Fixing Frege (Princeton Monographs in Philosophy)

Citation preview

(+:+0)(4')'                                 24+0%'610 /101)4#2*5 +0 2*+.1512*;

KK

(+:+0)(4')' ,QJP2$WTIGUU 24+0%'610 70+8'45+6; 24'55 24+0%'610 #0& 1:(14&

%QR[TKIJV l 

KKK (QT#NGZKCPF(QMKQP

KX

%QPVGPVU #EMPQYNGFIOGPVUVKZ  %*#26'4 (TGIG4WUUGNNCPF#HVGT   %*#26'4 2TGFKECVKXG6JGQTKGU   %*#26'4 +ORTGFKECVKXG6JGQTKGU   6CDNGUV  0QVGUV  4GHGTGPEGUV  +PFGZV

X

#EMPQYNGFIOGPVU (

14 51 UJQTV C YQTM CU VJG RTGUGPV QPG VJG NKUV QH RGTUQPU VQ YJQO VJG CWVJQT JCU VJG

RNGCUCPV FWV[ QH CEMPQYNGFIKPI FGDVU KU C NQPI QPG 2CWN $GPCEGTTCH 5QNQOQP (GHGTOCP -KV (KPG *CTXG[ (TKGFOCP # 2 *C\GP 4KEJCTF *GEM #MKJKTQ -CPCOQTK 5CWN -TKRMG 5VGHCP .GWGPDGTIGT «[UVGKP .KPPGDQ 2GPGNQRG /CFF[ 'FYCTF 0GNUQP %JCTNGU 2CTUQPU 0GKN 6GPPCPV -CK 9GJOGKGT CPF %TKURKP 9TKIJV &QWDVNGUU VJGTG JCXG DGGP QVJGTU YJQUG PCOGU JCXG GUECRGF O[ JKIJN[ HCNNKDNG OGOQT[ VQ YJQO + OWUV QHHGT CRQNQIKGU CU YGNN CU VJCPMU 5QOG QH VJQUG PCOGF QHHGTGF KPHQTOCVKQP CPF CUUKUVCPEG QP URGEKHKE RQKPVU QH YJKEJ RCTVKEWNCTU YKNN DG PQVGF KP VJG DQF[ QH VJG OQPQITCRJ 1VJGTU JCXG GZGTVGF C INQDCN KPVGNNGEVWCN KPHNWGPEG YJQUG EQPVTKDWVKQP VQ VJG YQTM KU PQV VJG NGUU KORQTVCPV HQT DGKPI JCTFGT VQ NQECNK\G ;GV QVJGTU RTQXKFGF QPG QT CPQVJGT HQTWO KP YJKEJ + EQWNF RTGUGPV XGTUKQPU QH UQOG QH VJG OCVGTKCN HQT ETKVKECN EQOOGPV CPF VQ VJKU NCUV ECVGIQT[ + OWUV PQY CFF *CTT[ (TCPMHWTV YJQO + OGPVKQP KP ITCVKVWFG HQT VJG KPXKVCVKQP VQ EQPVTKDWVG VQ VJG UGTKGU KP YJKEJ VJKU DQQM PQY CRRGCTU

 (TGIG4WUUGNNCPF#HVGT 6*' )4'#6 NQIKEKCP )QVVNQD (TGIG YTQVG VJTGG DQQMU GCEJ TGRTGUGPVKPI C UVCIG KP C ITCPF RTQITCO HQT RTQXKFKPI C NQIKECN HQWPFCVKQP HQT CTKVJOGVKE CPF JKIJGT OCVJGOCVKEU *KU

$GITKHHUUEJTKHV

JCNH QH JKU

 KPVTQFWEGF C EQORTGJGPUKXG U[UVGO QH U[ODQNKE NQIKE 6JG HKTUV

)TWPFNCIGP

 QHHGTGF C FGXCUVCVKPI ETKVKSWG QH RTGXKQWU CEEQWPVU QH VJG

HQWPFCVKQPU QH CTKVJOGVKE YJKNG VJG UGEQPF JCNH QHHGTGF CP QWVNKPG QH JKU QYP KPIGPKQWU RTQRQUGF HQWPFCVKQP 6JG VYQ XQNWOGU QH VJG

)TWPFIGUGV\G

  HKNNGF KP VJG

VGEJPKECN FGVCKNU QH JKU QWVNKPG WUKPI JKU NQIKECN U[ODQNKUO CPF GZVGPFGF VJG RTQLGEV HTQO CTKVJOGVKE VJG VJGQT[ QH VJG PCVWTCN PWODGTU VQ OCVJGOCVKECN CPCN[UKU VJG VJGQT[ QH VJG TGCN PWODGTU 7PHQTVWPCVGN[ LWUV CU VJG UGEQPF XQNWOG QH VJG

O

)TWPFIGUGV\G

YCU IQKPI VQ

RTGUU $GTVTCPF 4WUUGNN FKUEQXGTGF C EQPVTCFKEVKQP KP (TGIG U U[UVGO 5WDUGSWGPV YQTM KP NQIKE CPF HQWPFCVKQPU QH OCVJGOCVKEU NCTIGN[ D[RCUUGF VJG )TWPFIGUGV\G

