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Ŀ p↔ q ŀ 2bi p`�B bbB p ⇐⇒ q

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J�Bb BH MǶv � T�b /2 +2`+H2 pB+B2mt HQ;B[m2 , HǶ2tTHB+�iBQM [m2 MQmb �pQMb /QMMû2 /2 H� bB;MB}+�iBQM/2 H� Mû;�iBQM Ŀ ¬ ŀ M2 +QMi2M�Bi T�b H2 bB;M2 Ŀ ¬ ŀ [mǶQM ûi�Bi 2M i`�BM /Ƕ2tTHB[m2` U+2[mB +QMbiBim2`�Bi 2M 7�Bi mM +�b /2 +2`+H2 pB+B2mtVX LQmb 2tTHB[mQMb Ŀ ¬ ŀ 2M 2tTHQBi�Mi MQi`2+QKTûi2M+2 /�Mb mM H�M;�;2 /Bzû`2Mi Ĝ /m 7`�MÏ�Bb [mB MQmb b2`i /2 Kûi�H�M;�;2 Ĝ [mB TQm``�BiHmB@KāK2- bB QM 2M �p�Bi #2bQBM- āi`2 7Q`K�HBbû 2i û[mBTû /ǶmM2 bûK�MiB[m2 `B;Qm`2mb2 U+QKK2MQmb HǶ�pQMb 7�Bi �p2+ H2 H�M;�;2@Q#D2i- H2 H�M;�;2 /2 H� HQ;B[m2 /2b T?`�b2bVXR9

jXk G� p�HB/Biû 2i H� pû`Biû HQ;B[m2amTTQbQMb [mǶmM2 BM7û`2M+2 HQ;B[m2 �Bi H� 7Q`K2 bmBp�Mi2 ,

UjX3V pX 1i qX J�Bb �mbbB rX .QM+ sX

LQmb �pQMb pm [mǶQM T2mi K2ii`2 +2ii2 BM7û`2M+2 /�Mb H� 7Q`K2 bmBp�Mi2 ,

UjX3′Vpqrs

úi�Mi /QMMû H� bB;MB}+�iBQM /m +QMM2+i2m` Ŀ ∧ ŀ UĿ 2i ŀV- QM T2mi û;�H2K2Mi û+`B`2 HǶBM7û`2M+2UjX3′V +QKK2 bmBi ,

UjX3′′V p ∧ q ∧ rs

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dk jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

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UjX3′′′V aB Ŀ p ŀ- Ŀ q ŀ 2i Ŀ r ŀ bQMi p`�Bb- �HQ`b Ŀ s ŀ HǶ2bi û;�H2K2MiX

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UjXNV p→ ¬p¬p

2bi p�HB/2- BH 7�mi pQB` bǶBH 2bi TQbbB#H2 [m2 H� T`ûKBbb2 Ŀ p→ ¬p ŀ bQBi p`�B2 2i [mǶ2M KāK2 i2KTbH� +QM+HmbBQM Ŀ ¬p ŀ bQBi 7�mbb2X LQmb 2M 7�BbQMb /QM+ H2b i�#H2b /2 pû`Biû ,

p ¬p p→ ¬pV F FF V V

p ¬pV FF V

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G� p�HB/Biû 2i H� pû`Biû HQ;B[m2 dj

p ¬p p→ ¬p (p→ ¬p)→ ¬pV F F VF V V V

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p |4 q ⇐⇒ |4 p→ q

§ /`QBi2 /2 +2ii2 û[mBp�H2M+2- MQmb miBHBbQMb Ŀ |4 ŀ TQm` /B`2 [m2 Ŀ p→ q ŀ 2bi mM2 i�miQHQ;B2-+Ƕ2bi@¨@/B`2 MǶ� T�b #2bQBM /2 T`ûKBbb2b TQm` āi`2 p�HB/2X PM T�`H2 �HQ`b /ǶĿ BKTHB+�iBQM 7Q`K2HH2 ŀ ,

