1. 2 3 hx xx x is happythere is some x : who is happythere is someone is happysomeone
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INTRO LOGICINTRO LOGICDAY 17DAY 17
Translations in PLTranslations in PL33
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REVIEW of REVIEW of DAY 1 and DAY 2DAY 1 and DAY 2
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Existential QuantifierExistential Quantifier
Hxxx is happythere is some x :
who is happythere is someone
is happysomeone
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Existential-Negative QuantifierExistential-Negative Quantifier
Hxxx is not happythere is some x :
who is not happythere is someone
is unhappysomeone
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Universal QuantifierUniversal Quantifier
Hxx
x is happyno matter who x is
you are happyno matter who you are
is happyeveryone
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Universal-Negative QuantifierUniversal-Negative Quantifier
Hxx
x is not happyno matter who x is
you are not happyno matter who you are
is unhappyeveryone
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Negative-Existential QuantifierNegative-Existential Quantifier
Hxxx is happythere is no x :
who is happythere is no one
is happyno one
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Negative-Universal QuantifierNegative-Universal Quantifier
Hxx
x is happynot: no matter who x is
you are happynot: no matter who you are
is happynot everyone
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Quantifier-Specification – ‘some’Quantifier-Specification – ‘some’
some Freshman is Happy
Hx&Fxxx is Handx is Fthere is some x
there is someone who is Handwho is F
( )
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Quantifier-Specification – ‘no’Quantifier-Specification – ‘no’
no Freshman is Happy
Hx&Fxxx is Handx is Fthere is no x
there is no one who is Handwho is F
( )
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Quantifier-Specification – ‘every’Quantifier-Specification – ‘every’
every Freshman is Happy
HxFxxIF x is F THEN x is Hno matter who x is
no matter who you are IF you are F THEN you are H
)(
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new material new material for day 3for day 3
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Multiple QuantificationMultiple Quantificationsentences with more than sentences with more than
one quantifierone 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 1Example 1
everyone is FRIENDLY, but not everyone is HAPPY
x Hxx Fx
&
everyone is Hnotbut everyone is F
x Hxx Fx
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Example 2Example 2
every CAT is a PET, but not every PET is a CAT
x ( Px Cx )x ( Cx Px )
&
every P is Cnotbut every C is P
x ( Px Cx )x ( Cx Px )
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Example 3Example 3
if everyone is FRIENDLY, then everyone is HAPPY
xHxxFx
everyone is Htheneveryone is Fif
xHxxFx
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Example 4Example 4
if every STUDENT is FRIENDLY,then every STUDENT is HAPPY
x(Sx Hx)x(Sx Fx)
every S is Hthenevery S is Fif
x(Sx Hx)x(Sx Fx)
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‘‘Any’ versus ‘Every’ Any’ versus ‘Every’
Basic Principle
both ‘any’ and ‘every’ are universal quantifiers,
BUT
they are usually
not inter-changeable.
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Some Some times they times they areare interchangeableinterchangeable
every one can DancexDx
any one can Dance
if I can Dance, then every one can
Di xDx
if I can Dance, then any one can
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no one respects every one
if every one can fix your car, then I can
Jay doesn’t respect every one
is every one here?
UsuallyUsually, they , they areare notnot interchangeable interchangeable
no one respects any one
if any one can fix your car, then I can
Jay doesn’t respectany one
is any one here?
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Difference between ‘every’ and ‘any’ Difference between ‘every’ and ‘any’
the scope of ‘every’ is narrow
the scope of ‘any’ is wide
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Example — Not-Every Example — Not-Every
xRjx
not! Jay respects everyone
Jay doesn’t respect everyone
‘not’ () has wide scope
‘every’ () has narrow scope
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Example — Not-Any Example — Not-Any
‘any’ () has wide scope‘not’ () has narrow scope
RjxxJay does not respect xno matter who x is
Jay does not respect youno matter who you are
etc.
Rjcno!does Jay respect c?Rjbno!does Jay respect b?Rjano!does Jay respect a?
Jay doesn’t respect anyone
?Rjc?Rjb?Rja
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Not-Any Not-Any = = None None = = Not-Some Not-Some
RjxxJay respects xthere is no x :
whom Jay respectsthere is no one
Jay respects no one
Recall
Jay doesn’t respect anyone
=Jay respects
no one
x Rjx= x Rjx
=
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satan winseveryone fails
Example — IF-EVERY Example — IF-EVERY
‘every’ has narrow scope
‘if…then’ has wide scope
if everyone fails, then satan wins
WsxFx
thenif
WsxFx
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How do we SHOW such a formula?How do we SHOW such a formula?
: Ws(3)
As xFx(2)
CD: xFx Ws(1)
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Example — IF-ANY Example — IF-ANY
etc.
satan winsthenc failsif
satan winsthenb failsif
satan winsthena failsif
if anyone fails, then satan wins
satan winsthenanyone failsif
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In Other WordsIn Other Words
‘any’ has wide scope
‘if…then’ has narrow scope
satan winsthenyou failifno matter who you are
satan winsthenx failsifno matter who x is
WsFxx )(
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How do we SHOW such a formula? How do we SHOW such a formula?
UD = Universal Derivation (later!)
: Ws(4)
As Fa(3)
CD : Fa Ws(2)
UD: x ( Fx Ws )(1)
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Special NoteSpecial Note
Sometimes (but not always)
‘if-any’ = ‘if-some’
x ( Fx )
=
xFx
provided has no free occurrence of ‘x’
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THE ENDTHE END