               ,BOOK ,,II
                   #A

  ,WE H ALR EXPLA9$ ! NUMB] (! FIGURES1
! "* & NUMB] (! PREMISSES1 :5 & H[ A
SYLLOGISM IS =M$2 FUR!R :AT WE M/ LOOK =
:5 A REFUT+ & E/ABLI%+ PROPOSI;NS1 & H[
WE %D 9VE/IGATE A GIV5 PRO#M 9 ANY BRAN*
( 9QUIRY1 AL 0:AT M1NS WE % OBTA9
PR9CIPLES APPROPRIATE 6EA* SUBJECT4
,S9CE "S SYLLOGISMS >E UNIV]SAL1 O!RS
"PICUL>1 ALL ! UNIV]SAL SYLLOGISMS GIVE
M ?AN "O RESULT1 &( "PICUL> SYLLOGISMS !
A6IRMATIVE YIELD M ?AN "O1 ! NEGATIVE
YIELD ONLY ! /AT$ 3CLU.N4 ,= ALL
PROPOSI;NS >E 3V]TI# SAVE ONLY ! "PICUL>
NEGATIVE3 &! 3CLU.N /ATES "O DEF9ITE ?+
AB ANO!R DEF9ITE ?+4 ,3SEQU5TLY ALL
SYLLOGISMS SAVE ! "PICUL> NEGATIVE YIELD
M ?AN "O 3CLU.N1 E4G4 IF ,A HAS BE5
PROV$ 66ALL OR 6"S ;,B1 !N ;,B M/ 2L;G
6"S ,A3 & IF ,A HAS BE5 PROV$ 62L;G 6NO
;,B1 !N ;,B 2L;GS 6NO ,A4 ,? IS A DI6]5T
3CLU.N F ! =M]4 ,B IF ,A DOES N 2L;G 6"S
;,B1 X IS N NEC T ;,B %D N 2L;G 6"S ,A3
= X MAY POSSIBLY 2L;G 6ALL ,A4
  ,? !N IS ! R1SON -MON 6ALL SYLLOGISMS
:E!R UNIV]SAL OR "PICUL>4 ,B X IS POSSI#
6GIVE ANO!R R1SON 3C]N+ ^? : >E
UNIV]SAL4 ,= ALL ! ?+S T >E SUBORD9ATE
6! MI4LE T]M OR 6! 3CLU.N MAY 2 PROV$ 0!
SAME SYLLOGISM1 IF ! =M] >E PLAC$ 9 !
MI4LE1 ! LATT] 9 ! 3CLU.N2 E4G4 IF !
3CLU.N ;,,AB IS PROV$ "? ;,C1 :AT"E IS
SUBORD9ATE TO ;,B OR ;,C M/ A3EPT !
PR$ICATE ,A3 = IF ;,D IS 9CLUD$ 9 ;,B Z
9 A :OLE1 & ;,B IS 9CLUD$ 9 ;,A1 !N ;,D
W 2 9CLUD$ 9 ,A4 ,AG IF ;,E IS 9CLUD$ 9
;,C Z 9 A :OLE1 & ;,C IS 9CLUD$ 9 ;,A1
!N ;,E W 2 9CLUD$ 9 ,A4 ,SIMIL>LY IF !
SYLLOGISM IS NEGATIVE4 ,9 ! SECOND
FIGURE X W 2 POSSI# 69F] ONLY T : IS
SUBORD9ATE 6! 3CLU.N1 E4G4 IF ,A 2L;GS
6NO ;,B & 6ALL ;,C2 WE 3CLUDE T ;,B
2L;GS 6NO ,C4 ,IF !N ;,D IS SUBORD9ATE
TO ;,C1 CLE>LY ;,B DOES N 2L;G 6X4 ,B T
;,B DOES N 2L;G 6:AT IS SUBORD9ATE TO
;,A IS N CLE> 0M1NS (! SYLLOGISM4 ,& YET
;,B DOES N 2L;G TO ;,E1 IF ;,E IS
SUBORD9ATE TO ;,A4 ,B :ILE X HAS BE5
PROV$ "? ! SYLLOGISM T ;,B 2L;GS 6NO
;,C1 X HAS BE5 ASSUM$ )\T PRO( T ;,B
DOES N 2L;G TO ;,A1 3SEQU5TLY X DOES N
RESULT "? ! SYLLOGISM T ;,B DOES N 2L;G
TO ;,E4
  ,B 9 "PICUL> SYLLOGISMS "! W 2 NO
NECESS;Y ( 9F]R+ :AT IS SUBORD9ATE 6!
3CLU.N 7=A SYLLOGISM DOES N RESULT :5 ?
PREMISS IS "PICUL>71 B :AT"E IS
SUBORD9ATE 6! MI4LE T]M MAY 2 9F]R$1 N
H["E "? ! SYLLOGISM1 E4G4 IF ,A 2L;GS
6ALL ;,B & ;,B 6"S ,C4 ,NO?+ C 2 9F]R$
AB T : IS SUBORD9ATE TO ;,C2 "S?+ C 2
9F]R$ AB T : IS SUBORD9ATE TO ;,B1 B N
"? ! PREC$+ SYLLOGISM4 ,SIMIL>LY 9 ! O!R
FIGURES4 ,T : IS SUBORD9ATE 6! 3CLU.N _C
2 PROV$2 ! O!R SUBORD9ATE C 2 PROV$1
ONLY N "? ! SYLLOGISM1 J Z 9 ! UNIV]SAL
SYLLOGISMS :AT IS SUBORD9ATE 6! MI4LE
T]M IS PROV$ 7Z WE SAW7 F A PREMISS : IS
N DEMON/RAT$3 3SEQU5TLY EI A 3CLU.N IS N
POSSI# 9 ! CASE ( UNIV]SAL SYLLOGISMS OR
ELSE X IS POSSI# AL 9 ! CASE ( "PICUL>
SYLLOGISMS4

                   #B

  ,X IS POSSI# =! PREMISSES (! SYLLOGISM
6BE TRUE1 OR 6BE FALSE1 OR 6BE ! "O
TRUE1 ! O!R FALSE4 ,! 3CLU.N IS EI TRUE
OR FALSE NECESS>ILY4 ,F TRUE PREMISSES X
IS N POSSI# 6DRAW A FALSE 3CLU.N1 B A
TRUE 3CLU.N MAY 2 DRAWN F FALSE
PREMISSES1 TRUE H["E ONLY 9 RESPECT 6!
FACT1 N 6! R1SON4 ,! R1SON _C 2 E/ABLI%$
F FALSE PREMISSES3 :Y ? IS S W 2 EXPLA9$
9 ! SEQUEL4
  ,F/ !N T X IS N POSSI# 6DRAW A FALSE
3CLU.N F TRUE PREMISSES1 IS MADE CLE> 0?
3SID],N4 ,IF X IS NEC T ;,B %D 2 :5 ,A
IS1 X IS NEC T ,A %D N 2 :5 ;,B IS N4
,IF !N ,A IS TRUE1 ;,B M/ 2 TRUE3
O!RWISE X W TURN \ T ! SAME ?+ BO? IS &
IS N AT ! SAME "T4 ,B ? IS IMPOSSI#4
,LET X N1 2C ,A IS LAID D[N Z A S+LE
T]M1 2 SUPPOS$ T X IS POSSI#1 :5 A S+LE
FACT IS GIV51 T "S?+ %D NECESS>ILY
RESULT4 ,= T IS N POSSI#4 ,= :AT RESULTS
NECESS>ILY IS ! 3CLU.N1 &! M1NS 0: ? -ES
AB >E AT ! L1/ ?REE T]MS1 & TWO REL,NS (
SUBJECT & PR$ICATE OR PREMISSES4 ,IF !N
X IS TRUE T ,A 2L;GS 6ALL T 6: ;,B
2L;GS1 & T ;,B 2L;GS 6ALL T 6: ;,C
2L;GS1 X IS NEC T ,A %D 2L;G 6ALL T 6:
;,C 2L;GS1 & ? _C 2 FALSE3 = !N ! SAME
?+ W 2L;G & N 2L;G AT ! SAME "T4 ,S ,A
IS POSIT$ Z "O ?+1 2+ TWO PREMISSES TAK5
TGR4 ,! SAME HOLDS GD ( NEGATIVE
SYLLOGISMS3 X IS N POSSI# 6PROVE A FALSE
3CLU.N F TRUE PREMISSES4
  ,B F :AT IS FALSE A TRUE 3CLU.N MAY 2
DRAWN1 :E!R BO? ! PREMISSES >E FALSE OR
ONLY "O1 PROVID$ T ? IS N EI (!
PREMISSES 9DI6]5TLY1 IF X IS TAK5 Z
:OLLY FALSE3 B IF ! PREMISS IS N TAK5 Z
:OLLY FALSE1 X DOES N MATT] : (! TWO IS
FALSE4

7#A7 ,LET ,A 2L;G 6! :OLE ( ;,C1 B 6N"O
  (! ,BS1 NEI LET ;,B 2L;G TO ;,C4 ,? IS
  POSSI#1 E4G4 ANIMAL 2L;GS 6NO /"O1 NOR
  /"O 6ANY MAN4 ,IF !N ,A IS TAK5 62L;G
  6ALL ;,B & ;,B 6ALL ;,C1 ,A W 2L;G
  6ALL ;,C2 3SEQU5TLY ?\< BO? !
  PREMISSES >E FALSE ! 3CLU.N IS TRUE3 =
  E MAN IS AN ANIMAL4 ,SIMIL>LY )!
  NEGATIVE4 ,= X IS POSSI# T NEI ,A NOR
  ;,B %D 2L;G 6ANY ;,C1 AL? ,A 2L;GS
  6ALL ;,B1 E4G4 IF ! SAME T]MS >E TAK5
  & MAN IS PUT Z MI4LE3 = NEI ANIMAL NOR
  MAN 2L;GS 6ANY /"O1 B ANIMAL 2L;GS 6E
  MAN4 ,3SEQU5TLY IF "O T]M IS TAK5
  62L;G 6N"O ( T 6: X DOES 2L;G1 &! O!R
  T]M IS TAK5 62L;G 6ALL ( T 6: X DOES N
  2L;G1 ?\< BO? ! PREMISSES >E FALSE !
  3CLU.N W 2 TRUE4
7#B7 ,A SIMIL> PRO( MAY 2 GIV5 IF EA*
  PREMISS IS "PI,Y FALSE4
7#C7 ,B IF "O ONLY (! PREMISSES IS
  FALSE1 :5 ! F/ PREMISS IS :OLLY FALSE1
  E4G4 ;,,AB1 ! 3CLU.N W N 2 TRUE1 B IF
  ! PREMISS ,,BC IS :OLLY FALSE1 A TRUE
  3CLU.N W 2 POSSI#4 ,I M1N BY ,8:OLLY
  FALSE0' ! 3TR>Y (! TRU?1 E4G4 IF :AT
  2L;GS 6N"O IS ASSUM$ 62L;G 6ALL1 OR IF
  :AT 2L;GS 6ALL IS ASSUM$ 62L;G 6N"O4
  ,LET ,A 2L;G 6NO ;,B1 & ;,B 6ALL ,C4
  ,IF !N ! PREMISS ,,BC : ,I TAKE IS
  TRUE1 &! PREMISS ;,,AB IS :OLLY FALSE1
  VIZ4 T ,A 2L;GS 6ALL ;,B1 X IS
  IMPOSSI# T ! 3CLU.N %D 2 TRUE3 = ,A
  2L;G$ 6N"O (! ,C'S1 S9CE ,A 2L;G$
  6NO?+ 6: ;,B 2L;G$1 & ;,B 2L;G$ 6ALL
  ,C4 ,SIMIL>LY "! _C 2 A TRUE 3CLU.N IF
  ,A 2L;GS 6ALL ;,B1 & ;,B 6ALL ;,C1 B
  :ILE ! TRUE PREMISS ,,BC IS ASSUM$1 !
  :OLLY FALSE PREMISS ;,,AB IS AL
  ASSUM$1 VIZ4 T ,A 2L;GS 6NO?+ 6: ;,B
  2L;GS3 "H ! 3CLU.N M/ 2 FALSE4 ,= ,A W
  2L;G 6ALL ;,C1 S9CE ,A 2L;GS 6"EY?+ 6:
  ;,B 2L;GS1 & ;,B 6ALL ,C4 ,X IS CLE>
  !N T :5 ! F/ PREMISS IS :OLLY FALSE1
  :E!R A6IRMATIVE OR NEGATIVE1 &! O!R
  PREMISS IS TRUE1 ! 3CLU.N _C 2 TRUE4
7#D7 ,B IF ! PREMISS IS N :OLLY FALSE1 A
  TRUE 3CLU.N IS POSSI#4 ,= IF ,A 2L;GS
  6ALL ;,C & 6"S ;,B1 & IF ;,B 2L;GS
  6ALL ;,C1 E4G4 ANIMAL 6E SWAN & 6"S
  :ITE ?+1 & :ITE 6E SWAN1 !N IF WE TAKE
  Z PREMISSES T ,A 2L;GS 6ALL ;,B1 & ;,B
  6ALL ;,C1 ,A W 2L;G 6ALL ;,C TRULY3 =
  E SWAN IS AN ANIMAL4 ,SIMIL>LY IF !
  /ATE;T ;,,AB IS NEGATIVE4 ,= X IS
  POSSI# T ,A %D 2L;G 6"S ;,B & 6NO ;,C1
  & T ;,B %D 2L;G 6ALL ;,C1 E4G4 ANIMAL
  6"S :ITE ?+1 B 6NO SN[1 & :ITE 6ALL
  SN[4 ,IF !N "O %D ASSUME T ,A 2L;GS
  6NO ;,B1 & ;,B 6ALL ;,C1 !N W 2L;G 6NO
  ,C4
7#E7 ,B IF ! PREMISS ;,,AB1 : IS ASSUM$1
  IS :OLLY TRUE1 &! PREMISS ,,BC IS
  :OLLY FALSE1 A TRUE SYLLOGISM W 2
  POSSI#3 = NO?+ PREV5TS ,A 2L;G+ 6ALL
  ;,B & 6ALL ;,C1 ?\< ;,B 2L;GS 6NO ;,C1
  E4G4 ^! 2+ SPECIES (! SAME G5US : >E N
  SUBORD9ATE "O 6! O!R3 = ANIMAL 2L;GS
  BO? 6HORSE & 6MAN1 B HORSE 6NO MAN4
  ,IF !N X IS ASSUM$ T ,A 2L;GS 6ALL ;,B
  & ;,B 6ALL ;,C1 ! 3CLU.N W 2 TRUE1 AL?
  ! PREMISS ,,BC IS :OLLY FALSE4
  ,SIMIL>LY IF ! PREMISS ;,,AB IS
  NEGATIVE4 ,= X IS POSSI# T ,A %D 2L;G
  NEI 6ANY ;,B NOR 6ANY ;,C1 & T ;,B %D
  N 2L;G 6ANY ;,C1 E4G4 A G5US 6SPECIES
  ( ANO!R G5US3 = ANIMAL 2L;GS NEI
  6MUSIC NOR 6! >T ( H1L+1 NOR DOES
  MUSIC 2L;G 6! >T ( H1L+4 ,IF !N X IS
  ASSUM$ T ,A 2L;GS 6NO ;,B1 & ;,B 6ALL
  ;,C1 ! 3CLU.N W 2 TRUE4
7#F7 ,& IF ! PREMISS ,,BC IS N :OLLY
  FALSE B 9 "P ONLY1 EV5 S ! 3CLU.N MAY
  2 TRUE4 ,= NO?+ PREV5TS ,A 2L;G+ 6!
  :OLE ( ;,B &( ;,C1 :ILE ;,B 2L;GS 6"S
  ;,C1 E4G4 A G5US 6XS SPECIES & DI6];E3
  = ANIMAL 2L;GS 6E MAN & 6E FOOT$ ?+1 &
  MAN 6"S FOOT$ ?+S ?\< N 6ALL4 ,IF !N X
  IS ASSUM$ T ,A 2L;GS 6ALL ;,B1 & ;,B
  6ALL ;,C1 ,A W 2L;G 6ALL ;,C3 & ? EX
  HYPO!SI IS TRUE4 ,SIMIL>LY IF !
  PREMISS ;,,AB IS NEGATIVE4 ,= X IS
  POSSI# T ,A %D NEI 2L;G 6ANY ;,B NOR
  6ANY ;,C1 ?\< ;,B 2L;GS 6"S ;,C1 E4G4
  A G5US 6! SPECIES ( ANO!R G5US & XS
  DI6];E3 = ANIMAL NEI 2L;GS 6ANY WISDOM
  NOR 6ANY 9/.E ( ,8SPECULATIVE0'1 B
  WISDOM 2L;GS 6"S 9/.E (
  ,8SPECULATIVE0'4 ,IF !N X %D 2 ASSUM$
  T ,A 2L;GS 6NO ;,B1 & ;,B 6ALL ;,C1 W
  2L;G 6NO ;,C3 & ? EX HYPO!SI IS TRUE4
  ,9 "PICUL> SYLLOGISMS X IS POSSI# :5 !
  F/ PREMISS IS :OLLY FALSE1 &! O!R
  TRUE1 T ! 3CLU.N %D 2 TRUE2 AL :5 ! F/
  PREMISS IS FALSE 9 "P1 &! O!R TRUE2 &
  :5 ! F/ IS TRUE1 &! "PICUL> IS FALSE2
  & :5 BO? >E FALSE4
7#G7 ,= NO?+ PREV5TS ,A 2L;G+ 6NO ;,B1 B
  6"S ;,C1 & ;,B 6"S ;,C1 E4G4 ANIMAL
  2L;GS 6NO SN[1 B 6"S :ITE ?+1 & SN[
  6"S :ITE ?+4 ,IF !N SN[ IS TAK5 Z
  MI4LE1 & ANIMAL Z F/ T]M1 & X IS
  ASSUM$ T ,A 2L;GS 6! :OLE ( ;,B1 & ;,B
  6"S ;,C1 !N ! PREMISS ,,BC IS :OLLY
  FALSE1 ! PREMISS ,,BC TRUE1 &! 3CLU.N
  TRUE4 ,SIMIL>LY IF ! PREMISS ;,,AB IS
  NEGATIVE3 = X IS POSSI# T ,A %D 2L;G
  6! :OLE ( ;,B1 B N 6"S ;,C1 AL? ;,B
  2L;GS 6"S ;,C1 E4G4 ANIMAL 2L;GS 6E
  MAN1 B DOES N FOLL[ "S :ITE1 B MAN
  2L;GS 6"S :ITE2 3SEQU5TLY IF MAN 2
  TAK5 Z MI4LE T]M & X IS ASSUM$ T ,A
  2L;GS 6NO ;,B B ;,B 2L;GS 6"S ;,C1 !
  3CLU.N W 2 TRUE AL? ! PREMISS ;,,AB IS
  :OLLY FALSE4
7#H7 ,IF ! PREMISS ;,,AB IS FALSE 9 "P1
  ! 3CLU.N MAY 2 TRUE4 ,= NO?+ PREV5TS
  ,A 2L;G+ BO? TO ;,B & 6"S ;,C1 & ;,B
  2L;G+ 6"S ;,C1 E4G4 ANIMAL 6"S?+
  B1UTI;L & 6"S?+ GRT1 & B1UTI;L 2L;G+
  6"S?+ GRT4 ,IF !N ,A IS ASSUM$ 62L;G
  6ALL ;,B1 & ;,B 6"S ;,C1 ! PREMISS
  ;,,AB W 2 "PI,Y FALSE1 ! PREMISS ,,BC
  W 2 TRUE1 &! 3CLU.N TRUE4 ,SIMIL>LY IF
  ! PREMISS ;,,AB IS NEGATIVE4 ,=! SAME
  T]MS W S]VE1 & 9 ! SAME POSI;NS1
  6PROVE ! PO9T4
7#I7 ,AG IF ! PREMISS ;,,AB IS TRUE1 &!
  PREMISS ,,BC IS FALSE1 ! 3CLU.N MAY 2
  TRUE4 ,= NO?+ PREV5TS ,A 2L;G+ 6! :OLE
  ( ;,B & 6"S ;,C1 :ILE ;,B 2L;GS 6NO
  ;,C1 E4G4 ANIMAL 6E SWAN & 6"S BLACK
  ?+S1 ?\< SWAN 2L;GS 6NO BLACK ?+4
  ,3SEQU5TLY IF X %D 2 ASSUM$ T ,A 2L;GS
  6ALL ;,B1 & ;,B 6"S ;,C1 ! 3CLU.N W 2
  TRUE1 AL? ! /ATE;T ,,BC IS FALSE4
  ,SIMIL>LY IF ! PREMISS ;,,AB IS
  NEGATIVE4 ,= X IS POSSI# T ,A %D 2L;G
  6NO ;,B1 & N 6"S ;,C1 :ILE ;,B 2L;GS
  6NO ;,C1 E4G4 A G5US 6! SPECIES (
  ANO!R G5US & 6! A3ID5T ( XS [N
  SPECIES3 = ANIMAL 2L;GS 6NO NUMB] & N
  6"S :ITE ?+S1 & NUMB] 2L;GS 6NO?+
  :ITE4 ,IF !N NUMB] IS TAK5 Z MI4LE1 &
  X IS ASSUM$ T ,A 2L;GS 6NO ;,B1 & ;,B
  6"S ;,C1 !N ,A W N 2L;G 6"S ;,C1 : EX
  HYPO!SI IS TRUE4 ,&! PREMISS ;,,AB IS
  TRUE1 ! PREMISS ,,BC FALSE4
7#AJ7 ,AL IF ! PREMISS ;,,AB IS "PI,Y
  FALSE1 &! PREMISS ,,BC IS FALSE TOO1 !
  3CLU.N MAY 2 TRUE4 ,= NO?+ PREV5TS ,A
  2L;G+ 6"S ;,B & 6"S ;,C1 ?\< ;,B 2L;GS
  6NO ;,C1 E4G4 IF ;,B IS ! 3TR>Y ( ;,C1
  & BO? >E A3ID5TS (! SAME G5US3 =
  ANIMAL 2L;GS 6"S :ITE ?+S & 6"S BLACK
  ?+S1 B :ITE 2L;GS 6NO BLACK ?+4 ,IF !N
  X IS ASSUM$ T ,A 2L;GS 6ALL ;,B1 & ;,B
  6"S ;,C1 ! 3CLU.N W 2 TRUE4 ,SIMIL>LY
  IF ! PREMISS ;,,AB IS NEGATIVE3 =!
  SAME T]MS >RANG$ 9 ! SAME WAY W S]VE =
  ! PRO(4
7#AA7 ,AL ?\< BO? PREMISSES >E FALSE !
  3CLU.N MAY 2 TRUE4 ,= X IS POSSI# T ,A
  MAY 2L;G 6NO ;,B & 6"S ;,C1 :ILE ;,B
  2L;GS 6NO ;,C1 E4G4 A G5US 9 REL,N 6!
  SPECIES ( ANO!R G5US1 & 6! A3ID5T ( XS
  [N SPECIES3 = ANIMAL 2L;GS 6NO NUMB]1
  B 6"S :ITE ?+S1 & NUMB] 6NO?+ :ITE4
  ,IF !N X IS ASSUM$ T ,A 2L;GS 6ALL ;,B
  & ;,B 6"S ;,C1 ! 3CLU.N W 2 TRUE1 ?\<
  BO? PREMISSES >E FALSE4 ,SIMIL>LY AL
  IF ! PREMISS ;,,AB IS NEGATIVE4 ,=
  NO?+ PREV5TS ,A 2L;G+ 6! :OLE ( ;,B1 &
  N 6"S ;,C1 :ILE ;,B 2L;GS 6NO ;,C1
  E4G4 ANIMAL 2L;GS 6E SWAN1 & N 6"S
  BLACK ?+S1 & SWAN 2L;GS 6NO?+ BLACK4
  ,3SEQU5TLY IF X IS ASSUM$ T ,A 2L;GS
  6NO ;,B1 & ;,B 6"S ;,C1 !N ,A DOES N
  2L;G 6"S ,C4 ,! 3CLU.N !N IS TRUE1 B !
  PREMISSES >C FALSE4