BKTHB+�iBQM K�iû`B2HH2 /2 q T�` p ⇐⇒ Ŀ p→ q ŀ 2bi p`�BU Ŀ p b2mH2K2Mi bB q ŀV UbQBi Ŀ p ŀ 2bi 7�mt bQBi Ŀ q ŀ 2bi p`�BVBKTHB+�iBQM 7Q`K2HH2 /2 q T�` p ⇐⇒ Ŀ p→ q ŀ 2bi mM2 i�miQHQ;B2U Ŀ q ŀ 2bi mM2 +QMbû[m2M+2 bûK�MiB[m2 UBH MǶ2bi T�b HQ;B[m2K2Mi TQbbB#H2

/2 Ŀ p ŀV [m2 Ŀ p ŀ bQBi p`�B 2i Ŀ q ŀ 7�mtXV

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R3X CǶmiBHBb2 H2 bB;M2 Ŀ ! ŀ TQm` T�`H2` /m T`Q+2bbmb /2 7Q`K�HBb�iBQMX

d9 jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

Ŀ lM �`;mK2Mi /2 H� 7Q`K2

pqrs 2bi p�HB/2X ŀ ! Ŀ {p, q, r} |4 s ŀ

Ŀ lM �`;mK2Mi /2 H� 7Q`K2p ∧ q ∧ r

s 2bi p�HB/2X ŀ ! Ŀ {p ∧ q ∧ r} |4 s ŀ

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UjXRyV p |4 q

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UjXRy′′V Ŀ p ŀ |4 Ŀ q ŀ

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de jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

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Ç Ŀ J�`B2 ŀ 7�Bi T�`iB2 /2 φXÇ ψ +QMbBbi2 2M [m�i`2 KQibX

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UjXRy′′′V φ |4 ψ

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UjXRy′′′′V Ŀ p ∧ q ŀ |4 Ŀ q ŀ

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UjXRRV φ ∧ ψ |4 φ

MQmb M2 TQm``BQMb THmb bm#biBim2` Ŀ φ ŀ T�` ĿĿ a�K � 7�BK ŀŀ 2i Ŀ ψ ŀ T�` ĿĿ J�`B2 p2mi b2K�`B2` ŀŀ +�` /�Mb +2 +�b@H¨ MQmb �m`BQMb ,

UjXRR′V Ŀ a�K � 7�BK ŀ ∧ Ŀ J�`B2 p2mi b2 K�`B2` ŀ |4 Ŀ a�K � 7�BK ŀ

UjXRR′V 2bi mM MQM@b2Mb- +�` Ŀ ∧ ŀ `2HB2 /2b T?`�b2b 2i MQM T�b /2b MQKb /2 T?`�b2bXkj 1bb�vQMbH� 7Q`K�HBb�iBQM bmBp�Mi2 ,

UjXRR′′V Ŀ φ ∧ ψ ŀ |4 φ

kjX .�Mb H� iû`KBMQHQ;B2 /2 H� ;`�KK�B`2 +�iû;Q`B2HH2 U+7X TX k9R 2i bmBpXV H2b +QMM2+i2m`b T`QTQbBiBQMM2Hb bQMi/2b 2tT`2bbBQMb /m ivT2 a/aa- 2i MQM T�b /2b 2tT`2bbBQMb /m ivT2 a/LLX LQmb /û}MB`QMb 7Q`K2HH2K2Mi +2 [mǶ2bimM2 `2H�iBQM ¨ H� TX jkkX

G� [m�bB@+Bi�iBQM 2i H2b /2KB@+`Q+?2ib /2 ZmBM2 dd

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UjXRkV "φ ∧ ψ#

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Ç LQmb miBHBbQMb Ŀ "φ∧ψ# ŀ TQm` T�`H2` /2 iQmi2b H2b T?`�b2b +QMDQM+iBp2b- T�` 2tX Ŀ p∧ q ŀ-Ŀ (p ∨ q) ∧ s ŀ- Ŀ p ∧ (q → s) ŀ 2i+X c

Ç LQmb miBHBbQMb Ŀ "φ→ ψ# ŀ TQm` T�`H2` /2 iQmi2b H2b BKTHB+�iBQMb K�iû`B2HH2b- [m2H[m2 bQBiH2m` +QKTH2tBiû c

Ç LQmb miBHBbQMb Ŀ "(φ)# ŀ TQm` T�`H2` /2 iQmi2b H2b 2tT`2bbBQMb [mB +QKK2M+2Mi �p2+ mM2T�`2Mi?ĕb2 ;�m+?2 2i }MBbb2Mi T�` mM2 T�`2Mi?ĕb2 ;�m+?2 c

