syllogism；　and　therefore　a　fortiori　demonstration；　is　addressed　not　to
  the　spoken　word；　but　to　the　discourse　within　the　soul；　and　though　we
  can　always　raise　objections　to　the　spoken　word；　to　the　inward
  discourse　we　cannot　always　object。　That　which　is　capable　of　proof
  but　assumed　by　the　teacher　without　proof　is；　if　the　pupil　believes　and
  accepts　it；　hypothesis；　though　only　in　a　limited　sense　hypothesis…that
  is；　relatively　to　the　pupil；　if　the　pupil　has　no　opinion　or　a　contrary
  opinion　on　the　matter；　the　same　assumption　is　an　illegitimate
  postulate。　Therein　lies　the　distinction　between　hypothesis　and
  illegitimate　postulate：　the　latter　is　the　contrary　of　the　pupil's
  opinion；　demonstrable；　but　assumed　and　used　without　demonstration。
  The　definition…viz。　those　which　are　not　expressed　as　statements　that
  anything　is　or　is　not…are　not　hypotheses：　but　it　is　in　the　premisses
  of　a　science　that　its　hypotheses　are　contained。　Definitions　require
  only　to　be　understood；　and　this　is　not　hypothesis…unless　it　be
  contended　that　the　pupil's　hearing　is　also　an　hypothesis　required　by
  the　teacher。　Hypotheses；　on　the　contrary；　postulate　facts　on　the　being
  of　which　depends　the　being　of　the　fact　inferred。　Nor　are　the
  geometer's　hypotheses　false；　as　some　have　held；　urging　that　one　must
  not　employ　falsehood　and　that　the　geometer　is　uttering　falsehood　in
  stating　that　the　line　which　he　draws　is　a　foot　long　or　straight；
  when　it　is　actually　neither。　The　truth　is　that　the　geometer　does　not
  draw　any　conclusion　from　the　being　of　the　particular　line　of　which
  he　speaks；　but　from　what　his　diagrams　symbolize。　A　further　distinction
  is　that　all　hypotheses　and　illegitimate　postulates　are　either
  universal　or　particular；　whereas　a　definition　is　neither。
  11
  So　demonstration　does　not　necessarily　imply　the　being　of　Forms　nor　a
  One　beside　a　Many；　but　it　does　necessarily　imply　the　possibility　of
  truly　predicating　one　of　many；　since　without　this　possibility　we
  cannot　save　the　universal；　and　if　the　universal　goes；　the　middle
  term　goes　witb。　it；　and　so　demonstration　becomes　impossible。　We
  conclude；　then；　that　there　must　be　a　single　identical　term
  unequivocally　predicable　of　a　number　of　individuals。
  The　law　that　it　is　impossible　to　affirm　and　deny　simultaneously
  the　same　predicate　of　the　same　subject　is　not　expressly　posited　by　any
  demonstration　except　when　the　conclusion　also　has　to　be　expressed　in
  that　form；　in　which　case　the　proof　lays　down　as　its　major　premiss　that
  the　major　is　truly　affirmed　of　the　middle　but　falsely　denied。　It　makes
  no　difference；　however；　if　we　add　to　the　middle；　or　again　to　the　minor
  term；　the　corresponding　negative。　For　grant　a　minor　term　of　which　it
  is　true　to　predicate　man…even　if　it　be　also　true　to　predicate
  not…man　of　itstill　grant　simply　that　man　is　animal　and　not
  not…animal；　and　the　conclusion　follows：　for　it　will　still　be　true　to
  say　that　Calliaseven　if　it　be　also　true　to　say　that
  not…Calliasis　animal　and　not　not…animal。　The　reason　is　that　the
  major　term　is　predicable　not　only　of　the　middle；　but　of　something
  other　than　the　middle　as　well；　being　of　wider　application；　so　that　the
  conclusion　is　not　affected　even　if　the　middle　is　extended　to　cover　the
  original　middle　term　and　also　what　is　not　the　original　middle　term。
  The　law　that　every　predicate　can　be　either　truly　affirmed　or　truly
  denied　of　every　subject　is　posited　by　such　demonstration　as　uses
  reductio　ad　impossibile；　and　then　not　always　universally；　but　so　far
  as　it　is　requisite；　within　the　limits；　that　is；　of　the　genus…the
  genus；　I　mean　（as　I　have　already　explained）；　to　which　the　man　of
  science　applies　his　demonstrations。　In　virtue　of　the　common　elements
  of　demonstration…I　mean　the　common　axioms　which　are　used　as
  premisses　of　demonstration；　not　the　subjects　nor　the　attributes
  demonstrated　as　belonging　to　them…all　the　sciences　have　communion　with
  one　another；　and　in　communion　with　them　all　is　dialectic　and　any
  science　which　might　attempt　a　universal　proof　of　axioms　such　as　the
  law　of　excluded　middle；　the　law　that　the　subtraction　of　equals　from
  equals　leaves　equal　remainders；　or　other　axioms　of　the　same　kind。
  Dialectic　has　no　definite　sphere　of　this　kind；　not　being　confined　to　a