O

WPVKN C EQWRNG QH FGECFGU CIQ YJGP RJKNQUQRJGTU CPF NQIKEKCPU VQQM C PGY

NQQM CV (TGIG U KPEQPUKUVGPV U[UVGO CPF TGEQIPK\GF VJCV OQTG ECP DG UCNXCIGF HTQO KV VJCP JCF RTGXKQWUN[ DGGP VJQWIJV +P VJGUG NCUV [GCTU COGPFGF CPF RCTCFQZHTGG XGTUKQPU

O

QH (TGIG U U[UVGO JCXG DGGP RTQFWEGF UWDUVCPVKCN RQTVKQPU QH ENCUUKECN OCVJGOCVKEU JCXG DGGP FGXGNQRGF YKVJKP UWEJ U[UVGOU CPF C UEJQQN QH PGQ(TGIGCPU JCU CTKUGP ENCKOKPI RJKNQUQRJKECN DGPGHKVU HQT UWEJ CP CRRTQCEJ VQ VJG HQWPFCVKQPU QH OCVJGOCVKEU 6JG VJQWIJV WPFGTN[KPI VJG RTGUGPV OQPQITCRJ KU VJCV JQYGXGT YQPFGTHWN VJG RJKNQUQRJKECN DGPGHKVU QH PGQ(TGIGCP TGEQPUVTWEVKQPU QH OCVJGOCVKEU VJG CUUGUUOGPV QH VJG WNVKOCVG UKIPKHKECPEG QH CP[ PGQ(TGIGCP CRRTQCEJ OWUV CYCKV C FGVGTOKPCVKQP QH LWUV JQY

OWEJ

QH OCVJGOCVKEU ECP DG TGEQPUVTWEVGF YKVJQWV TGUQTV VQ

CF JQE

J[RQVJGUGU QP

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

O

(TGIG U QYP UVTCVGI[ VJCP FQ QVJGTU 0GQ(TGIGCPU UQOGVKOGU UWIIGUV VJCV VJG ENQUGT QPG UVC[U VQ (TGIG VJG ITGCVGT VJG RJKNQUQRJKECN DGPGHKVU +V KU PQV O[ CKO KP VJG RTGUGPV YQTM

VQ CTIWG HQT QT CICKPUV UWEJ ENCKOU 9JCV + FQ KPUKUV KU VJCV CP[ RJKNQUQRJKECN ICKPU OWUV DG YGKIJGF CICKPUV OCVJGOCVKECN NQUUGU (QT VJG UWTXG[ VQ HQNNQY UJQYU VJCV UQOG PGQ(TGIGCP CRRTQCEJGU [KGNF OWEJ OQTG QH OCVJGOCVKEU VJCP QVJGTU CPF KV QHVGP UGGOU VJCV VJG NGUU QPG MGGRU QH (TGIG VJG OQTG QPG IGVU QH OCVJGOCVKEU 0QPGVJGNGUU GXGP KP VJG NCUV U[UVGO VQ DG EQPUKFGTGF JGTG YJKEJ [KGNFU CNN QH QTVJQFQZ OCVJGOCVKEU CPF OQTG CNUQ VJGTG TGOCKPU QPG UOCNN DWV UKIPKHKECPV KPITGFKGPV QH (TGIGCP CPEGUVT[

O

#U C PGEGUUCT[ RTGNKOKPCT[ VQ VJG UWTXG[ QH CVVGORVU VQ TGRCKT (TGIG U U[UVGO VJCV U[UVGO OWUV KVUGNH DG TGXKGYGF 6JG WPFGTN[KPI NQIKE QH VJG CUUWORVKQP CFFGF VJGTGVQ KP VJG

)TWPFIGUGV\G

$GITKHHUUEJTKHV

VJG

HQT RWTRQUGU QH FGXGNQRKPI OCVJGOCVKEU

O

VJCV FGXGNQROGPV KVUGNH VJG RCTCFQZ 4WUUGNN HQWPF KP VJG U[UVGO CPF 4WUUGNN U QYP CVVGORVU VQ TGRCKT KV OWUV GCEJ DG DTKGHN[ GZCOKPGF CPF VJG OCVJGOCVKECN CPF RJKNQUQRJKECN IQCNU C PGQ(TGIGCP RTQLGEV QT HQT VJCV OCVVGT CP[ RTGUGPVFC[ HQWPFCVKQPCN RTQITCO OKIJV UGV KVUGNH DTKGHN[ UWTXG[GF

(TGIGOU.QIKE 9JKNG (TGIG KU JQPQTGF CU C HQWPFGT QH OQFGTP NQIKE JKU U[UVGO YKNN PQV NQQM CV CNN HCOKNKCT VQ RTGUGPVFC[ UVWFGPVU QH VJG UWDLGEV 6Q DGIKP YKVJ (TGIG WUGU C PQPNKPGCT PQVCVKQP VJCV PQ UWDUGSWGPV YTKVGT JCU HQWPF KV EQPXGPKGPV VQ CFQRV CPF VJCV YKNN PQV DG GPEQWPVGTGF KP CP[ OQFGTP VGZVDQQM 5KPEG VJG RTGUGPV YQTM KU CP[VJKPI DWV CP JKUVQTKECN