Ç LQmb miBHBbQMb Ŀ " S�`Bb # ŀ TQm` T�`H2` /m MQK /2 H� pBHH2 /2 S�`Bb- +Ƕ2bi@¨@/B`2 TQm` T�`H2`/2 Ŀ S�`Bb ŀ c +Ƕ2bi /QM+ mM MQK /2 H� KāK2 2tT`2bbBQM [m2 ĿĿ S�`Bb ŀŀ 5

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UjXRk′V Ŀ p ∧ q ŀ

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k8X LQmb `2iQm`M2`QMb- /�Mb H� b+iX RyXR- ¨ +2b MQKb +QKTH2t2b [mǶQM �TT2HH2`� Ŀ 7QM+iBQMb ŀX

d3 jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

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UjXRR′′′V "φ ∧ ψ# |4 φ

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p q p⊤qV V VV F VF V VF F V

p q p⊥qV V FV F FF V FF F F

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G2b i�miQHQ;B2b 2i H2b +QMi`�/B+iBQMb dN

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qp ∨ ¬p

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p ∧ ¬pq

q

¬(p ∧ ¬p)

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UjXRjV |4 p ∨ ¬p

*2+B bǶ�++Q`/2 �p2+ MQi`2 2tTHB+�iBQM /2 H� +QMbû[m2M+2 HQ;B[m2 , H� T?`�b2 Ŀ φ |4 ψ ŀ 2bi p`�B2bB 2i b2mH2K2Mi bǶBH 2bi p`�B TQm` MǶBKTQ`i2 [m2Hb φ 2i ψ [m2 , bB φ 2bi p`�B- �HQ`b ψ HǶ2bi �mbbBXaǶBH 2bi BKTQbbB#H2 iQmi +Qm`i [m2 ψ bQBi 7�mbb2- �HQ`b +2ii2 +QM/BiBQM 2bi iQmDQm`b `2KTHB2 , TQm`MǶBKTQ`i2 [m2H T?`�b2 Ŀ p ŀ- H� /BbDQM+iBQM Ŀ p∨¬p ŀ2bi p`�B2 Ĝ 2HH2 2bi mM2 +QMbû[m2M+2 HQ;B[m2/2 HǶ2Mb2K#H2 pB/2 /2 T`ûKBbb2bXSQm` /B`2 [m2 Ŀ p ∧ ¬p ŀ 2bi mM2 +QMi`�/B+iBQM- QM T2mi �{`K2` ,

UjXR9V |4 ¬(p ∧ ¬p)

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3y jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

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¬(p ∧ q) ⇐⇒ ¬p ∨ ¬q¬(p ∨ q) ⇐⇒ ¬p ∧ ¬q

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UjXR8V "¬(φ ∧ ψ)# ⇐⇒ "¬φ ∨ ¬ψ#"¬(φ ∨ ψ)# ⇐⇒ "¬φ ∧ ¬ψ#

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jRX *7X /2 JQ`;�M UR39d- RReVX .Ƕ�T`ĕb ƇmF�bB2rB+x URNj9�V- :mBHH�mK2 /ǶP+F?�K H2b �p�Bi /ûD¨ `2+QMMm2bX

jkX LQmb �pQMb #2bQBM /2b /2KB@+`Q+?2ib +�` H2b 2tT`2bbBQMb bmBp�Mi2b ,¬(φ ∧ ψ) ⇐⇒ ¬φ ∨ ¬ψ¬(φ ∨ ψ) ⇐⇒ ¬φ ∧ ¬ψ

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Ŀ ¬(φ ∧ ψ) ŀ ⇐⇒ Ŀ ¬φ ∨ ¬ψ ŀĿ ¬(φ ∨ ψ) ŀ ⇐⇒ Ŀ ¬φ ∧ ¬ψ ŀ

