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

BEWARE Zombie Logic! good PetsmakeDogsandCatsall It makes a good Pet THEN it is a Dog and it is a Cat IFfor any thing PxPx  DxDx  CxCx xx in other words every CAT-DOG is a PET ) )( (

No, we are not talking about CAT-DOGS !

Distributive Use of ‘And’ every Cat and Dog 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 )

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

Examples of Relational Predication every Grespectsevery F some Grespectsevery F every Grespectssome F some Grespectssome F Kayrespectsevery F Kayrespectssome F every FrespectsJay some FrespectsJay KayrespectsJay

Example 1 Jay respects x RjxRjx xx x is respected by Jaythere is some x : who is respected by Jaythere is someone Jay respects someone

Example 2 Rxk xx x respects Kayno matter who x is you respect Kayno matter who you are everyone respects Kay

Rxj xx x respects Jaythere is no x : who respects Jaythere is no one no-one respects Jay

Example 4 x respects everyone xx no matter who x is you respect everyoneno matter who you are everyone respects everyone  x  y Rxy Rxy yy x respects yno matter who y is x respects youno matter who you are

Example 5 x respects someone xx there is some x : who respects someonethere is someone someone respects someone  x  y Rxy Rxy yy x respects ythere is some y : whom x respectsthere is someone

x respects everyone  x x respects everyonethere is no x : who respects everyonethere is no one no one respects everyone  x  y Rxy Rxy yy x respects yno matter who y is x respects youno matter who you are

everyone respects x  x x is R’ed by everyonethere is no x : who is R’ed by everyonethere is no one no one is respected by everyone  x  y Ryx Ryx yy y respects xno matter who y is you respect xno matter who you are everyone respects x

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

What is the Difference? Situation 1JayKayElle does everyone respect someone? is there someone whom everyone respects? YES NO

What is the Difference? (cont.) Situation 2JayKayElle does everyone respects someone? is there someone whom everyone respects? YES NOTE : the latter implies the former

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

Example 9 everyone respects x xx there is some x : whom everyone respectsthere is some one everyone respects someone (the same)  x  y Ryx Ryx yy y respects xno matter who y is you respect xno matter who you are

Example 10 x respects no one xx there is some x : who respects no onethere is some one  x  y Rxy Rxy  y x respects ythere is no y : whom x respectsthere is no one

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

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