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1. EFQ: Ex Falso Quodlibet2. Equivalence of Contractins3. Disjunctive Syllogism4 EQT: Ex Quodlibet True5. De Morgan Negation6. Boolean Negation


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9. EFQ: Ex Falso Quodlibet10. Equivalence of Contractins11. Disjunctive Syllogism12 EQT: Ex Quodlibet True


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17_(D/Mendelson, Elliott, 1964, Introduction to Mathematical Logic, London, Champan & Hall, forth edition,1997, p. 34.


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D-$D,("HK+K*`'H("1L_"("yx+y"H_Rxy+Ax+gdy+]v"B+$gd+0y$A'(n9;=hR(

gd`D0B A+B Bx(!0]v"iB A$]v"8(o_f0"y"DRxy-DAy$gdBy+$gdB B$]v"

_f" 8(oy$ A'(j=` A D#5 8_ 0A ( B A )+A ( B B )x#(_,+$gd9rxRgd=L_H!D8_(

H0,,3!l+H`+8_)A`E=

&'(^"+mf(o0D^_#3!l.03!l_f$/R=/8_$/;8_f8;K*1

( A C ) ( B B )

,.B B+A C$Rr7"=^_#&'(`8_D`58_

+b` D9; - _f $ 3!lA C+`9;D,9;,rRgd_}-+0ZA'(B B0Zgd,r=

$$!$'/$lb^"+m8_`'.&'(L!R5Q0$=/"4rfDlb+$!$'+$lbL_"f5,

'/$$!$' Da/!5D$lb+ "DR0Zlb =hRgd$L!/$8_QRolbR_=DH!-9;+&'(",(oAx$gdA0y"DRxyRgdD=H0ZubD*8_"1

A Bx$gd0y+z"DRxyz-DAy$gdBzRgd=

(lH0t(_,+D4(_DH_flb$'8_"H0,H`0A ( B A )+A ( B B )3!lL!/$8_1L_"("yxD

y+z"H_Rxyz+Ax+DBy"0$gdDz(!gd=(


92/237

gdhR`D0B A+B Bx(!0]v"i0"8(o_fy+z"DRxyz-DBx$gdAz0"8(o}+Rgdy+z"D

Rxyz- DBx$ gdBzR gdj=` A D#5 8_ 0

A ( B A )+A ( B B )x_,+$gd9r#(xRgd=H0,,3!l+H`+8_)AÈL_H!D8_(=

/D`5R!-8;K*$L!/$_fR0Z`'lbÈ8/0R!-)AK*D?3*(("yH"HA C+BDD

x+y+(RgdBzR]v"=D,(",9;;+HH(";.&'(A'(n01"kAÈ

À ( B B )9_"H_RoDH_""v(l_A'(`8_B B

xH_]v"=.&'(L!/$(_04(_"!R!i9;+&'(+`nj,9;/0DD_};D08_A'(,9;L_;+(RA'(n(

,9;_$H0,,r &'(_k*gdH!$faL!/$D/0D0(((RA'(n_A'=,9;(_;D.&'(L!/$D+8_f0

(_0/,rD_}-A'(,9;L_D(("8l/DA'(n0_g"i=qZ/4D.&'(0,9;fD_}-A'(0/,rP

"=D_,!$("S;U/;:N=j

61$7%

$d%

.&'(L!/$R!Z$DiFramej$L__(S,g,R)3#n,r"$F,(0=!Z$};fL_0ub,("H_fm1

S

1@?74L&/Sl* _lb+l f$3isituation, set-upj"n_8/(

(R,9;};_,9;+D8/=lb+/43#n0K,#+(0,r/H$(0/0SH_=


93/237

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g+,_`6L5;+DL!/$8_,9;L_?*+,,rA'(_k*,!$fa

