Log. Univers.
c 2012 Springer Basel AG
DOI 10.1007/s11787-012-0051-z
Logica Universalis
Frege’s Ancestral and Its Circularities
Ignacio Angelelli
Abstract. After presenting the ordinary and the Fregean formulations of
the ancestral, I raise the question of what is their relationship, the natural candidate being that the Fregean version is an analysans intended
to improve upon, and replace, the common notion of ancestral (the analysandum). Next, two types of circles that arise in connection with the
Fregean ancestral are presented, and it is claimed that one of the circles
makes it impossible to maintain the just described (“replacement”) interpretation. A reference is made to Kerry, who was the first to point out
a circularity in Frege’s ancestral. Some of Frege’s remarks are examined
in order to tentatively sketch, an answer to the issue of the relationship
between ordinary and Fregean ancestral; the latter, if not as an analysans
replacing the common notion, can still be seen as a profound enrichment
of the former.
Mathematics Subject Classification. 03A05.
Keywords. Ancestral, analysans, analysandum, Frege, Kerry.
Van Heijenoort, shortly after participating in our History of Logic conference, Spring 1980,
at the University of Texas at Austin, donated a large amount of papers to the History of
Logic Collection I had been developing since the 1960s, as part of the Harry Ransom Humanities Research Center (HRC). Soon, however, a new administration of the HRC removed the
history of science and of ideas at large from its priorities, contrary to the plans of the creator
of the HRC, the great humanist Harry Ransom (for further background cf. [8]). The many
boxes with Van Heijenoort’s Nachlass had to be relocated somewhere else in the huge UT
campus. Van Heijenoort strongly complained, in a letter to the president of The University
of Texas at Austin, July 23, 1983, wondering if it would not be “best to have the papers
transferred to another institution”. UT Vice-president W. S. Livingston reassured (August
10, 1983) Van Heijenoort that his papers were “in good hands” and “well being cared for”.
As of now, the UT Library catalogue (http://www.lib.utexas.edu/) shows under “Jean Van
Heijenoort”, as item 7, “Van Heijenoort, Jean, Papers” including reprints, manuscripts, lectures, and research notes, as well as letters from other scholars. The materials are located
at the Briscoe Center for American History-Archives of American Mathematics (“remote
storage”, requiring 24-h notice).
I. Angelelli
Log. Univers.
1. Ordinary Ancestral and Fregean Ancestral
The ordinary understanding of the ancestral is that there is a “chain” from the
ancestor to the descendant, along the chosen relation. “Karl is an ancestor of
Fritz” is true if it can be shown that Karl is parent of someone who is parent of
. . . Fritz. For Frege, “Karl is an ancestor of Fritz” is true iff “For every property F , if F is hereditary (in the relation of being a parent of) and all children
of Karl have F , then Fritz has F too”.1 The distinction can be extended to
predicates: for example “Karl is an ordinary ancestor of x” expresses the property shared by the ordinary descendants of Karl, while “For every property
F , if F is hereditary (in the relation of being a parent of) and all children of
Karl have F , then x has F too”, expresses the property shared by the Fregean
descendants of Karl.
2. What is the Relationship Between the Two Ancestrals?
What is exactly the relationship between the ordinary and the Fregean ancestrals? In the preceding section the weak truth-functional “iff” was used to
describe that relationship. The phrase used by Frege: sei gleichbedeutend
(“let the Fregean ancestral have the same meaning as the common one”) is
much stronger, unless one dares to read gleichbedeutend as “having the same
truth-value” (anachronism?). While it seems difficult to exactly evaluate the
gleichbedeutend it should be clear that we are not dealing here with a definition, where the definiens is just a symbolically more convenient expression of
the “the same content” of the definiendum [4, §24]. In fact, Frege speaks of a
reduction (zurückführen, [4, Preface], [5, §108]) in the sense that the ordinary
following or preceding in a series, the common ancestral, is “reduced” to the
logical consequence (logische Folge). It seems natural to interpret the product of the reduction as intended to replace the initial notion. In alternative
terms, it seems natural to construe the Fregean ancestral as an analysans that
replaces, in Frege’s project, the analysandum (the common ancestral). Such
would be the analysis interpretation.
3. Circles (1) and (2)
Let us insert a bit of fiction. A certain Fritz, in the 1880s, encounters enormous difficulties in finding the documentation that would prove that Karl is his
ancestor, so that Fritz can inherit Karl’s possessions. Accidentally, Fritz runs
into a copy of a little new book titled Begriffsschrift, and becomes very excited
by seeing that much of it has to do with ancestors, descendants, and hereditary properties. As a final, desperate attempt, Fritz recites aloud, in court, the
1 The proposition “if every object to which x stands in the relation falls under the concept F ,
and if from the proposition that d falls under the concept F it follows universally, whatever
d may be, that every object to which d stands in the relation φ falls under the concept F ,
then y falls under the concept F , whatever concept F may be” is to mean the same [sei
gleichbedeutend] as “y follows in the φ-series after x” and again the same as “x comes in the
φ-series before y” [5, §79].
