Abstract
The present chapter begins with the notion of function as found in the Begriffsschrift of 1879 and some related questions (6.1–6.3). The distinction function-argument is first considered from the point of view of its being preferred by Frege to the traditional subject-predicate (6.1). Frege’s intention is to introduce a symbolism for relational propositions, which satisfies an old logical desideratum (6.11). In this context, there is occasion to mention Frege’s relation with a member of Brentano’s school, A. Marty (6.2). Finally, the notion of function in BG is presented (6.3).
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References
Frege explains his new terminology in BG, §§ 3, 9 (in fine).
Cf. Section 6.3.
Cf. BG, p. IV, § 3.
“Objekt, grammatisch res obiecta, das Ziel der Thätigkeit” (Grimm [1], “Objekt”). For the English equivalent, cf. Nesfield [1], § 136: “the word or words denoting that person or thing, to which the action of the verb is directed, are called the object of the verb”. For the present section, cf. also Reichenbach [2], p. 24.
BG, p. 3.
Cf. Grimm [1], or again Nesfield [I], § 2: “the word or words which say something about the person or thing denoted by the subject [chrwww(133)] are called the predicate”.
Of course “subject—object” could be maintained, but only with the meaning of “referent—relatum”.
BG, § 3, in fine.
Cf. Sections 6.42, 6.73, and notes 44 and 45.
“Mais il faut particulièrement remarquer ici que toutes les propositions composées de verbes actifs et de leur régime peuvent être appelées complexes, et qu’elles contiennent en quelque manière deux propositions” (Port-Royal [1], Partie II, chap. 5).
Port-Royal [1], Partie III, chap. 9.
A non-trivial syllogism is one involving relational propositions: “La loi divine commande d’honnorer les rois; Louis XIV est roi; FUNCTION Donc la loi divine commande d’honnorer Louis XIV”. (Port-Royal [1], Partie III, chap. 9).
Port-Royal [1], Partie II, chap. 2. In reference to the syllogism quoted in the preceding note, Port-Royal asks what “les rois” is; since it is not an attribute (because it is not said about anything), the authors agree that “il est le sujet d’une autre proposition, enveloppée dans celle-là” (Port-Royal [1], Partie III, Chap. 9).
BG, §§ 4, 7; GRG I; VERN, in fine; UZBG, p. 10. Cf. Port-Royal [1], Partie II, chap. 2.
Marty [1], p. 56.
Ibid., p. 55.
Ibid., p. 57, note 2. Frege gives Haus as an example of “unbeurtheilbarer Inhalt” in BG, p. 2, note.
Nevertheless, Marty realizes that relational propositions are a strong motivation in Frege’s philosophy (p. 57, note 2).
BG, § 3, in fine.
I am grateful to Professor O. Funke (University of Bern) for having informed me (in a letter dated January 22, 1961) that he did not recall having seen Frege’s name in the Nachlass of A. Marty.
BG, p. 16.
This applies to the definition in BG, p. 16, but it does not apply, of course, when Frege says that we may regard in several different ways the same conceptual content as a function of some argument (BG, p. 17).
In BG, p. 15, a sentence like “oxygen is lighter than carbon dioxide” is analysed into a function “being lighter than” and two arguments, “oxygen” and “carbon dioxide”, which is a “natural” order. But in BG, § 10 it is said that functions may be viewed as arguments.
BG, p. 17.
BG, § 1 (unsaturatedness might be seen as incidentally appearing in BG, p. 14, cf. 2.11).
BG, p. 17.
Ibid.
BG, p. 15.
Cf. Section 2.11.
Cf. Section 2.61.
GRL, p. X.
Prefixing the definite article is indeed a “transformation”. The concept “ceases to be” a concept; cf. text (4) quoted in Section 6.532.
Cf. text (10) quoted in Section 6.532.
