Kutyłowski M, Logika w5s

! !"#

$&%&')(+*-,/.102,/(+3542%&6879,;:=<&*5>@?

ACB=DE(GFIH1JLKMKONQP;RTSVUXWZYW\[^]O_à\b+Sdcfehg2i

jk2l1m5nGoqpsrtvuxwzy{}|~wM5{

SisfP;isRcfehsSERTs;]SPOieT]s`gS;g2Rh_`acfi _X¡zb¢S£sSV2`

}sss_¤dcW}¥¦fs_qVq§!_àMq§¨Sg2RhsTSV;Zeh©RTsV}iUª];OgVW-«TcfissiZ_àL¬S£\fc#`LRT¤£s;i

P®cf;°¯±v²z³X´W-µ¶P;RT`Lg;bªVO]@·VS£}_hSf¸2`L&acfi z_¡}iss¸!\_h];RV`Lq§cfRhV}}Uª]O;gVXTVi;SM;dc¹V;i

i;S£s;iSVRV)RTsVT©#;isRT¤cfOShiG·SRV)e_qLb `Li 0· 1W-º}RTs!P;RT`LeT_©LPOi;i¬;Sf;»;;iXiUªSRh!]®b ¼¶RTssO]"\P½SM©_hgL]¾];e_qVV!`QTgLieT`L¨¸½S¸½s;i!`];£`2c°VTVgVSeT`L!¸®SRhsVXiG·¿Uª];;giÀ·

eT_Lb¢`Lq§ï2isO2`Lq§@·OP@W

R1, R2, . . .· f1, f2, . . .· c, d, . . .· x, y, z, . . .W)Á@eV;OisSg£s\fgVScZs;XT

se_#;Sc¹SS\M·OcÂP;RVg_àLs9gS_gLe_]c#`L;igzEP;RT`LP½SRhs©V;gVSdc¹V;i!e`L¨¸½SSVÃP½Sehss¤`Mq§

Rh¤ÀWIÁ@cfehf;\VgQ]O£`2c°!`Rh VXi

= ;Ss;Vss;iRT¤cfOShi8_hsRT¤dcW

•

k2wzy Å"O»\;i]!`iO;];gV`d;iÆ

gzV£sM`eT`L¨¸½S8eT_Lb+X°i¿gzV£Òe`M!¸®SV5i;;1seT_f_hsRh·

sTi

e©_sRTViG·

t1, . . . , tk

f seT_e`M!¸®SVÈUª]O;gVXiG·\_hS_hsRhVi;se_Rh¤dcf;is£c#`MRhzÉ

£s;i

f (t1, t2, . . . , tk)W

• Á=SL;;i_q©Ê;»\;iT© Qe`L¨¸½SiIUª];;gi sin, cos, + SV£s\ ¸;]O;Sdc°V_qVg2iQP;Rh`Lg\b+V;SdcZ

_Rh`fTg

sin(2+cos(x))W-ËsVe;siV·SL;;iZZ;»\OiXT©_hsRh¬_hs¨P½SdcfiOi¸2`LZVP;ie2`

TVgS

sin(+(2, cos(x)))W

•

c°VOÆ8;i}sV£©V;Vih`^c;»\;ii2i;;];g`d;5¸`gzV£Ò^}eT`L¨¸½SiUª];;gVXi2iLbMSg2RhhS\©

¤VRT`Íis¸®VRT];s_h¤cWÎ¥¦isg2isÒÏ_qVg ehiÐ¸®Sdcfis

s`L;i¦ÑTgVSP;RT`Lg;bªV S£s_]

P½Seb¢];£`LeT`L¨¸½S

max·\g_¤R`Ss;Vss\XcfigLeT`VRhV];s_¤dcW

•

uxwzy¾{)|ÓÅ;»\;i]s`i;;]OgV`d;iVÆ

sTi

e©_RhViG·;_hSUªSVRh¨]½bªVi@eh©;;Sdc¹SV;Se`L¨¸½S]Rh VXi Ç

t1, . . . , tk

RÆ

R(t1, . . . , tk)·

t1 = t2

sTi

φ SVRV ψ eh©UªSRT¨]®b+ViG·M_SUªSRh!]®bªVi@eh©RT¤dcf;i£

φ ∧ ψ · φ ∨ ψ · ¬φ· φ ⇒ ψ · ∀x(ψ)· ∃x(ψ)

