Intensional First-Order Logic. From AI to New SQL Big Data

542 Pages·2022·7.574 MB·English

by  Zoran Majkić

ZoranMajkić
IntensionalFirst-OrderLogic
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Zoran Majkić
Intensional
First-Order Logic
|
From AI to New SQL Big Data
Author
ZoranMajkić
ViaPalestro13
00185Rome
Italy
[email protected]
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e-ISBN(EPUB)978-3-11-098146-9
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|
Dedicatedtomydaughters
VivianaandSofia
Calypso,thebeautifulgoddess,wasthefirsttospeak,andsaid:“SonofLaertes,sprungfrom
Zeus,Odysseusofmanydevices,would’stthouthenfarenowforthwithhometothydearnative
land!Yet,evensofaretheewell.Howbeitifinthyheartthouknewestallthemeasureofwoeit
isthyfatetofulfilbeforethoucomesttothynativelandthouwouldestabidehereandkeepthis
housewithme,andwouldestbeimmortal,forallthydesiretoseethywifeforwhomthoulongest
daybyday.SurelynotinferiortoherdoIdeclaremyselftobeeitherinformorstature,forinno
wiseisitseemlythatmortalwomenshouldviewithimmortalsinformorcomeliness.”
ThenOdysseusofmanywilesansweredher,andsaid:“Mightygoddess,benotwrothwithmefor
this.IknowfullwellofmyselfthatwisePenelopeismeanertolookuponthanthouincomeliness
andinstature,forsheisamortal,whilethouartimmortalandageless.ButevensoIwishandlong
daybydaytoreachmyhome,andtoseethedayofmyreturn.Andifagainsomegodshallsmite
meonthewine-darksea,Iwillendureit,havinginmybreastaheartthatenduresaffliction.For
erethisIhavesufferedmuchandtoiledmuchamidthewavesandinwar;letthisalsobeadded
untothat.”
Sohespoke,andthesunsetanddarknesscameon.
TheOdyssey,book5,byHomer.

Preface
In“ÜberSinnundBedeutung,”Fregeconcentratedmostlyonthesensesofnames,
holding that all names have a sense (meaning). It is natural to hold that the same
considerationsapplytoanyexpressionthathasanextension.Buttwogeneralterms
canhavethesameextensionanddifferentcognitivesignificance.So,generalterms,
predicates,andsentencesallhavesensesaswellasextensions.Thesamegoesforany
expressionthathasanextension,orisacandidateforextension.
Thesignificantaspectofanexpression’smeaningisitsextension.Wecanstip-
ulatethattheextensionofasentenceisitstruth-value,andthattheextensionofa
singulartermisitsreferent.Theextensionofotherexpressionscanbeseenasasso-
ciated entities that contribute to the truth-value of a sentence in a manner broadly
analogoustothewayinwhichthereferentofasingulartermcontributestothetruth-
valueofasentence.Inmanycases,theextensionofanexpressionwillbewhatwe
intuitivelythinkofasitsreferent,althoughthisneednotholdinallcases.WhileFrege
himselfisofteninterpretedasholdingthatasentence’sreferentisitstruth-value,this
claimiscounterintuitiveandwidelydisputed.Wecanavoidthatissueinthepresent
frameworkbyusingthetechnicalterm“extension.”Inthiscontext,theclaimthatthe
extensionofasentenceisitstruth-valueisastipulation.
“Extensional”ismostdefinitelyatechnicalterm.Saythattheextensionofaname
isitsdenotation,theextensionofapredicateisthesetofthingsitappliesto,andthe
extensionofasentenceisitstruthvalue.Alogicisextensionalifcoextensionalexpres-
sionscanbesubstitutedoneforanotherinanysentenceofthelogic“salvaveritate,”
thatis,withoutachangeintruthvalue.Theintuitiveideabehindthisprincipleisthat
inanextensionallogictheonlylogicallysignificantnotionofmeaningthatattaches
toanexpressionisitsextension.Anintensionallogicsisexactlyoneinwhichsubsti-
tutivitysalvaveritatefailsforsomeofthesentencesofthelogic.