Frege’s Ancestral and Its Circularities
Fregean version of his claim: “For every property F , if F is hereditary (in the
relation of being a parent of) and all children of Karl have F , then Fritz has F
too” (we kindly imagine that in the series from Karl to Fritz each parent has
only one child). Fritz does not fully understand what he is saying but hopes
that the audience will be intimidated by such a display of conceptual weaponry, and perhaps the case will not be dismissed. Alas, a smart opponent ruins
this plan by requesting Fritz to defend the Fregean version of his claim for the
predicate “Karl is an ordinary ancestor of x”. Fritz is dialogically forced to
assert the conditional: if (the property “Karl is an ordinary ancestor of x” is
hereditary and all children of Karl have it) then Karl is an ordinary ancestor of
Fritz. The opponent then attacks the conditional by asserting its antecedent,
which is a conjunction whose two conjuncts the opponent defends successfully.
Thus, poor Fritz is left with the obligation of defending the same statement
he wanted to shun: “Karl is an ordinary ancestor of Fritz”. Let us refer to this
situation as circle (1).
Continuing the fiction, let us imagine that another of the opposing lawyers asks Fritz to defend his Fregean statement for the predicate “For every
property F , if F is hereditary (in the relation of being a parent of) and all
children of Karl have F , then x has F too”, abbreviated: FFx. This dialogically
forces Fritz to assert the conditional: “if (the property FFx is hereditary and
all children of Karl have it) then FF(Fritz)” which the opponent attacks by
asserting its conjunctive antecedent, whose first conjunct: “FFx is hereditary”
is nothing less than theorem 109 of Frege’s Begriffsschrift, and the second
conjunct seems obvious. Fritz is obligated to defend the consequent of the conditional . . . which is the same Fregean ancestral statement he started with.
The esoteric statement, under the chosen attack, reproduces itself indefinitely.
Let us refer to this situation as circle (2).
4. Circle (1) Invalidates the Analysis Interpretation
Clearly, circle (1) ruins the possibility of construing the Fregean ancestral
as the result of an analysis intended to substitute for the common notion of
ancestral—with ordinary ancestral as the analysandum and Fregean ancestral
as the analysans: the analysandum reappears, untouched, in the analysans.
Thus, Frege’s speaking of a “reduction” (zurückführen) is misleading to the
extent that it may suggest that the old item is disposed of and replaced by the
product of the reduction.
It might be argued that circle (1) can be blocked by removing the predicate “Karl is an ordinary ancestor of x” from the list of possible attacks on
the quantifier “For all F ”. It might be said that Frege intended to create a
better language, where the imperfections of the ordinary language would be
eliminated. According to this plan the predicate “Karl is an ordinary ancestor of x” should be disposed of and replaced by the better Fregean version of
it. This defense, however, is not justifiable. When the ordinary language has
defects that need to be improved upon (e.g., the ambiguity of “is”), then the
replacement of the defective expressions by the better ones is in order. This is
I. Angelelli
Log. Univers.
precisely the point of Frege’s thesis, in anticipated opposition to the linguistic phenomenalism or naturalism that has proliferated in the second half of
the twentieth century. However, the ordinary ancestor, especially in the finite
case, does not show any defects that would call for ortholinguistic repairs. It
is a transparent, multiple conjunction: Karl is parent of . . . & . . . is parent
of Fritz, where each of the conjuncts is proof-definite in terms of documents
and other historical evidence. In sum, the attack with the predicate “Karl is
ordinary ancestor of x” seems legitimate, and the Fregean ancestral cannot be
interpreted as intended to replace the ordinary notion.
The difference between the two circles is that while circle (1) invalidates
the process of analysis itself, i.e., the transition from analysandum to analysans, circle (2) affects only the product of the analysis: the analysandum can
be challenged in a way that the only possible defense is to assert, again and
again, the analysandum itself, bringing the Fregean ancestral close to a debatable yet respectable territory of the philosophy of logic: impredicativity. When
Chisholm writes that “the analysis should not be circular” [3, p. 1] he means
circle (1), not (2). Surely, the word “analysis” can be construed in a much
weaker sense, compatible with circularity, corresponding in fact to what Frege
did with his ancestral (cf. below).