x seT_isO\©OW

•

¤dcf;Sdc°V£\;SV_VTOÆsVi Ve_gc°V_à2»\gz_hSR

∀ ÓUªSRQVÖ¡×]O£`2c°eTiÊ_hs£

·fsVi Ve_

∃ vØMieT_ea¡×f]O£`2c°eTi

¤£sOë©Rh¤dcf;is£gVSV2cZsOeT_cfi VOi Q\cfiVeh¤dcW¥¦z¸\VRT;si

sO;Ss;Vs2`¸`\b+¸`VP;ie

(∀x)(φ)W}Ù¹sh9\cfiVeh¤dc SPO];ehsEeTiSï¦gVS_hsg2eT_h]SV£s\

SL;s`L_VV·M;if_¦\cfi Ve`P®Sdcfi;`¸`LVWZÁ=¸`2_¹;]O£¨is¸\!\cfi VeT¤dc SM`2cfiTsi^¸\VRT;si

P;RTsseTsgzV;s!;i£#P½SV9SPO_àLs;i¦RTSsP½S\VV·cTVg2i;ehP½SeT¤¸UªSVRh¨]½bª9sSVeT_q2bª^s¸;]O;Sdc°V;OW

•

wu½Û-ÜªkMy Å)Ý1i_RhP;RT_VT9UªSRT¨]®bx¹g2c¹V_à2»\gz_SVRVi;\P®SV_àLgz9\9P;RTS¸;s`W&¾`MVV\

se_SeT_hRhS£;ShWIº}RT`Lg;bªV=Æ

∃x(x2 = 4) ⇒ ∃z∃x(z2 = 4 ∧ x2 = z)

´qàáßâ

y ´q¯±)ãzäI²åVæT²V±X´èçé±Xê

´q¯±&ãzäI²åVæT²V±a´çéáßâÓëhìdísîQåLêså²V±Àï\âÓ´

×W

ÕßÞ

º)RhsP;RTSc¹V;iih!`ð;VeT_hsP;]T©scf;iSehgSdc°VOiÆñV!`

x2 = y SVRV z2 = y WòÁ@z_s

x2 = z2 Wôó¶ x2 = z ·cfi¾SVeT_qz_ss;i z = z2 WÎ¾`L;igzLb¢S¸`õeT_©V@·£s z = 0, 1·fsS

V¸;eTS]O_;i;i!V;seTi^_àL·\£

z2 = 4W

s`LcfihiSV_RT`LV\ehPORhss;Shc#`L;ig\b+;iUªRhVehSV¸;i÷cZShi}cÎi2_hsRTP;Rh_qVXi)g2c¹V_à2»OÉ

gz_SVRh¤dcW

¤\VehV\OÆ

sis;OcfP;RTSdc°VOV;gc°2_à2»\gz_hSRhø_SìsçâÓ´ï\ï=Éê!ìsï½²Lëhìz´ï\â+ùá êdãV²Váßï\îzçW

ó¶¸`;SVg\b+V;;i¿s;»\;iSc¹V°SsS¦q§;SL;si2SLOc¹SLb+]!`ehiZOS¶P½SsIVeTi]¨gc¹V_à2»\gz_SRT¤dc ifgc°V_à2»\gz_hSR"cfiV£©ssVS\V;c#`LeT_q©2P;is;iÐsis;OWñúZScfis

_q¾eV¾si;\"c

Rh¤V£s`Lq§ÊiXehsVq§ÊUªSRT¨]®b `;i!];ehi8S\VV^_SeVS;û

•

r\ýnªLþ"ÿEr\mÅ=l1ÿMrduxwrÊ;»\;i]!eTii;;];g`d;iVÆ-hi8cñSg2RhsTSV;#UªSVRh¨]O^c#`LeT_hsP;]

gc°2_à2»\gz_hSRs·M_S¨P;_RT`L!`c g2_h¤RT`L

Ss;i¶;»\OiXi=i;;];g`d;¹sSVeT_q2b¿¤dc gc°V_àÉ

»\gz_hSR&;SPOiehV`W^OÈ;SVP;iehViT!`S^;S

φ·_S^sVehisis_hsS^gc¹V_à2»\gz_SRh#eT_}c9bªh;i

φW

c¹VOÆ5ehis°se_)cfi©V2È°Sg2RhhS`MÎc#`Me_q©2P;is;is

φW&µ¶9P;Rh`Lg\b+V;]@·c

UªSRT¨];