                   #C

  ,9 ! MI4LE FIGURE X IS POSSI# 9 E WAY
6R1* A TRUE 3CLU.N "? FALSE PREMISSES1
:E!R ! SYLLOGISMS >E UNIV]SAL OR
"PICUL>1 VIZ4 :5 BO? PREMISSES >E :OLLY
FALSE2 :5 EA* IS "PI,Y FALSE2 :5 "O IS
TRUE1 ! O!R :OLLY FALSE 7X DOES N MATT]
: (! TWO PREMISSES IS FALSE72 IF BO?
PREMISSES >E "PI,Y FALSE2 IF "O IS Q
TRUE1 ! O!R "PI,Y FALSE2 IF "O IS :OLLY
FALSE1 ! O!R "PI,Y TRUE4 ,=

7#A7 IF ,A 2L;GS 6NO ;,B & 6ALL ;,C1
  E4G4 ANIMAL 6NO /"O & 6E HORSE1 !N IF
  ! PREMISSES >E /AT$ 3TR>IWISE & X IS
  ASSUM$ T ,A 2L;GS 6ALL ;,B & 6NO ;,C1
  ?\< ! PREMISSES >E :OLLY FALSE !Y W
  YIELD A TRUE 3CLU.N4 ,SIMIL>LY IF ,A
  2L;GS 6ALL ;,B & 6NO ;,C3 = WE % H !
  SAME SYLLOGISM4
7#B7 ,AG IF "O PREMISS IS :OLLY FALSE1 !
  O!R :OLLY TRUE3 = NO?+ PREV5TS ,A
  2L;G+ 6ALL ;,B & 6ALL ;,C1 ?\< ;,B
  2L;GS 6NO ;,C1 E4G4 A G5US 6XS CO-
  ORD9ATE SPECIES4 ,= ANIMAL 2L;GS 6E
  HORSE & MAN1 & NO MAN IS A HORSE4 ,IF
  !N X IS ASSUM$ T ANIMAL 2L;GS 6ALL (!
  "O1 & N"O (! O!R1 ! "O PREMISS W 2
  :OLLY FALSE1 ! O!R :OLLY TRUE1 &!
  3CLU.N W 2 TRUE :I*"E T]M ! NEGATIVE
  /ATE;T 3C]NS4
7#C7 ,AL IF "O PREMISS IS "PI,Y FALSE1 !
  O!R :OLLY TRUE4 ,= X IS POSSI# T ,A %D
  2L;G 6"S ;,B & 6ALL ;,C1 ?\< ;,B 2L;GS
  6NO ;,C1 E4G4 ANIMAL 6"S :ITE ?+S & 6E
  RAV51 ?\< :ITE 2L;GS 6NO RAV54 ,IF !N
  X IS ASSUM$ T ,A 2L;GS 6NO ;,B1 B 6!
  :OLE ( ;,C1 ! PREMISS ;,,AB IS "PI,Y
  FALSE1 ! PREMISS ;,,AC :OLLY TRUE1 &!
  3CLU.N TRUE4 ,SIMIL>LY IF ! NEGATIVE
  /ATE;T IS TRANSPOS$3 ! PRO( C 2 MADE
  0M1NS (! SAME T]MS4 ,AL IF !
  A6IRMATIVE PREMISS IS "PI,Y FALSE1 !
  NEGATIVE :OLLY TRUE1 A TRUE 3CLU.N IS
  POSSI#4 ,= NO?+ PREV5TS ,A 2L;G+ 6"S
  ;,B1 B N TO ;,C Z A :OLE1 :ILE ;,B
  2L;GS 6NO ;,C1 E4G4 ANIMAL 2L;GS 6"S
  :ITE ?+S1 B 6NO PIT*1 & :ITE 2L;GS 6NO
  PIT*4 ,3SEQU5TLY IF X IS ASSUM$ T ,A
  2L;GS 6! :OLE ( ;,B1 B 6NO ;,C1 !
  PREMISS ;,,AB IS "PI,Y FALSE1 !
  PREMISS ;,,AC IS :OLLY TRUE1 &! 3CLU.N
  IS TRUE4
7#D7 ,& IF BO? ! PREMISSES >E "PI,Y
  FALSE1 ! 3CLU.N MAY 2 TRUE4 ,= X IS
  POSSI# T ,A %D 2L;G 6"S ;,B & 6"S ;,C1
  & ;,B 6NO ;,C1 E4G4 ANIMAL 6"S :ITE
  ?+S & 6"S BLACK ?+S1 ?\< :ITE 2L;GS
  6NO?+ BLACK4 ,IF !N X IS ASSUM$ T ,A
  2L;GS 6ALL ;,B & 6NO ;,C1 BO?
  PREMISSES >E "PI,Y FALSE1 B ! 3CLU.N
  IS TRUE4 ,SIMIL>LY1 IF ! NEGATIVE
  PREMISS IS TRANSPOS$1 ! PRO( C 2 MADE
  0M1NS (! SAME T]MS4 ,X IS CLE> AL T \R
  !SIS HOLDS 9 "PICUL> SYLLOGISMS4 ,=
7#E7 NO?+ PREV5TS ,A 2L;G+ 6ALL ;,B &
  6"S ;,C1 ?\< ;,B DOES N 2L;G 6"S ;,C1
  E4G4 ANIMAL 6E MAN & 6"S :ITE ?+S1 ?\<
  MAN W N 2L;G 6"S :ITE ?+S4 ,IF !N X IS
  /AT$ T ,A 2L;GS 6NO ;,B & 6"S ;,C1 !
  UNIV]SAL PREMISS IS :OLLY FALSE1 !
  "PICUL> PREMISS IS TRUE1 &! 3CLU.N IS
  TRUE4 ,SIMIL>LY IF ! PREMISS ;,,AB IS
  A6IRMATIVE3 = X IS POSSI# T ,A %D 2L;G
  6NO ;,B1 & N 6"S ;,C1 ?\< ;,B DOES N
  2L;G 6"S ;,C1 E4G4 ANIMAL 2L;GS 6NO?+
  LIFE.S1 & DOES N 2L;G 6"S :ITE ?+S1 &
  LIFE.S W N 2L;G 6"S :ITE ?+S4 ,IF !N X
  IS /AT$ T ,A 2L;GS 6ALL ;,B & N 6"S
  ;,C1 ! PREMISS ;,,AB : IS UNIV]SAL IS
  :OLLY FALSE1 ! PREMISS ;,,AC IS TRUE1
  &! 3CLU.N IS TRUE4 ,AL A TRUE 3CLU.N
  IS POSSI# :5 ! UNIV]SAL PREMISS IS
  TRUE1 &! "PICUL> IS FALSE4 ,= NO?+
  PREV5TS ,A FOLL[+ NEI ;,B NOR ;,C AT
  ALL1 :ILE ;,B DOES N 2L;G 6"S ;,C1
  E4G4 ANIMAL 2L;GS 6NO NUMB] NOR 6ANY?+
  LIFE.S1 & NUMB] DOES N FOLL[ "S LIFE.S
  ?+S4 ,IF !N X IS /AT$ T ,A 2L;GS 6NO
  ;,B & 6"S ;,C1 ! 3CLU.N W 2 TRUE1 &!
  UNIV]SAL PREMISS TRUE1 B ! "PICUL>
  FALSE4 ,SIMIL>LY IF ! PREMISS : IS
  /AT$ UNIV]S,Y IS A6IRMATIVE4 ,= X IS
  POSSI# T %D ,A 2L;G BO? TO ;,B & TO
  ;,C Z :OLES1 ?\< ;,B DOES N FOLL[ "S
  ;,C1 E4G4 A G5US 9 REL,N 6XS SPECIES &
  DI6];E3 = ANIMAL FOLL[S E MAN & FOOT$
  ?+S Z A :OLE1 B MAN DOES N FOLL[ E
  FOOT$ ?+4 ,3SEQU5TLY IF X IS ASSUM$ T
  ,A 2L;GS 6! :OLE ( ;,B1 B DOES N 2L;G
  6"S ;,C1 ! UNIV]SAL PREMISS IS TRUE1 !
  "PICUL> FALSE1 &! 3CLU.N TRUE4
7#F7 ,X IS CLE> TOO T ?\< BO? PREMISSES
  >E FALSE !Y MAY YIELD A TRUE 3CLU.N1
  S9CE X IS POSSI# T ,A %D 2L;G BO? TO
  ;,B & TO ;,C Z :OLES1 ?\< ;,B DOES N
  FOLL[ "S ,C4 ,= IF X IS ASSUM$ T ,A
  2L;GS 6NO ;,B & 6"S ;,C1 ! PREMISSES
  >E BO? FALSE1 B ! 3CLU.N IS TRUE4
  ,SIMIL>LY IF ! UNIV]SAL PREMISS IS
  A6IRMATIVE &! "PICUL> NEGATIVE4 ,= X
  IS POSSI# T ,A %D FOLL[ NO ;,B & ALL
  ;,C1 ?\< ;,B DOES N 2L;G 6"S ;,C1 E4G4
  ANIMAL FOLL[S NO SCI;E B E MAN1 ?\<
  SCI;E DOES N FOLL[ E MAN4 ,IF !N ,A IS
  ASSUM$ 62L;G 6! :OLE ( ;,B1 & N 6FOLL[
  "S ;,C1 ! PREMISSES >E FALSE B !
  3CLU.N IS TRUE4

                   #D

  ,9 ! LA/ FIGURE A TRUE 3CLU.N MAY -E
"? :AT IS FALSE1 ALIKE :5 BO? PREMISSES
>E :OLLY FALSE1 :5 EA* IS "PLY FALSE1 :5
"O PREMISS IS :OLLY TRUE1 ! O!R FALSE1
:5 "O PREMISS IS "PLY FALSE1 ! O!R :OLLY
TRUE1 & VICE V]SA1 & 9 E O!R WAY 9 : X
IS POSSI# 6ALT] ! PREMISSES4 ,=

7#A7 NO?+ PREV5TS NEI ,A NOR ;,B F 2L;G+
  6ANY ;,C1 :ILE ,A 2L;GS 6"S ;,B1 E4G4
  NEI MAN NOR FOOT$ FOLL[S ANY?+ LIFE.S1
  ?\< MAN 2L;GS 6"S FOOT$ ?+S4 ,IF !N X
  IS ASSUM$ T ;,A & ;,B 2L;G 6ALL ;,C1 !
  PREMISSES W 2 :OLLY FALSE1 B ! 3CLU.N
  TRUE4 ,SIMIL>LY IF "O PREMISS IS
  NEGATIVE1 ! O!R A6IRMATIVE4 ,= X IS
  POSSI# T ;,B %D 2L;G 6NO ;,C1 B ,A
  6ALL ;,C1 & T %D N 2L;G 6"S ;,B1 E4G4
  BLACK 2L;GS 6NO SWAN1 ANIMAL 6E SWAN1
  & ANIMAL N 6"EY?+ BLACK4 ,3SEQU5TLY IF
  X IS ASSUM$ T ;,B 2L;GS 6ALL ;,C1 & ,A
  6NO ;,C1 ,A W N 2L;G 6"S ;,B3 &!
  3CLU.N IS TRUE1 ?\< ! PREMISSES >E
  FALSE4
7#B7 ,AL IF EA* PREMISS IS "PLY FALSE1 !
  3CLU.N MAY 2 TRUE4 ,= NO?+ PREV5TS BO?
  ;,A & ;,B F 2L;G+ 6"S ;,C :ILE ,A
  2L;GS 6"S ;,B1 E4G4 :ITE & B1UTI;L
  2L;G 6"S ANIMALS1 & :ITE 6"S B1UTI;L
  ?+S4 ,IF !N X IS /AT$ T ;,A & ;,B 2L;G
  6ALL ;,C1 ! PREMISSES >E "PI,Y FALSE1
  B ! 3CLU.N IS TRUE4 ,SIMIL>LY IF !
  PREMISS ;,,AC IS /AT$ Z NEGATIVE4 ,=
  NO?+ PREV5TS ,A F N 2L;G+1 & ;,B F
  2L;G+1 6"S ;,C1 :ILE ,A DOES N 2L;G
  6ALL ;,B1 E4G4 :ITE DOES N 2L;G 6"S
  ANIMALS1 B1UTI;L 2L;GS 6"S ANIMALS1 &
  :ITE DOES N 2L;G 6"EY?+ B1UTI;L4
  ,3SEQU5TLY IF X IS ASSUM$ T ,A 2L;GS
  6NO ;,C1 & ;,B 6ALL ;,C1 BO? PREMISSES
  >E "PLY FALSE1 B ! 3CLU.N IS TRUE4
7#C7 ,SIMIL>LY IF "O (! PREMISSES ASSUM$
  IS :OLLY FALSE1 ! O!R :OLLY TRUE4 ,= X
  IS POSSI# T BO? ;,A & ;,B %D FOLL[ ALL
  ;,C1 ?\< ,A DOES N 2L;G 6"S ;,B1 E4G4
  ANIMAL & :ITE FOLL[ E SWAN1 ?\< ANIMAL
  DOES N 2L;G 6"EY?+ :ITE4 ,TAK+ ^! !N Z
  T]MS1 IF "O ASSUMES T ;,B 2L;GS 6!
  :OLE ( ;,C1 B ,A DOES N 2L;G TO ;,C AT
  ALL1 ! PREMISS ,,BC W 2 :OLLY TRUE1 !
  PREMISS ;,,AC :OLLY FALSE1 &! 3CLU.N
  TRUE4 ,SIMIL>LY IF ! /ATE;T ,,BC IS
  FALSE1 ! /ATE;T ;,,AC TRUE1 ! 3CLU.N
  MAY 2 TRUE4 ,! SAME T]MS W S]VE =!
  PRO(4 ,AL IF BO? ! PREMISSES ASSUM$ >E
  A6IRMATIVE1 ! 3CLU.N MAY 2 TRUE4 ,=
  NO?+ PREV5TS ;,B F FOLL[+ ALL ;,C1 &
  ,A F N 2L;G+ TO ;,C AT ALL1 ?\< ,A
  2L;GS 6"S ;,B1 E4G4 ANIMAL 2L;GS 6E
  SWAN1 BLACK 6NO SWAN1 & BLACK 6"S
  ANIMALS4 ,3SEQU5TLY IF X IS ASSUM$ T
  ;,A & ;,B 2L;G 6E ;,C1 ! PREMISS ,,BC
  IS :OLLY TRUE1 ! PREMISS ;,,AC IS
  :OLLY FALSE1 &! 3CLU.N IS TRUE4
  ,SIMIL>LY IF ! PREMISS ;,,AC : IS
  ASSUM$ IS TRUE3 ! PRO( C 2 MADE "? !
  SAME T]MS4
7#D7 ,AG IF "O PREMISS IS :OLLY TRUE1 !
  O!R "PLY FALSE1 ! 3CLU.N MAY 2 TRUE4
  ,= X IS POSSI# T ;,B %D 2L;G 6ALL ;,C1
  & ,A 6"S ;,C1 :ILE ,A 2L;GS 6"S ;,B1
  E4G4 BIP$ 2L;GS 6E MAN1 B1UTI;L N 6E
  MAN1 & B1UTI;L 6"S BIP$S4 ,IF !N X IS
  ASSUM$ T BO? ;,A & ;,B 2L;G 6! :OLE (
  ;,C1 ! PREMISS ,,BC IS :OLLY TRUE1 !
  PREMISS ;,,AC "PLY FALSE1 ! 3CLU.N
  TRUE4 ,SIMIL>LY IF (! PREMISSES ASSUM$
  ;,,AC IS TRUE & ,,BC "PLY FALSE1 A
  TRUE 3CLU.N IS POSSI#3 ? C 2 PROV$1 IF
  ! SAME T]MS Z 2F >E TRANSPOS$4 ,AL !
  3CLU.N MAY 2 TRUE IF "O PREMISS IS
  NEGATIVE1 ! O!R A6IRMATIVE4 ,= S9CE X
  IS POSSI# T ;,B %D 2L;G 6! :OLE ( ;,C1
  & ,A 6"S ;,C1 &1 :5 !Y >E S1 T ,A %D N
  2L;G 6ALL ;,B1 "!=E X IS ASSUM$ T ;,B
  2L;GS 6! :OLE ( ;,C1 & ,A 6NO ;,C1 !
  NEGATIVE PREMISS IS "PLY FALSE1 ! O!R
  PREMISS :OLLY TRUE1 &! 3CLU.N IS TRUE4
  ,AG S9CE X HAS BE5 PROV$ T IF ,A 2L;GS
  6NO ;,C & ;,B 6"S ;,C1 X IS POSSI# T
  ,A %D N 2L;G 6"S ;,C1 X IS CLE> T IF !
  PREMISS ;,,AC IS :OLLY TRUE1 &!
  PREMISS ,,BC "PLY FALSE1 X IS POSSI# T
  ! 3CLU.N %D 2 TRUE4 ,= IF X IS ASSUM$
  T ,A 2L;GS 6NO ;,C1 & ;,B 6ALL ;,C1 !
  PREMISS ;,,AC IS :OLLY TRUE1 &!
  PREMISS ,,BC IS "PLY FALSE4
7#E7 ,X IS CLE> AL 9 ! CASE ( "PICUL>
  SYLLOGISMS T A TRUE 3CLU.N MAY -E "?
  :AT IS FALSE1 9 E POSSI# WAY4 ,=! SAME
  T]MS M/ 2 TAK5 Z H BE5 TAK5 :5 !
  PREMISSES >E UNIV]SAL1 POSITIVE T]MS 9
  POSITIVE SYLLOGISMS1 NEGATIVE T]MS 9
  NEGATIVE4 ,= X MAKES NO DI6];E 6!
  SETT+ \ (! T]MS1 :E!R "O ASSUMES T :AT
  2L;GS 6N"O 2L;GS 6ALL OR T :AT 2L;GS
  6"S 2L;GS 6ALL4 ,! SAME APPLIES
  6NEGATIVE /ATE;TS4

