## Notes to Logicism and Neologicism

2. Benacerraf (1981: 21) claims that “If Frege was the first logicist, then he was also the last.” Naturally one’s assessment of this claim will turn on how broad is one’s conception of logicism.

3. Lanier Anderson (2004) argues that “Kant deploys a clear and defensible notion of concept containment” and that

[o]nce we understand it, that notion of containment provides resources for a compelling argument that arithmetic must be synthetic,

sensuKant. (p. 503)

But the logicist will be unmoved by this: the rejoinder will be that
the most appropriate sense of ‘analytic’, for the purposes
of establishing an interesting logicist thesis about mathematics, is
one that *departs* in carefully considered and well-motivated
ways from analyticity *sensu* Kant.

4. See Michael Friedman 1992 (p. 84). Another useful discussion of the role of the pure form of temporal intuition in Kant’s account of arithmetic is Parsons 1983.

5. In
his *Habilitationsschrift* of 1874, Frege wrote:

Die Elemente aller geometrischen Konstruktionen sind Anschauungen, und auf Anschauung verweist die Geometrie als Quelle ihrer Axiome. Da das Objekt der Arithmetik keine Anschaulichkeit hat, so können auch ihre Grundsätze aus der Anschauung nicht stammen.(Frege 1967: 50)

We offer the following translation:

The elements of all geometrical constructions are intuitions, and geometry points to intuition as the source of its axioms. Since the object of arithmetic lacks intuitiveness, so too are its basic laws unable to stem from intuition.

This early conviction of Frege’s to the effect that the ultimate
justification for the axioms of geometry was to be found in intuition,
was reprised in 1903, in “*Über die Grundlagen der
Geometrie*” (see Frege 1967: 262).

6. The date of 1853 in
the English translation Dedekind (1996b: 793) of the preface to the
first edition of Dedekind’s later work *Was sind und was sollen die
Zahlen?* is in error. The German original (Dedekind 1888: 339) has
1858.

7. This passage is one of the most important ‘position statements’ ever written on foundational matters, and is worth quoting here in full. This translation, due to W. W. Beman and extensively revised by William Ewald, is from Dedekind (1996a: 767). Emphases have been added:

In discussing the concept of the approach of a variable magnitude to a fixed limiting value—in particular, in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value—I took refuge in geometrical evidence. Even now I regard such invocation of geometric intuition [

Anschauung] in a first presentation of the differential calculus as exceedingly useful from a pedagogic standpoint, and indeed it is indispensable, if one does not wish to lose too much time. But no one will deny that this form of introduction into the differential calculus can make no claim to being scientific. For myself this feeling of dissatisfaction was so overpowering that I resolved to meditate on the question until I should find apurely arithmetical and perfectly rigorous foundation[Begründung] for the principles of infinitesimal analysis. The statement is frequently made that the differential calculus deals with continuous quantities, yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but they either appeal more or less consciously to geometric representations or to representations suggested by geometry, or they depend upon theorems which are never established in a purely arithmetical manner. Among these, for example, belongs the above-mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded as a more or less sufficient foundation for infinitesimal analysis. It only remained to discoverits true origin in the elements of arithmetic[sic], and thereby to secure a real definition of the essence of continuity. I succeeded on November 24, 1858 …

8. This English
translation is from Dedekind (1996a: 767). The German original, in
Dedekind (1872: 4) is: “*eine rein arithmetische und
völlig strenge Begründung der Prinzipien der
Infinitesimalanalysis*”.

9. The reason why we
speak here of ‘the Fregean’ is that Frege himself, in
the *Grundgesetze*, did not treat of addition and
multiplication on the natural numbers. He treated only zero and
successor.

10. It might be
objected that our decimal place-notation ‘12’ is really to
be regimented as (1 × 10) + 2—which involves
multiplication as well as addition. If *that* is how the
Kantian is to be accommodated, then the composite singular term for
‘12’ in the (0, *s*, +, ×)-language of Peano
arithmetic would be

(s0 ×ssssssssss0) +ss0,

and the derivation of ‘7+5=12’ as

sssssss0 +sssss0 = (s0 ×ssssssssss0) +ss0

would involve appeal to the recursion axioms for multiplication, which are

∀xx× 0 = 0;

∀x∀yx×sy= (x×y) +x.

The logicist will have derived these axioms too (whose variables
are understood as ranging over natural numbers) from deeper,
‘logical’ principles involving expressions not occurring
in the recursion axioms themselves. The same kind of reply, in
principle, would deal with the even more exigent requirement that the
place-notation numeral ‘12’ be regimented as (1 ×
(10^{1})) + (2 × (10^{0})), which now brings in
exponentiation as well.

