1. propositional logic

7/23/2019 1. Propositional Logic

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Propositional Logic

Rosen 5Rosen 5ththed., 1.1-1.2ed., 1.1-1.2


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Foundations of Logic: Overview

Proositiona! !ogic:Proositiona! !ogic:

"#asic definitions.#asic definitions.

"$%uiva!ence ru!es & derivations.$%uiva!ence ru!es & derivations.

Predicate !ogicPredicate !ogic

"Predicates.Predicates.

"'uantified redicate e(ressions.'uantified redicate e(ressions.

"$%uiva!ences & derivations.$%uiva!ences & derivations.


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Proositiona! Logic

Propositional LogicPropositional Logicis the !ogic of co*oundis the !ogic of co*ound

state*ents +ui!t fro* si*!er state*entsstate*ents +ui!t fro* si*!er state*ents

usingusingBooleanBooleanconnectives.connectives.

!ications:!ications:

esign of digita! e!ectronic circuits.esign of digita! e!ectronic circuits.

$(ressing conditions in rogra*s.$(ressing conditions in rogra*s.

'ueries to data+ases & search engines.'ueries to data+ases & search engines.


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efinition of aProposition

propositionproposition//pp,, qq,, rr, 0 is si*! a, 0 is si*! a

statementstatement //i.e.i.e., a dec!arative sentence, a dec!arative sentencewithwith

a definite meaninga definite meaning, having a, having a truth valuetruth valuethat3s eitherthat3s either truetrue/4 or/4 orfalsefalse/F //F /nevernever

+oth, neither, or so*ewhere in +etween.+oth, neither, or so*ewhere in +etween.

6n6nprobability theory,probability theory,we assignwe assign degrees of certaintydegrees of certaintyto roositions. For now: 4rue7Fa!se on!89to roositions. For now: 4rue7Fa!se on!89


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$(a*!es of Proositions

6t is raining.; /hina.;

1 ? 2 @ );1 ? 2 @ );

4he fo!!owing are4he fo!!owing are NOTNOTroositions:roositions:

Aho3s thereB; /interrogative, %uestionAho3s thereB; /interrogative, %uestion

La !a !a !a !a.; /*eaning!ess inter=ectionLa !a !a !a !a.; /*eaning!ess inter=ection Cust do it8; /i*erative, co**andCust do it8; /i*erative, co**and

Deah, 6 sorta dunno, whatever...; /vagueDeah, 6 sorta dunno, whatever...; /vague

1 ? 2; /e(ression with a non-true7fa!se va!ue1 ? 2; /e(ression with a non-true7fa!se va!ue


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nn operatoroperatororor connectiveconnectiveco*+ines one orco*+ines one or

*ore*ore operandoperand e(ressions into a !argere(ressions into a !arger

e(ression. /e(ression. /E.g.E.g., ?; in nu*eric e(rs., ?; in nu*eric e(rs.

UnaryUnaryoerators tae 1 oerand /oerators tae 1 oerand /e.g.,e.g.,-)G-)G

binarybinary oerators tae 2 oerands /oerators tae 2 oerands /egeg)) ..

PropositionalPropositionalororBooleanBooleanoerators oerate onoerators oerate on

roositions or truth va!ues instead of onroositions or truth va!ues instead of on

nu*+ers.nu*+ers.

Oerators 7 >onnectives


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4he Iegation Oerator

4he unar4he unar negation operatornegation operatorJ; /J; /N!N!

transfor*s a ro. into its !ogica!transfor*s a ro. into its !ogica!negationnegation..

E.g.E.g.6f6fpp@ 6 have +rown hair.;@ 6 have +rown hair.;

then Jthen Jpp@ 6 do@ 6 do notnothave +rown hair.;have +rown hair.;

!ruth table!ruth tablefor IO4:for IO4: p p

4 F

F 4


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4he >on=unction Oerator

4he +inar4he +inar con"unction operatorcon"unction operator; /; /#N$#N$

co*+ines two roositions to for* theirco*+ines two roositions to for* their

!ogica!!ogica! con"unctioncon"unction..

