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2.#Deductive#Arguments
!
A.#The#Concept#of#Deduction
!!!!!!!!!!!!The!first!type!of!argument!that!we!are!going!to!learn!about!is!deduction.!As!we!saw!in!the!previous
chapter,!deductive#arguments!attempt!to!support!their!conclusions!with!certainty.!What!is!certainty?
Certainty!is!the!highest!level!of!knowledge,!the!complete!lack!of!doubt:!If!a!deductive!argument!is
successful,!there!is!no!doubt!that!its!conclusion!is!true.!Deduction!is!the!gold!standard!for!arguments:!In
contrast!to!induction,!it!is!able!to!provide!the!maximum!amount!of!support!that!a!conclusion!can!receive.!If
inductive!arguments!are!inherently!inferior,!why!we!bother!with!them?!Why!don’t!we!limit!ourselves!to
deductive!arguments?!It!turns!out!that!in!certain!types!of!cases!deductive!arguments!are!inapplicable!or!it!is
much!more!difficult!to!find!premises!for!them.
!!!!!!!!!!!!How!do!successful!deductive!arguments!support!their!conclusions!with!certainty?!They!accomplish
this!by!providing!an!iron!clad!connection!between!the!premises!and!the!conclusion.!The!form!or!structure!of
the!argument!is!so!solid!that,!given!what!is!stated!in!the!premises,!the!conclusion!is!true*by*definition.
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!In
a!sense,!the!conclusion!is!already!found!in!the!premises:!Either!its!components!parts!are!located!in!different
premises!or!the!entire!conclusion!is!part!of!one!of!the!premises.!In!the!first!case,!the!argument!combines
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together!these!parts,!and!in!the!second!it!extracts!the!conclusion!from!the!larger!statement!it!is!part!of.
!!!!!!!!!!!!There!are!special!sets!of!terms!that!we!use!to!evaluate!deductive!and!inductive!arguments.!If!a
deductive!argument!possesses!a!proper!logical!form!–!the!second!condition!of!a!successful!argument!–!then
we!say!that!it!is!valid.!If!it!lacks!a!proper!logical!form,!we!say!that!it!is!invalid.!When!a!deductive!argument!is
valid,!it!is!impossible!for!its!premises!to!be!true!and!its!conclusion!to!be!false.!If!a!deductive!argument!is
successful,!supporting!its!conclusion!with!certainty,!we!say!that!it!is!sound.!A!deductive!argument!is!sound
when!all!of!its!premises!are!true!and!it!is!valid,!in!other!words,!when!it!satisfies!both!of!the!conditions!for!a
successful!argument.!If!one!or!more!of!these!conditions!is!missing!in!a!deductive!argument,!i.e.,!if!it!is
invalid!or!one!or!more!of!its!premises!is!doubtful,!then!it!is!unsound.!As!we’ll!see!in!the!next!chapter,!we
use!a!different!set!of!terms!to!evaluate!inductive!arguments.!Students!often!mix!up!the!terms!we!use!to
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evaluate!statements!–!truth!and!falsity!–!with!the!terms!we!use!to!evaluate!arguments.!Premises!and
conclusions!and!other!statements!are!true!or!false,!but!are!never!valid!or!sound.!Conversely,!deductive
arguments!are!valid!or!invalid,!sound!or!unsound,!but!are!never!true!or!false.!Be!careful!not!to!misapply
these!terms.
!
B.#Categorical#Syllogisms
!!!!!!!!!!!!There!are!two!types!of!deductive!arguments,!categorical!syllogisms!and!propositional!arguments.!As
the!names!indicate,!the!primary!difference!between!them!is!the!basic!unit!each!of!them!revolves!around.
Categorical!syllogisms!reason!on!the!basis!of!the!relationship!between!categories.!Categorical!syllogisms
have!two!premises!and!a!conclusion,!and!each!of!these!statements!contains!two!categories!–!its!subject!and
its!predicate.!For!example,!in!the!categorical!statement!“All!dogs!are!mammals”,!the!subject!is!‘dogs’!and
the!predicate!is!‘mammals’.!There!are!four!types!of!categorical!statements:
!
!
Universal#Affirmative####################################Particular#Affirmative
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All!S!are!P. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Some!S!are!P.
Example:!All!dogs!are!mammals!!!!!!!!!!!!!!!!!!!!Example:!Some!dishes!are!cold.
!
Universal#Negative########################################Particular#Negative
No!S!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Some!S!are!not!P.
Example:!No!snakes!are!mammals.!!!!!!!!!!!!!!!Example:!Some!dishes!are!not!cold.
We’ll!only!be!working!with!the!first!two!types,!universal!affirmative!and!particular!affirmative
statements.
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!The!two!best!known!categorical!syllogisms!are!called!Barbara!and!Darii:
!
Barbara###############################################Example:
1.!All!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!mammals!are!living!beings.
2.!All!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!All!dogs!are!mammals.
3.!Therefore,!all!S!are!P.!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!all!dogs!are!living!beings.
