Abstract
In his Prolegomena to Pure Logic Husserl works with Bolzano’s idea that the entire field of truths can be partitioned into several parts, each of which consists of all truths “of a certain kind,” that is, all truths that are germane to a certain homogeneous kind (Gattung) of objects.1 Husserl says that “it is not arbitrary where and how we delimit fields of truth,”2 “the domain of truth is not an unordered chaos,”3 but it is articulated in “natural provinces”4 that are also called “fields of knowledge (Erkenntnisgebiete)”5 or fields of experience (in a broad sense of this word which comports well with common mathematical usage, where, for instance, a system of abstract objects like e.g. the natural numbers equipped with certain functions and relations is said to be a “field of experience”). Each field of experience in Husserl’s sense can be viewed as “an independent reality with its own experimentally determined mathematical structure.”6 In this sense fields of “the purely mathematical sciences whose objects are numbers, manifolds (Mannigfaltigkeiten), etc., things thought of as mere bearers of ideal properties, independently from real being or not being,”7 are also to be considered fields of knowledge or of experience.8
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- 1.
Cp. the allusion to Bolzano’s WL in PR 29, PRe 73.
- 2.
PR 5, PRe 54.
- 3.
PR 15, PRe 62.
- 4.
PR 25, PRe 70.
- 5.
PR 19, PRe 65.
- 6.
Here I borrow the terminology from Webb 1980, 79.
- 7.
PR 11, PRe 69.
- 8.
Tieszen 2004 rightly stresses that “Husserl also says that among the eidetic sciences some are exact and some are inexact. Mathematics and logic are exact … mathematics and logic set the standard for what is clear, distinct and precise” (33–34). Our considerations in this chapter always refer (unless otherwise specified) to exact sciences.
- 9.
Bolzano 1810: “In the realm of truth, that is in the collection of all true judgements, reigns an objective interconnection that is independent of the contingent fact that we subjectively acknowledge it; it is in virtue of this that some of those judgements are the reasons of others and the latter the consequences of the former” (Part II, §2). Cp. Cavaillès 1938, 54–55.
- 10.
It would be more correct to translate “Begründung” as “grounding” and to reserve “foundation” for “Fundierung”. The former is Husserl’s version of Bolzano’s “Abfolge”, as we shall try to show. The latter is used in the third LU to signify one of the possible dependence relations between the parts of an object. However, for the sake of fluency of style we shall use both terms (accompanied by the German word in brackets).
- 11.
See for instance the title of §9 (PR 22, PRe 68): “Methodical ways of proceeding in the sciences – in part groundings, in part auxiliary devices towards groundings”.
- 12.
PR 15, PRe 62.
- 13.
“The task of the theory of science will therefore also be to deal with the sciences as systematic unities of this or that sort…” Each science “can be subsumed under the concept of method, so that the Wissenschaftslehre’s task is not merely to deal with the methods of knowledge in the sciences, but also with such methods as are themselves styled sciences” (PR 25, PRe 70).
- 14.
PR 25, PRe 70.
- 15.
A broadening insofar as, besides the problems of order, organization and systematization which pertain to the exposition (Darstellung) of a theory, we find in Husserl, from c.1896 onwards, the idea that not only mathematical but all theories insofar as they are formalized are to be made the object of the investigation. And formalization in this sense is not present in Bolzano. For more on this issue see next chapter.
- 16.
PR 12, PRe 60.
- 17.
PR 29, PRe 73. Actually Bolzano defines Wissenschaftslehre as “the aggregate (Inbegriff) of all those rules which we must follow when subdividing the entire realm of truth into single sciences and representing them in special textbooks, if we want to proceed in a useful way” (WL I, §1, 7).
- 18.
PR 32, PRe 76.
- 19.
See for instance PR 47, PRe 87: “it is … easy to see that each normative, and, a fortiori, each practical discipline, presupposes one or more theoretical disciplines as its foundations, in the sense namely, that it must have a theoretical content free from all normativity, which as such has its natural location in certain theoretical sciences …”
- 20.
