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Bernard Bolzano

Copyright c 2009 by the authorEdgar Morscher

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Bernard BolzanoFirst published Thu Nov 8, 2007

Bernard Bolzano (17811848) was a Catholic priest, a professor of thedoctrine of Catholic religion at the Philosophical Faculty of the Universityof Prague, an outstanding mathematician and one of the greatest logiciansor even (as some would have it) the greatest logician who lived in thelong stretch of time between Leibniz and Frege. As far as logic isconcerned, Bolzano anticipated almost exactly 100 years before Tarskiand Carnap their semantic definitions of logical truth and logicalconsequence; and in mathematics he is not only known for his famousParadoxes of the Infinite, but also for certain results that have becomeand still are standard in textbooks of mathematics such as the Bolzano-Weierstrass theorem. Bolzano also made important contributions to otherfields of knowledge in and outside of philosophy. Due to the versatility ofhis talents and the various fields to which he made substantialcontributions he became one of the last great polymaths in the history ofideas.

The presentation of Bolzano's personality that is given here would still beincomplete unless it were added that Bolzano was also a greatphilanthropist. This becomes evident not only by Bolzano's writings inethics and political philosophy, but it also manifests itself in practice byhis activities for the benefit of the poor, subjugated, discriminated anddisadvantaged people. Together with his friends and pupils he supportedactivities in favor of such things as poorhouses, homes for the blind, loanbanks for the working-class, and libraries and elementary schools in thecountryside. Bolzano's liberal intellectuality and his progressivetheological and political ideas, combined with his practical activities andhis enormous influence as a priest and as a university professor on peoplein general and the opinion leaders in Bohemia in particular, were a highlyexplosive mixture in the political and religious atmosphere in which

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explosive mixture in the political and religious atmosphere in whichBolzano lived. Bohemia and its capital Prague were at his time part of theAustrian empire. Due to Klemens Frst von Metternich, a very illiberaland repressive political system was established in the Austrian empire bymeans of police force and censorship. All kinds of liberal and nationalmovements were suppressed in this political system. No wonder thatBolzano's progressive political ideas and activities were found to beunacceptable to the political authorities. This situation in combinationwith personal intrigues resulted in January of 1820 in Bolzano's removalfrom his professorship by Emperor Franz (who signed personally alldecrees of appointment and dismissals of professors of all the universitiesin the empire). From that time on Bolzano was forbidden to teach, preach,or publish, and he had to sustain himself on a meager pension that wasgraciously granted to him by the emperor. It came as a blessing indisguise that Bolzano now exempted from teaching duties had allthe time he needed to elaborate and write his new foundation of logic. Itwas published in 1837 in four volumes as Theory of Science. After that,Bolzano took great pains to elaborate a new foundation of mathematics.The realization of this project was considerably developed but not yetcompleted when Bolzano died in 1848.

Small pieces of his voluminous philosophical and mathematical literaryremains have been published from time to time. The complete edition ofhis works that was planned several times had to wait until 1969 when thetwo most meritorious Bolzano scholars, Eduard Winter and Jan Berg,together with the publisher Gnther Holzboog started the Bernard-Bolzano-Gesamtausgabe, which due to the effort of the three persons became one of the most distinguished complete editions of the worksof a philosopher in our time.

Due to the peculiar circumstances of Bolzano's life and theirinterconnection with his scientific career, the biographical part of thisarticle is longer than usual and divided into two sections. The third

Bernard Bolzano

2 Stanford Encyclopedia of Philosophy

article is longer than usual and divided into two sections. The thirdsection of the article presents a survey of Bolzano's main writings, and thefollowing sections (4 to 13) are devoted to the different branches ofphilosophy to which Bolzano made contributions. In the final section (14)Bolzano's influence on the development of the sciences and on theintellectual life in Bohemia is considered.

1. Bolzano's Life and Scientific Career2. Bolzano's Removal from Office and the Bolzano Trial3. Bolzano's Main Writings4. Logic

4.1 Bolzano's Concept of Logic4.2 Bolzano's Conception of Logic4.3 The Basis of Bolzano's Logic and of his Whole Philosophy4.4 Bolzano's Analysis of Propositions (i.e., of his Sentences inThemselves)4.5 Bolzano's Theory of Ideas (i.e., of his Ideas inThemselves)4.6 Bolzano's Method of Idea-Variation4.7 Bolzano's Definition of Logical Truth4.8 Bolzano's Definition of Material Consequence and ofLogical Consequence4.9 Further Applications of the Method of Idea-Variation

5. Epistemology and Philosophy of Science5.1 Appearances of Propositions and Ideas in Human Minds5.2 Subjective Intuitions and Subjective Concepts5.3 Judgments A Priori and A Posteriori5.4 Immediate and Mediated Judgments5.5 The Entailment Relation

6. Ethics6.1 Critique of Kant's Categorical Imperative6.2 Bolzano's Highest Moral Law

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6.3 Bolzano's Ethics as a Mixed Normative Theory6.4 Bolzano's Deontic Logic

7. Aesthetics8. Political and Social Philosophy9. Philosophy of Religion and Theology10. Metaphysics11. Philosophy of Nature and of Physics12. Philosophy of Mathematics

12.1 Geometry12.2 Analysis12.3 Preparatory Writings in Set Theory12.4 New Foundations of Mathematics

13. Metaphilosophy and History of Philosophy14. The So-called Bolzano Circle and Bolzano's Influence on theDevelopment of the Sciences and on Intellectual History

14.1 The So-called Bolzano Circle14.2 Bolzano's Influence on the Development of the Sciencesand on Intellectual History

BibliographyBolzano's WritingsSecondary LiteratureWorks Without Reference to Bolzano, Cited in this Article

Other Internet ResourcesRelated Entries

1. Bolzano's Life and Scientific CareerBernard Bolzano was born on 5 October 1781 in Prague. His father camefrom Lombardy (hence the Italian surname), though he lived already fromchildhood in Bohemia; by profession he was a merchant. Bolzano'smother came from the German speaking family Maurer in Prague.Bernard was the fourth of twelve children altogether, most of whom died

Bernard Bolzano


Bernard was the fourth of twelve children altogether, most of whom diedyoung.

When he was ten years old, Bolzano entered the Gymnasium (i.e. a kindof high school) of the Piarists in Prague, which he attended from 1791 to1796. He subsequently began his philosophical studies at the Universityof Prague; they lasted three years, roughly corresponding to the higherlevel of high schools. Included in the philosophical studies, besidesphilosophy itself, were subjects such as history, languages, and biology,but above all also mathematics and physics.

After this, Bolzano wanted to study theology. In his intention to do so,however, he came into conflict with his father. Thus, for the purpose ofreflection, he spent the academic year of 1799/1800 attending courses inmathematics and philosophy. He used this time for thorough-goingreflection concerning his future, especially with regard to studies as wellas choice of profession. After mature and conscientious consideration,Bolzano began his study of theology at the University of Prague in theautumn of 1800. At that time such a course of studies lasted four years; hefinished it in the summer of 1804. The decision to pursue the study oftheology, however, did not yet coincide with the final decision also toenter the clergy; rather, he hoped to get clear about his choice ofprofession precisely by studying theology.

For a long time, Bolzano was tormented by doubts concerning faith. Upto the end of his study of theology, certain doctrines of the church seemedto him irreconcilable with reason or at least with historical facts. Onlywith painstaking effort was Bolzano able to overcome these doubts, dueto the help of a conception advocated by the pastoral professor MarianMika, whom he held in high regard: For the justification of a religiousdoctrine it suffices (according to Mika) to show that the belief in itguarantees certain moral advantages (Bolzano 1836, 27). In religionwhat is at issue is not at all what the nature of a matter is in itself, but

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what is at issue is not at all what the nature of a matter is in itself, butrather only what the most edifying presentation is for us (Bolzano 1836,17); the doctrines of religion and especially of (Catholic) Christianityserve primarily to advance virtue and happiness and thus the generalwelfare wherein Bolzano saw in harmony with utilitarianism thefinal goal of morality.

This conception rescued Bolzano from his religious doubts. It became thefoundation of his whole philosophy of religion and was the decisive factorthat prompted him to get ordained as priest finally on 7 April 1805. A fewdays later, on 17 April 1805, he received his doctorate of philosophy atthe University of Prague. Just two days later, on 19 April 1805, he tookup the newly established chair for religious doctrine in the PhilosophicalFaculty at the University of Prague, which had been granted to himprovisionally by the Austrian Emperor, Kaiser Franz, on 13 February1805. The definitive occupancy of this chair was followed by hisappointment as professor ordinarius on 23 September 1806. Due to poorhealth (hemoptysis) he did not teach for more than two years (from June1813 to October 1815), and only at the beginning of the academic year1815/16 did he return to teaching. In the academic year of 1818/19Bolzano was dean of the Philosophical Faculty. On 20 January 1820Bolzano was removed from his professorship definitely and was notallowed to engage in any further teaching and publishing. The moredetailed circumstances that led to these measures and their consequenceswill be explained in the following section.

