FREGE ON IDENTITY STATEMENTS

I am very pleased to be able to contribute this paper to a festschrift for Andrea

Bonomi. This is not however, the paper I really wanted to write; I would have much

rather have contributed a paper comparing the pianistic styles of Lennie Tristano and

Bill Eva n s, which I think Andrea would have found much more fascinating than an

essay devoted to an under standing of Frege’s thinking. But I do not totally despair.

Andrea’s first paper published in English was entitled “On the Concept of Logical

Form in Frege,” so perhaps I can maintain so me hope that this paper will appeal to

lingering interests that Andrea wrote of in the past. I would like to thank Johannes

Brandl, Ben Caplan, Bill Demopoulos, Bob Fiengo, Mark Kalderon, Patricia Marino,

Gila Sher, Michael Thau, Dan Vest and especially Aldo Antonelli for very helpful

discussion.

1 In what follows, I will reference Frege’s writings by name, with bibliographic listings

given at the rear of the paper. References to other works are given in the notes. I will

also for the most part utilize standard logical notations in lieu of Frege’s.

2 Included with the latter are statements with more prolix locutions for identity, such

as “is the same as” or “is coincident with.”

3 See, for example, W.V.O Quine, Philosophy of Logic (Englewood Cliffs: Prentice-

Hall, 1970), pp. 61 - 64.

FREGE ON IDENTITY STATEMENTS

Robert May

1. Looking at the sweep of Frege’s writings, one is struck by the

coheren c e of his philosophical perspective; virtually all of the basic

issues that animated his thinking are brought forth in his early work, and

much of his subsequent th ought can be seen as attempting to find the

most cogent and coherent packaging to express these ideas. 1 One place

where this can be seen most graphically is in Frege’s known remarks on

i dentity statements, a category into which he ultimately l umped

mathematical equalities, containing the equals sign, with sentences of

natural language like “Hesperus is Phosphorous” that contain a form of

the verb “be.” 2 With remarks spanning a period of 35 years from 1879

to 1914, Frege’s perception of the import ance of identity statements,

especially for his views of logic and number, rem ained constant,

stemming throughout from his understanding of the need for an identity

symbol in his “conceptual notation” ( Begriffssc hrift ) in order to obtain

the generality required of a logistic system. But the presence of this

symbol raised a problem that still perplexes us today. 3 On the one hand,

2

identity statem ents play a logical role, licensing substitutivity; but yet

they also express substantive propositions, to be proved or established.

Is the identity symbol to be a logical or non-log ical symbol? Both of

Frege’s approaches to identity statements, that of Be griffsschrift in

1879, and that of Grundgesetze de r Arithmetik and “On Sense and

Reference” in the early 1890’s, although born from somewhat disparate

considerations, bear strong similarities in the way the y attempt to

resolve this tension. Indeed, Frege toiled so diligently and profoundly at

this resolution that it resulted in one of his most important and enduring

contributions to philosophical thought, the distinction between sense and

reference.

In Begriffsschrift , as the title implies, Frege’s primary goal was

to present a logical theory; part of the brilliance of this theory was in the

way Frege saw how logic an d semantics were related, and the

importance of logic al form for understanding the intimacy of this

relation. Thus, Frege makes much of the logical form “ f ( a ),” composed

of a func tion-expression and an argument-expression, as transparently

representing the application of function to object. Where this tight

relation comes unstuck, as Frege saw it, was precisely at identity

stat ements. From the outset of his explicitly logical explorations in

Begriffsschrift , Frege t h ought it undeniable that among the basic

symbols must be one for identity; otherwise, logic could not suffice as a

general system of reasoning. To achieve this generali ty, Frege

understood that not only must this symbol appear in propositions that

can be true or false, it must also be a logical symbol; the truth of a

statement of identity allows for a transition between propositions by

substitution in the course of proof. Frege initially thought that he could

connect the logical and semantic aspects of identity statements by a

cla im about the constituents of the logical form of identity statements,

that they could be ana lyzed metalinguisticaly as relations between

expressions. But by the time of G rundlagen , Frege had desist ed from

justifying an “identity of content” sign; with the emergence of the

logicist project, considerations that did not weigh on Frege in

Begriffsschrift made it clear that a different analysis was called for, for

at the heart of this project were propositions that had to be construed as

identity statements holding of logical objects, the most basic sorts of

objects that logic concerns itself with, on Frege’s view. That the

analysis of the identity symbol as a sign of objectual identity was not a

b ar to identity statements playing their logical role was definitivel y

4 Frege never ex p licitly remarked on the foundational implications for logic of assuming

his more finely developed semantic views, never developing an intensional logic. This

task was most famously undertaken by Alonzo Church; see his “A Formulation of the

Logic of Sense and Denotation,” in P. Henle, H. M. Kallen, and S . K. Langer, eds.,

Structure, Method and Meaning: Essays in Honor of Henry M. Sheffer (New York: The

Liberal Arts Press, 1951), pp. 3 - 24 as well as subsequent papers revising the form-

ulation.

