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2

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I |= AX6VHp^�

I |= MS~DWOWU�S�E7{UEDcl]�LNOWS�E7R��6E7M

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I |= A��b�RT['OFLNS��I, β |= M :⇔

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2

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2

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A := ∀x∃y y ◦ x = e

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2

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2

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I, β�I, β |= A

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M ⊆ For�

AX6VHp^�

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I, β |= AX6VHp^�

I, β |= Mq

['R��6E7MPR?] ['OWOFG4$UB<"�� ��h2!G!dKR���!�2

|= AX6VHp^�

M |= AGPDWR�� �?DFE � ['Z6MPZ�E7DWU�E7R)q}�?DFE Q6R�G DWR>U�E7M�EHG�GPDFE7M�E7R)� J�S~O ��LNX/E7R��

Mgr |= AZ�E7D�c~Urq���[DcA['Q�G �6E7R w�LNMP{�E7OWR DWR

Mgr

]�LNOWS~U�3:Q6RT['X6Z�E'R6S~DWS�kmLNR E7DWR�E7{ �mLNR6�#M�E7U�E7RIX6VHp^�

I, βq

��� Z)���6E7M���LNR6�#M�E7U�E7R u MPQ6�6�/E��['RT['OFLNS�]��6M�� v M��/E!M q6JdE7�#U�LNMPM�E'Q6{�E^E7U�j~�M

� v R6R>U�E^['Q�jIZ u EHG�E7UPVrE�]��6MC ['U�E7R�GPUPMPQ6�#UPQ6M�E7R:E7R>UPZT['OWU�E7R)q�V�� �,�

∀x∀s first(prefix (x, s)) = x�

{S|~}(���Z�3|F�Z���Z�,�5�q��| �z�q���5��|��8���~�(�L�L�q�Y�8�q�Z|T� ��}5�5�������q|L������}5|~�����T���5��|

� �d����C^� � b�c�edn�`/- u � � � ���#c >�w e � ;

�V�Y-�wL-/A5!B&5!#-/) ��1�s Y�"�EA ∈ For �

Ek[AV�h2!�+FE�V

⇒ A"�� ��hU!G!dK ���

'�H�

A$UB<"�� ��h2!G!dKR���

⇒ AB�ES(�H2+ V�hU!o+FE�V

EGEk[AV�h2!�+FE�VXW

|= A[⇔ ∅ |= A

� 'Zt�'c!BA � DWD �∅ |= A ⇔

⇔ AS~�6OWUPDWS�� 2

2n$%&(p �/1 u Ya"�EA ∈ For

$UBQ'M ⊆ For �

�#��B�B V�E !o+M |= A

V2"�BQ��$b' ��B�B�

"�B�BM ∪ {¬A}$UB<"�� ��hU!G!dKR���#E %�+�� %,P�" )�ES"�!G!�E %�+

AV�h2!�+FE�V V&"�BQ��$O' ��B�B

"�B�B¬A

$2B<"��G��hU!G!�KR���#EG%�+�

� 'Zt�'c!BA � ]��6M}['OWOFEI, β : I, β |= M ⇒ I, β |= AS�E7RT['Q:��['R6R�p E7R6R

]��6M}['OWOFEI, β : I, β |= M ⇒ I, β 6|= ¬A

���dM�LN�/L~GPDWUPDFLNR � � � D �P�S�E7RT['Q:��['R6R�p E7R6RM ∪ {¬A}

DWG�U�Q6R�E7M�]4�6OWOWXT['Mr�2

`�!_V ����E + H2��EG%�(�H;" N�'&"n"��xJdE7M�G�Q�jIZ�E~q�E7M�]��6OWOFE7R��6EI, β

]��6M¬A

VHQ �mLNR�GPUPMPQ6DFE7M�E7R�xS�E7OWDWR6S~U��?DFEHGrq)DFG�UAR6DFjIZ>U S~�6OWUPDWS �4Q6R��\p�DWMKZT['X/E7R

I, β['OFG ��E!OFE7So��[�]4�6M;�!q�G�LNRTGPU DFGPUKEHGKS~�6OWUPDWS �4p E7R6R �6DWEsJ�E7M�GPQ�jIZ�E

S�EHE7DWS~R�E7U�GPDWR��T�!�

{S|~}(���Z�3|F�Z���Z�,�5�q��| �z�q���5��|��8���~�(�L�L�q�Y�8�q�Z|T� ��}5�5�������q|L������}5|~�����T���5��|