11. If, instead of
using the language of second-order logic, one were to have a
two-sorted first-order logic in which properties are one of the sorts
and in which there are singular terms *t* denoting properties,
then # could be used as a function symbol like *d*( ), so that
#*t* could be a singular term denoting the number of things
falling under the concept denoted by *t*. But we shall not be
discussing such variants in what follows.

12. We shall see in due
course
(§§2,3)
what abstraction principle(s) the neo-logicists have put forward in
their attempts to characterize the meaning of this number-abstraction
operator. Some, like Wright and Tennant, use the variable-binding
operator #*x*; others, like Heck and Zalta, use the variable-free
operator #, and apply it to either second-order variables or
predicates.

13. This displayed
pasigraph is just a useful abbreviation. Note that it has no free
variables. So there is no need, here, for quantifiers to be prefixed
in order to bind the variables *x* and *y*. In the
abbreviatory pasigraph given, the free occurrences of *x*
and *y* in the part
[*Fx*
1–1

↦

onto
*Gy*] can be thought of as bound by the initial
occurrence of *Rxy*. There are different but equivalent
pasigraphic (abbreviatory) notations in the literature for registering
the existence of a one-one, onto mapping between the *F*s and
the *G*s; any one of them will do. What matters rather is
the *definiens* which the pasigraph abbreviates.

14. The reader is put
on notice that the contrast here between single- and
double-abstraction principles is a contrast drawn within a purposely
narrowed class: abstraction principles in biconditional form that
have *identities* on their left-hand sides. With the
double-abstraction identity principles, the left-hand sides will be of
the form #*xFx* = #*xGx*; whereas
with single-abstraction identity principles they will be of the
form *t* = #*xFx*. There are of
course many other kinds of abstraction principle in the literature
which are correctly called ‘single-abstraction’
principles, but they are ones that do not deal with identities of the
latter form on their left-hand sides. (Indeed, some of them are not
even of biconditional form.) Examples of
‘single-abstraction’ principles of biconditional form
whose left-hand sides are not identity statements are

t∈{x|x∈A∧ φx} ↔ (t∈A∧ φt).

and β-conversion in the λ-calculus:

[λx_{1}…x_{n}φ(x_{1},…,x_{n})]t_{1}…t_{n}↔ φ(t_{1},…,t_{n}).

Such principles are being purposely excluded from consideration here as single-abstraction principles, because our focus is on the use made of single-abstraction principles by neo-Fregeans, which are all identity abstraction principles.

15. As early as
December 7, 1873 Cantor wrote to Dedekind that he had “found the
reason why the totality [of real numbers] … *cannot* be
correlated one-one with the totality [of natural numbers]” (see
Ewald 1996: 846). The proof in question (which did not use his famous
diagonal method) was published as Cantor 1874.

16. As Richard Heck (1997a) points out, Frege also developed the ‘Caesar’ objection to his doubly-abstractive definition of line-directions. The objection was that the definition would not enable us to distinguish England from the direction of the Earth’s axis.

17. Exercise: Show that this set-abstraction principle logically implies the Axiom of Extensionality,

∀

x∀y(∀z(z∈x↔z∈y) →x=y).

18. Courtesy of Zalta’s definition (1999: 630)

#

G=_{df}ιx(Ax∧ ∀F(xF↔Fis equinumerous withG))

19. Explicit use of the label ‘Bad Company’ has an interesting early history. Dummett (1991: 188–9) is quoted at length by Wright (1998: 344–5; §II, titled “Bad Company?”). Dummett’s complaint had been

[I]f the context principle, as expounded by Wright, is enough to validate the ‘contextual’ method of introducing the cardinality operator, it must be enough to validate a similar means of introducing the [class] abstraction operator.

Dummett (1998: 375; title: “Neo-Fregeans: In Bad Company?”), reprises the criticism thus:

In

Grundgesetze, value-ranges are introduced in a manner precisely analogous to that in which Wright argued, in his book, that Frege ought to have introduced cardinal numbers …: and yet it was so far from being justified as to lead to actual contradiction.