(∃y∀x(x = y)) ⇒ (∀x∀y(x = y))

UªSRT¨]®b+

x = y c#`LeT_hsP;]Ocf]OgLRTSV_Oi·) gzV£sM`M

Rss

eT_hSi¹PORhs;i© g2c¹V_à2»\gz_SVRsW

Á@VeTii

P;isRTcfeTsSÏgc¹V_àL»;gz_SRh

∀x eT_P;isRTcfeTs¾c#`Me_q©2P;is;i x = y ·¦VeTii

gc°2_à2»\gz_hSR

∀x ssTsi8UªSRh!]®b¢`P½Ss\gL]iPOigzViOeT_;Rh]Oi^c#`LeT_©LPOi;i x = y W

•

ynÀkLm5m5r

x pskLýEu½ÜGm8rc/\V`L

iXeh]UªSRh!]®b¢`V·shixiXehsf_S;i¦;V£`;Sehis]

£;;sVS"gc°2_à2»\gz_hSR

∀x· ∃xWÏ

PORhssi÷cf`M

c#`LP\V;g2]/i;\

x seT_ n Or\m5r=W

[cf;_à2»\gz_SRTs·8g_¤R`cfi©V£s_©Qsi;\©¨se_\z¸;i£seh`Ðgc°2_à2»\gz_hSRP®SVeT_qsi

∀x· ∃x

P½ScZ#eT_RTS;iSMiXehsO·\cñg_¤R`LÃc#`LeT_P;]!ScZS

• º)Rh`Lg\b+V@Æ sin x = z ∧ ∃x(x2 = z) sin x se_¦OSM;_;i+×qW

\SRh!]®bª_qeTg\b+V\QeTiE¨Oc¹¤Lq§ÐssTsiÀW¨º)isRTcfeT¤dcfi1£s!c°VR_STsi©

sin(x) seT_ z ·@;RT];

s·)£s

z eT_gc¹V;Rh_«P½cf;XEi;;X¨iss¸`VW - is¸\ S\VssV\eT_RT¤cfOi£P;Rhs

x Ñsis;\_qs\zO]ehi¶s;;VgcÎVeTi]gc°V_à2»\gz_hSR ∃x QP;RhsE_SOiQ;i

cfehP½¤;sVSi;\©cZS\©

x P;isRTcfeTsXfsshiGW

•

VgõeTi¾];e_Rhs¾_sVS _àLP;]

P;Rh`MP;V;gV¤dc

¥¦zP;RTSeT_hehÍRhsV]®bª

¤cfiÀ·^£scfP;RTSc¹V;szT©V

gc°2_à2»\gz_hSRT`gVSVRh`Le_qV`Ê!eTP®TV`Mq§Ði;`Mq§;is]O£`2c°2`Lq§Ðc¬i;`Lq§Ðsshi Vq§

UªSRT¨]®b `WC¥¶P;RT`Lg;bªV

SV£s\

gzV£sÒ/g2c¹V_i÷»\gz_hSR

Qx Ss\s`L¾iO;sg2eh

i9_sÂe

i;;sg2e-;SP;ieVI;Sf_àLq§!c#`LeT_VP;i

x·zg_¤RTZeh©fcfi©V;ZSdc#`Lðgc°V_à2»\gz_hSRhW

cfiV\

UªSRT¨]®b+PORh`L¸;is£¨P½SMOeT_qVÆ

sin x = z ∧ ∃1x1(x2 = z)

P½S¨PORhsi VOSc¹V;i]sis;`Lq§S£s\!SM`2cfiTsi9i;;sg2e°PORh`gc°2_à2»\gz_hSRh9SP;]Ohsiz×W

m5r;okMm5nªk ÿEr;mèÅ@l1ÿMrduxw

• Á@¶gzV£ÒL

RhVs·VÒ¨VÈSg2RTsehifc¹VR_ST°SVis\©^Sg2RTshS;X1UªSVRh¨]½b¢`!¼5RTs;]];eT_Vi

\Vs£`sVRh¤dcf;SÊvTcfi_¡q·)g2_h¤RhSÐS\;SV_àLs`ÊTVg¾i¹s;Vss;isis;`Lq§ÍcZS2`Lq§@W"¥¦i

ehP½SeT¤¸eT_¸®Sdcfis

¤dcfiSc¹VR_STsi}SVis;X\P;Rh`Lg\b+VUªSVRh¨]½b¢`

∃x(x2 = −1) ¸½s

Sg2Rhhs;iO·°`

q§;SL;iS¾is¸` Rhss`LcfieT_hV·#S¾is¸`ÍseTP®SS;·#ÐS£sseTssÊS¾iO;

ss2_àVW 1VgÏeVS

;i¾eTP®SVeh¤¸ P½ScfisOsisV·^`/UªSVRh¨]½bª

x = 1 seT_PORcf;si÷c¹Í¸½s

];e_qVs;ic°VR_STsi8;

