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Single Quantifier, Multiple Predicates xx there is some x there is some one some Student Respects Jay AND Rxj Sx )( who is a student who respects Jay AND x is a student x respects Jay x ( Sx Rxj )

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Single Quantifier, Multiple Predicates xx no matter who x is IF no matter who you are every Student Respects Kay THEN Rxk Sx ) ( you are a student you respect Kay IF THEN x is a student x respects Kay x ( Sx Rxk )

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Single Quantifier, Multiple Predicates xx there is no x there is no one Jay Respects no Student AND Rjx Sx ) ( who is a student whom Jay respects AND x is a student Jay respects x x ( Sx Rjx )

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Multiple Quantifiers, Multiple Predicates xx no matter who x is IF no matter who you are every Student Respects someone or other THEN x respects someone Sx ) ( you are a student you respect someone IF THEN x is a student x respects someone x ( Sx x respects someone )

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Multiple Quantifiers, Multiple Predicates x ( Sx y Rxy ) Rxy yy x R's ythere is some y whom x R'sthere is someone ) x R's someone x ( Sx

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Multiple Quantifiers, Multiple Predicates xx x respects every politician there is no x : who respects every politicianthere is no one no-one Respects every Politician x ( x respects every politician )

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Multiple Quantifiers, Multiple Predicates xx x y ( Py Rxy ) IF no matter who you are x Respects every Politician THEN you are P x R’s you IF no matter who y is THEN y is P x R’s y yy Rxy Py ) (

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xx there is some x there is some one there is a Politician whom no-one Respects AND no-one R's x Px ) ( who is a politician whom no-one respects AND x is a politician no-one respects x x ( Px no-one R's x )

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x ( Px y Ryx ) Ryx yy 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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xx there is some x there is some one there is a Politician who Respects every Citizen AND x respects every C Px ) ( who is a politician who respects every C AND x is a politician x respects every C x ( Px x respects every C )

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x ( Px y ( Cy Rxy ) ) yy no matter who you are ) x R's every C x ( Px IF THEN you are C x R’s you IF THEN y is C x R’s y no matter who y is Rxy Cy ) (

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xx no matter who x is IF no matter who you are every Citizen Respects some Politician (or other) THEN x respects some P Cx ) ( you are a citizen you respect some P IF THEN x is a citizen x respects some P x ( Cx x respects some P )

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x ( Cx y ( Py Rxy ) ) yy there is some y there is someone ) x R's some Politician x ( Cx AND who is a P whom x R's AND y is a P x R's y Rxy Py )(

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xx there is some x there is some one there is a Politician whom every Citizen Respects AND every C respects x Px )( who is a politician whom every C respects AND x is a politician every C respects x x ( Px every C respects x )

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x ( Px y ( Cy Ryx ) ) yy no matter who you are ) every C R's x x ( Px IF THEN you are C you R x IF THEN y is C y R’s x no matter who y is Ryx Cy ) (

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xx there is some x there is some one there is a Citizen who Respects no Politician AND x respects no P Cx ) ( who is a citizen who respects no P AND x is a citizen x respects no P x ( Cx x respects no P )

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yy there is no-one ) x R's no P x ( Cx AND who is P whom x R's AND y is P x R’s y there is no y Rxy & Py ) ( x ( Cx y ( Py Rxy ) )

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Chapter 1: The Foundations: Logic and Proofs

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