proposition truth valueharris/1106/part2outlines.pdfhowever, there are many other (non-preferred)...

1 P2M1–P2M5MGF1106LectureOutlinesSpring2019

Part 2 Logic

Part 2 Module 1 – Introduction to Logic InP2M1-P2M5wewillstudylogic.Logicisasystematicstudyofreasoning,ortheprocessofusing

___________________________.Wewillanalyzerelationshipsbetweeninformationandusingestablished

factsandassumptionsto_______________________.

Statements, quantif iers, negations

A orpropositionisadeclarativesentencethathastruthvalue.

Tosaythatasentencehastruthvaluemeansthat,whenwehearorreadthesentence,itmakessensetoask

whetherthesentenceis

Herearesomeexamplesofstatements:

Wordslike“all,”“some,”and“none”arecalled

Inlogic,theword“some”hasaspecificmeaning.Itmeans

“_________________________________________.”Unlikeineverydayusage,inlogic,“some”doesnot

necessarilyindicateplural.

QuantifiedStatements

Inlogic,termslike“all,”“some,”or“none”arecalled________________________________

Astatementbasedonaquantifieriscalledaquantifiedstatementor

______________________________________________

Herearesomeexamplesofquantifiedstatements:

“Allbadhairdaysarecatastrophes.”

“Noslugsarespeedy.”

“Someowlsarehooty.”

Quantifiedstatementsstatearelationshipbetweentwoormoreclassesof

_______________________________________________

Intheaboveexamples,thecategoriesmentionedwere:

Inthiscourse,asentencethatsoundslikeanopinionwillbetreatedasanacceptablestatement.Insucha

casewewillpretend,forthesakeofdiscussion,thatasubjective,value-ladentermlike“dishonest”hasbeen

preciselydefined.

Sentencesthataren’tstatements


Noteverysentenceisastatement.Herearesomeexamples:

Questionsarenotstatements.

Itdoesn’tmakesensetoaskwhetheraquestionistrueorfalse.

Commandsarenotstatements.

Itdoesn’tmakesensetoaskwhetheracommandistrueorfalse.

Theprevioussentenceisaparadox.Itisneithertruenorfalse,soitisn’t

astatement.

ExistentialStatements,anddiagrammingthemAstatementoftheform“SomeAareB”or“SomeAaren’tB”assertsthe

_______________________________element

Inlogic,theword“some”hasameaningof“__________________________________”.

Categoricalstatementshavingthoseformsarecalled__________________________

“Someowlsarehooty”

“Somewolverinesarenotcuddly”

Theseareexamplesof__________________________________________statements.

“Someowlsarehooty”assertsthatthereexistsatleastonethingthat___________________________.

Thatis,theintersectionofthecategories“owls”and“hootythings”is_______________________.Wecan

conveythatinformationbymakingamarkonaVenndiagram.

Weplacean“X”inaregionofaVenndiagramtoindicatethatthatregionmustcontainat

______________________.

owlshooty things

According to the statement “Some owls are hooty,” there must be at least one element in this region of the diagram.

XW C

According to the statement “Some W are not C,” there must be at least one element in this region of the diagram.

X


Theexistentialstatement“Somewolverinesarenotcuddly”assertsthatthere____________________

elementwhoisawolverine(W)but__________________

UniversalStatements,anddiagrammingthem

Astatementoftheform“NoAareB”iscalledanegativeuniversal.

ItassertsthatthereisnoelementincategoryAandcategoryBatthesametime.Thismeansthatthe

intersectionofthetwocategoriesis_______________.

Astatementoftheform“AllAareB”iscalledapositiveuniversalStatement

anditassertsthatthereisnoelementincategoryAthatisn’talsoinCategoryB.

“Allbadhairdaysarecatastrophes”

“Noslugsarespeedy”

Areexamplesofapositiveandanegativeuniversalstatement.

DiagrammingNegativeUniversalStatementsInlogic,weuseshadingtoindicatethatacertainregionofaVenndiagramisempty(containsnoelements).

Thenegativeuniversalstatement“Noslugsarespeedy”assertsthattheregionofthediagramwhere“Slugs”

and“Speedythings”intersectmustbeempty.

DiagrammingPositiveUniversalStatements

Thepositiveuniversalstatement“Allbadhairdaysarecatastrophes”assertsthatitisimpossibletobea

______________________________withoutalsobeing______________________________.

ThismeansthattheregionofthediagramthatisinsideBbut_____________________mustbe

______________.

According to the statement “No slugs are speedy,” this region of the diagram must be empty.


InterpretingVennDiagramsinLogicWewilluseVenndiagrams(typicallythree-circlediagrams)toconveytheinformationinpropositionsabout

relationshipsbetweenvariouscategories.

Shadingmeans“nothinghere…”

Inlogic,whenaregionofaVenndiagramisshaded,thistellsusthatthatregion

__________________________.Thatis,ashadedregionis__________________

SupposethatwearepresentedwiththemarkedVenndiagramshownbelowandonthefollowingslides.We

shouldbeabletointerpretthemeaningofthemarksonthediagram.

An“X”means“somethingishere…”

Inlogic,whenaregionofaVenndiagramcontainsan“X”,thistellsusthatthatregion

__________________________________________________

Inlogic,whenan“X”,appearsontheborderbetweentworegions,thistellsusthatthereis

_________________elementintheunionofthetworegions,butwearenotcertainwhetherthe

element(s)are

According to the statement “All B are C,” this region of the diagram must be empty.

These two regions containno elements.


Nomarkingmeans“uncertain…”

Inlogic,whenaregionoftheVenndiagramcontainsnomarkings,itis_____________________________

whetherornotthatregion__________________________________.

ExampleUseathree-circleVenndiagramtoconveyinformationabouttherelationshipsbetweenthesethree

categories:Angryapes(A);Blissfulbaboons(B);Churlishchimps(C).Selectthediagramwhosemarkings

correspondto“Noblissfulbaboonsareangryapes.”Assumethatwedonotknowofanyotherrelationships

betweencategories.

Names for statements

Wewilltendtouselowercaseletters,likep,q,r,andsoon,asnamesforstatements.

p:TodayisSaturday. q:TodayIhavemathclass.

XThis region contains at least one element.

X

X

There is at least one element in these two regions combined.

X

X

We donʼt know if these two regions contain any elements.

X

X


r:1+1=3

s:Somecatshavefleas.

u:Alllawyersaredishonest.

Letpbeanystatement.The____________________________________,denoted

isanotherstatementthatislogicallyoppositetop.

Thismeansthat~pwill___________________________________________________top.

• Inanysituationthatmakespatruestatement,~pwillbefalse.

• Inanysituationthatmakespafalsestatement,~pwillbetrue.

Foreachofthestatementsthatwerenamedatthebeginningofthisdiscussion,writethenegation.

Thereisaverystrongrelationshipbetweenanystatementpanditsnegation~p:

Itisimpossibletoconceiveofasituationwhere_____________________willhavethesame

______________________________________

ExampleSelectthecorrectnegationof“Somecatshavefleas.”

A.Allcatshavefleas.

B.Somecatsdon’thavefleas.

C.Nocatshavefleas.

D.Somefleashavecats.

Thecorrectnegationof“Somecatshavefleas”is____________________________

Fact:ifastatementhastheform“SomeAareB”,itsnegationwillhavetheform

“_____________________________________”Onewaytoverifythisfactisbydiagramming.

