conclusion　becomes　its　contradictory…i。e。　true。　Similarly　（ii）　if
  the　middle　is　taken　from　another　series　of　predication；　e。g。　suppose　D
  to　be　not　only　contained　within　A　as　a　part　within　its　whole　but
  also　predicable　of　all　B。　Then　the　premiss　D…B　must　remain
  unchanged；　but　the　quality　of　A…D　must　be　changed；　so　that　D…B　is
  always　true；　A…D　always　false。　Such　error　is　practically　identical
  with　that　which　is　inferred　through　the　'appropriate'　middle。　On　the
  other　hand；　（b）　if　the　conclusion　is　not　inferred　through　the
  'appropriate'　middle…（i）　when　the　middle　is　subordinate　to　A　but　is
  predicable　of　no　B；　both　premisses　must　be　false；　because　if　there
  is　to　be　a　conclusion　both　must　be　posited　as　asserting　the　contrary
  of　what　is　actually　the　fact；　and　so　posited　both　become　false：　e。g。
  suppose　that　actually　all　D　is　A　but　no　B　is　D；　then　if　these
  premisses　are　changed　in　quality；　a　conclusion　will　follow　and　both　of
  the　new　premisses　will　be　false。　When；　however；　（ii）　the　middle　D　is
  not　subordinate　to　A；　A…D　will　be　true；　D…B　false…A…D　true　because　A
  was　not　subordinate　to　D；　D…B　false　because　if　it　had　been　true；　the
  conclusion　too　would　have　been　true；　but　it　is　ex　hypothesi　false。
  When　the　erroneous　inference　is　in　the　second　figure；　both　premisses
  cannot　be　entirely　false；　since　if　B　is　subordinate　to　A；　there　can　be
  no　middle　predicable　of　all　of　one　extreme　and　of　none　of　the　other；
  as　was　stated　before。　One　premiss；　however；　may　be　false；　and　it　may
  be　either　of　them。　Thus；　if　C　is　actually　an　attribute　of　both　A　and
  B；　but　is　assumed　to　be　an　attribute　of　A　only　and　not　of　B；　C…A
  will　be　true；　C…B　false：　or　again　if　C　be　assumed　to　be　attributable
  to　B　but　to　no　A；　C…B　will　be　true；　C…A　false。
  We　have　stated　when　and　through　what　kinds　of　premisses　error　will
  result　in　cases　where　the　erroneous　conclusion　is　negative。　If　the
  conclusion　is　affirmative；　（a）　（i）　it　may　be　inferred　through　the
  'appropriate'　middle　term。　In　this　case　both　premisses　cannot　be　false
  since；　as　we　said　before；　C…B　must　remain　unchanged　if　there　is　to
  be　a　conclusion；　and　consequently　A…C；　the　quality　of　which　is
  changed；　will　always　be　false。　This　is　equally　true　if　（ii）　the　middle
  is　taken　from　another　series　of　predication；　as　was　stated　to　be　the
  case　also　with　regard　to　negative　error；　for　D…B　must　remain
  unchanged；　while　the　quality　of　A…D　must　be　converted；　and　the　type　of
  error　is　the　same　as　before。
  （b）　The　middle　may　be　inappropriate。　Then　（i）　if　D　is　subordinate　to
  A；　A…D　will　be　true；　but　D…B　false；　since　A　may　quite　well　be
  predicable　of　several　terms　no　one　of　which　can　be　subordinated　to
  another。　If；　however；　（ii）　D　is　not　subordinate　to　A；　obviously　A…D；
  since　it　is　affirmed；　will　always　be　false；　while　D…B　may　be　either
  true　or　false；　for　A　may　very　well　be　an　attribute　of　no　D；　whereas
  all　B　is　D；　e。g。　no　science　is　animal；　all　music　is　science。　Equally
  well　A　may　be　an　attribute　of　no　D；　and　D　of　no　B。　It　emerges；　then；
  that　if　the　middle　term　is　not　subordinate　to　the　major；　not　only　both
  premisses　but　either　singly　may　be　false。
  Thus　we　have　made　it　clear　how　many　varieties　of　erroneous　inference
  are　liable　to　happen　and　through　what　kinds　of　premisses　they　occur；
  in　the　case　both　of　immediate　and　of　demonstrable　truths。
  18
  It　is　also　clear　that　the　loss　of　any　one　of　the　senses　entails
  the　loss　of　a　corresponding　portion　of　knowledge；　and　that；　since　we
  learn　either　by　induction　or　by　demonstration；　this　knowledge　cannot
  be　acquired。　Thus　demonstration　develops　from　universals；　induction
  from　particulars；　but　since　it　is　possible　to　familiarize　the　pupil
  with　even　the　so…called　mathematical　abstractions　only　through
  induction…i。e。　only　because　each　subject　genus　possesses；　in　virtue　of
  a　determinate　mathematical　character；　certain　properties　which　can
  be　treated　as　separate　even　though　they　do　not　exist　in　isolation…it
  is　consequently　impossible　to　come　to　grasp　universals　except
  through　induction。　But　induction　is　impossible　for　those　who　have
  not　sense…perception。　For　it　is　sense…perception　alone　which　is
  adequate　for　grasping　the　particulars：　they　cannot　be　objects　of