• º)isRTcfeT`L

gLRTSg2i

se_¶];e_qVsOiSM;]=Æ¹s¸OiSRT]s_¤dc

`Li}ùLï\â+ä¹´æ¯ùMç×¹_Vg2£s

Rh iÀ·&Uª];;gVXi°iIsss_¤dcÃ];;i÷c¹Rheh]O

SMOP®SdcfiV\zT©V`Lq§ ]O£`2c°2`L

e`L¨¸½SSV«Rh iÀ·

Uª];;gi8i@e_qLb `Mq§=W

•

r\ÜÀ{5r\oqp ;V`2c¹V!`;SdcZS\©Uª]O;gVX¦P;RT`LP½SRhs©V;gVSdcf]T©©!i;`M

s_à];;i÷c¹RÉ

eh]OW

•

r\ÜÀ{5r\oqp

u yu2m8rwzux2ýsk2wÅm5r8uEu½ÜÀm5kdk2wzyÐÅ

_VgzRTSseTsRhsSV\©!c¹V];VSs;zÉ

s`L!`P;Rhs

U ×qWIµ¶»\;iTeT_i;;];g`d\OÆ

;ëT_Lb+X

c·LP;RTss U (c) S\VssV!`sss_#]O;icZsRTeh];

c#`M¸ORV`QO

c c P;isRcfeh`L

g2RhSg2]@·

;si;;X

x c¹VRT_hSh U (x) = u(x) se_cfeTgzVV;P;RTs!SR`Li;V;©c¹V]\V·

U (f (t1, t2)) se_¨;»\;iSdc°VOTVgS F (U (t1), U (t2))·8;i F eT_Uª];;gT©O·5g_¤RheT_

S\VssV\EP;RTs^eT`L¨¸½S

f W1¼a\Vs¹¤dcfi ©V·;\zPOiRTcÂc#`Mis!`Qc¹VRT_hShRh];MÉ

_¤dcÂUª];;gi5P®SV_hsé;Vg;bªV\!`\;i^Uª];;gVXV·xg_¤RT°eT`L¨¸½Ss

se_

f W

•

r;wduonªuEr;m-nªkUªSRh!]®b5¼}Rhs;];»\Oi]!ì;;];g`d;iVWIVR_STsiSdc°VOi^_SsV£`SL

c¹V];ViÀWIµ¶Sg;bªVO;iX·x; Ec¹V]\Vi U Æ

\SRh!]®bª

!c¹VRT_hSh

V]c¹V£s`·;£

t1 = t2

T ·hi=ss2_aÙ (t1) i U (t2) e©Rh¤dcf; Õ

£`Q_hSSLc°V]\VXi

U SRhVLb¢S£sSi2_hsRTP;Rh_qVXi¿eT`L¨¸½Si8Uª]O;gVXi5i¿e_qLb `Lq§@û÷×

\SRh!]®bª

R(t1, . . . , tk) c°RT_SVh T ·-hi¦_aÙ (t1), . . . , U (tk) e©cÈRhsVXi S\VsS;XEe`M!¸®SV

R P½S;Sdcf;iV·}c#`Mis!`

s;Vss;i_Rh¤dc

RhSc°£V`L

Tcfisii¿¸\V;V!`èh©SO9cñRh iÀ·\g2_h¤R©P;RT`dsiT!`;