"3{=3#n,9;8_g(=iL!/$,9;(o0,rf_DA'(0

t#+_}-,gj="$ub+gY#+$lb+L_}gS=R@%+6LR'lb+,p!(!$'"$lb$,00=lb+0x+

y+z'R-9(+-x+y}!DH0Dz(R=

$d%)5E Raaa l = Rabc Rbac ,A C= Rabcd Racbd ,AC :=

,A+C;\b_4Zrn0R(!0=

cBN/

!Z$!Z$L_!*+DL!/$`DR/3l0,rRf+R($0/R=Wf\L_(fD?#8/0Wf*l+0!0Y-

`H!00=Wf*\/0"(!00/gd._R,/0\*+Y-Wf0(("8l`0Wf(='_R8(oL_`L_Wf!3#

Wf/0`L_Wf_+(o/0_9(08_+-q0+3!D&_Q/8l`3!=

BNc/51e$/Wf\_ D!0Dlb+0D0!0i_9(+j+gdWf+f

3]Y"_lb+L_`0+$Rf+!/$L!/$Y#+0]v"_$gd=


94/237

UV

Wf\0V,Wf+f_lb+0!0"(0{1,0}($0=D_3Vlb+/4+0!/4Wf/400

=d,f1

V: Var S ValV: Var(iables) S(ituations) Val(ues)V: {P, Q, } {a, b, c, } {1,0}.

-lb+gd0!/4xWf0V/V(x)rH0,

"D_fhRhR$f|0ZDR1Rgxy V(x) V(y)

Z

\/0!Z$L_$`_#Wf0!0=d,f1

Model = (Frame , Valuation)M = (F , V) = (S, g, R, V)

8_`,!"Wf#"0Wf\_D"00!0

- 8l 0/ gd ._R " W!- 6=Wf ._R 8_ 0* Wf}0,r7_v+(H_fm=

BN?2"5//Wf*gd._R_0T+F('nlb+0/++

lb+L$7"&'("(!0d,/001

wV(T,w)=1 wV(F,w)=0 V(AB,w)=1 V(A,w)=1 V(B,w)=1V(AB,w)=1 V(A,w)=1 V(B,w)=1


95/237

U

Wf*0t+fA'(,9;(fgWf*+$'0/0(D+'Rf1

Vt V(t,w)=1 w = gVf V(f,w)=0 w = g*V V(AB,w)=1 xy(Rxyw V(A,x)=1 V(B,y)=1)V+ V(A+B,w)=1 xy [ Rxy*w* ( V(A,x*)=1 V(B,y)=1) ]V+ V(AB,w)=1 xy{Rwxy [V(A,x*)1 V(B,y)=1]}Vg V(AB,g)=1 x [V(A,x)=1 V(B,x)=1]V V(AB,w)=1 xy{Rwxy [V(A,x)=1 V(B,y)=1]}V V(AB,w)=1 xy{Rwxy [V(A,x)=1 V(B,y)=1] [V(B,x) V(A,y)]=1]}

gdhR_3+hz53$h"$r!$/D

$3E.&'(L!/$"4(_R"vH("=8_gdhR,9(,03R++00,rRgdh++rH_=fCVVCx

$+r_fhRD&1

V+ V(A+B,w)=1 xy [ Rxyw ( V(A,x)=1 V(B,y)=1) ]

8,/-8_DgdhR8_,?*D3E.&'(L!/$D&_"f$6"#5,?*D3E.&'(_/_$mf:i`lC:CVVj"c7d_fd&_gdhR1

Wow! I hadn't realized that I made that mistake. The condition should be

a|=A+B iff (b)(c)(b*|=A or c|=B).

$*D!3#"=

+_8/0D}D&_R,,9(,rgdhR+03R++0gdhR,r0t+rH_=

qWf._R"$0t+f("_fd_R$L!/$1


96/237

UC

Vt V(t,w)=1 xy {Rwxy pVariables [V(p,x)=1 V(p,y)=1]}Vf V(f,w)=1 xy (Rxyw pVariables [V(p,x)=1 V(p,y*)=1] )

"$8_8_?#hR!*+xy {Rwxy pVariables [V(p,x)=1 V(p,y)=1]}lb+wb_A!$R*",,

xy {Rwxy AFormulas [V(A,x)=1 V(A,y)=1]}

`Dd8_p ( p p )+tw(_,+0Zgdw+gR=

g*8!R/_$xZ#0`0Wf,,

3!+3!`&'("Q+xZ!Z$L_+xZ#L_0 .R`3!9_D?#8/0/8li&'(+!Z$D#jH(""v$(_l=H0à!$3!D8(oi&'(+!Z$D#jf}

lH0+H_2_H98_,("H_fm=3!2_l!"v#H("=

ZG$?5M>À$3!lAgRgd#,rf

à!$AB

$3!lBA

g

Rgd#,rf (l_x[V(B,x)=1 V(A,x)=1]