O

VTGCVKUG VJG PQVCVKQP YKNN DG TWVJNGUUN[ OQFGTPK\GF YJGP (TGIG U U[UVGO KU GZRQWPFGF JGTG

O

$WV GXGP YJGP VJG PQVCVKQP KU OQFGTPK\GF (TGIG U

JKIJGTQTFGT

NQIKE JCU C

ITCOOCT VJCV VJQWIJ UVKNN UKORNGT D[ HCT VJCP VJG ITCOOCT QH )GTOCP QT 'PINKUJ QT CP[ PCVWTCN NCPIWCIG KU CRRTGEKCDN[ OQTG EQORNGZ VJCP VJG ITCOOCT QH VJG

O

HKTUVQTFGT

NQIKE

QH RTGUGPVFC[ VGZVDQQMU 0QPGVJGNGUU CHVGT (TGIG U WPHCOKNKCT WPFGTN[KPI ITCOOCVKECN CPF QPVQNQIKECN CUUWORVKQPU JCXG DGGP GZRQWPFGF QPN[ C HGY HWTVJGT GZRNCPCVKQPU UJQWNF DG TGSWKTGF VQ GPCDNG VJG TGCFGT HCOKNKCT YKVJ HKTUVQTFGT NQIKE VQ WPFGTUVCPF JKIJGTQTFGT NQIKE

O

.GV WU DGIKP VJGP YKVJ (TGIG U ITCOOCT (QT (TGIG VJGTG CTG VYQ ITCOOCVKECN ECVGIQTKGU QT ITCOOCVKECN V[RGU QH YJCV JG ECNNU DG ECNNGF 0 QT VJG ECVGIQT[ QH

PCOGU

UKPIWNCT FGHKPKVG FGUETKRVKQPU UWEJ CU HTQO KPFGZKECNU CPF FGUKIPCVG CP

Q

R

VJKPI

UCVWTCVGF

GZRTGUUKQPU 6JG HKTUV JGTG VQ

KPENWFGU RTQRGT PCOGU UWEJ CU

Q

Q

VJG OQUV HCOQWU UVWFGPV QH 5QETCVGU

QDLGEV

R R

2NCVQ  DWV CNUQ VJCV CTG HTGG

YJKEJ OC[ DG C RGTUQP QT RNCEG TCVJGT VJCP C

KP C EQNNQSWKCN UGPUG KPFGRGPFGPVN[ QH EQPVGZV 6JG UGEQPF JGTG VQ DG ECNNGF 5

QT VJG ECVGIQT[ QH

UGPVGPEGU

KPENWFGU FGENCTCVKXG UGPVGPEGU UWEJ CU

Q

2NCVQ YCU VJG OQUV

R

HCOQWU UVWFGPV QH 5QETCVGU

VJCV CTG HTGG HTQO KPFGZKECNU CPF JCXG C

VTWVJXCNWG

VTWG QT

HCNUG KPFGRGPFGPVN[ QH EQPVGZV +P CFFKVKQP VJGTG CTG OCP[ V[RGU QH

WPUCVWTCVGF

GZRTGUUKQPU YKVJ QPG QT OQTG

ICRU VJCV KH HKNNGF KP YKVJ CP GZRTGUUKQP QT GZRTGUUKQPU QH CRRTQRTKCVG V[RG U YQWNF RTQFWEG CP GZRTGUUKQP QH V[RG 0 QT 5 6JQUG VJCV VJWU HKNNGF KP YQWNF RTQFWEG GZRTGUUKQPU QH V[RG 5 TCVJGT VJCP V[RG 0 YKNN DG QH OQUV KPVGTGUV JGTG CPF VJGUG OC[ DG ECNNGF

RTGFKECVGU

KP C DTQCF UGPUG +P C PQVCVKQP FGTKXGF HTQO VJG OWEJ NCVGT YTKVGTU

-CUKOKGT\ #LFWMKGYKE\ CPF ;GJQUJWC $CT*KNNGN CP GZRTGUUKQP YKVJ

W W

HKNNGF KP YKVJ GZRTGUUKQPU QH V[RGU 6 DG UCKF VQ DG C RTGFKECVG QH V[RG 56

M

DNCPMU KP KV VJCV KH

 6M YKNN RTQFWEG CP GZRTGUUKQP QH V[RG 5 OC[

6M 6JG UKORNGUV ECUG KU VJCV QH

RTGFKECVGU

KP VJG

PCTTQY UGPUG QH GZRTGUUKQPU QH V[RG 50 5QOG QVJGT ECUGU CTG UJQYP KP 6#$.' # 6JG NCDGN

TGNCVKQPCN RTGFKECVG

YKNN DG WUGF HQT VJG VYQRNCEG VJTGGRNCEG CPF

OCP[RNCEG V[RGU 500 5000 CPF UQ QP YKVJ VJG PWODGT QH RNCEGU DGKPI OGPVKQPGF GZRNKEKVN[ YJGP KV KU ITGCVGT VJCP VYQ 6JG NCDGN

JKIJGT RTGFKECVG

YKNN DG WUGF HQT VJG

UGEQPFNGXGN VJKTFNGXGN CPF JKIJGTNGXGN V[RGU 5 50 5 5 50 CPF UQ QP YKVJ VJG PWODGT QH VJG NGXGN DGKPI OGPVKQPGF GZRNKEKVN[ KH KV KU JKIJGT VJCP VJG UGEQPF 5KOKNCTN[ YKVJ VJG NCDGN