!
Darii!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Example:
1.!All!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!pine!trees!are!evergreens.
2.!Some!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!Some!trees!are!pine!trees.
3.!Therefore,!some!S!are!P.!!!!!!!!!!!!!!!!3.!Therefore,!some!trees!are!evergreen.
!
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As!you!can!see,!there!are!three!different!categories!in!every!categorical!syllogism,!a!subject!–!the!first!term
in!the!conclusion!which!is!also!found!in!the!second!premise!–!a!predicate!–!the!second!term!in!the
conclusion!which!is!also!found!in!the!first!premise!–!and!a!middle*term!–!the!term!found!in!both!of!the
premises.!It!turns!out!that!there!are!256!possible!varieties!of!categorical!syllogisms.!However,!only!15!of
them!are!valid,!including!Barbara!and!Darii.!Fortunately!for!students,!the!other!13!varieties!are!merely
variations!of!Barbara!and!Darii,!so!there’s!no!need!for!you!to!memorize!them.
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!!!!!!!!!!!!You!might!wonder!why!Barbara!and!Darii!are!valid!arguments,!i.e.,!why!the!conclusion!is!true!by
definition!given!what!is!stated!in!the!premises!–!the!strongest!possible!connection!between!the!premises
and!the!conclusion.!There!are!a!variety!of!methods!for!demonstrating!validity!which!you!can!learn!about!in
a!dedicated!logic!class.!But!there!is!an!easier!way!to!explain!this.!Aristotle!(384]322!BCE),!the!famous!Greek
philosopher!who!invented!logic,!defined!deduction!as!an!inference!from!general!to!particular,!and!induction
as!an!inference!from!particular!to!general.!These!definitions!are!outdated!because!we!have!since!discovered
forms!of!deduction!and!induction,!like!propositional!arguments,!that!don’t!follow!this!pattern.!But!it!does
help!clarify!the!nature!of!categorical!syllogisms.!A!valid!categorical!syllogism!establishes!a!connection
between!the!subject!of!the!conclusion!and!the!predicate!of!the!conclusion!by!means!of!a!middle!term.!In
Barbara,!the!predicate!is!the!largest!category,!the!subject!is!the!smallest!category,!and!the!middle!term!is
medium!sized.!Barbara!infers!that!the!subject!is!part!of!the!predicate!from!the!fact!that!the!middle!term!is
part!of!the!predicate!and!the!subject!is!part!of!the!middle!term.!In!the!example!above,!‘dogs’!are!part!of!the
larger!category!‘mammals’,!which!are!part!of!the!even!larger!category!‘living!beings’.!The!middle!term
connects!together!the!subject!and!the!predicate!so!solidly!that!we!know!with!certainty!that!the!subject!is
part!of!the!predicate.
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!!!!!!!!!!!!The!invention!of!categorical!syllogisms!allowed!the!ancient!Greeks!to!find!order!in!the!world!by
clarifying!the!categorical!subdivisions!that!different!things!fall!into.!It’s!no!accident!that!Aristotle!was!the
founder!of!biology!and!many!other!sciences!–!he!invented!the!notion!of!biological!classification!into!species,
genus,!family,!order,!etc.!Another!example!of!this!system!of!organization!is!found!in!libraries,!where!books
are!grouped!according!to!discipline,!subdiscipline,!etc.!If!books!were!randomly!thrown!together,!it!would!be
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very!difficult!to!find!anything!and!the!array!of!topics!would!seem!mind]boggling.!However,!when!we
organize!books!according!to!their!similarities!and!differences,!we!are!able!to!identify!patterns!and
understand!the!relations!between!the!different!topics!that!are!covered,!and!it!is!much!easier!to!locate!a
particular!book.
We!use!the!Barbara!argument!with!great!frequency,!to!apply!general!rules!or!concepts!to!particular
cases.!This!argument!is!used!often!in!virtually!all!academic!disciplines!and!everyday!activities,!e.g.,!in!the
sciences,!mathematics,!to!classify!artists!or!artworks,!and!to!formulate!moral!arguments.!However,!it!is
important!to!make!sure!that!you’re!using!the!right!form!since!there!are!arguments!that!look!very!similar!to
Barbara!and!Darii!that!are!invalid:
!
Barbara###################################Invalid#Arguments
1.!All!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!M!are!P.
2.!All!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!!2.!All!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!2.!All!M#are!S.
3.!Therefore,!all!S!are!P.!!!!!!!!!3.!Therefore,!all!S!are!P.!!!!!!!!!3.!Therefore,!all!S!are!P.
!
1.!All!P!are!M.
2.!All!M!are!S.
3.!Therefore,!all!S!are!P.
!
Darii########################################Invalid#Arguments
1.!All!M#are!P.!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!P!are!M.
2.!Some!S!are!M.!!!!!!!!!!!!!!!!!!!!2.!Some!S!are!M.!!!!!!!!!!!!!!!!!!!!2.!All!M!are!S.
