In FA §56 Frege immediately scraps the §55 definitions, at least in part because of the Julius Caesar problem, i.e., because of their failure to well-define the definienda. In §66 he dismisses for the same reason the proposal to define by abstraction a functor for the function that assigns to any concept its numerater. Ultimately he chooses instead to define the cardinality of a concept as the equivalence class of concepts in 1–1 correspondence with the concept. Whereas the Caesar problem is indeed fatal to definition by abstraction, Frege is not forced to abandon the §55 definitions. The discredited definitions are easily retrofitted to be made Caesar-proof.

Frege’s analogies of the notion of the cardinality of a concept to that of the directional orientation of a line and the shape of a triangle, although brilliant, also do a disservice to his program. The FA §68 identification of a line’s directional orientation, and of a triangle’s shape, with an equivalence class of things having that direction or shape in common is conceptually dissonant. Directional orientations and triangular shapes are properties , qualities , or features of things, not sets of the things with those properties. Analogously, the meaning of an expression is not its synonymy class. The situation is very different with regard to the natural numbers. Words like ‘three’ and ‘thirty-seven’ are unlike terms for directional orientations (e.g., ‘north–south’) or triangular shapes (‘equiangular’). Number-words are determiners . As such, they are naturally regarded as quantifiers. A quantifier is essentially a quantitative second-order predicate (typically monadic) combined with concept-abstraction (in Frege’s idiosyncratic sense of ‘concept’). The classical logicist program basically casts numerals as numerically-definite second-order predicates (‘nothing’, ‘exactly one thing’, ‘exactly two things’, etc.). The identification of natural numbers with equinumerous equivalence classes then simply follows.^{31}

The following modification of the

FA §55 definitions is decidedly Fregean in spirit. First a variable-binding term for a zero-concept is defined thus:

The operator ‘∄’ is, in effect, the negated-existential quantifier—in English, the ‘nothing’ in ‘Nothing is without mass’, the ‘none’ in ‘None deserves the fair’. Its semantic extension is the function that assigns truth to the empty concept (i.e., to the constant function to falsity) and falsity to everything else.

A special functor ‘I+’ can be defined for the basic arithmetical function that assigns to any quantificational concept Φ the quantificational concept that “immediately succeeds” it:

D I+ [I + Φ]a Fa = _{ df } ∃a (Fa & Φb [Fb & a ≠ b ]).

These definitions are purely logical. They determine unique semantic extensions for ‘∄’ and ‘I+’ respectively. Each of ‘∄’, ‘[I + ∄]’, ‘[I + [I + ∄]]’, etc. is a numerically-definite quantifier:

nothing ;

exactly one more thing than nothing ;

exactly one more thing than exactly one more thing than nothing ; etc. Let us abbreviate ‘[I + ∄]’ as ‘I’, ‘[I + [I + ∄]]’ as ‘II’, ‘[I + [I + [I + ∄]]]’ as ‘III’, etc. Thus: ⊦ I

x F

x ↔ ∃!

x F

x ; ⊦ II

x F

x ↔ ∃

y ∃

z (

y ≠

z & ∀

x [F

x ↔

x =

y ∨

x =

z ]), etc. Each of these numerically-definite quantifiers has its semantic extension the characteristic function of an equivalence class of equinumerous concepts. Those characteristic functions constitute

the quantifier -

numbers : ∄, I, II, III, IV, … .

Dℚ№ ℚ№ (Φ) = _{ df } ∀ℑ[ℑ(∄) & ∀Ψ(ℑ(Ψ) → ℑ[I + Ψ]) → ℑ(Φ)].

That is, a second-level concept Φ is a

quantifier -

number iff Φ falls under every (including the most restrictive) third-level concept ℑ under which ∄ falls and which is closed under I+.

The notation introduced by D ∄-Dℚ№ is well-defined. The definitions do not provide identity criteria for numbers, in the sense that previous interpreters think Frege requires, but the Caesar problem, as Frege means it, does not arise.^{32} That Caesar is not among the quantifier-numbers is no truth of logic—it is not an analytic truth—but D ∄ and D I+ together with the facts of logic and the facts about Caesar determine that ‘ℚ№(Caesar)’ is false.

For idiosyncratic reasons Frege evidently rejects as impossible any function like I+ whose values are themselves functions. (See note 23 above.) Frege’s scruple against functions to functions poses an inconvenience, but not an insurmountable obstacle. There is a straightforward workaround. In lieu of the offending function I+, we define instead the corresponding immediately-preceding relation between quantifier-numbers:

D ℘ ℘(Φ, Ψ) = _{ df } ∀F[Ψx Fx ↔ ∃a (Fa & Φb [Fb & a ≠ b ])].

