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INTRO LOGICINTRO LOGICDAY 18DAY 18

Translations in PLTranslations in PL44

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Chapter 7 Chapter 7 Polyadic Polyadic

Predicate Logic Predicate Logic

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Examples of Polyadic PredicationExamples of Polyadic Predication

every Grespectsevery F

some Grespectsevery F

every Grespectssome F

some Grespectssome F

Kayrespectsevery F

Kayrespectssome F

every FrespectsJay

some FrespectsJay

KayrespectsJay

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Example 1 Example 1

Jay respects x

Rjxx

x is respected by Jay there is some x :

who is respected by Jay there is someone

Jay respects someone

R is always read in “active voice”

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Example 1 - redone Example 1 - redone

Rjxx

Jay respects xthere is some x :

Jay respects him (her)there is someone :

whom Jay respects there is someone

Jay respects someone

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Example 2 Example 2

Rxkx

x respects Kay no matter who x is

you respect Kay no matter who you are

everyone respects Kay

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Example 3 Example 3

Rxjx

x respects Jay there is no x :

who respects Jay there is no one

no-one respects Jay

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Example 4 Example 4

x respects everyonex

x respects everyoneno matter who x is

you respect everyone no matter who you are

everyone respects everyone

x y Rxy

Rxyy

x respects yno matter who y is

x respects you no matter who you are

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Example 5 Example 5

x respects someonex

x respects someonethere is some x :

who respects someone there is someone

someone respects someone

x y Rxy

Rxyy

x respects ythere is some y :

whom x respects there is someone

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Example 6 Example 6

x respects everyonex

x respects everyonethere is no x :

who respects everyone there is no one

no one respects everyone

x y Rxy

Rxyy

x respects yno matter who y is

x respects youno matter who you are

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Example 7 Example 7

everyone respects xx

x is R’ed by everyonethere is no x :

who is R’ed by everyone there is no one

no one is respected by everyone

x y Ryx

Ryxy

y respects xno matter who y is

you respect xno matter who you are

everyone respects x

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A Scope AmbiguityA Scope Ambiguity

everyone respects someone

everyone respects someone or other

vs.

everyone respects someone (the same one)

=

there is someone whom everyone respects

is ambiguous

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What is the Difference?What is the Difference?

Situation 1

Jay Jay

Kay Kay

Elle Elle

does everyone respect someone?

is there someone whom everyone respects?

YES

NO

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What is the Difference? (cont.)What is the Difference? (cont.)

Situation 2

Jay Jay

Kay Kay

Elle Elle

does everyone respects someone?

is there someone whom everyone respects?

YES

YES

NOTE: the latter implies the former

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Example 8 Example 8

x respects someonex

x respects someoneno matter who x is

you respect someone no matter who you are

everyone respects someone (or other)

x y Rxy

Rxyy

x respects ythere is some y :

whom x respects there is someone

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Example 9 Example 9

everyone respects xx

everyone respects xthere is some x :

whom everyone respectsthere is some one

everyone respects someone (the same)

x y Ryx

Ryxy

y respects xno matter who y is

you respect xno matter who you are

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Example 10 Example 10

x respects no onex

x respects no onethere is some x :

who respects no onethere is some one

x y Rxy

Rxyy

x respects ythere is no y :

whom x respectsthere is no one

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Example 11 Example 11

no one respects xx

no one respects xthere is some x :

whom no one respectsthere is some one

x y Ryx

Ryxy

y respects xthere is no y :

who respects xthere is no one

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Example 12 Example 12

Jay does not respect xx

Jay does not respect xno matter who x is

Jay does not respect you no matter who you are

Jay doesn’t respect anyone

x Rjx

Rjx

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THE ENDTHE END