3.!Therefore,!some!S!are!P.!!!!3.!Therefore,!some!S!are!P.!!!!3.!Therefore,!some!S!are!P.
!
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Make!sure!that!you!can!tell!the!difference!–!it’s!easy!to!mix!them!up.!Other!common!mistakes!that!are
made!are!equivocation!–!using!different!categories!when!the!same!categories!should!be!used!–!and!leaving
out!the!copula!–!‘is’,!‘are’!or!other!cognates!of!the!verb!‘to!be’.!Equivocation!is!sometimes!due!to!ambiguity
–!using!different!senses!of!the!same!term!of!phrase.!For!example:
!
1.!All!forms!of!light!are!entities!that!make!other!things!visible.
2.!All!feathers!are!light.
3.!Therefore,!all!feathers!are!entities!that!make!other!things!visible.
!
The!word!‘light’!can!refer!to!something!that!makes!things!visible!or!the!property!of!having!little!weight.!The
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middle!term!of!the!argument!uses!different!senses!of!the!word!‘light’,!so!the!argument!fails!to!connect!the
subject!and!predicate!and!is!therefore!invalid.!Another!type!of!equivocation!occurs!when!a!category!is
somewhat!different!in!one!instance!than!the!other.
!
1.!All!undeserved!harm!is!bad.
2.!All!acts!of!violence!against!innocent!people!are!harmful!and!undeserved.
3.!Therefore,!all!acts!of!violence!are!bad.
!
In!this!argument,!the!subject!category!is!broader!in!the!conclusion!(‘acts!of!violence’)!and!narrower!in!the
second!premise!(‘acts!of!violence!against!innocent!people’).!Hence,!the!argument!is!invalid.!The!version!in
the!conclusion!includes!acts!of!violence!against!guilty!people,!which!would!throw!the!truth!of!the!second
premise!into!doubt.!Finally,!students!often!leave!out!the!copula,!which!makes!it!hard!to!tell!exactly!what!the
categories!mean.
!
1.!All!acts!that!decrease!energy!consumption!are!helpful!for!the!environment.
2.!All!use!of!public!transportation!can!decrease!energy!consumption.
3.!Therefore,!all!use!of!public!transportation!is!helpful!for!the!environment.
!
The!second!premise!uses!the!word!‘can’!instead!of!‘is’,!which!changes!the!meaning!of!the!second!category
from!‘acts!that!decreases!energy!consumption’!to!‘acts!that!might!decrease!energy!consumption’!–!an
equivocation.!If!a!premise!or!conclusion!leaves!out!the!copula,!that!doesn’t!necessarily!mean!that!there!is!a
hidden!equivocation!in!the!argument.!But!it!is!much!easier!to!detect!equivocations!when!you!include!the
copula,!which!is!why!you!should!always!use!copulas!when!constructing!categorical!syllogisms.
!
C.#Propositional#Arguments
!!!!!!!!!!!!The!second!type!of!deductive!reasoning!–!known!as!propositional!logic!–!was!discovered!in!the!19th
century.
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!As!the!name!indicates,!the!basic!unit!of!this!type!of!argument!is!the!proposition,!not!the
category.!‘Proposition’!is!a!synonym!for!statement!or!claim.!The!sentence!“All!dogs!are!mammals”!contains
two!categories!(‘dogs’!and!‘mammals’),!but!is!merely!one!statement.!However,!some!sentences!contain
more!than!one!statement,!for!example!“All!dogs!are!mammals,!and!all!cats!are!mammals!too”.
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Propositional!logic!studies!the!way!that!we!modify!or!combine!simple*statements!in!order!to!make!complex
statements,!and!how!to!determine!the!truth!and!falsity!of!complex!statements.!Complex!statements!are
constructed!using!logical*operators,!and!we!will!learn!about!four!of!them:!negation,!conjunction,
disjunction,!and!the!conditional.!The!first!is!expressed!using!words!like!‘no’,!‘not’,!and!‘neither’,!for
example,!in!the!complex!statement!‘I’m!not!going!to!the!beach’.!Negation!reverses!the!truth!value!of!the
statement!it!negates:!The!preceding!example!is!equivalent!to!saying!that!the!statement!‘I’m!going!to!the
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beach’!is!false.!If!a!statement!is!true,!its!negation!is!false;!and!if!a!statement!is!false,!its!negation!is!true.
Like!mathematical!negation,!a!double!negation!in!logic!cancels!itself!out:!‘It’s!not!the!case!that!I’m!not!going
to!the!beach’!is!equivalent!to!‘I’m!going!to!the!beach’.!In!contrast!to!negation,!the!other!logical!operators
are!connectives!–!they!join!together!two!statements.!Conjunction!is!expressed!using!words!such!as!‘and’,
‘but’,!or!‘however’,!for!example,!‘I’m!going!to!the!beach!and!I’ll!return!home!later!tonight’.!This!complex
statement!asserts!that!both!conjuncts!–!the!two!statements!joined!together!–!are!true.!In!other!words,!a
conjunction!is!only!true!when!both!of!its!conjuncts!are!true.!If!either!‘I’m!going!to!the!beach’!or!‘I’ll!return
home!later!tonight’!are!false,!or!both!of!them!are!false,!then!the!entire!conjunction!is!false.