E.g.E.g.6f6fpp@6 wi!! have sa!ad for !unch.; and@6 wi!! have sa!ad for !unch.; and

q%q%6 wi!! have stea for dinner.;, then6 wi!! have stea for dinner.;, then

ppqq@6 wi!! have sa!ad for !unch@6 wi!! have sa!ad for !unch andand

6 wi!! have stea for dinner.;6 wi!! have stea for dinner.;


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Iote that aIote that a

con=unctioncon=unction

pp11pp22 00 ppnn

ofof nnroositionsroositions

wi!! have 2wi!! have 2nnrowsrows

in its truth ta+!e.in its truth ta+!e. J andJ and oerations together are universa!,oerations together are universa!,

i.e., sufficient to e(ressi.e., sufficient to e(ress anyanytruth ta+!e8truth ta+!e8

>on=unction 4ruth 4a+!e

p q pq

F F F

F 4 F

4 F F

4 4 4


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4he is=unction Oerator

4he +inar4he +inar dis"unction operatordis"unction operator; /; /&&


!ogica!!ogica! dis"unctiondis"unction..

pp@4hat car has a +ad engine.;@4hat car has a +ad engine.;

q%q%4hat car has a +ad car+uretor.;4hat car has a +ad car+uretor.;

ppqq@$ither that car has a +ad engine,@$ither that car has a +ad engine, oror

that car has a +ad car+uretor.;that car has a +ad car+uretor.;


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Iote thatIote thatppqq *eans*eans

thatthatppis true, oris true, or qqisis

true,true, or bothor bothare true8are true8

No this oeration isNo this oeration is

a!so ca!!eda!so ca!!ed inclusive or,inclusive or,

+ecause it+ecause it includesincludesthetheossi+i!it that +othossi+i!it that +othppandand qqare true.are true.

J; and J; and ; together are a!so universa!.; together are a!so universa!.

is=unction 4ruth 4a+!e

p q pq

F F F

F 4 T

4 F T

4 4 4


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Ni*!e $(ercise

LetLetpp@6t rained !ast night;,@6t rained !ast night;,

qq@4he srin!ers ca*e on !ast night,;@4he srin!ers ca*e on !ast night,;

rr@4he !awn was wet this *orning.;@4he !awn was wet this *orning.;

4rans!ate each of the fo!!owing into $ng!ish:4rans!ate each of the fo!!owing into $ng!ish:

JJpp @@

rrJJpp @@

JJ rr ppq %q %

6t didn3t rain !ast night.;4he !awn was wet this *orning, andit didn3t rain !ast night.;$ither the !awn wasn3t wet this

*orning, or it rained !ast night, or

the srin!ers ca*e on !ast night.;


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4heE'clusive rOerator

4he +inar4he +inar e'clusive(or operatore'clusive(or operator; /; /)&)&


!ogica! e(c!usive or; /e(=unctionB.!ogica! e(c!usive or; /e(=unctionB.

pp@ 6 wi!! earn an in this course,;@ 6 wi!! earn an in this course,;

qq@@6 wi!! dro this course,;6 wi!! dro this course,;

ppqq @ 6 wi!! either earn an for this@ 6 wi!! either earn an for this

course, or 6 wi!! dro it /+ut not +oth8;course, or 6 wi!! dro it /+ut not +oth8;


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Iote thatIote thatppqq *eans*eans

thatthatppis true, oris true, or qqisis

true, +uttrue, +ut not bothnot both88

4his oeration is4his oeration is

ca!!edca!!ed e'clusive or,e'clusive or,

+ecause it+ecause it excludesexcludesthetheossi+i!it that +othossi+i!it that +othppandand qqare true.are true.

J; and J; and ; together are; together are notnotuniversa!.universa!.

$(c!usive-Or 4ruth 4a+!e

p q pq

F F F

F 4 4

4 F 4

4 4 F


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Iote thatIote that $ng!ish$ng!ishor; isor; is by itselfby itself a*+iguousa*+iguous

regarding the +oth; case8regarding the +oth; case8

Pat is a singer orPat is a singer or

Pat is a writer.; -Pat is a writer.; -

Pat is a *an orPat is a *an or

Pat is a wo*an.; -Pat is a wo*an.; -

Ieed conte(t to disa*+iguate the *eaning8Ieed conte(t to disa*+iguate the *eaning8

For this c!ass, assu*e or; *eansFor this c!ass, assu*e or; *eans inc!usiveinc!usive..

Iatura! Language is *+iguous

p q por q

F F FF 4 4

4 F 4

4 4undef.


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4he*mplicationOerator

4he4he implicationimplicationpp qqstates thatstates thatppi*!iesi*!ies q.q.

6t is FLN$6t is FLN$ on!on!in the case that is 4R$in the case that is 4R$

+ut % is FLN$.+ut % is FLN$.

