Download Module 4: Propositional Logic Proofs

Module4:Propositional
LogicProofs
Duedates
• Pre-classquiz#5isdueSundayJanuary29that
19:00
• Assignedreadingforthequiz:
• Epp,4thedition:3.1,3.3
• Epp,3rdedition:2.1,2.3
• Assignment#2isdueThursdayFebruary2nd at
4pm.
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Learninggoals:pre-class
• Bythestartofthisclassyoushouldbeableto
• Usetruthtablestoestablishorrefutethevalidity
ofaruleofinference.
• Givenaruleofinferenceandpropositionallogic
statementsthatcorrespondtotherule'spremises,
applytheruletoinferanewstatementimpliedby
theoriginalstatements.
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Learninggoals:in-class
• Bytheendofthismodule,youshouldbeableto
• Determinewhetherornotapropositionallogic
proofisvalid,andexplainwhyitisvalidorinvalid.
• Exploretheconsequencesofasetofpropositional
logicstatementsbyapplicationofequivalenceand
inferencerules,especiallyinordertomassage
statementsintoadesiredform.
• Deviseandattemptmultipledifferent,appropriate
strategiesforprovingapropositionallogic
statementfollowsfromalistofpremises.
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CPSC121:theBIG questions:
• Howcanweconvinceourselvesthatanalgorithm
doeswhatit'ssupposedtodo?
• Weneedtoprovethatitworks.
• Wehavedoneafewproofsinthelastweekorso.
• Nowwewilllearn
• Howtodecideifaproofisvalidinaformalsetting.
• (soon)HowtowriteproofsinEnglish.
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Module4 outline
• Proofsandtheirmeaning.
• Apropositionallogicproof.
• Theonnagata problem.
• Furtherexercises.
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Whatisaproof?
• Arigorousformalargumentthatdemonstrates
thetruthofaproposition,giventhetruthofthe
proof’spremises.
• Aproofisusedtoconvinceotherpeople(or
yourself)ofthetruthofaconditionalproposition.
• Writingaproof:
• Youdoitstepbystep.
• Makesurethatyoujustifyhoweachsteprelatesto
theprevioussteps.
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Thingswemightprove
• Wecanbuildacombinationalcircuitmatchingany
truthtable.
• Wecanbuildanycombinationallogiccircuitusing
only2-inputNORgates.
• Themaximumnumberofswapsweneedtoorder
nstudentsisn(n-1)/2.
• Nogeneralalgorithmexiststosortnvaluesusing
fewerthannlog2ncomparisons.
• Thereareproblemsthatnoalgorithmcansolve.
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Reviewquestions
• Howdoweuseatruthtabletoprovethatan
argumentisvalid?
• Ifapremiseisp->q,whataresomepossibleways
thatIcantransformthispremiseusingrulesof
inferenceorlogicalequivalencelaws?
• Ifp^qistrue,whatdoIknowaboutpandq?
• Ifpvqistrueandqisfalse,whatdoIknowabout
p?
Whatdoesitmeanforanargumenttobe
valid?
Supposethatyouproved
this:
Premise 1
...
Premise n
\ Conclusion
Doesitmean:
a. Ifallofthepremisesaretrue,
thentheconclusionistrue.
b. Theimplicationpremise1^…
^premisen->conclusionis
true.
c. Theimplicationpremise1^…
^premisen->conclusionisa
tautology.
d. 2of(a),(b),and(c)aretrue.
e. Allof(a),(b),and(c)aretrue.
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Whatdoesitmeanforanargumenttobe
valid?
Supposethatyouproved
this:
Premise 1
...
Premise n
\ Conclusion
Doesitmean:
a. Ifallofthepremisesaretrue,
thentheconclusionistrue.
b. Theimplicationpremise1^…
^premisen->conclusionis
true.
c. Theimplicationpremise1^…
^premisen->conclusionisa
tautology.
d. 2of(a),(b),and(c)aretrue.
e. Allof(a),(b),and(c)aretrue.
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Whatdoesitmeanforanargumenttobe
valid?
Supposethatyouproved
this:
Premise 1
...
Premise n
\ Conclusion
Doesitmean:
a. Allofthepremisesaretrue.
b. Theconclusionistrue.
c. Both(a)and(b)aretrue.
d. Neitherof(a)and(b)istrue.
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Whatdoesitmeanforanargumenttobe
valid?
Supposethatyouproved
this:
Premise 1
...
Premise n
\ Conclusion
Doesitmean:
a. Allofthepremisesaretrue.
b. Theconclusionistrue.
c. Both(a)and(b)aretrue.
d. Neitherof(a)and(b)istrue.
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Whatdoesitmeanforanargumenttobe
valid?
Supposethatyouproved
this:
Premise 1
...
Premise n
\ Conclusion
Doesitmean:
a. Itispossibleforthepremises
tocontradicteachother.
b. Itispossibleforthe
conclusiontobea
contradiction.
c. Both(a)and(b)aretrue.
d. Neitherof(a)and(b)istrue.
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Whatdoesitmeanforanargumenttobe
valid?
Supposethatyouproved
this:
Premise 1
...
Premise n
\ Conclusion
Doesitmean:
a. Itispossibleforthepremises
tocontradicteachother.
b. Itispossibleforthe
conclusiontobea
contradiction.
c. Both(a)and(b)aretrue.
d. Neitherof(a)and(b)istrue.
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Whatdoesitmeanforanargument
tobevalid?
• Ifanargumentisvalid,
• Weknowthatifallthepremisesaretrue,then
theconclusionistrue.
• Wedonotknow
• Whetheranypremiseistrue, or
• Whethertheconclusionistrue.