!!!!!!!!!!!!The!next!logical!operator!is!called!disjunction,!and!it!is!expressed!using!the!word!‘or’,!for!example,
‘I’m!going!to!the!beach!or!I’ll!be!doing!my!homework’.!This!complex!statement!asserts!that!at*least*one!of
the!disjuncts!–!the!two!statements!joined!together!–!is!true.!In!other!words,!a!disjunction!is!only!false!when
both!disjuncts!are!false.!If!either!‘I’m!going!to!the!beach’!or!‘I’ll!be!doing!my!homework’!are!true,!or!both!of
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them!are!true,!then!the!disjunction!as!a!whole!is!true.
!The!final!logical!operator!we’ll!be!studying!is!the
conditional.!It!is!expressed!using!‘if!…!then’!statements,!e.g.,!‘If!you!flip!the!light!switch,!then!the!lights!will
go!out’.!Conditional!statements!are!more!complicated!than!conjunctions!and!disjunctions:!The!two
statements!it!joins!together!have!different!functions,!and!hence!different!names.!The!first!one!–!the!one
that!comes!between!‘if’!and!‘then’!–!is!known!as!the!antecedent!–!in!the!example!above,!‘You!flip!the!light
switch’.!The!second!is!known!as!the!consequent,!and!follows!the!word!‘then’!–!‘the!lights!will!go!out’!in!the
example.!It!is!important!not!to!mix!them!up:!Reversing!the!order!of!the!conjuncts!in!a!conjunction!or!the
disjuncts!in!a!disjunction!doesn’t!make!any!difference,!but!switching!the!antecedent!and!consequent
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changes!the!meaning!of!a!conditional!statement.!A!conditional!statement!asserts!that!if!the!antecedent!is
true,!the!consequent!will!be!true!as!well.!In!other!words,!a!conditional!statement!is!only!false!if!the
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antecedent!is!true!and!the!consequent!is!false.!Otherwise!it!is!true.
!
!!!!!!!!!!!
Name##############Synonyms#######Parts#######################################Rule
Negation!!!!!!!!‘no’,!‘not’!!!!!!!!!Not!applicable!!!!!!!!!!!!!!!!!!!!!!!!!Reverses!the!truth!value!of!the!original
statement.
Conjunction!!!‘and,!‘but’!!!!!!!!Conjuncts!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Only!true!if!both!conjuncts!are!true.
Disjunction!!!!!‘or’!!!!!!!!!!!!!!!!!!!Disjuncts!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Only!false!if!both!disjuncts!are!false.
Conditional!!!!‘if!…!then!…’!!!!Antecedent,!consequent!!!!!!!!!Only!false!if!the!antecedent!is!true!and!the
consequent!is!false.
!!!!!!!!!!!!We!use!conditional!statements!to!express!the!relationship!between!causes!and!effects!(e.g.,!‘If!you
flip!the!light!switch,!then!the!lights!will!go!out’),!the!relationship!between!categories!(e.g.,!‘If!it!is!a!dog,!then
it!is!a!mammal’),!and!to!express!practical!rules!(e.g.,!‘If!you!graduated!from!SCSU!then!you!must!have
satisfied!your!math!requirement’).!Propositional!logic!has!a!multitude!of!applications:!For!example,!it!plays
an!essential!role!in!computers!and!other!electronic!devices.!The!logical!operators!–!negation,!conjunction,
disjunction,!and!conditional,!are!the!basis!for!all!computer!languages!and!electronics.!If!propositional!logic
hadn’t!been!discovered!by!philosophers!and!mathematicians!in!the!19th!century,!the!cell!phones,
computers,!and!other!electronic!devices!that!we!take!for!granted!today!wouldn’t!exist.!Let’s!illustrate!this
with!the!electronics!in!your!car:!There!is!a!buzzing!signal!when!the!driver’s!seat!belt!isn’t!buckled!(negation);
the!windshield!wipers!only!work!then!the!key!is!in!the!ignition!and!a!switch!is!turned!(conjunction);!pushing
either!of!two!pads!on!the!steering!wheel!makes!the!horn!honk!(disjunction);!and!pushing!on!the!brake!pedal
turns!on!the!break!lights!(conditional).
!!!!!!!!!!!!There!are!many!valid!propositional!arguments,!but!we!will!only!be!learning!four!of!them:
!
Modus#Ponens#######################Example
1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!you!flip!the!light!switch,!then!the!lights!will!go!out.
2.!A!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!You!flipped!the!light!switch.
3.!Therefore,!B.!!!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!the!lights!must!have!gone!out.
!
Modus#Tollens!!!!!!!!!!!!!!!!!!!!!!!Example
1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!you!flip!the!light!switch,!then!the!lights!will!go!out.
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!
2.!Not!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!The!lights!didn’t!go!out.