E.g.E.g.,,pp@6 a* e!ected.;@6 a* e!ected.;

qq@6 wi!! !ower ta(es.;@6 wi!! !ower ta(es.;

pp q %q % 6f 6 a* e!ected, then 6 wi!! !ower6f 6 a* e!ected, then 6 wi!! !ower

ta(es;ta(es; /e!se it cou!d go either wa/e!se it cou!d go either wa


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6*!ication 4ruth 4a+!e

pp qq isis falsefalseon! whenon! when

ppis true +utis true +ut qqisis notnottrue.true.

pp qq doesdoes notnot i*!i*!

thatthatppcausescausesqq88


thatthatpporor qqare ever trueare ever true88

E.g.E.g./1@M/1@M igs can f!; is 4R$8igs can f!; is 4R$8

p q pq

F F 4

F 4 44 F F

4 4 4


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$(a*!es of 6*!ications

6f this !ecture ends, then the sun wi!! rise6f this !ecture ends, then the sun wi!! rise

to*orrow.;to*orrow.; !rue!rueoror+alse+alseBB

6f 4uesda is a da of the wee, then 6 a* a6f 4uesda is a da of the wee, then 6 a* a

enguin.;enguin.; !rue!rueoror+alse+alseBB

6f 1?1@E, then


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6nverse, >onverse, >ontraositive

No*e ter*ino!og:No*e ter*ino!og:

4he4he inverseinverse ofofpp qq is: Jis: Jpp JJqq

4he4he converseconverseofofpp qq is:is: qq pp..

4he4he contrapositivecontrapositiveofofpp qq is: Jis: Jqq JJp.p.

One of these has theOne of these has thesame meaningsame meaning/sa*e/sa*etruth ta+!e astruth ta+!e asppqq. >an ou figure out. >an ou figure out

whichBwhichB


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ow do we now for sureB

Proving theProving the e%uiva!encee%uiva!enceofofpp qq and itsand its

contraositive using truth ta+!es:contraositive using truth ta+!es:

p q q p pq qp

F F 4 4 4 4F 4 F 4 4 4

4 F 4 F F F4 4 F F 4 4


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4he biconditionaloerator

4he4he biconditionalbiconditionalpp qq states thatstates thatppis trueis true ififand only ifand only if*++- q*++- qis true.is true.

6t is 4R$ when +oth6t is 4R$ when +othpp qq andand qq pp areare4R$.4R$.

pp @ 6t is raining.;@ 6t is raining.;

qq@@4he ho*e tea* wins.;4he ho*e tea* wins.;pp q %q % 6f and on! if it is raining, the ho*e6f and on! if it is raining, the ho*e

tea* wins.;tea* wins.;


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#iconditiona! 4ruth 4a+!e

pp qq *eans that*eans thatppandand qq

have thehave the samesametruth va!ue.truth va!ue.

Iote this truth ta+!e is theIote this truth ta+!e is the

e(acte(act oppositeoppositeofof 3s83s8

pp qq *eans J/*eans J/pp qq


ppandand qqare true, or cause each other.are true, or cause each other.

p q pq

F F 4

F 4 F

4 F F

4 4 4


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#oo!ean Oerations Nu**ar

Ae have seen 1 unar oerator / ossi+!eAe have seen 1 unar oerator / ossi+!e

and 5 +inar oerators /1E ossi+!e.and 5 +inar oerators /1E ossi+!e.

p q p pq pq pq pq pqF F 4 F F F 4 4F 4 4 F 4 4 4 F

4 F F F 4 4 F F4 4 F 4 4 F 4 4


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Precedence of Logica! Oerators

JJ 11

22

))

55

Oerator Precedence


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Iested Proositiona! $(ressions

se arentheses tose arentheses togroup sub(e'pressionsgroup sub(e'pressions::

6 =ust saw * o!d6=ust saw * o!dffriendriend, and either, and either he3she3s

ggrownrownoror 63ve63vesshrunhrun.; @.; @ff//ggss" //ffgg ss wou!d *ean so*ething different wou!d *ean so*ething different

" ffggss wou!d +e a*+iguous wou!d +e a*+iguous

# convention, J; taes# convention, J; taesprecedenceprecedenceoverover+oth +oth ; and ; and ;.;.

" JJss ff *eans /J *eans /Jssff ,, notnot J /J /ss ff


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No*e !ternative Iotations

Ia*e: not and or (or i*!ies iff

Proositiona! !ogic:

#oo!ean a!ge+ra: p pq ?

>7>??7Cava /wordwise: ! && || != ==

>7>??7Cava /+itwise: ~ & | ^

Logic gates:


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4auto!ogies and >ontradictions

tautologytautologyis a co*ound roosition that isis a co*ound roosition that is

truetrueno matter whatno matter whatthe truth va!ues of itsthe truth va!ues of its

ato*ic roositions are8ato*ic roositions are8

E'.E'.pp pp Ahat is its truth ta+!eB9 Ahat is its truth ta+!eB9

contradictioncontradiction is a co*. ro. that isis a co*. ro. that is falsefalse

no *atter what8no *atter what8 E'.E'.pp pp 4ruth ta+!eB94ruth ta+!eB9

Other co*. ros. areOther co*. ros. are contingenciescontingencies..