GRL, § 94f; GRL, p. 119; UFT, p. 97f. In a paragraph of GRL whose title is Begriffe sind von Gegenständen zu unterscheiden, it is obvious that Frege associates his principle of the sharp distinction of concepts and objects with the mistake of building definite descriptions where uniqueness and existence are not guaranteed. Cf. text (13) in 6.532.
“Round square” is not an empty name, but the name of an empty conceptchrwww(133) Frege says in KSCH (p. 454). For Frege, to use “the round square” implies that “there is an x such that it is round and it is a square and it is the only entity fulfilling this condition” (cf. 6.532), i.e. “the round square” is an empty name. But there is another use of the definite article in the philosophical tradition according to which “the round square” means rather the empty Fregean concept. This other use of the definite article might be called Meinong’s beard.
GRL, p. 77.
In BGGE, p. 198, line 12, Frege would correct this.
“zum Subjecte”. In GRL (p. 82 bottom) Frege speaks of the two subjects of a relation.
This could be held consistently along with the main thesis of GRL (i.e. concepts as subjects of statements of number); it would suffice to consider implicit higher predication (cf. 6.83).
Such warning is not trivial with respect to traditional predication theory. Cf. 3.2, note 20.
Cf. BG, chapter on function; GRL, p. 82.
With regard to the restriction to singular beurtheilbaren Inhalt, I think this is done in order to avoid having “whales” as object and “mammals” as concept in such sentences as “all whales are mammals”. Cf. GRL, pp. 60 and 100 above.
For this expression, well known in traditional philosophy, cf. for instance Gredt [1], I, 123, 1: “carentia divisions ex parte principiorum formalium, quae constituunt quidditatem”; in Fregean terms: the set of marks making up a concept. Frege also sees in the concept an Einheit (GRL, p. 66).
GRL, §§ 51, 52.
Cf. text quoted at the beginning of the preceding section; also, GRG II, § 56.
Kant [3], quoted in note 152.
Cf. Chapter 5, note 5. (Of course, “Sinn” here has no relation with Frege’s technical term).
GRL, p. 64; also: “Plural ist nur von Begriffswörtern möglich” (GRL, p. 49).
Traditional unsaturated entities did not require unsaturated names; cf. Section 6.77.
GRL, p. 66.
CANT, p. 270.
GRL, § 34: “Der Begriff Katze [chrwww(133)1 ist aber eben dadurch nur Einer.”
GRL, § 54.
GRL, § 48.
Cf. Section 6.83.
To use an expression of Bernays-Fraenkel [1], p. 58.
But this is the weak objectivity (cf. Section 2.51); Frege asks from concepts only that they not be subjective in the sense of empirical subjectivity (in modern terms) or in the sense of not being “conceptus subiectivus” (in earlier terminology). But then Frege’s demand is trivial.
GRL, § 49.
For instance, GRL, § 51.
For instance, GRL, p. 87. The idea of scharfe Begrenzung had already appeared in Frege’s dissertation (venia docendi): “Wenn wir in jedem Falle beurtheilen können, wann Gegenstände in einer Eigenschaft übereinstimmen, so haben wir offenbar den richtigen Begriff der Eigenschaft” (DISS2, p. 2). Any function (in particular, logical functions) must render a well determined value for every object; cf. GRG II, § 56.
GRL, § 95.
GRL, § 94; also § 95, third line, § 96 in fine and p. 119.
Kerry [1], p. 272.
Ibid.
For Cantor, cf. Cantor [1], p. 440–442. For Marty, cf. Section 6.2.
Kerry [1], p. 273 note.
Cf. Section 6.1.
Kerry [1], p. 281f.
Kerry [1], p. 265.
Aristotle [3], (Anal. Priora) A, 27.
For some explanation of this, cf. Section 6.54.
“Es zeigte sich, dass die Zahl, mit der sich die Arithmetik beschäftigt, nicht als ein unselbständiges Attribut, sondern substantivisch gefaßt werden muß” (GRL, p. 116).