  ,X IS CLE> !N T IF ! 3CLU.N IS FALSE1
! PREMISSES (! >GU;T M/ 2 FALSE1 EI ALL
OR "S ( !M2 B :5 ! 3CLU.N IS TRUE1 X IS
N NEC T ! PREMISSES %D 2 TRUE1 EI "O OR
ALL1 YET X IS POSSI#1 ?\< NO "P (!
SYLLOGISM IS TRUE1 T ! 3CLU.N MAY N"O !
LESS 2 TRUE2 B X IS N NECESSITAT$4 ,!
R1SON IS T :5 TWO ?+S >E S RELAT$ 6"O
ANO!R1 T IF ! "O IS1 ! O!R NECESS>ILY
IS1 !N IF ! LATT] IS N1 ! =M] W N 2 EI1
B IF ! LATT] IS1 X IS N NEC T ! =M] %D
BE4 ,B X IS IMPOSSI# T ! SAME ?+ %D 2
NECESSITAT$ 0! 2+ & 0! N-2+ (! SAME ?+4
,I M1N1 = EXAMPLE1 T X IS IMPOSSI# T ;,B
%D NECESS>ILY 2 GRT S9CE ,A IS :ITE & T
;,B %D NECESS>ILY 2 GRT S9CE ,A IS N
:ITE4 ,= :5"E S9CE ?1 ;,A1 IS :ITE X IS
NEC T T1 ;,B1 %D 2 GRT1 & S9CE ;,B IS
GRT T ;,C %D N 2 :ITE1 !N X IS NEC IF IS
:ITE T ;,C %D N 2 :ITE4 ,& :5"E X IS
NEC1 S9CE "O ( TWO ?+S IS1 T ! O!R %D
BE1 X IS NEC1 IF ! LATT] IS N1 T ! =M]
7VIZ4 ,A7 %D N BE4 ,IF !N ;,B IS N GRT
,A _C 2 :ITE4 ,B IF1 :5 ,A IS N :ITE1 X
IS NEC T ;,B %D 2 GRT1 X NECESS>ILY
RESULTS T IF ;,B IS N GRT1 ;,B XF IS
GRT4 7,B ? IS IMPOSSI#47 ,= IF ;,B IS N
GRT1 ,A W NECESS>ILY N 2 :ITE4 ,IF !N :5
? IS N :ITE ;,B M/ 2 GRT1 X RESULTS T IF
;,B IS N GRT1 X IS GRT1 J Z IF X 7 PROV$
"? ?REE T]MS4


                   #E

  ,CIRCUL> & RECIPROCAL PRO( M1NS PRO(
0M1NS (! 3CLU.N1 I4E4 03V]T+ "O (!
PREMISSES SIMPLY & 9F]R+ ! PREMISS : 0
ASSUM$ 9 ! ORIG9AL SYLLOGISM3 E4G4
SUPPOSE X HAS BE5 NEC 6PROVE T ,A 2L;GS
6ALL ;,C1 & X HAS BE5 PROV$ "? ;,B2
SUPPOSE T ,A %D N[ 2 PROV$ 62L;G TO ;,B
0ASSUM+ T ,A 2L;GS TO ;,C1 & ;,C TO ;,B-
S ,A 2L;GS TO ;,B3 B 9 ! F/ SYLLOGISM !
3V]SE 0 ASSUM$1 VIZ4 T ;,B 2L;GS TO ;,C4
,OR SUPPOSE X IS NEC 6PROVE T ;,B 2L;GS
TO ;,C1 & ,A IS ASSUM$ 62L;G TO ;,C1 : 0
! 3CLU.N (! F/ SYLLOGISM1 & ;,B 62L;G TO
;,A B ! 3V]SE 0 ASSUM$ 9 ! E>LI]
SYLLOGISM1 VIZ4 T ,A 2L;GS TO ;,B4 ,9 NO
O!R WAY IS RECIPROCAL PRO( POSSI#4 ,IF
ANO!R T]M IS TAK5 Z MI4LE1 ! PRO( IS N
CIRCUL>3 = NEI (! PROPOSI;NS ASSUM$ IS !
SAME Z 2F3 IF "O (! A3EPT$ T]MS IS TAK5
Z MI4LE1 ONLY "O (! PREMISSES (! F/
SYLLOGISM C 2 ASSUM$ 9 ! SECOND3 = IF
BO? ( !M >E TAK5 ! SAME 3CLU.N Z 2F W
RESULT3 B X M/ 2 DI6]5T4 ,IF ! T]MS >E N
3V]TI#1 "O (! PREMISSES F : ! SYLLOGISM
RESULTS M/ 2 UNDEMON/RAT$3 = X IS N
POSSI# 6DEMON/RATE "? ^! T]MS T ! ?IRD
2L;GS 6! MI4LE OR ! MI4LE 6! F/4 ,IF !
T]MS >E 3V]TI#1 X IS POSSI# 6DEMON/RATE
"EY?+ RECIPROC,Y1 E4G4 IF ;,A & ;,B &
;,C >E 3V]TI# ) "O ANO!R4 ,SUPPOSE !
PROPOSI;N ;,,AC HAS BE5 DEMON/RAT$ "?
;,B Z MI4LE T]M1 & AG ! PROPOSI;N ;,,AB
"? ! 3CLU.N &! PREMISS ,,BC 3V]T$1 &
SIMIL>LY ! PROPOSI;N ,,BC "? ! 3CLU.N &!
PREMISS ;,,AB 3V]T$4 ,B X IS NEC 6PROVE
BO? ! PREMISS ,,CB1 &! PREMISS ,,BA3 =
WE H US$ ^! AL"O )\T DEMON/RAT+ !M4 ,IF
!N X IS ASSUM$ T ;,B 2L;GS 6ALL ;,C1 &
;,C 6ALL ;,A1 WE % H A SYLLOGISM RELAT+
;,B TO ;,A4 ,AG IF X IS ASSUM$ T ;,C
2L;GS 6ALL ;,A1 & ,A 6ALL ;,B1 ;,C M/
2L;G 6ALL ,B4 ,9 BO? ^! SYLLOGISMS !
PREMISS ,,CA HAS BE5 ASSUM$ )\T 2+
DEMON/RAT$3 ! O!R PREMISSES _H EX
HYPO!SI BE5 PROV$4 ,3SEQU5TLY IF WE
SU3E$ 9 DEMON/RAT+ ? PREMISS1 ALL !
PREMISSES W H BE5 PROV$ RECIPROC,Y4 ,IF
!N X IS ASSUM$ T ;,C 2L;GS 6ALL ;,B1 &
;,B 6ALL ;,A1 BO? ! PREMISSES ASSUM$ H
BE5 PROV$1 & ;,C M/ 2L;G TO ;,A4 ,X IS
CLE> !N T ONLY IF ! T]MS >E 3V]TI# IS
CIRCUL> & RECIPROCAL DEMON/R,N POSSI#
7IF ! T]MS >E N 3V]TI#1 ! MATT] /&S Z WE
SD ABV74 ,B X TURNS \ 9 ^! AL T WE USE =
! DEMON/R,N ! V ?+ T IS 2+ PROV$3 = ;,C
IS PROV$ ( ;,B1 & ;,B ( 0ASSUM+ T ;,C IS
SD (& ;,C IS PROV$ ( ,A "? ^! PREMISSES1
S T WE USE ! 3CLU.N =! DEMON/R,N4
  ,9 NEGATIVE SYLLOGISMS RECIPROCAL PRO(
IS Z FOLL[S4 ,LET ;,B 2L;G 6ALL ;,C1 &
,A 6N"O (! ,BS3 WE 3CLUDE T ,A 2L;GS
6N"O (! ,C'S4 ,IF AG X IS NEC 6PROVE T
,A 2L;GS 6N"O (! ,BS 7: 0 PREVI\SLY
ASSUM$7 ,A M/ 2L;G 6NO ;,C1 & ;,C 6ALL
;,B3 ?US ! PREVI\S PREMISS IS REV]S$4
,IF X IS NEC 6PROVE T ;,B 2L;GS TO ;,C1
! PROPOSI;N ;,,AB M/ NO L;G] 2 3V]T$ Z
2F3 =! PREMISS ',B 2L;GS 6NO ,A0' IS
ID5TICAL )! PREMISS ',A 2L;GS 6NO ,B0'4
,B WE M/ ASSUME T ;,B 2L;GS 6ALL ( T
6N"O ( : L;GS4 ,LET ,A 2L;G 6N"O (! ,C'S
7: 0 ! PREVI\S 3CLU.N7 & ASSUME T ;,B
2L;GS 6ALL ( T 6N"O ( : ,A 2L;GS4 ,X IS
NEC !N T ;,B %D 2L;G 6ALL ,C4 ,3SEQU5TLY
EA* (! ?REE PROPOSI;NS HAS BE5 MADE A
3CLU.N1 & ? IS CIRCUL> DEMON/R,N1
6ASSUME ! 3CLU.N &! 3V]SE ( "O (!
PREMISSES1 & DEDUCE ! REMA9+ PREMISS4
  ,9 "PICUL> SYLLOGISMS X IS N POSSI#
6DEMON/RATE ! UNIV]SAL PREMISS "? ! O!R
PROPOSI;NS1 B ! "PICUL> PREMISS C 2
DEMON/RAT$4 ,CLE>LY X IS IMPOSSI#
6DEMON/RATE ! UNIV]SAL PREMISS3 = :AT IS
UNIV]SAL IS PROV$ "? PROPOSI;NS : >E
UNIV]SAL1 B ! 3CLU.N IS N UNIV]SAL1 &!
PRO( M/ />T F ! 3CLU.N &! O!R PREMISS4
,FUR!R A SYLLOGISM _C 2 MADE AT ALL IF !
O!R PREMISS IS 3V]T$3 =! RESULT IS T BO?
PREMISSES >E "PICUL>4 ,B ! "PICUL>
PREMISS MAY 2 PROV$4 ,SUPPOSE T ,A HAS
BE5 PROV$ ( "S ;,C "? ,B4 ,IF !N X IS
ASSUM$ T ;,B 2L;GS 6ALL ;,A &! 3CLU.N IS
RETA9$1 ;,B W 2L;G 6"S ;,C3 = WE OBTA9 !
F/ FIGURE & ,A IS MI4LE4 ,B IF !
SYLLOGISM IS NEGATIVE1 X IS N POSSI#
6PROVE ! UNIV]SAL PREMISS1 =! R1SON GIV5
ABV4 ,B X IS POSSI# 6PROVE ! "PICUL>
PREMISS1 IF ! PROPOSI;N ;,,AB IS 3V]T$ Z
9 ! UNIV]SAL SYLLOGISM1 I4;E ',B 2L;GS
6"S ( T 6"S ( : ,A DOES N 2L;G0'3
O!RWISE NO SYLLOGISM RESULTS 2C !
"PICUL> PREMISS IS NEGATIVE4

                   #F

  ,9 ! SECOND FIGURE X IS N POSSI#
6PROVE AN A6IRMATIVE PROPOSI;N 9 ? WAY1
B A NEGATIVE PROPOSI;N MAY 2 PROV$4 ,AN
A6IRMATIVE PROPOSI;N IS N PROV$ 2C BO?
PREMISSES (! NEW SYLLOGISM >E N
A6IRMATIVE 7=! 3CLU.N IS NEGATIVE7 B AN
A6IRMATIVE PROPOSI;N IS 7Z WE SAW7 PROV$
F PREMISSES : >E BO? A6IRMATIVE4 ,!
NEGATIVE IS PROV$ Z FOLL[S4 ,LET ,A 2L;G
6ALL ;,B1 & 6NO ;,C3 WE 3CLUDE T ;,B
2L;GS 6NO ,C4 ,IF !N X IS ASSUM$ T ;,B
2L;GS 6ALL ;,A1 X IS NEC T ,A %D 2L;G
6NO ;,C3 = WE GET ! SECOND FIGURE1 ) ;,B
Z MI4LE4 ,B IF ! PREMISS ;,,AB 0
NEGATIVE1 &! O!R A6IRMATIVE1 WE % H ! F/
FIGURE4 ,= ;,C 2L;GS 6ALL ;,A & ;,B 6NO
;,C1 3SEQU5TLY ;,B 2L;GS 6NO ,A3 NEI !N
DOES ,A 2L;G TO ;,B4 ,"? ! 3CLU.N1 "!=E1
& "O PREMISS1 WE GET NO SYLLOGISM1 B IF
ANO!R PREMISS IS ASSUM$ 9 A4I;N1 A
SYLLOGISM W 2 POSSI#4 ,B IF ! SYLLOGISM
N UNIV]SAL1 ! UNIV]SAL PREMISS _C 2
PROV$1 =! SAME R1SON Z WE GAVE ABV1 B !
"PICUL> PREMISS C 2 PROV$ :5"E !
UNIV]SAL /ATE;T IS A6IRMATIVE4 ,LET ,A
2L;G 6ALL ;,B1 & N 6ALL ;,C3 ! 3CLU.N IS
,,BC4 ,IF !N X IS ASSUM$ T ;,B 2L;GS
6ALL ;,A1 B N 6ALL ;,C1 ,A W N 2L;G 6"S
;,C1 ;,B 2+ MI4LE4 ,B IF ! UNIV]SAL
PREMISS IS NEGATIVE1 ! PREMISS ;,,AC W N
2 DEMON/RAT$ 0! 3V].N ( ;,,AB3 = X TURNS
\ T EI BO? OR "O (! PREMISSES IS
NEGATIVE2 3SEQU5TLY A SYLLOGISM W N 2
POSSI#4 ,B ! PRO( W PROCE$ Z 9 !
UNIV]SAL SYLLOGISMS1 IF X IS ASSUM$ T ,A
2L;GS 6"S ( T 6"S ( : ;,B DOES N 2L;G4

                   #G

  ,9 ! ?IRD FIGURE1 :5 BO? PREMISSES >E
TAK5 UNIV]S,Y1 X IS N POSSI# 6PROVE !M
RECIPROC,Y3 = T : IS UNIV]SAL IS PROV$
"? /ATE;TS : >E UNIV]SAL1 B ! 3CLU.N 9 ?
FIGURE IS ALW "PICUL>1 S T X IS CLE> T X
IS N POSSI# AT ALL 6PROVE "? ? FIGURE !
UNIV]SAL PREMISS4 ,B IF "O PREMISS IS
UNIV]SAL1 ! O!R "PICUL>1 PRO( (! LATT] W
"S"TS 2 POSSI#1 "S"TS N4 ,:5 BO? !
PREMISSES ASSUM$ >E A6IRMATIVE1 &!
UNIV]SAL 3C]NS ! M9OR EXTREME1 PRO( W 2
POSSI#1 B :5 X 3C]NS ! O!R EXTREME1
IMPOSSI#4 ,LET ,A 2L;G 6ALL ;,C & ;,B
6"S ;,C3 ! 3CLU.N IS ! /ATE;T ;,,AB4 ,IF
!N X IS ASSUM$ T ;,C 2L;GS 6ALL ;,A1 X
HAS BE5 PROV$ T ;,C 2L;GS 6"S ;,B1 B T
;,B 2L;GS 6"S ;,C HAS N BE5 PROV$4 ,&
YET X IS NEC1 IF ;,C 2L;GS 6"S ;,B1 T
;,B %D 2L;G 6"S ,C4 ,B X IS N ! SAME T ?
%D 2L;G 6T1 & T 6?3 B WE M/ ASSUME 2SS T
IF ? 2L;GS 6"S ( T1 T 2L;GS 6"S ( ?4 ,B
IF ? IS ASSUM$ ! SYLLOGISM NO L;G]
RESULTS F ! 3CLU.N &! O!R PREMISS4 ,B IF
;,B 2L;GS 6ALL ;,C1 & ,A 6"S ;,C1 X W 2
POSSI# 6PROVE ! PROPOSI;N ;,,AC1 :5 X IS
ASSUM$ T ;,C 2L;GS 6ALL ;,B1 & ,A 6"S
,B4 ,= IF ;,C 2L;GS 6ALL ;,B & ,A 6"S
;,B1 X IS NEC T ,A %D 2L;G 6"S ;,C1 ;,B
2+ MI4LE4 ,& :5"E "O PREMISS IS
A6IRMATIVE ! O!R NEGATIVE1 &! A6IRMATIVE
IS UNIV]SAL1 ! O!R PREMISS C 2 PROV$4
,LET ;,B 2L;G 6ALL ;,C1 & ,A N 6"S ;,C3
! 3CLU.N IS T ,A DOES N 2L;G 6"S ,B4 ,IF
!N X IS ASSUM$ FUR!R T ;,C 2L;GS 6ALL
;,B1 X IS NEC T ,A %D N 2L;G 6"S ;,C1
;,B 2+ MI4LE4 ,B :5 ! NEGATIVE PREMISS
IS UNIV]SAL1 ! O!R PREMISS IS N EXCEPT Z
2F1 VIZ4 IF X IS ASSUM$ T T 2L;GS 6"S (
T1 6"S ( : ? DOES N 2L;G1 E4G4 IF ,A
2L;GS 6NO ;,C1 & ;,B 6"S ;,C3 ! 3CLU.N
IS T ,A DOES N 2L;G 6"S ,B4 ,IF !N X IS
ASSUM$ T ;,C 2L;GS 6"S ( T 6"S ( : DOES
N 2L;G1 X IS NEC T ;,C %D 2L;G 6"S (!
,BS4 ,9 NO O!R WAY IS X POSSI# 03V]T+ !
UNIV]SAL PREMISS 6PROVE ! O!R3 = 9 NO
O!R WAY C A SYLLOGISM 2 =M$4
  ,X IS CLE> !N T 9 ! F/ FIGURE
RECIPROCAL PRO( IS MADE BO? "? ! ?IRD &
"? ! F/ FIGURE-IF ! 3CLU.N IS A6IRMATIVE
"? ! F/2 IF ! 3CLU.N IS NEGATIVE "? !
LA/4 ,= X IS ASSUM$ T T 2L;GS 6ALL ( T
6N"O ( : ? 2L;GS4 ,9 ! MI4LE FIGURE1 :5
! SYLLOGISM IS UNIV]SAL1 PRO( IS POSSI#
"? ! SECOND FIGURE & "? ! F/1 B :5
"PICUL> "? ! SECOND &! LA/4 ,9 ! ?IRD
FIGURE ALL PRO(S >E MADE "? XF4 ,X IS
CLE> AL T 9 ! ?IRD FIGURE & 9 ! MI4LE
FIGURE ^? SYLLOGISMS : >E N MADE "? ^?
FIGURES !MVS EI >E N (! NATURE ( CIRCUL>
PRO( OR >E IMP]FECT4