à!$A&

B1;;BnA

$3!lB1BnAgRgd#,rf(l_x[V(B1Bn,x)=1 V(A,x)=1]

0(l_x1D===Dxn+x[Rx1xnx V(B1,x1)=1 V(Bn,xn)=1] V(A,x)=1

à!$A2'

B1,,BnA

3!l$B1BnA

g

(l_Rgd#,rfx{V(B1,x)=1 V(Bn,x)=1] V(A,x)=1}


97/237

U:

!"2_l8_"$8_$/0}50gdBAg$#`llb$'*"R$1x[V(B,x)=1 V(A,x)=1]=H08_8!RDf3{

à!$!/$08/,9;ffgR0Z=H0k*D8(oABà!$`l

BA$!3,r,0Y#+-0Z+8/,9;ffg-=H0,0"YfL!/$*"_0Vg(!0f_!Z$._R39r

f$xRgxx_f*$$01

1 V(AB,g)=1 y2 xy{Rgxy [V(A,x)=1 V(B,y)=1]} V

3 y{Rgxy [V(A,x)=1 V(B,y)=1]} |Y54 Rgxx [V(A,x)=1 V(B,x)=1] |Y5

5 Rgxx !Z$hR

6 [V(A,x)=1 V(B,x)=1] A\b+

7 x[V(A,x)=1 V(B,x)=1] l

^,0$$=

$d%G$?5*+,>

!Z$L_à!$_`L_R$3!lR3!l,r0#/0=&'(à!$_`L_R$3!lR3!l0#/0=

""H>

(A (A B)) (A B)

,013{`3!gD$"(A (A B)) (A B)H("3{1

1 V(A(AB),a)=1 y2 xy{Raxy [V(A,x)=1 V(AB,y)=1]} V3 xy{Raxy [V(A,x)=1

vw(Ryvw (V(A,v)=1 V(B,w)=1)]}V


98/237

UK

4 xw{y(Raxy Ryxw) [V(A,x)=1 V(B,w)=1]} 0$&'(:5 xw{Raxxw [V(A,x)=1 V(B,w)=1]} 2_lRlb9oK6 Raxw y

7 Rxxx !Z$hRilj8 Raxw Rxxx +P9 Raxxw 2_lRlb9o10 Raxw Raxxw QRl11 xw{Raxw [V(A,x)=1 V(B,w)=1]} lN+V12 V(AB,a)=1

"H>( A ( B C )) ( A B C )

,r},r#$gdk*8_A"lb+0"H("{$"

^+$gdlb+1

1 V[ ( A ( B C )) , a ] = 1 A

2 xy{Raxy [V(A,x)=1 V(B C,y)=1]} 3!2_l

3 xy{Raxy [V(A,x)=1 vw(Ryvw (V(B,v)=1 V(C,w)=1)]} 3!2_lC

4 xy{Raxy vw [V(A,x)=1 (Ryvw (V(B,v)=1 V(C,w)=1)]} `/6&'(0:5 xyvw {Raxy [V(A,x)=1 (Ryvw (V(B,v)=1 V(C,w)=1)]} `/6&'(0K

6 xvw y {Raxy [V(A,x)=1 (Ryvw (V(B,v)=1 V(C,w)=1)]} `/6&'(0N

7 xvw{y(Raxy Ryvw) { [V(A,x)=1 V(B,v=1] V(C,w)=1)}} `/6&'(0

8 xvw{Raxvw { [V(A,x)=1 V(B,v)=1] V(C,w)=1)}} 2_lRlb9oP

9 V[ ( A B C ) , a ] = 1 A

10 xy{Raxy [V(A B,x)=1 V(C,y)=1]} 3!2_lU

11 xy{Raxy [vw(Rvwx V(A,v)=1 V(B,w)=1) V(C,y)=1]} 3!2_lV

12 xy{Raxy vw[(Rvwx V(A,v)=1 V(B,w)=1) V(C,y)=1]} (`/6&'0

13 xyvw {Raxy [(Rvwx V(A,v)=1 V(B,w)=1) V(C,y)=1]} `/6&'(0C

14 yvwx {Raxy [(Rvwx V(A,v)=1 V(B,w)=1) V(C,y)=1]} `/6&'(0:

