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