Table Of ContentOXFORDLOGICGUIDES
SeriesEditors
A.J.MACINTYRE A.J.WILKIE D.S.SCOTT
EmeritusEditors
D.M.GABBAY
JOHNSHEPHERDSON
OXFORDLOGICGUIDES
Forafulllistoftitlespleasevisit
http://www.oup.co.uk/academic/science/maths/series/OLG/
21. C.McLarty:ElementaryCategories,ElementaryToposes
22. R.M.Smullyan:RecursionTheoryforMetamathematics
23. PeterCloteandJanKrajícek:Arithmetic,ProofTheory,andComputationalComplexity
24. A.Tarski:IntroductiontoLogicandtotheMethodologyofDeductiveSciences
25. G.Malinowski:ManyValuedLogics
26. AlexandreBorovikandAliNesin:GroupsofFiniteMorleyRank
27. R.M.Smullyan:DiagonalizationandSelf-Reference
28. Dov M. Gabbay, Ian Hodkinson, and Mark Reynolds: Temporal Logic: Mathematical Foundations and
ComputationalAspects,Volume1
29. SaharonShelah:CardinalArithmetic
30. ErikSandewall:FeaturesandFluents,VolumeI:ASystematicApproachtotheRepresentationofKnowledgeabout
DynamicalSystems
31. T.E.Forster:SetTheorywithaUniversalSet:ExploringanUntypedUniverse,SecondEdition
32. AnandPillay:GeometricStabilityTheory
33. DovM.Gabbay:LabelledDeductiveSystems
34. R.M.SmullyanandM.Fitting:SetTheoryandtheContinuumProblem
35. AlexanderChagrovandMichaelZakharyaschev:ModalLogic
36. G.SambinandJ.Smith:Twenty-FiveYearsofMartin-LöfConstructiveTypeTheory
37. MaríaManzano:ModelTheory
38. DovM.Gabbay:FibringLogics
39. MichaelDummett:ElementsofIntuitionism,SecondEdition
40. D.M.Gabbay,M.A.ReynoldsandM.Finger:TemporalLogic:MathematicalFoundationsandComputational
Aspects,Volume2
41. J.M.DunnandG.Hardegree:AlgebraicMethodsinPhilosophicalLogic
42. H.Rott:Change,ChoiceandInference:AStudyofBeliefRevisionandNonmonotoicReasoning
43. Johnstone:SketchesofanElephant:AToposTheoryCompendium,Volume1
44. Johnstone:SketchesofanElephant:AToposTheoryCompendium,Volume2
45. DavidJ.PymAndEikeRitter:ReductiveLogicandProofSearch:ProofTheory,SemanticsandControl
46. D.M.GabbayAndL.Maksimova:InterpolationandDefinability:ModalandIntuitionisticLogics
47. JohnL.Bell:SetTheory:Boolean-ValuedModelsandIndependenceProofs,ThirdEdition
48. Laura Crosilla and Peter Schuster: From Sets And Types to Topology and Analysis: Towards Practicable
FoundationsforConstructiveMathematics
49. SteveAwodey:CategoryTheory
50. RomanKossakandJamesSchmerl:TheStructureofModelsofPeanoArithmetic
51. AndréNies:ComputabilityandRandomness
52. SteveAwodey:CategoryTheory,SecondEdition
53. ByunghanKim:SimplicityTheory
54. EttoreCasari:Bolzano’sLogicalSystem
Bolzano’s Logical System
ETTORE CASARI
3
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WilhelmAckermann,GisbertHasenjaeger
PREFACE
1.StudiesconcerningBernardBolzano’sthoughthaveseenremarkablegrowthin
recentdecades.Asisevidentevenbysimplyconsideringthecatalogueoftheexcel-
lentseriesoftheBeiträgezurBolzano-Forschung,suchstudieshaveexaminedallof
themultifacetedaspectsofhisthought:fromlogicandmathematicstoontology,
epistemology,andphilosophyoflanguage;frommoralphilosophytophilosophy
ofreligion,theology,politicalandsocialphilosophy,andaesthetics.Inaddition,a
greatdealofattentionhasbeendevotedtotheprocessesthroughwhichhisthought
matured.