                   #H

  ,63V]T A SYLLOGISM M1NS 6ALT] ! 3CLU.N
& MAKE ANO!R SYLLOGISM 6PROVE T EI !
EXTREME _C 2L;G 6! MI4LE OR ! MI4LE 6!
LA/ T]M4 ,= X IS NEC1 IF ! 3CLU.N HAS
BE5 *ANG$ 96XS OPPOSITE & "O (!
PREMISSES /&S1 T ! O!R PREMISS %D 2
DE/ROY$4 ,= IF X %D /&1 ! 3CLU.N AL M/
/&4 ,X MAKES A DI6];E :E!R ! 3CLU.N IS
3V]T$ 96XS 3TRADICTORY OR 96XS 3TR>Y4
,=! SAME SYLLOGISM DOES N RESULT :I*"E
=M ! 3V].N TAKES4 ,? W 2 MADE CLE> 0!
SEQUEL4 ,03TRADICTORY OPPOSI;N ,I M1N !
OPPOSI;N ( ,86ALL0' TO ,8N 6ALL0'1 &(
,86"S0' TO ,86N"O0'2 03TR>Y OPPOSI;N ,I
M1N ! OPPOSI;N ( ,86ALL0' TO ,86N"O0'1 &
( ,86"S0' TO ,8N 6"S0'4 ,SUPPOSE T ,A
BE5 PROV$ ( ;,C1 "? ;,B Z MI4LE T]M4 ,IF
!N X %D 2 ASSUM$ T ,A 2L;GS 6NO ;,C1 B
6ALL ;,B1 ;,B W 2L;G 6NO ,C4 ,& IF ,A
2L;GS 6NO ;,C1 & ;,B 6ALL ;,C1 ,A W
2L;G1 N 6NO ;,B AT ALL1 B N 6ALL ,B4 ,=
7Z WE SAW7 ! UNIV]SAL IS N PROV$ "? !
LA/ FIGURE4 ,9 A ^W X IS N POSSI#
6REFUTE UNIV]S,Y 03V].N ! PREMISS :
3C]NS ! MAJOR EXTREME3 =! REFUT,N ALW
PROCE$S "? ! ?IRD S9CE X IS NEC 6TAKE
BO? PREMISSES 9 REF];E 6! M9OR EXTREME4
,SIMIL>LY IF ! SYLLOGISM IS NEGATIVE4
,SUPPOSE X HAS BE5 PROV$ T ,A 2L;GS 6NO
;,C "? ,B4 ,!N IF X IS ASSUM$ T ,A 2L;GS
6ALL ;,C1 & 6NO ;,B1 ;,B W 2L;G 6N"O (!
,C'S4 ,& IF ;,A & ;,B 2L;G 6ALL ;,C1 ,A
W 2L;G 6"S ;,B3 B 9 ! ORIG9AL PREMISS X
2L;G$ 6NO ,B4
  ,IF ! 3CLU.N IS 3V]T$ 96XS
3TRADICTORY1 ! SYLLOGISMS W 2
3TRADICTORY & N UNIV]SAL4 ,= "O PREMISS
IS "PICUL>1 S T ! 3CLU.N AL W 2 "PICUL>4
,LET ! SYLLOGISM 2 A6IRMATIVE1 & LET X 2
3V]T$ Z /AT$4 ,!N IF ,A 2L;GS N 6ALL
;,C1 B 6ALL ;,B1 ;,B W 2L;G N 6ALL ,C4
,& IF ,A 2L;GS N 6ALL ;,C1 B ;,B 2L;GS
6ALL ;,C1 ,A W 2L;G N 6ALL ,B4 ,SIMIL>LY
IF ! SYLLOGISM IS NEGATIVE4 ,= IF ,A
2L;GS 6"S ;,C1 & 6NO ;,B1 ;,B W 2L;G1 N
6NO ;,C AT ALL1 B-N 6"S ,C4 ,& IF ,A
2L;GS 6"S ;,C1 & ;,B 6ALL ;,C1 Z 0
ORIG9,Y ASSUM$1 ,A W 2L;G 6"S ,B4
  ,9 "PICUL> SYLLOGISMS :5 ! 3CLU.N IS
3V]T$ 96XS 3TRADICTORY1 BO? PREMISSES
MAY 2 REFUT$1 B :5 X IS 3V]T$ 96XS
3TR>Y1 NEI4 ,=! RESULT IS NO L;G]1 Z 9 !
UNIV]SAL SYLLOGISMS1 REFUT,N 9 : !
3CLU.N R1*$ BY ;,O1 3V].N LACKS
UNIV]SAL;Y1 B NO REFUT,N AT ALL4
,SUPPOSE T ,A HAS BE5 PROV$ ( "S ,C4 ,IF
!N X IS ASSUM$ T ,A 2L;GS 6NO ;,C1 & ;,B
6"S ;,C1 ,A W N 2L;G 6"S ;,B3 & IF ,A
2L;GS 6NO ;,C1 B 6ALL ;,B1 ;,B W 2L;G
6NO ,C4 ,?US BO? PREMISSES >E REFUT$4 ,B
NEI C 2 REFUT$ IF ! 3CLU.N IS 3V]T$ 96XS
3TR>Y4 ,= IF ,A DOES N 2L;G 6"S ;,C1 B
6ALL ;,B1 !N ;,B W N 2L;G 6"S ,C4 ,B !
ORIG9AL PREMISS IS N YET REFUT$3 = X IS
POSSI# T ;,B %D 2L;G 6"S ;,C1 & %D N
2L;G 6"S ,C4 ,! UNIV]SAL PREMISS ;,,AB
_C 2 A6ECT$ 0A SYLLOGISM AT ALL3 = IF ,A
DOES N 2L;G 6"S (! ,C'S1 B ;,B 2L;GS 6"S
(! ,C'S1 NEI (! PREMISSES IS UNIV]SAL4
,SIMIL>LY IF ! SYLLOGISM IS NEGATIVE3 =
IF X %D 2 ASSUM$ T ,A 2L;GS 6ALL ;,C1
BO? PREMISSES >E REFUT$3 B IF ! ASSUMP;N
IS T ,A 2L;GS 6"S ;,C1 NEI PREMISS IS
REFUT$4 ,! PRO( IS ! SAME Z 2F4

                   #I

  ,9 ! SECOND FIGURE X IS N POSSI#
6REFUTE ! PREMISS : 3C]NS ! MAJOR
EXTREME 0E/ABLI%+ "S?+ 3TR>Y 6X1 :I*"E
=M ! 3V].N (! 3CLU.N MAY TAKE4 ,=!
3CLU.N (! REFUT,N W ALW 2 9 ! ?IRD
FIGURE1 & 9 ? FIGURE 7Z WE SAW7 "! IS NO
UNIV]SAL SYLLOGISM4 ,! O!R PREMISS C 2
REFUT$ 9 A MANN] SIMIL> 6! 3V].N3 ,I
M1N1 IF ! 3CLU.N (! F/ SYLLOGISM IS
3V]T$ 96XS 3TR>Y1 ! 3CLU.N (! REFUT,N W
2 ! 3TR>Y (! M9OR PREMISS (! F/1 IF 96XS
3TRADICTORY1 ! 3TRADICTORY4 ,LET ,A 2L;G
6ALL ;,B & 6NO ;,C3 3CLU.N ,,BC4 ,IF !N
X IS ASSUM$ T ;,B 2L;GS 6ALL ;,C1 &!
PROPOSI;N ;,,AB /&S1 ,A W 2L;G 6ALL ;,C1
S9CE ! F/ FIGURE IS PRODUC$4 ,IF ;,B
2L;GS 6ALL ;,C1 & ,A 6NO ;,C1 !N ,A
2L;GS N 6ALL ;,B3 ! FIGURE IS ! LA/4 ,B
IF ! 3CLU.N ,,BC IS 3V]T$ 96XS
3TRADICTORY1 ! PREMISS ;,,AB W 2 REFUT$
Z 2F1 ! PREMISS1 ;,,AC 0XS 3TRADICTORY4
,= IF ;,B 2L;GS 6"S ;,C1 & ,A 6NO ;,C1
!N ,A W N 2L;G 6"S ,B4 ,AG IF ;,B 2L;GS
6"S ;,C1 & ,A 6ALL ;,B1 ,A W 2L;G 6"S
;,C1 S T ! SYLLOGISM RESULTS 9 !
3TRADICTORY (! M9OR PREMISS4 ,A SIMIL>
PRO( C 2 GIV5 IF ! PREMISSES >E
TRANSPOS$ 9 RESPECT ( _! QUAL;Y4
  ,IF ! SYLLOGISM IS "PICUL>1 :5 !
3CLU.N IS 3V]T$ 96XS 3TR>Y NEI PREMISS C
2 REFUT$1 Z AL HAPP5$ 9 ! F/ FIGURE10'
IF ! 3CLU.N IS 3V]T$ 96XS 3TRADICTORY1
BO? PREMISSES C 2 REFUT$4 ,SUPPOSE T ,A
2L;GS 6NO ;,B1 & 6"S ;,C3 ! 3CLU.N IS
,,BC4 ,IF !N X IS ASSUM$ T ;,B 2L;GS 6"S
;,C1 &! /ATE;T ;,,AB /&S1 ! 3CLU.N W 2 T
,A DOES N 2L;G 6"S ,C4 ,B ! ORIG9AL
/ATE;T HAS N BE5 REFUT$3 = X IS POSSI# T
,A %D 2L;G 6"S ;,C & AL N 6"S ,C4 ,AG IF
;,B 2L;GS 6"S ;,C & ,A 6"S ;,C1 NO
SYLLOGISM W 2 POSSI#3 = NEI (! PREMISSES
TAK5 IS UNIV]SAL4 ,3SEQU5TLY ! PROPOSI;N
;,,AB IS N REFUT$4 ,B IF ! 3CLU.N IS
3V]T$ 96XS 3TRADICTORY1 BO? PREMISSES C
2 REFUT$4 ,= IF ;,B 2L;GS 6ALL ;,C1 & ,A
6NO ;,B1 ,A W 2L;G 6NO ;,C3 B X 0 ASSUM$
62L;G 6"S ,C4 ,AG IF ;,B 2L;GS 6ALL ;,C
& ,A 6"S ;,C1 ,A W 2L;G 6"S ,B4 ,! SAME
PRO( C 2 GIV5 IF ! UNIV]SAL /ATE;T IS
A6IRMATIVE4

                  #AJ

  ,9 ! ?IRD FIGURE :5 ! 3CLU.N IS 3V]T$
96XS 3TR>Y1 NEI (! PREMISSES C 2 REFUT$
9 ANY (! SYLLOGISMS1 B :5 ! 3CLU.N IS
3V]T$ 96XS 3TRADICTORY1 BO? PREMISSES
MAY 2 REFUT$ & 9 ALL ! MOODS4 ,SUPPOSE X
HAS BE5 PROV$ T ,A 2L;GS 6"S ;,B1 ;,C 2+
TAK5 Z MI4LE1 &! PREMISSES 2+ UNIV]SAL4
,IF !N X IS ASSUM$ T ,A DOES N 2L;G 6"S
;,B1 B ;,B 2L;GS 6ALL ;,C1 NO SYLLOGISM
IS =M$ AB ;,A & ,C4 ,NOR IF ,A DOES N
2L;G 6"S ;,B1 B 2L;GS 6ALL ;,C1 W A
SYLLOGISM 2 POSSI# AB ;,B & ,C4 ,A
SIMIL> PRO( C 2 GIV5 IF ! PREMISSES >E N
UNIV]SAL4 ,= EI BO? PREMISSES >RIV$ AT
0! 3V].N M/ 2 "PICUL>1 OR ! UNIV]SAL
PREMISS M/ REF] 6! M9OR EXTREME4 ,B WE
F.D T NO SYLLOGISM IS POSSI# ?US EI 9 !
F/ OR 9 ! MI4LE FIGURE4 ,B IF ! 3CLU.N
IS 3V]T$ 96XS 3TRADICTORY1 BO? !
PREMISSES C 2 REFUT$4 ,= IF ,A 2L;GS 6NO
;,B1 & ;,B 6ALL ;,C1 !N ,A 2L;GS 6NO
;,C3 AG IF ,A 2L;GS 6NO ;,B1 & 6ALL ;,C1
;,B 2L;GS 6NO ,C4 ,& SIMIL>LY IF "O (!
PREMISSES IS N UNIV]SAL4 ,= IF ,A 2L;GS
6NO ;,B1 & ;,B 6"S ;,C1 ,A W N 2L;G 6"S
;,C3 IF ,A 2L;GS 6NO ;,B1 & TO ;,C1 ;,B
W 2L;G 6NO ,C4
  ,SIMIL>LY IF ! ORIG9AL SYLLOGISM IS
NEGATIVE4 ,SUPPOSE X HAS BE5 PROV$ T ,A
DOES N 2L;G 6"S ;,B1 ,,BC 2+ A6IRMATIVE1
;,,AC 2+ NEGATIVE3 = X 0 ?US T1 Z WE
SAW1 A SYLLOGISM CD 2 MADE4 ,:5"E !N !
3TR>Y (! 3CLU.N IS ASSUM$ A SYLLOGISM W
N 2 POSSI#4 ,= IF ,A 2L;GS 6"S ;,B1 &
;,B 6ALL ;,C1 NO SYLLOGISM IS POSSI# 7Z
WE SAW7 AB ;,A & ,C4 ,NOR1 IF ,A 2L;GS
6"S ;,B1 & 6NO ;,C1 0 A SYLLOGISM POSSI#
3C]N+ ;,B & ,C4 ,"!=E ! PREMISSES >E N
REFUT$4 ,B :5 ! 3TRADICTORY (! 3CLU.N IS
ASSUM$1 !Y >E REFUT$4 ,= IF ,A 2L;GS
6ALL ;,B1 & ;,B TO ;,C1 ,A 2L;GS 6ALL
;,C3 B ,A 0 SUPPOS$ ORIG9,Y 62L;G 6NO
,C4 ,AG IF ,A 2L;GS 6ALL ;,B1 & 6NO ;,C1
!N ;,B 2L;GS 6NO ;,C3 B X 0 SUPPOS$
62L;G 6ALL ,C4 ,A SIMIL> PRO( IS POSSI#
IF ! PREMISSES >E N UNIV]SAL4 ,= ;,,AC
2COMES UNIV]SAL & NEGATIVE1 ! O!R
PREMISS "PICUL> & A6IRMATIVE4 ,IF !N ,A
2L;GS 6ALL ;,B1 & ;,B 6"S ;,C1 X RESULTS
T ,A 2L;GS 6"S ;,C3 B X 0 SUPPOS$ 62L;G
6NO ,C4 ,AG IF ,A 2L;GS 6ALL ;,B1 & 6NO
;,C1 !N ;,B 2L;GS 6NO ;,C3 B X 0 ASSUM$
62L;G 6"S ,C4 ,IF ,A 2L;GS 6"S ;,B & ;,B
6"S ;,C1 NO SYLLOGISM RESULTS3 NOR YET
IF ,A 2L;GS 6"S ;,B1 & 6NO ,C4 ,?US 9 "O
WAY ! PREMISSES >E REFUT$1 9 ! O!R WAY
!Y >E N4
  ,F :AT HAS BE5 SD X IS CLE> H[ A
SYLLOGISM RESULTS 9 EA* FIGURE :5 !
3CLU.N IS 3V]T$2 :5 A RESULT 3TR>Y 6!
PREMISS1 & :5 A RESULT 3TRADICTORY 6!
PREMISS1 IS OBTA9$4 ,X IS CLE> T 9 ! F/
FIGURE ! SYLLOGISMS >E =M$ "? ! MI4LE &!
LA/ FIGURES1 &! PREMISS : 3C]NS ! M9OR
EXTREME IS ALWAY REFUT$ "? ! MI4LE
FIGURE1 ! PREMISS : 3C]NS ! MAJOR "? !
LA/ FIGURE4 ,9 ! SECOND FIGURE
SYLLOGISMS PROCE$ "? ! F/ &! LA/
FIGURES1 &! PREMISS : 3C]NS ! M9OR
EXTREME IS ALW REFUT$ "? ! F/ FIGURE1 !
PREMISS : 3C]NS ! MAJOR EXTREME "? !
LA/4 ,9 ! ?IRD FIGURE ! REFUT,N PROCE$S
"? ! F/ &! MI4LE FIGURES2 ! PREMISS :
3C]NS ! MAJOR IS ALW REFUT$ "? ! F/
FIGURE1 ! PREMISS : 3C]NS ! M9OR "? !
MI4LE FIGURE4

                  #AA

  ,X IS CLE> !N :AT 3V].N IS1 H[ X IS
E6ECT$ 9 EA* FIGURE1 & :AT SYLLOGISM
RESULTS4 ,! SYLLOGISM P] IMPOSSIBILE IS
PROV$ :5 ! 3TRADICTORY (! 3CLU.N /AT$ &
ANO!R PREMISS IS ASSUM$2 X C 2 MADE 9
ALL ! FIGURES4 ,= X RESEM#S 3V].N1 DI6]+
ONLY 9 ?3 3V].N TAKES PLACE AF A
SYLLOGISM HAS BE5 =M$ & BO? ! PREMISSES
H BE5 TAK51 B A REDUC;N 6! IMPOSSI#
TAKES PLACE N 2C ! 3TRADICTORY HAS BE5
AGRE$ 6ALR1 B 2C X IS CLE> T X IS TRUE4
,! T]MS >E ALIKE 9 BO?1 &! PREMISSES (
BO? >E TAK5 9 ! SAME WAY4 ,= EXAMPLE IF
,A 2L;GS 6ALL ;,B1 ;,C 2+ MI4LE1 !N IF X
IS SUPPOS$ T ,A DOES N 2L;G 6ALL ;,B OR
2L;GS 6NO ;,B1 B 6ALL ;,C 7: 0 ADMITT$
6BE TRUE71 X FOLL[S T ;,C 2L;GS 6NO ;,B
OR N 6ALL ,B4 ,B ? IS IMPOSSI#3
3SEQU5TLY ! SUPPOSI;N IS FALSE3 XS
3TRADICTORY !N IS TRUE4 ,SIMIL>LY 9 !
O!R FIGURES3 = :AT"E MOODS ADMIT ( 3V].N
ADMIT AL (! REDUC;N P] IMPOSSIBILE4
  ,ALL ! PRO#MS C 2 PROV$ P] IMPOSSIBILE
9 ALL ! FIGURES1 EXCEPT+ ! UNIV]SAL
A6IRMATIVE1 : IS PROV$ 9 ! MI4LE & ?IRD
FIGURES1 B N 9 ! F/4 ,SUPPOSE T ,A 2L;GS
N 6ALL ;,B1 OR 6NO ;,B1 & TAKE 2SS ANO!R
PREMISS 3C]N+ EI (! T]MS1 VIZ4 T ;,C
2L;GS 6ALL ;,A1 OR T ;,B 2L;GS 6ALL ;,D2
?US WE GET ! F/ FIGURE4 ,IF !N X IS
SUPPOS$ T ,A DOES N 2L;G 6ALL ;,B1 NO
SYLLOGISM RESULTS :I*"E T]M ! ASSUM$
PREMISS 3C]NS2 B IF X IS SUPPOS$ T ,A
2L;GS 6NO ;,B1 :5 ! PREMISS ,,BD IS
ASSUM$ Z WELL WE % PROVE SYLLOGI/IC,Y
:AT IS FALSE1 B N ! PRO#M PROPOS$4 ,= IF
,A 2L;GS 6NO ;,B1 & ;,B 2L;GS 6ALL ;,D1
,A 2L;GS 6NO ,D4 ,LET ? 2 IMPOSSI#3 X IS
FALSE !N ,A 2L;GS 6NO ,B4 ,B ! UNIV]SAL
A6IRMATIVE IS N NECESS>ILY TRUE IF !
UNIV]SAL NEGATIVE IS FALSE4 ,B IF !
PREMISS ,,CA IS ASSUM$ Z WELL1 NO
SYLLOGISM RESULTS1 NOR DOES X D S :5 X
IS SUPPOS$ T ,A DOES N 2L;G 6ALL ,B4
,3SEQU5TLY X IS CLE> T ! UNIV]SAL
A6IRMATIVE _C 2 PROV$ 9 ! F/ FIGURE P]
IMPOSSIBILE4
  ,B ! "PICUL> A6IRMATIVE &! UNIV]SAL &
"PICUL> NEGATIVES C ALL 2 PROV$4
,SUPPOSE T ,A 2L;GS 6NO ;,B1 & LET X H
BE5 ASSUM$ T ;,B 2L;GS 6ALL OR 6"S ,C4
,!N X IS NEC T ,A %D 2L;G 6NO ;,C OR N
6ALL ,C4 ,B ? IS IMPOSSI# 7= LET X 2
TRUE & CLE> T ,A 2L;GS 6ALL ,C73
3SEQU5TLY IF ? IS FALSE1 X IS NEC T ,A
%D 2L;G 6"S ,B4 ,B IF ! O!R PREMISS
ASSUM$ RELATES TO ;,A1 NO SYLLOGISM W 2
POSSI#4 ,NOR C A 3CLU.N 2 DRAWN :5 !
3TR>Y (! 3CLU.N IS SUPPOS$1 E4G4 T ,A
DOES N 2L;G 6"S ,B4 ,CLE>LY !N WE M/
SUPPOSE ! 3TRADICTORY4
  ,AG SUPPOSE T ,A 2L;GS 6"S ;,B1 & LET
X H BE5 ASSUM$ T ;,C 2L;GS 6ALL ,A4 ,X
IS NEC !N T ;,C %D 2L;G 6"S ,B4 ,B LET ?
2 IMPOSSI#1 S T ! SUPPOSI;N IS FALSE3 9
T CASE X IS TRUE T ,A 2L;GS 6NO ,B4 ,WE
MAY PROCE$ 9 ! SAME WAY IF ! PROPOSI;N
,,CA HAS BE5 TAK5 Z NEGATIVE4 ,B IF !
PREMISS ASSUM$ 3C]NS ;,B1 NO SYLLOGISM W
2 POSSI#4 ,IF ! 3TR>Y IS SUPPOS$1 WE % H
A SYLLOGISM & AN IMPOSSI# 3CLU.N1 B !
PRO#M 9 H& IS N PROV$4 ,SUPPOSE T ,A
2L;GS 6ALL ;,B1 & LET X H BE5 ASSUM$ T
;,C 2L;GS 6ALL ,A4 ,X IS NEC !N T ;,C %D
2L;G 6ALL ,B4 ,B ? IS IMPOSSI#1 S T X IS
FALSE T ,A 2L;GS 6ALL ,B4 ,B WE H N YET
%[N X 6BE NEC T ,A 2L;GS 6NO ;,B1 IF X
DOES N 2L;G 6ALL ,B4 ,SIMIL>LY IF ! O!R
PREMISS TAK5 3C]NS ;,B2 WE % H A
SYLLOGISM &A 3CLU.N : IS IMPOSSI#1 B !
HYPO!SIS IS N REFUT$4 ,"!=E X IS !
3TRADICTORY T WE M/ SUPPOSE4
  ,6PROVE T ,A DOES N 2L;G 6ALL ;,B1 WE
M/ SUPPOSE T X 2L;GS 6ALL ;,B3 = IF ,A
2L;GS 6ALL ;,B1 & ;,C 6ALL ;,A1 !N ;,C
2L;GS 6ALL ;,B2 S T IF ? IS IMPOSSI#1 !
HYPO!SIS IS FALSE4 ,SIMIL>LY IF ! O!R
PREMISS ASSUM$ 3C]NS ,B4 ,! SAME RESULTS
IF ! ORIG9AL PROPOSI;N ,,CA 0 NEGATIVE3
= ?US AL WE GET A SYLLOGISM4 ,B IF !
NEGATIVE PROPOSI;N 3C]NS ;,B1 NO?+ IS
PROV$4 ,IF ! HYPO!SIS IS T ,A 2L;GS N
6ALL B 6"S ;,B1 X IS N PROV$ T ,A 2L;GS
N 6ALL ;,B1 B T X 2L;GS 6NO ,B4 ,= IF ,A
2L;GS 6"S ;,B1 & ;,C 6ALL ;,A1 !N ;,C W
2L;G 6"S ,B4 ,IF !N ? IS IMPOSSI#1 X IS
FALSE T ,A 2L;GS 6"S ;,B2 3SEQU5TLY X IS
TRUE T ,A 2L;GS 6NO ,B4 ,B IF ? IS
PROV$1 ! TRU? IS REFUT$ Z WELL2 =!
ORIG9AL 3CLU.N 0 T ,A 2L;GS 6"S ;,B1 &
DOES N 2L;G 6"S ,B4 ,FUR!R ! IMPOSSI#
DOES N RESULT F ! HYPO!SIS3 = !N !
HYPO!SIS WD 2 FALSE1 S9CE X IS IMPOSSI#
6DRAW A FALSE 3CLU.N F TRUE PREMISSES3 B
9 FACT X IS TRUE3 = ,A 2L;GS 6"S ,B4
,3SEQU5TLY WE M/ N SUPPOSE T ,A 2L;GS
6"S ;,B1 B T X 2L;GS 6ALL ,B4 ,SIMIL>LY
IF WE %D 2 PROV+ T ,A DOES N 2L;G 6"S
;,B3 = IF ,8N 62L;G 6"S0' & ,862L;G N
6ALL0' H ! SAME M1N+1 ! DEMON/R,N ( BO?
W 2 ID5TICAL4
  ,X IS CLE> !N T N ! 3TR>Y B !
3TRADICTORY "\ 6BE SUPPOS$ 9 ALL !
SYLLOGISMS4 ,= ?US WE % H NECESS;Y (
9F];E1 &! CLAIM WE MAKE IS "O T W 2
G5],Y A3EPT$4 ,= IF ( "EY?+ "O OR O!R (
TWO 3TRADICTORY /ATE;TS HOLDS GD1 !N IF
X IS PROV$ T ! NEG,N DOES N HOLD1 !
A6IRM,N M/ 2 TRUE4 ,AG IF X IS N ADMITT$
T ! A6IRM,N IS TRUE1 ! CLAIM T ! NEG,N
IS TRUE W 2 G5],Y A3EPT$4 ,B 9 NEI WAY
DOES X SUIT 6MA9TA9 ! 3TR>Y3 = X IS N
NEC T IF ! UNIV]SAL NEGATIVE IS FALSE1 !
UNIV]SAL A6IRMATIVE %D 2 TRUE1 NOR IS X
G5],Y A3EPT$ T IF ! "O IS FALSE ! O!R IS
TRUE4