15 yvw {x(Raxy Rvwx) [ V(A,v)=1 V(B,w)=1) V(C,y)=1]} `/6&'(0K

16 yvw {Ravwy [ V(A,v)=1 V(B,w)=1) V(C,y)=1]} 2_lRlb9oN


99/237

UN

H0;0'$f+H0D0'$f+U$$r_r=

%"H> (A B) (A (A B))

,013{3!l"Dg`,3$(P Q) (P (P Q))H("3{

+gdg`a!$A,rlb+L_"H0,#L_$"8_3{+R]v",r#=

lb+9o"_q!Z$gDaDb+cHR!R+R!R1Rabc+P+Qc$+]v"lb+_(Rgd0=D,("P Qa$]v"iH__f

Rabc+Pb#+$gdQc$]v"j="4rfPa#+$gdP Qa8_($]v"P (P Q)g$]v"="$#58_P Qg

$gdib+0_f"!lP$gdQ$gd}jY#+P Q/P (P Q)04!=

H0f}!$+00_f$"D3{D.&'(L!/$Rf?5

7kl0$1

R(ab)cd R(ba)cd R(ac)bd R(ca)bd R(bc)ad R(cb)ad Ra(bc)d Rb(ac)d Ra(cb)d Rc(ab)d Rb(ca)d Rc(ba)d .

Rabc R(aa)bc, Rabc Rb(aa)c, Rabc R(bb)ac, Rabc Ra(bb)c,Rabc Ra(ab)c, Rabc R(ba)ac, Rabc Rb(ba)c, Rabc R(ab)bc,Rabc R(ab)ac, Rabc Ra(ba)c, Rabc R(ba)bc, Rabc Rb(ab)c,

Rabc R(ab)cc, Rabc R(ba)cc, Rabc Rc(ba)c, Rabc Rc(ab)c,Rabc Ra(bc)c, Rabc Rb(ac)c, Rabc R(cb)ac, Rabc R(ca)bc,Rabc R(bc)ac, Rabc R(ac)bc, Rabc Ra(cb)c, Rabc Rb(ca)c,

R(ab)cd R(ac)(bc)d, R(ab)cd R(ca)(cb)d, R(ab)cd R(cb)(ca)d,R(ab)cd R(ac)(cb)d, R(ab)cd R(ca)(bc)d, R(ab)cd R(cb)(ac)d,


100/237

U

V2

*+,R

,*+)A^*(!0A'(H5L_f?R+)A^1

( A B ) ( ~ B ~ A ) )A^,*1

A B iff ~ B ~ A )A^*1

,r)A^gdDQR`0gdfD)A^,*($,

,r)A^,k*DQR`L_,k*f9(D)A^*()A^ 9( +$ _k* +6 D)A^ * D_ 3 $,

$k*D_k*=DgdL_)A^"$8/D*8_(D$3R3gd

H("+6_k*)A^?d-r2lbA'(fR,r"H$RH=

R= R+ + DN A ~ ~ A+ Cont

A B iff ~ B ~ A

&'(f2lb&'(/;fD.0B*+?d8/09(D)AD

=D?d+8_L/"8_D8/0+"3{,-8*,)*,-)*=)*8_0-2lb,-)*})*(

.&'(Rgd,r)A^?d"+("*,-)*R=


101/237

UP

*+,R

?@":2O5EfCgKhi

_l2Q2_l?d2_lA

B =\ ~ ( A ~ B )

A B =\ ~ ( ~ A ~ B )

QRb`d_;

limj(A (B C)) (B (A C))(A B) ((C A) (C B))

CB

)*b`d

2k)Al2k)A|Y5

)A^-f$l

A ~ ~ A~ ~ A A(A B) (~ B ~ A)A ~ ( A ~ A )