Thefirstconceptionofintensionalentities(orconcepts)isbuiltintothepossible-
worlds treatment of Properties, Relations and Propositions (PRP)s. This conception
iscommonlyattributedtoLeibniz,andunderliesAlonzoChurch’salternativeformu-
lationofFrege’stheoryofsenses(“AformulationoftheLogicofSenseandDenota-
tion”inHenle,KallenandLanger,3–24,and“OutlineofaRevisedFormulationofthe
LogicofSenseandDenotation”intwoparts,Nous,VII(1973),24–33,andVIII,(1974),
135–156).ThisconceptionofPRPsisideallysuitedfortreatingthemodalities(neces-
sity,possibility,etc.)andtoMontague’sdefinitionofintensionofagivenvirtualpred-
icateϕ(x1,...,xk)(aFOLopensentencewiththetupleoffreevariables(x1,...xk)),as
amappingfrompossibleworldsintoextensionsofthisvirtualpredicate.Amongthe
possible worlds, we distinguish the actual possible world. For example, if we con-
siderasetofpredicates,ofagivendatabase,andtheirextensionsindifferenttime-
instances,thentheactualpossibleworldisidentifiedbythecurrentinstanceofthe
time.
https://doi.org/10.1515/9783110981438-201
VIII | Preface
ThesecondconceptionofintensionalentitiesistobefoundinRussell’sdoctrine
oflogicalatomism.Inthisdoctrine,itisrequiredthatallcompletedefinitionsofin-
tensional entities be finite as well as unique and noncircular: it offers an algebraic
wayfordefinitionofcomplexintensionalentitiesfromsimple(atomic)entities(i.e.,
algebraofconcepts),conceptionalsoevidentinLeibniz’sremarks.Inapredicatelog-
ics,predicatesandopen-sentences(withfreevariables)expressesclasses(properties
andrelations),andsentencesexpresspropositions.Notethatclasses(intensionalen-
tities)arereified,i.e.,theybelongtothesamedomainasindividualobjects(partic-
ulars).Thisendowstheintensionallogicswithagreatdealofuniformity,makingit
possibletomanipulateclassesandindividualobjectsinthesamelanguage.Inpar-
ticular, when viewed as an individual object, a class can be a member of another
class.
The distinction between intensions and extensions is important (as in lexicog-
raphy [1]), considering that extensions can be notoriously difficult to handle in an
efficientmanner.Theextensionalequalitytheoryofpredicatesandfunctionsunder
higher-ordersemantics(e.g.,fortwopredicateswiththesamesetofattributesp=qis
trueiffthesesymbolsareinterpretedbythesamerelation),i.e.,thestrongequational
theoryofintensions,isnotdecidable,ingeneral.Forexample,thesecond-orderpred-
icatecalculusandChurch’ssimpletheoryoftypes,bothunderthestandardseman-
tics,arenotevensemidecidable.Thus,separatingintensionsfromextensionsmakesit
possibletohaveanequationaltheoryoverpredicateandfunctionnames(intensions)
thatisseparatefromtheextensionalequalityofrelationsandfunctions.
Relevantrecentworkabouttheintension,anditsrelationshipwithFOL,hasbeen
presentedin[2]intheconsiderationofrigidandnonrigidobjects,w.r.t.thepossible
worlds,wheretherigidobjects,like“GeorgeWashington,”andarethesamethings
frompossibleworldtopossibleworld.Nonrigidobjects,like“theSecretary-Generalof
UnitedNations,”arevaryingfromcircumstancetocircumstanceandcanbemodeled
semantically by functions from possible worlds to the domain of rigid objects, like
intensionalentities.