Example:Selectthecorrectnegationof“Alllawyersaredishonest.”

A.Alllawyersarehonest.

B.Somelawyersarehonest.

C.Nolawyersaredishonest.

D.Somelawyersaredishonest.

Fact:Ifastatementhastheform“AllAareB”thenitsnegationwillhavetheform

“_________________________________________.”

Again,onewaytoverifythisfactisbydiagramming.


Negations, alternative phrasing

Wehaveseenthatthecorrectnegationof“Alllawyersaredishonest”is

______________________________________.However,therearemanyother(non-preferred)waysto

correctlystatethenegationof“Alllawyersaredishonest.”Eachofthefollowingstatementsisacorrect

negationof“Alllawyersaredishonest.”

“Itisnotthecasethatalllawyersaredishonest.”

“Itisnottruethatalllawyersaredishonest.”

“Alllawyersaredishonest…NOT!

ExampleSelectthenegationof“Nobeetlesfightbattles.”A.Allbeetlesfightbattles.

B.Somebeetlesfightbattles.

C.Somebeetlesdon’tfightbattles.

D.Nobeetlesswingpaddles

ExampleSelectthenegationof“Somepoodlesdon’tleappuddles.”

A.Somepoodlesleappuddles.

B.Nopoodlesleappuddles.

C.Allpoodlesleappuddles.

D.Noneoftheabove.

Compound statements

Acompoundstatementisformedbyjoining______________________________,usingspecialconnecting

wordsorstructuressuchas“and,”“or,”or“if…then.”

1+1=2or4<3 isanexampleofacompoundstatement.

Alllawyersaredishonestandsomecatshavefleasisanotherexample.

Logical connectives

Wordsorphrasessuchas“and,”“or,”or“if…then,”___________________arecalledlogicalconnectives.

TheConjunction

Letp,qbeanystatements.Theirconjunctionisthecompoundstatementhavingtheform

“_______________________.”Thisisdenoted


Inorderforaconjunctiontobetrue,bothterms______________________.

TheDisjunction

Letp,qbeanystatements.Theirdisjunctionisthecompoundstatementhavingtheform

“____________________________.”Thisisdenoted

Inorderforadisjunctiontobetrue,__________________________________termsmustbetrue.

Adisjunctionisfalseonlyinthecasewhere___________________________.

Symbolic statements

Supposeprepresentsthestatement“Ihaveadime,”andqrepresentsthestatement“Ihaveanickel.”

Thesymbolicstatement

~p∨q correspondsto

“____________________________________________________________.”

Thesymbolicstatement

p∧~q correspondsto

“__________________________________________________________.”

Thislaststatementcanalsobereadas

“IhaveadimebutIdon’thaveanickel.”

ExampleLet p:youarenice q:youarefunny

Symbolizethecompoundstatement“Youaren'tniceoryouarefriendly.”

Symbolize“Itisn'tthecasethatyouareniceoryouarefriendly.”

Symbolize“Youaren'tniceandfunny.”

Finding Truth values of Compound Statements

ExampleSupposeprepresentsatruestatement,whileq,rrepresentfalsestatements.Findthetruthvalue

of (~r∧ p)∧ ~(~q∨r)


ExampleSupposep,qrepresentfalsestatements,whilerrepresentsatruestatement.Findthetruthvalue

of ~[q∨(~p∧ r)]

ExampleSupposep,qrepresentfalsestatements,whilerrepresentsatruestatement.Findthetruthvalue

of ~[~r∧ (p∨~q)]

ExampleSupposeprepresentsafalsestatementandqrepresentsafalsestatement.Findthetruthvalueof

~(~p∧ q)

SummaryoftheConjunctionandDisjunction

Conjuntion

AandB

A∧ B

A∧ Bisistrueonlywhen

Disjunction

AorB

A∨B

A∨Bisfalseonlywhen


Truth tables

Atruthtableisadevicethatallowsustoanalyzeandcomparecompoundlogicstatements.

Consider,forexample,thesymbolicstatementp∨~q.

Whetherthisstatementturnsouttobetrueorfalsewilldependuponwhetherpistrueorfalse,whetherqis

trueorfalse,andthewaythe“∨”and“~”operatorswork.

Atruthtablewillshowallthepossibilities.

Asanintroductiontoconstructingandfillingintruthtables,wewillmakeatruthtableforthestatementp∧

qandatruthtableforthestatementp∨q.

ExampleMakeatruthtableforthestatementp∨~q


Example

Referringtothetruthtableshownbelow,answerTrueorFalsebasedonwhetherthetruthtablerepresents

thecorrectresultsforthegivenstatement.Beawarethatthevaluesintherightmostcolumnmaynotbe

correct.Insertintermediatecolumnsasneededwhereyousee(“????”),fillinthetruthtable,anddecide

whethertherightmostcolumniscorrectlyfilledinasshown.

A. Yes,therightmostcolumniscorrectlyfilledin.

B. No,thevaluesintherightmostcolumnarenotallcorrect.

A tautology

Thetruthtablecolumnforthestatementq∨~(p∧q)shows___________________

Thismeansthatitisneverpossibleforthatstatement________________________.

Thestatementisalways_____________________,duetoitslogicalstructure.

Astatementthatcan______________________iscalledatautology.

Atautologyisastatementthatcanneverbefalse,duetoitslogicalstructure.Todecidewhetherasymbolic

statementisatautology,makeatruthtablehavingacolumnforthatstatement.

Ifthetruthtablecolumnshowsonly___________________________,thenthestatementisatautology.

Otherwise,thestatementisnotatautology.


Example

Decidewhetherthissymbolicstatementisatautology:(p∧ ~q)∨ (~p∨ q)

A.Yes,thisstatementisatautology.B.No,thisstatementisn’tatautology.

Negation of a compound statement

Selectthecorrectnegationof

A.I’malumberjackandI’mnotokay.

B.I’mnotalumberjackandI’mnotokay.

C.I’mnotalumberjackorI’mnotokay.

D.Noneofthese.

DeMorgan’s Laws

Thepreviousexamplesuggeststhefollowingfacts,knownasDeMorgan’sLawsforLogic:

~(p∧ q)≡~p∨~q

~(p∨q)≡~p∧ ~q

Tonegateaconjunctionordisjunction,negatebothtermsandswitchtheconnectivetotheother.

(Thethree-barredequalssignmeans“isequivalentto”inlogic.)

DeMorgan’sLawsshowusaneconomicalwaytostatethenegationof

______________________________________________________________________________.

Forexample,insteadofusingtheawkwardsentence

“ItisnotthecasethatIhavebothadimeandanickel”

wecanusethemuchsimplerform

“Idon’thaveadimeorIdon’thaveanickel.”

InPart2Module1wehavenowseenfourrulesfornegations.

Statement Negation

SomeAareB NoAareB

AllAareB SomeAaren’tB

p∧ q ~p∨ ~q

p∨ q ~p∧ ~q


Example

Selectthestatementthatisthenegationof"TodayisMondayanditisn'training."

A.Todayisn'tMondayanditisn'training.

B.Todayisn'tMondayoritisn'training.

C.Todayisn'tMondayoritisraining.

D.Todayisn'tMondayanditisraining.

E.TodayisFridayanditissnowing.

Example

Selectthestatementthatisthenegationof"I'mcarefulorImakemistakes."

A.I'mnotcarefulandIdon'tmakemistakes.

