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

BEWARE Zombie Logic!good PetsmakeDogsandCatsall

It makes a good Pet

THENit is

a Dogandit is a Cat

IFfor any thing

PxDxCxx

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-DogsIS

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

Rjxx

x is respected by Jay there is some x :

who is respected by Jay there is someone

Jay respects someone

Example 2

Rxkx

x respects Kay no matter who x is

you respect Kay no matter who you are

everyone respects Kay

Example 3

Rxjx

x respects Jay there is no x :

who respects Jay there is no one

no-one respects Jay

Example 4

x respects everyonex

x respects everyoneno matter who x is

you respect everyone no matter who you are

everyone respects everyone

x y Rxy

Rxyy

x respects yno matter who y is

x respects you no matter who you are

Example 5

x respects someonex

x respects someonethere is some x :

who respects someone there is someone

someone respects someone

x y Rxy

Rxyy

x respects ythere is some y :

whom x respects there is someone

Example 6

x respects everyonex

x respects everyonethere is no x :

who respects everyone there is no one

no one respects everyone

x y Rxy

Rxyy

x respects yno matter who y is

x respects youno matter who you are

Example 7

everyone respects xx

x is R’ed by everyonethere is no x :

who is R’ed by everyone there is no one

no one is respected by everyone

x y Ryx

Ryxy

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 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?

is there someone whom everyone respects?

YES

YES

NOTE: the latter implies the former

Example 8

x respects someonex

x respects someoneno matter who x is

you respect someone no matter who you are

everyone respects someone (or other)

x y Rxy

Rxyy

x respects ythere is some y :

whom x respects there is someone

Example 9

everyone respects xx

everyone respects xthere is some x :

whom everyone respectsthere is some one

everyone respects someone (the same)

x y Ryx

Ryxy

y respects xno matter who y is

you respect xno matter who you are

Example 10

x respects no onex

x respects no onethere is some x :

who respects no onethere is some one

x y Rxy

Rxyy

x respects ythere is no y :

whom x respectsthere is no one

Example 11

no one respects xx

no one respects xthere is some x :

whom no one respectsthere is some one

x y Ryx

Ryxy

y respects xthere is no y :

who respects xthere is no one

Example 12

Jay does not respect xx

Jay does not respect xno matter who x is

Jay does not respect you no matter who you are

Jay doesn’t respect anyone

x Rjx

Rjx