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2"((*2/*#)$%,#*%("#$%(=&'"($%7&$%*(2")*%/)&1/-$%*/%)-7-/"($%4&%7)"2-0#)*< \*7)-$*(/-*%$28-+&/1,#*K +*(/. 4&%7)"2-0#)*%*$/%0"(2%4&%$#1'&(/* B
J< ]$/K2*%,#=14%S%&%0*$%2"((*2/*#)$%)*41&(/%0*$%78)&$*$%2"+74I/*$ c g7741,#"($%4*$%)I94*$%7"#)%4*$
2"((*2/*#)$<
b< ]$/K2*%,#=14%S%&%0*$%>")+#4*$%,#1%2"++*(2*(/%7&)%L∃M%"#%7&)%L¬∀M c g7741,#"($%4*$%)I94*$%2")K )*$7"(0&(/*$%7"#)%"#')1)%0*%("#'*44*$%:)&(28*$%7"#)%/"#/*$%4*$%2"($/&(/*$%*/%#(*%("#'*44*<
h< ]$/K2*%,#=14%S%&%0*$%>")+#4*$%,#1%2"++*(2*(/%7&)%L∀M%"#%7&)%L¬∃M c g7741,#"($%4*$%)I94*$%2")K )*$7"(0&(/*$%7"#)%>&1)*%0*$%1($/&(21&/1"($%7"#)%/"#/*$%4*$%2"($/&(/*$<
i< ]$/K2*%,#=14%S%&%0*$%2"((*2/*#)$%)*41&(/%0*$%78)&$*$%2"+74I/*$ c \*/"#)("($%A%4=-/&7*%J6<
Z< ]$/K2*%,#=14%S%&%0*$%>")+#4*$%,#1%2"++*(2*(/%7&)%L∃M%"#%7&)%L¬∀M c \*/"#)("($%A%4=-/&7*%b6<
j< ]$/K2*%,#=14%S%&%0*$%>")+#4*$%,#1%2"++*(2*(/%7&)%L∀M%"#%7&)%L¬∃M c \*/"#)("($%A%4=-/&7*%h6<
H=&):)*%(=*$/%*(/1I)*+*(/%0-'*4"77-%,#1%$1%4*$%/)"1$%2"(01/1"($%$#1'&(/*$%$"(/%/"#/*$%)*+741*$ B J< #(*%)I94*%7"#)%0*$%2"((*2/*#)$%(=*$/%&77412&:4* U
b< ("#$%&'"($%2)-*$%0*%("#'*44*$%:)&(28*$%7"#)%/"#/*$%4*$%>")+#4*$%2"++*(W&(/%7&)%L∃M%"#%7&) L¬∀M%7"#)%/"#/*$%4*$%2"($/&(/*$%$#)%4*$%:)&(28*$%2"))*$7"(0&(/*$ U
h< ("#$%&'"($%&;"#/-%/"#/*$%4*$%1($/&(21&/1"($%0*$%>")+#4*$%2"++*(W&(/%7&)%L∀M%"#%7&)%L¬∃M%7"#) /"#/*$%4*$%2"($/&(/*$%$#)%4*$%:)&(28*$%2"))*$7"(0&(/*$<
@(%&):)*%*(/1I)*+*(/%0-'*4"77-%$*%>*)+*%$1%("#$%/)"#'"($%0*$%>")+#4*$%*/%4*#)$%(-9&/1"($%$#)%/"#/*$
4*$%:)&(28*$ U $1%2*4&%(=*$/%7&$%4*%2&$. 14%*$/%"#'*)/<
**2< +(0)$'(4F(0)$#')B#),-&"./()/(0)#'1'(0
X"($10-)"($. 7&)%*?*+74*. 4&%>")+#4*%L∃x∀y(Rxy)∧ ∃x∀y¬(Rxy)M< !"#$%2"++*(W"($%A%2"($/)#1)*
$"(%&):)*%*(%&7741,#&(/%4&%)I94*%7"#)%4&%2"(;"(2/1"( B
# ∃x∀y(Rxy)∧ ∃x∀y¬(Rxy)
∃x∀y(Rxy)
∃x∀y¬(Rxy)
