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INTRO LOGIC
DAY 16
Translations in PL
2
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Overview
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Exam 1:
Exam 2:
Exam 3:
Exam 4:
Exam 5:
Exam 6:
Sentential Logic
Sentential Logic
Predicate Logic
Predicate Logic
(finals)
(finals)
Translations (+)
Derivations
Translations
Derivations
very similar to Exam 3
very similar to Exam 4
When computing your final grade,
I count your four highest scores.
(A missed exam counts as a zero.)
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REVIEW of DAY 1
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Existential Quantifier
original sentence
paraphrase
someone is happy
there is someone who is happy
pronoun  variable there is some x (s.t.) x is happy
formula
x Hx
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Universal Quantifier
original sentence
paraphrase
pronoun  variable
formula
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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Negative-Existential Quantifier
original sentence
paraphrase
pronoun  variable
formula
no one is happy
there is no one who is happy
there is no x (s.t.) x is happy
x Hx
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Negative-Universal Quantifier
original sentence
paraphrase
pronoun  variable
formula
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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Equivalences
xHx
not-everyone
is happy
someone
is un-happy
xHx
xHx
no-one
is happy
everyone
is un-happy
xHx

=


=

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new material for day 2
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Quantifier Specification
Generic Quantifier versus Specific Quantifier
every one is H
every F is H
not-every one is H
not-every F is H
every one is un-H
every F is un-H
some one is H
some F is H
no one is H
no F is H
some one is un-H
some F is un-H
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Example 1
original sentence some Freshman is Happy
paraphrase
there is someone who …
there is someone who is F and who is H
there is some x
x is F
x ( Fx
and
&
x is H
Hx )
DON’T FORGET PARENTHESES
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Example 2
original sentence
no Freshman is Happy
paraphrase
there is no one who …
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 )
DON’T FORGET PARENTHESES
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Example 3
original sentence
every Freshman is Happy
paraphrase
no matter who you are … you
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 )
DON’T FORGET PARENTHESES
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Arrow versus Ampersand
Rule of Thumb (not absolute)
the connective immediately “beneath”
a universal quantifier ()
is usually a conditional ()
(  )
the connective immediately “beneath”
an existential quantifier ()
is usually a conjunction (&)
( & )
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Summary of Quantifier Specification
everyone is H
xHx
every F is H
x(Fx  Hx)
not-everyone is H xHx
not-every F is H x(Fx  Hx)
everyone is un-H xHx
every F is un-H x(Fx  Hx)
someone is H
xHx
some F is H
x(Fx & Hx)
no-one is H
xHx
no F is H
x(Fx & Hx)
someone is un-H xHx
some F is un-H x(Fx & Hx)
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Conjunctive Predicate-Combinations
x is an American Biker
=
x is an American, and x is a Biker
x is an AB
=
[ Ax & Bx ]
every AMERICAN BIKER is CLEVER
every AB is C
x ( [ Ax & Bx ]  Cx )
some AMERICAN BIKER is CLEVER
some AB is C
x ( [ Ax & Bx ] & Cx )
no AMERICAN BIKER is CLEVER
no AB is C
x ( [ Ax & Bx ] & Cx )
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Non-Conjunctive Predicates
Combinations
alleged criminal
imitation
expectant
experienced
small
large
deer
racecar
leather
mother
sailor
whale
shrimp
hunter
driver
woman racecar driver
baby whale killer
dandruff shampoo
productivity software
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Ambiguous Examples
Bostonian
Bostonian
cab driver
attorney
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A Pitfall
Compare the following:
every Bostonian Attorney is Clever
every BA is C
vs.
every Bostonian and Attorney is Clever
every B and A is C
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Zombie Logic
every
x
{(
for any
IF
thing
Cat
and
Dog
Cx
&
Dx
it is
a Cat
and
it is
a Dog
)
is
a Pet

Px
THEN
}
it is
a Pet
in other words
every CAT-DOG is a PET
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WHAT EXACTLY IS A CAT-DOG?
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Another Candidate
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“Distributive” Use of ‘And’
every Cat and Dog is a Pet
every Cat and every Dog is a Pet
every Cat is a Pet, and every Dog is a Pet
x ( Cx  Px )
&
x ( Dx  Px )
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“Plural” Use of ‘And’
every member of the class Cats-and-Dogs is a Pet
no matter who x is
if x is a member of the class Cats-and-Dogs,
then x is a 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 )
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‘Only’ as a Quantifier
only  are 
examples
only Citizens are Voters
only Men play NFL football
employees only
members only
cars only
right turn only
only Employees are Allowed
only Members are Allowed
only Cars are Allowed
only Right turns are Allowed
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Recall ‘only if’
 only IF 
not 
IF
not 
IF not  THEN not 
  
‘only’ is an implicit double-negative modifier
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One Rendering of ‘only’
only  are 
only if you are , are you 
(no matter who you are)
you are  only if you are 
(no matter who you are)
x is  only if x is 
(no matter who x is)
x is not  if x is not 
(no matter who x is)
if x is not , then x is not 
(no matter who x is)
x ( x  x )
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Alternative Rendering of ‘only’
only = no non
only  are 
no non- are 
no one who is not  is 
there is no one who is not  but who is 
there is no x ( x is not  but x is  )
x (x & x )
=
x ( x  x )
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THE END
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