3.!Therefore,!not!A.!!!!!!!!!!!!!!!!3.!Therefore,!you!must!not!have!flipped!the!light!switch.
!
Hypothetical#Syllogism#########Example
1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!you!graduated!from!SCSU,!then!you!must!have!taken!math.
2.!If!B!then!C.!!!!!!!!!!!!!!!!!!!!!!!!!!2.!If!you’ve!taken!math,!you!must!have!learned!about!binomials.
3.!Therefore,!if!A!then!C.!!!!!!!!3.!Therefore,!if!you!graduated!from!SCSU,!you!must!have!learned!about
binomials.
!
Disjunctive#Syllogism####################################Example
1.!A!or!B!!!!!!!!!!Or:!!!!!!!1.!A!or!B!!!!!!!!!!!!!!!!!!!!!!1.!She’s!going!to!the!beach!or!she’s!doing!her!homework.
2.!Not!A.!!!!!!!!!!!!!!!!!!!!!!2.!Not!B.!!!!!!!!!!!!!!!!!!!!!!2.!She!didn’t!go!to!the!beach.
3.!Therefore,!B.!!!!!!!!!!!3.!Therefore,!A.!!!!!!!!!!!3.!Therefore,!She!must!have!been!doing!her!homework.
As!you!can!see,!the!letters!in!propositional!arguments!stand!for!statements!not!categories,!as!is!the!case!in
categorical!syllogisms.!Also,!Modus!Ponens!and!Hypothetical!Syllogism!arguments!use!conditional
statements,!Modus!Tollens!arguments!use!conditional!statements!and!negations,!and!Disjunctive!Syllogism
arguments!use!disjunctions!and!negations.
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!You!might!wonder!why!these!are!valid!arguments,!why!they
establish!the!strongest!possible!connection!between!the!premises!and!the!conclusion.!Just!like!categorical
syllogisms,!there!are!a!variety!of!methods!for!demonstrating!the!validity!of!propositional!arguments,!and
you!can!learn!about!them!in!a!dedicated!logic!class.!But!there!is!a!relatively!straightforward!way!to!explain
why!the!conclusion!is!true!by!definition!given!the!meaning!of!the!statements!in!the!premises:!As!we!saw
above,!the!rule!for!conditional!statements!is!that!they!are!only!false!if!the!antecedent!is!true!and!the
consequent!is!false.!Assuming!that!a!conditional!statement!is!true,!this!implies!two!things:!(1)!If!the
antecedent!is!true,!the!consequent!must!be!true!as!well.!And!(2)!If!the!consequent!is!false,!the!antecedent
must!be!false!as!well.!The!first!implication!is!the!basis!for!Modus!Ponens!and!the!second!is!the!basis!for
Modus!Tollens.!As!with!categorical!syllogisms,!there!are!invalid!argument!forms!that!look!very!similar!–
make!sure!that!you!can!tell!the!difference!between!them:
!
Modus#Ponens###########Affirming#the#Consequent#(Invalid)#############Example
1.!If!A!then!B.!!!!!!!!!!!!!!1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!you!flip!the!light!switch,!the!light
will!go!out.
2.!A.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!The!light!went!out.
3.!Therefore,!B.!!!!!!!!!!!3.!Therefore,!A.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!you!must!have!flipped
the!light!switch.
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!
!
Modus#Tollens###########Denying#the#Antecedent#(Invalid)###############Example
1.!If!A!then!B.!!!!!!!!!!!!!!1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!you!flip!the!light!switch,!the!light
will!go!out.
2.!Not!B.!!!!!!!!!!!!!!!!!!!!!!2.!Not!A.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!You!didn’t!flip!the!light!switch.
3.!Therefore,!not!A.!!!!3.!Therefore,!not!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!the!lights!must!not
have!gone!out.
The!reason!why!these!arguments!are!invalid!is!because!conditional!statements!don’t!have!any!implications
when!the!consequent!is!true!(i.e.,!the!antecedent!could!be!either!true!or!false)!or!when!the!antecedent!is
false!(i.e.,!the!consequent!could!be!either!true!or!false).!Consequently,!the!conclusions!of!these!arguments
don’t!follow!with!certainty!from!the!premises.!We!can!illustrate!this!with!the!examples!above:!The!lights
going!out!doesn’t!guarantee!that!the!light!switch!was!flipped,!because!the!conditional!statement!doesn’t
rule!out!other!possible!causes!(e.g.,!the!bulbs!burning!out,!a!power!outage,!or!rats!chewing!through!the
wires).!For!the!same!reason,!the!fact!that!you!didn’t!flip!the!light!switch!doesn’t!guarantee!that!the!lights
didn’t!go!out.
!!!!!!!!!!!!Hypothetical!Syllogism!is!valid!because!it!establishes!a!sequence!of!conditions,!connecting!the
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antecedent!in!the!first!premise!to!the!consequent!in!the!second.
!Think!of!a!chain!of!causes!–!one!thing
leads!to!another,!which!leads!to!another:
!