g 2*%$/&0*. ("#$%&7741,#"($%4&%)I94*%7"#)%4*%,#&(/132&/*#)%*?1$/*(/1*4%A%4&%7)*+1I)*%0*%("$%0*#?%>")+#4*$
*?1$/*(/1*44*+*(/%,#&(/13-*$. 2=*$/KAK01)*%A%L∃x∀y(Rxy)M< X"++*%("/)*%$*#4*%:)&(28*%(*%2"(/1*(/%&#K 2#(*%2"($/&(/*%1(01'10#*44*. ("#$%(=&'"($%,#=A%1(/)"0#1)*%#(*%("#'*44*%2"($/&(/*. ,#*%("#$%&:)-9*"($
7&)%LaM B
>*= **2 +#),-&"./()/(0)#'1'(0
# ∃x∀y(Rxy)∧ ∃x∀y¬(Rxy)
a#∃x∀y(Rxy)
∃x∀y¬(Rxy)
∀y(Ray)
C"#)%("#$%$"#'*(1)%,#*%("#$%&'"($%1($/&(21-%4&%,#&(/132&/1"(%*?1$/*(/1*44*%&'*2%4&%2"($/&(/*%LaM. ("#/
4=-2)1'"($%&#%0*$$#$%0#%2)"28*/%,#1%+&),#*%4&%>")+#4*%2"++*%/)&1/-*<
Y=&7)I$%("$%)I94*$%0*%7)"2-0#)*. ("#$%0*'"($%/)&1/*)%4*%0*#?1I+*%,#&(/132&/*#)%*?1$/*(/1*4%L∃x∀y¬(Rxy)M
&'&(/%0=&7741,#*)%4&%)I94*%7"#)%4*%,#&(/132&/*#)%#(1'*)$*4<!"#$%1(/)"0#1$"($%&4")$%#(*%("#'*44*%2"($/&(/*.
LbM%*/%1($/&(21"($%4&%0*#?1I+*%,#&(/132&/1"(%*?1$/*(/1*44*%-9&4*+*(/%&'*2%("/)*%&(21*((*%2"($/&(/*
LaM. ":/*(&(/%0*#?%:)&(28*$ B
# ∃x∀y(Rxy)∧ ∃x∀y¬(Rxy)
a#∃x∀y(Rxy)
a,b#∃x∀y¬(Rxy)
∀y(Ray)
%%%
&
&
&
∀y¬(Ray) ∀y¬(Rby)
g7)I$%&'"1)%/)&1/-%4*$%,#&(/132&/1"($%*?1$/*(/1*44*$. ("#$%7"#'"($%+&1(/*(&(/%1($/&(21*)%4*$%,#&(/13K 2&/1"($%#(1'*)$*44*$< X*21%("#$%":419*%0*%2"($10-)*)%/"#/*$%4*$%2"($/&(/*$%,#1%&77&)&1$$*(/%$#)%4*$%0*#?
:)&(28*$%)*$7*2/1'*$<
V#)%4&%:)&(28*%9*. ("#$%(=&'"($%,#=#(*%$*#4*%2"($/&(/*. LaM B 4*$%0*#?%1($/&(21&/1"($%0*$%,#&(/13K 2&/1"($%#(1'*)$*44*$%L∀y(Ray)M%*/%L∀y¬(Ray)M%("#$%0"((*(/%0"(2%LP aaM%*/%L¬P aaM%^%4&%:)&(28*
$*%>*)+*< g 0)"1/*. ("#$%&'"($%0*#?%2"($/&(/*$%*/%":/*("($%LP aaM%*/%LP abM%0*%4&%7)*+1I)*%,#&(/13K 2&/1"(%#(1'*)$*44*%L∀y(Ray)M%*/%L¬P baM%*/%L¬P bbM%0*%4&%0*#?1I+*. L∀y¬(Ray)M%^%4&%:)&(28*%(*%$*
>*)+*%7&$< !"#$%$"++*$%&))1'-$%A%4=&):)*%$#1'&(/ B
# ∃x∀y(Rxy)∧ ∃x∀y¬(Rxy)
a#∃x∀y(Rxy)
a,b#∃x∀y¬(Rxy)
a#∀y(Ray)a,b#
%%%
&
&
&
a#∀y¬(Ray) ∀y¬(Rby)a,b#
Raa Raa
Rab
¬Raa
¬Rba
¬Rbb
!