The following relationships obtain: ℘(∄, I); ℘(I, II);℘(II, III), etc. It follows immediately from

D ℘ that ℘(Φ, Ψ) & ℘(Φ, Χ) → ∀F(Ψ

x F

x ↔ Χ

y F

y ).

^{33} As noted above, on the one hand Frege appears to be content in

FA to identify numbers with the quantifier-numbers. On the other hand, he sees concepts as functions and therefore not objects, and he is very clear that numbers are objects and therefore not functions. I do not share Frege’s view on this issue—the quantifier-numbers are numbers enough for the purposes of classical logicism—but to a significant extent his preference can be accommodated.

^{34} Insofar as Frege insists that numbers be objects, the numeral ‘0’ and the functor ‘

s ’ are easily introduced as the extension-oriented analogs of ∄ and I+. Terms for these analogs are defined using Frege’s variable-binding extension-abstraction operator ‘ἐ’, which he regards as a device of pure logic. For good measure, the familiar dyadic predicate ‘∈’ of set theory is first introduced as a purely logical term for the binary relation between an object

x and the extension

y of a concept G under which

x falls.

D∈ x ∈ y = _{ df } ∃G[Gx & y = ἐGɛ].

D0 0 = _{ df } ἐ∄a ɛ(a ).

Ds s (n ) = _{ df } ἐ∃a (ɛ(a ) & λb [ɛ(b ) & a ≠ b ] ∈ n ).

The FA -numbers , to be distinguished from the quantifier-numbers, are exactly the elements of the sequence: 0, s (0), s [s (0)], s (s [s (0)]), … . These are the numeraters of FA , the objects that numerically “belong” to concepts.

With

D0 and

Ds in play, Frege’s

D0num ,

Dsnum , and

D ~

num may be pressed into service to provide a legitimate recursive definition for the relational predicate ‘numerate’ as a term for a particular third-level relation (and without pretending also to define thereby ‘0’ and a successor-functor—see note 8). The recursive definition is easily converted into a fourth-order “explicit” definition:

Dnum n numerates F = _{ df } ∀ϕ [∀G[ϕ (0, G) ↔ ∀x ~Gx ] & ∀m ∀H(ϕ [s (m ), H] ↔ ∃a [Ha & ϕ (m , λb [Hb & a ≠ b ])]) → ϕ (n , F)].

That is, a class

n of first-level concepts is said to

numerate (or to

be a number that belongs to ) a first-level concept F iff

n stands to F in every (including the most restrictive) third-level binary relation

ϕ such that (

i ) 0 stands in

ϕ to just the vacuous concepts; and (

ii ) the successor

s (

m ) of a class

m of concepts stands in

ϕ to a concept H iff there is an object

a that falls under H and such that

m stands in

ϕ to the concept λ

b [H

b &

a ≠

b ]. One third-level binary relation that satisfies (

i ) and (

ii ) is the Julius-numerating relation. But

D ~

num quickly dispels any fear on this score. The most restrictive relation satisfying (

i ) and (

ii ) is that of an

FA -number

n to a concept F of there being exactly

n F’s. Caesar Julius-numerates but he does not numerate.

The implicit definition

D № is retained intact. (Alternatively,

Dℚ № may be imitated for the

FA -numbers.) The cardinality functor ‘#’ is defined in terms of ‘numerate’:

This completes the system of purely logical definitions that replaces the discredited

FA §55 definitional system as well as

FA ’s key definition (§68) of

the number of F’s as the class of concepts in one-to-one correspondence with F. That Caesar is not among the

FA -numbers is no truth of arithmetic, but the replacement definitions together with the relevant facts determine that ‘№(Caesar)’ is false. The notation is well-defined. The Caesar problem does not arise.

^{35} None of the definitions directly employs the notion of one-to-one correspondence. For Frege’s purposes, HP must be derivable from them as an analytic truth. Moreover, the use of ‘#’ is justified if it can be proved, as a matter of pure logic, that ∀n ∀m (n numerates F & m numerates F → n = m ). Here Basic Law V rears its ugly head. Contrary to a widely held view, the logical inconsistency of Frege’s program is not due to his identification of the natural numbers with the FA -numbers (equivalence classes of equinumerous concepts). An exactly analogous inconsistency awaits the philosophically more natural identification of the natural numbers with the quantifier-numbers. Inconsistency results if the concept-calculus is untyped and the classical λ-conversion rule of λ-expansion is applied across the board directly to unreduced paradoxical constructions like ‘λF[~F(F)]’, which purports to designate the Russellian putative concept, concept that does not fall under itself . (Cf . the derivation of HP in FA §73.) The disease is borne not by the identifications but by naïve comprehension, whether right-to-left Basic Law V or unrestricted λ-expansion in an untyped Begriffsschrift .^{36}