!!!!!!!!!!!!1.!If!you!flip!the!light!switch,!then!the!lights!will!go!out.
!!!!!!!!!!!!2.!If!the!lights!go!out,!someone!will!trip!and!hurt!themselves.
!!!!!!!!!!!!3.!Therefore,!if!you!flip!the!light!switch,!someone!will!trip!and!hurt!themselves.
!
Keep!in!mind!that!there!are!invalid!propositional!arguments!that!look!very!similar!to!Hypothetical!Syllogism
since!both!premises!and!the!conclusion!are!conditional!statements,!and!are!easily!confused!with!it:
!
Hypothetical#Syllogism#########Invalid#Arguments
1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!B!then!A.
2.!If!B!then!C.!!!!!!!!!!!!!!!!!!!!!!!!!!2.!If!C!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!2.!If!B!then!C.
3.!Therefore,!if!A!then!C.!!!!!!!!3.!Therefore,!if!A!then!C.!!!!!!!!3.!Therefore,!if!A!then!C.
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!B!then!A.!!!!!!!!!!!!!!!!!!!!!!!!!!1.!If!A!then!B.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!If!C!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!2.!If!D!then!C.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!if!A!then!C.!!!!!!!!3.!Therefore,!if!A!then!C.
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!
These!arguments!are!invalid!because!they!fail!to!establish!a!connection!between!condition!A!and!condition
C.!Make!sure!you!are!able!to!distinguish!them!from!Hypothetical!Syllogism.
!!!!!!!!!!!!What!about!Disjunctive!Syllogisms?!Why!are!they!valid?!Disjunctive!Syllogism!is!a!simple!version!of
the*process*of*elimination,!which!states!that!there!are!a!certain!number!of!possibilities,!and!eliminates!all!of
them!except!for!one,!proving!that!the!remaining!one!must!be!true.!We!often!use!this!line!of!reasoning!when
troubleshooting!a!problem:!For!example,!your!car!won’t!start!and!it’s!either!because!the!battery!is!dead!or
the!gas!tank!is!empty.!But!when!you!turn!the!key,!the!lights!go!on!and!the!engine!turns!over.!Since!the
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battery!isn’t!dead,!the!only!possible!cause!remaining!is!the!gas!tank!being!empty.
!As!with!the!other
valid!arguments,!there!are!invalid!propositional!arguments!that!look!very!similar!to!Disjunctive!Syllogism
and!are!easily!confused!with!it:
!
!
Disjunctive#Syllogism########################Invalid#Arguments
1.!A!or!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!A!or!B.!!!!!!!!!!!!!!!!!!!!!1.!A!or!B.
2.!Not!A.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!A!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!A
3.!Therefore,!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!B.!!!!!!!!!!!3.!Therefore,!not!B.
!
1.!A!or!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!A!or!B.!!!!!!!!!!!!!!!!!!!!!1.!A!or!B.
2.!Not!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!B!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!B
3.!Therefore,!A.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!3.!Therefore,!A.!!!!!!!!!!!3.!Therefore,!not!A.
As!with!Modus!Ponens!and!Modus!Tollens,!Disjunctive!Syllogism!follows!from!the!implication!of!one!of!its
premises:!Assuming!that!a!disjunction!is!true,!if!one!of!its!disjuncts!is!negated!(i.e.,!is!false),!the!other!one
must!be!true.!However,!if!one!of!its!disjuncts!is!true,!this!doesn’t!tell!us!anything!about!the!other!disjunct
since!it!could!be!either!true!or!false.!Make!sure!that!you!can!distinguish!Disjunctive!Syllogism!from!these
invalid!propositional!arguments.
!!!!!!!!!!!!Finally,!be!aware!that!there!are!other!common!mistakes!students!make!when!working!with
propositional!arguments.!Just!as!with!categorical!syllogisms,!equivocation!or!not!using!the!proper!forms!of
complex!statements!will!make!an!argument!invalid.!If!a!statement!changes!meaning!in!different!places!in!an
argument,!then!it!isn’t!really!the!same!statement,!and!argument!isn’t!following!the!proper!form.!Here’s!an
example!of!an!equivocal!Hypothetical!Syllogism:
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###########
!
1.!If!the!police!raid!a!gambling!den,!then!they!are!undertaking!a!criminal!action.
2.!If!they!are!undertaking!a!criminal!action,!then!they!are!doing!something!illegal.
3.!Therefore,!if!the!police!raid!a!gambling!den,!then!they!are!doing!something!illegal.
The!ambiguous!term!is!‘criminal!action’:!In!the!first!premise!it!means!‘an!action!undertaken!to!fight!crime
and!enforce!the!law’,!and!in!the!second!it!means!‘an!act!that!violates!the!law’.!Also!be!careful!to!avoid
phrasing!complex!statements!in!a!confusing!manner!since!this!can!make!it!difficult!to!determine!what
logical!operator!was!intended.!Let’s!consider!a!few!examples:
!
!
