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INTRO LOGICINTRO LOGICDAY 16DAY 16

Translations in PLTranslations in PL22

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OverviewOverviewExam 1: Sentential Logic Translations (+)

Exam 2: Sentential Logic Derivations

Exam 3: Predicate Logic Translations

Exam 4: Predicate Logic Derivations

Exam 5: (finals) very similar to Exam 3

Exam 6: (finals) very similar to Exam 4

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When computing your final grade,I count your four highest scoresfour highest scores.

(A missedmissed exam counts as a zerozero.)

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REVIEW of DAY 1REVIEW of DAY 1

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Existential QuantifierExistential Quantifier

Hxxformula

x is happythere is some x (s.t.)pronoun variable

who is happythere is someoneparaphrase

is happysomeoneoriginal sentence

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Universal QuantifierUniversal Quantifier

Hxxformula

x is happyno matter who x ispronoun variable

you are happyno matter who you areparaphrase

is happyeveryoneoriginal sentence

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Negative-Existential QuantifierNegative-Existential Quantifier

Hxxformula

x is happythere is no x (s.t.)pronoun variable

who is happythere is no oneparaphrase

is happyno oneoriginal sentence

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Negative-Universal QuantifierNegative-Universal Quantifier

Hxxformula

x is happynot: no matter who x ispronoun variable

you are happynot: no matter who you areparaphrase

is happynot everyoneoriginal sentence

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EquivalencesEquivalences

xHx no-one is happy

xHxsomeone is un-happy

everyoneis un-happy xHx

not-everyoneis happyxHx

=

=

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new material for day 2new material for day 2

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Quantifier SpecificationQuantifier Specification

some F is un-Hsome one is un-H

no F is Hno one is H

some F is Hsome one is H

every F is un-Hevery one is un-H

not-every F is Hnot-every one is H

every F is H

versus

every one is H

Specific QuantifierGeneric Quantifier

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Example 1Example 1

there is someone who …paraphrase

some Freshman is Happyoriginal sentence

Hx&Fxx

x is Handx is Fthere is some x

there is someone who is Handwho is F

( )

DON’T FORGET PARENTHESES

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Example 2Example 2

there is no one who …paraphrase

no Freshman is Happyoriginal sentence

Hx&Fxx

x is Handx is Fthere is no x

there is no one who is Handwho is F

( )

DON’T FORGET PARENTHESES

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Example 3Example 3

no matter who you are … you paraphrase

every Freshman is Happyoriginal sentence

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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DON’T FORGET PARENTHESES

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Arrow Arrow versusversus Ampersand 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 SpecificationSummary of Quantifier Specification

x(Fx & Hx)some F is un-HxHxsomeone is un-H

x(Fx & Hx)no F is HxHxno-one is H

x(Fx & Hx)some F is HxHxsomeone is H

x(Fx Hx)every F is un-HxHxeveryone is un-H

x(Fx Hx)not-every F is HxHxnot-everyone is H

x(Fx Hx)every F is HxHxeveryone is H

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Conjunctive Predicate-CombinationsConjunctive Predicate-Combinations

x ( [ Ax & Bx ] Cx )every AB is C

every AMERICAN BIKER is CLEVER

x is an American Biker=

x is an American, and x is a Biker

x is an AB=

[ Ax & Bx ]

x ( [ Ax & Bx ] & Cx )some AB is C

some AMERICAN BIKER is CLEVER

x ( [ Ax & Bx ] & Cx )no AB is C

no AMERICAN BIKER is CLEVER

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Non-Conjunctive Predicates Non-Conjunctive Predicates CombinationsCombinationsalleged criminal

imitation leatherexpectant mother

experienced sailorsmall whalelarge shrimpdeer hunter

racecar driver

baby whale killer

dandruff shampoo

productivity software

woman racecar driver

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Ambiguous ExamplesAmbiguous Examples

Bostonian cab driverBostonian attorney

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A Pitfall 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 LogicZombie Logica PetisDogandCatevery

it is a Pet

THENit is

a Dogandit is

a CatIF

for any thing

PxDx&Cxx

in other words

every CAT-DOG is a PET

})({

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WHAT EXACTLY IS A CAT-DOG?WHAT EXACTLY IS A CAT-DOG?

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Another CandidateAnother Candidate

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““Distributive” Use of ‘And’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’Plural” Use of ‘And’every member of the class Cats-and-Dogs is a 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 is a Pet

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‘‘Only’ as a QuantifierOnly’ as a Quantifier

onlyonly are

onlyonly Citizens are Voters

onlyonly Men play NFL football

employees onlyonly

members onlyonly

cars onlyonly

right turn onlyonly

onlyonly Employees are Allowed

onlyonly Members are Allowed

onlyonly Cars are Allowed

onlyonly Right turns are Allowed

examples

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Recall ‘only if’Recall ‘only if’

‘onlyonly’ is an implicit double-negative modifier

onlyonly IFIF notnot IFIF notnot

IFIF notnot THENTHEN notnot

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One Rendering of ‘only’ 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’ 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 ENDTHE END