GRL, p. X; this is the text referred to by Kerry in order to support the first part of the thesis that Frege introduced in GRL an “absolute” separation of concepts and objects.
Section 6.41.
Cf. Kerry’s text in Section 6.51. I quote the passage of GRL in Section 6.42.
GRL, p. 68.
GRL, § 66.
GRL, p. 82.
Cf. Section 6.75.
Section 6.42. I mean not necessarily in Kerry’s way.
Section 6.41.
“Ich bin nun zwar der Meinung, daß beide Einwände gehoben werden können; aber das möchte hier zu weit führen” (GRL, p. 80 note).
Our text (3) should be compared with GRL, § 34, where it is stressed that the process of abstraction leads to nur Einer (concept).
GRL, § 54.
Kerry [1], p. 273.
Ibid: “chrwww(133) so z.B. bedeutet in dem Satze ”der Begriff, von dem ich jetzt eben spreche, ist ein Individualbegriff“, der aus den ersten acht Wörtern bestehende Name, wiewohl er mit dem bestimmten Artikel versehen ist (”der Begriff“), sicherlich einen Begriff und - falls ein Begriff nicht zugleich auch Gegenstand sein kann - keinen Gegenstand.”
“Man kann nämlich auch positiv zeigen daß Begriffe zugleich Gegenstände -und hierin liegt ja auch das Umgekehrte involvirt - sein können. Betrachtet man auch den Fall des Begriffes von einem Begriffe [chrwww(133)] und den Fall einander subordinirter Begriffe als verschieden von den Verhältnissen, wie es zwischen Begriff und Gegenstand besteht, und läßt sie darum hier nicht in Frage kommen, so muß doch, um gleich ein Beispiel zu geben, eingeräumt werden, daß vermöge des Urteiles: ”Der Begriff `Pferd’ ist ein leicht gewinnbarer Begriff“ der Begriff ”Pferd“ auch Gegenstand ist und zwar einer der Gegenstände, die unter den Begriff ”leicht gewinnbarer Begriff“ fallen. Ebenso fallen unter Begriffe, wie der von Frege in seiner Darlegung [chrwww(133)1 benutzte: ”gleichzahlig dem Begriffe F“ einer ist, lauter Begriffe als Gegenstände” (Kerry [1], p. 274).
“Wenn daher Frege nicht müde wird zu betonen, daß die 0, die 1, u.s.w. nur als Gegenstände anzusehen seien - Begriffe seien sie jedenfalls nicht - so wäre dem gegenüber vorerst das Resultat der obigen Ausführungen: daß Gegenständlichkeit und Begrifflichkeit einander nicht ausschließen, geltend zu machenchrwww(133)” (Kerry [1], p. 274).
Cf. Section 6.41.
BGGE, p. 193.
Cf. Chapter 3, note 20.
Cf. note 88.
BGGE, pp. 202–204.
BGGE, p. 195.
Cf. 6.531.
Moreover, it should be observed: (a) References to the definite article continue to concern primarily the passage from ç,(x) to (?x)(Tx) and not the magic creation of non-concepts out of concepts. Cf. FUB, p. 5; BGGE, p. 204, 1.10–13. (b) Frege’s references (in BGGE) to his previous works are quite questionable. They are: (bl) p. 195, note 2, (b2) p. 197, note 1, (b3) p. 199. The first concerns the “criterion” of the definite article; as we mention in the present section, Frege refers to texts (4), (7) and (8) of our list in 6.532. Frege should at least have taken into account the fact that these texts hardly provide a clear criterion in the sense of BGGE. The second reference to GRL (b2) amounts to saying that the idea of a “transformation” (verändern) of concepts into objects in GRL, p. X was exactly what Kerry understood it to be. The third reference (b3) concerns GRL, p. 80, note; the same remarks apply here as in the case of (bl).
BGGE, p. 196.
BGGE, pp. 196–197.
Ibid.