WL I, §13, 53.
- 21.
PR 12, PRe 60.
- 22.
Cp. Tieszen 2004, 28.
- 23.
PR 246, PRe 239.
- 24.
Loc. cit.
- 25.
For this conception see also Jan Berg, BGA, vol. 11/1, 18.
- 26.
- 27.
Casari 1987, 330. The epithet “etiological” alludes to the Greek word ‘aitía’ which means whatever is specified in an answer to a why-question. (Aristotle’s famous theory of the “four aitíai” is a theory of four kinds of because, rather than a theory of four kinds of cause.)
- 28.
Op. cit., 331–332.
- 29.
Bolzano, WL II, §198, 341. Cp. Aristotle, An. Post. I, 13; Aquinas, Summa Theologiae I, quaestio 2, art. 2.
- 30.
Bolzano, WL IV, §525, 261–2. Cp. Bolzano 1834, I, §3, No. 2.
- 31.
17, II, §12; cp. Bolzano 1834, I, §3, No. 2.
- 32.
Op. cit., §26 d). The editor of this volume of the Husserliana neither gets the title of Bolzano’s booklet right nor the name of the editor of the 1926 edition (Heinrich Fels). Husserl says that this early work of Bolzano had been “nearly unavailable”. This does not imply that he himself only came across it in 1926. Another Brentanist, Benno Kerry, had referred to it already in the eighties of the nineteenth century: see Künne 2009, 327.
- 33.
PR 228; PRe 225.
- 34.
PR 15; PRe 62.
- 35.
The relation of derivability is characterized by rules of inference that exclusively concern the form of the involved propositions.
- 36.
See Appendix 4 to this chapter for some explanation of these notions.
- 37.
See below.
- 38.
PR, §63, 231.
- 39.
PR, §64, 233; PRe 229.
- 40.
In this respect, see the following quotation from PR, Chapter XI.
- 41.
PR 235; PRe 229.
- 42.
Bolzano, WL II, §200, 346–48. But cp. WL II, §221 note, 388.
- 43.
Bolzano, WL II, §199, 345. For further references see Künne 2008, 400.
- 44.
See below, Appendix 4.
- 45.
There is a formal treatment of these notions in Appendix 4.
- 46.
For example, see LV96, 246.
- 47.
PR 232; PRe 228.
- 48.
See footnote 49 below.
- 49.
See for instance the calculus of an axiomatic-synthetic kind presented in LV96 (Appendix 5). One of the principles that must precede the constitution of every deductive theory is the modus ponens of traditional logic, that is, the rule: A, A → B/B. This is a typical inference rule of the kind Ableitbarkeit.
- 50.
The inference: a → b, b → c, c → d/a → d can for instance be decomposed into the simpler inferences: a → b, b → c/a → c and a → c, c → d/a → d. That Husserl thinks this way clearly emerges, as we said above, from his characterization of Begründungen as schemata “If (the Begründungen) were formless and lawless, if it were not a fundamental truth that all Begründungen have certain inherent ‘forms’, not peculiar to the inference set before us hic et nunc, but typical of a whole class of inference …” (PR 20; PRe 66). Here it is quite natural to think that with these ‘forms of inference’ that act as patterns for a whole class of particular inferences we are looking for forms of inferences of the most elementary kind.
- 51.
All quotations in this paragraph are from PR 15–16; PRe 62–63.
- 52.
PR 228; PRe 225.
- 53.
Cp., for instance, this passage: “…whether it is the domain of cardinal numbers (Anzahlengebiet) or some other conceptual domain (Begriffsgebiet), that the general arithmetic … governs” (PoA 7; PdA 7). The notion of a “conceptual determination (Begriffsbestimmung) that delimits a field of knowledge” derives from Bolzano. See e.g., WL I, § 2, 9: “… the field of a science that we obtain by means of this conceptual determination (das Gebiet der Wissenschaft, die wir durch diese Begriffsbestimmung erhalten)”.
