Abstract
Frege’s notion of Begriffsumfang (which preceded the introduction of Wertverlauf) is considered first (8.1). The notion of Wertverlauf was certainly intended to preserve the traditional Begriffsumfang as an interpretation of the “something common” which is yielded by definitions by abstraction. But at the same time Frege wishes to keep his Wertverläufe as unintuitive and semantically undetermined as possible (8.21). Thus it is necessary to consider two Fregean approaches to the notion of class. The first may be called formal; Frege views Wertverläufe simply as a result of definitions by abstraction (8.22). The second approach is that according to which the Wertverlauf signs are intended to denote the traditional extensions and then one may ask Frege what he means by “class”.
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References
UZBG.
GRL, § 68 note, § 107.
GRL, p. 59, § 74.
GRL, § 28; UFT, pp. 96–97.
“Vielheit”, “Mehrheit” (ibid.).
“... les unités sont à part et l’entendement les prend ensemble quelques dispersées qu’elles soyent” (Leibniz [1], V, p. 132 (= Nouv. Ess. 2, XII). “Quis autem credat unitatem, quae est in India, uniri per se cum unitate, quae est in Hispania, ut constituat binarium vel aliquem alium numerum?” Answer: .. dicitur, quod unitates inter se distantes ad rationem numeri eodem modo uniuntur atque indistantes, quia cum supponantur divisae et separatae, perinde est, quod multum vel parum distent“ (Ioannes a Sto Thoma [1], I, p. 562–563). Frege’s objection against the term ”Menge“, or rather the notion of Menge as a definition of number (i.e., the Euclidean definition), is partially based on his conception of ”set“ as implying ”räumliches Zusammensein“. Frege rejects as well Mill’s ”physical“ conception of number (i.e, a number is a concrete perceptible phenomenon), by pointing out that it would imply that a horse in Germany and another horse in America would not make two (horses) (GRL, § 25).
UFT, pp. 96–97.
FUB, pp. 16–17.
Literally: “range of values of a function”.
JOURD, p. 251; letter to Russell in Sluga [1], p. 199.
JOURD, ibid.: “because by that many simplifications could be reached”. Cf. Section 8.22, point 6.
This is, of course, my interpretation; cf. 2.26 and 2.62.
Cf. note 2.
Of course, since GRL Frege’s system imposes some qualifications upon the traditional notion, in particular that the Begriffsumfang should not ambiguously include both subordinate concepts and individuals (this is a consequence of the distinction UF—UO, made, for instance, in UZBG, p. 3). But this is a minor correction, occurring within the familiar notion of the extension of a concept. Frege has often suggested that he intended to identify traditional Begriffsumfänge with his Wertverläufe of logical functions (FUB, p. 16; GRG I, p. VII (bottom), p. X top; § 3, § 9 (p. 14), p. 18, note 1, second line; etc.) JOURD, p. 251 (dated 1910) shows that Frege wished to use the notion of “extent of a concept” in the way in which that notion had “for a long time been spoken of in logic”. Frege’s later adoption of the term “Klasse” (as synonymous with “Begriffsumfang” and therefore with “Wertverlauf” of a logical function) also indicates that the set of arguments satisfying the propositional function is the intended interpretation. (For instance, Nachwort to GRG II, also GRG II, § 161; at the same time he introduces “Relation” to designate the extension of a Beziehung, ibid., § 162.) If Frege had changed the meaning of the term “Begriffsumfang” (by identifying it with some other possible “something in common” provided by the definitions by abstraction), he would have remembered that in GRL he had taken “Begriffs-umfang” in the sense of “his readers” and he would have explicitly explained the new terminology. In fact, in the Nachwort to GRG II Frege shows regret that his “way out” of the contradiction discovered by Russell is a departure from the “traditional” (p. 260 bottom) or the “usual” (p. 257 second column) sense of “Begriffsumfang”. The same text corroborates the fact that Frege had intended to fix in his Axiom V that traditional and usual sense (ibid., pp. 260–261). For all of this see Sluga [1], where some fragments of the Frege-Russell correspondence are published (especially pp. 201–202). When the Wertverlauf is not assigned to a propositional or logical function, it must be interpreted in some other way, for example, as graph of a curve (FUB, pp. 8–9; GRG I, p. 36).
GRG II, § 146.
Hausdorff [1] (III, § 1 and Anhang, p. 453) points out some confusion in the use of definitions by abstraction which would apply to Frege unless our proposed interpretation is adopted. As for the term “formal” in “formal approach”, I follow Hausdorff [1], p. 47.
GRG I, § 3. Also FUB, pp. 9–10, 16.