                  #AB

  ,X IS CLE> !N T 9 ! F/ FIGURE ALL
PRO#MS EXCEPT ! UNIV]SAL A6IRMATIVE >E
PROV$ P] IMPOSSIBILE4 ,B 9 ! MI4LE &!
LA/ FIGURES ? AL IS PROV$4 ,SUPPOSE T ,A
DOES N 2L;G 6ALL ;,B1 & LET X H BE5
ASSUM$ T ,A 2L;GS 6ALL ,C4 ,IF !N ,A
2L;GS N 6ALL ;,B1 B 6ALL ;,C1 ;,C W N
2L;G 6ALL ,B4 ,B ? IS IMPOSSI# 7=
SUPPOSE X 6BE CLE> T ;,C 2L;GS 6ALL ,B73
3SEQU5TLY ! HYPO!SIS IS FALSE4 ,X IS
TRUE !N T ,A 2L;GS 6ALL ,B4 ,B IF !
3TR>Y IS SUPPOS$1 WE % H A SYLLOGISM &A
RESULT : IS IMPOSSI#3 B ! PRO#M 9 H& IS
N PROV$4 ,= IF ,A 2L;GS 6NO ;,B1 & 6ALL
;,C1 ;,C W 2L;G 6NO ,B4 ,? IS IMPOSSI#2
S T X IS FALSE T ,A 2L;GS 6NO ,B4 ,B ?\<
? IS FALSE1 X DOES N FOLL[ T X IS TRUE T
,A 2L;GS 6ALL ,B4
  ,:5 ,A 2L;GS 6"S ;,B1 SUPPOSE T ,A
2L;GS 6NO ;,B1 & LET ,A 2L;G 6ALL ,C4 ,X
IS NEC !N T ;,C %D 2L;G 6NO ,B4
,3SEQU5TLY1 IF ? IS IMPOSSI#1 ,A M/ 2L;G
6"S ,B4 ,B IF X IS SUPPOS$ T ,A DOES N
2L;G 6"S ;,B1 WE % H ! SAME RESULTS Z 9
! F/ FIGURE4
  ,AG SUPPOSE T ,A 2L;GS 6"S ;,B1 & LET
,A 2L;G 6NO ,C4 ,X IS NEC !N T ;,C %D N
2L;G 6"S ,B4 ,B ORIG9,Y X 2L;G$ 6ALL
;,B1 3SEQU5TLY ! HYPO!SIS IS FALSE3 ,A
!N W 2L;G 6NO ,B4
  ,:5 ,A DOES N 2L;G 6AN ;,B1 SUPPOSE X
DOES 2L;G 6ALL ;,B1 & 6NO ,C4 ,X IS NEC
!N T ;,C %D 2L;G 6NO ,B4 ,B ? IS
IMPOSSI#3 S T X IS TRUE T ,A DOES N 2L;G
6ALL ,B4 ,X IS CLE> !N T ALL !
SYLLOGISMS C 2 =M$ 9 ! MI4LE FIGURE4

                  #AC

  ,SIMIL>LY !Y C ALL 2 =M$ 9 ! LA/
FIGURE4 ,SUPPOSE T ,A DOES N 2L;G 6"S
;,B1 B ;,C 2L;GS 6ALL ;,B3 !N ,A DOES N
2L;G 6"S ,C4 ,IF !N ? IS IMPOSSI#1 X IS
FALSE T ,A DOES N 2L;G 6"S ;,B2 S T X IS
TRUE T ,A 2L;GS 6ALL ,B4 ,B IF X IS
SUPPOS$ T ,A 2L;GS 6NO ;,B1 WE % H A
SYLLOGISM &A 3CLU.N : IS IMPOSSI#3 B !
PRO#M 9 H& IS N PROV$3 = IF ! 3TR>Y IS
SUPPOS$1 WE % H ! SAME RESULTS Z 2F4
  ,B 6PROVE T ,A 2L;GS 6"S ;,B1 ?
HYPO!SIS M/ 2 MADE4 ,IF ,A 2L;GS 6NO
;,B1 & ;,C 6"S ;,B1 ,A W 2L;G N 6ALL ,C4
,IF !N ? IS FALSE1 X IS TRUE T ,A 2L;GS
6"S ,B4
  ,:5 ,A 2L;GS 6NO ;,B1 SUPPOSE ,A 2L;GS
6"S ;,B1 & LET X H BE5 ASSUM$ T ;,C
2L;GS 6ALL ,B4 ,!N X IS NEC T ,A %D 2L;G
6"S ,C4 ,B EX HYPO!SI X 2L;GS 6NO ;,C1 S
T X IS FALSE T ,A 2L;GS 6"S ,B4 ,B IF X
IS SUPPOS$ T ,A 2L;GS 6ALL ;,B1 ! PRO#M
IS N PROV$4
  ,B ? HYPO!SIS M/ 2 MADE IF WE >E PROVE
T ,A 2L;GS N 6ALL ,B4 ,= IF ,A 2L;GS
6ALL ;,B & ;,C 6"S ;,B1 !N ,A 2L;GS 6"S
,C4 ,B ? WE ASSUM$ N 6BE S1 S X IS FALSE
T ,A 2L;GS 6ALL ,B4 ,B 9 T CASE X IS
TRUE T ,A 2L;GS N 6ALL ,B4 ,IF H["E X IS
ASSUM$ T ,A 2L;GS 6"S ;,B1 WE % H ! SAME
RESULT Z 2F4
  ,X IS CLE> !N T 9 ALL ! SYLLOGISMS :
PROCE$ P] IMPOSSIBILE ! 3TRADICTORY M/ 2
ASSUM$4 ,& X IS PLA9 T 9 ! MI4LE FIGURE
AN A6IRMATIVE 3CLU.N1 & 9 ! LA/ FIGURE A
UNIV]SAL 3CLU.N1 >E PROV$ 9 A WAY4


                  #AD

  ,DEMON/R,N P] IMPOSSIBILE DI6]S F
O/5SIVE PRO( 9 T X POSITS :AT X WI%ES
6REFUTE 0REDUC;N 6A /ATE;T ADMITT$ 6BE
FALSE2 ":AS O/5SIVE PRO( />TS F ADMITT$
POSI;NS4 ,BO?1 9DE$1 TAKE TWO PREMISSES
T >E ADMITT$1 B ! LATT] TAKES !
PREMISSES F : ! SYLLOGISM />TS1 ! =M]
TAKES "O ( ^!1 AL;G )! 3TRADICTORY (!
ORIG9AL 3CLU.N4 ,AL 9 ! O/5SIVE PRO( X
IS N NEC T ! 3CLU.N %D 2 "KN1 NOR T "O
%D SUPPOSE 2FH& T X IS TRUE OR N3 9 !
O!R X IS NEC 6SUPPOSE 2FH& T X IS N
TRUE4 ,X MAKES NO DI6];E :E!R ! 3CLU.N
IS A6IRMATIVE OR NEGATIVE2 ! ME?OD IS !
SAME 9 BO? CASES4 ,"EY?+ : IS 3CLUD$
O/5SIVELY C 2 PROV$ P] IMPOSSIBILE1 & T
: IS PROV$ P] IMPOSSIBILE C 2 PROV$
O/5SIVELY1 "? ! SAME T]MS4 ,:5"E !
SYLLOGISM IS =M$ 9 ! F/ FIGURE1 ! TRU? W
2 F.D 9 ! MI4LE OR ! LA/ FIGURE1 IF
NEGATIVE 9 ! MI4LE1 IF A6IRMATIVE 9 !
LA/4 ,:5"E ! SYLLOGISM IS =M$ 9 ! MI4LE
FIGURE1 ! TRU? W 2 F.D 9 ! F/1 :AT"E !
PRO#M MAY BE4 ,:5"E ! SYLLOGISM IS =M$ 9
! LA/ FIGURE1 ! TRU? W 2 F.D 9 ! F/ &
MI4LE FIGURES1 IF A6IRMATIVE 9 F/1 IF
NEGATIVE 9 ! MI4LE4 ,SUPPOSE T ,A HAS
BE5 PROV$ 62L;G 6NO ;,B1 OR N 6ALL ;,B1
"? ! F/ FIGURE4 ,!N ! HYPO!SIS M/ H BE5
T ,A 2L;GS 6"S ;,B1 &! ORIG9AL PREMISSES
T ;,C 2L;GS 6ALL ;,A & 6NO ,B4 ,= ?US !
SYLLOGISM 0 MADE &! IMPOSSI# 3CLU.N
R1*$4 ,B ? IS ! MI4LE FIGURE1 IF ;,C
2L;GS 6ALL ;,A & 6NO ,B4 ,& X IS CLE> F
^! PREMISSES T ,A 2L;GS 6NO ,B4
,SIMIL>LY IF HAS BE5 PROV$ N 62L;G 6ALL
,B4 ,=! HYPO!SIS IS T ,A 2L;GS 6ALL ;,B2
&! ORIG9AL PREMISSES >E T ;,C 2L;GS 6ALL
,A B N 6ALL ,B4 ,SIMIL>LY TOO1 IF !
PREMISS ,,CA %D 2 NEGATIVE3 = ?US AL WE
H ! MI4LE FIGURE4 ,AG SUPPOSE X HAS BE5
PROV$ T ,A 2L;GS 6"S ,B4 ,! HYPO!SIS "H
IS T IS T ,A 2L;GS 6NO ;,B2 &! ORIG9AL
PREMISSES T ;,B 2L;GS 6ALL ;,C1 & ,A EI
6ALL OR 6"S ;,C3 = 9 ? WAY WE % GET :AT
IS IMPOSSI#4 ,B IF ;,A & ;,B 2L;G 6ALL
;,C1 WE H ! LA/ FIGURE4 ,& X IS CLE> F
^! PREMISSES T ,A M/ 2L;G 6"S ,B4
,SIMIL>LY IF ;,B OR ,A %D 2 ASSUM$ 62L;G
6"S ,C4
  ,AG SUPPOSE X HAS BE5 PROV$ 9 ! MI4LE
FIGURE T ,A 2L;GS 6ALL ,B4 ,!N !
HYPO!SIS M/ H BE5 T ,A 2L;GS N 6ALL ;,B1
&! ORIG9AL PREMISSES T ,A 2L;GS 6ALL
;,C1 & ;,C 6ALL ;,B3 = ?US WE % GET :AT
IS IMPOSSI#4 ,B IF ,A 2L;GS 6ALL ;,C1 &
;,C 6ALL ;,B1 WE H ! F/ FIGURE4
,SIMIL>LY IF X HAS BE5 PROV$ T ,A 2L;GS
6"S ;,B3 =! HYPO!SIS !N M/ H BE5 T ,A
2L;GS 6NO ;,B1 &! ORIG9AL PREMISSES T ,A
2L;GS 6ALL ;,C1 & ;,C 6"S ,B4 ,IF !
SYLLOGISM IS NEGATIVE1 ! HYPO!SIS M/ H
BE5 T ,A 2L;GS 6"S ;,B1 &! ORIG9AL
PREMISSES T ,A 2L;GS 6NO ;,C1 & ;,C 6ALL
;,B1 S T ! F/ FIGURE RESULTS4 ,IF !
SYLLOGISM IS N UNIV]SAL1 B PRO( HAS BE5
GIV5 T ,A DOES N 2L;G 6"S ;,B1 WE MAY
9F] 9 ! SAME WAY4 ,! HYPO!SIS IS T ,A
2L;GS 6ALL ;,B1 ! ORIG9AL PREMISSES T ,A
2L;GS 6NO ;,C1 & ;,C 2L;GS 6"S ;,B3 =
?US WE GET ! F/ FIGURE4
  ,AG SUPPOSE X HAS BE5 PROV$ 9 ! ?IRD
FIGURE T ,A 2L;GS 6ALL ,B4 ,!N !
HYPO!SIS M/ H BE5 T ,A 2L;GS N 6ALL ;,B1
&! ORIG9AL PREMISSES T ;,C 2L;GS 6ALL
;,B1 & ,A 2L;GS 6ALL ;,C2 = ?US WE % GET
:AT IS IMPOSSI#4 ,&! ORIG9AL PREMISSES
=M ! F/ FIGURE4 ,SIMIL>LY IF ! DEMON/R,N
E/ABLI%ES A "PICUL> PROPOSI;N3 !
HYPO!SIS !N M/ H BE5 T ,A 2L;GS 6NO ;,B1
&! ORIG9AL PREMISSES T ;,C 2L;GS 6"S
;,B1 & ,A 6ALL ,C4 ,IF ! SYLLOGISM IS
NEGATIVE1 ! HYPO!SIS M/ H BE5 T ,A 2L;GS
6"S ;,B1 &! ORIG9AL PREMISSES T ;,C
2L;GS 6NO ;,A & 6ALL ;,B1 & ? IS ! MI4LE
FIGURE4 ,SIMIL>LY IF ! DEMON/R,N IS N
UNIV]SAL4 ,! HYPO!SIS W !N 2 T ,A 2L;GS
6ALL ;,B1 ! PREMISSES T ;,C 2L;GS 6NO
;,A & 6"S ;,B3 & ? IS ! MI4LE FIGURE4
  ,X IS CLE> !N T X IS POSSI# "? ! SAME
T]MS 6PROVE EA* (! PRO#MS O/5SIVELY Z
WELL4 ,SIMIL>LY X W 2 POSSI# IF !
SYLLOGISMS >E O/5SIVE 6REDUCE !M AD
IMPOSSIBILE 9 ! T]MS : H BE5 TAK51 :5"E
! 3TRADICTORY (! 3CLU.N (! O/5SIVE
SYLLOGISM IS TAK5 Z A PREMISS4 ,=!
SYLLOGISMS 2COME ID5TICAL ) ^? : >E
OBTA9$ 0M1NS ( 3V].N1 S T WE OBTA9 IMMLY
! FIGURES "? : EA* PRO#M W 2 SOLV$4 ,X
IS CLE> !N T E !SIS C 2 PROV$ 9 BO?
WAYS1 I4E4 P] IMPOSSIBILE & O/5SIVELY1 &
X IS N POSSI# 6SEP>ATE "O ME?OD F ! O!R4