*2b 7Q`KmH�iBQMb TQb2Mi û;�H2K2Mi mM ;`�p2 T`Q#HĕK2 /m 7�Bi /2 M2 T�b āi`2 ;ûMû`�H2bX G2b 2tT`2bbBQMb ¨ ;�m+?22i ¨ /`QBi2 /2 H� ~ĕ+?2 /Qm#H2 Ŀ ⇐⇒ ŀ M2 /ûMQi2Mi [m2 +2b 2tT`2bbBQMb T�`iB+mHBĕ`2bX *2HH2 2M ?�mi ¨ ;�m+?2-

G2b i�miQHQ;B2b 2i H2b +QMi`�/B+iBQMb 3R

Mû;�iBQMb /2p�Mi /2b +QMDQM+iBQMb 2i /2b /BbDQM+iBQMb ¨ HǶBMiû`B2m` /2 +2b 7Q`KmH2b- 2M û+?�M;2�MiH2 +QMM2+i2m` Ŀ ∧ ŀ Qm Ŀ ∨ ŀ �p2+ bQM +QMM2+i2m` Ł QTTQbû ł Ŀ ∨ ŀ Qm Ŀ ∧ ŀXjj

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UjXReV "φ→ ψ# :⇐⇒ "¬φ ∨ ψ#

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RX aB HǶQM M2 T`2M�Bi [m2 Ŀ ¬ ŀ 2i Ŀ ∧ ŀ +QKK2 T`BKBiB7b U+QKK2 "`2Mi�MQ UR3d9V 2i CQ?MbQMURN9dVV- MQmb TQm``BQMb /û}MB` H2b �mi`2b +QKK2 bmBi ,"φ ∨ ψ# :⇐⇒ "¬(¬φ ∧ ¬ψ)#"φ→ ψ# :⇐⇒ "¬(φ ∧ ¬ψ)#"φ↔ ψ# :⇐⇒ "¬(φ ∧ ¬ψ) ∧ ¬(¬φ ∧ ψ)#

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G2 +�``û /2b QTTQbBiBQMb 38

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p ∧ q ¬p ∧ ¬q

p ∨ q p|q- H� #�``2 /2 a?2z2`- /û}MB2 T�` ,

⎛

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p q p|qV V FV F VF V VF F V

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⎟⎟⎟⎟⎟⎠

+QMi`�B`2

bm#+QMi`�B`2

bm#�Hi2`M2 bm#�Hi2`M2

+QMi`�/B+iQB`2

+QMi`�/B+iQB`2

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3e jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

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UjXkjVp

p→ qq

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UjXkj′Vp

p→ q(p ∧ (p→ q))→ q

q

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UjXkj′′V pp→ q

q

pp→ q

(p ∧ p→ q)→ qq

pp→ q

(p ∧ (p→ q))→ q(p ∧ (p→ q) ∧ ((p ∧ (p→ q))→ q))→ q

q . . .

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UjXk8V S(p|q) := S(p∧q)S(q)