; b.>,�6�#b u edn�`/- u � � � =

� � ���/�-� � �� ���� �D���

� ��� � ���� � � ��� # ������� & �� ��� $�� $'&� ����#� ) �!#'ZA('c` OP$ w,!B&('*+ GIE7D

P = P0 = {p0, p1, p2, ...} � Q6R�� � = � DFGPU k�E7MPX/LNU�E7R)��edG UPM�E7U�E7R�DWR�w�LNM�{�E!OWR['OFG�L*�mE7DWR�E�cxE7MP{�E ['Q?]�q#�6E7{�RT[NjIZ�['Q�jIZ �mE7DWR�E�J ['MPDt['X6OFE7R)�PEG%�+ �6E7{�RT[NjIZ��?DFE�h:E7R6S�E,�6E7M�['U�LN{l['M�E7R

wTLNMP{�E7OWR)q��>Q6MPV�`7+,���Z"H�� E7OFE7S~Q6R6S�E7RsGPDWR��KDWMPM�E7OFE7k~['R>UrqT��� Z)�

I |= AS�E7RT['Q:��['R6R�p E7R6R

I, β |= A�

-&� e���S~DWX6U^EHG,�mE7DWR�E�J�['MPDt['X6OFE7R)q)��E7OFE7S~Q6R6S�E7R)q∀xq Q6R��

F = ∅ EHG,S~DWX6U ['OFG�LKR>Q6M�

P0,→,¬�a�iR>U�E7M�g

�6M�E7UI['UPDFLNR�ZT['X/E7R:�6E7{�RT[NjIZ:�?DFE^w�LNMP{I : P → {T, F}�YDWMd� v R�R�E7Ra['OWOFE.� R>U�E7MP�6M�E7UI['UPDFLNR�E7R �4X6VHp^�~DWZ6M�E7RlM�E7OFE7kN['R>U�E7RacxE7DWO � {�DWUPU�E7OFG��f�DH&��H;"�EF+ %�+,� ��"�!��?Q6M�jIZ?g

�6M�LNX6DFE7M�E7R��S�E7RT['Q ��['R6R)q p�E7R6R ['OWOFE�e�DWR>UPM�E'S�E�E7DWR�E7M^wTLNMP{�E7OTGPDWR���q/DFG�U �?DFE�wTLNMP{�E7O�S~�6OWUPDWS�� u ��O g

UPDWS~�mE7DWU�DFGPU}['OFG�Lp"�BC+ %�(�H;"�ES'&KR���!�� 'c!BA5w,!#'*+�>[1J�

A ≡ p → (q → p)

B ≡ (¬p → ¬q) → (q → p)����LNR>UPMI['�/L~GPDWUPDFLNR �

C ≡ (¬p → ¬q) → (p → q)

p q q → p A ¬p → ¬q B p → q C

T T T T T T T T

T F T T T T F F

F T F T F T T T

F F T T T T T T

AQ6R��

BGPDWR��KS~��O_U�D_S�q

CR6DFjIZ>Ur�

2

e�DWR�E�wTLNMP{�E7O DFG�U,E7M�]��6OWOWXT['M S�E7RT['Q ��['R6R)q p E7R6R {�DWR��6EHG�U�E7R�G^E7DWR e�DWR>UPMI['S�DWR �6E7M,�z['Z6MPZ�E7DWU�GPUI[�]�E7OTDFG�Ur�6b}OFGIL&DFG�U�e�M�]4�6OWOWXT['MP�mE7DWU�E7R>U�G�jIZ�E7DF�?XT['Mr�

� 'c!BA5w,!#'*+#'-� � �CDWR���E7DFGP�6DFE7O7;?� � DFGPU�E7M�]��6OWOWXT['Mr�

;?��� E7D#k>DFE7OFE7R&b�U�LN{�E7R �n��DFGPUy�?DFEHGxJdE7M�] ['Z6M�E7R&GIE7Z6M ['Q?]4p E7R��?DWS��

2n k/E7DWOFE7R ��'E7k#UPO � � S�E7VHDFE7OWU�E��#Q�jIZ�E � qV�� �,���� E7DW{ b}R>UPM;['S�['Q?] ��[NG,JdLNM��6D_��OWLN{ GPU�E7Z�E7R E7DWR p E7D�c�EHGrq�E7DWR M�LNU�EHG,Q6R�� E7DWR X6Ot['Q�EHG wTLNMP{*Q6Ot['M VHQ6M

�z['Z6O ��C�DFE^��MP�?]�Q6R6S�G�LNM��?R>Q6R6S E7R>UPZ�E'OWU�]�LNOWS�E7R��6E J�LNM�G�jIZ6MPD_]�U�E7R��[��

w → ¬b�S�iR � LNMPU�E7R��6� E7M�E7DWR�p E7D�c�EHG}wTLNMP{*Q6Ot['M}E7DWR6M�E7DFjIZ>Urq ��['M�] ��E7DWR�X6Ot['Q�EHG}E7DWR6M�E7DFjIZ�E7R)� �

X �w ∨ rj �r → b ∧ w

b}R6S�E7R�LN{�{�E7Rr = T

q6�T['R6Rb}OFG�L

r = Fq6��['{�DWU�j �

Xp E7S�E7Rp E7S�E7Re�MPS~DWX6U��?DFE E7DWR6VHDWS�E E7M�]��6OWOFE7R��6E4� R>U�E7MP�6M�E7UI['UPDFLNR)��iR:�6E7M�b}Q�GIGI['S�E7R6OFLNS~DW�KZ�E7D�c~U �

=||=� E7D_R6]�[NjIZ � ZT['X/E7R:�?DFEHGIE7OWX/E7R:hsL>�6E7OWOFE � /['Q�jIZ:��[NG}� v R6R�E7R�p�DWM

{�DWU��z['Z6MPZ�E7DWU�GPUI[�]�E7OWR��?Q6M�jIZ6�6M�LNX6DFE7MIE7R)����������������! "�$#&%'%'(*)+��%'%-,'�.���0/1 �2.��,'3.45)+�.67(�8�9�4.:�,')+�!%�;

{S|~}(���Z�3|F�Z���Z�,�5�q��| �z�q���5��|��8���~�(�L�L�q�Y�8�q�Z|T� ��}5�5�������q|L������}5|~�����T���5��|

; b.>,�6�#b u edn�`/- u � � � j

� 'c!BA5w,!#'*+�>[1 >p → (q → r) =||= (p ∧ q) → r

p q r q → rOWDWR6�#G

p ∧ qM�EHjIZ>U�G

� � �T T F F F T F

T F T T T F T� � �� � �

F T F F T F T� � �C,[��?DFE*�>�T['OWU�E7R�VHQ�OWDWR6�?G�Q6R�� M�EHjIZ>U�G�S~OFE7DFjIZ�GPDWR���q?]�LNOWS~U}�?DFE � �mQ6DWk~['OFE7R6V��

2

A ≡ p → (q → p)['Q�G���E7DFGP�6DFE7OD;?� � DFGPU�R#Q6M�E7DWR6VrE7OWR�E�wTLNMP{�E7O, � RT['UP�6M�O_DFjIZ � S~DWOWUl['Q�jIZ

