INTRO LOGIC
DAY 17
Translations in PL
3
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REVIEW of
DAY 1 and DAY 2
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Existential Quantifier
someone is happy
there is someone who is happy
there is some x : x is happy
x Hx
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Existential-Negative Quantifier
someone is unhappy
there is someone who is not happy
there is some x : x is not happy
x Hx
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Universal Quantifier
everyone is happy
no matter who you are you are happy
no matter who x is x is happy
x Hx
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Universal-Negative Quantifier
everyone is unhappy
no matter who you are you are not happy
no matter who x is x is not happy
x Hx
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Negative-Existential Quantifier
no one is happy
there is no one who is happy
there is no x : x is happy
x Hx
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Negative-Universal Quantifier
not everyone is happy
not: no matter who you are you are happy
not: no matter who x is x is happy
x Hx
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Quantifier-Specification – ‘some’
some Freshman is Happy
there is someone who is F and who is H
there is some x
x is F
x ( Fx
and
&
x is H
Hx )
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Quantifier-Specification – ‘no’
no Freshman is Happy
there is no one who is F and who is H
there is no x
x is F
x ( Fx
and
&
x is H
Hx )
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Quantifier-Specification – ‘every’
every Freshman is Happy
no matter who you are IF you are F THEN you are H
no matter who x is
IF x is F
x ( Fx
THEN

x is H
Hx )
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new material
for day 3
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Multiple Quantification
sentences with more than
one quantifier
GENERAL STRATEGY
(1) Count the number of quantifiers in original sentence.
(2) Determine the overall structure of the sentence.
(3) Work on constituents separately.
(4) Substitute constituents back into overall formula.
(5) Count the number of quantifiers in final formula.
(6) Compare (5) with (1).
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Example 1
everyone is FRIENDLY, but not everyone is HAPPY
everyone is F
but not
everyone is H
x Fx
& 
x Hx
x Fx
x Hx
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Example 2
every CAT is a PET, but not every PET is a CAT
every C is P
but not
every P is C
x ( Cx  Px ) &  x ( Px  Cx )
x ( Cx  Px )
x ( Px  Cx )
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Example 3
if everyone is FRIENDLY,
then everyone is HAPPY
if
everyone is F then everyone is H
xFx
xFx

xHx
xHx
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Example 4
if every STUDENT is FRIENDLY,
then every STUDENT is HAPPY
if
every S is F
then
every S is H
x(Sx  Fx)

x(Sx  Hx)
x(Sx  Fx)
x(Sx  Hx)
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‘Any’ versus ‘Every’
Basic Principle
both ‘any’ and ‘every’
are universal quantifiers,
BUT
they are usually
not inter-changeable.
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Some times they are
interchangeable
any one can Dance
xDx
every one can Dance
if I can Dance,
then any one can
if I can Dance,
then every one can
Di  xDx
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Usually, they are
not interchangeable

is any one here?
Jay doesn’t respect
every one

Jay doesn’t respect
any one
if every one can fix your
car, then I can

if any one can fix your
car, then I can
no one respects
every one

no one respects
any one
is every one here?
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Difference between ‘every’ and ‘any’
the scope of ‘every’ is narrow
the scope of ‘any’ is wide
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Example — Not-Every
Jay doesn’t respect everyone
not!
Jay respects everyone
xRjx
‘not’ () has wide scope
‘every’ () has narrow scope
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Example — Not-Any
Jay doesn’t respect anyone
does Jay respect a?
does Jay respect b?
does Jay respect c?
?Rja
?Rjb
?Rjc
no!
no!
no!
Rja
Rjb
Rjc
etc.
no matter who you are Jay does not respect you
no matter who x is Jay does not respect x
x  Rjx
‘any’ () has wide scope
‘not’ () has narrow scope
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Not-Any = None = Not-Some
Jay respects no one
there is no one whom Jay respects
there is no x : Jay respects x
x Rjx
Recall
  
=
  
 x Rjx
=
x  Rjx
Jay respects
Jay doesn’t
=
no one
respect anyone
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Example — IF-EVERY
if everyone fails, then satan wins
if
everyone fails then
xFx
xFx

satan wins
Ws
Ws
‘every’ has narrow scope
‘if…then’ has wide scope
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How do we SHOW such a formula?
(1) : xFx  Ws CD
(2)
xFx
(3)
: Ws
As
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Example — IF-ANY
if anyone fails, then satan wins
if
a fails
then
satan wins
if
b fails
then
satan wins
if
c fails
then
satan wins
etc.
if anyone fails then
satan wins
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In Other Words
no matter who you are if
no matter who x is
x
if
you fail
then satan wins
x fails
then satan wins
( Fx

Ws
)
‘any’ has wide scope
‘if…then’ has narrow scope
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How do we SHOW such a formula?
(1) : x ( Fx  Ws )
(2)
: Fa  Ws
(3)
Fa
(4)
: Ws
UD
CD
As
UD = Universal Derivation
(later!)
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Special Note
Sometimes (but not always)
‘if-any’ = ‘if-some’
x ( Fx   )
=
xFx  
provided  has no free
occurrence of ‘x’
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THE END
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