.�Mb HǶBMi2`T`ûi�iBQM #�vûbB2MM2 /2 H� T`Q#�#BHBiû- H� T`Q#�#BHBiû +QM/BiBQMM2HH2 S(p|q) 2tT`BK2HǶBM+Q`TQ`�iBQM /ǶmM2 MQmp2HH2 BM7Q`K�iBQM U[m2 qV 2M pm2 /2 H� T`Q#�#BHBiû [m2 p , �v�Mi �TT`Bb[m2 q- MQmb /2p`BQMb i2MB` +QKTi2 /2 +2ii2 MQmp2HH2 BM7Q`K�iBQM /�Mb MQi`2 2biBK�iBQM /2 H�T`Q#�#BHBiû /2 p 2i �HQ`b HmB �ii`B#m2` H� T`Q#�#BHBiû +QM/BiBQMM2HH2XSQm` H2b T`Q#�#BHBiûb +QM/BiBQMM2HH2b- BH 2bi bQmp2Mi i`ĕb BKTQ`i�Mi /2 /BbiBM;m2` H� +QM/BiBQM /m+QM/BiBQMMû , 2M ;ûMû`�H S(p|q)- MǶ2bi T�b /m iQmi H� KāK2 +?Qb2 [m2 S(q|p)X amTTQbQMb [m2 MQmb�pQMb mM i2bi }�#H2 TQm` H� /2M;m2 , S(T2`bQMM2 A � ûiû i2biû2 TQbBiB7 TQm` H� /2M;m2 | T2`bQMM2A � H� /2M;m2) = 0.97X G� }�#BHBiû /ǶmM i2H i2bi bB;MB}2 [mǶBH v �m`� T2m /2 Ł 7�mt Mû;�iB7b ł , mM`ûbmHi�i TQbBiB7 2bi }�#H2 T�`+2 [mǶBH 2bi i`ĕb `�`2 U[m2 /�Mb 3W /2b +�bV [mǶmM2 T2`bQMM2 �v�Mi H�/2M;m2 M2 bQBi T�b /B�;MQbiB[mû TQbBiBp2K2Mi �m i2bi /2 H� /2M;m2X �mi`2K2Mi /Bi- BH MǶv � [m2T2m /ǶBM72+iûb [mB T�bb2Mi BM�T2`Ïmb �m i2biX GǶ2tBbi2M+2 /ǶmM i2H i2bi 2bi i`ĕb #B2M +QKT�iB#H2�p2+ mM MQK#`2 �bb2x ûH2pû /2 Ł 7�mt TQbBiB7b ł , 2M 2z2i- BH 2bi bQmp2Mi `�BbQMM�#H2 /ǶmiBHBb2` mMi2bi Ł bm`@b2MbBiB7 ł- [mB T`Q/mBi /2b `ûbmHi�ib TQbBiB7b �mbbB /�Mb [m2H[m2b +�b Qɍ H� T2`bQMM2 2bi 2M2z2+i b�BM2X LQmb TQmpQMb �HQ`b �pQB` S( T2`bQMM2 A � H� /2M;m2 | T2`bQMM2 A � ûiû i2biû TQbBiB7TQm` H� /2M;m2) = 0.5- +Ƕ2bi@¨@/B`2 �pQB` mM2 bBim�iBQM Qm mM `ûbmHi�i TQbBiB7 /ǶmM T`2KB2` i2biM2 MQmb BM/B[m2 [mǶmM2 T`Q#�#BHBiû /2 50W [m2 H� T2`bQMM2 2M [m2biBQM �Bi `û2HH2K2Mi �ii`�TûH� /2M;m2 Ĝ /Ƕ�mi`2b i2bib- THmb +QKTHB[mûb- b2`QMi �HQ`b Mû+2bb�B`2bX �mi`2K2Mi /Bi- H2 i2bi +`û2

8yX G2 HBp`2 /2 �/�Kb Ŀ h?2 GQ;B+ Q7 *QM/BiBQM�HbX �M �TTHB+�iBQM Q7 S`Q#�#BHBiv iQ .2/m+iBp2 GQ;B+ ŀ URNd8V2i HǶ�`iB+H2 /2 .�pB/ G2rBb Ŀ S`Q#�#BHBiB2b Q7 *QM/BiBQM�Hb �M/ *QM/BiBQM�H S`Q#�#BHBiB2b ŀ URNdeV QMi BMBiBûb 2i7Q`i2K2Mi BM~m2M+ûb +2 /û#�iX

Ny jX _2H�iBQMb HQ;B[m2b 2i BM7û`2M+2b HQ;B[m2b

/2 7�mbb2b �H�`K2b- K�Bb �m pm /2 H� T`Q#�#BHBiû S(p|q)- H� 7�B#H2 T`Q#�#BHBiû S(q|p) 2bi KQBMbBM/ûbB`�#H2 , MQmb T`û7û`QMb �pQB` THmb /Ƕ�H�`K2b +Qmp`�Mi Nd W /2b +�b /ǶBM72+iBQMb- THmiƬi [m2/Ƕ�pQB` KQBMb /Ƕ�H�`K2b 2i [m2 THmb /2 T2`bQMM2b BM72+iû2b T�bb2Mi BM�T2`Ïm2bX8R

AH 2tBbi2- +2T2M/�Mi- mM2 `2H�iBQM K�i?ûK�iB[m2 i`ĕb BKTQ`i�Mi2 2Mi`2 H2b /2mt ivT2b /2 T`Q#�@#BHBiûb +QM/BiBQMM2HH2b- �TT2Hû2 H2 Ŀ i?ûQ`ĕK2 /2 "�v2b ŀ ,