After this, especially important for Bolzano's life was his acquaintancewith Anna Hoffmann, whom he first met in 1823 thus three years afterhis removal from office at the deathbed of her daughter. From then onit was his habit to live during the summer on the estate of the Hoffmannfamily in Tchobuz (Southern Bohemia) and during the winter with hisbrother, who was a merchant in Prague where he had taken over hisfather's firm; from 1830, when the Hoffmann family stayed in Tchobuz,

Bernard Bolzano


father's firm; from 1830, when the Hoffmann family stayed in Tchobuz,Bolzano took up permanent residence there and only went to Prague forshort visits with his brother.

Bolzano himself did not at all feel his removal from office to be amisfortune (see Bolzano 1836, 49). He tried instead to see it in the bestpossible light. (Cf. Bolzano 1836, 72 ff., Bolzano 1965, 28 and 406.) Nowhe could devote himself completely to the study of logic and mathematics,for which he previously had only little time due to his strict adherence tohis duties as an academic teacher. Thus the removal had also advantagesfor Bolzano as a scientist and writer, as well as for his health, as hehimself stated (Wisshaupt 1850, 30).

One of the main difficulties that Bolzano's removal caused for hisscientific work was of course that he was forbidden not only to teach, butalso to publish. Thus great effort was required on the part of Bolzano andhis pupils and friends in order to arrange for his writings to appear at leastanonymously and outside of Austria. In 1834 the Lehrbuch derReligionswissenschaft (Bolzano 1834b) was edited anonymously in fourvolumes from Bolzano's lecture notes without his knowledge by hispupils; it appeared in a publishing house in Bavaria. In 1837, Dr. BernardBolzanos Wissenschaftslehre (Bolzano 1837a) appeared, again in fourvolumes, in the same publishing house. As many others of Bolzano'swritings, this book was published in such a way that it appeared that ithappened without Bolzano's intention. Later his hope (expressed inBolzano 1836, 79) was fulfilled to be allowed to publish at least writingswhich were not of religious or political content without playing hide-and-seek. In this enormous endeavor which was needed for eventually findingpublishers for the most important works and having them printed,Bolzano was well supported by his pupils and friends. They managed topublish a good deal of accompanying literature (introductions andprefaces to editions, essays in journals, reviews etc.), very often alsoanonymously. In this regard especially Michael Josef Fesl and Franz

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anonymously. In this regard especially Michael Josef Fesl and FranzPhonsk were of great merit, though they were more or less lacking inphilosophical and generally scientific talent.

Mrs. Hoffmann, who always took good care of the sickly Bolzano andwas a great help and good friend to him, became seriously ill herself in1841. Thus she moved with her family to Prague. Bolzano also moved toPrague and lived there with his brother from the end of 1841 until the endof his life. Mrs. Hoffmann died in 1842.

In Prague Bolzano could also be more active again in the RoyalBohemian Society of the Sciences as lecturer and administrator. In 1815Bolzano already became a regular member of this prestigious scientificsociety, and he was its director in 1819; he took up this office once againin 1843. In addition, he was director of its mathematical section from theend of 1841 to 1845, and director of the section for philosophy and puremathematics in 1842 and from the end of 1845 up to his death.

Bolzano suffered from attacks of coughing up blood, apparently due tohaving had tuberculosis. When he had once again overcome such anattack in the beginning of December 1848, he fell into extreme chills,from which he never recovered. On 18 December 1848 he died at 10:05AM; the cause of death was said to be (in the medical terminology of thetime) a lung ailment. On 21 December 1848 Bolzano was buried at theWolschan Cemetery in Prague, where his grave can still be found today.

2. Bolzano's Removal from Office and the BolzanoTrialIn taking up the chair for religious doctrine, Bolzano had achieved thegoal to which he strove upon entering the station of clergyman: The goalof this station he found in the advancement of virtue and an inwardhappiness dependent on this by means of the doctrines of Christianity

Bernard Bolzano


happiness dependent on this by means of the doctrines of Christianity(Bolzano 1853a, 37). This goal, he believed, could best be achieved byteaching at the university (or also at a Gymnasium). He wanted to educatethe youth in the spirit of the Enlightenment with respect to religion aswell as morals and politics. In this regard, however, Bolzano pursued agoal that was, unbeknownst to him, exactly opposed to what the emperorhad in mind upon establishing such chairs by royal decree of 3 February1804 at universities and secondary educational institutions (Jaksch 1830,589593, Unger 1840, 542544). Emperor Franz was appalled by theFrench Revolution and its disastrous consequences, for which heultimately blamed the Enlightenment for the most part; he thereforeabsolutely rejected every Enlightenment-related tendency. The chairs forreligious doctrine were for him an instrument in the struggle against thenegative political effects which, on his view, resulted from theEnlightenment. By means of such chairs the students were to bereligiously grounded and trained in conscientious action and that meantconformity with the state (Jaksch 1830, 590); they were consequently tobe shaped into good Christians and law-abiding citizens (Jaksch 1830,593). Religion and morals had the aim of supporting the state, accordingto the emperor, and the afore-mentioned chairs for religious doctrine,whose first occupant at the University of Prague was Bolzano, wereestablished as a means to this end.

According to the decree of 3 February 1804, the chairs for religiousdoctrine were established for the sake of improving religious instruction(Jaksch 1830, 589). Connected with these professorships was theimportant assignment to deliver the Sunday homilies, also calledexhortations or Erbauungsreden (edifying addresses), to the students(Bolzano gave them voluntarily also on holidays); the effectiveness ofthese chairs (whose occupants were therefore also called catechists cf. Bolzano 1836, 31, Jaksch 1830, 589) was thereby essentiallyenhanced.

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From the very beginning, already before he was definitively assigned asoccupant of the chair for religious doctrine, Bolzano confronted obstaclesand opposing forces. Attacks against Bolzano were brought up again andagain from religious as well as from political standpoints. At first all theseintrigues seemed to produce no effect. However, the decisive blow againstBolzano was eventually delivered. In this the personal physician ofEmperor Franz, Freiherr Andreas Josef von Stifft, played an importantrole. Stifft presented a report to the emperor on 2 February 1819concerning Bolzano's edifying addresses, a selection of which had alreadyappeared in 1813 as a book. In this report, Stifft managed to draw theemperor's attention to the following passage, which could only haveimpressed him as Stifft intended: There will come a time when thethousand different kinds of orders of rank and of barriers among humanbeings, which cause so much evil, will be put back into their properconfines, when everyone will deal with his neighbor as a brother ought todeal with his brother! There will come a time when one will introduceconstitutions which will no longer be so terribly open to abuse as ourpresent ones are (Bolzano 1813, 99, slightly modified and with a detailedfootnote in the 2nd edition, p. 36).

Bolzano himself, who confessed to having never read newspapers(Bolzano 1836, 42) or having read too few of them (Bolzano 1956, 250),presumably was not aware of the impression which such views weremaking at that time among the state officials and especially in theemperor; moreover, when he delivered (and also published) this homily,he could not foresee that some day it was to be presented to theauthorities to be used against him.

In a decree which the State Counsel authorized on 27 November 1819 andthe emperor signed on 24 December 1819, Bolzano's removal from hischair was finalized and a reprimand was issued to him. The officiallystated reason for Bolzano's removal from his office was that in his

Bernard Bolzano


stated reason for Bolzano's removal from his office was that in his(already published) edifying addresses and in his lecture notes (that werelater published as Textbook of the Science of Religion), he severelyviolated the duty of the priest, the teacher of religion to the youth, and thegood citizen of the state. There were thus in the foreground here allegedinstances of misconduct against the state rather than erroneous religiousteachings. The decree in which Bolzano's removal was finalized wasgiven to him on 20 January 1820, and on this date he was not onlyremoved from his chair, but also forbidden to teach, preach or publish anymore (cf. Bolzano 1836, 48 f., and Bolzano 1965, 28).

As regards the question concerning Bolzano's orthodoxy, the archbishopwas given the task of conducting his own investigation. The ecclesialinvestigations took a few years since one wanted to demand from Bolzanoa public retraction such as the one to which his pupil Fesl was compelled,though under essentially more unfavorable conditions, namelyconfinement to a monastery. Bolzano steadfastly refused to make such apublic retraction, since he himself was convinced that he had neveradvocated views that diverged from the teachings of the Catholic Church.It was finally seen to be satisfactory for Bolzano to submit a declarationof his orthodoxy before the archbishop on 31 December 1825 and toconfess the Trinitarian faith in the presence of the archbishop and theinvestigating committee.

3. Bolzano's Main WritingsBolzano's uncommonly versatile work culminated in three extensive mainwritings in three different areas of knowledge: (1) in theology his fourvolume Textbook of the Science of Religion (Bolzano 1834b), (2) inphilosophy the four volume Theory of Science (Bolzano 1837a), whichprovides a new foundation for logic and is at the same time an extensivemanual of logic, and (3) in mathematics the Theory of Quantities,conceived of as a monumental work, but not completed.

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conceived of as a monumental work, but not completed.