**2< +(0)$'(4F(0)$#')B#),-&"./()/(0)#'1'(0 >*G
!"#$%(*%7"#'"($%74#$%&7741,#*)%#(*%)I94*%A%4&%:)&(28*%0)"1/*%*/%("#$%&'"($%/)"#'-%#(%+"0I4*%7"#) 4&%78)&$*%L∃x∀y(Rxy)∧ ∃x∀y¬(Rxy)M%^%#(*%$/)#2/#)*A. "ORA)*41*aAa*/aAb*/%)1*(%0=&#/)*<
@(%+"0I4*%7"#)%4&%78)&$*%1(1/1&4*%$*)&1/%0"(2%7&)%*?*+74*%#(*%$/)#2/#)*%,#1%2"($1$/*%0=#(%#(1'*)$%0*
01$2"#)$%0*%0*#?%7*)$"((*$. V&+%*/%Q&)1*. */%0=#(*%)*4&/1"(%*?7)1+-*%7&)%Lx&1+*yM%,#1%*$/%/*44*%,#*
V&+%&1+*%$"1K+R+*%*/%Q&)1*. */%Q&)1*%(*%$=&1+*%7&$%*/%(=&1+*%7&$%("(%74#$%V&+<
X"++*%0&($%4*%2&$%0*%4&%4"91,#*%7)"7"$1/1"((*44*. ("#$%7"#'"($%-9&4*+*(/%#/141$*)%4&%+-/8"0*%0*$
&):)*$%7"#)%'-)13*)%4&%'&4101/-%0=#(*%7)"7"$1/1"( B #(*%78)&$*%$*)&%'&410*%$1%*/%$*#4*+*(/%$1%$&%(-9&/1"(
*$/%1(2"($1$/&(/*. 2=*$/KAK01)*%$1%/"#/*$%4*$%:)&(28*$%0*%4=&):)*%7"#)%2*//*%(-9&/1"(%$*%>*)+*(/<
k-)13"($. 7&)%*?*+74*. 4&%'&4101/-%0*%4&%78)&$*%L∀x(Rxx→ ∃yRxy)M< X*//*%78)&$*%(*%2"(/1*(/%#(*
2"($/&(/*%1(01'10#*44*%^%2=*$/%7"#),#"1%("#$%2"++*(W"($%7&)%1($/&(21*)%$&%(-9&/1"(%&'*2%#(*%("#'*44*
2"($/&(/*%LaM%*/%("#$%&7741,#"($%4&%)I94*%7"#)%4*$%>")+#4*$!¬(φ→ψ)"B
a#¬∀x(Rxx→ ∃yRxy)
#¬(Raa→ ∃yRay)
a#¬∃yRayRaa
X"++*%LaM%*$/%4&%$*#4*%2"($/&(/*. 4=1($/&(21&/1"(%("#$%0"((*%LP aaM%*/%("#$%7"#'"($%>*)+*)%4&%5$*#4*6
:)&(28* B a#¬∀x(Rxx→ ∃yRxy)
#¬(Raa→ ∃yRay)
a# ¬∃RaayRay
¬Raa
!