UjXkeV S(p|q) := S(p)·S(q|p)S(q))

*2 7�K2mt i?ûQ`ĕK2 /Bi [m2 H� T`QTQ`iBQM 2Mi`2 H2b T`Q#�#BHBiûb +QM/BiBQMM2HH2 S(p|q) 2i S(q|p)2bi H� KāK2 [m2 H� T`QTQ`iBQM 2Mi`2 S(p) 2i S(q)X.�Mb #2�m+QmT /2 +�b- MQmb M2 TQmpQMb T�b /ûi2`KBM2` /B`2+i2K2Mi H� T`Q#�#BHBiû BM+QM/BiBQM@M2HH2 /2 H� +QM/BiBQM S(q) Ĝ /�Mb H2 +�b Qɍ BH bǶ�;Bi /ǶmM `ûbmHi�i /2 i2bi TQbBiB7- T�` 2t2KTH2- 2HH2/2p`�Bi āi`2 +�H+mHû2 bm` iQmi2 H� TQTmH�iBQM- i2biû2 Qm MQMX S�`+2 [m2 +2ii2 T`Q#�#BHBiû S(q) 2biH� bQKK2 /2 H� T`Q#�#BHBiû /2 Ŀ q ŀ /�Mb H2 +�b [m2 p 2i H� T`Q#�#BHBiû /2 Ŀ q ŀ /�Mb H2 +�b [m2¬p , S(q) = S(q ∧ p) + S(q ∧ ¬p) 2i T�`+2 [m2 +2b /2mt /2`MB2`b bQMi û;�H2b ¨ S(p) · S(q|p) 2i ¨S(¬p) · S(q|¬p) `2bT2+iBp2K2Mi- ;`�+2 ¨ H� /û}MBiBQM /2 H� T`Q#�#BHBiû +QM/BiBQMM2HH2- MQmb �pQMb

UjXkdV S(p|q) := S(p)·S(q|p)S(p)·S(q|p)+ S(¬p)·S(q|¬p)

.�Mb MQi`2 2t2KTH2- +2ii2 7Q`KmH2 MQmb T2`K2i /2 +�H+mH2` H� T`Q#�#BHBiû [m2 [m2H[mǶmM i2biûTQbBiBp2K2Mi 2bi `û2HH2K2Mi K�H�/2 US(/2M;m2 | TQbBiB7)V bm` H� #�b2 /2 H� T`Q#�#BHBiû /2 H� K�H�/B2/�Mb H� TQTmH�iBQM i2biû2 US(d) = 1WV ,

S(d|+) =S(d) · S(+|d)

S(d) · S(+|d) + S(¬d) · S(+|¬d)

=0.1 · · · 0.97

0.1 · 0.97 + 0.9 · 0.03 =0.097

0.097 + 0.027=

0.097

0.124= 0.78226

lM2 T2`bQMM2 i2biû2 TQbBiBp2K2Mi �m`� /QM+ mM2 T`Q#�#BHBiû /2 THmb /2 78W /2 `û2HH2K2Mi āi`2K�H�/2X

jXd G� +QMbû[m2M+2 HQ;B[m2LQmb TQmpQMb K�BMi2M�Mi MQi2` THmbB2m`b T`QT`Bûiûb /2 H� `2H�iBQM /2 +QMbû[m2M+2 HQ;B[m2 [mǶQM� �TT2Hû2 Ŀ |4 ŀX8k

S`2KBĕ`2K2Mi- `2K�`[mQMb [mǶmM2 �bb2`iBQM /2 +QMbû[m2M+2 HQ;B[m2 Ŀ φ |4 ψ ŀ`2pB2Mi ¨ /B`2 [m2iQmi2 �ii`B#miBQM /2 p�H2m`b /2 pû`Biû [mB `2M/ φ p`�B `2M/ û;�H2K2Mi ψ p`�BX �BMbB- H� +QMbû[m2M+2HQ;B[m2 2bi �z�B`2 /2 p�HB/Biû- TmBb[m2 b2HQM iQmi2 TQbbB#BHBiû HQ;B[m2- bB φ 2bi p`�B- �HQ`b ψ 2bi p`�Bû;�H2K2MiX *2ii2 T`QT`Bûiû M2 H� /BbiBM;m2 +2T2M/�Mi T�b /2 HǶBKTHB+�iBQM K�iû`B2HH2- 2i BH MQmb

