every Politician Mx  xx { xx Mis  every  x {  y(Py  Rxy)  Mx } a moronis one who respects every P every } you are Mthen if you are   you

15  x {  y(Py  Rxy)   y(Py  Ryx) } respected by every P every one who respects every Pol is respected by every Pol  y ( Py  Ryx )  y ( Py  Rxy ) x is R’ed by every Px R’s every P xx  xx { xx  is  every } one who R’s every Peveryis you are  then if you are   you

16 xx   x {  y ( Py  Rxy ) & Rxx } who is  and who is  there is no one no one who respects every pol respects him/herself Rxx  y (Py  Rxy ) x R’s him/herself x R’s every P } & xx { xx x is  x is  there is no x :  is  no respects him/herselfone who R’s every Pno

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