Certainly,animportantmotivationfortherecentinterestinBolzano’sworkis
thelarge-scaleprojecttoproduceacompleteeditionofhisworks.Thissubstantial
enterprisebeganattheendofthe1960sandhasalreadyreachedsome90ofthe
roughly130volumesplannedintotal.
While such scholarly interest has succeeded in bringing to light Bolzanian
thoughtasawhole,aswellascontinuingtoprovidenewinsightsintohismany-
sidedwork,thereisnodoubtthatthemostimportantaspectofhisthoughtwas
that concerned with logical problems. The wide scope of research carried out
on this aspect of Bolzano’s work, especially after the publication in 1962 of Jan
Berg’sBolzano’sLogic(stillafundamentaltextforthissubject),confirmsHusserl’s
judgementthat:“wemustcounthimasoneofthegreatestlogiciansofalltime”.
One of the foundations for this assessment no doubt is Bolzano’s monumental
Wissenschaftslehreof1837.Inthiswork,especiallythesecondandthirdparts,we
can appreciate the mature core of his reflection on logic in its broadest sense,
includingthetheoryofknowledge.
Asiswellknown,theWissenschaftslehreisdividedintofiveparts.Thefirstpart,
‘TheoryofFundamentals’(Fundamentallehre)focusesontheconceptualbasisof
thelogicalsystemanditsjustification. Itdoesthis,ontheonehand,byintrodu-
cingthefundamentaldistinctionbetweenthesphereoflogicalobjectivity(truthsin
themselvesand,moregenerally,propositionsinthemselves)andthesphereoflo-
gicalsubjectivity(judgementsandknowledge),and,ontheotherhand,byrefuting
the respective sceptical objections, by firstly demonstrating that truths in them-
selves,evenaninfinityofthem,reallyarethere,and,secondly,byarguingthattruths
ofthiskindcanalsoberecognized.
viii | PREFACE
Thesecondpart,the‘TheoryofElements’(Elementarlehre),developsthethe-
oryoflogicalobjectivity,firstofallbyanalysingobjectiverepresentationsandthen
gradually propositions in themselves, followed by truths in themselves, and con-
cluding with logical truths. Husserl observed that this part of Bolzano’s writing
represents“aworkwhich,inhistreatmentofthelogical‘theoryofelements’,far
surpasses everything that world literature has to offer in the way of a systematic
sketchoflogic”.
The third part of the Wissenschaftslehre, the ‘Theory of Knowledge’
(Erkenntnißlehre), on the other hand, develops the theory of logical subjectivity,
discussingfirstofallsubjectiverepresentations,thenjudgements,thenthetruthof
judgements,and,finally,thetrustweplaceinourjudgements.
The last two parts, which deserve far greater attention than has been given
themtodate,bycontrast,haveanexplicitlydidacticaim.Thefourthsection,the
‘Heuristics’ (Erfindungskunst), aims to demonstrate “how one must proceed in
the search for truth” (as far as this may depend on logic). The fifth part of his
work,instead,exploreswhatBolzanoregardedasthe‘ProperTheoryofScience’
(EigentlicheWissenschaftlehre),thatis,thepresentationofthe“generalrulesaccord-
ingtowhichonemustproceedinthedeterminationofthedomainsofthevarious
sciencesaswellasintheworkingoutofsuchsciencesintreatises”.
2.Asalreadymentioned,manyspecialiststudieshaveanalysedmanyoftheques-
tions that arise from the first three parts of Bolzano’s work, as well as providing
comprehensive expositions of them, often very successfully. Yet, it seems to us
thatthereremainsroomforamoresystematicreconsiderationofBolzano’slogical
thought.
Thisbookisconcernedpreciselywiththisaim.Inundertakingthistask,thebook
isintendedasanexploration,notsomuchofthemorespecificallydiscursiveas-
pectsofBolzano’slogicalthought—alreadyamplystudied—asmuchasoneaimed
atidentifyingthesingularlycoherentandsystematicnatureofthelogicpresented
intheWissenschaftslehre.