DNIDNEContCI

;+(o|#+q1

q?d+2Qfn|#+q?d+2Q(_`+$.&'(0=DD|#+?d+2QD.&'(&+&_0/+(]b

D\/;++,D(0=D+8_f+2_l?*|#+qD,re_(l?d+2Q(_+

}-?d+2Qb`dH__H(""v-;=L|#+b`d(_$|#+Sb?d+"(l8__Ym

/QR/0?d3{D/(("3{QRd_k*/0_9((

A A$2k)A8*f!$(f=?d3{+0S+Wl?d(fD-f$CI$D=

?d(_9oCI-_-A4! Z)ARgd/0=}-"?dD8_($-f$+)*(!ZRZ)A}!CI}-_-"

$9_d8_+$-f$(l_-f$$gd=9o!f(_H4(m&(l,|#+}$_1}--A)A

-}B,8_04!"0A+B+_bL_0+-f$H03

Q_L_0k5"Q$,r\/4#l("=}--D+8_fA}-)AB$k+-f$(l8_04!=D^m2_lA B

~ ( A ~ B )-f$(lA+B$_L_3=(lD&D8_($-f$=


102/237

U

$"D|#+}&_(l,+r$(_HR2k)A,*H("`/,r2_l1

A B =\ ~ ( ~ A ~ B )=\ ~ ~ ( ~ A ~ ~ B )

= ~ A B

D+8_f&(lm,1,/0'QR-+}#QRH_*&'(

DH_H9z63;,QRf7}&"-4!DH_*&'(Z\`@,/0|#+($H_*&'(=

2?@":fj,9%gkVist?3*0QRb`dD"L?d+2QDH_="4rf

`D|#+-f$l?d0S+W4!9rf+0DR4!f_)*`df?d8_$"S+W;*`d+H("8_}_

,(9(`dkAH(""v.&'((st1

Sb?d$2k)Al2k)A|Y5

)A^

A ~ ~ A~ ~ A A

(A B) (~ B ~ A)