5 I have in mind here Frege’s strong attachment, at least in his earlier work, to the

central importance of the Kantian picture of the categories of judgements. For

discussion of Kantian influences on Frege, see Philip Kitcher, “Frege’s Epistemology,”

Philosophical Review , LXXXVIII, 2, (1979): 235 - 262.

3

clarified by Frege in Grundgesetze . But now the semantic issue of how

identity statements could be truly, and not tri vially, about something,

loomed large, for something fundamental was presently a t stake for

Frege, the nature of number. In particular, Frege archly felt the need to

meet and deflect the views and criticism leveled by the mat hematical

formalists, which of comp eting views most directly threatened Frege’s

foundational assumptions. Thus, from the 1890''s on, the discussion

shifts to the semantic analysis of identity stateme nts, with center stage

belong ing to the justification of the central semantic relation, deter-

mination of reference, or as it is now called, Sinn . 4

While Frege’s initial analysis of identity statements was perhaps

too heavil y governed by views that seem anachronistic today, it was a

well-considered view, as was his depart ure to a view intimate with the

great later developments in his thinking. As noted, Frege strived to pre-

sent a coherent philosophical picture; finding the right analysis of

identity statements that fit within his philosophical and mathematical

innovations, as well as his prejudices, 5 was essential to maintaining that

coherence. Thus, to the extent that fundamental notions remained

constant in his thought, we find similarity in his views of identity

statements; to the extent that they evolve, discordance. In the course of

the remarks to follow, I shall try to track the developments I have been

describing so as to illuminate these similarities and discordances, with

an eye to the central importance of identity statements in Frege’s

thinking. Our route will be roughly historical; after a brief discussion of

rudimentary aspects of the system of B egriffsschrift (§2), we will turn

to the identity of content analysis presented there (§§3-4). §§5 and 6

will explore the considerations that led Frege away from this account, to

4

the objectual identity account; §7, the criticism this account engendered

from the mathematical formalists. Frege’s answer to this challenge, via

his theory of thought and judgement, is the topic of §8. The role of

identity statements in the theory of number of Grundg esetze and Basic

Law V is the concern of §9, and finally, in §10, we turn to Frege’s

ultimate meditation on identity statements, his most famous essay, “On

Sense and Reference.”

2. The task that Frege posed for himself in B egriffsschrift he sets out in

the sub-title of the book; it is to pr esent “a formula language, modeled

upon that of arithmetic.” The notation that Frege develops in B egriff-

sschrift, his “conceptual notation,” carried through this project as a for-

malization and generalization of the way that mathematical reasoning,

stripped of fool ish assumptions and mistakes, is carried out. It consti-

tuted a substantial advance in logic for reasons that are well-known: for

the first time it became possible to have a formal system in which rigor-

ous, gapless proofs could be carried out. Frege’s formalism for carry-

ing through these proofs, although famously unwieldily, does have cer-

tain virtues absent from more familiar notations. In Frege’s system, the

formulas can be parsed into two fundamental parts, the strokes and the

symbols . The strokes and symbols each have their own particular

modes of combination. Basic for the joining of strokes are condition-

alization and scope; for symb ols, application of function to argument.

These distinctions in junction, according to Frege , are themselves for-

mal, or syntactic, but they map onto a distinction in content. The con-

tent of the strokes is given by their logical role, as specified by what

a mounts to a system of truth-tables; their significance lays in the rela-

tions holding among the (complete) symbols they connect together. The

content of the symbols, on the other hand, is semantic, or as Frege

would put it, “conceptual”; their significance resides in their designation

of concepts and objects. Thus, the graphically distinct presentations of

strokes and symbols mirr ored distinct roles they play in the overall lo-

gistic system, thereby justifying the orthography of the system. Via the

strokes, Frege characterized what we would call today the logical terms;

6 To be clear, Frege did not have a theory of logical terms; he presented no necessary

and sufficient conditions whereby we could classify terms as logical or non-logical. My

point is just that his system embedded the d istinction, and the notation graphically

distinguished the terms.

7 There are two complete translation of Begriffsschrift into English, one by Bynum, the

other by Bauer-Mengelberg. There are also partial translations by Geach and by

Beaney. Unless otherwise noted, quotations are from the Bynum translation.

8 The system I am describing is that of Begriffs schrift . Frege adopts different

terminology in G rundgesetze , re -labeling the “content-stroke” as the “horizontal.”

