Table Of ContentPerspectives in Formal Induction, Revision and Evolution
Wei Li · Yuefei Sui
R-Calculus,
II: Many-Valued
Logics
Perspectives in Formal Induction, Revision and
Evolution
Editor-in-Chief
WeiLi,Beijing,China
SeriesEditors
JieLuo,Beijing,China
YuefeiSui,InstituteofComputingTechnology,ChineseAcademyofSciences,
Beijing,China
PerspectivesinFormalInduction,RevisionandEvolutionisabookseriesfocusing
onthelogicsusedincomputerscienceandartificialintelligence,includingbutnot
limited to formal induction, revision and evolution. It covers the fields of formal
representation, deduction, and theories or meta-theories of induction, revision and
evolution,wheretheinductionisofthefirstlevel,therevisionisofthesecondlevel,
andtheevolutionisofthethirdlevel,sincetheinductionisattheformulastratum,
therevisionisatthetheorystratum,andtheevolutionisatthelogicstratum.
Inhisbook“TheLogicofScientificDiscovery”,KarlPopperarguesthatascien-
tificdiscoveryconsistsofconjecture,theory,refutation,andrevision.Somescientific
philosophersdonotbelievethatareasonableconjecturecancomefrominduction.
Hence, induction, revision and evolution have become a new territory for formal
exploration. Focusing on this challenge, the perspective of this book series differs
fromthatoftraditionallogics,whichconcernsconceptsanddeduction.
The series welcomes proposals for textbooks, research monographs, and edited
volumes,andwillbeusefulforallresearchers,graduatestudents,andprofessionals
interestedinthefield.
Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/16789
·
Wei Li Yuefei Sui
R-Calculus, II: Many-Valued
Logics
WeiLi YuefeiSui
BeihangUniversity InstituteofComputingTechnology
Beijing,China ChineseAcademyofSciences
Beijing,China
ISSN2731-3689 ISSN2731-3697 (electronic)
PerspectivesinFormalInduction,RevisionandEvolution
ISBN978-981-16-9293-2 ISBN978-981-16-9294-9 (eBook)
https://doi.org/10.1007/978-981-16-9294-9
JointlypublishedwithSciencePress,Beijing,China
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Preface to the Series
Classicalmathematicallogics(propositionallogic,first-orderlogicandmodallogic)
andappliedlogics(temporallogic,dynamiclogic,situationcalculus,etc.)concern
deduction, a logical process from universal statements to particular statements.
Descriptionlogicsconcernconceptswhichanddeductioncomposeofgenerallogics.
Induction and belief revision are the topics of general logics and philosoph-
icallogics,andevolutionisanewresearchareaincomputerscience.Toformalize
induction,revisionandevolutionisagoalofthisseries.
Revisionisomnipresentinsciences.Anewtheoryusuallyisarevisionofanold
theory or several old theories. Copernicus’ heliocentric theory is a revision of the
Tychonicsystem;thetheoryofrelativityisarevisionoftheclassicaltheoryofmove-
ment;thequantumtheoryisarevisionofclassicalmechanics;etc..TheAGMpostu-
lates is a set of conditions a reasonable revision operator should satisfy. Professor
Li proposed a calculus for first-order logic, called R-calculus, which is sound and
completewithrespecttomaximalconsistentsubsets.R-calculushasseveralvariants
which can be used in other logics, such as nonmonotonic logics, can propose new
problemsinclassicallogics,andwillbeusedinbigdata.
Popper proposed in his book The Logic of Scientific Discovery that a scien-
tificdiscoveryconsistsoffouraspects:conjecture,theory,refutation,andrevision.
Somescientificphilosophersrefutedthatareasonableconjectureshouldcomefrom
induction. Hence, induction, revision and evolution becomes a new territory to be
discoveredinaformalway.
An induction process is from particular statements to universal statements. A
typical example is the mathematical induction, which is a set of nontrivial axioms
in Peano arithmetic which makes Peano arithmetic incomplete with respect to the
standardmodelofPeanoarithmetic.Alogicforinductionisneededtoguidedata-
mininginartificialintelligence.Data-miningisacanonicalinductionprocess,which
minesrulesfromdata.
In biology, Evolution is change in the heritable traits of biological populations
oversuccessivegenerations.Insciences,Darwin’sevolutiontheoryisanevolutionof
intuitivetheoriesofplantsandanimals.Inlogic,anevolutionisageneratingprocess
ofcombiningtwologicsintoanewlogic,wherethenewlogicshouldhavethetraits
v
vi PrefacetotheSeries
oftwologics.Hence,weshoulddefinethecorrespondingheritabletraitsoflogics,
sets of logics, and sequences of logics. Simply speaking, predicate modal logic is
an evolution of propositional modal logic and predicate logic, and there are many
new problems in predicate modal logic to be solved, such as the constant domain
semantics,thevariantdomainsemantics,etc..