BGGE, p. 204: “indem ein Gegenstand genannt wird, wo ein Begriff gemeint ist”. Cf. BGGE, p. 197, note 3.
“Die Sprache aber ist, wie wir sehen, das Wahrhaftere; in ihr widerlegen wir selbst unmittelbar unsere Meinungchrwww(133)” (Hegel [1], A, 1).
If it is at all possible to say that what we mean by “the concept F” is a concept.
BGGE, p. 197.
Ibid.
BGGE, p. 198.
BGGE, p. 199.
Ibid.
BGGE, p. 205.
BGGE, p. 199.
BGGE, p. 200–201. It is important to observe that sentences (4), (5) and (7) are said to express the same Gedanke (BGGE, p. 196 note).
Ibid., p. 201.
FUB, p. 19; GRG 1, p. 7 (at the end of § 2).
Frege says “vertreten”, BGGE, p. 197.
For the present section cf. Wells [1], p. 550.
DISS5, p. 2; also DISSI, introduction, “eindeutige Zuordnung”.
BG, pp. 18, 57.
Cf. Section 6.1.
Cf. for instance Section 6.42. “Predicability” is the name for unsaturatedness in the paradigmatic case of concepts; BGGE, p. 197, note.
To this extent I am inclined to disagree with Prof. Bergmann’s interesting argument (Bergmann [2]). As to the opposition of concept and function in Cassirer [1], there seems to be some misunderstanding and unclearness involved. For instance, on page 28 of Cassirer’s book the “advantage” of functions consists in that they (their names!) have variables, and that thereby the Inbegriff, the class of entities falling under the concept, is somehow preserved. This is what enables Cassirer to hold that in the new approach abstraction does not lead to ever more impoverished contents, in fact, the poorer the concept, the wider the class. Cassirer seems to oppose one-place to many-place predicates, rather than exemplification to mapping. In any case the Fregean synthesis of the latter appears not to be reached by Cassirer’s argument.
FUB, p. 28f; also GRG I, § 72.
As possible antecedents, there is perhaps the conception of concepts as rules (Gassendi, Kant); cf. Aaron [1], pp. 112, 93.
GRL’s reference to unsaturatedness, on the contrary, is not preceded by that semantic distinction (GRL, § 70).
Examples are abundant in Frege’s works. The first (apart from GRL, § 70) are to be found in FUB, pp. 17, 18, 19, 27.
This is also a constantly recurring theme in Frege’s works. Apart from GRL, § 70, the first references appear in FUB, pp. 6, 7–8.
Cf. below, Section 6.771.
Cf. ibid., note 4.
Frege would prefer this symbolism rather than “2x”, in order to express the function “the double of a number x”. The reason is that “x” is used also in order to express universality (“any”).
FUB, p. 7–8. The image of open and closed intervals seems to have played a dominant role in Frege’s insight into unsaturatedness; see the last page of BGGE.
This is again a frequently recurring theme in Frege’s works, after FUB.
This is the most which Frege’s published or unpublished works may allow one to say; the term “analogy” is mine. Cf. note 135 below. Also, Stenius [1], p. 138.
BGGE, p. 193 note; for “grammatical predicate” cf. 3.3. Carnap [2], § 38 erroneously associates “incomplete symbols” with unsaturated symbols.
This a priori follows from the fact that grammatical predicates in ordinary language are viewed as names of functions (names of concepts), and therefore they must include empty argument places. And Frege, in fact, stresses that the unsaturatedness should be expressed either by empty brackets (not “sin x” but, better, “sin()”, WIF, p. 664), or by points (“chrwww(133)”, VERN, p. 155) or by a special letter (“r, GRG II, § 63). This for natural language or mathematics; for the description of his Begriffsschrift Frege uses a special symbolism for unsaturatedness.
BGGE, p. 197, note 3; also UBR, p. 9.
