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Today’s Topics

Explore multiple quantification and expansions of relational predicates in logic. Understand the distributive and plural use of 'and' in predicate statements. Examples of relational predication and the dangers of ambiguity.

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Today’s Topics

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  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 Jay Respects someone who respects him there is some one who is respected by Jay AND who respects Jay there is some x Jay respects x AND x respects Jay x Rjx (  Rxj ) x ( RjxRxj )

  9. Example 3 everyone respects Kay no matter who you are you respect Kay no matter who x is x respects Kay x Rxk

  10. Example 4 no-one respects Jay there is no one who respects Jay there is no x : x respects Jay ~x Rxj

  11. Example 5 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

  12. Example 6 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

  13. Example 7 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yRxy

  14. Example 8 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

  15. 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

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

  17. 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

  18. Example 9 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

  19. Example 10 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

  20. Example 11 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

  21. Example 12 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

  22. Example 13 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

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