Bolzano's teaching was concerned exclusively with fundamental topics oftheology; in addition he worked mainly in logic. Nevertheless, hisscientific development began in mathematics. It was mathematics thatwas the starting point for his scientific work and to which he ultimatelyreturned in order to create a new foundation on which mathematics as awhole could be built; he succeeded in doing this, however, only in bitsand pieces. For a professional career in mathematics Bolzano hadprerequisites and also opportunities no less than he had for his teaching ofthe science of religion, which was the course he actually chose andinvolved so many difficulties. Already in 1804 Bolzano's first scientificpublication, namely Considerations on Some Objects of ElementaryGeometry (Bolzano 1804) appeared. Further mathematical monographsfollowed (Bolzano 1810, Bolzano 1816, Bolzano 1817a and Bolzano1817b), thereafter only posthumously. When Bolzano applied for thechair of religious doctrine he also applied for a chair in mathematics at theUniversity of Prague and underwent the examination (or Konkurs as itwas called). His qualification for the chair of mathematics was obviousand accordingly certified by the committee. It was a matter of chance thathe was finally appointed not to the chair of mathematics but to that ofreligious doctrine.

Bolzano's research in philosophy, and in particular in logic, served him asa basis for his work in theology as well as in mathematics.

4. LogicAlthough it is logic in which Bolzano's greatest philosophicalaccomplishments can be found, his interest in logic was nevertheless,strictly speaking, secondary in nature: Bolzano came upon the idea ofdeveloping a new logic because he noticed already in his earlymathematical investigations that the logic at that time did not suffice for

Bernard Bolzano


mathematical investigations that the logic at that time did not suffice forthe treatment of mathematics. Likewise, Bolzano's investigations in thescience of religion and theology compelled him to engage in logicalstudies, since he also wanted to construct the science of religion andtheology on a solid logical foundation, as can be seen in sections 1014 ofthe first part of his Textbook of the Science of Religion (RW I, 3241).For there are in theology no less than in mathematics and philosophy asignificant number of concepts which must be developed, very lengthyseries of inferences which must be properly executed, and very many,extremely deceptive fallacies whose total lack of soundness must bediscovered (RW I, 10; cf. also Bolzano 1838b, 334). The decisiveimpulse to write a textbook of logic, however, resulted astonishinglyenough from Bolzano's ethical investigations: in his attempt toestablish the highest moral law, he found that the logic of his time wasdeficient and in need of a fundamental renewal (Bolzano 1968a, 30 f.).

4.1 Bolzano's Concept of Logic

The term logic was understood in Bolzano's time, as also by Bolzanohimself, not in the narrow sense of formal logic, as it is commonly usednowadays, but rather in the broad sense which includes besides formallogic also both epistemology and the philosophy of science. ThusBolzano used instead of logic also the term theory of science. Bylogic or theory of science Bolzano means that discipline or sciencewhich formulates rules, according to which we must proceed in thebusiness of dividing the entire realm of truth into single sciences and inthe exposition thereof in special textbooks if we want to proceed in a trulyexpedient fashion (WL I, 7; cf. also WL I, 19 and 56).

This definition of logic makes a strange impression at first glance and wasalso often misunderstood. By putting it forward, Bolzano ascribes to logica task that is generally not included in philosophy at all, but rather in thetechnique of scientific procedure. Bolzano, however, considered this task

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technique of scientific procedure. Bolzano, however, considered this taskto be important enough to subject it to a scientific treatment. Since he didnot see how it could otherwise be brought within the purview of ascience, he assigned this task to logic which seemed to him to be the bestcandidate for this. What Bolzano mentions in his definition is, in his ownview, by no means the only task of logic. In order to avoid superfluouscriteria, Bolzano stated only the most concrete purpose in his definition,for the other tasks are entailed by it. Bolzano's logic is a composite of thetheory of foundations, the theory of elements, the theory of knowledge, theart of discovery (heuristics), and the theory of science proper. Thus, logicis, for Bolzano, an encompassing philosophical discipline, and the theoryof science proper is only a sort of appendix of his logic.

This assessment of the different parts of the Theory of Science coincideswith Bolzano's own judgment, according to which the philosophicalgain is only in the first three (indeed, first two) volumes (Bolzano 1965,232): the theory of elements is due to the originality of its views, themost important, noteworthy and meritorious part of the Theory of Science(Bolzano 1838b, 355, 340).

4.2 Bolzano's Conception of Logic

Bolzano's logic was based upon a fundamental view that was the veryopposite of the common view of his time. Whereas it was quite commonat his time to mix logical with psychological investigations, Bolzanomade every effort to separate them. For him logical concepts are conceptsof their own, and their definition therefore must be kept free from anypsychological admixture (WL I, 6166). Bolzano's approach to logic was long before Frege and Husserl unmistakably antipsychologistic,even if he did not yet use this term. In order to overcome psychologismand to achieve a strict separation of logic from psychology, Bolzanoopened up for logic a realm or world of its own, different from theworld of material objects as well as from the world of mental phenomena,

Bernard Bolzano


world of material objects as well as from the world of mental phenomena,a World 3 to speak with Karl Popper. Bolzano's motive for postulatingsuch a logical realm of its own obviously was the erroneous belief thatlogical properties (such as logical truth) and logical relations (such aslogical consequence) need purely logical objects as their bearers in orderto remain purely logical and free from any psychological admixturethemselves. What is more, Bolzano had an unshakeable intuition thatthere are and must be such purely logical objects, namely objectivesentences or sentences in themselves (Stze an sich), and their parts,i.e., objective ideas or ideas in themselves (Vorstellungen an sich). Intoday's terminology, Bolzano's sentences in themselves are calledpropositions; this term (without any epithet) will be used for them inwhat follows. (The term sentence without epithet, however, will be usedin its linguistic sense for certain strings of words.) Moreover, followingBolzano's practice, we will use the term idea (Vorstellung) withoutepithet for idea in itself (Vorstellung an sich). Propositions and ideasare the objects which can be grasped by mental phenomena (subjectivepropositions, in particular judgments, and subjective ideas) and expressedin language, but despite their close connection to their mental andlinguistic correlates they must be rigorously distinguished from them.

Due to his conception of logic, Bolzano was in need of propositions andideas and therefore postulated that there are such genuinely logicalobjects. He himself, however, was convinced that he need not postulatethem but can undoubtedly prove that there must be propositions and ideas.

4.3 The Basis of Bolzano's Logic and of his Whole Philosophy

Something can be true even nobody knows that it is. We therefore need aconcept of truth that does not require of every truth that someone knowsit. For this concept Bolzano introduced the term Wahrheit an sich(truth in itself). If something is a truth and no human being knows that itis, then for Bolzano it must be a truth in itself; being true but not

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is, then for Bolzano it must be a truth in itself; being true but notknown to be true is for him a distinguishing characteristic of a truth initself. A truth in itself (Wahrheit an sich) is nothing but a trueproposition, i.e., a proposition that has the property of being true. Bolzanotakes the proof that there is at least one truth in itself to be fundamentalnot only for logic and for philosophy in general, but for any science.Bolzano offers therefore several arguments such as the following ones forthe claim that there must be truths in themselves: (i) There areobviously truths that are unknown and therefore (so Bolzano) truths inthemselves. E.g. one of the two propositions There are winged snakesor There are no winged snakes must be true, but we do not know whichone (WL I, 108); and one of the propositions stating that a specific tree ata certain moment bears a certain number of blossoms must be true, evennobody knows which one (WL I, 112). (ii) The Pythagorean theorem orthe Copernican discovery that the earth rotates around the sun have notbecome true by their discoveries but have always been true, i.e.(according to Bolzano), they are truths in themselves (Zimmermann1847, 71 f., 136). (iii) If no thinking being existed, it were true that thereexists no thinking being, but this (according to Bolzano) could only be atruth in itself (Bolzano 1839, 150). There is one proof for the existenceof a truth in itself or a true proposition, however, which Bolzano takesto be decisive. It is an improved version of the traditional refutation ofscepticism by self-application (RW I, 35, WL I, 145). Before we presentand discuss this proof, a terminological remark seems to be in order: Theword truth (as well as the German word Wahrheit) is ambiguousinsofar as it denotes the property of being true on the one hand and abearer of this property (a true proposition, a true sentence, a truejudgment etc.) on the other hand (WL I, 108 f.). Only the latter can bemeant, however, as soon as the word truth is preceded by the indefinitearticle (a truth) or if it is put into the plural form (truths). It is in suchclear cases only that we will allow the word truth(s) without epithet tobe used in the sense of Bolzano's truth in itself or truths in themselves,

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be used in the sense of Bolzano's truth in itself or truths in themselves,but otherwise we will use true proposition(s) instead to avoid confusion.In general we will therefore reserve the word truth for the property ofbeing true. That being said, we can now turn to Bolzano's alleged prooffor the existence of true propositions:

1. There is no true proposition (assumption of reductio).2. (1) expresses a proposition.3. The proposition expressed by (1) is not true (from (1) and (2)).4. It is not the case that there is no true proposition (from (3) in

contradiction to (1)).5. There is a (i.e., at least one) true proposition (from (1)(4) by

reductio).

The improvement of Bolzano's version of this traditional proof consists inBolzano's explicitly mentioning premise (2) as indispensable for thededuction (WL I, 145, 151; cf. also WL IV, 282 f.), thus making clear thatthe proof would not go through if this premise were not accepted. Theinference of (4) from (3) is vindicated by Bolzano's explication ofpropositional negation (see section 4.4). Bolzano is not content withhaving proven (allegedly) that there is at least one true proposition; hewants even to prove by mathematical induction that there are more andeven infinitely many true propositions (see section 12.3).