JKIJGT TGNCVKQPCN RTGFKECVG

YJKEJ GXGP YKVJ LWUV VYQ RNCEGU CPF GXGP CV

LWUV VJG UGEQPF NGXGN EQXGTU C XCTKGV[ QH V[RGU KPENWFKPI PQV QPN[ 5 50 50 CU UJQYP KP VJG VCDNG DWV CNUQ HQT KPUVCPEG 5 500 500 CPF UWEJ OKZGF V[RGU CU 5 50 500 CPF GXGP 50 50 +V KU CP KPUVTWEVKXG GZGTEKUG VQ NQQM HQT PCVWTCN NCPIWCIG GZCORNGU KNNWUVTCVKPI UWEJ RQUUKDKNKVKGU

O

+P (TGIG  YJKEJ CHVGT JKU VJTGG DQQMU KU KVU CWVJQT U OQUV HCOQWU YQTM (TGIG KPVTQFWEGF C FKUVKPEVKQP DGVYGGP VJG

UGPUG

GZRTGUUGF CPF VJG

TGHGTGPV

FGPQVGF D[

CP GZRTGUUKQP 6JG TGCFGT YKNN PQV IQ HCT YTQPI YJQ VJKPMU QH YJCV (TGIG ECNNU VJG

Q

R

UGPUG

QH CP GZRTGUUKQP QH YJCVGXGT V[RG CU TQWIJN[ GSWKXCNGPV VQ YJCV QVJGT

RJKNQUQRJGTU YQWNF ECNN KVU

Q

R

OGCPKPI  9JCV VJG

Q

R

TGHGTGPV

QH CP GZRTGUUKQP KU VQ DG

WPFGTUVQQF VQ DG XCTKGU HTQO ITCOOCVKECN V[RG VQ ITCOOCVKECN V[RG +P VJG ECUG QH C RTQRGT PCOG QT UKPIWNCT FGHKPKVG FGUETKRVKQP QH V[RG 0 VJG TGHGTGPV KU VJG QDLGEV FGUKIPCVGF VJG VJKPI DGCTKPI VJG PCOG QT CPUYGTKPI VQ VJG FGUETKRVKQP %NGCTN[ VYQ GZRTGUUKQPU QH V[RG 0 HQT KPUVCPEG CPF

Q

R

VJG OQUV HCOQWU VGCEJGT QH #TKUVQVNG

JCXG VJG UCOG TGHGTGPV

M

Q

ECP JCXG FKHHGTGPV UGPUGU GXGP VJQWIJ VJG[

KP VJKU KPUVCPEG 2NCVQ 'ZRTGUUKQPU YKVJ FKHHGTGPV UGPUGU DWV VJG

UCOG TGHGTGPV RTQXKFG FKHHGTGPV

Q

R

OQFGU QH RTGUGPVCVKQP

6JG UGPUG QH C UGPVGPEG QH V[RG 5 (TGIG ECNNU C HCT YTQPI YJQ VJKPMU QH YJCV (TGIG ECNNU C RJKNQUQRJGTU ECNN C

Q

R

VJG OQUV HCOQWU UVWFGPV QH 5QETCVGU

R

Q

R

VJQWIJV

QH VJG UCOG QDLGEV

VJQWIJV

CPF VJG TGCFGT YKNN PQV IQ

CU TQWIJN[ GSWKXCNGPV VQ YJCV QVJGT

RTQRQUKVKQP  6JG TGHGTGPV QH C UGPVGPEG QH V[RG 5 (TGIG VCMGU VQ DG

UKORN[ KVU VTWVJXCNWG 1DXKQWUN[ VYQ UGPVGPEGU QH V[RG 5 HQT KPUVCPEG

R Q R Q M

HCOQWU UVWFGPV QH 5QETCVGU KU C HGCVJGTNGUU DKRGF

CPF

VJG[ JCXG VJG UCOG TGHGTGPV

CPF

Q

2NCVQ KU VJG OQUV

R Q

2NCVQ KU VJG OQUV HCOQWU VGCEJGT QH #TKUVQVNG  QT

R

2NCVQ

2NCVQ KU C TCVKQPCN CPKOCN  ECP JCXG FKHHGTGPV UGPUGU VJQWIJ KP VJGUG KPUVCPEGU VJG VTWVJXCNWG

VTWG

5Q OWEJ HQT VJG TGHGTGPVU QH EQORNGVG QT UCVWTCVGF GZRTGUUKQPU #U HQT VJG

W

TGHGTGPVU QH KPEQORNGVG QT WPUCVWTCVGF GZRTGUUKQPU QH V[RGU 0

QT 5

W

 VJG[ CTG

UWRRQUGF VQ DG KPEQORNGVG NKMG VJG GZRTGUUKQPU VJGOUGNXGU EQPVCKPKPI ICRU VJCV YJGP CRRTQRTKCVGN[ HKNNGF KP YKNN RTQFWEG CP QDLGEV QT C VTWVJXCNWG 6JG TGHGTGPVU QH GZRTGUUKQPU QH V[RG 0 5