- 54.
PoA 414 ff. PdA 434: “If we understand by cardinal number the answer to the question ‘How many?’ then the number series is the closed manifold of particularizations that are possible in the sphere of the concept how many.”
- 55.
PoA 238 ff., PdA 226 ff.
- 56.
PoA 412, PdA 433.
- 57.
Cp. for instance PoA 414, PdA 435: “operations which are grounded in the Idea of the cardinal number” or LV96, 241: “the general logical laws divide in several groups: in laws which have their roots in the concept of proposition, in laws which have their roots in the concept of concept, in laws which have their roots in the concept of object”.
- 58.
Fine 1994 maintains that as regards explaining the concept of essence the classical doctrine of real definitions is superior to an account in terms of necessity. Every general law that affirms an essential relation among the objects of a certain field is a necessary truth, but essence cannot be explained by modal notions. Fine appeals to a conception of essence that is admittedly inspired by the Husserlian one, that is the notion of nature/essence of an object is primitive. “[E]ach class of objects, be they concepts or individuals or entities of some other kind, will give rise to its own domain of necessary truths, the truths which flow from the nature of the objects in question …” An important development of this conception of “to be grounded in the essence of” can be found in Mulligan 2004.
- 59.
PR 15; PRe 62.
- 60.
PR 228; PRe 225–226.
- 61.
PR 236–237; PRe 232.
- 62.
PR 241; PRe 235.
- 63.
PR 240; PRe 234.
- 64.
Concerning our choice to refer to non-propositional components of propositions as notions see below Appendix 4, footnote 96.
- 65.
PR 239; PRe 234.
- 66.
Cp. Casari 1999; Ortiz Hill 2002, 87–88; Tieszen 2004, 26–34. As Tieszen rightly stresses, in FTL the third level of Husserl’s stratification of ‘objective formal logic’ is constituted by what Husserl calls ‘truth logic’ (Wahrheitslogik), an attempt to identify material conditions of truths for judgements that are already established to be consistent. We will not consider this level here.
- 67.
Null & Simons 1982, 448.
- 68.
Loc. cit.
- 69.
See LV96, 135–141. “The three forms of connection discussed above (sc. conjunction, disjunction and implication) are the only elementary ones for propositions in general” (140). Negation is considered as an operation, since it takes only one argument: “The operation of negation is applicable to any proposition, that is to every proposition corresponds its denial (negation). This is a proposition that has the original proposition as its topic (Subjekt) and denies its truth.” Husserl also regards the affirmation of a proposition, that is the passage from ‘S is p’ to ‘[That] S is p is true’, as an operation and writes: “Both affirmation and negation of a proposition … [are] propositions about (über) propositions” (135).
- 70.
PR 244; PRe 237. Cp. also FTL [ed. 1929], 77–78.
- 71.
All quotations in this paragraph are from PR 245–246; Pre 237–239.
- 72.
PR 245–246; PRe 238–239.
- 73.
For a formal presentation of Husserl’s theory of propositional inferences, see below Appendix 5.
- 74.
PR 246; PRe 239.
- 75.
PR 246; PRe 239.
- 76.
PR 247; PRe 239.
- 77.
PR 248; PRe 241. Cp. also Ortiz Hill 2002, 88.
- 78.
Cp. Casari 2000. Ortiz Hill 2000a rightly stresses Husserl’s distinction between pure sets in Cantor’s sense and manifolds: “[Husserl] had come to clearly distinguish his manifolds from Cantor’s Mannigfaltigkeiten or sets …” (173), and in 2002 she writes: “Husserl’s manifolds are not aggregates of elements without relations. It is precisely the relations that are essential and serve to distinguish a manifold from a mere aggregate.… Husserl saw manifolds as aggregates of elements that are not just combined into a whole, but are continuously interdependent and ordered …” (97). Apparently Ortiz Hill, too, assumes that manifolds are sets provided with some topological structure. Husserl clearly marks the relevant distinction when he says in FTL: “From [a] particular field of objects [we obtain] the form of a field or, as the mathematician says, a manifold. It is not a mere manifold, for that would be the same of a mere set … Rather it is a set whose special feature is … that it is conceived as ‘a’ field which is determined by [a] complete group … of axioms-forms…” (81).
