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INTRO LOGICINTRO LOGICDAY 19DAY 19

Translations in PLTranslations in PL55

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Review – 1 Quantifier, 1 PredicateReview – 1 Quantifier, 1 Predicate

x Rxkno-one respects Kay

x Rxkeveryone respects Kay

x RjxJay respects someone

x RjxJay respects everyone

x RjxJay respects no one

x Rxksome one respects Kay

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Review – 2 Quantifiers, 1 PredicateReview – 2 Quantifiers, 1 Predicate

x y Ryxthere is someone whom no-one respects

x y Rxythere is someone who respects no-one

x y Ryxthere is someone whom everyone R’s

x y Rxyeveryone respects someone (or other)

x y Ryxno-one is respected by everyone

x y Rxyno-one respects everyone

x y Rxysomeone respects someone

x y Rxyeveryone respects everyone

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1 quantifier, 2 predicates1 quantifier, 2 predicates2 quantifiers, 2 predicates2 quantifiers, 2 predicates2 quantifiers, 3 predicates 2 quantifiers, 3 predicates

New Material New Material

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Example 1 Example 1

( Sx Rxk )x

IF x is a student, THEN x respects Kay

no matter who x is

IF you are a student, THEN you respect Kay

no matter who you are

every student respects Kay

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Example 2 Example 2

( Sx & Rxj )x

x is a S and x R's jthere is no x :

who is a S and who R's j there is no-one

no student respects Jay

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Example 3 Example 3

(Sx & Rjx)x

x is a S and j R's xthere is someone x

who is a S and whom j R's

who is a S (whom) j respects there is someone

Jay respects some (at least one) student

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Example 4 Example 4

x respects every politicianx

x respects every politicianthere is no x :

who respects every politicianthere is no one

no-one respects every politician

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Example 4b Example 4b

IF y is P, THEN x R’s yno matter who y isx

x y ( Py Rxy )

y ( Py Rxy )

Py Rxyy

IF you are P, THEN x R’s youno matter who you are

x respects every politician

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Example 5 Example 5

x ( Sx x R's someone )

Sx x R's someonex

IF x is S, THEN x R's someoneno matter who x is

IF you are S, THEN you R someoneno matter who you are

every Student respects someone or other

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Example 5b Example 5b

x ( Sx y Rxy )

Rxyy

x R's ythere is some y

whom x R's

who is R'ed by xthere is someone

)

x R's someone

x ( Sx

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Example 6 Example 6

Px & no-one R's xx

x is P AND no-one R's xthere is some x :

who is P AND whom no-one R's

who is a P whom no-one R‘sthere is someone

there is a politician whom no-one respects

x ( Px & no-one R's x )

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Example 6b Example 6b

x ( Px & y Ryx )

Ryxy

y R's xthere is no y

who R’s xthere is no one)

no one R's x

x ( Px &

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Example 7 Example 7

Px & every C R's xx

x ( Px & every C R's x )

x is P AND every C R's xthere is some x

who is P AND whom every C R's

who is a P whom every C R'sthere is someone

there is a Politician whom every Citizen Respects

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Example 7b Example 7b

x ( Px & y ( Cy Ryx ) )

y ( Cy Ryx )

Cy Ryxy

if y is C then y R’s xno matter who y is

if you are C then you R xno matter who you are

)

every C R's x

x ( Px &

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Example 8 Example 8

Cx x R's some Px

x ( Cx x R's some P )

if x is C, then x R's some Pno matter who x is

if you are C, then you R some Pno matter who you are

every Citizen Respects some Politician (or other)

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Example 8b Example 8b

x ( Cx y ( Py & Rxy ) )

y ( Py & Rxy )

Py & Rxyy

y is P AND x R's ythere is some y

who is P AND whom x R's

who is a P AND who is R'ed by x

who is a P R'ed by x

there is someone

)

x R's some P

x ( Cx

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Example 9 Example 9

Cx & x R's no Px

x ( Cx & x R's no P )

x is a C AND x R's no Pthere is some x :

who is a C AND who R's no P

who is a C who R's no Pthere is someone

there is a Citizen who Respects no Politician

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Example 9b Example 9b

x ( Cx & y ( Py & Rxy ) )

y ( Py & Rxy )

Py & Rxyy

y is P AND x R's y

y is P AND y is R’ed by xthere is no y :

who is a P AND who is R'ed by x

there is no one

)

x R's no P

x ( Cx &

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THE ENDTHE END