R×qW

µ¶UªSRT¨]®bv·1g_¤RTP½SdcfeT_q2b¢`ÊP½SSMs©eTP®¤Oig¤cCSis2`Lq§

φ i ψ ;»\;i]s`

c¹VRT_hShSiss;©V;VSVis;i°TVgQ;EUªSRh!]®b-s\;iSdc#`Lq§@W

¼aeT_hSV_`Lðss_ eT_-s;»\;iSdc°VOi°c¹VRT_hShiMO fUªSRh!]®b;Igc°2_à2»\gz_hSRiÀW)µ¶

UªSRT¨]®b `

∀xφ P®Se_P;]s`\Ve_sPO]T©VsSOÆ

;g£s;Ss_]

s ];;i÷c¹Rheh]OÆ

∗ cUªSRT¨]; φ isOi V`!cfeT`LeT_hg2i¹cZS;Zc#`Le_q©2P;i;i x P;RhsZeT_q2bª©fSs\sT©V©

sss_

s·

∗ S¸;iV`c°RT_SVhSVis\©SV_hRh`LV;X#UªSRT¨]®b `;c¹V]\VXi U W

sTi®¨gzV£ÒL

RhVs

SV_RT`LVih`c¹VRT_hSh

T ·2_S!UªSRh!]®bª ∀x φ !c°RT_SVh T ;

c¹V];Vi

U W

P½SM;SV¸;;iP®Se_P;]s`¾;

∃x φÆ^Rh¤£;iP®Ss;_àL·-£c°VR_ST T \V\s!`V·

Ò;q§;SL°s;OsS

s SV_hRh`LViT!`c¹VRT_hSh T W

• Á®cfRT¤M!`]c°VVV·;£¦cõ_RhVgVi¦c°VR_STsiSdc°VOi ÜªSRT¨]®b@¶gc¹V_àL»;gz_SRhVi=sis;OZcfi©zÉ

;Oi\V\s!`c¹VRT_hShi)c#`MOigzzT©V`Lq§c¹V]\ViÀ·1sRTSc¹V£sV!`c gVS_hsg2hi

gc°2_à2»\gz_hSRM·½g_h¤Rh°^cfi ©£sVW

8r\m5nÀr pskL-m®Å¿y

ÿEr\mÅ=l1ÿMrduxwkMy

º-SL;S¸O;i¶Tg;QRq§L]O;g2]\ @·@ieT_h;iT©Q_qVg2iEUªSRÉ

¨]®b `¼¹RTs;]@·\g_¤RTe©ehP½zb¢;iSV;; ;SdcZS;#c¹V]\VXi¿i@;Sdc¹SV;SSL;]É}s`Li¿_q]O_SSiVW

s;;Vgi;Vs;i£cøP;Rh`LP\V;g2]ÏRhVq§2];;g2]/\ ÏOise_S£i÷cZÊehPORcf;s;iÊ_S_SL\©

sRhSÉsÒL;gSc¹©OÆIcñRT9c¹q§;SL;i5OiehgVS ;ss;i¨cfis!SL;si¿i@c°]\iÀW

• P;RT`Lg;bªV=Æ ¬((∃x)φ(x)) ⇔ (∀x)(¬φ(x))W

sRT`2»\gzVXTO·¹£eT__S_q]O_SSiO·}SP\RT_qQse_S;iVRhVW

`Leh]zs!` cfi _sÆs_à

Sdc¹SQcfi _;i ©eTi¨\OcfiEgz_hsSRTi^ÉZ_¨g_h¤Rh!ehP½zb+Oi zT©

φ i¿_!g_¤RT¨;iëTP®b+;izT©

φW

c¹eT_hRhS;Rh¤dcf;Sdc¹V£sOShi1c¹VRT_hSh

T ·½VÒ (∃x)φ(x) c¹VR_ST F WfÁ@_hsø\

P½SM;e_qcfiO»\;ii\c#_hsÒV·MM`¨assTfTcfi_q¡fsss_¤dc eTP®b+;izT©V`Mq§

φ seT_IP;]OeT_qMW

P;Rhdc¹eT_hRhS\RT¤cfOSc¹V£;Shi}Qc¹VRT_hSh

T ·¿ÒÊ;gzV£;sVSs_] s RhSVc¹zÉ

£;sSaTcfi_¡QUªSVRh¨]½bª

¬φ(s) Êc°RT_SVh T ·&s`Li¹UªSVRh¨]½bª φ(s) Êc¹VRT_hSh F W

Á@_hs

c#_hsÒV·xÒsLb¢`vTcfi_¡^ehg;bª\eTiss2_h¤dcð;ieTP®b+;izT©V`Mq§

φW

YiS;·®£!c¬SV¸;]ÊP;RT`LP\VOgq§ÐSV_RT`LVih`ÊRT¤£s`Li-ORhSi8RT¤£s;i¨¸ORhsi©Vsc°Rh];;g2iÀ·

e©S;^RT¤dcf;Sdc°V£;V·®_hsøRTSc¹V£\UªSVRh¨]½bª¶se__V]O_hSSVi ©MW

º)Rh`Mg;b+VÍ¸`\bf_R`2cfi V`"É\eT_P;;SgzV]T© ehi;i_Vg¾SM`2cfie_i¹VehVi#e© ¸®b¢s;;i