‘You!flip!the!light!switch,!then!the!lights!go!out’
‘The!lights!will!go!out!if!you!flip!the!light!switch.’
‘I’m!going!to!the!beach!or!I’m!doing!homework!and!I’m!doing!the!dishes.’
The!first!statement!looks!like!a!conditional!(though!it!is!missing!the!‘If’),!but!in!fact!it!is!closer!to!a
conjunction.!In!the!second!statement,!the!order!of!the!antecedent!and!the!conclusion!are!reversed,!so!what
it!actually!means!is!‘If!you!flip!the!light!switch,!then!the!lights!will!go!out’.!In!the!third!statement,!it!isn’t
clear!whether!the!disjunction!or!the!conjunction!takes!precedence.!In!other!words,!is!it!saying!‘I’m!going!to
the!beach,!or!I’m!doing!homework!and!I’m!doing!the!dishes’!or!is!it!saying!‘I’m!going!to!the!beach!or!I’m
doing!homework,!and#I’m!doing!the!dishes’?!The!meaning!is!different!in!each!case.!Make!sure!to!avoid
these!types!of!errors,!to!avoid!equivocations!and!phrasing!that!will!confuse!your!readers.
!
[1]
!I’m!drawing!an!analogy!here!between!successful!deductive!arguments!and!analytic!truths.!A!statement!is!true!analytically!if!its
subject!is!already!contained!in!its!predicate.!Consider!the!statement!“Two!is!a!number.”!The!definition!of!two!includes!the!concept!of
number.!Hence!the!statement!is!true*by*definition.!What!about!statements!in!which!the!predicate!isn’t!contained!within!the
definition!of!the!subject,!such!as!“My!book!is!blue”?!A!statement!like!this!is!synthetic!because!it!adds!something!new!to!the!subject.
If!this!statement!is!true,!it’s!because!it!is!consistent!with!contingent!facts!–!it!happens!to!match!the!world,!though!the!world!is
capable!of!changing!and!making!the!statement!false.
[2]
!The!argument!form!‘addition’!is!an!exception.!Here’s!an!example!of!this!type!of!argument:!“I’m!going!to!the!store.!Therefore,!I’m
going!to!the!store!or!I’m!going!to!the!gym.”!This!argument!form!exploits!the!truth!functionality!of!disjunctions!(something!we!will
learn!about!in!section!C!of!this!chapter)!–!the!fact!that!they!are!never!false!when!one!of!their!disjuncts!is!true.!It!seems!like!a!trivial
exception!because!the!disjunct!added!is!logically!inert!in!the!sense!that!we!can’t!using!disjunctive!syllogism!to!break!it!off!as!an
independent!statement!without!contradicting!the!premise!of!the!addition!argument.
[3]
!The!‘all’!can!be!left!out!of!universal!affirmative!statements!–!in!other!words,!“Dogs!are!mammals”!means!the!same!thing!as!“All
dogs!are!mammals”.
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[4]
!Using!the!rule!of!obversion,!universal!affirmative!statements!can!be!translated!into!universal!negative!statements!and!vice!versa
(“All!S!are!P”!=!“No!S!are!non]P”),!and!particular!affirmative!statements!can!be!translated!back!and!forth!into!particular!negative
statements!(“Some!S!are!P”!=!“Some!S!are!not!non]P”).!Another!translation!rule!is!conversion,!which!enables!us!to!change!“No!S!are
P”!to!“No!P!are!S”,!and!“Some!S!are!P”!to!“Some!P!are!S”.!Conversion!doesn’t!apply!to!the!other!two!types!of!categorical!statements,
“All!S!are!P”!and!“Some!S!are!not!P”.
[5]
!Using!obversion!and!conversion,!the!rules!mentioned!in!the!previous!footnote,!Barbara!can!be!translated!into!4!variants,!and
Darii!into!9:
!
Variants#of#Barbara
Camestres!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Calemes#################################Celarent################################Cesare
1.!All!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!1.!All!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!1.!No!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!1.!No!P!are!M.
2.!No!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!2.!No!M!are!S.!!!!!!!!!!!!!!!!!!!!!!!2.!All!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!2.!All!S!are!M.
3.!Therefore,!no!S!are!P.!!!!3.!Therefore,!no!S!are!P.!!!!3.!Therefore,!no!S!are!P.!!!!3.!Therefore,!no!S!are!P.
!
Variants#of#Darii
Datisi######################################################Baroco####################################################Ferio
1.!All!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!No!M!are!P.
2.!Some!M!are!S.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!Some!S!are!not!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!Some!S!are!M.
3.!Therefore,!some!S!are!P.!!!!!!!!!!!!!!!3.!Therefore,!some!S!are!not!P.!!!!!!!!3.!Therefore,!some!S!are!not!P.
!
Ferison###################################################Festino###################################################Fresison
1.!No!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!No!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!No!P!are!M.
2.!Some!M!are!S.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!Some!S!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!Some!M!are!S.