The above derivation is an illustrative example of Bolzano's importationof proof methods from mathematics such as the methods of indirectproof and of mathematical induction into philosophy. (For certainreservations concerning these methods cf. WL IV, 269285.)Nevertheless, this alleged proof for the existence of a true propositiondoes not reach its goal for a simple reason: The proof is in no waypeculiar for truths in the sense of true propositions as Bolzano needs tohave it. If successful at all, it would work in the same way also forproving the existence of true judgments or true sentences (understood as

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proving the existence of true judgments or true sentences (understood aslinguistic entities). A proof for the existence of at least one truth in itself,i.e. a true proposition, requires premise (2) (as well as premise (3)). Inpremise (2), however, the existence of a proposition is alreadypresupposed since it is shorthand for what can be explicitly stated as:There is a (i.e., at least one) proposition expressed by (1). Thus, if wetake the derivation as a proof for the existence of at least one trueproposition, we will succumb to an obvious petitio principii. By correctly requiring the addition of premise (2) to be necessary for theformal correctness of the proof, Bolzano unwittingly displayed its failuredue to an informal fallacy. Bolzano himself, however, was convinced thathe had correctly and successfully proven that there are truths inthemselves. From this it follows that there must be propositions, sincetruths in themselves are a certain kind of propositions (WL I, 111 f.).And since in every proposition ideas are contained as its parts, a furtherconsequence is that there are also ideas. For every proposition there isalso another proposition that is its propositional negation. Thepropositional negation of a true proposition is a false proposition.Therefore, among propositions there must also be false ones. They areoften overlooked because a special name such as falsehood in itself wasnever introduced officially for them and is at least not customary.

As far as the ontological status of propositions and ideas is concerned,Bolzano stresses again and again their objectivity, i.e. their independencefrom thinking in general and from the minds and mental phenomena ofthinking beings by which they are grasped in particular. Bolzano doesnot only accentuate the objectivity of propositions and ideas, but he alsolays particular emphasis on their being not real (wirklich), whereby realmeans, in Bolzano's terminology, the same as efficacious (wirksam, fromwhich etymologically wirklich is derived). The realm of reality includeseverything in space and time and herewith all material objects and eventsof the physical world (i.e., Popper's World 1) as well as all mental (orpsychical) phenomena of our inner world (i.e., World 2). (In addition,

Bernard Bolzano


psychical) phenomena of our inner world (i.e., World 2). (In addition,Bolzano includes in his realm of reality also God who is outside of spaceand time.) Propositions and ideas belong to a third realm (World 3)outside of Bolzano's realm of reality which encompasses World 1 andWorld 2. Unfortunately, Bolzano uses the nouns Existenz (existence)and Sein (being) synonymously with Wirklichkeit (reality); hetherefore states again and again that there are (es gibt) propositions andideas but they do not exist and they do not have being. This peculiar ifnot odd terminology has caused numerous misunderstandings, not onlyabout Bolzano's views but also for Bolzano himself, e.g., in his discussionabout Kant's dictum that being is not a real predicate. (Despite Bolzano'sterminological convention, we will here use the English word existencein general in the broad sense of Bolzano's es gibt.)

It is not only for this terminological reason that Bolzano's characterizationof the ontological status of propositions and ideas remains in the lastanalysis nebulous and has therefore repeatedly evoked criticism. Onemust of course add, for the sake of doing justice to Bolzano, that no otheradvocate of such a World 3 including Husserl, Popper and even Frege has done better than he had done and has succeeded in saying anythingclearer about this than what can already be found in his work. Despite allthese deficiencies, Bolzano's unproven assumption or postulation ofpropositions and ideas turned out to be extremely fruitful for his ownresearch.

4.4 Bolzano's Analysis of Propositions (i.e., of his Sentences inThemselves)

Although Bolzano contributed many highly interesting and valuableinsights to the analysis of propositions, they all are shaped in the subject-predicate scheme. We must explain some of these insights before wediscuss Bolzano's main contributions to logic in the sections 4.64.9.

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In what follows brackets will be used for the denotation of propositionsand ideas. Thus [Socrates has wisdom] will denote the propositionexpressed by the words Socrates has wisdom, and [Socrates] and[wisdom] will denote the ideas expressed by the words Socrates andwisdom, respectively. (Such a notation is not without problems, but thisis not the place to discuss them.) Bolzano starts his analysis ofpropositions by proclaiming the traditional subject-predicate view as adogma: Despite the variety of their linguistic expressions, all propositionsare of the form [A has b] and therefore have exactly three parts: a subjectidea [A], a predicate idea [b], and the copula [has], i.e. the idea expressedby the word has or another form of to have (WL II, 917). Bolzanoprefers this copula to the copula expressed by a form of to be for thefollowing reason: In everyday language we try to avoid abstracts such aswisdom and prefer saying Socrates is wise; but in doing so weattribute a property namely wisdom to Socrates. The logicalstructure of the proposition is therefore best displayed says Bolzano by the words Socrates has wisdom. Due to the stylistic preference ofadjectives over abstract nouns, everyday language very often does noteven provide us with an abstract noun corresponding to an adjective; insuch cases we therefore use the adjective at hand, although we couldalways easily introduce a corresponding artificial noun into our language.

Since every proposition has the same copula, two propositions can bedifferent only if they have either different subject ideas or differentpredicate ideas or both. This results in Bolzano's identity criterion forpropositions: Two propositions [A1 has b1] and [A2 has b2] are identicaliff (i.e., if and only if) [A1] = [A2] and [b1] = [b2]. In order for [A has b]to be a proposition, it will suffice that the predicate idea [b] be anarbitrary idea, in a way at least pretending to be an idea of an attribute(WL II, 1618). In order for [A has b] to be true, however, it is necessarythat the predicate idea [b] is an idea of an attribute (Beschaffenheit). Anattribute can be an inner attribute, i.e., a property (Eigenschaft) of anobject, or an outer attribute, i.e., a relation (Verhltnis) among objects.

Bernard Bolzano


object, or an outer attribute, i.e., a relation (Verhltnis) among objects.Examples of properties are wisdom or omnipotence, examples of relationsare friendship to so-and-so, fatherhood to so-and-so, being twice as longas such-and-such (WL II, 378389). In his careful analysis of thedifference between inner and outer attributes, Bolzano already came closeto what in our day is sometimes called a mere Cambridge property (WL I,388 f.).

The main trouble resulting from the traditional subject-predicate view ingeneral and from Bolzano's uniform [A has b] structure of everyproposition is that under its subject idea [A] two different cases areconcealed insofar as it can be singular (as in the case of [God] or [thesun] or [Bernard Bolzano]) or general (as in the case of [man] or [animal]or [planet]). Due to this duality, Bolzano has to add that [A has b] or [Ahave b] is to be understood (when [A] is general) in the sense of [Every Ahas b] or [All A have b], e.g. [Animals have sensitivity] = [Every animalhas sensitivity] = [All animals have sensitivity] (WL II, 24 f.).

In order to confirm his thesis that every proposition has the form [(Every)A has b], Bolzano shows of all kinds of verbal forms of significantsentences how to transfer them into his standard form (cf. WL II, 38 ff.and 211 ff.). Here are some of the rather important examples of Bolzano'sanalysis.

i. Predicate negation (i.e., inner negation): The lack of an attribute b(such as the lack of omnipotence) is itself a property (WL II, 47) thatwe can denote by the negation non-b (non-omnipotence).Negative propositions of the form [A has non-b], e.g. [BernardBolzano has non-omnipotence (i.e., lack of omnipotence)], thereforeshare the general form [A has b] with all other propositions (WL II,4452).

ii. Propositional negation (i.e., external negation): From propositionsof the form [A has non-b] we have to distinguish propositions in

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of the form [A has non-b] we have to distinguish propositions inwhich another proposition is denied. We can express suchpropositions by It is not the case that A has b. According toBolzano, such a proposition is about another proposition and statesthat this proposition is false, i.e., not true. Its subject idea is thereforean idea of a proposition and its predicate idea is the idea of falsity orlack of truth, i.e., non-truth. Their form is therefore best displayed as[[A has b] has non-truth] (WL II 6264).

iii. Subjunctive and disjunctive propositions: The subjunction of twopropositions s1 and s2 is explained by Bolzano as a proposition ofthe form [s2 is a consequence of s1] (WL II, 198 f., 224226;Bolzano proposes here also alternative interpretations of if sosentences). An inclusive or exclusive disjunction of two propositionss1 and s2 is interpreted by Bolzano as a proposition which attributesto the idea [a true proposition belonging to the collection consistingof s1 and s2] the property of being non-empty or singular,respectively (WL II, 204 f., 228 f.).

iv. Particular propositions and there-is propositions: In view ofBolzano's interpretation of [A has b] as [Every A has b] whenever[A] is general, it is of special interest how he deals with particularpropositions expressed by Some A have b. Bolzano transforms suchsentences into there-is sentences of the form There is at least one Athat has b. But what about such there-is sentences? In a sentence ofthe form There is at least one A we attribute, according to Bolzano,not a property to A itself but to [A], i.e., the idea of A, namely theproperty of being non-empty. The form of the correspondingpropositions is therefore best given as [[A] has non-emptiness] (WLII, 5254, 214218). This analysis is completely in accordance withKant's and of course even more so with Frege's and Russell's lateron , although Bolzano never stopped criticizing Kant for hisdictum that existence is not a real predicate. Bolzano took this dictumto be about existence in his own narrow sense of existence or

Bernard Bolzano


to be about existence in his own narrow sense of existence orbeing, i.e., in the sense of reality, and not as he should have in the broad sense of his es gibt. His disagreement with Kant onthis point is therefore merely verbal in nature. For Bolzano'sapproach, a true negative existential sentence such as There is noround square does not pose a problem any more; the propositionexpressed by this sentence is [[a round square] has emptiness] (WLII, 54 f.).