W

W

(TGIG ECNNU

HWPEVKQPU

CPF VJG TGHGTGPVU QH GZRTGUUKQPU QH V[RG

 VJCV KU VJG TGHGTGPVU QH RTGFKECVGU JG ECNNU

EQPEGRVU

%QTTGURQPFKPI VQ VJG FKHHGTGPV

ITCOOCVKECN V[RGU QH RTGFKECVGU CTG FKHHGTGPV QPVQNQIKECN V[RGU QH EQPEGRVU KPENWFKPI EQPEGRVU QH VJG PCTTQYGUV HKTUVNGXGN QPGRNCEG MKPF DWV CNUQ

TGNCVKQPCN EQPEGRVU

CPF

JKIJGT EQPEGRVU

(QT C EQPEGRV QH V[RG 50 KH HKNNKPI KV KP YKVJ C EGTVCKP QDLGEV RTQFWEGU VJG VTWVJXCNWG

VTWG

VJGP VJG QDLGEV KU UCKF VQ

HCNN WPFGT

VJG EQPEGRV (QT KPUVCPEG CUUWOKPI

HQT VJG UCMG QH GZCORNG VJCV 2NCVQ OC[ VTWN[ DG ECNNGF YKUG UKPEG VJG EQPEGRV FGPQVGF D[

QW

R Q

KU YKUG  YJKEJ KU VQ UC[ VJG EQPEGRV QH DGKPI YKUG YJGP HKNNGF KP D[ VJG QDLGEV

FGPQVGF D[

Q

R

2NCVQ  YJKEJ KU VQ UC[ VJG QDLGEV 2NCVQ RTQFWEGU VJG VTWVJXCNWG FGPQVGF D[

R

2NCVQ KU YKUG  YJKEJ KU VQ UC[ VJG VTWVJXCNWG

VTWG

KV HQNNQYU VJCV VJG QDLGEV 2NCVQ HCNNU

WPFGT VJG EQPEGRV QH DGKPI YKUG 6JG UCOG VGTOKPQNQI[ KU WUGF KP EQPPGEVKQP YKVJ QVJGT V[RGU QH RTGFKECVGU UKPEG 5QETCVGU VCWIJV 2NCVQ VJG RCKT 5QETCVGU CPF 2NCVQ KP VJCV QTFGT HCNN WPFGT VJG TGNCVKQPCN EQPEGRV QH JCXKPI VCWIJV UKPEG 2NCVQ KU CP GZCORNG QH UQOGQPG YJQ KU YKUG VJG EQPEGRV QH DGKPI YKUG HCNNU WPFGT VJG JKIJGT EQPEGRV QH DGKPI GZGORNKHKGF D[ 2NCVQ 6YQ EQPEGRVU CTG ECNNGF

EQGZVGPUKXG

KH VJG[ CRRN[ VQ VJG UCOG KVGOU QT KP QVJGT

YQTFU KH YJCVGXGT HCNNU WPFGT GKVJGT HCNNU WPFGT VJG QVJGT 0QY VJG ETWEKCN FKHHGTGPEG DGVYGGP VJG

TGHGTGPVU

QH RTGFKECVGU YJKEJ KU VQ UC[ EQPEGRVU CPF VJG

UGPUGU

QH RTGFKECVGU

KU VJCV CEEQTFKPI VQ (TGIG EQGZVGPUKXG EQPEGRVU CTG VJG UCOG 6JWU KH GXGT[ HGCVJGTNGUU DKRGF KU C TCVKQPCN CPKOCN CPF XKEG XGTUC VJGP VJQWIJ VJG UGPUGU QH

R QW

DKRGF

CPF

KU C TCVKQPCN CPKOCN

R

QW

KU C HGCVJGTNGUU

CTG FKHHGTGPV VJG EQPEGRV QH DGKPI C HGCVJGTNGUU

DKRGF CPF VJG EQPEGRV QH DGKPI C TCVKQPCN CPKOCN CTG VJG UCOG +V UQWPFU QFF VQ UC[ UQ CPF VJG FGITGG QH QFFKV[ KU C OGCUWTG QH VJG FGITGG QH

O

FGRCTVWTG QH (TGIG U VGEJPKECN WUCIG QH

Q

R

EQPEGRV

HTQO VJG QTFKPCT[ WUCIG QH

Q

R

EQPEGRV 

YJKEJ VGPFU VQ UWIIGUV VJG UGPUG TCVJGT VJCP VJG TGHGTGPV QH C RTGFKECVG 6JG NCDGN

Q

R

EQPEGRV

YCU KP HCEV EJQUGP D[ (TGIG YGNN DGHQTG JG TGEQIPK\GF VJG KORQTVCPEG QH

U[UVGOCVKECNN[ FKUVKPIWKUJKPI UGPUG CPF TGHGTGPEG $[ JKPFUKIJV KV UGGOU JG OKIJV JCXG

FQPG YGNN CHVGT TGEQIPK\KPI VJG KORQTVCPEG QH VJCV FKUVKPEVKQP VQ TGXKUG JKU VGTOKPQNQI[ + YCU VGORVGF VQ UWDUVKVWVG

Q

R Q R

ENCUUKHKECVKQP

Q

GZRQUKVKQP DWV JCXG UVWEM YKVJ

EQPEGRV

HQT

R

EQPEGRV

KP VJG HQTGIQKPI UJQTV

DGECWUG KV KU UVKNN WUGF D[ OQUV QH VJG

PGQ(TGIGCPU YKVJ YJQUG XKGYU + YKNN DG EQPEGTPGF +P (TGIGCP VGTOKPQNQI[ VJGP VQ VJG ITCOOCVKECN ECVGIQTKGU QH PCOGU UGPVGPEGU CPF RTGFKECVGU VJGTG EQTTGURQPF VJG QPVQNQIKECN ECVGIQTKGU QH QDLGEVU VTWVJXCNWGU CPF