B.I'mnotcarefulorIdon'tmakemistakes.

C.I'mnotcarefulandImakemistakes.

D.I'mnotcarefulorImakemistakes.

E.Inevermakemisteaks.☺

A three-variable truth table

Supposeweneedtomakeatruthtableforastatementinvolvingthreevariables(p,q,r),suchas

(r∨~q)∧(~p∨q).

Thismore-complicatedstatementwillrequireamore-complicatedtruthtableskeleton.

Ifastatementinvolvesthreevariables,thenitstruthtableskeletonrequireseightrows,notfour,andbegins

withacolumnsforp,q,andr,filledinasshownbelow.

p q r T T T T T F T F T T F F F T T F T F F F T F F F


Example

Referringtothepartially-completedtruthtableshownbelow,addintermediatecolumnsasneeded,fill

everythingin,andthenselectthechoicethatshowsthecolumnfor(r∨~q)∧(~p∨q)correctlyfilled-in.

p q r ???????? (r∨~q)∧ (~p∨q) T T T T T F T F T T F F F T T F T F F F T F F F

A. B. C. D. None of these (r∨~q)∧ (~p∨q) (r∨~q)∧ (~p∨q) (r∨~q)∧ (~p∨q)

T T T F F T F F T T F T T T T F F T T T T F T T


Example

'Butwaitabit,'theOysterscried,Beforewehaveourchat;

Forsomeofusareoutofbreath,Andallofusarefat!'

"Nohurry!'saidtheCarpenter.Theythankedhimmuchforthat.

Selectthestatementthatisthenegationof"Someofusareoutofbreath,andallofusarefat."

A.Someofusaren'toutofbreathornoneofusisfat.

B.Someofusaren'toutofbreathandnoneofusisfat.

C.Noneofusisoutofbreathandsomeofusaren'tfat.

D.Noneofusisoutofbreathorsomeofusaren'tfat.

ExampleSelectthestatementthatisthenegationofthefollowingstatement(overheardinthecrowdattheLittle

LeagueballparkinWoodville,Florida.)

"Allofmyhusbandsaredeadorinjail."

A.Noneofmyhusbandsisdeadornoneofmyhusbandsisinjail.

B.Noneofmyhusbandsisdeadandnoneofmyhusbandsisinjail.

C.Atleastoneofmyhusbandsisnotdeadandisnotinjail.

D.Atleastoneofmyhusbandsisnotdeadorisnotinjail.

E.Noneofthese.

Another connective Considerthefollowingdisjunction,whichmaybeawarningissuedtoayoungchild:

“Youwillbehave,oryouwillgetpunished.”

Canyouthinkofanotherwaytoconveyexactlythesamewarningwithoutusingthe“or”connectiveorthe

“and”connective?

Howabout:

“Ifyoudon’tbehave,thenyouwillgetpunished.”

Thisisanexampleofaconditionalstatement.

Aconditionalstatementhastheform“Ifp,thenq.”

TheconditionalstatementisthetopicinPart2Module2.


Part 2 Module 2 - The Conditional Statement, Exclusive Disjunction and IFF

Aconditionalstatementisastatementoftheform _____________________________________

denoted_____________________________

Example

Letprepresent:"YoudrinkDr.Pepper."

Letqrepresent:"Youarehappy."

Inthiscase__________________isthestatement:"IfyoudrinkDr.Pepper,thenyouarehappy."

Intheconditionalstatement“IfyoudrinkDr.Pepper,thenyouarehappy,”thesimplestatement“Youdrink

Dr.Pepper”iscalledthe___________________________

andthesimplestatement“Youarehappy”iscalledthe ____________________________ .

Variations on the conditional statement

Foraconditionalstatementsuchas“IfyoudrinkDr.Pepper,thenyouarehappy,”therearethreesimilar-

soundingconditionalstatementsthathavespecialnames:

Variations:theConverse

Supposeastatementhastheform___________________suchas“IfyoudrinkDr.Pepper,thenyouare

happy.”(Wewillrefertothisasthe .)

Therelatedstatement_________________________iscalledtheconverse.

“Ifyouarehappy,thenyoudrinkDr.Pepper”istheconverseof“IfyoudrinkDr.Pepper,thenyouarehappy.”

Wecanalsosaythatthosetwostatementsareconversesofeachother.

Variations:theInverseSupposethedirectstatementhastheform___________,suchas“IfyoudrinkDr.Pepper,thenyouare

happy.”

Therelatedstatement______________________iscalledtheinverse.

“Ifyoudon’tdrinkDr.Pepper,thenyouaren’thappy”istheinverseof“IfyoudrinkDr.Pepper,thenyouare

happy.”Wecanalsosaythatthosetwostatementsareinversesofeachother.

Variations:theContrapositive

Supposethedirectstatementhastheform____________,suchas“IfyoudrinkDr.Pepper,thenyouare

happy.”Therelatedstatement____________________________iscalledthecontrapositive.“Ifyouaren’t


happy,thenyoudon’tdrinkDr.Pepper”isthecontrapositiveof“IfyoudrinkDr.Pepper,thenyouarehappy.”

Wecanalsosaythatthosetwostatementsarecontrapositivesofeachother.

Example

Selectthestatementthatistheinverseto‘Ifyouaren'tawhale,thenyoudon'tliveinthebrinydeep.’

A.Ifyoudon'tliveinthebrinydeep,thenyouaren'tawhale.

B.Ifyouareawhale,thenyouliveinthebrinydeep.

C.Ifyouliveinthebrinydeep,thenyouareawhale.

D.Ifyouareawhale,thenyoudon’tliveinthebrinydeep.

E.Noneofthese.

Truthtablefor____________________

p q p →q

T T

T F

F T

F F


The Fundamental Property of the Conditional Statement

TheonlysituationinwhichaconditionalstatementisFALSEiswhenthe________________isTRUEwhilethe

_______________________isFALSE.

Anyotherconfigurationyields_______________.

Example

Supposepistrue,qistrue,andrisfalse.Findthetruthvalueof

IFF operator

The“iff”operator,alsocalled”__________________________________”isanoperatorconnectingtwostatementssuchthatthenewstatementformedholdstruewhenbothstatementsarefalse,orbothstatementsaretrue..Notationforexclusivedisjunctionis___________________________andthefollowingtruthtabledemonstratesthetruthvalueofcompoundstatementsformedusingthisoperator.

p q

T T

T F

F T

F F


Exclusive Disjunction Operator The“exclusivedisjunction”operator,alsocalled“____________________________”isanoperatorconnectingtwostatements,suchthatthenewstatementformedholdstrueonlyinthecaseswhere_____________________________________.Inotherwords,thestatementformedistrueifandonlyifoneistrueandtheotherisfalse.Notationforexclusivedisjunctionis___________________________andthefollowingtruthtabledemonstratesthetruthvalueofcompoundstatementsformedusingthisoperator.

p q

T T

T F

F T

F F


Example

Supposepistrue,qistrue,andrisfalse.Findthetruthvalueof

[~q∧(~p⨁q)]↔(p⨁q)

Example

Createatruthtabletodemonstratethetruthvalueofthefollowingstatement:

~r↔(p⨁q)


Truth tables, tautologies

Example

Decideifthefollowingstatementisatautology:

[~q∧(~p→q)]→p

A.Yes,thisstatementisatautology.

B.No,thisstatementisn’tatautology.