H&%:)&(28*%#(1,#*%*$/%>*)+-* B 4&%78)&$*%*$/%1(2"($1$/&(/*%*/%$&%(-9&/1"(%*$/%'&410*<
g'*2%4&%+-/8"0*%0*$%&):)*$. ("#$%7"#'"($%-9&4*+*(/%/*$/*)%4&%'&4101/-%0=#(*%1(>-)*(2*<C)*("($%4=&)9#+*(/
,#1%&%2"++*%7)-+1$$*$%L∀x(F x → Gx)M%*/%L¬∃x(Gx)M. */%&%L¬∃x(F x)M%2"++*%2"(24#$1"(< C"#) '-)13*)%$1%"#1%"#%("(%4&%2"(24#$1"(%*$/%#(*%2"($-,#*(2*%4"91,#*%0*$%7)-+1$$*$%5$1%"#1%"#%("(%14%*$/
4*%2&$%,#*{∀x(F x → Gx),¬∃x(Gx)} |l ¬∃x(F x)6. ("#$%2"($/)#1$"($%4=&):)*%7"#)%4&%(-9&/1"(%0*
4=1+7412&/1"(%2"))*$7"(0&(/*< !"#$%&7741,#"($%0=&:")0%4*$%)I94*$%7"#)%4*$%2"((*2/*#)$ B
# ¬((∀x(F x→Gx)∧ ¬∃x(Gx))→ ¬∃x(F x))
#∀x(F x→Gx)∧ ¬∃x(Gx)
# ¬¬∃x(F x)
∀x(F x→Gx)
¬∃x(Gx)
∃x(F x)
>>A **2 +#),-&"./()/(0)#'1'(0 X"++*%("#$%(=&'"($%#(*%2"($/&(/*%*/%,#=#(*%$*#4*%,#&(/132&/1"(%*?1$/*(/1*44*%. ("#$%1(/)"0#1$"($
#(*%$*#4*%("#'*44*%2"($/&(/*. &'*2%4&,#*44*%("#$%1($/&(21"($%4*$%0*#?%,#&(/132&/1"(%#(1'*)$*44*$ B
# ¬((∀x(F x→Gx)∧ ¬∃x(Gx))→ ¬∃x(F x))
#∀x(F x→Gx)∧ ¬∃x(Gx)
# ¬¬∃x(F x)
a#∀x(F x→Gx)
a#¬∃x(Gx)
a#∃x(F x)
F a
#F a→Ga
¬Ga
%%
%
&
&
&
¬F a
! Ga
!
@(%&):)*%/)I$%$1+14&1)*%7)"#'*%4&%'&4101/-%0*%4=1(>-)*(2*%L∀x(F x→Gx))(∀x(F x)→ ∀x(Gx))M B
# ¬(∀x(F x→Gx)→(∀x(F x)→ ∀x(Gx)))
a# ∀x(F x→Gx)
# ¬(∀x(F x)→ ∀x(Gx))
a#∀x(F x)
a#¬∀x(Gx)
¬Ga
#F aF a→Ga
%%
%
&
&
&
¬F a
! Ga
!
C"#)%7)"#'*)%,#*%L∃x(Gx∧Hx)M%$=*($#1/%0*$%0*#?%7)-+1$$*$%L(∀x(F x→Gx)M%*/%L∃x(F x∧Hx))→M.
("#$%2"++*(W"($%-9&4*+*(/%&'*2%4=1($/&(/1&/1"(%0*%4&%7)-+1$$*%*?1$/*(/1*44* B
# ¬((∀x(F x→Gx)∧ ∃x(F x∧Hx))→ ∃x(Gx∧Hx))
∀x(F x→Gx)
#a ∃x(F x∧Hx)
¬∃x(Gx∧Hx)
#F a∧Ha
**2? +(0)#'1'(0)%565%0)(&)45()0%,$B%6D#&%.5)/(0)'8CB(0 >>*
!"#$%7"#'"($%+&1(/*(&(/%1($/&(21*)%4*$%,#&(/132&/1"($%#(1'*)$*44*$%*/%2"($/)#1)*%4=&):)*%$#1'&(/ B
# ¬((∀x(F x→Gx)∧ ∃x(F x∧Hx))→ ∃x(Gx∧Hx))
#a ∀x(F x→Gx)
#a ∃x(F x∧Hx)
#a ¬∃x(Gx∧Hx)
#F a∧Ha
#F a→Ga
#¬(Ga∧Ha)
!!