- 79.
Cp. Null & Simons 1982, where an interpretation of manifolds as certain well-defined classes of relational structures is developed.
- 80.
For a precise account of these notions see next chapter.
- 81.
PR 247; PRe 240.
- 82.
For Bolzano, too, syncategorematic expressions are meaningful, but Bolzano does not distinguish between dependent and independent meanings.
- 83.
This is an application to the field of meaning of the notion of dependence that was explained in the third Investigation with respect to objects in general. Dependent are “contents not able to exist alone, but only as parts of more comprehensive wholes” (LU IV, §7, 311; LI 506).
- 84.
LU IV, §10, 318–319; LI 511.
- 85.
LU IV, §10, 319; LI 511–512. Cp. Tieszen 2004, 26 ff.
- 86.
The categorization of linguistic expressions that is invoked here could be usefully compared to the typification of entities in Russell’s theory of types.
- 87.
LU IV, §10, 317; LI 510.
- 88.
LU IV, §14, 338; LI 526.
- 89.
LU IV, §14, 336; LI 524.
- 90.
For Bolzano objectuality is primarily a property of notions (Vorstellungen an sich): a notion is objectual if and only if there is an object that falls under it, and objectless otherwise.
- 91.
LU, IV, 294; LI, 493.
- 92.
“One must, of course, distinguish the … incompatibilities to which the study of syncategorematica has introduced us, from the other incompatibilities illustrated by the example ‘a round square’” (LU IV, §12, 326; LI 516).
- 93.
All quotations in this paragraph and the next two are from LU, 3, §12 (‘Basic determinations concerning analytic and synthetic propositions’), 254–256; LI II, 457. On Bolzano’s account of analyticity see Morscher 2008, 60–63, 161–167, Künne 2008, 233–304 and, for a comparison with Husserl, Künne 2009, §§3–4.
- 94.
LU VI, §14, 334; LI 523.
- 95.
WL II, §154, 100.
- 96.
Notions are either (objective) concepts or (objective) intuitions in themselves (Anschauungen). Intuitions are said to be notions that are simple and have exactly one object, and concepts are defined as notions that are not intuitions and do not contain any intuition as part. An intuition is expressed in an utterance of ‘this’ if the demonstrative is used to refer to something perceptually given. Cp. WL I, §§72–78, 325–360.
- 97.
WL I, §108, 513–515. It was invoked for the first time in WL I, §66, 299–300 where the topic of indexicality is briefly touched: the notion that is now expressed by “a presently living human” is replaced by another notion when this phrase is uttered at a different time, since the time-specifying component of the former notion is varied.
- 98.
Nowadays this terminology is prone to cause a misunderstanding, since it has become customary to use this term in a purely syntactical sense. ‘Deducibility’ would have a strong syntactical connotation, too; whereas for Bolzano derivability is a (quasi-)semantical relation. Nevertheless, we have preferred to stay close to Bolzano`s wording.
- 99.
According to Bolzano all propositions can be expressed by instances of the schema ‘A has b’ where ‘A’ expresses any notion (of whatever complexity), while ‘b’ expresses a notion of a property (Beschaffenheit). Bolzano’s copula ‘has’ expresses the notion of exemplification which is a logical notion. In WL II, §148, 84 Bolzano maintains that there is no sharp line of demarcation between logical and extra-logical notions, but he leaves no doubt that his copula expresses a logical notion.
- 100.
WL II, §155, 113–114.
- 101.