Truth tables and equivalencies

Example

Selectthestatementthatisequivalentto“Ifyouareadog,thenyouwagyourtailwhenyouarehappy.”

A.Ifyouwagyourtailwhenyouarehappy,thenyouareadog.

B.Youaren’tadog,oryouwagyourtailwhenyouarehappy.

C.Youareadog,andyoudon’twagyourtailwhenyouarehappy.

D.Ifyouaren’tadog,thenyoudon’twagyourtailwhenyouarehappy.

Basedonthedefinitionsofp,qabove,herearethesymbolicrenditionsofeachmultiple-choiceanswer.

A.q→p B.~p∨q C.p∧~q D.~p→~q

Atruthtablewillshowwhichofthesechoicesisequivalenttop→q.


Facts

Thereareseveralgeneralizationsthatfollowfromthetruthtableinthepreviousexercise.

Notethatthecolumnforp→qisdifferentfromthecolumnforq→p:

1.AconditionalstatementisNOTequivalenttoitsconverse.

Notethatthecolumnforp→qisdifferentfromthecolumnfor~p→~q:

2.AconditionalstatementisNOTequivalenttoitsinverse.

Notethatthecolumnforp→qisthesameasthecolumnfor~p∨q:

3.p→qisequivalentto~p∨q

Notethatthecolumnforp→qisexactlytheoppositeofthecolumnforp∧~q:

4.Thenegationofp→qisp∧~q

Anequivalencyfor“ifp,thenq”

Thetruthtableinthepreviousexampleconfirmsthefollowingfact:

p→q≡~p∨q

Thatis,youcanchangeaconditionalstatementintoanequivalent“or”statement,by

______________________ andswitchingthe_______________________

Example

Selectthatstatementthatislogicallyequivalentto:"Ifyoudon'tcarryanumbrella,you'llgetsoaked."

A.Youcarryanumbrellaandyouwon'tgetsoaked.

B.Youcarryanumbrellaoryougetsoaked.

C.Youdon'tcarryanumbrellaandyougetsoaked.

D.Youdon'tcarryanumbrellaoryougetsoaked.

E.Youleaveyourumbrellaintheclassroom,soyougetsoakedanyway.


Negation of a conditional statement

Basedonthetruthtableweconstructedinanearlierexercise,wehavealreadymadeanobservationabout

thecorrectformforthenegationofaconditionalstatement.

Wecanalsousethisequivalency:

p→q≡~p∨q

tofindthecorrectnegationofp→q

Thenegationof p→q is p∧~q.

Noticethatthenegationofan“if…then”statementdoesn’thaveany“ifs”or“thens.”

Negations: Summary

InPart2Modules1and2wehaveseenfiverulesfornegations.Heretheyare.

Statement Negation

SomeAareB NoAareB.

AllAareB. SomeAaren’tB.

p∧ q ~p∨~q

p∨q ~p∧ ~q

p→q p∧ ~q

Example

Selectthestatementthatisthenegationof"Ifadogwagsitstail,thenitdoesn'tbite."

A.Adogwagsitstailanditbites.

B.Adogwagsitstailanditdoesn'tbite.

C.Adogdoesn'twagitstailoritbites.

D.Ifadogdoesn'twagitstail,thenitbites.

E.Noneofthese.

Anotherequivalency


Selectthestatementthatistheequivalentto"IfIamacloud,thenIhaveasilverlining."

A.IfIhaveasilverlining,thenIamacloud.

B.IfIamnotacloud,thenIdon’thaveasilverlining.

C.IfIdon’thaveasilverlining,thenIamnotacloud.

D.A,B,Careallequivalenttothegivenstatement.

E.Noneoftheseiscorrect.

Wewilluseatruthtabletoanswerthisquestion.

Equivalency

Thetruthtableinthepreviousexerciseestablishesthefollowingfact:

p→q≡~q→~p

Thatis,aconditionalstatementisequivalenttoitscontrapositive,butnotequivalenttoitsconverseor

inverse.

Wenowhavetworulesforequivalency:

1.p→q≡~p∨q

2.p→q≡~q→~p


Summary: The conditional statement

LetA→Bbeanyconditionalstatement.

Aistheantecedent.Bistheconsequent.

FundamentalRule

TheonlysituationthatmakesA→BfalseiswhenAistruewhileBisfalse.

Negation

ThenegationA→BofisA∧~B

TwoEquivalencies

1.A→B≡~A∨B

2.A→B≡~B→~A

Variations

Converse:B→A Inverse:~A→~B Contrapositive:~B→~A


Part 2 Module 4 - Categorical Syl logisms and Diagramming

Somelawyersarejudges.

Somejudgesarepoliticians.

Therefore,somelawyersarepoliticians.

ThisisanexampleofaCATEGORICALSYLLOGISM,whichisanargumentinvolving

_______________________,bothofwhich(alongwiththeconclusion)are

________________________________

.

Categoricalstatementsarepropositionsoftheform__________________________,

__________________________,

_________________________________,or________________________________.

Rememberthatthevalidityofanargumenthasnothingtodowithwhethertheconclusionsoundstrueor

reasonableaccordingtoyoureverydayexperience.Thepreviousargumentis

__________________________________.

Onewaytoseethattheargumenthasaninvalidstructureistoreplace“lawyers”with“alligators,”replace

“judges”with“gray(things),”andreplace“politicians”with“cats.”Then,theargument

_____________________________

Somealligatorsaregray.

Somegraythingsarecats.

Therefore,somealligatorsarecats.

Wewillintroduceaformaltechniquetodealwithcategoricalsyllogisms.

Duringthemiddleages,scholasticphilosophersdevelopedanextensiveliteratureonthesubjectofcategorical

syllogisms.Thisincludedaglossaryofspecialtermsandsymbols,aswellasaclassificationsystemidentifying

andnamingdozensofforms.ThiswashundredsofyearsbeforethebirthofJohnVennandthesubsequent

inventionofVenndiagrams.ThroughtheuseofVenndiagrams,analysisofcategoricalsyllogismsbecomesa

processofcalculation,likesimplearithmetic.

Diagramming categorical syl logisms

Hereisasynopsisofthediagrammingmethodthatwillbedemonstratedindetailinthefollowingexercises.

ItissimilartothemethodofdiagrammingUniversal-Particulararguments.


1.Totestthevalidityofacategoricalsyllogism,useathreecircleVenndiagram.

2.Markthediagramsothatitconveystheinformationinthetwopremises.Alwaysstartwitha

universalpremise.(Ifthereisnotatleastoneuniversalpremise,theargumentisinvalid;nofurther

workneeded.)

3.Ifthemarkeddiagramshowsthattheconclusionistrue,thentheargumentisvalid.

4.Ifthemarkeddiagramshowsthattheconclusionisfalseoruncertain,thentheargumentisinvalid.

Example

Somebulldogsareterriers.

Noterriersaretimid.

Therefore,somebulldogsarenottimid.

A.Valid

B.Invalid

1.Avalidcategoricalsyllogismmusthaveatleastoneuniversalpremise.Ifbothpremisesareexistential

statements(“Someare…,”“Somearen’t…”)thentheargumentisinvalid,andwearedone.

Noterriersaretimid.

2.Assumingthatonepremiseisuniversalandonepremiseisexistential,drawathree-circleVenndiagram

andmarkittoconveytheinformationintheuniversalpremise.Thiswillalwayshaveeffectofshadingout

tworegionsofthediagram,becauseauniversalstatementwillalwaysassert,eitherdirectlyorindirectly,that

somepartofthediagrammustcontainnoelements.