“Gegenstand ist Alles, was nicht Function ist, dessen Ausdruck also keine leere Stelle mit sich führt” (FUB, p. 18). Apart from the peculiar Fregean view that objects are characterized by their names, this definition may be compared with a definition of individual (Gegenstand) given by Hering [I]: “.. Alles, ausgenommen die Wesenheit and ihre Dienerin die Washaftigkeit” (p. 510, n. 3). Cf. GRG I, §§ 1, 2, 4.
Cf. GRG II, p. 148, note 2; also a letter to Russell, dated 1904, in Bartlett [1], p. 56; or VERN, p. 155, where it is said that names should “reflect” the unsaturatedness of entities. But I would not affirm that Frege always thought in this way. In the Nachwort (GRG II) both kinds of unsaturatedness appear, but is it clear which is the first? Or in WIF (p. 665) is not the unsaturatedness of names perhaps the prior one?
BGGE, p. 196, note 2.
BGGE, pp. 195, 196 (and note), 198 (bottom), 204, 205.
Ibid., p. 196.
BGGE, p. 196, note 2. We cannot name grammatical predicates unless by means of unsaturated signs. This is why Carnap’s solution (cf. Geach [1], p. 146), consisting in recourse to language about language, cannot be accepted.
Professor Geach points out two ways of escaping the paradox. These seem to be intended to preserve Frege’s system. The first (Geach [1], p. 156) is to be found also in the Nachlaß (“was..” _ “whatchrwww(133)”), but it seems to be a concession which could be only momentarily taken into account. In fact, the proposed expression would have no passport enabling it to enter Frege’s world, where it is asked: Begriffswort oder Eigenname? (GRG II, p. XII, top; the “oder” is exclusive). If concept-word, the expression must have an empty place; if proper name it must have no empty place. The “was” solution is an ad hoc solution, resorted to in order to formulate in ordinary language a criterion of identity of meaning for concept words (cf. Bartlett [1], p. 50). Frege could not consistently assume it; and on the other hand he gives only a tentative and hesitating formulation, en passant and ad hoc. The criterion of identity for concepts is qualified in such a way that Frege is obliged to speak of an un-eigentlichen Gebrauch des Wortes “dasselbe”, and one wonders whether this is not the same as saying that the two “was..” expressions at both sides of “_” are not capable of ausfüllen (GRG I, p. 37) the two gaps of the function () dasselbe (). I think that the second solution proposed by Professor Geach fails because of the same reasons. Dummett [1] mentions “two ways of escape” (from the paradox). Unfortunately, the second does not seem to be quite clear. As to the first, I think that it is in agreement with Frege’s implicit higher predication (cf. Section 6.8), but I believe that no Fregean name of concept can fill the empty place of “the concept `()’ is realized”. In my view, there is no way out of the paradox, if one wishes to stay within Frege’s system. Outside Frege’s system solutions have been available for many centuries. Frege himself clearly indicates that there is no real way out. In Über Logik and Mathematik (Dated 1914. I am grateful to the Library of the University of Münster for having provided a microfilm of this long unpublished text), Frege continues to maintain, as in BGGE (p. 204 bottom) that one has to take such expressions as “the functionchrwww(133) ’ cum grano salis. ”Cum grano salis“ indicates that we should take into account what we mean and not what we say. Is this perhaps the simplest way out?
Martinus de Dacia [1], Modi significandi, n. 3. Cf. Tomas de Erfurt [1].
Ibid., n. 32.
Ibid., Quaest. sup. lib. Porph.,q. 19 (p. 144). Frege would say that “homo” and “humanitas” have different reference because they are not meaningfully interchangeable (BGGE, p. 201, top).
“Humanitas autem, licet significat illud idem [as ”homo“] cum tarnen non eodem modo, sed de suo modo intelligendi habet modum significandi in abstractione, quasi absolvendo banc formam, quae est humanitas, ab omni materiachrwww(133)” (Martinus de Dacia [1], Quaest. sup. lib. Porph., q. 19).