5. Kerry and Others
Benno Kerry forcefully criticized, as early as 1887, the Fregean ancestral.2
Here is the most important passage (my translation):
Now, this criterion is to begin with of dubious value because there
is not a catalogue of such properties, hence one is never sure that
one has examined the totality of them. Moreover, there is the crucial fact that, as the author himself has proved [in a footnote Kerry
cites Begriffsschrift, p. 71 Theorem 97], of the properties that are
hereditary in the f -series is also the following: to follow x in the
f -series. Thus, the determination of whether y follows x in the
f -series, according to the definition given for this concept, depends
on whether, in addition to a lot of other things on hereditary properties in general, one knows, in particular, about the hereditary property “being a descendant of x”, that y has it or not. It is clear that
this circle should totally prevent from saying, in Frege’s sense, that
any y follows x in an f -series3 (italics mine).
Kerry does not seem to distinguish between circle (1) and circle (2) but
he seems to have in mind circle (2) because of his reference to Frege’s theorem
2 After having discussed Kerry–Frege in [1], but not with regard to the ancestral, I have
over the years encouraged many students (Buenos Aires and Texas) to look into Kerry’s
circularity criticism of Frege, as well as to work towards a new edition of the very short
corpus of this Austrian philosopher—a desideratum not yet fulfilled.
3 Nun ist dieser Kriterium schon darum von zweilfelhaften Werthe, weil kein Katalog solcher
Eigenschaften existirt, man also nie sicher ist, den Inbegriff derselben erschöpft zu haben.
Hiezu kommt aber als ausschlaggebend noch der Umstand, dass, wie unser Autor selbst
Frege’s Ancestral and Its Circularities
97 in [4]. The same applies to Russell and Carnap, who tried to defend Frege
from Kerry’s objection [2, §44], [9, p. 158], [10, § 496]. It should be interesting
to determine to what extent Russell or Carnap assume that the Fregean ancestral is an analysis or explication of the common notion, which would raise the
issue of circle (1). Aside from this central issue, it may be observed that the
quoted Kerry passage starts with an interesting hint at constructivism and
ends with a not too fortunate expression: “this circle should totally prevent
from saying, in Frege’s sense, that any y follows x in a f -series”—the author
probably meant “proving” rather than “saying”.
6. Attempting to Understand Frege’s “Remarks”
on His Ancestral
Leaving aside the analysis interpretation of Frege’s ancestral, one should rather
focus on what Frege himself has to say, [5, §80], on the just stated formulation of the ancestral (end of §79). Here, as in other cases, it is helpful to look
at the titles of the sections of Die Grundlagen der Arithmetik [5], displayed
separately at the beginning of the volume; such titles often complete or even
expand the information given in the section; the title of §80 is: Bemerkungen
hierzu, meaning “remarks on the ancestral”. These “remarks” may be read as
involving two recommendations.
First (first paragraph of [5, §80]), Frege wants that, in any example of
ancestral the peculiar individual features of the example be distinguished from
what is essential. This applies to the underlying relation of the ancestral. In
one case, e.g., in genealogical considerations, time and space are pertinent, in
another case, e.g., in the philosophy of arithmetic, spatial and temporal references are utterly inappropriate. In [5, §76] Frege had defined the relation from
a number to its immediate successor as independent of space or time, according to his campaign of exorcising4 pure intuition (Kantian reine Anschauung)
from the foundations of arithmetic. In sum, spatio-temporal connotations are
Footnote 3 continued
nachgewiesen hat [footnote: Begriffsschrift, S. 71 (Formel 97)], eine der in der f -Reihe sich
vererbenden Eigenschaften auch die ist: in der f -Reihe auf x zu folgen.
Hienach hängt die Entscheidung darüber, ob y auf x in der f -Reihe folge, laut der für
diesen Begriff gegebenen Definition davon ab, dass man, nebst sehr vielem Anderen über
vererbende Eigenschaften überhaupt, speciell von der vererbenden Eigenschaft: auf x zu folgen, Das wisse, ob y sie besitze oder nicht. Es ist klar, dass der hier vorliegende Cirkel es
durchweg verhindern muss, dass im Sinne Freges von irgend einem y gesagt werde, es folge
auf x in einer f -Reihe (Kerry [7], p. 295).
4 A passage from Kant’s [6] (B15) exemplifies the devil that Frege wants to exorcise: for
example, in performing the addition of 5 and 7, Kant says that we must “call in the aid of
the intuition which corresponds to one of them, our five fingers, for instance. . . .” Intuition
will reappear, far more forcefully and systematically, in the subsequent constructivist-intuitionistic authors and schools, where the natural numbers are introduced, for example, by the
production of successive strokes, either in space (e.g., on paper, a new stroke to the right of
the previous ones) or in time (ticktack of a clock). The Fregean path from an ancestor in the
number series, say 5, to one of its descendants, say 100, unfolds through pure concepts—no
strokes or ticktacks, no reine Anschauung at all.