Q

R

O

EQPEGRVU  6JG HQTOCN NCPIWCIG QH (TGIG U JKIJGTQTFGT NQIKE KU OQTG EQORNGZ VJCP VJG

HQTOCN NCPIWCIGU QH VJG HKTUVQTFGT NQIKE GZRQWPFGF KP RTGUGPVFC[ VGZVDQQMU KP VJCV KV OCMGU RTQXKUKQP HQT RTGFKECVGU QH CNN V[RGU FGPQVKPI EQPEGRVU QH CNN V[RGU

5Q OWEJ HQT VJG ITCOOCT DGJKPF VJG NQIKE 6WTPKPI VQ NQIKE KVUGNH OQFGTP VGZVDQQMU KPVTQFWEG VJG UVWFGPV VQ VJG PQVKQPU QH C HKTUVQTFGT  HQT KFGPVKV[ CPF WUWCNN[ QVJGT

PQPNQIKECN

RNCEG QPGU QT UGPVGPEG U[ODQNU CPF QT EQPUVCPVU #NUQ KPVTQFWEGF CTG

HWPEVKQP U[ODQNU KPENWFKPI RNCEG QPGU

TWNGU QH HQTOCVKQP

CPF VJG PQVKQP QH C

HTQO XCTKCDNGU CPF EQPUVCPVU WUKPI HWPEVKQP U[ODQNU RWVVKPI VGTOU KP VJG RNCEGU QH TGNCVKQP U[ODQNU CPF

PQVKQPU QH

HTGG

CPF

DQWPF

XCTKCDNGU CPF

YKVJ VJG U[ODQN

U[ODQNU PRNCEG TGNCVKQP U[ODQNU KPENWFKPI

PRNCEG

HQTOWNCU WUKPI VJG NQIKECN QRGTCVQTU ` 

NCPIWCIG

CVQOKE HQTOWNC

HQTOWNC

VGTO

DWKNV WR

QDVCKPGF D[

DWKNV WR HTQO CVQOKE

∨ → ↔ ∀ ∃ CNQPI YKVJ VJG CPEKNNCT[

QRGP

CPF

ENQUGF

HQTOWNCU YKVJ UQOG XCTKCDNGU

HTGG CPF YKVJ CNN XCTKCDNGU DQWPF TGURGEVKXGN[ #NUQ KPVTQFWEGF EGTVCKP KP FKHHGTGPV DQQMU UWEJ PCOGU CU

Q

M

TWNGU QH FGFWEVKQP

YJKEJ OC[ VCMG YKFGN[ FKHHGTGPV HQTOU

VJG FKHHGTGPV HQTOCVU HQT FGFWEVKQPU WUGF KP FKHHGTGPV DQQMU IQKPI D[

R Q

*KNDGTVUV[NG

CPF

R Q

)GPV\GPUV[NG

CPF

(KVEJUV[NG

DQQMU VQ GSWKXCNGPV PQVKQPU QH YJCV KV KU HQT QPG HQTOWNC VQ DG JGPEG VQ GSWKXCNGPV PQVKQPU QH C

VJGQT[

RM

DWV NGCF KP CNN

FGFWEKDNG

HTQO QVJGTU CPF

EQPUKUVKPI QH CNN VJG HQTOWNCU ECNNGF

VJGQTGOU

QH

VJG VJGQT[ VJCV CTG FGFWEKDNG HTQO EGTVCKP URGEKHKGF HQTOWNCU ECNNGF VJG PQPNQIKECN CZKQOU

QH VJG VJGQT[ (COKNKCTKV[ YKVJ CNN VJGUG PQVKQPU OWUV DG RTGUWRRQUGF JGTG DWV EGTVCKP

EQPXGPVKQPU VJCV CTG PQV EQXGTGF KP CNN VGZVDQQMU OC[ DG DTKGHN[ TGXKGYGF 6Q DGIKP YKVJ KV RTQXGU EQPXGPKGPV KP RTCEVKEG YJGP YTKVKPI QWV HQTOWNCU VQ OCMG EGTVCKP FGRCTVWTGU HTQO YJCV KP RTKPEKRNG QWIJV VQ DG YTKVVGP +P RCTVKEWNCT HQT VJG UCMG QH EQPEKUGPGUU CPF ENCTKV[ EGTVCKP

FGHKPGF

U[ODQNU CTG CFFGF VQ VJG

UKORNGUV ECUG KU VJG KPVTQFWEVKQP QH



Z

l

[

↔`Z[

l

HQT

RTKOKVKXG

FKUVKPEVPGUU

QT QHHKEKCN U[ODQNU 6JG

D[ VJG WUWCN FGHKPKVKQP CU HQNNQYU

l Q

9JCV KV OGCPU VQ UC[ VJCV

KU

R

FGHKPGF

D[  KU VJCV QHHKEKCNN[

l O

KUP V RCTV QH VJG

PQVCVKQP CV CNN CPF VJCV VJG NGHV UKFG QH  KU VQ DG VCMGP UKORN[ CU CP WPQHHKEKCN CDDTGXKCVKQP HQT VJG TKIJV UKFG QH  6JG