The series should focus on formal representation, deduction, theories or meta-
theoriesofinduction,revisionandevolution,wheretheinductionisofthefirst-level,
the revision is of the second-level, and the evolution is of the third level, because
theinductionisattheformulastratum,therevisionisatthetheorystratum,andthe
evolutionisatthelogicstratum.
The books in the series differ in level: some are overviews and some highly
specialized.Here,thebooksofoverviewsareforundergraduatedstudents;andthe
highlyspecializedonesareforgraduatestudentsandresearchersincomputerscience
andmathematicallogic.
Beijing,China LiWei
October2016 LuoJie
SongFangming
SuiYuefei
WangJu
ZhuWujia
Preface
R-calculusisaGentzen-typedeductionsystemthatisnonmonotonicandisaconcrete
beliefrevisionoperatorwhichisprovedtosatisfytheAGMpostulatesandtheDP
postulates. This book is to show that in many-valued logics, each R-calculus is a
combinationofonemonotonictableauproofsystemandonenonmonotonicone.
WeconsiderR-calculiinpropositionallogic,descriptionlogic,three-valuedlogic,
andB2-valuedpropositionallogic.Foreachvalue∗,
2
• thereisamonotonictableauproofsystemT∗ andanonmonotonictableauproof
systemS∗,whichcomposeofanR-calculusR∗,and
• thereisamonotonictableauproofsystemT∗ andanonmonotonictableauproof
systemS∗,whichcomposeofanR-calculusT∗.
Hence,eachR-calculusisnonmonotonic.
Wealsoconsidertheproductandsumoftwologics.Forproduct,wewillshow
that
(i) atthevalue-level,B2-valuedPListheproductoftwoB -valuedPL,
2 2
(ii) atthelogicallevel,eachtableauproofsysteminB2-valuedPListheproduct
2
oftwotableauproofsystemsinB -valuedPL,and
2
(iii) at the revision level, each R-calculus in B2-valued PL is the product of two
2
R-calculiinB -valuedPL.
2
Forsumwithoutcommonelement,wewillshowthat
(i) L -valuedPListhesumoftwoB -valuedPL,
3 2
(ii) each tableau proof system in L -valued PL is the sum of two tableau proof
3
systemsinB -valuedPL,and
2
(iii) eachR-calculusinL -valuedPListhesumoftwoR-calculiinB -valuedPL;
3 2
andforsumwithcommonelement,wewillshowthat
(iv) L -valuedPListhesumoftwoB -valuedPL,
4 2
(v) each tableau proof system in L -valued PL is the sum of two tableau proof
4
systemsinB -valuedPL,and
2
(vi) eachR-calculusinL -valuedPListhesumoftwoR-calculiinB -valuedPL.
4 2
vii
viii Preface
Weconsiderthealgebraicalandlogicalpropertiesoftableauproofsystemsand
R-calculiinmany-valuedlogics.
Wewishthatthecontentsinthisbookwillinspirenewideasinthelogicalresearch
ofthiscentury.
Beijing,China WeiLi
October2016 YuefeiSui
Contents
1 Introduction .................................................. 1
1.1 R-Calculus .............................................. 1
1.2 Many-ValuedLogics ...................................... 2
1.3 ContentsintheFirstVolume ............................... 3
1.4 ContentsinThisVolume .................................. 6
1.5 Notations ............................................... 10
References .................................................... 11
2 R-CalculusforPL ............................................. 13
2.1 BasicDefinitions ......................................... 14
2.2 MonotonicTableauProofSystems .......................... 15
2.2.1 TableauProofSystemTf ......................... 16
2.2.2 TableauProofSystemTt ......................... 18
2.3 NonmonotonicTableauProofSystems ...................... 20
2.3.1 TableauProofSystemSt ......................... 21
2.3.2 TableauProofSystemSf ......................... 23
2.4 R-Calculi ............................................... 24
2.4.1 R-CalculusRt .................................. 25
2.4.2 R-CalculusRf .................................. 29
2.5 ProjectingR-CalculitoTableauProofSystems ............... 30
2.6 Notes ................................................... 31
References .................................................... 31
3 R-CalculusforDescriptionLogic ............................... 33
3.1 BasicDefinitions ......................................... 34
3.2 MonotonicTableauProofSystems .......................... 36
3.2.1 TableauProofSystemT ......................... 36
t
3.2.2 TableauProofSystemT ......................... 39
f
3.3 NonmonotonicTableauProofSystems ...................... 40
3.3.1 TableauProofSystemSt ......................... 40
3.3.2 TableauProofSystemSf ......................... 43
3.4 R-Calculi ............................................... 44
ix