Hilbert-Ackermann [1], p. 114; also p. 50. Russell [4], § 49 is a quite “traditional” reply to Frege’s Ungesättigtheit: properties of concepts are called “external relations” and the marks of concepts are described as the “intrinsic nature” of concepts. Russell (ibid.) tries to show that unsaturatedness leads to contradictions, but Frege will say that these are scheinbare Widersprüche (UGGI, p. 372). One of the three points of disagreement between Russell and Frege is precisely that Russell rejects the Fregean unsaturatedness (ibid., § 475). But there seems to be a grave misunderstanding involved in Russell’s consideration of unsaturatedness; § 481 seems to read BGGE, pp. 195 and 197 (not 196!) as if “the concept F” were the name of a name, rather than the name of an object as Frege holds. Russell [4] (§ 481) is obscure on this point, but I presume that the author was misled by the reference to inverted commas in BGGE, p. 197, or perhaps also by a letter of Frege dated 1902, where the latter apparently suggests that the paradoxes of unsaturatedness do not pass into metalanguage (against our Section 6.74 in fine). In any case, “Frege’s theory” (on unsaturatedness) is not the one found in Russell [4] (§ 483). For a recent criticism of Frege’s doctrine, cf. Kneale [1], especially p. 586 (also: pp. 621, 622, 667, 671).
When Frege talks about das Wesen der Function, this seems to concern only designata, not signs (FUB, p. 3, GRG I, § 1). Still, in a letter to Russell (edited by Bartlett [1], p. 57), Frege declares: “Die Isolierung des Funktionszeichens widerspricht dem Wesen der Funktion.” Does the essence of a function consist in being named in some particular way? In the present section I suggest that the answer is trivially affirmative, for functions are also names (perhaps) for Frege. In any case, his stress upon the distinction between signs and designata (for instance: GRL, p. 107; UFT, p. 97 last line; FUB, p. 4 top; GRG II, p. 72) is hardly compatible with someone so concerned with names, unless one grants that for him names themselves are the object of research.
Cf. Section 2.12.
Cf. Carnap [3], Introduction to Semantics, p. 233.
Aristotle [2], Z, 1, 1028a, 20–27. Also Aristotle [3], (An. Post. A, 22) 83b, 17–23; the idea of prosthesis, for instance, points to a similar fact (cf. Ross in Aristotle [3], p. 582). These ideas have been developed by the philosophical tradition. Cf. Aquinas [1], n. 1255; Caietanus [1], p. 244f. Pacius [2] will say that an accident connotat subiectum.
Martinus de Dacia [1], Modi significandi, n. 41 and 43, and Quaest. sup. lib. Porph., q. 19 (p. 144). Our author uses the term indeterminatio, which seems to be reserved for the dimension singularity—universality.
Aristotle [4], B, 2, 193b, 35. Frege, on the contrary, will say that concepts can be unterschieden, but not abgeschieden from their relation to objects (UGGI, p. 372).
As Scholz [5] (pp. 15–16) has pointed out. Cf. the following passage: “Begriffe aber beziehen sich, als Pr delicate möglicher Urteile, auf irgend eine Vorstellung von einem noch unbestimmten Gegenstand” (Kant [3], p. 86).
FUB, p. 7.
BGGE, in fine.
Is not the name of what we mean by saying “UF”. For, again, “UF” (if taken as a saturated name) would fail to designate a relation.
GRG I, p. 39.
As presented, for instance, by Suarez [1], VII, sect. 1, n. 18, 19, etc.
But we find, again, in classical ontology, the individuality of accidents, and even modes: “haec inhaerentia numero” (ibid.).