In his analysis of propositions Bolzano distinguished clearly differentlevels in his realm of propositions and ideas. He even introduced a specialname for ideas of ideas, such as the idea of the idea [A], i.e., [[A]]; hecalled them symbolic ideas (WL I, 426 ff.). In his efforts to show that allpropositions can be shaped into [A has b], Bolzano makes extensive usageof such symbolic ideas (and also of ideas of propositions) as subject ideasof propositions, as it is exemplified under iii and iv above.

Bolzano took pains to systematize his attempts of shaping all propositionsinto [A has b]. His attempts, however, remained only on the level ofexamples, since he missed a key for their systematization such as Frege'sfunction-argument scheme.

Bolzano combines his doctrine that the form [A has b] is common to allpropositions with a correspondence theory of truth, whereby he, likeAristotle, avoids speaking of correspondence or adequation. A proposition[A has b] is true, according to Bolzano, iff A has (in fact) b (WL I, 112).There is an important qualification, however, that Bolzano has thereby inmind, namely that there is at least one A; a proposition [A has b] can onlybe true, according to Bolzano, if it is about something and if its subjectidea [A] is therefore non-empty (WL II, 16, 328330, 399 ff.). Formulatedmore carefully, Bolzano's truth condition must therefore be stated asfollows (WL I, 112, 121224, WL II, 26 f., 328330):

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Due to this definition of truth and due to Bolzano's doctrine that everyproposition has the form [A has b], every proposition has existentialimport for Bolzano. (Existential must here be taken in the sense ofBolzano's there is; and it must be kept in mind that Bolzano'sinterpretation transfers many propositions to a meta-level and that in sucha case the existential import concerns [A] rather than A itself.) Thispeculiar kind of an existential presupposition of Bolzano's logic makes histheory of syllogisms (which he himself saw as a mere section of his wholelogic) an intermediate system between Aristotle and Venn: WhereasAristotle's theory of categorical syllogisms does not allow empty terms atall, Bolzano's logic does so, but they cannot be the subject ideas of truepropositions in Bolzano's logic. As a consequence, the so-called conclusioad subalternatam is logically valid also for Bolzano, i.e., [Some A have b]follows logically from [All A have b] (WL II, 114, 399 ff.), but [All Ahave non-b] is not convertible, i.e., [All B have non-a] does not logicallyfollow from [All A have non-b] (WL II, 401 f., 526). Therefore exactlytwo of the 24 valid Aristotelian syllogisms (namely the modi CAMENES or CALENTES in Bolzano's terminology and CAMENOP of formIV) are invalid in Bolzano's logic as he himself proved by means ofcounter-examples (WL II, 415, 558), whereas all other Aristotelian modi(including the weakened ones) are logically valid also in Bolzano's logic.

4.5 Bolzano's Theory of Ideas (i.e., of his Ideas inThemselves)

The three immediate parts of a proposition are its subject idea, itspredicate idea and the copula [has]. In further analyzing the subject andpredicate idea of a proposition, we will find out, that in special cases (as,e.g., in the case of the idea [the judgment that God is omnipotent]), a

[A has b] is true (or: has truth) iff [A] is non-empty, and for everyx that is an object of [A] there is a y that is an object of [b] suchthat x has y.

Bernard Bolzano


e.g., in the case of the idea [the judgment that God is omnipotent]), acomplete proposition will turn out to be a part of an idea (WL I, 221). Ingeneral, however, the parts of an idea are themselves ideas. After carefulconsideration Bolzano decided against the view to define a proposition assomething constructed out of ideas (i.e., a connection of two arbitraryideas by means of the copula [has]) (WL II, 18); he rather suggested thatwe define ideas as those parts of a proposition which are not themselvespropositions (WL I, 216, WL II, 18). In this sense he granted priority topropositions over ideas, thereby anticipating Frege and Wittgenstein.There is a clear demarcation between propositions and ideas: Whereaseach proposition is either true or false (WL II, 7), an idea cannot be trueor false (WL I, 239 ff.).

There are two dimensions to be distinguished in each idea: its internaldimension of being divided into parts, and its external dimension ofeventually being directed towards objects.

As far as the inner structure of ideas is concerned (WL I, 243 ff.),Bolzano distinguishes simple from complex ideas: A simple idea has noproper parts, whereas a complex idea has. The sum (Summe) of properparts of a complex idea is called its content (Inhalt). Due to Bolzano'speculiar usage of the term sum that is restricted like his usage ofcollection in general to sets with at least two members, he could notapply his concept of content to all ideas, but only to complex ideas. Inorder to simplify matters, we will use here the modern concept of a set,allowing for a set to be a singleton (i.e., containing only one singlemember) or even to be empty (i.e., containing no member at all). In whatfollows we will therefore take the content of an arbitrary idea to be the setof all of its parts (including improper ones, i.e., including itself). Thecontent of a simple idea i is then the singleton {i} containing i itself as itsonly member. Two ideas i1 and i2 can have the same content, i.e., thesame parts, without themselves being identical, because the common partsof i1 and i2 can be arranged in different ways in i1 and i2. Bolzano's

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of i1 and i2 can be arranged in different ways in i1 and i2. Bolzano'sfavorite example is: [an erudite son of an unerudite father] has the samecontent, but is not identical with [an unerudite son of an erudite father];the same holds for [35] and [53] (WL I, 244). In analyzing an idea, wewill in all cases eventually come upon simple ideas (WL I, 263265).Without explicitly expressing it, Bolzano obviously held the view thatevery idea is recursively constructed out of simple ideas. Two ideas aretherefore identical iff they are constructed out of the same simple ideas inthe same way. In order to be able to apply this general idea precisely inconcrete cases, we would have to be able to identify the simple ideas andthe formation rules involved. Unfortunately, Bolzano informs us aboutboth only by hinting at examples here and there. As examples of simpleideas he mentions [something] (WL I, 447), [has] (WL I, 380, WL II, 18),[non] (WL II, 415), [Wirklichkeit], i.e., [reality] (WL II, 60), and [Sollen],i.e., [ought] (WL II, 69, WL IV, 489).

With respect to its external dimension, an idea can have several (maybe even infinitely many) objects, exactly one object, or no object at all.An idea that has no object at all is an empty idea; Bolzano calls itgegenstandlos (objectless). Bolzano puts forward the thesis withparticular emphasis that there are empty ideas; his standard examples areideas such as [nothing], [golden mountain] (WL I, 304 f., WL II, 329) or[winged horse] (WL III, 24). A special kind of empty ideas, viz.contradictory ideas (or, as Bolzano usually prefers to call them,imaginary ideas) cannot even have an object (WL I, 315 ff., WL III, 405f.), examples being [a round polygon], [a round square], [a triangle that isquadrangular], [a regular pentagon], [a wooden iron tool], [an equilateralrectangular triangle] (WL I, 305, 315, 317, 321, 324, WL II, 329).

Non-empty ideas are called gegenstndlich (objectual) by Bolzano.They can be singular as, e.g., [the philosopher Socrates], [the city ofAthens], [the fixed star Sirius] (WL I, 306), [an even integer between 4and 8] (WL III, 407), [God] (WL III, 408), or general; if general, they can

Bernard Bolzano


and 8] (WL III, 407), [God] (WL III, 408), or general; if general, they canhave a finite number of objects, such as [a heir of Genghis-Khan'sEmpire] (WL I, 299) or [an integer between 1 and 10] (WL I, 308), or aninfinite number of objects, such as [a line] or [an angle] (WL I, 298). Fornon-empty ideas (and only for them) Bolzano defines their extension(Umfang) (WL I, 297 f.); by using again the modern concept of a set (aswe already did with Bolzano's definition of the content of an idea), wecan extend his definition to all ideas including empty ones; the extensionof an arbitrary idea i (or Ext(i), as an abbreviation) is then nothing but theset of all objects of i.

By crossing the internal with the external dimension of ideas we canget new and interesting creations. Combining, e.g., the smallest contentwith the smallest extension of a non-empty idea results in a new type ofidea, viz. in an intuition in itself or, as we may say for the sake ofbrevity, an intuition (Anschauung). An intuition is an idea which issimple, i.e., has no proper part, and at the same time singular, i.e., hasonly one single object (WL I, 325330). If an idea is neither itself anintuition nor contains any intuition as a proper part, it is called Begriff(concept) by Bolzano; examples of concepts are the simple idea[something] and the complex idea [God] whereby, for Bolzano, [God] =[the real being that has no cause of its being real]. A mixed idea is acomplex idea which contains at least one intuition as a proper part (WL I,330 f.). The distinction between intuitions and concepts plays animportant role in Bolzano's epistemology (cf. section 5.2 where we willalso present examples of intuitions).