!!
"
"
"
"
¬F a
! Ga
'' '' (( ((
¬Ga
! ¬Ha
!
**2? +(0)#'1'(0)%565%0)(&)45()0%,$B%6D#&%.5)/(0)'8CB(0
H&%)I94*%0*%:)&(28*+*(/%7"#)%4*%,#&(/132&/*#)%*?1$/*(/1*4%5*/%4&%(-9&/1"(%0#%,#&(/132&/*#)%#(1'*)$*46
*/%4&%)I94*%0=1($/&(21&/1"(%7"#)%4*%,#&(/132&/*#)%#(1'*)$*4%5*/%4&%(-9&/1"(%0#%,#&(/132&/*#)%*?1$/*(/1*46 7*#'*(/%1(/*)>-)*)%4=#(*%&'*2%4=&#/)*. 0*%$")/*%,#*%4&%7)*+1I)*%("#$%>")2*%A%1(/)"0#1)*%0*%("#'*44*$
2"($/&(/*$%,#*%4&%$*2"(0*%("#$%>")2*%0=#/141$*)%0&($%0*$%1($/&(21&/1"($%,#1%m 7)"0#1$*(/%n 0*%("#'*44*$
,#&(/132&/1"($%*?1$/*(/1*44*$%,#1%0"((*(/%41*#%A%0*$%("#'*44*$%2"($/&(/*$%,#1%0"1'*(/%A%4*#)%/"#)%R/)*
#/141$-*$%7"#)%1($/&(21*)%4*$%,#&(/132&/1"($%#(1'*)$*44*$%*/%&1($1%0*%$#1/*<
X"($10-)"($%4&%78)&$*%L∀x∃yRxy∧GaM%*/%>&1$"($%$"(%&):)*<!"#$%)*+&),#"($%,#*%("#$%(=*(%&))1'*)"($
;&+&1$%&#%:"#/ B
# ∀x∃y Rxy∧Ga
a#∀x∃y Rxy Ga
a,b#∃yRay
!!
!!
"
"
"
"
a,b,c#∃RabyRby Raa
'' '' (( ((
####
###
a,b,c,d#∃yRcyRbc Rbb Rba
'' '' (( ((
####
###
))))))
))))
Rcc
Rcd Rcb Rca
∃yRdy
<<
<
>>> **2 +#),-&"./()/(0)#'1'(0
H=&):)*%,#1%*(%)-$#4/*%*$/%1(3(1%0&($%0*#?%01)*2/1"( B 14%2"(/1*(/%#(*%:)&(28*%1(3(1*%52*44*%/"#/%A%9*6
*/%14%2"(/1*(/%#(%("+:)*%1(3(1%0*%:)&(28*$< !"#$%":$*)'"($%&#$$1%,#*%4*$%:)&(28*$%7&)/&(/%A%0)"1/*
(=&;"#/*(/%)1*( B 4&%("#'*44*%2"($/&(/*%LbM. 7&)%*?*+74*. 1(/)"0#1/*%&7)I$%4*%7)*+1*)%:)&(28*+*(/. 7*#/
("#$%$*)'1)%-9&4*+*(/%7"#)%)*7)-$*(/*)%4&%'1*44*%2"($/&(/*%LaM%^%$1%("#$%4*%'"#4"($. ("#$%$"++*$%41:)*$
0*%4=1(/*)7)-/*)%7&)%4*%+R+*%1(01'10#%,#*%("#$%#/141$"($%7"#)%1(/*)7)-/*)%LaM< !"#$%&))1'"($%0"(2%&#?