Q

R

FGHKPKVKQP

DWV OGTGN[ CP CDDTGXKCVKQP HQT C VCWVQNQI[ QH VJG HQTO

 KU VJWU PQV C UWDUVCPVKXG CZKQO R

↔ R 6JG UNCUJ PQVCVKQP HQT

PGICVKQP OC[ CNUQ DG WUGF YKVJ EGTVCKP PQPNQIKECN TGNCVKQP U[ODQNU YJGP VJGKT UJCRG RGTOKVU #PQVJGT UNKIJVN[ NGUU UKORNG ECUG QHVGP PQV EQXGTGF KP VGZVDQQMU KU VJCV QH WPKSWG GZKUVGPEG

∃ HQT

YKVJ VYQ QH VJG UGXGTCN GSWKXCNGPV WUWCN FGHKPKVKQPU DGKPI CU HQNNQYU

C

∃Zφ Z ↔ ∃Zφ Z  ∀Z∀Z φ Z  φ Z → Z  Z

D

∃Zφ Z ↔ ∃Z∀[ φ [ ↔ [  Z

*GTG VJG HKTUV CPF UGEQPF EQPLWPEVU QH VJG EQPLWPEVKQP QP VJG TKIJV KP C CTG ECNNGF VJG GZKUVGPEG

CPF

WPKSWGPGUU

ENCWUGU TGURGEVKXGN[

;GV CPQVJGT CPF NGUU UKORNG ECUG OC[ CTKUG YJGP QPG JCU CUUWOGF CU CP CZKQO QT

Q

∃φ Z +V OC[ VJGP DG EQPXGPKGPV VQ IKXG C VJKU Z D[ KPVTQFWEKPI C EQPUVCPV E CPF CUUWOKPI φ E 6JG CUUWORVKQP φ E KU

FGFWEGF CU C VJGQTGO UQOGVJKPI QH VJG HQTO PCOG VQ

R

VJGP ECNNGF CP HQTOWNC

FGHKPKVKQP CPF

∃φ Z VJG RTGUWRRQUKVKQP QH VJCV FGHKPKVKQP #P[

ψ E ECP VJGP DG TGICTFGF KPFKHHGTGPVN[ CU CDDTGXKCVKPI GKVJGT QH VJG HQNNQYKPI

∀Z ∀[ φ [ ↔ [  Z → ψ Z

C

+ UC[

KORNKEKV

Q

R

KPFKHHGTGPVN[

D

∃Z ∀[ φ [ ↔ [  Z  ψ Z

DGECWUG C ECP DG FGFWEGF HTQO D CPF XKEG XGTUC WUKPI VJG

∃φ Z 0QVG VJCV φ E GZRCPFU KPVQ C NQIKECN EQPUGSWGPEG QH VJCV RTGUWRRQUKVKQP 0QVG CNUQ VJCV KH ψ E KU QH VJG HQTO `θ E KV KU TGCNN[ C OCVVGT QH RTGUWRRQUKVKQP

KPFKHHGTGPEG YJGVJGT QPG HKTUV WPRCEMU VJG CDDTGXKCVKQP KP

θ E CPF VJGP CRRNKGU PGICVKQP

QT HKTUV CRRNKGU PGICVKQP CPF VJGP WPRCEMU VJG CDDTGXKCVKQP KP `θ E (QT VJG VYQ TGUWNVU YJKEJ TGCF CU HQNNQYU

C

`∀Z ∀[ φ [

↔ [  Z → θ Z

D

∀Z ∀[ φ [ ↔ [  Z → `θ Z

CTG FGFWEKDNG HTQO GCEJ QVJGT CU YGTG C CPF D CPF VJG UCOG JQNFU HQT QVJGT EQORQWPFU VJCP PGICVKQP 5VKNN [GV CPQVJGT CPF GXGP NGUU UKORNG ECUG KU VJG OCP[RNCEG CPCNQIWG QH VJG CDDTGXKCVQT[ EQPXGPVKQP LWUV FKUEWUUGF 9JGP QPG JCU CUUWOGF QT FGFWEGF QPG OC[

Q

IKXG C PCOG VQ

R

VJKU

Z

D[ KPVTQFWEKPI C

HWPEVKQP U[ODQN H

∀[∃Zψ [ Z

CPF CUUWOKPI

∀[ψ [ H [ 6JG WPRCEMKPI VQ GNKOKPCVG H RTQEGGFU CPCNQIQWUN[ VQ VJG WPRCEMKPI VQ GNKOKPCVG

KP VJG QPGRNCEG ECUG #PF YJCV JCU LWUV DGGP UCKF CDQWV C QPGRNCEG

E

HWPEVKQPU[ODQN

H

CRRNKGU CNUQ VQ OCP[RNCEG HWPEVKQPU[ODQNU

O

$GTVTCPF 4WUUGNN U HCOQWU

VJGQT[ QH FGUETKRVKQPU

YJKEJ + YKNN YTKVG CU KQVC ι VJCV CVVCEJGU VQ C XCTKCDNG XCTKCDNG VQ HQTO C VGTO VJCV DGJCXGU NKMG CP PWODGT QH HTGG XCTKCDNGU QVJGT VJCP U[ODQN