                  #AE

  ,9 :AT FIGURE X IS POSSI# 6DRAW A
3CLU.N F PREMISSES : >E OPPOS$1 & 9 :AT
FIGURE ? IS N POSSI#1 W 2 MADE CLE> 9 ?
WAY4 ,V]B,Y F\R K9DS ( OPPOSI;N >E
POSSI#1 VIZ4 UNIV]SAL A6IRMATIVE
6UNIV]SAL NEGATIVE1 UNIV]SAL A6IRMATIVE
6"PICUL> NEGATIVE1 "PICUL> A6IRMATIVE
6UNIV]SAL NEGATIVE1 & "PICUL> A6IRMATIVE
6"PICUL> NEGATIVE3 B RE,Y "! >E ONLY
?REE3 =! "PICUL> A6IRMATIVE IS ONLY
V]B,Y OPPOS$ 6! "PICUL> NEGATIVE4 ,(!
G5U9E OPPOSITES ,I CALL ^? : >E UNIV]SAL
3TR>IES1 ! UNIV]SAL A6IRMATIVE &!
UNIV]SAL NEGATIVE1 E4G4 ,8E SCI;E IS
GD0'1 ,8NO SCI;E IS GD0'2 ! O!RS ,I CALL
3TRADICTORIES4
  ,9 ! F/ FIGURE NO SYLLOGISM :E!R
A6IRMATIVE OR NEGATIVE C 2 MADE \ (
OPPOS$ PREMISSES3 NO A6IRMATIVE
SYLLOGISM IS POSSI# 2C BO? PREMISSES M/
2 A6IRMATIVE1 B OPPOSITES >E1 ! "O
A6IRMATIVE1 ! O!R NEGATIVE3 NO NEGATIVE
SYLLOGISM IS POSSI# 2C OPPOSITES A6IRM &
D5Y ! SAME PR$ICATE (! SAME SUBJECT1 &!
MI4LE T]M 9 ! F/ FIGURE IS N PR$ICAT$ (
BO? EXTREMES1 B "O ?+ IS D5I$ ( X1 & X
IS A6IRM$ ( "S?+ ELSE3 B S* PREMISSES >E
N OPPOS$4
  ,9 ! MI4LE FIGURE A SYLLOGISM C 2 MADE
BO? O,LCONTRADICTORIES &( 3TR>IES4 ,LET
,A /& = GD1 LET ;,B & ;,C /& = SCI;E4
,IF !N "O ASSUMES T E SCI;E IS GD1 & NO
SCI;E IS GD1 ,A 2L;GS 6ALL ;,B & 6NO
;,C1 S T ;,B 2L;GS 6NO ;,C3 NO SCI;E !N
IS A SCI;E4 ,SIMIL>LY IF AF TAK+ ,8E
SCI;E IS GD0' "O TOOK ,8! SCI;E ( M$IC9E
IS N GD0'2 = ,A 2L;GS 6ALL ;,B B 6NO
;,C1 S T A "PICUL> SCI;E W N 2 A SCI;E4
,AG1 A "PICUL> SCI;E W N 2 A SCI;E IF ,A
2L;GS 6ALL ;,C B 6NO ;,B1 & ;,B IS
SCI;E1 ;,C M$IC9E1 & ,A SUPPOSI;N3 = AF
TAK+ ,8NO SCI;E IS SUPPOSI;N0'1 "O HAS
ASSUM$ T A "PICUL> SCI;E IS SUPPOSI;N4
,? SYLLOGISM DI6]S F ! PREC$+ 2C !
REL,NS 2T ! T]MS >E REV]S$3 2F1 !
A6IRMATIVE /ATE;T 3C]N$ ;,B1 N[ X 3C]NS
,C4 ,SIMIL>LY IF "O PREMISS IS N
UNIV]SAL3 =! MI4LE T]M IS ALW T : IS
/AT$ NEGATIVELY ( "O EXTREME1 &
A6IRMATIVELY (! O!R4 ,3SEQU5TLY X IS
POSSI# T 3TRADICTORIES MAY L1D 6A
3CLU.N1 ?\< N ALW OR 9 E MOOD1 B ONLY IF
! T]MS SUBORD9ATE 6! MI4LE >E S* T !Y >E
EI ID5TICAL OR RELAT$ Z :OLE 6"P4
,O!RWISE X IS IMPOSSI#3 =! PREMISSES _C
ANYH[ 2 EI 3TR>IES OR 3TRADICTORIES4
  ,9 ! ?IRD FIGURE AN A6IRMATIVE
SYLLOGISM C N"E 2 MADE \ ( OPPOSITE
PREMISSES1 =! R1SON GIV5 9 REF];E 6! F/
FIGURE2 B A NEGATIVE SYLLOGISM IS POSSI#
:E!R ! T]MS >E UNIV]SAL OR N4 ,LET ;,B &
;,C /& = SCI;E1 ;,A = M$IC9E4 ,IF !N "O
%D ASSUME T ALL M$IC9E IS SCI;E & T NO
M$IC9E IS SCI;E1 HE HAS ASSUM$ T ;,B
2L;GS 6ALL ;,A & ;,C 6NO ;,A1 S T A
"PICUL> SCI;E W N 2 A SCI;E4 ,SIMIL>LY
IF ! PREMISS ,,BA IS N ASSUM$ UNIV]S,Y4
,= IF "S M$IC9E IS SCI;E & AG NO M$IC9E
IS SCI;E1 X RESULTS T "S SCI;E IS N
SCI;E1 ,! PREMISSES >E 3TR>Y IF ! T]MS
>E TAK5 UNIV]S,Y2 IF "O IS "PICUL>1 !Y
>E 3TRADICTORY4
  ,WE M/ RECOGNIZE T X IS POSSI# 6TAKE
OPPOSITES 9 ! WAY WE SD1 VIZ4 ,8ALL
SCI;E IS GD0' & ,8NO SCI;E IS GD0' OR
,8"S SCI;E IS N GD0'4 ,? DOES N USU,Y
ESCAPE NOTICE4 ,B X IS POSSI# 6E/ABLI%
"O "P (A 3TRADIC;N "? O!R PREMISSES1 OR
6ASSUME X 9 ! WAY SU7E/$ 9 ! ,TOPICS4
,S9CE "! >E ?REE OPPOSI;NS 6A6IRMATIVE
/ATE;TS1 X FOLL[S T OPPOSITE /ATE;TS MAY
2 ASSUM$ Z PREMISSES 9 SIX WAYS2 WE MAY
H EI UNIV]SAL A6IRMATIVE & NEGATIVE1 OR
UNIV]SAL A6IRMATIVE & "PICUL> NEGATIVE1
OR "PICUL> A6IRMATIVE & UNIV]SAL
NEGATIVE1 &! REL,NS 2T ! T]MS MAY 2
REV]S$2 E4G4 ,A MAY 2L;G 6ALL ;,B & 6NO
;,C1 OR 6ALL ;,C & 6NO ;,B1 OR 6ALL (!
"O1 N 6ALL (! O!R2 "H TOO ! REL,N 2T !
T]MS MAY 2 REV]S$4 ,SIMIL>LY 9 ! ?IRD
FIGURE4 ,S X IS CLE> 9 H[ _M WAYS & 9
:AT FIGURES A SYLLOGISM C 2 MADE 0M1NS (
PREMISSES : >E OPPOS$4
  ,X IS CLE> TOO T F FALSE PREMISSES X
IS POSSI# 6DRAW A TRUE 3CLU.N1 Z HAS BE5
SD 2F1 B X IS N POSSI# IF ! PREMISSES >E
OPPOS$4 ,=! SYLLOGISM IS ALW 3TR>Y 6!
FACT1 E4G4 IF A ?+ IS GD1 X IS PROV$ T X
IS N GD1 IF AN ANIMAL1 T X IS N AN
ANIMAL 2C ! SYLLOGISM SPR+S \ (A
3TRADIC;N &! T]MS PRESUPPOS$ >E EI
ID5TICAL OR RELAT$ Z :OLE & "P4 ,X IS
EVID5T AL T 9 FALLACI\S R1SON+S NO?+
PREV5TS A 3TRADIC;N 6! HYPO!SIS F
RESULT+1 E4G4 IF "S?+ IS ODD1 X IS N
ODD4 ,=! SYLLOGISM [$ XS 3TR>IETY 6XS
3TRADICTORY PREMISSES2 IF WE ASSUME S*
PREMISSES WE % GET A RESULT T 3TRADICTS
\R HYPO!SIS4 ,B WE M/ RECOGNIZE T
3TR>IES _C 2 9F]R$ F A S+LE SYLLOGISM 9
S* A WAY T WE 3CLUDE T :AT IS N GD IS
GD1 OR ANY?+ ( T SORT UN.S A SELF-
3TRADICTORY PREMISS IS AT ONCE ASSUM$1
E4G4 ,8E ANIMAL IS :ITE & N :ITE0'1 & WE
PROCE$ ,8MAN IS AN ANIMAL0'4 ,EI WE M/
9TRODUCE ! 3TRADIC;N 0AN A4I;NAL
ASSUMP;N1 ASSUM+1 E4G41 T E SCI;E IS
SUPPOSI;N1 & !N ASSUM+ ,8,M$IC9E IS A
SCI;E1 B N"O ( X IS SUPPOSI;N0' 7: IS !
MODE 9 : REFUT,NS >E MADE71 OR WE M/
>GUE F TWO SYLLOGISMS4 ,9 NO O!R WAY ?AN
?1 Z 0 SD 2F1 IS X POSSI# T ! PREMISSES
%D 2 RE,Y 3TR>Y4

                  #AF

  ,6BEG & ASSUME ! ORIG9AL "Q IS A
SPECIES ( FAILURE 6DEMON/RATE ! PRO#M
PROPOS$2 B ? HAPP5S 9 _M WAYS4 ,A MAN
MAY N R1SON SYLLOGI/IC,Y AT ALL1 OR HE
MAY >GUE F PREMISSES : >E LESS "KN OR
EQU,Y UN"KN1 OR HE MAY E/ABLI% !
ANTEC$5T 0M1NS ( XS 3SEQU5TS2 =
DEMON/R,N PROCE$S F :AT IS M C]TA9 & IS
PRIOR4 ,N[ BE7+ ! "Q IS N"O ( ^!3 B S9CE
WE GET 6"K "S ?+S NATUR,Y "? !MVS1 & O!R
?+S 0M1NS ( "S?+ ELSE 7! F/ PR9CIPLES "?
!MVS1 :AT IS SUBORD9ATE 6!M "? "S?+
ELSE71 :5"E A MAN TRIES 6PROVE :AT IS N
SELF-EVID5T 0M1NS ( XF1 !N HE BEGS !
ORIG9AL "Q4 ,? MAY 2 D"O 0ASSUM+ :AT IS
9 "Q AT ONCE2 X IS AL POSSI# 6MAKE A
TRANSI;N 6O!R ?+S : WD NATUR,Y 2 PROV$
"? ! !SIS PROPOS$1 & DEMON/RATE X "? !M1
E4G4 IF ,A %D 2 PROV$ "? ;,B1 & ;,B "?
;,C1 ?\< X 0 NATURAL T ;,C %D 2 PROV$ "?
,A3 = X TURNS \ T ^? :O R1SON ?US >E
PROV+ ,A 0M1NS ( XF4 ,? IS :AT ^? P]SONS
D :O SUPPOSE T !Y >E 3/RUCT+ P>ALLEL
/RAI<T L9ES3 = !Y FAIL 6SEE T !Y >E
ASSUM+ FACTS : X IS IMPOSSI# 6DEMON/RATE
UN.S ! P>ALLELS EXI/4 ,S X TURNS \ T ^?
:O R1SON ?US M]ELY SAY A "PICUL> ?+ IS1
IF X IS3 9 ? WAY "EY?+ W 2 SELF-EVID5T4
,B T IS IMPOSSI#4
  ,IF !N X IS UNC]TA9 :E!R ,A 2L;GS TO
;,C1 & AL :E!R ,A 2L;GS TO ;,B1 & IF "O
%D ASSUME T ,A DOES 2L;G TO ;,B1 X IS N
YET CLE> :E!R HE BEGS ! ORIG9AL "Q1 B X
IS EVID5T T HE IS N DEMON/RAT+3 = :AT IS
Z UNC]TA9 Z ! "Q 6BE ANSW]$ _C 2 A
PR9CIPLE (A DEMON/R,N4 ,IF H["E ;,B IS S
RELAT$ TO ;,C T !Y >E ID5TICAL1 OR IF !Y
>E PLA9LY 3V]TI#1 OR ! "O 2L;GS 6! O!R1
! ORIG9AL "Q IS BE7$4 ,= "O MI<T EQU,Y
WELL PROVE T ,A 2L;GS TO ;,B "? ^? T]MS
IF !Y >E 3V]TI#4 ,B IF !Y >E N 3V]TI#1 X
IS ! FACT T !Y >E N T PREV5TS S* A
DEMON/R,N1 N ! ME?OD ( DEMON/RAT+4 ,B IF
"O 7 6MAKE ! 3V].N1 !N HE WD 2 DO+ :AT
WE H DESCRIB$ & E6ECT+ A RECIPROCAL PRO(
) ?REE PROPOSI;NS4
  ,SIMIL>LY IF HE %D ASSUME T ;,B 2L;GS
TO ;,C1 ? 2+ Z UNC]TA9 Z ! "Q :E!R ,A
2L;GS TO ;,C1 ! "Q IS N YET BE7$1 B NO
DEMON/R,N IS MADE4 ,IF H["E ;,A & ;,B >E
ID5TICAL EI 2C !Y >E 3V]TI# OR 2C ,A
FOLL[S ;,B1 !N ! "Q IS BE7$ =! SAME
R1SON Z 2F4 ,= WE H EXPLA9$ ! M1N+ (
BE7+ ! "Q1 VIZ4 PROV+ T : IS N SELF-
EVID5T 0M1NS ( XF4
  ,IF !N BE7+ ! "Q IS PROV+ :AT IS N
SELF-EVID5T 0M1NS ( XF1 9 O!R ^WS FAIL+
6PROVE :5 ! FAILURE IS DUE 6! !SIS 6BE
PROV$ &! PREMISS "? : X IS PROV$ 2+
EQU,Y UNC]TA91 EI 2C PR$ICATES : >E
ID5TICAL 2L;G 6! SAME SUBJECT1 OR 2C !
SAME PR$ICATE 2L;GS 6SUBJECTS : >E
ID5TICAL1 ! "Q MAY 2 BE7$ 9 ! MI4LE &
?IRD FIGURES 9 BO? WAYS1 ?\<1 IF !
SYLLOGISM IS A6IRMATIVE1 ONLY 9 ! ?IRD &
F/ FIGURES4 ,IF ! SYLLOGISM IS NEGATIVE1
! "Q IS BE7$ :5 ID5TICAL PR$ICATES >E
D5I$ (! SAME SUBJECT2 & BO? PREMISSES D
N BEG ! "Q 9DI6]5TLY 7IN A SIMIL> WAY !
"Q MAY 2 BE7$ 9 ! MI4LE FIGURE71 2C !
T]MS 9 NEGATIVE SYLLOGISMS >E N 3V]TI#4
,9 SCI5TIFIC DEMON/R,NS ! "Q IS BE7$ :5
! T]MS >E RE,Y RELAT$ 9 ! MANN]
DESCRIB$1 9 DIALECTICAL >GU;TS :5 !Y >E
AC 6-MON OP9ION S RELAT$4

                  #AG

  ,! OBJEC;N T ,8? IS N ! R1SON :Y !
RESULT IS FALSE0'1 : WE FREQU5TLY MAKE 9
>GU;T1 IS MADE PRIM>ILY 9 ! CASE (A
REDUCTIO AD IMPOSSIBILE1 6REBUT !
PROPOSI;N : 0 2+ PROV$ 0! REDUC;N4 ,=
UN.S A MAN HAS 3TRADICT$ ? PROPOSI;N HE
W N SAY1 ,8,FALSE CAUSE0'1 B URGE T "S?+
FALSE HAS BE5 ASSUM$ 9 ! E>LI] "PS (!
>GU;T2 NOR W HE USE ! =MULA 9 ! CASE (
AN O/5SIVE PRO(2 = "H :AT "O D5IES IS N
ASSUM$ Z A PREMISS4 ,FUR!R :5 ANY?+ IS
REFUT$ O/5SIVELY 0! T]MS ,,ABC1 X _C 2
OBJECT$ T ! SYLLOGISM DOES N DEP5D ON !
ASSUMP;N LAID D[N4 ,= WE USE ! EXPRES.N
,8FALSE CAUSE0'1 :5 ! SYLLOGISM IS
3CLUD$ 9 SPITE (! REFUT,N ( ? POSI;N2 B
T IS N POSSI# 9 O/5SIVE PRO(S3 S9CE IF
AN ASSUMP;N IS REFUT$1 A SYLLOGISM C NO
L;G] 2 DRAWN 9 REF];E 6X4 ,X IS CLE> !N
T ! EXPRES.N ,8FALSE CAUSE0' C ONLY 2
US$ 9 ! CASE (A REDUCTIO AD IMPOSSIBILE1
& :5 ! ORIG9AL HYPO!SIS IS S RELAT$ 6!
IMPOSSI# 3CLU.N1 T ! 3CLU.N RESULTS
9DI6]5TLY :E!R ! HYPO!SIS IS MADE OR N4
,! MO/ OBVI\S CASE (! IRRELEV.E ( AN
ASSUMP;N 6A 3CLU.N : IS FALSE IS :5 A
SYLLOGISM DRAWN F MI4LE T]MS 6AN
IMPOSSI# 3CLU.N IS 9DEP5D5T (! HYPO!SIS1
Z WE H EXPLA9$ 9 ! ,TOPICS4 ,= 6PUT T :
IS N ! CAUSE Z ! CAUSE1 IS J ?3 E4G4 IF
A MAN1 WI%+ 6PROVE T ! DIAGONAL (! SQU>E
IS 9COMM5SURATE )! SIDE1 %D TRY 6PROVE
,Z5O'S !OREM T MO;N IS IMPOSSI#1 & S
E/ABLI% A REDUCTIO AD IMPOSSIBILE3 =
,Z5O'S FALSE !OREM HAS NO 3NEXION AT ALL
)! ORIG9AL ASSUMP;N4 ,ANO!R CASE IS ": !
IMPOSSI# 3CLU.N IS 3NECT$ )! HYPO!SIS1 B
DOES N RESULT F X4 ,? MAY HAPP5 :E!R "O
TRACES ! 3NEXION UPW>DS OR D[NW>DS1 E4G4
IF X IS LAID D[N T ,A 2L;GS TO ;,B1 ;,B
TO ;,C1 & ;,C TO ;,D1 & X %D 2 FALSE T
;,B 2L;GS TO ;,D3 = IF WE ELIM9AT$ ;,A &
ASSUM$ ALL ! SAME T ;,B 2L;GS TO ;,C &
;,C TO ;,D1 ! FALSE 3CLU.N WD N DEP5D ON
! ORIG9AL HYPO!SIS4 ,OR AG TRACE !
3NEXION UPW>DS2 E4G4 SUPPOSE T ,A 2L;GS
TO ;,B1 ;,E TO ;,A & ;,F TO ;,E1 X 2+
FALSE T ;,F 2L;GS TO ;,A4 ,9 ? WAY TOO !
IMPOSSI# 3CLU.N WD RESULT1 ?\< ! ORIG9AL
HYPO!SIS 7 ELIM9AT$4 ,B ! IMPOSSI#
3CLU.N "\ 6BE 3NECT$ )! ORIG9AL T]MS3 9
? WAY X W DEP5D ON ! HYPO!SIS1 E4G4 :5
"O TRACES ! 3NEXION D[NW>DS1 ! IMPOSSI#
3CLU.N M/ 2 3NECT$ ) T T]M : IS PR$ICATE
9 ! HYPO!SIS3 = IF X IS IMPOSSI# T ,A %D
2L;G TO ;,D1 ! FALSE 3CLU.N W NO L;G]
RESULT AF ,A HAS BE5 ELIM9AT$4 ,IF "O
TRACES ! 3NEXION UPW>DS1 ! IMPOSSI#
3CLU.N M/ 2 3NECT$ ) T T]M : IS SUBJECT
9 ! HYPO!SIS3 = IF X IS IMPOSSI# T ;,F
%D 2L;G TO ;,B1 ! IMPOSSI# 3CLU.N W
4APPE> IF ;,B IS ELIM9AT$4 ,SIMIL>LY :5
! SYLLOGISMS >E NEGATIVE4
  ,X IS CLE> !N T :5 ! IMPOSSIBIL;Y IS N
RELAT$ 6! ORIG9AL T]MS1 ! FALSE 3CLU.N
DOES N RESULT ON A3.T (! ASSUMP;N4 ,OR
P]H EV5 S X MAY "S"TS 2 9DEP5D5T4 ,= IF
X 7 LAID D[N T ,A 2L;GS N TO ;,B B TO
;,K1 & T ;,K 2L;GS TO ;,C & ;,C TO ;,D1
! IMPOSSI# 3CLU.N WD / /&4 ,SIMIL>LY IF
"O TAKES ! T]MS 9 AN ASC5D+ S]IES4
,3SEQU5TLY S9CE ! IMPOSSIBIL;Y RESULTS
:E!R ! F/ ASSUMP;N IS SUPPRESS$ OR N1 X
WD APPE> 6BE 9DEP5D5T ( T ASSUMP;N4 ,OR
P]H WE "\ N 6"U/& ! /ATE;T T ! FALSE
3CLU.N RESULTS 9DEP5D5TLY (! ASSUMP;N1 9
! S5SE T IF "S?+ ELSE 7 SUPPOS$ !
IMPOSSIBIL;Y WD RESULT2 B R WE M1N T :5
! F/ ASSUMP;N IS ELIM9AT$1 ! SAME
IMPOSSIBIL;Y RESULTS "? ! REMA9+
PREMISSES2 S9CE X IS N P]H ABSURD T !
SAME FALSE RESULT %D FOLL[ F S"EAL
HYPO!SES1 E4G4 T P>ALLELS MEET1 BO? ON !
ASSUMP;N T ! 9T]IOR ANGLE IS GRT] ?AN !
EXT]IOR & ON ! ASSUMP;N T A TRIANGLE
3TA9S M ?AN TWO "R ANGLES4

                  #AH

  ,A FALSE >GU;T DEP5DS ON ! F/ FALSE
/ATE;T 9 X4 ,E SYLLOGISM IS MADE \ ( TWO
OR M PREMISSES4 ,IF !N ! FALSE 3CLU.N IS
DRAWN F TWO PREMISSES1 "O OR BO? ( !M M/
2 FALSE3 = 7Z WE PROV$7 A FALSE
SYLLOGISM _C 2 DRAWN F TWO PREMISSES4 ,B
IF ! PREMISSES >E M ?AN TWO1 E4G4 IF ;,C
IS E/ABLI%$ "? ;,A & ;,B1 & ^! "? ;,D1
;,E1 ;,F1 & ;,G1 "O ( ^! HI<] PROPOSI;NS
M/ 2 FALSE1 & ON ? ! >GU;T DEP5DS3 = ;,A
& ;,B >E 9F]R$ 0M1NS ( ;,D1 ;,E1 ;,F1 &
,G4 ,"!=E ! 3CLU.N &! ]ROR RESULTS F "O
( !M4

                  #AI

  ,9 ORD] 6AVOID HAV+ A SYLLOGISM DRAWN
AG/ U WE M/ TAKE C>E1 :5"E AN OPPON5T
ASKS U 6ADMIT ! R1SON )\T ! 3CLU.NS1 N
6GRANT HM ! SAME T]M TWICE OV] 9 8
PREMISSES1 S9CE WE "K T A SYLLOGISM _C 2
DRAWN )\T A MI4LE T]M1 & T T]M : IS /AT$
M ?AN ONCE IS ! MI4LE4 ,H[ WE "\ 6WAT* !
MI4LE 9 REF];E 6EA* 3CLU.N1 IS EVID5T F
\R "K+ :AT K9D ( !SIS IS PROV$ 9 EA*
FIGURE4 ,? W N ESCAPE U S9CE WE "K H[ WE
>E MA9TA9+ ! >GU;T4
  ,T : WE URGE M5 62W>E ( 9 _! ADMIS.NS1
!Y "\ 9 ATTACK 6TRY 63C1L4 ,? W 2 POSSI#
F/1 IF1 9/1D ( DRAW+ ! 3CLU.NS (
PRELIM9>Y SYLLOGISMS1 !Y TAKE ! NEC
PREMISSES & L1VE ! 3CLU.NS 9 ! D>K2
SECONDLY IF 9/1D ( 9VIT+ ASS5T
6PROPOSI;NS : >E CLOSELY 3NECT$ !Y TAKE
Z F> Z POSSI# ^? T >E N 3NECT$ 0MI4LE
T]MS4 ,= EXAMPLE SUPPOSE T ,A IS 6BE
9F]R$ 6BE TRUE ( ;,F1 ;,B1 ;,C1 ;,D1 &
;,E 2+ MI4LE T]MS4 ,"O "\ !N 6ASK :E!R
,A 2L;GS TO ;,B1 & NEXT :E!R ;,D 2L;GS
TO ;,E1 9/1D ( ASK+ :E!R ;,B 2L;GS TO
;,C2 AF T HE MAY ASK :E!R ;,B 2L;GS TO
;,C1 & S ON4 ,IF ! SYLLOGISM IS DRAWN "?
"O MI4LE T]M1 HE "\ 62G9 ) T3 9 ? WAY HE
W MO/ LIKELY DCV 8 OPPON5T4