  single　genus。　Otherwise　its　method　would　not　be　interrogative；　for　the
  interrogative　method　is　barred　to　the　demonstrator；　who　cannot　use　the
  opposite　facts　to　prove　the　same　nexus。　This　was　shown　in　my　work　on
  the　syllogism。
  12
  If　a　syllogistic　question　is　equivalent　to　a　proposition　embodying
  one　of　the　two　sides　of　a　contradiction；　and　if　each　science　has　its
  peculiar　propositions　from　which　its　peculiar　conclusion　is　developed；
  then　there　is　such　a　thing　as　a　distinctively　scientific　question；　and
  it　is　the　interrogative　form　of　the　premisses　from　which　the
  'appropriate'　conclusion　of　each　science　is　developed。　Hence　it　is
  clear　that　not　every　question　will　be　relevant　to　geometry；　nor　to
  medicine；　nor　to　any　other　science：　only　those　questions　will　be
  geometrical　which　form　premisses　for　the　proof　of　the　theorems　of
  geometry　or　of　any　other　science；　such　as　optics；　which　uses　the
  same　basic　truths　as　geometry。　Of　the　other　sciences　the　like　is　true。
  Of　these　questions　the　geometer　is　bound　to　give　his　account；　using
  the　basic　truths　of　geometry　in　conjunction　with　his　previous
  conclusions；　of　the　basic　truths　the　geometer；　as　such；　is　not　bound
  to　give　any　account。　The　like　is　true　of　the　other　sciences。　There
  is　a　limit；　then；　to　the　questions　which　we　may　put　to　each　man　of
  science；　nor　is　each　man　of　science　bound　to　answer　all　inquiries　on
  each　several　subject；　but　only　such　as　fall　within　the　defined　field
  of　his　own　science。　If；　then；　in　controversy　with　a　geometer　qua
  geometer　the　disputant　confines　himself　to　geometry　and　proves
  anything　from　geometrical　premisses；　he　is　clearly　to　be　applauded；　if
  he　goes　outside　these　he　will　be　at　fault；　and　obviously　cannot　even
  refute　the　geometer　except　accidentally。　One　should　therefore　not
  discuss　geometry　among　those　who　are　not　geometers；　for　in　such　a
  company　an　unsound　argument　will　pass　unnoticed。　This　is
  correspondingly　true　in　the　other　sciences。
  Since　there　are　'geometrical'　questions；　does　it　follow　that　there
  are　also　distinctively　'ungeometrical'　questions？　Further；　in　each
  special　science…geometry　for　instance…what　kind　of　error　is　it　that
  may　vitiate　questions；　and　yet　not　exclude　them　from　that　science？
  Again；　is　the　erroneous　conclusion　one　constructed　from　premisses
  opposite　to　the　true　premisses；　or　is　it　formal　fallacy　though　drawn
  from　geometrical　premisses？　Or；　perhaps；　the　erroneous　conclusion　is
  due　to　the　drawing　of　premisses　from　another　science；　e。g。　in　a
  geometrical　controversy　a　musical　question　is　distinctively
  ungeometrical；　whereas　the　notion　that　parallels　meet　is　in　one
  sense　geometrical；　being　ungeometrical　in　a　different　fashion：　the
  reason　being　that　'ungeometrical'；　like　'unrhythmical'；　is
  equivocal；　meaning　in　the　one　case　not　geometry　at　all；　in　the　other
  bad　geometry？　It　is　this　error；　i。e。　error　based　on　premisses　of
  this　kind…'of'　the　science　but　false…that　is　the　contrary　of
  science。　In　mathematics　the　formal　fallacy　is　not　so　common；　because
  it　is　the　middle　term　in　which　the　ambiguity　lies；　since　the　major
  is　predicated　of　the　whole　of　the　middle　and　the　middle　of　the　whole
  of　the　minor　（the　predicate　of　course　never　has　the　prefix　'all'）；　and
  in　mathematics　one　can；　so　to　speak；　see　these　middle　terms　with　an
  intellectual　vision；　while　in　dialectic　the　ambiguity　may　escape
  detection。　E。g。　'Is　every　circle　a　figure？'　A　diagram　shows　that
  this　is　so；　but　the　minor　premiss　'Are　epics　circles？'　is　shown　by　the
  diagram　to　be　false。
  If　a　proof　has　an　inductive　minor　premiss；　one　should　not　bring　an
  'objection'　against　it。　For　since　every　premiss　must　be　applicable
  to　a　number　of　cases　（otherwise　it　will　not　be　true　in　every　instance；
  which；　since　the　syllogism　proceeds　from　universals；　it　must　be）；　then
  assuredly　the　same　is　true　of　an　'objection'；　since　premisses　and
  'objections'　are　so　far　the　same　that　anything　which　can　be　validly
  advanced　as　an　'objection'　must　be　such　that　it　could　take　the　form　of
  a　premiss；　either　demonstrative　or　dialectical。　On　the　other　hand；
  arguments　formally　illogical　do　sometimes　occur　through　taking　as
  middles　mere