Talking about the external dimension of ideas, we made followingBolzano intensive usage of a certain relation R between ideas and theirobjects that is basic in Bolzano's theory of ideas. For i R x we usedphrases such as i is an idea of x or x is an object of i; other expressionsfor it are i represents x, x is subsumed under i or x falls under i (WLI, 298). The domain of R is the set of non-empty ideas, its range being the

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I, 298). The domain of R is the set of non-empty ideas, its range being theset of all objects; moreover, R has the following properties: it is neitherreflexive nor irreflexive (the latter due to counterexamples such as: [idea]R [idea]; cf. WL I, 461), it is neither symmetric nor asymmetric, it isneither transitive nor intransitive, and it is neither one-many nor many-one. Since, according to our definition, Ext(i) = {x | i R x}, we canexpress i R x also in terms of extension as x Ext(i).

Bolzano defines a variety of relations between ideas concerning theirextensions, such as the following ones: An idea i1 is compatible with anidea i2 iff i1 and i2 share a common object, i.e. Ext(i1) Ext(i2) ;and i1 is included in i2 (or: i2 includes i1) iff i1 is compatible with i2 andevery object of i1 is also an object of i2, i.e., Ext(i1) Ext(i2) , andExt(i1) Ext(i2). In Bolzano's theory of ideas, precise correlates areavailable of basic concepts of set theory such as the empty set as well asthe membership relation and the relation of inclusion between sets.Unfortunately, the clear distinction between membership and inclusion inhis theory of ideas vanished in his theory of propositions due to thecommon form [A has b] of all propositions whereby [A] can be not only asingular, but also a general (or an empty) idea.

Already in his theory of ideas Bolzano used a method which he was veryproud to have invented: the method of idea-variation. He made the mostfruitful usage of this method, however, by applying it to wholepropositions.

4.6 Bolzano's Method of Idea-Variation

In his analysis of propositions, Bolzano did not break through thetraditional paradigms. In another respect, however, namely concerning thedefinition of basic semantic concepts, he opened wide the gates to modernlogic. The main instrument to do so was the method of idea-variation thathe invented. He himself took it to be his main contribution to logic that

Bernard Bolzano


he invented. He himself took it to be his main contribution to logic thatwas for himself who certainly did not suffer from arrogance ofepoch making importance (Bolzano 1838b, 150).

The basic insight underlying Bolzano's method of idea-variation is quitesimple (WL II, 77 ff.). Let us take as our first example S1 the proposition[Kant is a German philosopher]. (In order to simplify matterslinguistically, we do not adhere to Bolzano's formulation A has b butwill allow also formulations of the kind A is (a) B. Moreover, we willtake in what follows words like German, French, European,American etc. in the sense of born in Germany, born in France,born in Europe, born in America etc.) We consider one or more extra-logical ideas that are parts of S1, i.e. the ideas [Kant], [German], or[philosopher], to be variable in the sense that we think that they arereplaced by another idea fitting to the former one (i.e., belonging to thesame category). In this way the idea [Kant] can be varied in S1 andreplaced, e.g., by [Hegel]; in other words, we can substitute [Hegel] for[Kant] in S1. The variation in question is a kind of replacement orsubstitution. It results in a new (or better: in another) proposition, viz.in the true proposition [Hegel is a German philosopher]; we will say that[Hegel is a German philosopher] is the [Hegel]/[Kant]-variant of S1. Afalse [Kant]-variant of S1 is its [Sartre]/[Kant]-variant [Sartre is a Germanphilosopher]. Similarly, [Kant is an European philosopher] is a true and[Kant is an American philosopher] is a false [German]-variant of S1. Theoperation of replacement (or substitution) can also be performed on twoor more parts of a proposition simultaneously: Replacing [Kant] and[philosopher] in S1 simultaneously by, e.g., [Gauss] and [mathematician],results in the true ([Gauss], [mathematician]/[Kant], [philosopher])-variant [Gauss is a German mathematician]. A false ([Kant],[philosopher])-variant of S1 is, e.g., [Sartre is a German musician]. Wecan also replace all extra-logical parts of S1 simultaneously: A true([Kant], [German], [philosopher])-variant of S1 is [Mozart is an Austrian

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([Kant], [German], [philosopher])-variant of S1 is [Mozart is an Austriancomposer], whereas [Sartre is a Greek mathematician] is a false one.Following Bolzano, we will here use this generalized operation ofsimultaneous replacement (or variation). Given an arbitrary proposition sand two sequences i1, i2,, in and j1, j2,, jn of ideas, a proposition s isuniquely determined by this operation; due to this operation, for each k (1 k n), the idea ik is replaced in s uniformly (i.e., wherever it occursin s) by the corresponding idea jk. The resulting proposition s is the (j1,j2,, jn/i1, i2,, in)-variant of s, or briefly put the j/i-variant of s.(Thereby we take i = and j = . Moreover, weare using here s, s, s1, s2, as variables for propositions, i1,i2,, j1, j2, as variables for ideas, and i and j as variables forsequences of ideas.) The close relationship between a variant of aproposition and a substitution instance of a sentence is quite obvious. Inorder for the i/j-variant of an arbitrary proposition s to be uniquelydetermined and to fulfill certain criteria of adequacy, however, severalrestrictions are required: (i) Each of the iks (1 k n) must be simple orat least relatively simple (in the sense that in each particular contextunder consideration they are not further analyzed into parts but taken tobe simple); (ii) each of the iks is an extra-logical idea; (iii) the iks arepair-wise distinct; moreover (iv), in order to keep the result of thereplacement operation well-formed, i.e., a genuine proposition, we mustrequire that each jk fits the corresponding ik, i.e., is of the samesemantic category; and finally (v), we must also require that at least oneof the ideas ik must be contained in s as one of its parts so that theoperation of replacement is never performed vacuously.

Instead of saying that a j/i-variant of a proposition s is true or false (orthat it is a true or false variant of s), Bolzano prefers to say: j macht shinsichtlich i wahr bzw. falsch (cf. e.g., WL II, 79, 113 ff.), i.e.: j verifiesor falsifies s with respect to i (or, more literally: j makes s true or falsewith respect to i).

Bernard Bolzano


As far as our first example S1 is concerned, there are true as well as falsevariants of it with respect to every single extra-logical part of it and alsowith respect to every sequence of such parts. But now let us consider thefollowing expamle S2: [Every German philosopher is European]. It hastrue as well as false [German]-variants, [European]-variants, ([German],[philosopher])-variants, ([German], [European])-variants, ([philosopher],[European])-variants and also ([German], [philosopher], [European])-variants. But obviously, all [philosopher]-variants of S2 must be true provided, says Bolzano, that their subject idea is non-empty. This provisois a typical feature of Bolzano's approach and mentioned by him againand again, since according to his truth condition a proposition withan empty subject idea is trivially false. By variation of the subject idea ofa proposition or of parts of it, however, we will in most cases also havevariants with empty subject ideas, and only in exceptional cases allvariants of a proposition could therefore turn out to be true. If all i-variants of a proposition s with an non-empty subject idea are true,Bolzano will say that s is universally valid with respect to i. Hereby wehave to take into account, that for Bolzano also in his meta-languagewords such as all, every or each have existential import and thattherefore his definition must explicitly stated as follows (WL II, 82):

Analogously we can define what it is for a proposition to be universallycontravalid:

[Every German philosopher is American] is an example of a propositionthat is universally contravalid with respect to [philosopher].

A proposition s is universally valid (allgemeingltig) with respectto a sequence i of ideas iff there is at least one true i-variant of s,and every i-variant of s with a non-empty subject idea is true.

A proposition s is universally contravalid (allgemeinungltig) withrespect to a sequence i of ideas iff every i-variant of s is false.

Edgar Morscher


If a proposition is universally valid or universally contravalid with respectto i, Bolzano says that it is analytic with respect to i, otherwise, that it issynthetic with respect to i. If a proposition is analytic or synthethic withrespect to at least one sequence i, Bolzano calls it analytic or synthetic,respectively, without further qualification (WL II, 8389, 331338).Herewith Bolzano starts a new tradition of usage of the term analytic asopposed to that from Kant up to Carnap and Quine: Whereas in this lattertradition the term analytic includes exclusively true propositions, inBolzano's terminology also all universally contravalid propositions aresubsumed under this term; and even a universally valid proposition couldbe false if it has an empty subject idea but all of its variants with non-empty subject ideas are true. (On this point, however, Bolzano is notalways consistent.)