DNIDNE

Cont

D8(/0H("!$_f?dLfD?d$;H1

Sb?dL_8?d(~ A ~ ~ ~ B) (B A)Bacon

?dLDL$7"&'()A^|Y5Cont_D( ~ A ~ B) (B A)D

_"L$7"&'((st".&'(D3E^"+m,A?#DL8_?d/4!2k)A,*W+f_DH_oD8_(08_D_f0

H(",3;A1


103/237

UU

=BaconD(~ A ~ ~ ~ B) (B A)DD_9(D8?dD.&'(04!`d_$=

C=Bacon2D( ~ ~ ~ A ~ B) (B A)D?d+D_9(D8&'(

D.04!`d_$=:=Cont_D(~ A ~ B) (B A)D)A^|Y5DDNID2k)AlD

A ~ ~ A=K=Cont_D(~ A ~ B) (B A)D)A^|Y5DDNED2k)A|Y5D

~ ~ A A=N=ContLD(~ A B) (~ B A)D)A^oDDNID2k)AlD

A ~ ~ A=

=ContRD(A ~ B) (B ~ A)D)A^$DDNED)A|Y52kD~ ~A A=

P=Cont+D(A B) (~ B ~ A))A^lDDDNIDDNED+l|Y52k)ADA ~ ~ A+~ ~ A A=

Y51

)AS"2_l]Y"+gdS,1

A = \ A F~ A = \ A f

_fb`d_Dd8_F+fHR+}q1F A( ( A f ) f ) A

"4rfF A

D$* A

`lF A

$=^m A

(l]v"7" ,A#5$~ A(lv"],A$=)AS+8__k*fZ"_A_fH_1


104/237

VV

2k)AlA AA ~ ~ ADNI2k)A|Y5 A A~ ~ A ADNE

)A^(A B) ( B A)(A B) (~ B ~ A)Cont

3E^"+m A ( A B ) A ( A B )Kz#{&RQA AA ~ Az#{&RQ2lb A A~ A ~ ~ A

`Z_f.Z$9r,3k*(0,D0=

=9RQR9R)*"__k*/0)*'QR=9Rn?dD8_+7 A A3{?*.&'(8_

)*8_D+8_f$,/)*9R=C=D2k)A|Y53!?# A A$8_D8_.&'(D

RH0ZD,r3!d"1A A_$gd_}-0(l_7"]v"=}-"4rf(D(!]v"7""H_v"0$m+,_D

`A A(!03!l2k)A|Y5+=:=?dD`58_ A Asmk*fD2_l(_f$3{?*!-

( A F ) ( ( A F ) F )$_r=

*+,CR

++_UP:!Z$A'(D!/$q57$D"z6+0$!t=&'(8_9rCRiClassical Relevant Logicj=CRst,/0

3ER+_k*)A^*+2k)A/0$EFQ=

CR = R+

+ DN A ~ ~ A+ RCont A B iff ~ B ~ A

+ EFQ A ~ A B

CR = R + EFQ A ~ A B


105/237

V

3{$rD)A^*D,-8*"(";R=#8_H0 3{fd(0_ CR"+=d 8_(0_f

H_+r1

d++_(UP:xNP1

CR = R+ + DNE ~ ~ A A 2k)A|Y5+ Antilogism A B ~ C iff A C ~ B *b

H03{d+8_f*b$"D(d*+D`+(

+2k)Al+)A^EFQdH("3{+(1

*b,0d`+(1

A B ~ C

~ ~ C ~ ( A B )

C ~ ~ C

C ~ ( A B )

C ( ~ A ~ B )

A C A ( ~ A ~ B )

A C ( A ~ A ) ~ B A C ~ B ~ B

A C ~ B

,0EFQd+(1 ~ A ~ B ~ A

~ A A ~ ~ B

~ A A B

d2k)Al*,0+(1 A ~ A ~ A

A A ~ ~ A

A ~ ~ A

d)A^*,0+(1


106/237

VC

A B

A ~ ~ B

T A ~ ~ B

T ~ B ~ A

~ B ~ A

d#$8_(1CR = R+ + EQT A B ~ B

+ EFQ A ~ A B

H03{d8_fd("Dd(+2kA,*$H("3{)A^*1

DN A ~ ~ ARCont A B iff ~ B ~ A

,01

1. ~ ~ A ~ ~ A2. ~ ~ A ~ A A EQT3. ~ ~ A ~ ~ A ( ~ A A ) +C4. ~ ~ A ( ~ ~ A ~ A ) A 2lbstm:5. ~ ~ A A A EFQK6. ~ ~ A A N

1. A A2. A ~ A ~ ~ A EQT3. A A ( ~ A ~ ~ A ) +C4. A ( A ~ A ) ~ ~ A 2lbstm:5. A ~ ~ A ~ ~ A EFQK6. A ~ ~ A N

A B iff ~ B ~ A

1. A B |2. ~ B ~ B


107/237

V:

3. A ~ B B ~ B +C4. A ~ B ~ A EFQ:5. ~ A ~ A6. ~ A ( A ~ B ) ~ A K+N7. ~ B ~ B8. ~ B A ~ A EQT9. ~ B ~ B ( A ~ A ) P+10. ~ B ( ~ B A ) ~ A 2lbstmU11. ~ B ~ A ( A ~ B ) _4;V12. ~ B ~ A QRl+

d_+,+3;(118CR = R+ + A ( B C ~ C )

+ ~ ( A B ) ~ A B+ EFQ A ~ A B

d8_q4D($(!0_f`+?dA++?d_f=

*+,CR*

"!R!&'(CRD,r"$8_;+!R)AS+DHHR1)*b`d_CRb`d_+R=+_H58_

+UPK+"c'D8#+&*&'(Dc7'df$CR)AS+CR*=-)*R(Rr/)*+CR/rH0,

RH0Z1CR* =\ R

~ + CR

CR* =\R+ + DN A ~ ~ A

+ Cont (A B) (~ B ~ A)

+ DN A A

18 S. Giambrone and R. K. Meyer, 1989 Completeness and conservative extension results for someBoolean relevant logics, Studia Logica, vol. XLVIII