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Proositiona! $%uiva!ence

4wo4wosyntacticallysyntactically//i.e.,i.e., te(tua!! differentte(tua!! different

co*ound roositions *a +eco*ound roositions *a +e

semanticallysemantically identica! /identica! /i.e.,i.e., have the sa*ehave the sa*e*eaning. Ae ca!! the**eaning. Ae ca!! the* equivalentequivalent. Learn:. Learn:

QariousQarious equivalence rulesequivalence rules ororlawslaws..

ow toow toproveprovee%uiva!ences usinge%uiva!ences usingsymbolicsymbolicderivationsderivations..


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Proving $%uiva!ences

>o*ound roositions>o*ound roositionsppandand qq are !ogica!!are !ogica!!

e%uiva!ent to each othere%uiva!ent to each other IFFIFFppandand qq containcontain

the sa*e truth va!ues as each other inthe sa*e truth va!ues as each other in a!!a!!rows of their truth ta+!es.rows of their truth ta+!es.

>o*ound roosition>o*ound roositionppisis logicallylogically

equivalentequivalent to co*ound roositionto co*ound roosition qq,,writtenwrittenppqq,, IFFIFFthe co*oundthe co*ound

roositionroositionppqq is a tauto!og.is a tauto!og.


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E'.E'.Prove thatProve thatppqq//pp qq..

p q ppqq pp qq ppqq ppqq

F F

F 4

4 F

4 4

Proving $%uiva!ence

via 4ruth 4a+!es

F4

44

4

4

4

44

4

FF F

F

FF

FF

44


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$%uiva!ence Laws

4hese are si*i!ar to the4hese are si*i!ar to the arith*etic identitiesarith*etic identities

ou *a have !earned in a!ge+ra, +ut forou *a have !earned in a!ge+ra, +ut for

roositiona! e%uiva!ences instead.roositiona! e%uiva!ences instead. 4he rovide a4he rovide aattern or te*!ateattern or te*!atethat canthat can

+e used to *atch *uch *ore co*!icated+e used to *atch *uch *ore co*!icated

roositions and to find e%uiva!ences forroositions and to find e%uiva!ences forthe*.the*.


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$%uiva!ence Laws - $(a*!es

*dentity*dentity:: ppTT p pp pFF pp

$omination$omination:: ppTT TT ppFF FF

*dempotent*dempotent:: pppp p pp ppp pp

$ouble negation$ouble negation pp pp

/ommutative p/ommutative p

qq

qq

p pp p

qq

qq

pp

#ssociative#ssociative //ppqqrrpp//qqrr

/ /ppqqrrpp//qqrr


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ore $%uiva!ence Laws

$istributive$istributive:: pp//qqrr //ppqq//pprr

pp//qqrr //ppqq//pprr

$e 0organ1s$e 0organ1s::

//ppqq pp qq

//ppqq pp qq


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ore $%uiva!ence Laws

#bsorption#bsorption::

pp//ppqq pp

pp //pp qq pp

!rivial tautology2contradiction!rivial tautology2contradiction::

ppppTT ppppFF


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efining Oerators via $%uiva!ences

sing e%uiva!ences, we cansing e%uiva!ences, we can definedefineoeratorsoerators

in ter*s of other oerators.in ter*s of other oerators.

6*!ication:6*!ication: ppqq pp qq

#iconditiona!:#iconditiona!:ppqq //ppqq//qqpp

ppqq //ppqq

$(c!usive or:$(c!usive or: ppqq//ppqq//ppqq

ppqq//ppqq//qqpp


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n $(a*!e Pro+!e*

>hec using a s*+o!ic derivation whether>hec using a s*+o!ic derivation whether//pp qq //pprr pp qqrr..

//pp qq //pprr$(and definition of$(and definition of 99 //pp qq //pprr

efn. ofefn. of 99 //pp qq ////pprr //pprr

eorgan3s Law9eorgan3s Law9

//ppqq////pprr //pprr


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$(a*!e >ontinued...

//pp qq ////pprr //pprrco**utes9co**utes9

//qqpp////pprr //pprrassociative9associative9

qq//pp////pprr //pprr distri+. distri+. overover 99qq//////pp//pprr //pp//pprr

assoc.9assoc.9 qq//////pppp rr //pp//pprr

trivia! taut.9trivia! taut.9 qq////TTrr //pp//pprr

do*ination9do*ination9qq//TT//pp//pprr

identit9identit9 qq//pp//pprrcont.cont.


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$nd of Long $(a*!e

qq//pp//pprr

eorgan3s9eorgan3s9 qq//pp//pprr

ssoc.9ssoc.9 qq////pppp rr

6de*otent96de*otent9 qq//pprr

ssoc.9ssoc.9 //qqpp rr

>o**ut.9>o**ut.9 pp qqrr

3.E.$. quod erat demonstrandum-3.E.$. quod erat demonstrandum-

1. propositional logic

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