Clearly, at the time of those writings of theirs quoted above, Tennant, Boolos, Dummett and Wright construed the ‘bad company’ for HP as consisting of BLV alone; and this construal endured through the late 1980s and even into the early 1990s. Subsequently, further formal discoveries were made, of yet other (double-)abstraction principles that are individually consistent, but jointly inconsistent (and some of them inconsistent with HP). The seminal paper in this regard is Boolos 1997. The problem highlighted by the proliferation of conflicting principles has been given the label ‘Embarrassment of Riches’ by Weir (2003). But this label has not caught on. Instead, the ‘Bad Company’ label was happily extended so as to emphasize the worsened delinquency of the group with its newly admitted companions. This has induced a shift in our collective understanding of what comprises ‘Bad Company’, and what the Bad Company problem amounts to. An informative collection of essays on these more recent developments is Linnebo, ed. (2009). As Linnebo puts it in the abstract of his editorial Introduction,

…the acceptable abstraction principles are surrounded by unacceptable (indeed often paradoxical) ones. This is the ‘bad company problem.’

20. The reader is
reminded that the variables *z* and *w* are bound in the
right-hand side of this biconditional. See footnote
13.

21. See Definition Z at
the end of §40 of Volume I of the *Grundgesetze*, at
p. 57.

22. We have been using
the mathematician’s ‘relation-slash’
notation *r* ∉ *r* instead of
the logician’s ‘sentence-prefix’
notation ¬(*r* ∈ *r*).

23. Fortunately, this
has recently been remedied. The Arché project at St. Andrews
has brought out an English translation of the *Grundgesetze*. See
the Bibliographic reference to Frege 1893, and its citation to Ebert
and Rossberg translation.

24. Russell’s
“Mathematical Logic as Based on the Theory of Types”
(1908) is an accessible presentation of these ideas. The official,
full development is *Principia Mathematica* (Whitehead and
Russell 1910).

25. The modern
set-theoretic conception of an ordinal number is due to von
Neumann. The very first ordinal is the empty set ∅. If α
is an ordinal, then α ∪ {α} is
the *successor* of α. Beginning with ∅, and taking
only successors, one obtains the finite ordinals. These are the usual
set-theoretic surrogates for the natural numbers. The first
transfinite ordinal, ω, is the set of all finite ordinals. Every
ordinal has a successor. But not every ordinal *is* a
successor. Ordinals which are like ω in that they are not
successors are called *limit* ordinals. Every ordinal is the
set of all ordinals that precede it. A *cardinal* number is an
ordinal that is not in 1–1 correspondence with any preceding
ordinal. So a cardinal number is the first ordinal of ‘its
size’. The finite ordinals are the finite cardinals. But in the
transfinite, of course—ω and beyond—the cardinals
are more thinly sprinkled among the ordinals. *Qua* cardinal
number, ω is called ℵ_{0}. It is *countably
infinite*. The first *uncountable* cardinal is
ℵ_{1}, which is the first ordinal after
ℵ_{0} not in 1–1 correspondence with any ordinal
preceding it.

26. For this latter observation, the author is indebted to John MacFarlane.

27. We do not say the
finite von Neumann ordinals are *the* set-theoretic surrogates
for the natural numbers, because of the well-known ‘Benacerraf
point’ that there are other recursive progressions within the
universe of (hereditarily finite) pure sets that could serve just as
well—Zermelo’s finite ordinals, for example (see Benacerraf
1965). Interestingly, though, Boolos has argued that von Neumann’s
finite ordinals are the most natural representatives, within a theory
of extensions, of Frege’s finite cardinals (see Boolos 1987; see also
Demopoulos 1998).

28. Cantor’s theorem (that every set has strictly more subsets than members) has the special case that ℘(ω) has more members than ω does, hence is uncountable.

29. This was an unpublished lecture delivered in Cambridge, Massachusetts, at a joint meeting of the Mathematical Association of America and the American Mathematical Society, 29–30 December 1933. See Feferman’s introductory note to Gödel 1993/1995 (p. 36).

30. John MacFarlane
suggested that a “perhaps superior option” would be to
require explicit domains *D* in the abstracts, thus:
{*x* ∈ *D* | Φ(*x*)}.
Then a free logic would not be needed, because, by Separation, the
denotation for any such term would exist.

31. Quine (1960: 267) advocates strongly for “the power of the notion of class to unify our abstract ontology”. He goes on to assert

To surrender this benefit and face the old abstract objects again in all their

primeval disorderwould be a wrench …. [Emphasis added]

32. Snapper (1979: 208) contends that logicism survived the shift from type theory to set theory. Here is how he viewed matters:

Of course, instead of

Principia, one can use any other formal set theory just as well. Since today the formal set theory developed by Zermelo and Fraenkel (ZF) is so much better known thanPrincipia, we shall from now on refer to ZF instead ofPrincipia. ZF has only nine axioms and, although several of them are actually axiom schemas, we shall refer to all of them as “axioms”. The formulation of the logicist’s program now becomes: Show that all nine axioms of ZF belong to logic.