                  #BJ

  ,S9CE WE "K :5 A SYLLOGISM C 2 =M$ &
H[ XS T]MS M/ 2 RELAT$1 X IS CLE> :5
REFUT,N W 2 POSSI# & :5 IMPOSSI#4 ,A
REFUT,N IS POSSI# :E!R "EY?+ IS 3C$$1 OR
! ANSW]S ALT]NATE 7"O1 ,I M1N1 2+
A6IRMATIVE1 ! O!R NEGATIVE74 ,= Z HAS
BE5 %[N A SYLLOGISM IS POSSI# :E!R !
T]MS >E RELAT$ 9 A6IRMATIVE PROPOSI;NS
OR "O PROPOSI;N IS A6IRMATIVE1 ! O!R
NEGATIVE3 3SEQU5TLY1 IF :AT IS LAID D[N
IS 3TR>Y 6! 3CLU.N1 A REFUT,N M/ TAKE
PLACE3 =A REFUT,N IS A SYLLOGISM :
E/ABLI%ES ! 3TRADICTORY4 ,B IF NO?+ IS
3C$$1 A REFUT,N IS IMPOSSI#3 = NO
SYLLOGISM IS POSSI# 7Z WE SAW7 :5 ALL !
T]MS >E NEGATIVE3 "!=E NO REFUT,N IS
POSSI#4 ,= IF A REFUT,N 7 POSSI#1 A
SYLLOGISM M/ 2 POSSI#2 AL? IF A
SYLLOGISM IS POSSI# X DOES N FOLL[ T A
REFUT,N IS POSSI#4 ,SIMIL>LY REFUT,N IS
N POSSI# IF NO?+ IS 3C$$ UNIV]S,Y3 S9CE
! FIELDS ( REFUT,N & SYLLOGISM >E DEF9$
9 ! SAME WAY4
                  #BA

  ,X "S"TS HAPP5S T J Z WE >E DCVD 9 !
>RANGE;T (! T]MS1 S ]ROR MAY >ISE 9 \R
?"\ AB !M1 E4G4 IF X IS POSSI# T ! SAME
PR$ICATE %D 2L;G 6M ?AN "O SUBJECT
IMMLY1 B AL? "K+ ! "O1 A MAN MAY =GET !
O!R & ?9K ! OPPOSITE TRUE4 ,SUPPOSE T ,A
2L;GS TO ;,B & TO ;,C 9 VIRTUE ( _!
NATURE1 & T ;,B & ;,C 2L;G 6ALL ;,D 9 !
SAME WAY4 ,IF !N A MAN ?9KS T ,A 2L;GS
6ALL ;,B1 & ;,B TO ;,D1 B ,A 6NO ;,C1 &
;,C 6ALL ;,D1 HE W BO? "K & N "K ! SAME
?+ 9 RESPECT (! SAME ?+4 ,AG IF A MAN 7
6MAKE A MISTAKE AB ! MEMB]S (A S+LE
S]IES2 E4G4 SUPPOSE ,A 2L;GS TO ;,B1 ;,B
TO ;,C1 & ;,C TO ;,D1 B "S "O ?9KS T ,A
2L;GS 6ALL ;,B1 B 6NO ;,C3 HE W BO? "K T
,A 2L;GS TO ;,D1 & ?9K T X DOES N4 ,DOES
HE !N MA9TA9 AF ? SIMPLY T :AT HE "KS1
HE DOES N ?9K8 ,= HE "KS 9 A WAY T ,A
2L;GS TO ;,C "? ;,B1 S9CE ! "P IS 9CLUD$
9 ! :OLE2 S T :AT HE "KS 9 A WAY1 ? HE
MA9TA9S HE DOES N ?9K AT ALL3 B T IS
IMPOSSI#4
  ,9 ! =M] CASE1 ": ! MI4LE T]M DOES N
2L;G 6! SAME S]IES1 X IS N POSSI# 6?9K
BO? ! PREMISSES ) REF];E 6EA* (! TWO
MI4LE T]MS3 E4G4 T ,A 2L;GS 6ALL ;,B1 B
6NO ;,C1 & BO? ;,B & ;,C 2L;G 6ALL ,D4
,= X TURNS \ T ! F/ PREMISS (! "O
SYLLOGISM IS EI :OLLY OR "PI,Y 3TR>Y 6!
F/ PREMISS (! O!R4 ,= IF HE ?9KS T ,A
2L;GS 6"EY?+ 6: ;,B 2L;GS1 & HE "KS T
;,B 2L;GS TO ;,D1 !N HE "KS T ,A 2L;GS
TO ;,D4 ,3SEQU5TLY IF AG HE ?9KS T ,A
2L;GS 6NO?+ 6: ;,C 2L;GS1 HE ?9KS T ,A
DOES N 2L;G 6"S ( T 6: ;,B 2L;GS2 B IF
HE ?9KS T ,A 2L;GS 6"EY?+ 6: ;,B 2L;GS1
& AG ?9KS T ,A DOES N 2L;G 6"S ( T 6:
;,B 2L;GS1 ^! 2LIEFS >E :OLLY OR "PI,Y
3TR>Y4 ,9 ? WAY !N X IS N POSSI# 6?9K2 B
NO?+ PREV5TS A MAN ?9K+ "O PREMISS ( EA*
SYLLOGISM ( BO? PREMISSES ( "O (! TWO
SYLLOGISMS3 E4G4 ,A 2L;GS 6ALL ;,B1 &
;,B TO ;,D1 & AG ,A 2L;GS 6NO ,C4 ,AN
]ROR ( ? K9D IS SIMIL> 6! ]ROR 96: WE
FALL 3C]N+ "PICUL>S3 E4G4 IF ,A 2L;GS
6ALL ;,B1 & ;,B 6ALL ;,C1 ,A W 2L;G 6ALL
,C4 ,IF !N A MAN "KS T ,A 2L;GS 6"EY?+
6: ;,B 2L;GS1 HE "KS T ,A 2L;GS TO ;,C4
,B NO?+ PREV5TS 8 2+ IGNORANT T ;,C
EXI/S2 E4G4 LET ,A /& = TWO "R ANGLES1
;,B = TRIANGLE1 ;,C =A "PICUL> DIAGRAM (
A TRIANGLE4 ,A MAN MI<T ?9K T ;,C DID N
EXI/1 ?\< HE KNEW T E TRIANGLE 3TA9S TWO
"R ANGLES2 3SEQU5TLY HE W "K & N "K !
SAME ?+ AT ! SAME "T4 ,=! EXPRES.N ,86"K
T E TRIANGLE HAS XS ANGLES EQUAL 6TWO "R
ANGLES0' IS AMBIGU\S1 M1N+ 6H ! K EI (!
UNIV]SAL OR (! "PICUL>S4 ,?US !N HE "KS
T ;,C 3TA9S TWO "R ANGLES )A K (!
UNIV]SAL1 B N )A K (! "PICUL>S2
3SEQU5TLY 8 K W N 2 3TR>Y 6HIS IGNOR.E4
,! >GU;T 9 ! ,M5O T LE>N+ IS RECOLLEC;N
MAY 2 CRITICIZ$ 9 A SIMIL> WAY4 ,= X N"E
HAPP5S T A MAN />TS )A =E"KL$GE (!
"PICUL>1 B AL;G )! PROCESS ( 2+ L$ 6SEE
! G5]AL PR9CIPLE HE RCVS A K (!
"PICUL>S1 0AN ACT 7Z X W]E7 ( RECOGNI;N4
,= WE "K "S ?+S DIRECTLY2 E4G4 T !
ANGLES >E EQUAL 6TWO "R ANGLES1 IF WE "K
T ! FIGURE IS A TRIANGLE4 ,SIMIL>LY 9
ALL O!R CASES4
  ,0A K (! UNIV]SAL !N WE SEE !
"PICUL>S1 B WE D N "K !M 0! K9D ( K : IS
PROP] 6!M2 3SEQU5TLY X IS POSSI# T WE
MAY MAKE MISTAKES AB !M1 B N T WE %D H !
K & ]ROR T >E 3TR>Y 6"O ANO!R3 R WE H !
K (! UNIV]SAL B MAKE A MISTAKE 9
APPREH5D+ ! "PICUL>4 ,SIMIL>LY 9 ! CASES
/AT$ ABV4 ,! ]ROR 9 RESPECT (! MI4LE T]M
IS N 3TR>Y 6! K OBTA9$ "? ! SYLLOGISM1
NOR IS ! ?"\ 9 RESPECT ( "O MI4LE T]M
3TR>Y 6T 9 RESPECT (! O!R4 ,NO?+ PREV5TS
A MAN :O "KS BO? T ,A 2L;GS 6! :OLE (
;,B1 & T ;,B AG 2L;GS TO ;,C1 ?9K+ T ,A
DOES N 2L;G TO ;,C1 E4G4 "K+ T E MULE IS
/]ILE & T ? IS A MULE1 & ?9K+ T ? ANIMAL
IS ) FOAL3 = HE DOES N "K T ,A 2L;GS TO
;,C1 UN.S HE 3SID]S ! TWO PROPOSI;NS
TGR4 ,S X IS EVID5T T IF HE "KS ! "O &
DOES N "K ! O!R1 HE W FALL 96]ROR4 ,& ?
IS ! REL,N ( K (! UNIV]SAL 6K (!
"PICUL>4 ,= WE "K NO S5SI# ?+1 ONCE X
HAS PASS$ 2Y ! RANGE ( \R S5SES1 EV5 IF
WE HAPP5 6H P]CVD X1 EXCEPT 0M1NS (!
UNIV]SAL &! POSSES.N (! K : IS PROP] 6!
"PICUL>1 B )\T ! ACTUAL EX]CISE ( T K4
,= 6"K IS US$ 9 ?REE S5SES3 X MAY M1N EI
6H K (! UNIV]SAL OR 6H K PROP] 6! MATT]
9 H& OR 6EX]CISE S* K3 3SEQU5TLY ?REE
K9DS ( ]ROR AL >E POSSI#4 ,NO?+ !N
PREV5TS A MAN BO? "K+ & 2+ MISTAK5 AB !
SAME ?+1 PROVID$ T 8 K & 8 ]ROR >E N
3TR>Y4 ,& ? HAPP5S AL 6! MAN ^: K IS
LIMIT$ 6EA* (! PREMISSES & :O HAS N
PREVI\SLY 3SID]$ ! "PICUL> "Q4 ,= :5 HE
?9KS T ! MULE IS ) FOAL HE HAS N ! K 9 !
S5SE ( XS ACTUAL EX]CISE1 NOR ON ! O!R
H& HAS 8 ?"\ CAUS$ AN ]ROR 3TR>Y 6HIS K3
=! ]ROR 3TR>Y 6! K (! UNIV]SAL WD 2 A
SYLLOGISM4
  ,B HE :O ?9KS ! ESS;E ( GD IS ! ESS;E
( BAD W ?9K ! SAME ?+ 6BE ! ESS;E ( GD &
! ESS;E ( BAD4 ,LET ,A /& =! ESS;E ( GD
& ;,B =! ESS;E ( BAD1 & AG ;,C =! ESS;E
( GD4 ,S9CE !N HE ?9KS ;,B & ;,C
ID5TICAL1 HE W ?9K T ;,C IS ;,B1 &
SIMIL>LY T ;,B IS ;,A1 3SEQU5TLY T ;,C
IS ,A4 ,= J Z WE SAW T IF ;,B IS TRUE (
ALL ( : ;,C IS TRUE1 & ,A IS TRUE ( ALL
( : ;,B IS TRUE1 ,A IS TRUE ( ;,C1
SIMIL>LY )! ^W ,8?9K0'4 ,SIMIL>LY AL )!
^W ,8IS0'2 = WE SAW T IF ;,C IS ! SAME Z
;,B1 & ;,B Z ;,A1 ;,C IS ! SAME Z ,A4
,SIMIL>LY "!=E ) ,8OP9E0'4 ,P]H !N ? IS
NEC IF A MAN W GRANT ! F/ PO9T4 ,B
PRESUMABLY T IS FALSE1 T ANY "O CD
SUPPOSE ! ESS;E ( GD 6BE ! ESS;E ( BAD1
SAVE 9CID5T,Y4 ,= X IS POSSI# 6?9K ? 9
_M DI6]5T WAYS4 ,B WE M/ 3SID] ? MATT]
BETT]4

                  #BB

  ,:5"E ! EXTREMES >E 3V]TI# X IS NEC T
! MI4LE %D 2 3V]TI# ) BO?4 ,= IF ,A
2L;GS TO ;,C "? ;,B1 !N IF ;,A & ;,C >E
3V]TI# & ;,C 2L;GS "EY?+ 6: ,A 2L;GS1
;,B IS 3V]TI# ) ;,A1 & ;,B 2L;GS 6"EY?+
6: ,A 2L;GS1 "? ;,C Z MI4LE1 & ;,C IS
3V]TI# ) ;,B "? ,A Z MI4LE4 ,SIMIL>LY IF
! 3CLU.N IS NEGATIVE1 E4G4 IF ;,B 2L;GS
TO ;,C1 B ,A DOES N 2L;G TO ;,B1 NEI W
,A 2L;G TO ;,C4 ,IF !N ;,B IS 3V]TI# )
;,A1 ;,C W 2 3V]TI# ) ,A4 ,SUPPOSE ;,B
DOES N 2L;G TO ;,A2 NEI !N W ;,C3 = EX
HYPO!SI ;,B 2L;G$ 6ALL ,C4 ,& IF ;,C IS
3V]TI# ) ;,B1 ;,B IS 3V]TI# AL ) ;,A1 =
;,C IS SD ( T ( ALL ( : ;,B IS SD4 ,& IF
;,C IS 3V]TI# 9 REL,N TO ;,A & TO ;,B1
;,B AL IS 3V]TI# 9 REL,N TO ;,A4 ,= ;,C
2L;GS 6T 6: ;,B 2L;GS3 B ;,C DOES N 2L;G
6T 6: ,A 2L;GS4 ,& ? AL"O />TS F !
3CLU.N2 ! PREC$+ MOODS D N D S Z 9 !
A6IRMATIVE SYLLOGISM4 ,AG IF ;,A & ;,B
>E 3V]TI#1 & SIMIL>LY ;,C & ;,D1 & IF ,A
OR ;,C M/ 2L;G 6ANY?+ :AT"E1 !N ;,B &
;,D W 2 S* T "O OR O!R 2L;GS 6ANY?+
:AT"E4 ,= S9CE ;,B 2L;GS 6T 6: ,A 2L;GS1
& ;,D 2L;GS 6T 6: ;,C 2L;GS1 & S9CE ,A
OR ;,C 2L;GS 6"EY?+1 B N TGR1 X IS CLE>
T ;,B OR ;,D 2L;GS 6"EY?+1 B N TGR4 ,=
EXAMPLE IF T : IS UNCR1T$ IS 9CORRUPTI#
& T : IS 9CORRUPTI# IS UNCR1T$1 X IS NEC
T :AT IS CR1T$ %D 2 CORRUPTI# & :AT IS
CORRUPTI# %D H BE5 CR1T$4 ,= TWO
SYLLOGISMS H BE5 PUT TGR4 ,AG IF ,A OR
;,B 2L;GS 6"EY?+ & IF ;,C OR ;,D 2L;GS
6"EY?+1 B !Y _C 2L;G TGR1 !N :5 ;,A &
;,C >E 3V]TI# ;,B & ;,D >E 3V]TI#4 ,= IF
;,B DOES N 2L;G 6"S?+ 6: ;,D 2L;GS1 X IS
CLE> T ,A 2L;GS 6X4 ,B IF ,A !N ;,C3 =
!Y >E 3V]TI#4 ,"!=E ;,C & ;,D 2L;G TGR4
,B ? IS IMPOSSI#4 ,:5 ,A 2L;GS 6! :OLE (
;,B & TO ;,C & IS A6IRM$ ( NO?+ ELSE1 &
;,B AL 2L;GS 6ALL ;,C1 X IS NEC T ;,A &
;,B %D 2 3V]TI#3 = S9CE ,A IS SD ( ;,B &
;,C ONLY1 & ;,B IS A6IRM$ BO? ( XF &(
;,C1 X IS CLE> T ;,B W 2 SD ( "EY?+ ( :
,A IS SD1 EXCEPT ,A XF4 ,AG :5 ;,A & ;,B
2L;G 6! :OLE ( ;,C1 & ;,C IS 3V]TI# )
;,B1 X IS NEC T ,A %D 2L;G 6ALL ;,B3 =
S9CE ,A 2L;GS 6ALL ;,C1 & ;,C TO ;,B
03V].N1 ,A W 2L;G 6ALL ,B4
  ,:51 ( TWO OPPOSITES ;,A & ;,B1 ,A IS
PREF]A# TO ;,B1 & SIMIL>LY ;,D IS
PREF]A# TO ;,C1 !N IF ;,A & ;,C TGR >E
PREF]A# TO ;,B & ;,D TGR1 ,A M/ 2
PREF]A# TO ;,D4 ,= ,A IS AN OBJECT (
DESIRE 6! SAME EXT5T Z ;,B IS AN OBJECT
( AV].N1 S9CE !Y >E OPPOSITES3 & ;,C IS
SIMIL>LY RELAT$ TO ;,D1 S9CE !Y AL >E
OPPOSITES4 ,IF !N ,A IS AN OBJECT (
DESIRE 6! SAME EXT5T Z ;,D1 ;,B IS AN
OBJECT ( AV].N 6! SAME EXT5T Z ;,C 7S9CE
EA* IS 6! SAME EXT5T Z EA*-! "O AN
OBJECT ( AV].N1 ! O!R AN OBJECT (
DESIRE74 ,"!=E BO? ;,A & ;,C TGR1 & ;,B
& ;,D TGR1 W 2 EQU,Y OBJECTS ( DESIRE OR
AV].N4 ,B S9CE ;,A & ;,C >E PREF]A# TO
;,B & ;,D1 ,A _C 2 EQU,Y DESIRA# ) ;,D2
= !N ;,B AL;G ) ;,D WD 2 EQU,Y DESIRA# )
,A AL;G ) ,C4 ,B IF ;,D IS PREF]A# TO
;,A1 !N ;,B M/ 2 LESS AN OBJECT ( AV].N
?AN ;,C3 =! LESS IS OPPOS$ 6! LESS4 ,B !
GRT] GD & LESS] EVIL >E PREF]A# 6! LESS]
GD & GRT] EVIL3 ! :OLE ,,BD !N IS
PREF]A# 6! :OLE ;,,AC4 ,B EX HYPO!SI ?
IS N S4 ,A !N IS PREF]A# TO ;,D1 & ;,C
3SEQU5TLY IS LESS AN OBJECT ( AV].N ?AN
,B4 ,IF !N E LOV] 9 VIRTUE ( 8 LOVE WD
PREF] ;,A1 VIZ4 T ! 2LOV$ %D 2 S* Z
6GRANT A FAV\R1 & YET %D N GRANT X 7= :
;,C /&S71 6! 2LOV$'S GRANT+ ! FAV\R
7REPRES5T$ BY ;,D7 )\T 2+ S* Z 6GRANT X
7REPRES5T$ BY ;,B71 X IS CLE> T ,A 72+ (
S* A NATURE7 IS PREF]A# 6GRANT+ ! FAV\R4
,6RCV A6EC;N !N IS PREF]A# 9 LOVE
6SEXUAL 9T]C\RSE4 ,LOVE !N IS M DEP5D5T
ON FR%IP ?AN ON 9T]C\RSE4 ,& IF X IS MO/
DEP5D5T ON RCVG A6EC;N1 !N ? IS XS 5D4
,9T]C\RSE !N EI IS N AN 5D AT ALL OR IS
AN 5D RELATIVE 6! FUR!R 5D1 ! RCVG (
A6EC;N4 ,& 9DE$ ! SAME IS TRUE (! O!R
DESIRES & >TS4