  scientific　knowledge；　because　neither　can　universals　give　us　knowledge
  of　them　without　induction；　nor　can　we　get　it　through　induction　without
  sense…perception。
  19
  Every　syllogism　is　effected　by　means　of　three　terms。　One　kind　of
  syllogism　serves　to　prove　that　A　inheres　in　C　by　showing　that　A
  inheres　in　B　and　B　in　C；　the　other　is　negative　and　one　of　its
  premisses　asserts　one　term　of　another；　while　the　other　denies　one　term
  of　another。　It　is　clear；　then；　that　these　are　the　fundamentals　and
  so…called　hypotheses　of　syllogism。　Assume　them　as　they　have　been
  stated；　and　proof　is　bound　to　follow…proof　that　A　inheres　in　C　through
  B；　and　again　that　A　inheres　in　B　through　some　other　middle　term；　and
  similarly　that　B　inheres　in　C。　If　our　reasoning　aims　at　gaining
  credence　and　so　is　merely　dialectical；　it　is　obvious　that　we　have　only
  to　see　that　our　inference　is　based　on　premisses　as　credible　as
  possible：　so　that　if　a　middle　term　between　A　and　B　is　credible
  though　not　real；　one　can　reason　through　it　and　complete　a
  dialectical　syllogism。　If；　however；　one　is　aiming　at　truth；　one　must
  be　guided　by　the　real　connexions　of　subjects　and　attributes。　Thus：
  since　there　are　attributes　which　are　predicated　of　a　subject
  essentially　or　naturally　and　not　coincidentally…not；　that　is；　in　the
  sense　in　which　we　say　'That　white　（thing）　is　a　man'；　which　is　not
  the　same　mode　of　predication　as　when　we　say　'The　man　is　white'：　the
  man　is　white　not　because　he　is　something　else　but　because　he　is　man；
  but　the　white　is　man　because　'being　white'　coincides　with　'humanity'
  within　one　substratum…therefore　there　are　terms　such　as　are
  naturally　subjects　of　predicates。　Suppose；　then；　C　such　a　term　not
  itself　attributable　to　anything　else　as　to　a　subject；　but　the
  proximate　subject　of　the　attribute　Bi。e。　so　that　B…C　is　immediate；
  suppose　further　E　related　immediately　to　F；　and　F　to　B。　The　first
  question　is；　must　this　series　terminate；　or　can　it　proceed　to
  infinity？　The　second　question　is　as　follows：　Suppose　nothing　is
  essentially　predicated　of　A；　but　A　is　predicated　primarily　of　H　and　of
  no　intermediate　prior　term；　and　suppose　H　similarly　related　to　G　and　G
  to　B；　then　must　this　series　also　terminate；　or　can　it　too　proceed　to
  infinity？　There　is　this　much　difference　between　the　questions：　the
  first　is；　is　it　possible　to　start　from　that　which　is　not　itself
  attributable　to　anything　else　but　is　the　subject　of　attributes；　and
  ascend　to　infinity？　The　second　is　the　problem　whether　one　can　start
  from　that　which　is　a　predicate　but　not　itself　a　subject　of　predicates；
  and　descend　to　infinity？　A　third　question　is；　if　the　extreme　terms　are
  fixed；　can　there　be　an　infinity　of　middles？　I　mean　this：　suppose　for
  example　that　A　inheres　in　C　and　B　is　intermediate　between　them；　but
  between　B　and　A　there　are　other　middles；　and　between　these　again　fresh
  middles；　can　these　proceed　to　infinity　or　can　they　not？　This　is　the
  equivalent　of　inquiring；　do　demonstrations　proceed　to　infinity；　i。e。
  is　everything　demonstrable？　Or　do　ultimate　subject　and　primary
  attribute　limit　one　another？
  I　hold　that　the　same　questions　arise　with　regard　to　negative
  conclusions　and　premisses：　viz。　if　A　is　attributable　to　no　B；　then
  either　this　predication　will　be　primary；　or　there　will　be　an
  intermediate　term　prior　to　B　to　which　a　is　not　attributable…G；　let
  us　say；　which　is　attributable　to　all　B…and　there　may　still　be
  another　term　H　prior　to　G；　which　is　attributable　to　all　G。　The　same
  questions　arise；　I　say；　because　in　these　cases　too　either　the　series
  of　prior　terms　to　which　a　is　not　attributable　is　infinite　or　it
  terminates。
  One　cannot　ask　the　same　questions　in　the　case　of　reciprocating
  terms；　since　when　subject　and　predicate　are　convertible　there　is
  neither　primary　nor　ultimate　subject；　seeing　that　all　the
  reciprocals　qua　subjects　stand　in　the　same　relation　to　one　another；
  whether　we　say　that　the　subject　has　an　infinity　of　attributes　or
  that　both　subjects　and　attributes…and　we　raised　the　question　in　both
  cases…are　infinite　in　number。　These　questions　then　cannot　be
  asked…unless；　indeed；　the　terms　can　reciprocate　by　two　different
  modes；　by　accidental　predication　i