$1+74132&/1"($%$#1'&(/*$%7"#)%("$%)I94*$%0*%:)&(28*+*(/ B
!∃x(φ(x))"
!φ(b)" 7"#)%#(*%("#'*44*%2"($/&(/*%LbM
!¬∀x(φ(x))"
!¬φ(b)" 7"#)%#(*%("#'*44*%2"($/&(/*%LbM
g'*2%2*$%("#'*44*$%)I94*$. ("/)*%&):)*%(=*$/%1(3(1%,#*%0&($%4&%01)*2/1"(%'*)/12&4* B
b,c,d,e,...#∀x∃y(Rxy)
b#∃yRay
c#∃yRbyRab
d#∃yRcyRbc
e#∃yRdyRcd
f#∃yReyRde
<<
<
C"#)%7)"#'*)%4&%'&4101/-%0=#(*%7)"7"$1/1"(. 4&%("#'*44*%+-/8"0*%*$/%&#$$1%*e2&2*%,#*%4=&(21*((*< X*21
*$/%0o%&#%>&1/%$#1'&(/ B
7"-.'8,( 9=2 !#(%&'(CE+6)'(6(%&(3"-F.'(-Bn#&-#0#-%)> 6."+)('@'(6(6%))#(%&(3"-F.'(-Bn+1#&-#0#-%)9 CDEFGE C&)/&(/%0=#(%+"0I4*%0*n1(01'10#%)*(0&(/%')&1*%4&%78)&$*%*(%,#*$/1"(. ("#$%)&;"#/"($%$1+74*K +*(/%#(%1(01'10#%0*%74#$%*/%$/17#4"($%,#*%4*$%+R+*$%7)-012&/$%$"(/%')&1*$%0*%4#1%,#*%4*%$"(/%0=#(%0*$n
1(01'10#$%0&($%4=&(21*(%+"0I4*< !
**2E +(0)$'-04$$.0-0)(H%0&(5&%(B0)/()B#)B.C%34()/(0)$'-/%D#&0 >><
H&%2"('*)$1"(%0*%2*%/8-")I+*%01/%,#*%$1%#(*%78)&$*%(=&%7&$%0*%+"0I4*%0*n+ 11(01'10#$. *44*%(=&#)&
7&$%0*%+"0I4*%0*n1(01'10#$< H*$%:)&(28*$%9-(-)-*$%7&)%4*$%)I94*$%0#%,#&(/132&/*#)%*?1$/*(/1*4%5*/%0*
4&%(-9&/1"(%0#%,#&(/132&/*#)%#(1'*)$*46%0*%4=&(21*((*%+-/8"0*%)*7)-$*(/&1*(/%0*%/*44*$%+"0I4*$ B ("#$
7)-$*)'"($%4=#/141/-%0*%4&%+-/8"0*%7"#)%7)"#'*)%0*$%78)&$*$%$1%("#$%4&1$$"($%/"+:*)%2*$%28*+1($%0*
'-)1/-%,#1%2"))*$7"(0&(/%A%0*$%+"0I4*$%2"(/*(&(/%+"1($%0=1(01'10#$<
X*//*%$1+74132&/1"(%0*$%)I94*$%("#$%7*)+*/%0=-2"("+1$*)%0*$%:)&(28*$%-9&4*+*(/%0&($%4*$%7)*#'*$%m
")01(&1)*$%n . 2"++*%4*%+"(/)*%4=*?*+74*%$#1'&(/< k"121%#(%&):)*%7"#)%L∀x(F x→Ga)→(∃x(F x)→ Ga))M. 7&)%4=&(21*((*%+-/8"0* B
# ¬(∀x(F x→Ga)→(∃x(F x)→Ga))
#a,b ∀x(F x→Ga)
#¬(∃x(F x)→Ga)
#a,b∃x(F x)
¬Ga
!!
!!
"
"
"
"
#F a→F aGa F b
#F b→Ga
F a→Ga
!!
!!
"
"
"
"
¬F a
! Ga
!
!!
!!
"
"
"
"
¬F b
! Ga
!
g'*2%4&%("#'*44*%+-/8"0*%("#$%":/*("($ B
# ¬(∀x(F x→Ga)→(∃x(F x)→Ga))
#b ∀x(F x→Ga)
#¬(∃x(F x)→Ga)
#b∃x(F x)
¬Ga
#F bF b→Ga
!!
!!
"
"
"
"
¬!F b !Ga
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####
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!!
!!
"
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####
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!!
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"
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$$
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