108/237

VK

+ RCont A B iff B A

+ EFQ A A B

/Db`d8_$,-)*+/#)*(= "4rf/+8_D*++$*#+)A^?d*,-)*+2lb#)*!-}i =`!$CVVVxP+8_

D",-)*+,-)*=j/3$_)AS+8_fL_0k5"$8_"8_!RR

_s_}__k*"=&'(,-)*`d,+}D_3CR0#fCR"CR/k*?_33k*)*`d,+}+("#&'(

R

#0fR

"R

/k*?_33k*("="Dc7'dR

+CR

w!53"$L_0W!-D_3R+CRCR*w!5R!-$=,(o,-)*+#)*,;?#!8__D_H0Z0l"

=H("}_/!H0f L$7"(o$$(4(_=L$7"QR"H

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CR* ~ A ~ t

CR* t CR* ~ A -*

`,#@*D.&'($$0`E+$D_k*,

)A=_k*,#@*-*,*8_)A`E,A+_k*-*W_Ym(=

P=-ArR#QR+#)**

CR* ~ A

CR* A

,01D?3**L/"


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=-ArR#QR+#)**

CR* ~ A

CR* A

,01 CR* ~ A yCR* A f

CR* A f

CR* A f

CR* A f

CR* A f

CR* ~ f

CR* A -*

U=,-+#)*,L$7"}!$1

CR* A ~ A

CR* ~ A A

,01$$=

517@O5E57oO5E

RQR,+}D0D+DD+0""2_l}_QRS+,9r+(!0/bQR/kQR+/b+QR0/D+

",_H_1

/bQRA B =\ ( A t ) B/kQRA B =\ A ( B f )

/b+QRA B =\ ( A t ) ( B f )


131/237

CP

_D8#+D/b+QRUPVxC$"l=D/kQRl#$8_DtR=

O5E3b517@/QR8_9rfZ_f"50H_fm1

=QR8_/02lb0"(l8_D(!0'QRf

R ( A B ) ( A B )R ( A B ) ( A B )

R ( A B ) ( A B )R ( A B ) ( A B )

R ( A B ) ( A B )R ( A B ) ( A B )

C=2lbD/b+QRDd8/0+QR_$f$01

R ( A B ) ( A B )R ( A B ) ( A B )

R ( A B ) ( A B )R ( A B ) ( A B )

R ( A B ) ( A B )R ( A B ) ( A B )

R ( A B ) ( A B )R ( A B ) ( A B )

ZH5,0i$8m5,0fR!3"jk*8_D!DR;

("1

R ( A B ) ( A C B D )

ZH5,0D,("+1

R ~ A ( A f )1

R_k*R ( A f ) ( A t B f )2

f+CR ~ A ( A t B f )3

R ~ A ( A B )4

R B ( t B )5

R_k*R ( t B ) ( A t B f )6

fN+R B ( A t B f )7

R B ( A B )8


132/237

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fK+R (~ A B ) ( A B )9

R ( A B ) ( A B )10

d"H0,$"D,-QR/b+QRf}!$3{

?Q7"4!D}!$8_!$f FH$12Z,0yR ( A B ) ( A B )1

R t ( f f )2

R f ( t f )3R_k*R ( t f ) ( ~ A t A f )4

R ( t f ) ( ~ A

A )5

f:+NR f ( ~ A A )6f+R f ( ~ A A )7R f ( ~ ~ A A )8R f A9

8;/UR f F10,0,_m=

:=k*+/b0*/k+k*f0i,-j"(l8_D(!0

If R A B then R A B

If R A B then R A B

$$,0=

K=k*+(!0,_D/b++,-01

If R A B then R A B

,01yR A B1.

2_lR A t B f2.


133/237

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}!$R ~ ( A t ) ( B f )3.,-R ~ A ~ t B f4.2_lfR ~ t ~ A B ~ t5.

R ~ t ~ A B6.-*R ~ A B7.

2_lR A B8.

N=QR8__k*,D8_($*_f.+D01

=

=

=QR8_Z,$H"5_.+D01

P=DAA5+*/k+/b*_k*,-)A^*k*+/b_k*L_,-)A^D/k_H_(l_D_1

R A B

R ~ B ~ A

R A B

R ~ B ~ A

,01)A^",-+QR8_2_l+/b0=

7/29/20

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6.11.86 nagheze mantegh.pdf