¬p →

(q ∧ r → ¬p)�6b�OWOWS�E7{�E7DWR��

2n$%&(py>n1�G Ya"�EB ∈ For

V�hU!o+FE�V��fE +<`�+,��� "�Bp0, p1, ..., pn

$2BQ'4%"�ES"�BA0, ..., An ∈ For . B′

"�BQ+ %�+M"�H;"��$2%

B �

EGB<'&"�� �Z��B�L�"R'&"�%pi

'�$2� (�HAi

"���%�"�+*)�+�����B�BpEG%�+

B′V�h2!�+FE�V

� 'c` 1 � C�DFEHG}S~DWOWU�['RT['OFLNS ['Q�jIZ)qTp E7R6RA0, ..., An

wTLNMP{�E7O_Rs�6E7M�['OWOWS�E7{�E7DWR�E7R .i� s '�E *D� +M"�B�!��RV�E * G�DWR��aB′

Z�E7D�c~U}��['R6R ��$2%n%�� V&"�B�!��nV�E %�(�H V�hU!�+FE�V~�2

� 'Zt�'c!BA � � E7MPU�E7R p�DWMB′

Q6R>U�E7M*E7DWR�E7M�� R>U�E7MP�6M�E7UI['UPDFLNRI ′['Q�Grq)X/EHGPUPDW{�{�E7Rzp�DWMrq�LNX

I ′ |= Ai

L>�6E7MR6DFjIZ>Ur�

I ′ |= B′Z�E'R6S~U�R6DFjIZ>U�kmLNR

Ai

q�G�LNR��6E7MPR R>Q6M�kmLNR �?DFEHG�E7Roe�MPS�E7X6R6DFG�G�E7RY['X)q��?DFE {�DWU→

Q6R��¬�mLN{*X6DWR6DFE7MPU�p E7M��6E7R)�/C�� Z)�

I ′ |= B′S�E7RT['Q ��['R6R)q p E7R6R

I |= BqTp LNX/E7D

pIi = T

S�E7RT['Qs��['R6Rsp E7R6RI ′ |= Ai

�C,[

I |= BDW{�{�E7M}S~DWOWUrq6DFGPU

B′S~��O_U�D_S��

]������Z��!d"��U��YDWM�X/E7UPMI[NjIZ>U�E7R�R>Q6M}�?DFE^e�M�G�E7UPVHQ6R6SaE7DWR�EHG}b�U�LN{�G

p ∈ P0 �?Q6M�jIZ A ∈ For�

� 'c!BA5w,!#'*+�� edM�G�E7UPVrEp�?Q6M�jIZ

p0 → p1DWR

(p → p0) → ¬p�

{S|~}(���Z�3|F�Z���Z�,�5�q��| �z�q���5��|��8���~�(�L�L�q�Y�8�q�Z|T� ��}5�5�������q|L������}5|~�����T���5��|

; b.>,�6�#b u edn�`/- u � � �(r

�>UPMPQ6�>UPQ6M�E7OWOFE C,E���R6DWUPDFLNR�kmLNR��i��%�"�+*)�$2BUV��/k QB ∈ For

G�E7DB′

�6E���R6DFE7MPU}p�DFE,]�LNOWS~U�

D �B ≡ q ∈ P

[��q ≡ p : B′ :≡

X �q 6≡ p : B′ :≡

DWD �B ≡ ¬B1 : B′ :≡ ¬(B′

1)