Wemarkourdiagramaccordingtothepremise“Noterriersaretimid.”

Somebulldogsareterriers.

3.Nowmarkthediagramsothatitconveystheinformationintheotherpremise.

Typically,thiswillbeanexistentialstatement,anditwillhavetheeffectofplacingan“X”somewhereonthe

diagram,becauseanexistentialstatementalwaysassertsthatsomepartorthediagrammustcontainatleast


oneelement.Payattentiontowhetherthe“X”sitsdirectlyinoneregionofthediagram,orontheborder

betweentworegions.

Therefore,somebulldogsarenottimid.

4.Nowthatwehavemarkedthediagramsothatitconveystheinformationinthetwopremises,wecheckto

seeifthemarkeddiagramshowsthattheconclusionistrue.Ifthemarkeddiagramshowsthattheconclusion

istrue,thentheargumentisvalid.Ifthemarkeddiagramshowsthattheconclusionisfalseoruncertain,then

theargumentisinvalid.Forthisargumenttobevalid,the“X”shouldbeinside“bulldogs”butoutside“timid”.

5.Inpresentingthistechnique,wehaveassumedthatonepremiseisauniversalstatement,andtheother

premiseisanexistentialstatement.

Thetechniqueworksinthecasewherebothpremisesareuniversalstatements,too.


Example

Usediagrammingtotestthevalidityofthisargument.

Someusefulthingsareinteresting.

Allwidgetsareinteresting.

Therefore,somewidgetsareuseful.

A.Valid

B.Invalid

Step1:Isthereauniversalpremise?

Step2:Markuniversalpremisesfirst.

Step3:Marktheotherpremise.

Step4:Istheconclusionshowntobetrue?

Example

Testthevalidityofthisargument.

Allmean-lookingdogsaregoodwatchdogs.

Allbulldogsaremean-lookingdogs.

Therefore,allbulldogsaregoodwatchdogs.

A.ValidB.Invalid


Part 2 Module 3 - Arguments and deductive reasoning

Logicisaformalstudyoftheprocessofreasoning,orusingcommonsense.

Deductivereasoninginvolvestakinginandanalyzinginformation,andrecognizingwhenacollectionoffacts

andassumptionscanleadtonewfactsandnewassumptions.Logicandreasoningformthefoundationfor

mathematics,science,scholarlyresearch,law,andeffectivecommunication,amongotherthings.

An____________________________inlogicisasimplemodelthatillustrateseithercorrect,logical

reasoning,orincorrect,illogicalattemptsatreasoning.

Formally,anargumenttypicallyinvolvestwoormorepropositions,called____________________________

followedbyanotherproposition,calledthe___________________________________.

Inanyargument,weareinterestedinthelogicalrelationshipbetweenthepremisesandtheconclusion.

Twosimplearguments

Herearetwoexamplesofshortarguments,suchasaprosecutormightmakeinsummarizinghis/hercaseto

thejuryattheendofatrial.


Argument#1

ThepersonwhorobbedtheMini-Martdrivesas1999Corolla.

Gomerdrivesa1999Corolla.

Therefore,GomerrobbedtheMini-Mart.

Argument#2

Thepersonwhodrankmycoffeeleftthesefingerprintsonthecup.

Gomeristheonlypersonintheworldwhohasthesefingerprints.

Therefore,Gomerdrankmycoffee.

Whenwereadthefirstargument,weprobablyrecognizethatthereasoningis_________________________

becausemanypeopledrive1999Corollas.Noticethatargumenttwodoesn’tsharethedefectofthefirst.In

thisargument,ifwebelievethetwopremises,wehaveto___________________________________.

Fromamoregeneralperspective,thisargumentisillogical(invalid)becauseitispossibleforusto

_________________________________________,evenifweaccept_____________________________.

Also,anargumentiswell-structured(valid)ifitis_________________________,assumingthatwe

____________________________________.

Valid arguments

Wearealwaysinterestedinthelogicalrelationshipbetweenthepremisesandtheconclusionofanargument.

Anargumentisvalidifitis______________________________________________________oruncertain

when

everypremise_________________________________________.

Notethatwhetheranargumentisvalidhasnothingtodowithwhetherthestatementsintheargumentsound

____________________________Validityisdeterminedentirelybyhowthestatementsintheargument

relatetooneanother,regardlessofwhetherthosestatementsseemreasonabletous.


Inval id arguments

Anargumentisinvalidifitis____________________________________________atthesametimethat

everypremiseis__________________________________________.

Aninvalidargumentisamodelofincorrectorillogicalattemptsatreasoning.

Techniques for analyzing arguments

Inthiscoursewewilllearnseveraldifferenttechniquesforanalyzingshortarguments.

Thesetechniquesarebaseduponthedefinitionofavalidargument:

Anargumentisvalidifit

__________________________________________________________________________whenevery

premiseis_____________________________________.

DiagrammingUniversal-Particulararguments

ThesimpleststyleofnontrivialargumentiscalledaUniversal-Particularargument.

AUniversal-Particularargumentisa__________________________________________inwhichonepremise

isauniversalproposition(“Allare…,”“Noneare…”),whiletheotherpremise,andtheconclusion,are

propositionsthatrelatea_________________________________________tothecategoriesintheuniversal

premise.The_________________________________willalsobereferredtoasthemajorpremise.

The______________________________willalsobereferredtoastheminorpremise.


ExamplesofU.P.arguments

Allcatshaverodentbreath.

Whiskersdoesn'thaverodentbreath.

Thus,Whiskersisn'tacat.

Gomerisnotarascal.

Norascalsarereliable.

Therefore,Gomerisreliable

DiagrammingaU.P.argumentOnewaytotestthevalidityofaUniversal-Particularargumentistouseamethodbaseduponthe

diagrammingtechniquesthatwereintroducedinPart2Module1.Inanutshell,themethodworkslikethis:

1.First,markthediagramaccordingtothecontentoftheuniversalpremise.

Iftheuniversalpremiseispositive,wewill“shadeout”acrescent-shapedregion.Iftheuniversal

premiseisnegative,wewill“shadeout”afootball-shapedregion.

Theshadingshowsthataregionmusthavenoelements.

2.Next,placea“X”onthediagramaccordingtothecontentoftheparticularstatement,bearingin

mindthemeaningoftheshadingalreadyonthediagram.(The“X”representstheparticularindividual

whoisthesubjectoftheargument.)

Ifitisuncertainwhichoftworegionsshouldreceivethe“X,”thenplacethe“X”ontheboundary

betweenthetworegions.

3.Ifthemarkeddiagramshowsthattheconclusionistrue,thentheargumentisvalid.

Ifthemarkeddiagramshowsthattheconclusionisfalseoruncertain,thentheargumentisinvalid.


Example

UsediagrammingtotestthevalidityofthefollowingU-Pargument:

Allcatshaverodentbreath.

Whiskersdoesn'thaverodentbreath.

Thus,Whiskersisn'tacat.

A.Valid

B.Invalid

Example

Usediagrammingtotestthevalidityofthisargument.

Gomerisnotarascal.

Norascalsarereliable.

Therefore,Gomerisreliable.

A.Valid

B.Invalid


Diagrammingconventions

ThefollowingisareminderofP2M1content.