Against this, it could be maintained that the move
from *Principia* to ZF represented an abandonment of the
aspirations of logicism, and an acknowledgement that the foundations
of mathematics can at best be provided within just one branch of
mathematics itself, namely set theory.

Cook 2015 is a subtle and technically demanding study that supports a principled pessimism about the prospects for a suitably modified neologicist approach to a theory of sets (or value-ranges) that might avoid the disaster that befell Frege’s Basic Law V, and deliver enough in the way of sets to allow the neologicist to develop a version of Frege’s account of cardinal numbers. Cook limits himself to a consideration of only double-abstraction principles; but his conclusion must give this kind of neologicist serious pause. Cook 2016 pursues the topic further, revealing a virtually uninhabitable neologicist no-man’s land between Hume’s Principle (which is consistent) and Basic Law V (which of course is not). These negative reflections on the potential reach of neologicism continue a theme from the earlier work Cook 2002, which concluded that ‘the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary’, and accordingly epistemologically suspect.

Boccuni 2013 has introduced a new twist in attempted
reconstructions of Fregean foundations for number theory, with a
system called ‘*Plural Grundgesetze*’, based on the
relational concept *x*η*X* (‘the
individual *x* is among the *X*s’). Its
‘Fregean devices’ include ‘the infamous Basic Law
V’, along with a *Plural Comprehension Principle*

∃X∀x(xηX↔ φx) (where φ does not containXfree)

Contradiction is presumed to be avoided by replacing George
Boolos’ plural semantics with Enrico Martino’s *Acts of Choice
Semantics*. It is too early to judge whether these novel resources
will prove to be a consistent mix.

33. Although Frege more
or less explicitly proved Frege’s Theorem in the *Grundlagen*
(and proved it fully explicitly in the *Grundgesetze*), Frege
himself never actually put his finger explicitly on the
‘it’ in question.

34. For useful discussion, see MacFarlane 2002 (pp. 40–42), which concludes, in effect, that Frege was deterred by worry about the truth of HP (i.e., of Principle (A)), because of it similarity to Basic Law V. For Frege did not have the benefit of knowing that second-order logic with HP is consistent if Real Analysis is. So perhaps Frege himself deserves credit for being the first thinker to appreciate the force of the ‘Bad Company’ objection (in the sense it enjoyed when first deployed with that label—see footnote 19).

35. The author is aware that the notion of a ‘significant part’ of a branch of mathematics requires further explication. For the time being, it is enough to note that what Quine called virtual set theory is a significant part of ZFC set theory; and PA is a significant part of Th(ℕ).

36. The incompleteness
phenomena affect *provability*. One could of course adopt a
second-order axiomatization that secures every truth (about, say,
ℕ) as a (second-order) logical consequence; but then one would
have to live with the drawback that second-order logical consequence
is not axiomatizable. See Rayo 2005.

37. Wright appears,
with hindsight, to have been rather too exclusively focused on
the *Grundlagen*, and somewhat underinfluenced by Frege’s own
logical maneuvers in the *Grundgesetze*. We note this
assessment from Dummett (1991: 123):

Crispin Wright devotes a whole section of his book … to demonstrate that, if we were to take the equivalence in question as an implicit or contextual definition of the cardinality operator, we would still derive the same theorems as Frege does. He could have achieved the same result with less trouble by observing that Frege himself gives just such a derivation of those theorems. He derives them all from that equivalence, with no further appeal to his explicit definition.

“[J]ust such a derivation”, of course, appears only in
the *Grundgesetze*.

38. As pointed out by Tennant (1987: 236–7), Boolos’s model for FA works only for FA taken by itself; and that model

will not serve its intended purpose when [FA] is embedded within a wider theory—such as the theory of sets—calling for models in which there are distinct infinities of objects.

The “theory of number …, like logic, is to apply to all subject matters.” The consistency of a logicist theory of number “is not to depend upon the particular subject matter over which numerical notions are deployed.” Boolos (1997: 260) agrees:

The worry is that … Frege Arithmetic … is incompatible with Zermelo–Fraenkel set theory plus standard definitions, on the usual and natural readings of the non-logical expressions of both theories.

(Note that Boolos is not a logicist; he admits only the standard logical operators as logical expressions.)

39. As Dummett (1993: 441) explains,

an indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it.

