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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