I. Angelelli
Log. Univers.
not excluded, but not necessary either when considering, in the abstract or per
se, the underlying relation of the ancestral. Briefly, in considering the ancestral
in general Frege tells us that the underlying relation must be left undetermined
(unbestimmt).
Secondly (second–fourth paragraph of [5, §80]), Frege wants our consideration of the series of items between an ancestor and a chosen descendant not
to resemble a touristic hiking—a Wanderung as Frege puts it nicely. Of course
it is not forbidden (so I read Frege) to enjoy a Wanderung, for instance when
one “hikes around” the history of a fascinating series of great kings or queens
or popes. However, in addition to this Wanderung, directed by the subjectivity
of our paying attention (Aufmerksamkeit) to one or other event, there is a not
less important search for the objective laws, if any, that govern the ancestral.
Frege has found one such law: the transitivity of the ancestral (fifth paragraph
of §80).
Now, how or where to find such objective or formal laws governing the
ancestral? Frege obviously discovered one source of such laws in the phenomenon of hereditary properties. First, it must have appeared to him as quite
plausible that, a descendant has all the hereditary properties (in the underlying relation) that the ancestor has. The converse of course is not plausible at
all, but it becomes plausible by means of a little trick: if an object b has all
the hereditary properties that all children of a have, then a is ancestor of b
(because any child of a has the hereditary property of having a as ancestor).
Again, it seems plausible that if a is ancestor of b then b has all the hereditary
properties that all children of a have. Thus we have the biconditional: “a is
(ordinary) ancestor of b iff b has all the hereditary properties that all children
of a have”.
Frege in fact views his formulation of the ancestral as what makes it
possible to establish the transitivity of the ancestral, proved five years earlier [4, Theorem 98]: “thus we can infer, according to our definition” ([5, §80]
fifth paragraph), and then finally to accomplish the logicist conquest of the
Bernouillische Induction (cf. [4, Theorem 81]), which he affirms to be possible
only by means of his formulation of the ancestral (sixth paragraph of [5, §80],
also §108). Notice that this is stronger than what he said about transitivity:
the latter was obtained by his ancestral, the induction only by his ancestral.
7. Enrichment and Generalization
In sum, the Fregean ancestral appears to be both an enrichment and a generalization of the common notion. The enrichment occurs through the “discovery”
of the property of being hereditary that many properties have. In particular,
some predicates generated from the ordinary ancestral (“Karl is ancestor of
x”) express hereditary properties. This enrichment includes the focusing on,
and helps towards the demonstration of the formal properties of the ordinary
ancestral, e.g., transitivity, which as Frege points out is what leads to the
logical understanding of arithmetical induction. The generalization is accomplished in that the ancestral’s underlying relation as such is conceived in a
Frege’s Ancestral and Its Circularities
most abstract fashion. While all this happens, the ordinary, common ancestral
has not been chased away (as an analysandum that can and should be forgotten) but remains untouched in the center of the new scenario. Thus while what
Frege does cannot be viewed as an analysis or reduction proper, it certainly
involves a profound, illuminating insight into the concept in question.
References
[1] Angelelli, I.: Studies on Gottlob Frege and Traditional Philosophy. Reidel, Dordrecht (1967)
[2] Carnap, R.: The Logical Syntax of Language. Kegan Paul Trench, Trubner &
Co., London (1937)
[3] Chisholm, R.M., Potter, R.C.: The paradox of analysis: a solution. Metaphilosophy 12, 1–6 (1981)
[4] Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache
des reinen Denkens. L. Nebert, Halle (1879)
[5] Frege, G.: Die Grundlagen der Arithmetik. Wilhelm Koebner, Breslau (1884)
[6] Kant, I.: Critique of Pure Reason, English translation by N. Kemp Smith.
St Martin’s Press, New York (1965)
[7] Kerry, B.: Ueber Anschauung und ihre psychische Verarbeitung. Vierter Artikel,
Vierteljahrsschrift für wissenschaftliche Philosophie 11, 249–307 (1887)
[8] Kolata, G.: Math archive in disarray. Science (New Series) 219(4587), 940 (1983)
[9] Linsky, B.: Russell’s notes on Frege for Appendix A of “The Principles of Mathematics”. Russell J. Bertrand Russell Stud. 24, 133–172 (2004–2005)
[10] Russell, B.: The Principles of Mathematics, 2nd edn. Allen and Unwin, London
(1956)
Ignacio Angelelli
Philosophy Department
The University of Texas at Austin
Austin
USA
e-mail: [email protected]
Received: April 28, 2012.
Accepted: May 2, 2012.