Today’s Topics

1. Today’s Topics Relational predicates Multiple quantification Expansions of relational predicates

2. BEWARE Zombie Logic! all Cats and Dogs make good Pets • in other words x ( ( Cx  Dx )  Px ) for any thing IF it is a Cat and it is a Dog THEN It makes a good Pet every CAT-DOG is a PET

3. No, we are not talking about CAT-DOGS!

4. Distributive Use of ‘And’ • every Cat andDog makes a good Pet • every Cat and every Dog makes a good Pet • every Cat makes a good Pet, and every Dog makes a good Pet  x ( CxPx ) x ( DxPx )

5. Plural Use of ‘And’ every member of the classCats-and-Dogs makes a good Pet no matter who x is if x is a member of the class Cats-and-Dogs, then x makes a good Pet to be a member of the class Cats-and-Dogs IS to be a Cat or a Dog x is a member of the class Cats-and-Dogs, = x is a Cat or x is a Dog = [CxDx] x ( [ CxDx ] Px )

6. Examples of Relational Predication Jay respects Kay Jay respects some F Jay respects every F some F respects Kay every F respects Kay some F respects some G some F respects every G every F respects some G every F respects every G

7. Example 1 Jay respects someone there is someone who is respected by Jay there is some x : x is respected by Jay Jay respects x x Rjx

8. Example 2 everyone respects Kay no matter who you are you respect Kay no matter who x is x respects Kay x Rxk

9. Example 3 no-one respects Jay there is no one who respects Jay there is no x : x respects Jay x Rxj

10. Example 4 everyone respects everyone no matter who you are you respect everyone no matter who x is x respects everyone x x respects everyone no matter who you are x respects you no matter who y is x respects y y Rxy x y Rxy

11. Example 5 someone respects someone there is someone who respects someone there is some x : x respects someone x x respects someone there is someone whom x respects there is some y : x respects y y Rxy x y Rxy

12. Example 6 no one respects everyone there is no one who respects everyone there is no x : x respects everyone x x respects everyone no matter who you are x respects you no matter who y is x respects y y Rxy x y Rxy

13. Example 7 no one is respected by everyone there is no one who is R’ed by everyone there is no x : x is R’ed by everyone everyone respects x x everyone respects x no matter who you are you respect x no matter who y is y respects x y Ryx x y Ryx

14. A Dangerous Ambiguity • everyone respects someone is ambiguous everyone respects someone or other vs. everyone respects someone (the same one) = there is someone whom everyone respects

15. What is the Difference? • Situation 1 • Jay Jay • Kay Kay • Elle Elle does everyone respect someone? YES is there someone whom everyone respects? NO

16. What is the Difference? (cont.) • Situation 2 • Jay Jay • Kay Kay • Elle Elle does everyone respects someone? YES is there someone whom everyone respects? YES NOTE: the latter implies the former

17. Example 8 everyone respects someone (or other) no matter who you are you respect someone no matter who x is x respects someone x x respects someone there is someone whom x respects there is some y : x respects y y Rxy x y Rxy

18. Example 9 everyone respects someone (the same) there is some one whom everyone respects there is some x : everyone respects x x everyone respects x no matter who you are you respect x no matter who y is y respects x y Ryx x y Ryx

19. Example 10 there is some one who respects no one there is some x : x respects no one x x respects no one there is no one whom x respects there is no y : x respects y y Rxy x y Rxy

20. Example 11 there is some one whom no one respects there is some x : no one respects x x no one respects x there is no one who respects x there is no y : y respects x y Ryx x y Ryx

21. Example 12 Jay doesn’t respect anyone no matter who you are Jay does not respect you no matter who x is Jay does not respect x x Jay does not respect x Rjx x  Rjx