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8kX SQm` 7�+BHBi2` HǶ2tTQbBiBQM- MQmb M2 +QMbB/û`QMb B+B [m2 /2b �`;mK2Mib [mB MǶQMi [mǶmM MQK#`2 }MB /2 T`ûKBbb2bXJ�Bb H2b T`QT`Bûiûb /2 H� p�HB/Biû 2M [m2biBQM p�H2Mi �mbbB TQm` /2b �`;mK2Mib [mB 2M QMi mM MQK#`2 BM}MBX LQmb`2pB2M/`QMb ¨ +2b T`QT`Bûiûb /2 `2H�iBQMb ¨ H� TX jkk �m +?X R8X

G� +QMbû[m2M+2 HQ;B[m2 NR

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*QKK2 BH 2bi +QmiȿK2 2M K�i?ûK�iB[m2b- MQmb i2MQMb +QKTi2 /2 +2ii2 KmHiBTHB+Biû 2M +QMbB/û`�MiH2b T`ûKBbb2b +QKK2 mM 2Mb2K#H2 2i H� +QMbû[m2M+2 HQ;B[m2 +QKK2 mM `2H�iBQM 2Mi`2 mM 2Mb2K#H2/2 T`ûKBbb2b 2i mM2 +QM+HmbBQMX1M ;ûMû`�H- H2b `2H�iBQMb T2mp2Mi TQbbû/2` /Bzû`2Mib ivT2b /2 T`QT`BûiûbX S�` `�TTQ`i ¨ H� `2H�iBQM/2 +QMbû[m2M+2 HQ;B[m2- Ŀ |4 ŀ- MQmb `2K�`[mQMb- iQmi /Ƕ�#Q`/- [m2 HǶQ`/`2 /2b T`ûKBbb2b MǶ2biT�b T2`iBM2Mi TQm` ûp�Hm2` mM �`;mK2Mi +QKK2 p�HB/2 Qm MQM p�HB/2X *2b T`ûKBbb2b bQMi /QM+T2`Kmi�#H2b ,

UT2`KV {p1, p2, . . . pn} |4 r 4⇒ {p2, p3, p1, . . . pn, . . . p7, . . . } |4 r

*2 MǶ2bi T�b 2M +?�M;2�Mi HǶQ`/`2 /2b T`ûKBbb2b [m2 HǶQM T2mi `2M/`2 MQM p�HB/2 mM �`;mK2Mip�HB/2X *2ii2 T`QT`Bûiû /2 T2`Kmi�#BHBiû /2 H� `2H�iBQM /2 +QMbû[m2M+2 HQ;B[m2 MQmb �TT`2M/ [m2H2b TQmpQB`b BM7û`2MiB2Hb /ǶmM2 KmHiBim/2 /2 T`ûKBbb2b `2bi2Mi BM+?�M;û bQmb /2b T2`Kmi�iBQMb /2+2ii2 KmHiBim/2X89