In the quoted passage, Bolzano explicitly requires compatibility also for the consequences M, N, O,…. This, however, already follows from the definition of derivability. We say that certain propositions follow from certain others when (i) there is a substitution that makes the premises true (compatibility); (ii) all substitutions that make the premises true, also make the conclusions true. From (i) and (ii) follows: (iii) there is a substitution that makes both the premises as well as the conclusions true.
- 102.
WL II, §155, 122–123.
- 103.
- 104.
Cp. George 1983.
- 105.
LV96, 238–239.
- 106.
LV96, 239.
- 107.
WL II, §155, 123.
- 108.
Cp. Paoli 1991, 233–234.
- 109.
Compare the proof given above.
- 110.
WL II, 125–126.
- 111.
- 112.
WL II, §162, 191.
- 113.
WL II, §177, 221–222; cp. §168, 207.
- 114.
WL II, §162, 192.
- 115.
Loc. cit.
- 116.
WL II, §203, 352; §209, 362.
- 117.
WL II, §205, 357; cp. §212, 370, §214, 374. More on this topic in Künne 2003, 46, 151–152.
- 118.
WL II, §204, 356.
- 119.
Cp., for example, Frege, Der Gedanke (1918), 61 (original pagination).
- 120.
WL II, §213, 371.
- 121.
WL II, §206, 359.
- 122.
WL II, §206, 359–360, though it can happen that different reasons share some partial consequences.
- 123.
WL II, §214, 374–376.
- 124.
WL II, §207, 360.
- 125.
See also the incipit of §201, 349: “If the relation of consecutivity is not a species of the relation of derivability, one cannot hope to explain the former in terms of the latter; hence one must look for other cognate concepts”.
- 126.
WL II, §200, 348.
- 127.
WL II, §168, 208; §201, 349–350.
- 128.
Cp. Casari 1987, 332.
- 129.
Leibniz 1704, Book IV, Chapter xvii, §3.
- 130.
WL II, §202, 351.
- 131.
Introduced in G. Gentzen, Untersuchungen über das logische Schließen, 1935. Comparing this with WL II, §220, 380–383, one should keep in mind that Gentzen’s notion of a “normal proof” is a syntactical concept.
- 132.
WL II, §§217–219, 377–380.
- 133.
This problem is central for an important family of non-classical logics, so called substructural logics, which includes (among the others) linear logic and many-valued logics. Indeed, linear logic is both resource-conscious and attentive to the problem of relevance, while many-valued logics are (usually) resource-conscious but not attentive to the problem of relevance. Bolzano’s logic is sensitive to the latter problem, but it is not, as we just saw, resource-conscious.
- 134.
WL II, §221, 384.
- 135.
WL II, §221, 388 note.
- 136.
WL IV, §530.
- 137.
Beyträge II §28, WL II, §223, 391–395.
- 138.
WL II, §221 note, 388.
- 139.
An analogous calculus is developed in the Logikvorlesung of 1902.
- 140.
This contrast in current logic corresponds to that between a Frege-Russell-Hilbert style calculus and a Gentzen style one.
- 141.
Cp. PoA 480; PdA 476.
- 142.
Here the symbol ‘→’ stands for the relation of conditionality (Bedingtheit), or ‘inferability’ between propositions. Cp. LV96, 254. See also Section 2.8.2.2 below.
- 143.
LV96, 243–245.
- 144.
Repr. in: Husserl, Aufsätze und Rezensionen, 3–43.
- 145.
LV96, 247.
- 146.
Cp. Schröder Review and LV96, 242–248.
- 147.
In a formal proof (understood as a finite sequence of formulae that satisfies the previously mentioned requirements) the first formula has to be an axiom. By contrast, the second and all subsequent formulae are either axioms or derived from the previous ones by application of a rule of inference. But if in the proof we admit – as Husserl seems to do – the possibility of appealing to a previously proven theorem (without proving it again), the second formula can also be obtained through an instance of the first, the third through an instance of the first or the second, and so on.