Materials for this: GRG I, p. 16 note 1; FUB, p. 11 above. Definitions imply equality of Bedeutung and Sinn (GRG I, § 27), but the statement identifying the meaning of the two phrases in question here is not a definition but a Festsetzung (GRG II, § 146). On the other hand, this Festsetzung becomes, in the object language, Axiom V.
GRG I, p. VII, § 9, etc. Also FUB.
GRG I, § 9.
The content of this paragraph (GRG I, § 10) is not invalidated by the Nachwort (GRG II, p. 263).
GRG I, p. 16, line 11 from the bottom, p. 17, line 5.
Our Wertverläufe and the two truth-values.
GRG I, p. 16, lines 6–11.
GRG I, p. 18 note 1. Cf. Whitehead-Russell [1] *51. These authors refer to KSCH (pp. 444–446); cf. also GRG I, p. 2 (middle).
GRG I, § 34.
GRG I, p. 240.
GRG I, § 35.
GRG II, p. 254.
GRG I, § 36.
GRG I, p. 49.
Cf. Section 8.21.
Melanchton, Hospinianus.
Which means that it is a Socratic discovery. Scholz has suggested that the threefold analysis of sentences (S-is-P) should be correlated with a class-interpretation of the predicate symbols, while the twofold analysis (nomen -}- verbum) corresponds to the predicate calculus. Cf. for instance Scholz-Hasenjaeger [1], p. 125, note, Scholz [2], p. 144, Scholz [3], p. 88, note 3. (I owe this reference to Prof. A. R. Raggio.) But this is meaningless if intended as a criterion for detecting the notion of class in the philosophical past. The notion of class can exist independently of one or another analysis of sentences. “Qui quidem notionis complexus et ambitus iam ab Aristotele discernuntur in Analyt. Post. I, 4, quamvis non peculiari nomine” (Trendelenburg [2], § 58). For instance, Olympiodorus [1], p. 49, distinguishes the ßâtoç and the zc tdxoç of predicates. Cf. also Chapter 4, note 3.
That certain forms of ordinary Latin did imply existence is another question (for instance, sentences beginning with “omnes”). “Omnis autem essentia vel quidditas potest intelligi sine hoc quod aliquid intelligatur de esse suo; possum enim intelligere quid est homo vel fenix et tarnen ignorare an esse habeat in rerum natura” (Aquinas [3], cap. IV). “Omnis sol lucet, igitur iste sol lucet”, is valid because there is only one sun (Paulus Venetus [2], f. 111). “Quia quaedam universalia sunt, quae non continent sub se nisi singulare, sicut sol et luna” (Aquinas [1], quoted in Gredt [1], ad 115).
Cf. Section 6.43.
Cantor [1], p. 146: “Es ist ferner zweckmässig, ein Zeichen zu haben, welches die Abwesenheit von Punkten ausdrückt, wir wählen dazu den Buchstaben O; P - O bedeutet also, das die Menge P keinen einzigen Punkt enthält, also streng genommen als solche gar nicht vorhanden ist.” Hausdorff [1], I, § 1, betrays a certain artificiality in his introduction of the idea of empty class. After having described what happens when the elements of a Menge disappear by saying that die Menge... verschwindet, he tries to qualify this revealing expression in a footnote, unsuccessfully I think.
Mauthner [1], III, p. 282: “... der Begriffsumfang oder der Assoziationsbereich des Begriffs ist immer konkret. Und mit diesem einzig Wirklichen am Begriff...” Ibid., p. 323: “Die Satzbildung oder das Urteilen braucht aber nicht immer vom Inhalt des Begriffs auszugehen; der Ausgangspunkt kann auch der Umfang des Begriffs sein, also etwas, was der Wirklichkeit näher liegt.” Cantor [1], p. 379: “Jede Menge wohlunterschiedener Dinge kann als ein einheitliches Ding für sich angesehen werden, in welchem jene Dinge Bestandteile oder konstitutive Elemente sind” (italics ours).
For instance, GRL, §§ 22, 48, 54.
HUSS, p. 322.
HUSS, p. 320 note.
GRG II (p. 86) divides objects into logical and physical ones according to their capacity of being perceived by our senses. Classes are explicitly defined as logical objects (and are excluded from physical objects) in a text published by Bartlett [1], p. 44. Cf. Quine [2], § 22, Bernays-Fraenkel [1], p. 56, Carnap [5], § 37.
Only a whole (Ganze) of material things is itself material. Frege says that he is not sitting upon the class of atoms of his chair (Ibid., as in note 42).
Cf. Section 2.25.
Cf. GRL, §§ 22, 48, 54.
