Today’s Topics Relational predicates Multiple quantification Expansions of relational predicates.
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Transcript of Today’s Topics Relational predicates Multiple quantification Expansions of relational predicates.
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Today’s Topics•Relational predicates•Multiple quantification•Expansions of relational predicates
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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
))((
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No, we are not talking about CAT-DOGS!
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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 )
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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
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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
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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
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Example 2
Rxkx
x respects Kay no matter who x is
you respect Kay no matter who you are
everyone respects Kay
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Example 3
Rxjx
x respects Jay there is no x :
who respects Jay there is no one
no-one respects Jay
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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
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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
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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
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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
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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
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What is the Difference?
Situation 1
Jay Jay
Kay Kay
Elle Elle
does everyone respect someone?
is there someone whom everyone respects?
YES
NO
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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
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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
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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
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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
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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
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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