Logical properties of a proposition are according to a classical view of a formal character, i.e., they are primarily properties of the form of aproposition rather than of the proposition itself. This allows us to presentBolzano's view also in an alternative way. Bolzano himself identifiesexplicitly the form of a proposition with a set of propositions (WL I, 48,WL II, 82): The form of a proposition s with respect to a sequence i ofideas or (as an abbreviation) the i-form of s is the set of all i-variants of s,provided that at least one of the iks is contained in s. (For cases in whichthis proviso is not met, neither an i-variant of s nor the i-form of s isdefined.) A propositional i-form can therefore be defined as the i-form ofat least one proposition s; and a propositional form is a propositional i-form with respect to at least one sequence i. Due to the provisomentioned, a propositional form as defined before can never be empty ora singleton. We can now define universal validity and universalcontravalidity first for a propositional form and subsequently for aproposition in the following way:

Bernard Bolzano


4.7 Bolzano's Definition of Logical Truth

The result of applying the operation of variation to a proposition dependsessentially on our choice of ideas to be varied in the proposition inquestion. And it can depend on matters of fact whether or not aproposition is universally valid (or contravalid) with respect to a sequencei of ideas. Thus, e.g., it is due to the fact that every German is Europeanthat the proposition [Every German philosopher is European] isuniversally valid with respect to [philosopher]. From a logical point ofview, the most interesting results will turn up if all extra-logical parts ofa proposition, which are simple (or as explained before relativelysimple), are taken to be variable (WL II, 84). To simplify matters, we willassume for what follows that for any proposition s there is always fixed acertain alphabetic order of all extra-logical simple ideas contained in it;thereby, for every proposition s, a sequence is of all extra-logical simpleideas contained in s is uniquely determined. It would appear that we nowcould define the concepts of logico-universal validity and logico-universal

A propositional form F is universally valid iff at least one memberof F is true, and every member of F with a non-empty subject ideais true; F is universally contravalid iff every member of F is false.

A proposition s is universally valid (or universally contravalid,respectively) with respect to a sequence i of ideas iff there is apropositional form F such that F is a propositional i-form which isuniversally valid (or contravalid, respectively), and s is a memberof F.

s is analytic with respect to i iff s is universally valid oruniversally contravalid with respect to i; s is synthetic with respectto i iff s is not analytic with respect to i.

s is analytic (or synthetic, respectively) iff s is analytic (orsynthetic, respectively) with respect to at least one sequence i.

Edgar Morscher


could define the concepts of logico-universal validity and logico-universalcontravalidity in the following way: A proposition s is logico-universallyvalid, or put briefly logically true, iff s is universally valid withrespect to is; s is logico-universally contravalid, or put briefly logically false, iff s is universally contravalid with respect to is; s islogically analytic iff s logically true or logically false; and s is logicallysynthetic iff s is not logically analytic.

In proceeding this way, however, we would have to face serious problemsconcerning purely logical propositions, i.e., propositions all of whoseparts are purely logical ideas. Due to our requirements (ii) and (v) aboveconcerning Bolzano's replacement operation, the is-variant of a purelylogical proposition is not defined; in consequence the precedingdefinitions of logical truth, logical falsity and logical analyticity are notapplicable to purely logical propositions, since they do not contain anyextra-logical idea. Consider, however, the following three purely logicalpropositions: [There is something, or it is not the case that there issomething], [There is something, and it is not the case that there issomething], and [There is something]. The first of these three propositionsis obviously logically true, the second one logically false, and the thirdone is neither logically true nor logically false. Giving up requirement (v),as some would like to have it, results in a purely logical proposition beingits only own is-variant; that, however, would turn every purely logicalproposition either into a logical truth or into a logical falsity according toour definition, in contrast to the fact that the proposition [There issomething] is not logically analytic.

A posssible way out of this dilemma is to choose the alternativeprocedure sketched above by taking these logical properties primarily tobe properties of propositional forms. Thereby the logical form of aproposition s is identified with the set of all of its is-variants; F is alogical propositional form therefore iff it is the logical form of at leastone proposition. (Please note that according to this approach, the logical

Bernard Bolzano


one proposition. (Please note that according to this approach, the logicalform of a purely logical proposition is not even defined; nevertheless, apurely logical proposition can be a member of a purely logicalpropositional form.) We will define first the relevant properties forpropositional forms:

We then define the corresponding properties for single propositions:

It is easy to find an example of each of these kinds of propositions: theproposition [Every German philosopher is German] is logically true, theproposition [Kant is German and Kant is non-German] is logically false,and the proposition [Kant is a German philosopher] is logically synthetic.

4.8 Bolzano's Definition of Material Consequence and ofLogical Consequence

A propositional form F is logically valid iff F is a logicalpropositional form that is universally valid, i.e., at least one of itsmembers is true, and all of its members with non-empty subjectideas are true.

F is logically contravalid iff F is a logical propositional form, andall of its members are false.

A proposition s is logically true iff there is a propositional form Fsuch that F is logically valid and s is a member of F.

s is logically false iff there is a propositional form F such that F islogically contravalid and s is a member of F.

s is logically analytic iff s is logically true or logically false.

s is logically synthetic iff s is not logically analytic.

Edgar Morscher


Bolzano's logical World 3 includes in addition to ideas and propositionsalso what we will call arguments. Bolzano is dealing with arguments atlengths in his Theory of Science (cf., e.g., WL II, 113 ff., 391 ff.), but hedoes not introduce a name for them. Following the general line of histerminology, he could have called them Schlsse an sich (derivations inthemselves or inferences in themselves), but he did not; he rather usedthis term for a certain kind of propositions, namely for propositionsstating that a proposition follows (or in his words is ableitbar, i.e.,derivable) from a set of propositions (WL I, 213, WL II, 200; for anotherusage of the same term by Bolzano cf. section 5.4). A Bolzanianargument consists of two sets of propositions: the set of its premises andthe set of its conclusions. In order to simplify matters, we will assumehere that an argument has always a single proposition (rather than a wholeset of propositions) as its conclusion, and we will identify an argumentwith an ordered pair consisting of a set of propositions (i.e., theset of premises) and a single proposition s (i.e., the conclusion).

Bolzano first explains what it means that a single proposition s isderivable (ableitbar) from a set of propositions with respect to a certainsequence i of ideas (WL II, 113 ff., 198 ff.). Since Bolzano's termAbleitbarkeit (derivability) is used nowadays in a purely syntacticalsense, we use here instead the more common phrases s follows from with respect to i or s is a consequence of with respect to i.

We have explained Bolzano's method of idea-variation in section 4.6 withrespect to single propositions. In order to apply it also to arguments, wehave now to extend our original definitions to whole sets of propositions.In applying the operation of variation to a set of propositions, eachmember of is replaced by its corresponding variant: The j/i-variant of aset of propositions is the set of all the j/i-variants of the members of .We will say of a set of propositions that it is true iff each of itsmembers is true; and we will say that a sequence j of ideas verifies a set of propositions with respect to i iff j verifies each member of with

Bernard Bolzano


of propositions with respect to i iff j verifies each member of withrespect to i, i.e., iff the j/i-variant of each member of is true. Aproposition s follows from a set of propositions (or, in other words, s isa consequence of ) with respect to a sequence i of ideas iff everysequence j of ideas that verifies with respect to i verifies s as well withrespect to i.

In transferring this formulation into a formal definition, we have to bearin mind again that every has existential import also in Bolzano's meta-language. The formal definition has therefore to be stated as follows:

In this sense, e.g., the proposition [Kant is European] follows from the set{[Kant is a philosopher], [Every philosopher is German]} with respect tothe idea [philosopher]. The conclusion follows, so to speak, materiallyfrom the premises, or is a material consequence of the premises, due tothe fact that every German is European. This is not enough, of course, foran argument to be logically correct. In order to be logically correct, theconclusion s of an argument must logically follow from , i.e., smust be a logical consequence of (WL II, 391395; the similarity withthe distinction between material and formal consequence in Tarski 1956,419, is obvious). A simple definition of logical consequence seems tosuggest itself: s follows logically from (or: s is a logical consequence of) iff s follows from with respect to the sequence i{s} of all extra-logical simple ideas contained in or s.

This simple answer, however, faces the same problems as thecorresponding answer with respect to logical truth (cf. section 4.7). In asimilar way as with logical truth, we must also here give priority to thelogical form of an argument and then proceed by this means to define the

s follows from with respect to i (or: s is a consequence of withrespect to i) iff there is a sequence j of ideas such that j verifies with respect to i, and every sequence j of ideas which verifies with respect to i, verifies s likewise with respect to i.

Edgar Morscher


logical form of an argument and then proceed by this means to define theconcept of logical consequence for particular arguments. This, however,will not be elaborated on here, but rather will be developed in thefollowing supplementary document, in which also the material of sections4.6 and 4.7 is reconstructed in a more technical way:

4.9 Further Applications of the Method of Idea-Variation

Beyond the usage of the method of idea-variation for his truly pioneeringdefinitions of logical truth and logical consequence and related concepts(such as the concepts of satisfiability and compatibility), Bolzano madeuse of this method also for a series of other purposes, above all in hisdevelopment of the theory of probability (WL II, 7782, 171191, 509514, WL III, 136138, 263288, 559568, RW II, 3949, 5761, 6671).Bolzano's theory of probability is based on his distinction of differentdegrees of validity of a proposition s with respect to a sequence i of ideas.This degree of validity of s with respect to i is representable as a fractionalnumber m/n where n is the number of all possible variants of s withrespect to i and m is the number of true variants of s with respect to i. Ifall variants of s with respect to i with non-empty subject ideas are true, m= n and m/n = 1, i.e., s is universally valid with respect to i; if all variantsare false, m = 0 and m/n = 0, i.e., s is universally contravalid with respectto i (WL II, 81 f.).