DWDWD �B ≡ B1 → B2 : B′ :≡

�#E7DI ′X/E7OWDFE7X6DWS�E#� R>U�E7MP�6M�E7UI['UPDFLNR��p�DWM}�6E���R6DFE7M�E7R

Iq?DWR��6E7{�p�DWM�]��6M}['OWOFE

q ∈ PG�E7UPVrE7R��

qI :=

qI′ p E7R6Rq 6≡ p

Tp E7R6R

q ≡ pQ6R��

I ′ |= A

FG�LNR�GPU

� E7Z)� �6wy�6M�['OWOFEB ∈ For

S~DWOWU�I |= B ⇔ I ′ |= B′

� E7p E7DFG �6�?Q�M�jIZ�GPUPMPQ6�#UPQ6M�E7OWOFE4� R��?Q6�>UPDFLNR��6X/E7MB

D � [��B ≡ p : I |= B ⇔ pI = T ⇐⇒ ������� I � . . . ⇐⇒ ��������� ������������������ I ′ |= B′

X �B ≡ q 6≡ p : I |= B ⇔ qI = T ⇔ . . . ⇔ I ′ |= B′

DWD �B ≡ ¬B1 : I |= B ⇔ I 6|= B1 . . . ⇔ I ′ |= ¬(B′

1) ⇔ I ′ |= B′

DWDWD � ['RT['OFLNS�]��6M→

8�E7UPVHU�GIE7Z�E7R p�DWM�?wy�6M�['OWOFEI ′S~DWOWU

I ′ |= B′S�E7RT['Q ��['R6R)q?p E7R6R ]4�6M�['OWOWE

I ′S~DWOWU

I |= B�;� GPU�['OFG�L

BS~�6OWUPDWS�qTG�L�['Q�jIZB′�

2

� 'c!BA5w,!#'*+�>[1J� 1�� -���&(A('Z&(pZ47)7.�2 �Tn,[NjIZ:�?['UPV�;?� =�S~DWOWU�]��6M}['OWOFEA, B ∈ For

�A → (B → A)

b}RT['OFLNS VHQ ;?�<=�]�LNOWS~U�['QTGB1 =||= B2

['Q�jIZB′

1 =||= B′2

�yn�[NjIZ���E7DFGP�6DFE7O ;?�<;:DFGPU�['OFG�L�]4�6M�['OWOFEA, B, C ∈ For

�A → (B → C) =||= A ∧ B → C

�?DFEsVHp E7DWU�E wTLNMP{�E7O,DFGPU p LNZ6O,X/EHG�G�E7M VHQkmE7M�GPU�E7Z�E7R)�

{S|~}(���Z�3|F�Z���Z�,�5�q��| �z�q���5��|��8���~�(�L�L�q�Y�8�q�Z|T� ��}5�5�������q|L������}5|~�����T���5��|

; b.>,�6�#b u edn�`/- u � � �@

� ��������

� EF+M"�� ��!�" �p,¬p

{�DWUp ∈ P0

e#!���$&%"�!a�l1 ∨ · · · ∨ ln

{�DWU}`�DWU�E7MI['OFE7Rli�

e�DWR�E wTLNMP{�E7O>DFG�U�DWR*X6VHp^��E7DWR�E.*D��BL�$UB)*9+FE qD"�WM�M[��4�����Z��! ������� �4��LNR )H� n�wd�!qrp E7R6R�GPDFE�E7DWR�E ��LNR )�Q6R6�#UPDFLNRkmLNR �,Ot['Q�G�E7OWR DFGPU V�� �,�

(p1 ∨ ¬p2 ∨ p3) ∧ (¬p1 ∨ p3) ∧ (p2 ∨ ¬p3)����e�M�]��6OWOWXT['MP�mE7DWU DFGPU G�jIZ>p E7M,VHQ

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T['X��

� 'c!BA5w,!#'*+��p ∨ q → ¬q =||=

D

p q p ∨ q ¬q •

T T T F F

T F T T T

F T T F F

F F F T T

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nb�U�LN{�E7R

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;�iR?]�LNM�g

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2

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A

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I |= Ai

2

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2

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M 6|= A�

2

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M ` AQ6R��

M ` ¬Aq���['R6R�S~DWOWU}p�S��/�?['UPV�;?� �~� ['Q�jIZ

M |= AQ6R��

M |= ¬AQ6R�����['{�DWU

I |= AQ6R��

I |= ¬A3a�YDF�6E7M�GP�6MPQ�jIZ�t

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2

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��EF+M ′ |= A �

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2

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ga��[Dc )�EH�6EHG�E7R��?OWDFjIZ�EM ′ ⊆ M

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