Inthiscase,thediagrammingrulesarestatedintermsofatwo-circleVenndiagram,becauseaU-Pargument

willinvolvetwocategories,notthree.

Also,thissummarywillinvolveasimplestkindofexistentialstatement–namely,aparticularstatement,which

proposestheexistenceofasingle,namedindividual,ratherthanasub-categorythatcouldconceivably

encompassmanyindividuals.

ThisstuffwillgetmorecomplicatedwhenwediscusscategoricalsyllogismsinPart2Module4.

Wediagramauniversalpremises(“allare…”,“noneare..”)byusingshadingto_________________________

theregion(s)ofthediagramthatcontradicttheuniversalstatement.

Inotherwords,weuseshadingtoindicatethattheshadedregionmustcontain

_________________________.

ExampleConsidertheuniversalstatement“Noelephantsaretiny”inthecontextofthistwo-circleVenn

diagram.Erepresentsthesetofelephants,andTrepresentsthesetoftinythings.

Shading“Noelephantsaretiny.”Accordingtothestatement“Noelephantsaretiny,”theregionwhereE

intersectsTmustbeempty.ThisisbecauseanyelementthatisintheintersectionofEwithTisbothan

elephantandtiny,contractingthestatementthat“Noelephantsaretiny.”Weshadethatregionofthe

diagram,toindicatethatit______________________________.

E TAccording to the statement —No elephants are tiny,“ this region must be empty.


Diagramminganegativeuniversalpremise

Diagramming“Allpoodlesareyappy.”

WewillmarktheVenndiagramtoconveytheinformationinthepositiveuniversalstatement“Allpoodlesare

yappy.” Prepresentsthesetofpoodles,andYrepresentsthesetofyappythings.Accordingtothe

statement“Allpoodlesareyappy,”anyregionofthediagramthatshowspoodleswhoaren’tyappymustbe

_________________.Soweneedto______________________________thisregion.

P YAny element in this region of the diagram is a poodle who isn‘t yappy.According to the statement —All poodlesare yappy,“ this region must be empty .


Diagramming“Allare…”

Generally,diagrammingastatementoftheform“allare…”,suchas“AllAareB”or“AllBareA,”willhavethe

effectofshadingacrescent-shapedregion.

Theshadingalwaysindicatesthattheshadedregionisempty.

DiagrammingaparticularstatementRecallthataparticularstatementisastatementthatrelatesanindividualtoacategory,suchas“Gomerisa

firefighter”or“Whiskersdoesn’thaverodentbreath.”

Todiagramaparticularstatement,weusean“X”torepresenttheparticularpersonwhoisthesubjectofthe

statement,andwhenplacethe“X”onthediagramaccordingtothecontentofthestatement.

Ifthe“X”canbeplacedineitheroftworegions,thenweplacethe“X”ontheboundarybetweenthetwo

regions.

Example:Diagramminga“Gomerisafirefighter.”

Supposethatthediagrambelowreferstothecategories“Firefighters”(F)and“Heroes”(H).

Markthediagramtoconveytheinformation“Gomerisafirefighter.”

Let“X”representGomer.

F H

—Gomer is a firefighter“ means that the —X“ representing Gomer could go in either of these two regions.


Notethattherearetworegionsofthediagraminwhichthe“X”canbeplacedtosatisfythestatement

“Gomerisafirefighter.”Thereforeweplacedthe“X”

____________________________________________Example:Diagramminga“Whiskersdoesn’thave

rodentbreath.”

SupposetheVenndiagrambelowrelatestothecategories“Cats”(C)and“thingswithRodentBreath”(R).

Markthediagramtoconveytheinformationintheparticularstatement“Whiskersdoesn’thaverodent

breath.Wewillusean“X”torepresenttheparticularindividual“Whiskers.”Notethattherearetworegions

ofthediagraminwhichthe“X”couldbeplacedtosatisfythethecondition“Whiskersdoesn’thaverodent

breath.”Thereforeweplacethe“X”______________________________________________________

C R

—Whiskers doesn‘t have rodent breath.“ Whiskers could go in either of these two regions.


P2M3 Contiued – Truth Tables to test for Valid arguments

Avalidargumenthasthefollowingproperty:

Itisimpossiblefor_________________________________,ifweassumethateverypremiseis

____________________

Inavalidargument,“thetruthofthepremises_____________________________oftheconclusion.”

Using a Truth Table to analyze argument val idity

1.Symbolize(consistently)allofthepremisesandtheconclusion.

2.Makeatruthtablehavingacolumnforeachpremiseandfortheconclusion.

3.Ifthereisarowinthetruthtablewhereeverypremisecolumnistruebuttheconclusioncolumnisfalse(a

counterexamplerow)thentheargumentisinvalid.Iftherearenocounterexamplerows,thentheargument

isvalid.

Whydoesthismethodwork?

Whenwehavefilledinthetruthtable,wearecheckingtoseeifthereisarowwheretheconclusionis

______________________________________everypremiseis_______________________

Ifthereissucharow,thenthetruthtablehasshownthatitis________________________________forthe

conclusiontobefalseatthesametimethat__________________________________________:thisisexactly

the

definitionofan_____________________________________________

ExampleUseatruthtabletotestthevalidityofthefollowingargument.

IfIenterthepoodleden,thenIwillcarrymyelectricpoodleprodormycanofmace.

Iamcarryingmyelectricpoodleprodbutnotmycanofmace.

Therefore,Iwillenterthepoodleden.

A.ValidB.Invalid


Example

Testthevalidityoftheargument.

Idon’tlikemuskrats.

IfIownabadgerorIdon’townawolverine,thenIlikemuskrats.

Therefore,IownawolverineandIdon’townabadger.

A.ValidB.Invalid

ExampleTheArgue-mentor,Part3hasmoreexampleslikethisone.Testthevalidityofthisargument.

~p∨ ~q

q

∴ ~p

A. Valid

B.Invali


P2M3 Continued – Common Logical Forms

Studythefollowingfourarguments.

IftodayisTuesday,thenIhavemathclass.TodayisTuesday.Therefore,Ihavemathclass.

Idon’townabadger.IfIdon’townabadger,thenIownatortoise.Therefore,Iownatortoise.

Ifthatanimalisawolverine,thenitisn’tcuddly.Thatanimalisawolverine.Therefore,thatanimalisn’t

cuddly.

Idon’tlikelivingbelowground.IfIdon’tlikelivingbelowground,thenI’mnotapotato.Therefore,I’mnota

potato.

Doyouseethatallfourargumentshavethesamestructure?

.

Eachofthesefourargumentscanbecharacterizedasfollows:

Onepremiseisa_________________________________________;theotherpremise

____________________________________

oftheconditionalpremise(“affirmstheantecedent”);theconclusion

________________________________________

oftheconditionalpremise(“affirmstheconsequent”).

Becauseallfourargumentshavethesamestructure,ifoneofthemisvalid,theotherthreeshouldalsobe

valid;ifoneofthemisinvalid,theotherthreeshouldalsobeinvalid.

IftodayisTuesday,thenIhavemathclass.TodayisTuesday.Therefore,Ihavemathclass.

p→q

p

∴ q


premisepremiseconclusion

pqp→q p q

TTTT T

TFFT F

FTTF T

FFTF F

ThetruthtableshowsthattheargumentisVALID.

Sincetheotherthreeargumentsonthepreviousslidehavethesamestructureasthisargument,theymust

alsobevalid.Wedon’tneedtomakethreemoretruthtables.