RhSVs];iV;P;Rhs ig¤cW

• P;RT`Lg;bªV_qV]M_SSiiGÆ ¬((∀x)φ(x)) ⇔ (∃x)(¬φ(x))

• P;RT`Lg;bªV_qV]M_SSiiGÆ (∃x)(φ(x) ∨ ψ(x)) ⇔ (∃x)φ(x) ∨ (∃x)ψ(x)

• \V;i (∃x)(φ(x) ∧ ψ(x)) ⇔ (∃x)φ(x) ∧ (∃x)ψ(x) ;i#eT_f_q]O_SSi©

P;RT`Lg;bªV@Æ

φ =aieT_h;iis¸\g2_h¤R9se_;SM;_;i¡q·

ψ =aieT_h;iis¸\g2_h¤R¶eT_P\VRT`LeT_¡èW

P;RT`Lg;bªVOsi¨_àL

PORc°eT_RTS\^seT_¶P;Rcf;ic¹O·=c¹UÀLb+eT`2c¹

;i!is¸2`V·@g_¤Rh

se_)s;;SLssT;i!P\VRT`LeT_i¿;isP;VRh`Me_qL×W

• P;RT`Lg;bªV_qV]M_SSiiGÆ (∀x)(φ(x) ∧ ψ(x)) ⇔ (∀x)φ(x) ∧ (∀x)ψ(x)

º)Rh`Mg;b+V@ÆIP®SOScfOi;isq§

φ =aieT_OiX!is¸\g_¤Rh¶seT_;SL\_h;i¡è·

ψ =¡XieT_h;iis¸\g2_h¤R¶eT_P\VRT`LeT_¡èW

PORh`Lg;bª;si¦_àL

c¹!eT_hRhS;#eT_IP;Rcf;ic¹

g£s\is¸\!;_];RhV\#eT_IP\VRT`LeT_!];¸

;isP\Rh`Le_qL×·OP;Rc¹UÀLb¢eh`Lc¹

8r\m5nÀrð !Eu½y"r¬ÿ¨r\mèÅ@l-ÿOrduxwr\yn

ó¶¸èhiÊSRhis_Sdc¹VÊcÈ\Vss;i]/\ /g2igVS

Rh¤£;isisO`s;Vss;isÈRh¤V£s`Lq§Ê¸;SgV¤dc¬gc¹V_à2»\gz_SRT¤dcW

• UªSRT¨]®b+E_àMPO] ∀x∃y φ(x, y)·

P;RT`Lg;bªV@Æ

∀x∃y y3 = x·;_GW1v;gzV£;°is¸`ieT_h;iP;isRTcfiVe_sg#"eT_hSP;;i¡qW

µ¶iVRhV

;¾Æ

xÉvì yÉvgLi1RhsPORhss_]zs!`P;RTssSMO;sis;Sc¹VVW

¾gzV£sOsSsss_]

x R`Leh]!`eT_RTLb¢gV;Ss_]

_Vg2iS¸`sVq§;SL;si+b+S

∀x∃y φ(x, y)WIº-SdcfeT_V¨!];ehi

RT`LeT];;sgx·½\g_¤R`MÃg£sÒ_

x EaP;VRT_h;sRh¡9c

s¸;iSRTs

yÉvgV¤dcW

• UªSRT¨]®b+E_àMPO] ∃x∀y φ(x, y)·

P;RT`Lg;bªV@Æ¦ie_;isOiss2_h];zXcfig2ehsS;Æ

∃x∀y y ≤ x·@ieT_Oi;i_]Vg2eT`LV÷É

;S;Æ

∃x∀y ¬(x < y)