Inordertorenderasvisibleaspossiblethesystematicnatureofthatlogic,Ihave
decidedtopresentitwithinaformalsystem.Despitebeingsurprisingeventome,it
hasbecomeclearthatinpursuingthisaim,itissufficienttoadopttheapproachof
thepredicatecalculuswithidentityandchoiceoperator,thatis,enlistingthewell-
knownHilbert’sepsiloncalculus.Asthisbookreveals,theformalizationofBolzano’s
logicinthiscalculusemergesquiteeffortlessly.
3.Asisnormalforareconstructiveworkofthisnature—aprocessthatIbegan
some30yearsago—thestateofscholarshiphasevolvedsuchastoenablebetter
accesstoanumberofrelevantquestions.Afewparticularlysignificantexamplesof
thisdevelopmentareinorderhere.
PREFACE | ix
ThefirstrelatestoaremarkthatImadeinmyearlieststudyofBolzanomany
yearsago,whichatthetimegaverisetosomeobjections,butnowseemstobeac-
ceptedgenerally:Iarguedthattheobjectiverepresentationsofqualitymaybenot
onlyemptyorsingular,astraditionallyagreed,butalsoplural,thatis,referringto
morethanonequality.ArelevantconsequenceofthisobservationisthatBolzano’s
authenticdefinitionofthetruthofapropositionoftheformAhasB,resultsas:each
objectfallingunderAenjoysatleastoneofthequalitiesfallingunderB.Another
importantconsequenceisthatinthiswayBolzano’sextraordinaryinsightintothe
strictparallelismbetweenthenon-emptinessofanobjectiverepresentationandthe
truthofapropositioninitselfbecomesmuchclearer.
Another relevant example here concerns Bolzano’s view on the interpretation
of the logical complexity of propositions. We are accustomed to analysing this
complexityintermsofoperations:asoperationsonpropositionsorpropositional
schemata (as in the usual logics of the Fregean tradition) or, alternatively, as op-
erations on concepts (as in the logics of the algebraic tradition: for example, in
Quine’sfunctoriallogic).Bolzano,onthecontrary,analysesthisconceptintermsof
semanticpredication:emptinessornon-emptinessofanobjectiverepresentation;
truthorfalsityofapropositioninitself.Abriefexampleillustratesthispointmore
clearly:whatweanalyseasapropositionobtainedfromthetwopropositionspand
qbymeansoftheoperationofdisjunctionisseenbyBolzanoastheassertionof
non-emptinessoftherepresentationofatruepropositionbetweenpandq.Among
the main consequences of accepting this daring and singular viewpoint is that it
ispossibletounderstandthereasonofBolzano’sprimafaciesurprisingconviction
thatallpropositionshavetheformAhasB,oratleastareequivalenttopropositions
ofthisform.
AnotherobservationworthyofmentionconcernsapointthatBolzanoraises—
while not explicitly in the Wissenschaftslehre, quite clearly in his Athanasia—
wherebyheoutlinesaparticularlyinterestingtreatmentofthosesingularqualities
thattodayoftenarecalled‘tropes’;indeed,heconsiderstropesoftropes,tropesof
tropesoftropes,andsoon,anddiscussesreductionproceduresinrelationtothem.
Finally,itseemsappositetomentiontheapparentlyignoredfactthatBolzanode-
velopedatheoryofsigns,whichincludes,inparticular,atheoryoflinguisticsigns,
whereheprovides,withgreataccuracy,thefirstformulationinthemodernageof
thedistinctionbetweensenseandreference.
4.Inaccordancewiththekeythemestobeexploredinthisstudy,thebookis
dividedintoninechapters.
The first chapter defines the Bolzanian logical universe as being inhabited by
objects and qualities, and introduces the basic concepts that determine its struc-
ture: belonging; part, variation; existence; objective representation (abbreviated to
