Polyadic Translation Exercise I - Selected Answers

1. Everybody loves somebody (or other).
   
   Lxy: x loves y
   
   (x)(∃y)Lxy

2. Someone loves someone.
   
   Lxy: x loves y
   
   (∃x)(∃y)Lxy

3. Someone loves everyone.
   
   Lxy: x loves y
   
   (∃x)(y)Lxy

4. There's someone who everybody loves.
   
   Lxy: x loves y
   
   (∃y)(x)Lxy

5. Everyone loves Ignat.
   
   Lxy: x loves y
   
   a: Ignat
   (x)Lxa

6. If someone loves Ignat, then everyone loves someone.
   
   Lxy: x loves y
   
   (∃x)Lxa  ⊃  (x)(∃y)Lxy

7. If someone loves Ignat, then everyone does.

8. There is someone who is smarter than everyone.
   
   Sxy: x is smarter than y
   (∃x)(y)Sxy

9. Someone doesn't love Ignat.

10. Someone doesn't love someone.
    
    Lxy: x loves y
    (∃x)(∃y)~Lxy

11. Someone doesn't love everyone.

12. Someone loves everyone who loves themself.
    
    Lxy: x loves y
    (y)(Lyy ⊃ (∃x)Lxy)

13. There is someone whom everyone doesn't love.

14. There is someone whom everyone doesn't love.
    
    Lxy: x loves y
    (∃y)(x)~Lxy

15. Fred eats fish.

16. Simon wants a camera.
    
    Wxy: x wants y
    Cx: x is a camera
    a: Simon
    (∃x)(Cx & Wsx)
    Note: This isn't a perfect translation, since it's truth depends on the existence of one or more camera, and one can want a camera even when there are no cameras! (The difficulty has to do with "wants" which makes this an intentional context.)
    

17. Simon wants all cameras.
    Wxy: x wants y
    Cx: x is a camera
    a: Simon
    (x)(Cx ⊃ Wsx)
    Note: This could be the translation of "Simon wants a camera" if that were understood as "Simon wants cameras."
    

18. 
    

19. All philosophers who drive desire a green tomato.
    
    Px: x is a philosopher
    Dx: x drives
    Exy: x desires y
    Tx: x is a tomato
    Gx: x is green.
    
    (x)(y))((Px & Dx) & (Ty & Gy)) ⊃ Exy)

20. Fred eats everything that doesn't move.