For example Maimon [1], p. 56: “Ein allgemeiner Begriff wird als etwas auf mehr als einerlei Art Bestimmbares gedacht. Er muß daher durch zwei mit einander verbundene Zeichen ausgedrückt werden. Ein Zeichen für das (gegebene) Bestimmbare, und das andere für die unbestimmte Bestimmung. Er kann also durch ax bezeichnet werden, a bedeutet (so wie in der Algebra) das gegebene Bestimmbare, und x eine jede mögliche Bestimmung desselben.”
Church [1], note 32; Scholz—Hasenjaeger [1], p. 127.
Whitehead—Russell [1], Introd. 1st. ed., Ch. II.
Ibid.
“When we say that ”qqx“ ambiguously denotes qua, q9b, qçc, etc., we mean that ”çpx“ means one of the objects qxa, qb, qpc, etc., though not a definite one, but an undetermined one” (p. 39). “We may regard the function itself as that which ambiguously denotes, while an undetermined value of the function is that which is ambiguously denoted” (p. 40). “The function itself, 0, is the single thing which ambiguously denotes its many values; while qox, where x is not specified, is one of the denoted objects, with the ambiguity belonging to the manner of denoting” (p. 40).
P. 47.
“This [the formulation of the rules of types] arises from the fact that a function is essentially an ambiguity, and that, if it is to occur in a definite proposition, it must occur in such a way that the ambiguity has disappeared and a wholly unambiguous statement has resulted” (p. 47). “But it is obvious that we cannot substitute for the function something which is not a function” (p. 47). . when a function can occur significantly as argument, something which is not a function cannot occur significantly as argument. But conversely, when something which is not a function can occur significantly as argument, a function cannot occur significantly. (p. 48). . a function is a mere ambiguity awaiting determinationchrwww(133) and in order that it may occur significantly it must receive the necessary determination, which it obviously does not receive if it is merely substituted for something determinate in a proposition“ (p. 48). Finally, p. 48, note, there is a reference to a rule according to which ”the elimination of the unctional ambiguity is necessary to significance“. All this should be compared with Frege’s statement: . erkennen wir daß ein Functionsname niemals die Stelle eines Eigennamens einnehmen kann, weil er leere Stellen entsprechend der Ungesättigtheit der Function mit sich führtchrwww(133) Als Argument der Function X() kann also niemals selbst wieder eine Function auftreten, wohl aber der Wert einer Function für ein Argumentchrwww(133)” (GRG I, § 21).
Cf., for instance, p. 49, note.
Cf. note 160.
This would especially concern the distinction Sinn—Bedeutung. Cf. Section 2.23.
BG, p. 17. UGG1, p. 371 note 2 would confirm that “Rang” in BG anticipates the later Stufen.
GRL, p. 65 bottom.
But Frege kindly admits that one could have interpreted GRL in a sense other than Kerry’s. BGGE, p. 199.
Perhaps the best Fregean page for this is GRG I, p. 37; some passages have been quoted in note 165.
Cf. note 132.
Section 6.6.3.
Cf. for instance Fitch [1], 17.1, attributes implicitly or explicitly assigned. This is a terminology familiar in mathematics.
BGGE, p. 199.
Cf. 6.771, note 151.
Cf. 6.75, note 145.
Frege’s symbolism for existential quantification is of course different. Cf. FUB, p. 26.
Implicit higher predication may have been suggested to Frege by mathematical analysis; in integrals, for instance, there is a reference to the variable of the integrated function. Cf. FUB, in fine,GRG I, pp. 38–39; JOURD, p. 266.
GRG I, § 22.
BGGE, p. 201.
GRG I, p. 39.
GRG I, § 24.
Cf. Sluga [1], p. 197.
With respect to being saturated, see below in this section.
Cf. FUB, in fine.
GRG I, p. 41. Cf. Sluga [1]. Frege should be defended against Carnap [2], § 38, where it is overlooked that Frege did classify expressions into meaningless and meaningful (the latter being either true or false); cf. for instance BGGE, p. 200.
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Angelelli, I. (1967). Function. In: Studies on Gottlob Frege and Traditional Philosophy. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3175-1_7
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