The logical degree of validity of s is then the degree of validity of s withrespect to is, i.e., with respect to a certain sequence of all simple extra-logical ideas contained in s. In order to be able to apply this notion in auseful way, we have to explain how to count the variants of a proposition,since for each variant of a proposition (as for each proposition in general)there are infinitely many others that are logically equivalent to it (e.g., due

A Formal Reconstruction of Bolzano's Definitions of LogicalTruth and Logical Consequence

Bernard Bolzano


there are infinitely many others that are logically equivalent to it (e.g., dueto replacing a part [b] of the proposition by [non-non-b], [non-non-non-non-b] etc.). If we would count all of them, the resulting fractionalnumber would not be very informative. Bolzano was completely aware ofthis problem (WL II, 79 f.), and he was very creative in developingmethods for solving it as well as many other puzzling problems.

Bolzano is not content with this concept of probability simpliciter butrather continues to develop an even more important relative concept ofprobability. What is at stake here is the probability of a proposition srelative to a set of propositions with respect to a sequence i of ideas,and in particular with respect to i{s}, i.e., the sequence of all simpleextra-logical ideas contained in s or . Its degree can again be representedby a fractional number 0 m/n 1, where n is the number of cases where comes out true and m is the number of cases where {s} comes outtrue (WL II, 171191; for a careful reconstruction of Bolzano's theory ofprobability see Berg 1962, 148150; cf. also Dorn 1987, and Berg 1992b).

In his Tractatus (5.15) Ludwig Wittgenstein came so close to Bolzano'sdefinition of probability that Georg Henrik von Wright felt it to beappropriate to speak of one definition of probability here and call it theBolzano-Wittgenstein definition (Wright 1982, pp. 144 f.).

Bolzano's work on probability was not only of purely theoretical interestto him but also had interesting practical consequences with respect toproblems in the philosophy of science (cf. section 5) and in particular alsowith respect to religious questions (cf. section 9).

5. Epistemology and Philosophy of ScienceBolzano strove for objectivity in pure logic. In applied logic, inparticular in epistemology, however, we have to take into account also thereal, i.e., empirical, conditions of the human mind and thinking according

Edgar Morscher


real, i.e., empirical, conditions of the human mind and thinking accordingto Bolzano (WL I, 66 f.). Nevertheless, he defined the basic concepts ofepistemology primarily on the level of ideas and propositions. This gaverise to the misunderstanding that his investigations are worthless forepistemology proper or even that there is no epistemology proper at all inBolzano's work. The fact that this is by no means the case, will hopefully be shown by the following examples taken from Bolzano'sepistemology.

5.1 Appearances of Propositions and Ideas in Human Minds

In epistemology, we are primarily and directly not concerned withpropositions and ideas, but rather with their appearances (Erscheinungen)in the minds of thinking beings (im Gemt von geistigen bzw. denkendenWesen). One and the same proposition or idea can, as Bolzano says,appear in the minds of different thinking beings or also at different timesin the mind of one and the same thinking being, without thereby beingmultiplied (WL I, 217, WL III, 13, 112). Bolzano says, in such a case,that the thinking being and its mind grasp the proposition or idea inquestion (the corresponding German word being erfassen orauffassen). What happens in such a case is that in the mind of thethinking being under consideration there is (or appears) a mentalphenomenon, or a mental process takes place in it, that is called byBolzano a Gedanke (thought); it can be a subjective idea or asubjective proposition.

In contrast to ideas and propositions (i.e., objective ideas andobjective propositions), subjective ideas and subjective propositionsbelong to the real world, in particular to World 2. For Bolzano, asubjective idea as well as a subjective proposition is a real property (oradherence) of the thinking being in whose mind it appears, or rather ofthis being's mind or soul itself (WL III, 6, 10 f., 109). A subjective ideais a mental phenomenon i.e., an attribute of a mind that grasps an

Bernard Bolzano


is a mental phenomenon i.e., an attribute of a mind that grasps anidea (in the objective sense of the word); and a mental phenomenonthat grasps a proposition is called Urteil (judgment) by Bolzano(WL III, 108). In addition to judgments, there is a second kind ofsubjective propositions, viz. propositions that are merely thought (blogedacht); a merely thought proposition is in fact a subjective idea of aproposition (WL I, 155). Merely having a subjective idea of a propositiondoes not require that we assert that this proposition is true, whereas ajudgment is an act (Handlung) of asserting that the proposition graspedby the judgment is true (WL III, 108, 199).

When we say of a subjective idea or a judgment that it grasps an idea orproposition, respectively, the word grasp is used in a more restrictivesense than before, where it was a thinking being or its mind of which itwas said that it grasps an idea or proposition. This stricter relationbetween subjective ideas and ideas, and between judgments andpropositions is fundamental for Bolzano's epistemology. Instead of sayingin this sense p grasps o, Bolzano will synonymously also use the phraseo is the material (Stoff) of p. Both formulations, however, are asBolzano emphasizes to be understood merely metaphorically (Bolzano1935, 84 f.). The corresponding relation is introduced as a primitiveconcept in Bolzano's epistemology. Let us use the symbol G for thisrelation, whereby p G o is to be read as p grasps o or, alternatively, ois the material of p.

In terms of G, i.e., the strict relation of grasping, the weaker relationof grasping between a thinking being and an idea or proposition can bedefined as follows: a thinking being x grasps (in the weaker sense ofthis word) an idea or proposition o iff there is a subjective idea or ajudgment p in xs mind such that p G o. The relation G is the linkbetween the mental phenomena of World 2 on the one hand and theWorld 3 of ideas and propositions on the other hand. Via relation G,items of World 3 can have a certain, non-causal influence on the mental

Edgar Morscher


items of World 3 can have a certain, non-causal influence on the mentalphenomena of World 2, and these on their part stand in causal relations tothe physical phenomena of World 1.

Due to the domain and the range of G being disjunctive sets, the relationG has the following formal features: it is irreflexive, asymmetric, andtransitive; moreover, G is many-one, but not one-many. Due to Gs beingmany-one, by each subjective idea and each judgment an idea or aproposition, respectively, is uniquely determined as its material (WL III, 8f., 108). Most of the important properties, relations and distinctions,which were defined by Bolzano primarily for ideas and propositions, cantherefore easily be transferred from the sphere of ideas and propositionsto the sphere of subjective ideas and judgments. We therefore will say ofa subjective idea that it is simple or complex, empty, singular or general,an intuition, a concept or mixed according as the idea grasped by it hasthe corresponding property; and the same goes for judgments.

5.2 Subjective Intuitions and Subjective Concepts

An intuition is for Bolzano an idea that is simple and singular (cf. section4.5). The question arises immediately whether such an intuition existsafter all and if so what it can contribute to epistemology. Bothquestions are answered by Bolzano at once: In order to show that thereare intuitions, he hints at subjective ideas (ideas in our minds) that graspobjective intuitions as defined before; and since these subjectiveintuitions exist in our actual world, the corresponding objectiveintuitions must exist (in the sense of es gibt) in the logical realm ofWorld 3. What are the examples of subjective intuitions, however, thatBolzano can put forward in support of his claim? It is subjective ideassuch as the subjective idea of the change in our mind that is theimmediate reaction on an outer object (such as a rose) that stimulates oursenses. In everyday language we usually express such an idea only by theword that (dieses or in Bolzano's old-fashioned orthography

Bernard Bolzano


word that (dieses or in Bolzano's old-fashioned orthography die). Bolzano's rather long-winded explication (in WL I, 326 f.)reveals subjective ideas of a particular sensation or sense-datum as hisfavorite examples of subjective intuitions (cf. also WL III, 21 f.). Theobject and the cause of a subjective intuition of this kind is itself a mentalphenomenon such as a subjective idea or a judgment (WL III, 85). If asubjective intuition is directly caused by an outer object, Bolzano callsit an outer intuition. Also in the case of an outer intuition, its properobject is not the outer cause of it but an inner mental event; humanbeings are capable only of having subjective intuitions whose properobject is a change in their mind (WL III, 89, 145). In other passagesBolzano seems to be less cautious and claims also of an idea such as[Vesuvius] (WL II, 38) or [Socrates] (WL I, 260, in explicit contradictionto WL III, 89) to be a pure intuition. This would obviously result in takingeach rigid designator (in today's terminology) to express an intuition.

In the same way in which a subjective intuition is defined as a subjectiveidea that grasps (in the sense of G) an objective intuition, we can alsodefine subjective concepts and mixed subjective ideas (WL III, 2123).

5.3 Judgments A Priori and A Posteriori

Using the distinction between intuitions and other ideas, Bolzano is nowable to draw an important epistemological distinction among propositionsand in particular also among true propositions: A proposition is aconceptual proposition iff it does not contain any intuition but consistsexclusively of concepts, such as the propositions [God is omnipotent],[Gratefulness is a duty], or [2 is irrational]; all the other propositions arecalled empirical propositions (or also perceptual propositions) byBolzano, e.g.

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