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2. 2 6 derivations in Predicate Logic 15 points each, plus 10 free points 1.universal derivation[Exercise Set C] 2.existential-out[Exercise Set D] 3.negation rules[Exercise Set E] 4.multiple quantifiers[Exercise Set F] 5.polyadic quantifiers[Exercise Set G] 6.polyadic quantifiers[Exercise Set G]

3. 3 (10) (9) (8) (7) (6) (5) (4) (3) (2) (1) there is someone whom everyone R’s / everyone R’s someone or other 8,9,  7,  Rab 6, Rab 4,  y  Ray 1,  yRyb DD  :  As   yRay  D (ID)  :  yRay UD  :  x  yRxy Pr  x  yRyx II OO OO OO OO

4. 4 (2) (1) there is a F who R’s every G / every G is R’ed by some F or other  :  x(Gx   y(Fy & Ryx)) Pr  x(Fx &  y(Gy  Rxy))

5. 5 (13) (17) (16) (15) (14) (12) (11) (10) (8) (9) (7) (6) (5) (4) (3) (2) (1) 11, Ga  Rba 15,16,  10,14  Rba 4,13, Rba 12, Fb   Rba 8,  (Fb & Rba)  y(Gy  Rby) 9, Fb 6,  y  (Fy & Rya) 1, Fb &  y(Gy  Rby) DD  :  As   y(Fy & Rya)  D (ID)  :  y(Fy & Rya) As Ga CD  : Ga   y(Fy & Rya) UD  :  x(Gx   y(Fy & Ryx)) Pr  x(Fx &  y(Gy  Rxy)) OO II OO OO  &O OO &O OO OO

6. 6 (2) (1) there is a G who R’s no F / every F is dis-R’ed by at least one G  :  x(Fx   y(Gy &  Ryx)) Pr  x(Gx &   y(Fy & Rxy))

7. 7 (19) (18) (17) (16) (15) (14) (13) (12) (11) (10) (9) (8) (7) (6) (5) (4) (3) (2) (1) 17,18,  4,16,  Rba 10,1,  Rba 15, Fa   Rba 13,  (Fa & Rba) 12, Gb   Rba 11,  y  (Fy & Rby) 9,  (Gb &  Rba)   y(Fy & Rby) 8, Gb 6,  y  (Gy &  Rya) 1, Gb &   y(Fy & Rby) DD  :  As   y(Gy &  Rya)  D (ID)  :  y(Gy &  Rya) As Fa CD  : Fa   y(Gy &  Rya) UD  :  x(Fx   y(Gy &  Ryx)) Pr  x(Gx &   y(Fy & Rxy)) II OO OO  &O OO OO OO &O OO OO

8. 8 GOOD LUCK ON THE FINALS! HAVE A GREAT SUMMER! GOOD LUCK ON EXAM 4! GO SOX!