3.!Therefore,!some!S!are!not!P.!!!!!!!!3.!Therefore,!some!S!are!not!P.!!!!!!!!3.!Therefore,!some!S!are!not!P.
!
Disamis##################################################Dimatis##################################################Bocardo
1.!Some!M!are!P.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!Some!P!are!M.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!Some!M!are!not!P.
2.!All!M!are!S.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!All!M!are!S.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!All!M!are!S.
3.!Therefore,!some!S!are!P.!!!!!!!!!!!!!!!3.!Therefore,!some!S!are!P.!!!!!!!!!!!!!!!3.!Therefore,!some!S!are!not!P.
#
For!a!challenge,!try!to!translate!Barbara!and!Darii!into!these!variations!using!the!rules!obversion!and!conversion.
!
[6]
!The!claim!that!categorical!syllogisms!move!from!general!to!particular!is!also!applicable!to!Darii,!though!to!a!lesser!extent:!The
predicate!is!a!larger!category!than!the!middle!term.!Though!the!subject!can!be!larger!than!both!of!them!since!it!is!isn’t!completely
encompassed!by!either!one.!This!is!illustrated!by!the!example!above:!‘Evergreens’!is!a!broader!category!than!‘pine!trees’,!but!‘trees’
is!broader!than!both!of!them.
[7]
!The!argument!that!was!left!out!(Datisi)!–
1.!All!M#are!P.!!!!!!
2.!Some!M!are!S.
3.!Therefore,!some!S!are!P.
–!is!in!fact!valid.!Why?!According!to!the!rule!of!conversion,!‘Some!M!are!S’!is!equivalent!to!‘Some!S!are!M’.!In!other!words,!Datisi!is
equivalent!to!Darii,!as!we!already!saw!in!footnote!5.
!
[8]
!‘Rediscovered’!would!be!more!accurate,!for!this!type!of!logic!was!first!discovered!by!the!ancient!Greek!stoics!in!the!3rd!century
BCE.
[9]
!Logical!negation!is!similar!to!negations!in!mathematics,!but!there!is!no!neutral!truth!value!equivalent!to!zero:!True!and!false!are
the!only!options!for!the!truth!value!of!statements.
[10]
!This!may!be!confusing!because!we!often!use!the!word!‘or’!in!an!exclusive!sense,!i.e.,!in!scenarios!in!which!it!is!only!possible!for
one!of!the!disjuncts!to!be!true.!For!example,!‘The!coin!will!land!heads!or!it!will!land!tails’.!However,!in!logic,!we!use!disjunction!in!an
inclusive!sense,!which!leaves!open!the!possibility!that!both!disjuncts!are!be!true.
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!The!fact!that!a!conditional!is!true!whenever!the!antecedent!is!false!might!seem!strange:!Like!disjunctions,!the!way!we!use
conditional!statements!in!ordinary!language!sometimes!differs!from!the!way!we!use!them!in!logic.
[12]
!
!
[13]
!We!won’t!be!memorizing!any!arguments!with!conjunctions,!but!in!case!you!want!to!learn!some,!here!are!a!couple!of!examples:
Conjunction##########################Example
1.!A!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!It’s!Tuesday.
2.!B!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!It’s!noon.
3.!Therefore!A!and!B.!!!!!!!!!!3.!Therefore,!It’s!Tuesday!and!it’s!noon.
!
Simplification########################################################Example
1.!A!and!B.!!!!!!!!!!!!!Or:!1.!A!or!B!!!!!!!!!!!!!!!!!!!!!!!!!!1.!It’s!Tuesday!and!it’s!noon.
2.!Therefore,!A.!!!!!!!!!!!2.!Therefore,!B.!!!!!!!!!!!!!2.!Therefore,!it’s!Tuesday.
!In!fact,!statement!‘B’!in!Hypothetical!Syllogisms!is!much!like!the!middle!term!in!the!Barbara!argument!–!the!resemblance!is
striking,!except!that!the!order!of!the!premises!is!reversed:
!
Hypothetical#Syllogism#######################Barbara
1.!If!A!then!B.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1.!All!M!are!P.
2.!If!B!then!C.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2.!All!S!are!M.
3.!Therefore,!if!A!then!C.!!!!!!!!!!!!!!!!!!!!3.!Therefore,!all!S!are!P.
!
As!it!turns!out,!Barbara!and!Darii!can!both!be!translated!into!propositional!arguments,!Barbara!into!Hypothetical!Syllogism!and!Darii
into!Modus!Ponens.!(The!reverse!isn’t!necessarily!the!case!because!propositional!logic!is!more!versatile!than!categorical!syllogisms:
For!example,!it!can!include!additional!types!of!statements!besides!categorical!ones,!and!can!employ!operators!that!can!be!difficult!to
translate!into!categorical!logic,!such!as!negation!and!conjunction.)
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!However,!if!there!are!other!possibilities!that!aren’t!considered!in!the!argument,!it!is!a!bad!or!fallacious!argument!known!as!false
dilemma.!We!will!learn!about!fallacies!in!Chapter!4.
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