}`MøRhVsé;iVRhVéP®Se_qV9viSV_hzb+g2iÖ¡èÆ}SLP½SÒL;ssSsss_]

bªss;i;Scfeh`LeT_g2iq§Ðsss_h¤c

y W

• Á@Vs£;STVÆ ∃x ∀y φ(x, y) ⇒ ∀y ∃x φ(x, y) P;RT`Lg;bªV@Æ

φ(x, y) _S x ≥ y

c¹VMÆ1iPOigzVTSMOcfRTSV_;Oi)se_ZeTP®b+;iS\Mû1`Le_qVRTs`cÍ_àM

s]RTSsP\z_Rhsis¸`

2b+gVSdcfi÷_Æ¹;Oiq§ÊUªSVRh¨]½bª

∀y ∃x x ≥ y se_¶eTP®b+;iS\O·x\z_SiVeT_fUªSRT¨]®b+ ∃x ∀y x ≥ y

;i°seT_eTP®b+;iS\OW

u½wzy{)|ÓÅ #wzkLy"rÐÿEr\mèÅ@l-ÿOrduxwzr;yn Ò

• ∀x ∃y ∀z (y|x ∧ (z|y ⇒ (z = 1 ∨ z = y))) UªSVRh¨]½bª_qSs\sagzV£\iss¸;;iOig Õ

P;isRcfeh`¡h×

\VR`Leh]°;i RVÃ; ;iû

• ∃u ∀v ∃w v = uz mod 7 vie_;i;sRh_SVRRTseT_SM;]OS%$d¡×

\VR`Leh]°;i RVÃ; ;iû

eT_ie_SV_h;i!RT¤£s`SLP®SPORhs;;isSOW

&u'dr\oqpr

Ù¹seT_hSP;iehs`

∀x ∈ A φ(x)

TVgVSehg2RT¤V_;

∀x(x ∈ A ⇒ φ(x))W

Á=¸\V\ih`

UªSRh!]®bªOW

gs]ehiV·)£sie_SV_h;eT_E_àLgVSÐ`

φ(x) eT_EeTP®b+;iS;; xÉa¤dc

AWÊµ¶ÊP½SsSVeT_q2b¢`Lq§

xÉ

¤dc

P½SP;RTs;;igiP;igzVi8se_E;iseTP®b+;iS`·IÊs_

iP;igzVTseT_EehP½zb+OiS;OWÐÁ@_hs

ehg2RT¤V_

SMOP®SdcfiV\\VeT`Løi_h];iXSW

ynªr\m8rr\ýnª2þ®{ ÿ¨r\mèÅ@l-ÿOrduxwr

gc°V_à2»\gz_hSROW9ÁRT];¸;ehseV\_q¤dcfiÀ·=£¨gc°2_à2»\gz_hSR9S£\P;RTssOSehiE\Vsshi iÀ·

Q Ss\sEOScZS`gc°V_à2»OÉ

• Qx (φ ∧ ψ(x)) ⇔ (φ ∧ Qx ψ(x))·;Si x ;i^c#`LeT_P;]fTVgVSsis;;cZS\c ψ

• Qx(φ ∨ ψ(x)) ⇔ (φ ∨ Qx ψ(x))·\Si x ;i^c#`LeT_hsP;]zTVgSsi;\c¹S\c ψ W

wkLm8u®ýkMm5nªk m8kLþxr\oqpsn

¬ (∃x φ(x)) ⇔ ∀x ¬φ(x)

ehP½zb¢;i;i"cZiP;RcZ

eT_hRhS`½Æ¨c¹e_RTS\Ss\sM·-£;i\eTiehP½zb¢;i

∃x φ(x)·1_SS\VssO·5£scfeh`Le_g2i

φ(x)W

º-SP;RhdcZ°eT_hRhS;iVP;ie;°seT_s·\£sgzV£sÒQsss_¦ehP½zb+Oi

¬φ(x)W

¬ (∀x ψ(x)) ⇔ ∃x ¬ψ(x)

_à2»\gz_SVRh¤dcWI¥¶P;Rh`Mg;b+V

¬ (∀x ∃y ∀z φ(x, y, z))

⇔ ∃x ¬ (∃y ∀z φ(x, y, z))

⇔ ∃x ∀y ¬ (∀z φ(x, y, z))

⇔ ∃x ∀y ∃z ¬ (φ(x, y, z))

(

Þ/

ìsçâÓ´ï\â6+^ä1¯hìsîd¯±+ãzâÓ´^ãzäI²Vï\±Gî10&ãV²V±XêzæèîäÂísá êdãzùï½²)½æTìz´qëâ+ä)ï=´W

• -Vg2iP;Rhseh]O;isi;siIÊ¸;Sggc°2_à2»\gz_hSRh¤dc

];S£si÷cfi sP;eTsi_];i`d;RhS]MÉ

i;iV·xsS;sSdc¹V\UªSRh!]®bªEcñieT_hSMiS\VOW