Cf., for instance, GRL § 54.
In KSCH, p. 451, Frege states that all empty concepts have the same extension. For relations, cf. GRG II, p. 160. Frege uses “leere Klasse” perhaps for the first time GRG I, Einleitung. (There he also uses “leerer Begriff”, which is perhaps the first occurrence of such a phrase, but of course the notion of empty concepts was most familiar to the author of GRL.)
But in Frege there are no indications as to how the empty class should be viewed intuitively.
On the contrary, Frege wants to preserve a “proportion” between the empty class and the other classes. Cf. the following text: “Ein Begriff, unter welchen nur ein Gegenstand fällt, hat ebenso einen bestimmten Umfang, wie ein Begriff, unter welchen kein Gegenstand fällt, oder ein Begriff, unter welchen unendlich viele Gegenstände fallen, in welchen Fällen es nach Herrn Husserl gar keinen Inbegriff giebt” (HUSS, p. 322).
GRG II (p. 86, § 147, p. 253, p. 265) include numbers among logical objects. And of course numbers are, since GRL, non-perceptible entities. In GRL numbers were classes of concepts; in GRG they are classes of classes. Thus, in GRG, if a class of two stones were “stony”, number 2 would be somehow contaminated. Cf. Carnap [5], § 40 (Carnap works with the “aut abstract auf concrete”, “aut class auf whole”).
CANT, p. 270.
Cf. Section 10.3.
Cf. Section 8.22.
Dedekind, Schröder in GRG I, Einleitung, Schröder particularly in KSCH.
If the work of art = its raw material, the “idea” (causa formalis, causa exemplaris) vanishes. Then the leere Klasse has to be an Erdichtung (GRG I, p. 2–3). Frege wishes to control die Grenzen dieser Erdichtungswillkür (ibid.) and I understand that he achieves this by the formal introduction of Wertverläufe: “Dann hat es kein Bedenken, von der Klasse der Gegenstände die b sind, auch zu sprechen, wenn es kein b giebt” (KSCH, p. 451).
After discussing Dedekind’s contention that classes are made up of their individuals, and when it is time for giving his own view, Frege says that “die Merkmale den Bestand des Begriffes ausmachen, nicht die unter den Begriff fallende Gegenstände” (GRG I, p. 3). Is this not curiously irrelevant as a rejection of Dedekind’s view? In opposition to the view that classes bestehen aus individuals, Frege affirms that classes do not have their Bestand in the individuals but in the concepts (KSCH, 451). Apparently this means that classes have their Halt (ibid., p. 455) in the concepts, which is of course not satisfactory as a negation of the thesis that classes bestehen aus individuals.
“Dagegen sind das, was den Bestand des Begriffes - oder seines Umfanges [italics ours] - ausmacht, nicht die Gegenstände, die unter ihn fallen, sondern seine Merkmale...” (GRG II, § 150). Here “Bestand” would be used not as “Halt” but in the sense of “der Bestand von meiner Kaß ist nicht des zählen werth” (Lessing, quoted in Grimm [1], article “Bestand”) or of “der Bestand der Bibliothek beträgt 10.000 Bände”. For this ambiguity of “Bestand” cf. Grimm [1], ibid.
Cf. Chapter 4.
For instance, Slupecki [1], in fine.
Frege would use, for example, the term Abgrenzung (CANT, p. 270).
GRG II, p. 260.
“... der Begriffsumfang im hergebrachten Sinne...” (GRG II, p. 260 bottom).
For an ambiguity in the term “class” similar to classF-classT, cf. Professor Geach’s interesting remark about Russell’s use of that term (Geach [1], p. 157).
Quine [5], Geach [3], Resnik [1], etc. Le§niewski’s analysis, in Sobocinski [1], is sui generis.
For Frege’s texts on the nature of axioms, cf. Steiner [1], p. 182.
Scholz-Schweitzer [5], Anhang 1.
“Ein Streit kann hierbei, soviel ich sehe, nur um mein Grundgesetz der Wertverläufe (V) entbrennen, das von den Logikern vielleicht noch nicht eigens ausgesprochen ist, obwohl man danach denkt, z.B. wenn man von Begriffsumfängen redet. Ich halte es für rein logisch. Jedenfalls ist hiermit die Stelle bezeichnet, wo die Entscheidung fallen muß” (GRG I, p. VII). This presentation offers some similarities with the way in which Cantor introduces the Wohlordnungssatz (Cantor ([1], p. 169).
GRG I, § 9.
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Angelelli, I. (1967). Wertverlauf. In: Studies on Gottlob Frege and Traditional Philosophy. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3175-1_9
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