Today’s Topics
•Relational predicates
•Multiple quantification
•Expansions of relational predicates
BEWARE Zombie Logic!
all
x
for any thing
Cats
((
IF
Cx
it is
a Cat
and

and
Dogs
Dx
it is
a Dog
make good Pets
)

THEN
in other words
every CAT-DOG is a PET
Px
)
It makes a good
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
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 )
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
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
Example 2
everyone respects Kay
no matter who you are you respect Kay
no matter who x is x respects Kay
x Rxk
Example 3
no-one respects Jay
there is no one who respects Jay
there is no x : x respects Jay
x Rxj
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
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
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
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
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
What is the Difference?
Situation 1
Jay
Jay
Kay
Kay
Elle
Elle
does everyone respect someone?
is there someone whom everyone respects?
YES
NO
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
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
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
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
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
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