Thereareseveralformsofshort,validarguments,andcorrespondinginvalidforms,thatoccursooftenthatit

ishelpfultobeabletorecognizeandnamethem.

Wewillencounternamessuchas

Four Common Logical Forms

VALIDforms INVALIDforms

DirectReasoning FallacyoftheConverse

ContrapositiveReasoning FallacyoftheInverse

ExamplesofContrapositiveReasoning

Eachoftheseargumentsisvalid,becauseofContrapositiveReasoning:

IfIhaveahammer,thenIwillhammerinthemorning.

Idon’thammerinthemorning.

Therefore,Idon’thaveahammer.

Idon’thavetowork.

IftodayisThursday,thenIhavetowork.

Therefore,todayisn’tThursday.


Iownabadger.

Ifdon’townawolverine,thenIdon’townabadger.

Therefore,Iownawolverine.

ExamplesofFallacyoftheInverse

Eachoftheseargumentsisinvalid,becauseofFallacyoftheInverse:

IftodayisWednesday,thenIhavemathclass.

Todayisn’tWednesday.

Therefore,todayIdon’thavemathclass.

Iownabike.

IfIdon’townabike,thenIhavemathclass.

Therefore,Idon’thavemathclass.

Example

Testthevalidityofthisargument:

I’mnotoutofbananasorIwon’tfeedmymonkeys.

Iwillfeedmymonkeys.

Therefore,I’mnotoutofbananas.

A.Valid B.Invalid

DisjunctiveSyllogism

DisjunctiveSyllogismisamethodthatturns_________________________________into

___________________argument,asfollows.

Anyargumenthavingoneoftheseformsisvalid:

A∨B A∨B

~A ~B

∴ B ∴ A

ThiscommonformiscalledDisjunctiveSyllogism.

ExamplesofDisjunctiveSyllogism


Eachofthefollowingargumentsisvalid,becauseitisadisjunctivesyllogism.Notethatthisformis

characterizedasfollows:

onepremiseisa_____________________________________,

theotherpremise_____________________________________________,whilethe

conclusion___________________________________

Argument1

IownabadgerorIownawolverine.

Idon’townabadger.

Therefore,Iownawolverine.

Argument2

IownabadgerorIownawolverine.

Idon’townawolverine.

Therefore,Iownabadger.


DisjunctiveFallacy

Inordertoturnan“or”premiseintoavalidargument,theminorpremise

__________________________________ofthemajor(“or”)premises.

Iftheminorpremiseaffirmsoneofthetermsofthe“or”premise,thenwehavethestructure

_______________________________________.Anyargumenthavingoneoftheseformsis

_____________________________.

A∨B A∨B

A B

∴ ~B ∴ ~A

ThiscommonformiscalledDisjunctiveFallacy.

Example

Testthevalidityoftheargument.

IfIgetelected,I'llreducetaxes.

IfIreducetaxes,theeconomywillprosper.

Thus,ifIgetelected,theeconomywillprosper.

A.Valid

B.Invalid

TransitiveReasonongThisisanexampleofTransitiveReasoning,avalidforminwhich__________________________________are

connected,sotospeak,inordertoarriveata____________________________________________.

Anyargumentthatcanbereducedtotheform

A→B

B→C

∴ A→C

isVALID.

WerefertothiscommonformasTransitiveReasoning.

Thefollowingargumentisvalid,becauseitisanexampleofTransitiveReasoning.


IfIeatmyspinach,thenI'llbecomemuscular.

IfIbecomemuscular,thenI'llbecomeaprofessionalwrestler.

IfIbecomeIprofessionalwrestler,thenI'llbleachmyhair.

IfIbleachmyhair,thenI'llwearsequinedtights.

IfIwearsequinedtights,thenI'llberidiculous.

Therefore,ifIeatmyspinach,thenI'llberidiculous.

ThepreviousexampleillustratesanimportantpropertyofTransitiveReasoning:Thismethodofreasoning

____________________________________________________

Weeasilycanconstructvalidargumentsthathaveasmany"if...then"premisesaswewish,aslongasthe

fundamentalpatterncontinues:namely,the

_________________________________________________________withthe

__________________________________________________________.

NotTransitiveReasoning

ThefollowingargumentlookssimilartoTransitiveReasoning,buttherelationshipbetweentermsisn’tquite

right.

IfIgetelected,I'lltakelotsofbribes.

IfIgetelected,I'llreducetaxes.

Thus,ifItakelotsofbribes,thenI'llreducetaxes.

Thisisanexampleofacommonfallacy,calledaFalseChain.

FalseChains

AnyargumentthatcanbereducedtooneoftheseformsisINVALID.

A→B A→B

A→C C→B

∴ B→C ∴ A→C

WerefertothesecommonfallaciesasFalseChains.


Examples

Argument1

IftodayisFriday,thenIhavemathclass.

IfIhavemathclass,thenIwrite.

Therefore,iftodayisFriday,thenIwrite.

Argument2


IftodayisFriday,thenIwashthedog.

Therefore,ifIhavemathclass,thenIwashthedog.

Argument3


IftodayisWednesday,thenIhavemathclass.

Therefore,iftodayisFriday,thentodayisWednesday

Althoughtheysoundsimilar,youshouldrecognizethat

_____________________________________________andArguments2and3are

_________________________________________.

Example

Testthevalidityofthisargument:

Somelawyersarejudges.

Somejudgesarepoliticians.

Therefore,somelawyersarepoliticians.

A.Valid

B.Invalid


Part 2 Module 5 - Analyzing premises, forming conclusions

Common Forms: InPart2Module3,weidentifiedanumberofcommonformsofvalidarguments,and

commonfallacies.ForourworkinPart2Module5,itwillbeespeciallyhelpfulifweareabletorecognize

thesecommonforms.

Direct Reasoning, Fal lacy of the Converse

Valid Invalid

A→B A→B

A B

∴ B ∴ A

IftodayisWednesday, IftodayisWednesday,

thenIhavemathclass. thenIhavemathclass.

TodayisWednesday. Ihavemathclass.

Therefore, Therefore,

Ihavemathclass. TodayisWednesday.

Contraposit ive Reasoning, Fal lacy of the Inverse

Valid Invalid

A→B A→B

~B ~A

∴ ~A ∴ ~B

IftodayisWednesday, IftodayisWednesday,

thenIhavemathclass. thenIhavemathclass.

Idon’thavemathclass. Todayisn’tWednesday.

Therefore, Therefore,

Todayisn’tWednesday. Idon’thavemathclass.


Transit ive Reasoning, False Chains

Valid Invalid Invalid

A→B A→B A→B

B→C A→C C→B

∴ A→C ∴B→C ∴A→C

Disj inctive Syl logism ( val id)

A∨B Iownacat,orIownadog.

~A Idon’townacat.

∴ B Therefore,Iownadog.


~B Idon’townadog.

∴ A Therefore,Iownacat.

Disjunctive fal lacy ( inval id)


A Iownacat.

∴ ~B Therefore,Idon’townadog


B Iownadog.

∴ ~A Therefore,Idon’townacat.

Example

Selectthestatementthatisavalidconclusionfromthefollowingpremises,ifavalidconclusioniswarranted.

IusemycomputerorIdon'tgetanythingdone.

Igetsomethingdone.

A.Iusemycomputer.

B.Idon'tusemycomputer.