                  #BC

  ,X IS CLE> !N H[ ! T]MS >E RELAT$ 9
3V].N1 & 9 RESPECT ( 2+ 9 A HI<] DEGREE
OBJECTS ( AV].N OR ( DESIRE4 ,WE M/ N[
/ATE T N ONLY DIALECTICAL & DEMON/RATIVE
SYLLOGISMS >E =M$ 0M1NS (! A=ESD
FIGURES1 B AL RHETORICAL SYLLOGISMS & 9
G5]AL ANY =M ( P]SUA.N1 H["E X MAY 2
PRES5T$4 ,= E 2LIEF -ES EI "? SYLLOGISM
OR F 9DUC;N4
  ,N[ 9DUC;N1 OR R ! SYLLOGISM : SPR+S \
( 9DUC;N1 3SI/S 9 E/ABLI%+ SYLLOGI/IC,Y
A REL,N 2T "O EXTREME &! MI4LE 0M1NS (!
O!R EXTREME1 E4G4 IF ;,B IS ! MI4LE T]M
2T ;,A & ;,C1 X 3SI/S 9 PROV+ "? ;,C T
,A 2L;GS TO ;,B4 ,= ? IS ! MANN] 9 : WE
MAKE 9DUC;NS4 ,= EXAMPLE LET ,A /& =
L;G-LIV$1 ;,B = BILE.S1 & ;,C =! "PICUL>
L;G-LIV$ ANIMALS1 E4G4 MAN1 HORSE1 MULE4
,A !N 2L;GS 6! :OLE ( ;,C3 = :AT"E IS
BILE.S IS L;G-LIV$4 ,B ;,B AL 7,8N
POSSESS+ BILE0'7 2L;GS 6ALL ,C4 ,IF !N
;,C IS 3V]TI# ) ;,B1 &! MI4LE T]M IS N
WID] 9 EXT5.N1 X IS NEC T ,A %D 2L;G TO
;,B4 ,= X HAS ALR BE5 PROV$ T IF TWO ?+S
2L;G 6! SAME ?+1 &! EXTREME IS 3V]TI# )
"O ( !M1 !N ! O!R PR$ICATE W 2L;G 6!
PR$ICATE T IS 3V]T$4 ,B WE M/ APPREH5D
;,C Z MADE UP ( ALL ! "PICUL>S4 ,=
9DUC;N PROCE$S "? AN ENUM],N ( ALL !
CASES4
  ,S* IS ! SYLLOGISM : E/ABLI%ES ! F/ &
IMM PREMISS3 = ": "! IS A MI4LE T]M !
SYLLOGISM PROCE$S "? ! MI4LE T]M2 :5 "!
IS NO MI4LE T]M1 "? 9DUC;N4 ,& 9 A WAY
9DUC;N IS OPPOS$ 6SYLLOGISM3 =! LATT]
PROVES ! MAJOR T]M 62L;G 6! ?IRD T]M
0M1NS (! MI4LE1 ! =M] PROVES ! MAJOR
62L;G 6! MI4LE 0M1NS (! ?IRD4 ,9 ! ORD]
( NATURE1 SYLLOGISM "? ! MI4LE T]M IS
PRIOR & BETT] "KN1 B SYLLOGISM "? 9DUC;N
IS CLE>] 6U4


                  #BD

  ,WE H AN ,8EXAMPLE0' :5 ! MAJOR T]M IS
PROV$ 62L;G 6! MI4LE 0M1NS (A T]M :
RESEM#S ! ?IRD4 ,X "\ 6BE "KN BO? T !
MI4LE 2L;GS 6! ?IRD T]M1 & T ! F/ 2L;GS
6T : RESEM#S ! ?IRD4 ,= EXAMPLE LET ,A 2
EVIL1 ;,B MAK+ W> AG/ NEI<B\RS1 ;,C
,A!NIANS AG/ ,!BANS1 ;,D ,!BANS AG/
,PHOCIANS4 ,IF !N WE WI% 6PROVE T 6FI<T
)! ,!BANS IS AN EVIL1 WE M/ ASSUME T
6FI<T AG/ NEI<B\RS IS AN EVIL4 ,EVID;E (
? IS OBTA9$ F SIMIL> CASES1 E4G4 T ! W>
AG/ ! ,PHOCIANS 0 AN EVIL 6! ,!BANS4
,S9CE !N 6FI<T AG/ NEI<B\RS IS AN EVIL1
& 6FI<T AG/ ! ,!BANS IS 6FI<T AG/
NEI<B\RS1 X IS CLE> T 6FI<T AG/ ! ,!BANS
IS AN EVIL4 ,N[ X IS CLE> T ;,B 2L;GS TO
;,C & TO ;,D 7= BO? >E CASES ( MAK+ W>
^U "O'S NEI<B\RS7 & T ,A 2L;GS TO ;,D
7=! W> AG/ ! ,PHOCIANS DID N TURN \ WELL
=! ,!BANS73 B T ,A 2L;GS TO ;,B W 2
PROV$ "? ,D4 ,SIMIL>LY IF ! 2LIEF 9 !
REL,N (! MI4LE T]M 6! EXTREME %D 2
PRODUC$ 0S"EAL SIMIL> CASES4 ,CLE>LY !N
6>GUE 0EXAMPLE IS NEI L R1SON+ F "P
6:OLE1 NOR L R1SON+ F :OLE 6"P1 B R
R1SON+ F "P 6"P1 :5 BO? "PICUL>S >E
SUBORD9ATE 6! SAME T]M1 & "O ( !M IS
"KN4 ,X DI6]S F 9DUC;N1 2C 9DUC;N />T+ F
ALL ! "PICUL> CASES PROVES 7Z WE SAW7 T
! MAJOR T]M 2L;GS 6! MI4LE1 & DOES N
APPLY ! SYLLOGI/IC 3CLU.N 6! M9OR T]M1
":AS >GU;T 0EXAMPLE DOES MAKE ? APPLIC,N
& DOES N DRAW XS PRO( F ALL ! "PICUL>
CASES4

                  #BE

  ,0REDUC;N WE M1N AN >GU;T 9 : ! F/ T]M
CLE>LY 2L;GS 6! MI4LE1 B ! REL,N (!
MI4LE 6! LA/ T]M IS UNC]TA9 ?\< EQU,Y OR
M PROBA# ?AN ! 3CLU.N2 OR AG AN >GU;T 9
: ! T]MS 9T]M$IATE 2T ! LA/ T]M &! MI4LE
>E FEW4 ,= 9 ANY ( ^! CASES X TURNS \ T
WE APPROA* M NE>LY 6K4 ,= EXAMPLE LET ,A
/& = :AT C 2 TAU<T1 ;,B = K1 ;,C =
JU/ICE4 ,N[ X IS CLE> T K C 2 TAU<T3 B X
IS UNC]TA9 :E!R VIRTUE IS K4 ,IF N[ !
/ATE;T ,,BC IS EQU,Y OR M PROBA# ?AN
;,,AC1 WE H A REDUC;N3 = WE >E NE>] 6K1
S9CE WE H TAK5 A NEW T]M1 2+ S F> )\T K
T ,A 2L;GS TO ;,C4 ,OR AG SUPPOSE T !
T]MS 9T]M$IATE 2T ;,B & ;,C >E FEW3 =
?US TOO WE >E NE>] K4 ,= EXAMPLE LET ;,D
/& = SQU>+1 ;,E = RECTIL9E> FIGURE1 ;,F
= CIRCLE4 ,IF "! 7 ONLY "O T]M 9T]M$IATE
2T ;,E & ;,F 7VIZ4 T ! CIRCLE IS MADE
EQUAL 6A RECTIL9E> FIGURE 0! HELP (
LUNULES71 WE %D 2 NE> 6K4 ,B :5 ,,BC IS
N M PROBA# ?AN ;,,AC1 &! 9T]M$IATE T]MS
>E N FEW1 ,I D N CALL ? REDUC;N3 NOR AG
:5 ! /ATE;T ,,BC IS IMM3 = S* A /ATE;T
IS K4

                  #BF

  ,AN OBJEC;N IS A PREMISS 3TR>Y 6A
PREMISS4 ,X DI6]S F A PREMISS1 2C X MAY
2 "PICUL>1 B A PREMISS EI _C 2 "PICUL>
AT ALL OR N 9 UNIV]SAL SYLLOGISMS4 ,AN
OBJEC;N IS BR"\ 9 TWO WAYS & "? TWO
FIGURES2 9 TWO WAYS 2C E OBJEC;N IS EI
UNIV]SAL OR "PICUL>1 0TWO FIGURES 2C
OBJEC;NS >E BR"\ 9 OPPOSI;N 6! PREMISS1
& OPPOSITES C 2 PROV$ ONLY 9 ! F/ & ?IRD
FIGURES4 ,IF A MAN MA9TA9S A UNIV]SAL
A6IRMATIVE1 WE REPLY )A UNIV]SAL OR A
"PICUL> NEGATIVE2 ! =M] IS PROV$ F ! F/
FIGURE1 ! LATT] F ! ?IRD4 ,= EXAMPLE LET
/& = "! 2+ A S+LE SCI;E1 ;,B = 3TR>IES4
,IF A MAN PREMISES T 3TR>IES >E SUBJECTS
(A S+LE SCI;E1 ! OBJEC;N MAY 2 EI T
OPPOSITES >E N"E SUBJECTS (A S+LE SCI;E1
& 3TR>IES >E OPPOSITES1 S T WE GET ! F/
FIGURE1 OR T ! "KA# &! UN"KA# >E N
SUBJECTS (A S+LE SCI;E3 ? PRO( IS 9 !
?IRD FIGURE3 = X IS TRUE ( ;,C 7! "KA# &
! UN"KA#7 T !Y >E 3TR>IES1 & X IS FALSE
T !Y >E ! SUBJECTS (A S+LE SCI;E4
  ,SIMIL>LY IF ! PREMISS OBJECT$ 6IS
NEGATIVE4 ,= IF A MAN MA9TA9S T 3TR>IES
>E N SUBJECTS (A S+LE SCI;E1 WE REPLY EI
T ALL OPPOSITES OR T C]TA9 3TR>IES1 E4G4
:AT IS H1L?Y & :AT IS SICKLY1 >E
SUBJECTS (! SAME SCI;E3 ! =M] >GU;T
ISSUES F ! F/1 ! LATT] F ! ?IRD FIGURE4
  ,9 G5]AL IF A MAN URGES A UNIV]SAL
OBJEC;N HE M/ FRAME 8 3TRADIC;N ) REF];E
6! UNIV]SAL (! T]MS TAK5 0HIS OPPON5T1
E4G4 IF A MAN MA9TA9S T 3TR>IES >E N
SUBJECTS (! SAME SCI;E1 8 OPPON5T M/
REPLY T "! IS A S+LE SCI;E ( ALL
OPPOSITES4 ,?US WE M/ H ! F/ FIGURE3 =!
T]M : EMBRACES ! ORIG9AL SUBJECT 2COMES
! MI4LE T]M4
  ,IF ! OBJEC;N IS "PICUL>1 ! OBJECTOR
M/ FRAME 8 3TRADIC;N ) REF];E 6A T]M
RELATIVELY 6: ! SUBJECT ( 8 OPPON5T'S
PREMISS IS UNIV]SAL1 E4G4 HE W PO9T \ T
! "KA# &! UN"KA# >E N SUBJECTS (! SAME
SCI;E3 ,83TR>IES0' IS UNIV]SAL
RELATIVELY 6^!4 ,& WE H ! ?IRD FIGURE3 =
! "PICUL> T]M ASSUM$ IS MI4LE1 E4G4 !
"KA# &! UN"KA#4 ,PREMISSES F : X IS
POSSI# 6DRAW ! 3TR>Y 3CLU.N >E :AT WE
/>T F :5 WE TRY 6MAKE OBJEC;NS4
,3SEQU5TLY WE BR+ OBJEC;NS 9 ^! FIGURES
ONLY3 = 9 !M ONLY >E OPPOSITE SYLLOGISMS
POSSI#1 S9CE ! SECOND FIGURE _C PRODUCE
AN A6IRMATIVE 3CLU.N4
  ,2SS1 AN OBJEC;N 9 ! MI4LE FIGURE WD
REQUIRE A FULL] >GU;T1 E4G4 IF X %D N 2
GRANT$ T ,A 2L;GS TO ;,B1 2C ;,C DOES N
FOLL[ ,B4 ,? C 2 MADE CLE> ONLY 0O!R
PREMISSES4 ,B AN OBJEC;N "\ N 6TURN (F
96O!R ?+S1 B H XS NEW PREMISS Q CLE>
IMMLY4 ,= ? R1SON AL ? IS ! ONLY FIGURE
F : PRO( 0SIGNS _C 2 OBTA9$4
  ,WE M/ 3SID] LAT] ! O!R K9DS (
OBJEC;N1 "NLY ! OBJEC;N F 3TR>IES1 F
SIMIL>S1 & F -MON OP9ION1 & 9QUIRE :E!R
A "PICUL> OBJEC;N _C 2 ELICIT$ F ! F/
FIGURE OR A NEGATIVE OBJEC;N F ! SECOND4

                  #BG

  ,A PROBABIL;Y &A SIGN >E N ID5TICAL1 B
A PROBABIL;Y IS A G5],Y APPROV$
PROPOSI;N3 :AT M5 "K 6HAPP5 OR N 6HAPP51
6BE OR N 6BE1 =! MO/ "P ?US & ?US1 IS A
PROBABIL;Y1 E4G4 ,8! 5VI\S HATE0'1 ,8!
2LOV$ %[ A6EC;N0'4 ,A SIGN M1NS A
DEMON/RATIVE PROPOSI;N NEC OR G5],Y
APPROV$3 = ANY?+ S* T :5 X IS ANO!R ?+
IS1 OR :5 X HAS -E 962+ ! O!R HAS -E
962+ 2F OR AF1 IS A SIGN (! O!R'S 2+ OR
HAV+ -E 962+4 ,N[ AN 5?YMEME IS A
SYLLOGISM />T+ F PROBABILITIES OR SIGNS1
&A SIGN MAY 2 TAK5 9 ?REE WAYS1
CORRESPOND+ 6! POSI;N (! MI4LE T]M 9 !
FIGURES4 ,= X MAY 2 TAK5 Z 9 ! F/ FIGURE
OR ! SECOND OR ! ?IRD4 ,= EXAMPLE ! PRO(
T A WOMAN IS ) * 2C %E HAS MILK IS 9 !
F/ FIGURE3 = 6H MILK IS ! MI4LE T]M4
,LET ,A REPRES5T 6BE ) *1 ;,B 6H MILK1
;,C WOMAN4 ,! PRO( T WISE M5 >E GD1 S9CE
,PITTACUS IS GD1 -ES "? ! LA/ FIGURE4
,LET ,A /& = GD1 ;,B = WISE M51 ;,C =
,PITTACUS4 ,X IS TRUE !N 6A6IRM BO? ;,A
& ;,B ( ;,C3 ONLY M5 D N SAY ! LATT]1 2C
!Y "K X1 ?\< !Y /ATE ! =M]4 ,! PRO( T A
WOMAN IS ) * 2C %E IS PALE IS M1NT 6-E
"? ! MI4LE FIGURE3 = S9CE PALE;S FOLL[S
WOM5 ) * & IS A 3COMITANT ( ? WOMAN1 P
SUPPOSE X HAS BE5 PROV$ T %E IS ) *4
,LET ,A /& = PALE;S1 ;,B = 2+ ) *1 ;,C =
WOMAN4 ,N[ IF ! "O PROPOSI;N IS /AT$1 WE
H ONLY A SIGN1 B IF ! O!R IS /AT$ Z
WELL1 A SYLLOGISM1 E4G4 ,8,PITTACUS IS
G5]\S1 S9CE AMBITI\S M5 >E G5]\S &
,PITTACUS IS AMBITI\S40' ,OR AG ,8,WISE
M5 >E GD1 S9CE ,PITTACUS IS N ONLY GD B
WISE40' ,9 ? WAY !N SYLLOGISMS >E =M$1
ONLY T : PROCE$S "? ! F/ FIGURE IS
IRREFUTA# IF X IS TRUE 7= X IS
UNIV]SAL71 T : PROCE$S "? ! LA/ FIGURE
IS REFUTA# EV5 IF ! 3CLU.N IS TRUE1 S9CE
! SYLLOGISM IS N UNIV]SAL NOR
CORRELATIVE 6! MATT] 9 "Q3 = ?\<
,PITTACUS IS GD1 X IS N "!=E NEC T ALL
O!R WISE M5 %D 2 GD4 ,B ! SYLLOGISM :
PROCE$S "? ! MI4LE FIGURE IS ALW REFUTA#
9 ANY CASE3 =A SYLLOGISM C N"E 2 =M$ :5
! T]MS >E RELAT$ 9 ? WAY3 = ?\< A WOMAN
) * IS PALE1 & ? WOMAN AL IS PALE1 X IS
N NEC T %E %D 2 ) *4 ,TRU? !N MAY 2 F.D
9 SIGNS :AT"E _! K9D1 B !Y H ! DI6];ES
WE H /AT$4
  ,WE M/ EI DIVIDE SIGNS 9 ! WAY /AT$1 &
AM;G !M DESIGNATE ! MI4LE T]M Z ! 9DEX
7= P CALL T ! 9DEX : MAKES U "K1 &!
MI4LE T]M ABV ALL HAS ? "*71 OR ELSE WE
M/ CALL ! >GU;TS DERIV$ F ! EXTREMES
SIGNS1 T DERIV$ F ! MI4LE T]M ! 9DEX3 =
T : IS PROV$ "? ! F/ FIGURE IS MO/ G5],Y
A3EPT$ & MO/ TRUE4
  ,X IS POSSI# 69F] "* F F1TURES1 IF X
IS GRANT$ T ! BODY &! S\L >E *ANG$ TGR
0! NATURAL A6EC;NS3 ,I SAY ,8NATURAL0'1
= ?\< P]H 0LE>N+ MUSIC A MAN HAS MADE "S
*ANGE 9 8 S\L1 ? IS N "O ( ^? A6EC;NS :
>E NATURAL 6U2 R ,I REF] 6PAS.NS &
DESIRES :5 ,I SP1K ( NATURAL EMO;NS4 ,IF
!N ? 7 GRANT$ & AL T = EA* *ANGE "! IS A
CORRESPOND+ SIGN1 & WE CD /ATE ! A6EC;N
& SIGN PROP] 6EA* K9D ( ANIMAL1 WE % 2
A# 69F] "* F F1TURES4 ,= IF "! IS AN
A6EC;N : 2L;GS PROP]LY 6AN 9DIVIDUAL
K9D1 E4G4 C\RAGE 6LIONS1 X IS NEC T "!
%D 2 A SIGN ( X3 = EX HYPO!SI BODY & S\L
>E A6ECT$ TGR4 ,SUPPOSE ? SIGN IS !
POSSES.N ( L>GE EXTREMITIES3 ? MAY 2L;G
6O!R K9DS AL ?\< N UNIV]S,Y4 ,=! SIGN IS
PROP] 9 ! S5SE /AT$1 2C ! A6EC;N IS
PROP] 6! :OLE K9D1 ?\< N PROP] 6X AL"O1
AC 6\R USUAL MANN] ( SP1K+4 ,! SAME ?+
!N W 2 F.D 9 ANO!R K9D1 & MAN MAY 2
BRAVE1 & "S O!R K9DS ( ANIMAL Z WELL4
,!Y W !N H ! SIGN3 = EX HYPO!SI "! IS "O
SIGN CORRESPOND+ 6EA* A6EC;N4 ,IF !N ?
IS S1 & WE C COLLECT SIGNS ( ? SORT 9 ^!
ANIMALS : H ONLY "O A6EC;N PROP] 6!M-B
EA* A6EC;N HAS XS SIGN1 S9CE X IS NEC T
X %D H A S+LE SIGN-WE % !N 2 A# 69F] "*
F F1TURES4 ,B IF ! K9D Z A :OLE HAS TWO
PROP]TIES1 E4G4 IF ! LION IS BO? BRAVE &
G5]\S1 H[ % WE "K : (! SIGNS : >E XS
PROP] 3COMITANTS IS ! SIGN (A "PICUL>
A6EC;N8 ,P]H IF BO? 2L;G 6"S O!R K9D ?\<
N 6! :OLE ( X1 & IF1 9 ^? K9DS 9 : EA*
IS F.D ?\< N 9 ! :OLE ( _! MEMB]S1 "S
MEMB]S POSSESS "O (! A6EC;NS & N ! O!R3
E4G4 IF A MAN IS BRAVE B N G5]\S1 B
POSSESSES1 (! TWO SIGNS1 L>GE
EXTREMITIES1 X IS CLE> T ? IS ! SIGN (
C\RAGE 9 ! LION AL4 ,6JUDGE "* F
F1TURES1 !N1 IS POSSI# 9 ! F/ FIGURE IF
! MI4LE T]M IS 3V]TI# )! F/ EXTREME1 B
IS WID] ?AN ! ?IRD T]M & N 3V]TI# ) X3
E4G4 LET ,A /& = C\RAGE1 ;,B = L>GE
EXTREMITIES1 & ;,C = LION4 ;,B !N 2L;GS
6"EY?+ 6: ;,C 2L;GS1 B AL 6O!RS4 ,B ,A
2L;GS 6"EY?+ 6: ;,B 2L;GS1 & 6NO?+ 2SS1
B IS 3V]TI# ) ;,B3 O!RWISE1 "! WD N 2 A
S+LE SIGN CORRELATIVE ) EA* A6EC;N4