Module4 outline
• Proofsandtheirmeaning.
• Apropositionallogicproof.
• Theonnagata problem.
• Furtherexercises.
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Apropositionallogicproof
• Apropositionallogicproofisasequenceof
propositions,whereeachpropositionisoneof
• Apremise
• Theresultofapplyingalogicalequivalenceora
ruleofinferencetooneormoreearlier
propositions.
• andwhoselastpropositionistheconclusion.
• Thesearegoodstartingpoint,becausetheyare
simplerthanthemorefree-formproofswewill
discusslater
• Onlyalimitednumberofchoicesateachstep.
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A propositionallogicproof
Prove that the following argument is valid:
1 ~(𝑞 ∨ 𝑟)
2 (𝑢 ∧ 𝑞) ⟷ 𝑠
3 ~𝑠 → ~𝑝
\ ~𝑝
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Proofstrategies
• Workbackwardsfromtheend
• Playwithalternateformsofpremises
• Identifyandeliminateirrelevantinformation
• Identifyandfocusoncriticalinformation
• Step back from the problem frequently to think about
assumptions you might have wrong or other
approachesyou could take.
• Ifyoudon’tknowwhethertheargumentisvalidor
not,alternatebetween
• tryingtoproveit,and
• tryingtodisproveitbyfindingacounterexample.
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Limitationsoftruthtables
Whycanwenotjustusetruthtablestoprove
propositionallogictheorems?
a. Noreason;truthtablesareenough.
b. Truthtablesscalepoorlytolargeproblems.
c. Rulesofinferencecanprovetheoremsthat
cannotbeprovenwithtruthtables.
d. Truthtablesrequireinsighttouse,whilerulesof
inferencecanbeappliedmechanically.
▷
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Limitationsoflogicalequivalences
Whynotuselogicalequivalencestoprovethatthe
conclusionsfollowfromthepremises?
a. Noreason;logicalequivalencesareenough.
b. Logicalequivalencesscalepoorlytolarge
problems.
c. Rulesofinferencecanprovetheoremsthat
cannotbeprovenwithlogicalequivalences.
d. Logicalequivalencesrequireinsighttouse,while
rulesofinferencecanbeappliedmechanically.
▷
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Module4 outline
• Proofsandtheirmeaning.
• Apropositionallogicproof.
• Theonnagata problem.
• Furtherexercises.
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Theonnagata problem
• Critiquethefollowingargument,drawnfroman
articlebyJulianBaggini onlogicalfallacies.
• Premise1:Ifwomenaretooclosetofemininityto
portraywomenthenmenmustbetoocloseto
masculinitytoplaymen,andviceversa.
• Premise2:Andyet,iftheonnagata arecorrect,
womenaretooclosetofemininitytoportraywomen
andyetmenarenottooclosetomasculinitytoplay
men.
• Conclusion:Therefore,theonnagata areincorrect,and
womenarenottooclosetofemininitytoportray
women.
• Note:onnagata aremaleactorsportrayingfemale
charactersinkabukitheatre.
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Whichdefinitionsshouldweuse?
Onnagata:whichdefinitionsshouldweuse?
a.w = women, m = men, f = femininity, m = masculinity, o =
onnagata, c = correct
b.w = women are too close to femininity, m = men are too
close to masculinity, pw = women portray women, pm =
men portray men, o = onnagata are correct
c.w = women are too close to femininity to portray women,
m = men are too close to masculinity to portray men, o =
onnagata are correct
d.None of these, but another set of definitions works well.
e.None of these, and this problem cannot be modeled well
with propositional logic.
▷
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TranslatingEnglishintopropositional
logicexpressions
• Premise1:Ifwomenaretooclosetofemininityto
portraywomenthenmenmustbetoocloseto
masculinitytoplaymen,andviceversa.
• Premise2:Andyet,iftheonnagata arecorrect,
womenaretooclosetofemininitytoportray
womenandyetmenarenottoocloseto
masculinitytoplaymen.
• Conclusion:Therefore,theonnagata areincorrect,
andwomenarenottooclosetofemininityto
portraywomen.
▷
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Dothepremisescontradicteachother?
• Dothetwopremisescontradicteachother(that
is,isp1^p2alwaysfalse)?
a. Yes
b. No
c. Notenoughinformationtotell
▷
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Whatcanweprove?
• WecanprovethattheOnnagata arewrong.
• Wecannotprovethatwomenarenottooclose
tofemininitytoportraywomen.
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Onelastquestion
Consider the following:
Alice is rich
If Alice is rich then she will pay your tuition
\ Alice will pay your tuition.
Is this argument valid?
Should you pay your tuition, or should you
assume that Alice will pay it for you? Why?
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Module4 outline
• Proofsandtheirmeaning.
• Apropositionallogicproof.
• Theonnagata problem.
• Furtherexercises.
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Module4.3:Furtherexercises
Provethatthefollowingargumentisvalid:
p®q
q ® (r ^ s)
~r v (~t v u)
p^t
\u
Given the following, what is everything you can prove?
p®q
p v ~q v r
(r ^ ~p) v s v ~p
~r
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Module4.3:Furtherexercises
Furtherexercises
Hercule Poirot has been asked by Lord Maabo to find
out who closed the lid of his piano after dumping the
cat inside. Poirot interrogates two of the servants,
Pearrh and Dr. Utuae. One and only one of them put
the cat in the piano. Plus, one always lies and one
never lies.
Dr. Utuae: I did not put the cat in the piano. Tgahaa
gave me less than $60 to help her study.
Pearrh: Dr. Utuae did it. Tgahaa paid him $50 to
help her study.
Who put the cat in the piano?
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