1M b2+QM/ HB2m- H� +QMbû[m2M+2 HQ;B[m2 2bi mM2 `2H�iBQM KQMQiQM2 ,

UKQMV {p1, p2, . . . pn} |4 r 4⇒ {p1, p2, . . . pn, q1, q2, . . . qm} |4 r

aB pQmb i2M2x mM �`;mK2Mi p�HB/2- pQi`2 BMi2`HQ+mi2m` M2 T2mi T�b pQmb +QMi`2/B`2 2M HmB �DQm@i�Mi mM2 T`ûKBbb2 bmTTHûK2Mi�B`2X �mi`2K2Mi /Bi- �DQmi2` /2b T`ûKBbb2b {q1, q2, . . . qm} /�MbHǶ2t2KTH2 +B@/2bbmb M2 T2mi T�b BMp�HB/2` H� +QMbû[m2M+2 bûK�MiB[m2 Q#i2Mm2 �p2+ H2b T`ûKBbb2b{p1, p2, . . . pn}X lM i2H �DQmi 2bi �TT2Hû Ŀ �z�B#HBbb2K2Mi ŀ /2 HǶBM7û`2M+2 [mB TQm`i�Mi `2bi2 p�HB/2XU.�Mb H2 +�b Qɍ HǶQM �DQmi2 mM2 T`ûKBbb2 [mB +QMi`2/Bi mM2 T`ûKBbb2 /ûD¨ T`ûb2Mi2- Ŀ r ŀ bǶ2MbmBiT�`+2 [m2 MǶBKTQ`i2 [m2HH2 T?`�b2 2bi mM2 +QMbû[m2M+2 HQ;B[m2 /ǶmM2 +QMi`�/B+iBQM- +Ƕ2bi@¨@/B`2T�` H2 b+?ûK� /ǶBM7û`2M+2 �TT2Hû Ŀ 2t 7�HbQ [mQ/HB#2i ŀXV88 *2ii2 T`QT`Bûiû bǶ�TT2HH2 Ŀ KQMQiQ@MB2 ŀ T�`+2 [mǶ2HH2 bB;MB}2 [m2 H� 7QM+iBQM [mB T`2M/ +QKK2 �`;mK2Mi mM +2`i�BM 2Mb2K#H2 /2T`ûKBbb2b 2i /QMM2 +QKK2 p�H2m` H2m`b +QMbû[m2M+2b 2bi KQMQiQM2 /�Mb H2 b2Mb K�i?ûK�iB[m2/2 +2 i2`K2 , `2T`ûb2Miû +QKK2 +Qm`#2- H� HB;M2 M2 #�Bbb2`� T�bXG� p�HB/Biû 2bi i`�MbBiBp2 ,

Ui`�MbV {p1, . . . pn} |4 r � {r, q1, . . . qn} |4 s 4⇒ {p1, . . . pn, q1, . . . qm} |4 s

aB Ŀ r ŀ bǶ2MbmBi /2b T`ûKBbb2b Ŀ p1 ŀ- Ŀ p2 ŀ- 2i+X 2i bB Ŀ s ŀ bǶ2MbmBi /2 +2ii2 T`ûKBbb2 BMi2`Kû/B�B`2Ŀ r ŀ 2i /2b T`ûKBbb2b Ŀ q1 ŀ- Ŀ q2 ŀ- 2i+X- �HQ`b QM T2mi /B`2+i2K2Mi BM7û`2` Ŀ s ŀ Ub�Mb �pQB`#2bQBM /ǶBM7û`2` /Ƕ�#Q`/ Ŀ r ŀ /2 Ŀ p1 ŀ- Ŀ p2 ŀ 2i+XVX8e

8jX *2+B MǶ2bi T�b p`�B /2 iQmi2b H2b HQ;B[m2b- +QKK2 BH 2tBbi2Mi /2b HQ;B[m2b ¨ +QM+HmbBQMb KmHiBTH2bX

89X *QKK2 H2b 2Mb2K#H2b bQMi BMb2MbBiB7b ¨ HǶQ`/`2 /2 H2m`b ûHûK2Mib- +2 +�`�+iû`BbiB[m2 DmbiB}2 MQi`2 +?QBt /2KQ/ûHBb�iBQM /2 H� `2H�iBQM /2 +QMbû[m2M+2 HQ;B[m2 +QKK2 `2H�iBQM 2Mi`2 mM 2Mb2K#H2 /2 T?`�b2b 2i mM2 T?`�b2X

88X G� `ĕ;H2 /ǶBM7û`2M+2 Ŀ 2t 7�HbQ [mQ/HB#2i ŀ +Q``2bTQM/ ¨ H� i�miQHQ;B2 Ŀ p→ (¬p→ q) ŀ /QMi MQmb T�`H2`QMb¨ H� TX Rk9X

8eX PM Q#iB2Mi mM2 7Q`KmH�iBQM THmb 7�+BH2K2Mi `2+QMM�Bbb�#H2 +QKK2 +2HH2 /ǶmM T`BM+BT2 /2 i`�MbBiBpBiû bB QM�bbmK2 [mǶBH MǶv � T�b /2 T`ûKBbb2b bmTTHûK2Mi�B`2b Ŀ q ŀ 2i mM2 b2mH2 T`ûKBbb2 BMBiB�H2 Ŀ p ŀ , p |4 r � r |4s 4⇒ p |4 sX

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