40. Some philosophers
of mathematics do not share this intuition; and some of those who do
might be reluctant to make it a *sine qua non* of any
successful logicist account. Thanks to both Julian Cole and Stewart
Shapiro for raising this ‘structuralist’ point. The point
is made also by Carnap (1931: 93) in what was intended to be a
sympathetic exposition of the aims and partial achievements of
logicism. Here is the English translation taken from Carnap 1983
(p. 43):

The natural numbers do not constitute a subset of the fractions but are merely correlated in obvious fashion with certain fractions. Thus the natural number 3 and the fraction 3/1 are not identical but merely correlated with one another. Similarly we must distinguish the fraction 1/2 from the real number correlated with it.

Carnap’s view here is deferential to the similar view of Russell,
who in turn had inherited from Frege this need to resort to a form of
structuralism. The inability to answer the inclusion question in turn
derives from not taking seriously enough the need to explain how the
various numbers are canonically *applied* in our theorizing not
only about numbers but also about concrete things. A correct solution
to the applicability problem could well point the way to an answer to
the inclusion question. Real numbers are used for measuring continuous
magnitudes in terms of some unit of measurement. When we say that a
rod is 3 units-of-length (say, meters) long, we are in effect saying
that the length of the rod, in meters, is 3. Equivalently: the number
of units-of-length that comprise the length of the rod is 3. The
latter is the familiar natural number 3.

41. It is well known
that by Gödel’s Second Incompleteness Theorem, no consistent,
sufficiently strong theory *T* of arithmetic can prove its own
consistency-statement Con_{T}. So the extended
theory *T* + Con_{T} is of
higher consistency-strength than the theory *T* itself. In
general, for consistent theories *T*, *T* ′
containing a sufficiently strong fragment of arithmetic,
theory *T* ′ is of higher consistency strength than
theory *T* just in case *T* ′ proves
that *T* is consistent. Usually this is done in one of two
ways. Either *T* ′ proves Con_{T}
(the sentence of arithmetic expressing the consistency of *T*);
or *T* ′ proves the existence of a model for (the
axioms of) *T*. A vast range of mathematical theories can be, and
have been, compared according to their consistency-strengths. These
comparisons have involved fragments of arithmetic, fragments of real
analysis, set theories, and type theories. The two ways in which
strength can be increased in second-order theories of arithmetic is by
allowing broader classes of substituends (complexity classes of
formulae) in their axiom-scheme of comprehension, and in their
axiom-scheme of mathematical induction. The main way in which strength
can typically be increased in set theory is by postulating the
existence of ever-larger cardinal numbers. The deep and puzzling
phenomenon that has emerged is that consistency-strengths
are *linearly* ordered. The modern program of Reverse
Mathematics, due to Harvey Friedman (see Friedman 1975, 1976), is the
major source of such insights into the relative strengths of various
mathematical theories. The best avenue into the relevant literature is
Simpson 1999. For a thorough investigation of the question of
consistency-strengths of various foundational theories that are
pertinent to logicism, see Burgess 2005, especially Table E.

42. It has been Michael Dummett, especially, who has advanced this interpretation.

43. The normalization theorem is due to Prawitz (see Prawitz 1965).

44. For a cogent
critique of Carnap’s lack of appreciation of certain metamathematical
subtleties in holding to his position that both logical and
mathematical truths were analytic *in* *Carnap’s explicating
sense*, see Koellner (Carnap ms in Other Internet Resources).

45. By contrast, Wright’s proof-sketch “stops short of being a fully rigorous deduction” (Burgess 2005: 147).

46. The use of free logic for logicist purposes has also been explored more recently by Shapiro and Weir (2000).

47. The condition of adequacy involving Schema N was put forward in Tennant 1984.

48. Note that an
affirmative answer to this question invites the reflection that, if
Logic commits one to the existence of any thing or kind of thing, then
such existence will be *necessitated*. The things in question
will be *necessary* existents.

49. This is properly
classified as a demarcation problem because of a particular pre-formal
view with which such a logicist methodology would be in tension. The
view in question combines two main theses. The first thesis is that
there is some clearly identifiable body of *mathematical*
theorizing, employing mathematical concepts, whose eligibility for
logicist reduction is in question. The second thesis is that the
problem for the logicist would be to show how to
‘logicize’ this theorizing by (i) defining those
mathematical concepts in purely logical terms, and then (ii) deriving
as logical theorems the *translations* induced by those
definitions of the erstwhile mathematical theorems.

50. This label is used here in its more recently acquired sense, not its original one. See footnote 19.

Copyright © 2017 by

Neil Tennant
<*tennant.9@osu.edu*>