C.Iuseanabacus.

D.Noneoftheseiswarranted.


Example


Ifwestrive,thenweexcel.

Wedidn'tstrive.

A.Weexcelled.

B.Wedidn'texcel.

C.Wedidn'tinhale.


Guidelines Inthiscourse,whenwearetryingto

_______________________________________________________________________,if

wehavethe___________________________________________________________________,thecorrect

choicewillalwaysbe“Noneoftheseiswarranted.”

Thisisbecauseitisneverpossibletoturnanillogicalpremiseset-upintoanon-trivialvalidargument.

Moreover,ifwehavethepremise__________________________________________________,thecorrect

answerwillneverbe“Noneoftheseiswarranted.”

ExampleSelectthestatementthatisavalidconclusionfromthefollowingpremises,ifavalidconclusioniswarranted.

Ifmycardoesn'tstart,thenI'llbelateforwork.

I'mnotlateforwork.

A.Mycarstarted.

B.Irodethebus.

C.I'mlateforwork.



Example


Nokittensarefierce.

Fluffyisn'tfierce.

A.Fluffyisakitten.

B.Fluffyhasfleas.

C.Fluffyisn'takitten.


Example


Allpoliticiansarepromisemakers.

Gomerisnotapolitician.

A.Gomerisnotapromisemaker.

B.Gomerisapolitician.

C.Allpromisemakersarepoliticians.



Ifyouwantabettergrade,thenyoubringanapplefortheteacher.

Ifyoubringanapplefortheteacher,thenyouexposetheteachertodangerousagriculturalchemicals.

A.Ifyouexposetheteachertodangerousagriculturalchemicals,thenyouwantabettergrade.

B.Ifyoudon'texposetheteachertodangerousagriculturalchemicals,thenyoudon'twantabetter

grade.

C.Youwantabettergrade.


Inwords,the__________________________________________is“Ifyouwantabettergrade,thenyou

exposetheteachertodangerousagriculturalchemicals.”Thisis

_____________________________________________________.


Using Transit ive Reasoning

InordertoseethatwecanuseTransitiveReasoningtoarriveatavalidconclusion,itmaybenecessary

to___________________________________statements

________________________________________________.

Wecanneverreplaceastatementwith___________________________________.


Ifyouaren’tbitey,thenyouaren’tawolverine.

Ifyouarebitey,thenyouaren’tcuddly

A.Ifyouaren’tawolverine,thenyouarecuddly.

B.Ifyouarecuddly,thenyouareawolverine.

C.IfyournameisDudley,thenyouarecuddly.

D.Ifyouarecuddlythenyouaren’tawolverine.

E.Noneoftheseiswarranted.

RecognizingcommonformsThepresenceofacommonlogicalformmaynotbeobviouswhenyoufirstreadthepremisesofanargument.

Tohelprecognizetheoccurrenceofacommonform,wecanalways:

1.Rearrangetheorderinwhichthepremisesarepresented;

2.Replacestatementswithequivalentstatements;

Inparticular,wecanalwaysreplaceaconditionalstatementwithitscontrapositive.


Example


IfIinvestwisely,thenIwon'tlosemymoney.

IfIdon'tinvestwisely,thenIbuyjunkbonds.

IfIreadInvestor'sWeekly,thenIwon'tbuyjunkbonds.

A.IfIinvestwisely,thenIreadInvestor'sWeekly.

B.IfIbuyjunkbonds,thenIdon'tinvestwisely.

C.IfIlosemymoney,thenIdon'treadInvestor'sWeekly.

D.IfIeatjunkfood,thenIinvestweakly.

E.Noneoftheseiswarranted.

Using Transit ive Reasoning

InordertoseethatwecanuseTransitiveReasoning,itmaybenecessarytorearrangetheorderinwhichthe

premisesarelisted.Wewantthefirst“if…then”premisetobeginwithatermthatappearsonlyonetimein

thepremisescheme.Continuerearrangingtheorderofthepremises,andperhapsreplacingpremiseswith

theircontrapositives,sothattheantecedentofeachsuccessivepremisematchestheconsequentofthe

precedingpremise.Whenwehaveusedeverypremiseinthismanner,wecanformachainofreasoningto

stateavalidconclusionthatuseseverypremise(amajorvalidconclusion).

Ifatanypointitis_________________________________________ofpremises,thentheargumentinvolves

a______________.Inthiscase,thecorrectanswerwill_______________________________________.

Universalstatements(“Allare…”“Noneare…”)canbewrittenas_____________________________.

“Allpoodlesareyappy”means“____________________________________________________.”

“Noporcupinesarecuddly”means“___________________________________________________.”

“AllAareB”isequivalenttoA→B.

“NoAareB”isequivalenttoA→~B.

Particularstatementscanalsobewrittenas________________________________________

“Gomerisajudge”means“__________________________________________________________”

“Homerisn’talawyer”means“_____________________________________________________.”


Existentialstatements(“Someare…”“Somearen’t…”)

__________________________________________________,sodon’teventry.

Example

Shouldwetryanexample?

Express“Somelawyersarejudges”asa________________________

Sorry_________________________________cannotbewrittenasan

________________________________________

Example


Allpeoplewhogetmanyticketsareuninsurable.

Allcarelessdriversgetmanytickets.

Allpeoplewhoareuninsurablehavebadcreditratings.

A.Allcarelessdrivershavebadcredit

ratings.

B.Ifyourcarisrepossessedbecauseyou

havebedcredit,thenyouareacar-less

driver.

C.Allpeopleareuninsurablegetmany

tickets.



Example


Ifyouaren'tagoodstirrer,thenyouaren'thandywithaswizzlestick.

IfyouareagraduateofBillyBob'sBigBoldSchoolofMixology,thenyouareabartender.

Nogoodstirrershaveweakwristmuscles.

Ifyoudon'thaveweakwristmuscles,thenyouhaveafirmhandshake.

Allbartendersarehandywithaswizzlestick.

A.IfyouareagraduateofBillyBob'sBig

BoldSchoolofMixology,thenyoudon't

haveafirmhandshake.

B.Ifyoudon'thaveafirmhandshake,then

youaren'tagraduateofBillyBob'sBigBold

SchoolofMixology.

C.Ifyouhaveafirmhandshake,thenyou

areagraduateofBillyBob'sBigBoldSchool

ofMixology.


Example


Sylvesterisn'taparakeet.

Elephantsneversquawk.

Allparakeetssquawk.

Noelephantsaretiny.

A.Sylvesterisanelephant.

B.Sylvesterisn'ttiny.

C.Allparakeetsaretiny.



Further discussion

Focusonthetwomiddlepremises.Ignorethefirstpremiseandthelastpremise.

1.Sylvesterisn'taparakeet.

2.Elephantsneversquawk.

3.Allparakeetssquawk.

4.Noelephantsaretiny.

Thisisanexampleofaminorvalidconclusion(avalidconclusionthatdoesn’trequiretheuseofevery

premise)Foraproblemlikethis,inthiscourse,aminorvalidconclusionwillneverbelistedamongthe

multiplechoiceoptions.Therightanswerwillalwaysbeamajorvalidconclusion(avalidconclusionthat

requirestheuseofeverypremise),

or,ifamajorvalidconclusionisnotpossible,therightanswerwillbe“Noneofthese…”

proposition truth valueharris/1106/part2outlines.pdfhowever, there are many other (non-preferred)...

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