Anotherapproachusedinintensionallogicprogrammingisanewformoflogic
programmingbasedonintensionallogicandpossibleworldssemantics,andisawell-
definedpracticeinusingtheintensionalsemantics[3].Intensionallogicallowsusto
uselogicprogrammingtospecifynonterminatingcomputationsandtocapturethe
dynamic aspects of certain problems in a natural and problem-oriented style. The
meanings of formulas of an intensional first-order language are given according to
intensionalinterpretationsandtoelementsofasetofpossibleworlds.Neighborhood
semanticsisemployedasanabstractformulationofthedenotationsofintensionalop-
erators.Themodel-theoreticandfixed-pointsemanticsofintensionallogicprograms
aredevelopedintermsofleast(minimum)intensionalHerbrandmodels.Intensional
logicprogramswithintensionaloperatordefinitionsareregardedasmetatheories.
Some of the important questions about intensional First-order Logic (FOL) was
enouncedbyMelvinFittinginhis2003preprint[2]:
Preface | IX
“Whatisfirst-ordermodallogicfor?Sincethisisobviouslynotasimplequestion;perhapsweshould
beginbyasking,whatispropositionalmodallogicfor?Hereweareonwell-exploredground.With
propositionalmodallogic,anditsrelationalsemantics,wewanttoexplicatevariousconstructs
fromnaturallanguage,andexplorenuancesofcertainconceptsarisinginphilosophicalinvestiga-
tions.Wewanttomodelknowledge,atleastinanidealsense.Wewanttoreasonaboutaction.And
thereisanotherpurposeaswell,onethathasbecomeclearerovertheyears.Instudyingproposi-
tionalmodallogics,–primarilythosecharacterizedbyclassesofframes,–wearealsostudying
fragmentsofclassicalfirst-order(andhigher-order)logic.Thisisknownascorrespondencetheory.
Forthispurposeaxiomatizability(ornot)isacentralissue.Inaddition,axiomsystemsallowthe
constructionofcanonicalmodels,whichprovidesametamathematicalmethodologythatisuniform
acrossmanylogics.Detailsmatteragreatdeal,ofcourse,butthebroadoutlinesofpropositional
modallogicshavebeenstandardizedforsometime.
Buttheoriginalquestionabovewas,whatisfirst-ordermodallogicfor?Whatdoquantifiersaddto
themix?”
Butinhisapproach,differentlyfromthisone,Fittingchangesalsothesyntaxofthe
FOL,byintroducingan“extensionof”theoperator,↓,inordertodistinguishtheinten-
sionalentity“grossdomesticproductofDenmark,”anditsusein“thegrossdomestic
productofDenmarkiscurrentlygreaterthangrossdomesticproductofFinland.”Inhis
approach,ifxisanintensionalvariable,↓ xisextensional,while↓isnotapplicable
toextensionalvariables,differentfromours,whereeachvariable(concept)hasboth
intensionalandextension.Moreover,inhisapproachtheproblemarisesbecausethe
actionoflettingxdesignate,i.e.,evaluating↓x,andtheactionofpassingtoanalter-
nativepossibleworld,thatisofinterpretingtheexistentialmodaloperator⬦,arenot
actionsthatcommute.Todisambiguatethis,onemorepieceofmachineryisneeded
aswell,whichsubstantiallyandadhocchangesthesyntaxandsemanticsofFOL,in-
troducesthehigher-ordermodallogics,andisnotaconservativeextensionofTarski’s
semantics.Inthemostrecentworkin[4,5]isgivenanintensionalversionoffirst-order
hybridlogic,whichisalsoahybridizedversionofFitting’sintensionalFOL,byakind
ofgeneralizedmodels;thus,isdifferentfromourapproachtoaconservativeextension
ofTarski’ssemanticstointensionalFOL.
AnotherrecentrelevantworkispresentedbyI-logicin[6],whichcombinesboth
approachestosemanticsofintensionalobjectsofMontagueandFitting.