CDQXG YQWNF DG

E

Z

PRNCEG

Z

RTQXKFGU C IGPGTCN PQVCVKQP

CPF C HQTOWNC JCXKPI

HWPEVKQP U[ODQN YJGTG

P

Z

CU C HTGG

KU VJG

+P VJKU PQVCVKQP QWT EQPUVCPV QT RNCEG HWPEVKQP

ιZφ Z YJKNG H [ YQWNF DG ιZψ [ Z +P DQVJ Z EQWPVU CU C

DQWPF XCTKCDNG %QPVGZVU EQPVCKPKPI ιVGTOU CTG GZRCPFGF CU KPFKECVGF CDQXG %GTVCKP CDDTGXKCVQT[ EQPXGPVKQPU CRRN[ URGEKHKECNN[ VQ VJG FKURNC[KPI QH NCYU

CZKQOU CPF VJGQTGOU QH C VJGQT[ 6JGUG CICKP OC[ DG KNNWUVTCVGF D[ VJG ECUG QH KFGPVKV[ YJGTG YG JCXG VJG NQIKECN NCY QH

KPFKUEGTPKDKNKV[ QH KFGPVKECNU

YJKEJ KP OCP[ VGZVDQQM

RTGUGPVCVKQPU KU VCMGP CU C NQIKECN CZKQO CPF KP QVJGTU CRRGCTU CU C NQIKECN VJGQTGO +PFKUEGTPKDKNKV[ DG UVCVGF CU HQNNQYU



Z



[

→ φ(Z ↔ φ [

1PG QH VJG EQPXGPVKQPU KNNWUVTCVGF D[ VJKU YC[ QH UVCVKPI VJG NCY KU VJCV VJQWIJ YG CNNQY QWTUGNXGU VQ URGCM KP VJG UKPIWNCT QH TGCNN[ JCU JGTG KU CP CZKQO

UEJGOG

Q R VJG

CZKQO QT VJGQTGO QH KPFKUEGTPKDKNKV[ YJCV QPG

OGCPKPI C TWNG VQ VJG GHHGEV VJCV CNN HQTOWNCU QH C

EGTVCKP HQTO CTG VQ DG EQWPVGF CU CZKQOU QT C VJGQTGO

UEJGOG

OGCPKPI C TGUWNV VQ VJG

GHHGEV VJCV CNN HQTOWNCU QH C EGTVCKP HQTO CTG VJGQTGOU 9JCV KU FKURNC[GF KP  KU PQV

Q R VJG

NCY DWV VJG IGPGTCN HQTO QH CP

KPUVCPEG

QH VJG UEJGOG YJGTGKP

φ OC[ DG CP[

HQTOWNC #EVWCNN[  FQGU PQV [GV HWNN[ FKURNC[ VJG IGPGTCN HQTO QH CP KPUVCPEG QH VJG NCY UKPEG KV KU EQPXGPVKQPCN KP FKURNC[KPI NCYU VQ QOKV KPKVKCN WPKXGTUCN SWCPVKHKGTU 6JWU YJCV CP KPUVCPEG TGCNN[ NQQMU NKMG KU VJKU

C

∀Z∀[(Z  [ → φ(Z ↔ φ [

#PF CEVWCNN[ GXGP C UVKNN FQGU PQV [GV HWNN[ GZJKDKV VJG IGPGTCN HQTO QH CP KPUVCPEG QH VJG NCY UKPEG KV KU EQPXGPVKQPCN KP FKURNC[KPI NCYU VQ QOKV XCTKCDNGU VJCV OC[ DG RTGUGPV 6JWU YJCV CP KPUVCPEG

D

∀W

W∀

∀Z∀[(Z  [ → φ(Z W

WM

W



WM

RCTCOGVGTU

TGCNN[

QT CFFKVKQPCN HTGG

NQQMU NKMG KU VJKU

↔ φ [ W

W



WM

.KMG CNN VJG QVJGT EQPXGPVKQPU VJCV JCXG DGGP FGUETKDGF UQ HCT VJGUG EQPXGPVKQPU CDQWV FKURNC[KPI CZKQOU CPF VJGQTGOU QH HKTUVQTFGT NQIKE YKNN CRRN[ CNUQ VQ JKIJGTQTFGT NQIKE

6JGUG EQPXGPVKQPU JCXG DGGP DTKGHN[ FGUETKDGF JGTG + UC[ DGECWUG PQV CNN VGZVDQQMU EQXGT VJGO 6JGTG KU QPG HWTVJGT KORQTVCPV VQRKE VJCV XGT[ HGY VGZVDQQMU EQXGT VJCV QH

HKTUVQTFGT NCPIWCIGU CPF VJGQTKGU # OCP[UQTVGF NCPIWCIG KU LWUV

OCP[UQTVGF

NKMG CP QTFKPCT[ QPG GZEGRV VJCV VJGTG KU OQTG VJCP QPG UV[NG QH XCTKCDNGU (QT KPUVCPEG KP C IGQOGVTKECN VJGQT[ CDQWV RQKPVU CPF NKPGU KV OC[ DG EQPXGPKGPV VQ JCXG QPG UV[NG QH XCTKCDNG : ;