- 148.
LV96, 250.
- 149.
LV96, 249.
- 150.
LV96, 249–250.
- 151.
LV96, 250.
- 152.
LV96, 251.
- 153.
Loc. cit.
- 154.
Loc. cit.
- 155.
LV96, 251 f.
- 156.
Most of the time we shall not use quotation-marks in our explanations when talking about symbols. Husserl’s prose is very loose in this respect, too, but hopefully no confusions will be caused by this sloppiness.
- 157.
€ is the symbol Schröder uses to signify the relation of inclusion among classes.
- 158.
In this way logical laws with implicative form for Husserl also play the role of rules of inference. This explains the terminological wavering between “theory of propositional inferences” and “theory of propositional laws”.
- 159.
LV96, 254.
- 160.
Loc. cit.
- 161.
LV96, 253.
- 162.
LV96, 254.
- 163.
Loc. cit.
- 164.
LV96, 254 f.
- 165.
LV96, 255.
- 166.
Loc. cit.
- 167.
As is the case for example in the Principia Mathematica of Russell and Whitehead.
- 168.
Loc. cit.
- 169.
We reproduce them here in Husserlian notation as well as in a transcription using current symbolism. With ∀[A] we indicate the universal closure of A.
- 170.
In the text we find Π(A + B) € ΠA+ΠB, but this is clearly a mistake or a trasnscription error; besides, Husserl explicitly points out: “only one half is valid” (LV96, 259).
- 171.
This is what we find in the text, and surely it is a logical law. However, it could be, as above, a mistake or transcription error (per the logical law Π(A0) € (ΣA)0).
- 172.
We do not use Husserl’s notation here; moreover we correct where necessary some errors that are present in the text regarding the progressive numeration (errors that indicate various stages of rewriting and rethinking these pages).
- 173.
The text erroneously has a biconditional instead of the main conditional.
- 174.
LV96, 258.
- 175.
LV96, 259–260 (the lower-case letters are in the text).
- 176.
It is unclear what is meant by ‘β’ in ‘XIβ’.
- 177.
A∨(B∧¬B) ↔ A is not listed among the theorems.
- 178.
LV96, 260.
- 179.
Cp. Bolzano on Pilate’s “What I have written I have written”: WL II, §148, 85, end of note 1.
- 180.
LV96, 260.
- 181.
In his last published paper, “Compound Thoughts” (1923), Frege maintains that A∨A and A∧A express the same thought as the plain A and that A→A expresses the same thought ¬(A∧¬A): see p 49 and 50 (original pagination).
- 182.
LV96, 261.
- 183.
LV96, 262 (my italics).
- 184.
LV96, 163–165.
- 185.
WL II, §127, 9.
- 186.
Aristotele, De Int. 12: 21b9–10.
- 187.
WL II, §127, 10.
- 188.
Frege 1892, 194.
- 189.
LV96, 144.
- 190.
Frege 1892, 197 n.
- 191.
Frege 1892, 193 and n.
- 192.
Cp. Künne 2003, 65 f.
- 193.
Husserl emphasizes that, even if Γ is considered as an indeterminate object (an Etwas), ‘Γε a’ is not to be confused with the existential statement “Something is a”, for which he suggests the formal notation ‘Σa’.
- 194.
LV96, 263.
- 195.
Husserl’s use of his notation is somewhat unstable: instead of ‘AB’ he sometimes writes ‘AB’.
- 196.
Schröder 1877, 1.
- 197.
Frege 1881, 15; transl. 14.
- 198.
Frege 1882, 100.
- 199.
Boole 1847, 58–59.
- 200.
For the last two points cp. Künne 2009, Chapter 5 and the literature registered there.
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Centrone, S. (2009). The Idea of Pure Logic. In: Logic and Philosophy of Mathematics in the Early Husserl. Synthese Library, vol 345. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3246-1_2
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