Inhisapproach,FittingfollowedtheMontaguetradition,differentfrommywider
approach provided in this book that uses both Montague tradition and algebraic
Bealer’s approach, and enriched them by a more detailed investigation of modal
logicsofFOL,andbythewayintroducednaturallytheintensionalsemanticsintotra-
ditionalextensionalFOLwithTarski’ssemanticsand,moreover,howtointroducein
suchminimalintensionalFOLandthe“higherlevel”modaloperators.Mostrelevant
formypersonalresearchhasbeentwosignificantapproachestointensionalFOL:
1. Montague’sapproachbasedonpossibleworldrepresentation[7,8,9,10,11],the
intensionofapropositionisafunctionfrompossibleworlds𝒲totruth-values,and
propertiesandfunctionsfrom𝒲tosetsofpossible(usuallynotactual)objects.
X | Preface
2. Bealer’sapproachin[12]totheintensionallogicthefundamentalentitiesarein-
tensionalabstractsorso-called,“that-clauses”andhisintroductionofintensional
algebras.
Thesetwoapproachesareunifiedinmyapproach,withprovidingaconservativeex-
tensionofTarski’ssemanticstointensionalFOLaswell.
Quovadislogic-basedAI?
Why this title? I will try to explain it by my relevant research history from 2003 to
2020. At “La Sapienza,” Roma, Italy, we had a number of good professors and re-
searchesinlogic-basedAI,andespeciallyintheresearchgroupofmyPhDadvisor,
ProfessorM.Lenzerini(from94to98),whointhatperiodwaschiefofthePhDpro-
gramsandresearchinknowledgebasisandAI.Inthatperiod,thebook[13]ofthelogic
basedapproachtoAIwasareferenceformyintroductiontothisfield,andmybook
insomewayisacontinuationofthatapproach.Ireturnedagaintohisdepartmentin
2002,4yearsafterIfinishedmyPhDthesis,toworkfortheEuropeaninteruniversity
projectofSemanticWeb,calledSEWASIEprojectIST-2001-34825.Ihavewritten,with
initialhelpofLenzerini,threeorfourresearchpapersforthisEuropeanresearchpro-
gram.Inthatperiod,Lenzeriniwasthechiefresearchleaderindataintegrationand
itwasalsotheverybeginningoftheP2Pdataintegrationsystems,basedonthefirst-
orderlogicanditssecond-orderextensionsfortheinterdatabasemappings.Thus,the
wholeframeworkwasjustinstandardextensionallogics,whichdemonstratedalotof
theoreticalproblemsaboutmutuallyinconsistentinformationcomingfromdifferent
sources.
Ofcourse,allmyworkinthisprojectwasjustinthisworkinglogicframework,but
Itriedtoconsidermany-valuedlogicstoovercometheseproblemsabouttheinconsis-
tencesandtofindamorerobustdataintegrationP2Psystem,tryingtoovercomethe
strong(extensional)mappingbetweendatabases.WhenIorganizedafirstshortpaper
withthesenewideas,Lenzeriniacceptedtojoinwithmetowritethefinalversion,but
fromthefactthathepublishedpaperswithhisworkinggroupofotherprofessorsand
researches,heaskedmetowaitonthedecisionofthisgroupiftheywouldacceptme
asnewmemberofthisgroup,andhenceauthorizedtopublishtheresearchpapersfor
journalsorconferencestogether.
Unfortunatelyforme,somebodyinhisgroupdidnotlikemyparticipationand
Lenzerini was sorry, but promised that if my papers were accepted for the confer-
encesorjournals,thatdepartmentwouldsupportallexpenses(traveling,participa-
tioncosts,etc.)forpresentations.Theirdecisionexplainswhy,duringthe3yearswork-
ingwithhisresearchgroupforthisEuropeanproject,Ihavenoanypublicationwith
theminjournalsorconferences,andwhyallsuchworkwasdonebymeonly.With
economicalsupportbythedepartmentofLenzerini,Ipublishedadozenpapersby