This makes clearer the p urely notational role in the logic played by this stroke; it is

there, so to speak, in order for other strokes to hang off of, but it has no interpretation

as a logical term itself. Frege must have realized, given the redundancy of “— A ” and

“ A, ” that his talk in Begriffsschrift of the content-stroke unifying a content added little

in way of clarification, and even less so in light of the more sophisticated notion of

content he presupposes in Grundgesetze , where he maintains that both of these denote

(name) the same tru th-value (§5). (In G rundgesetze , forms with the addition of the

vertical line are assertions of truth-values, not judgements, in line with a changes in

5

via the symbols, the non-logical. 6 Thus, we can rightly see Frege’s pri-

mary concern in Begriffsschrif t as being to give a theory of logical

form, given with the appropriate formal rigor to support proof-theoretic

certainty, conjoined with a semant ic theory, which, given Frege’s con-

cerns, is presented at a more informal level.

The most basic sort of logical form in the Begriffsschrift system

is that mad e up of the “content stroke” followed by a symbol for a

content:

—— A

Frege says of such forms that he takes the content stroke “to mean that

the content is unified” into a proposition; the “content stroke,” Frege

tells us, “serves also to relate any sign to the whole formed by the sym-

bols that follow the stroke.” ( Begriffsschrift , p. 112.) 7 The content

stroke is, if you will, the root stroke, to which others can be appended,

so as to connect to other contents. Primary among the connected forms

is that for the material conditional, from which, along with negation,

Frege specifies the other truth-fu nctional connectives. If a content,

whatever the complexity of its connected parts, is prefaced by the con-

tent stroke, one further additional stroke may be added to the content

stroke, the vertical “judgement stroke”: 8

Frege’s understanding of judgement; see discussion in §9 below.)

9 Frege’s term is beurtheilbarer Inhalt ; see T. W. Bynum, “On the Life and Work of

Gottlob Frege,” in Gottlob Frege, Conceptual Notation and Related Articles , translated

and edited by Terrell Ward Bynum, (Oxford: Clarendon Press, 1 972), p. 79 - 80 for

discussion of the various translations of this term.

10 While Frege when justifying judgement often slips into a psychological idiom, I think

it is unfair to accuse him, as Kenny does, of “a confusion between logic and what may

broadly be called psychology.” (Anthony Kenny, F rege , (Harmondsworth: Penguin

Books, 1995), p. 36.) Kenny comes to this conclusion because he thinks that Frege in

introducing the judgement symbol defines it in psychological terms. (p. 35) But this

is to confuse defining and justifying - at no point does Frege define any logical symbol

in “psychological” terms, although throughout his work he often justifies in such terms

the need to define a symbol. That Frege would talk in this way is not too surprising,

as the stated goal of B egriffsschrift is to model the fundamental properties of thought,

“freeing thought from that which only the nature of the linguistic means of expression

attaches to it,” (p. 106) and one aspect of thought to be so liberated was judging. There

are, as well, purely logical grounds justifying the judgement stroke, as Geach observes

in his “Frege,” in G.E.M Anscombe and P.T. Geach, Three Ph ilosophers , (Ithaca:

Cornell University Press, 1961), p. 133.

6

|––— A .

The judgement stroke may be added just in case a content possesses the

characteristic of being a possible content of judgement, or to use o ther

terminology, it must be an assertable content . 9 We can proceed for

instance from:

—— f ( a )

to:

|––— f ( a ),

where we have symbols for function and argument because what follows

the content stroke qualifies as a possible content of judgement; we could

not comparably proceed if only one or the other of these symbols oc-

curred without the other. While Frege mentions a variety of reasons for

the importance of distinguishing judged and unjudged proposition s, the

important point to recognize here is that in Begriffsschrift , a judgement

is a type of logical form ; (or to be more precise, there is an orthographi-

cally unique type of logical form that expresses a judgement). 10 Adding

11 This elucidation is found in a footnote describing the s ymbols of the conceptual

notation in “Boole’s logical Calculus and the Concept-script,” p. 11.

7

a vertica l judgement stroke to a horizontal content stroke thus t rans-

forms one sort of logical form, representing a mere conceptual content,

into another sort, representing a content of judgment. Or, as Frege puts

it, the judgement stroke “converts the content of possible judgement into

a judgement.” 11

Now Frege remarks with respect to contents of judgements that

there may be various ways of pars ing complex expressions as function

and argument; for example, “Wittge nstein admired Frege” may be

parsed as containing three di fferent functions, depending upon whether

“Frege,”“Wittgenstein” or both denote arguments of the function. But

whichever way the parse is made, “we can . . . apprehend th e same

conceptual content”; these differences in parse have “nothing to do with

conceptual content, but only with our way of viewing it.”

( Begriffsschrift , p. 126.) Frege is insistent on this point as part of

showing that there are parses as function and argument other than

subject/predicate. But the remark is also revealing of Frege’s view of

the relation of form and content. Content is something that is v iewed

through form; it is through form that content is accessible to reasoning.

It is because a content can be viewed that it can be judged. When so

viewed it may be a content of a judgement, and the transition made to a

logical form to which the judgement stroke has been added to the content

stroke. If we ask why in a judgement

|––— A

is the conceptual content viewable, the answer is that there is a certain

conceptual analysis phrased as function and argument that makes it so.

A judgement is a viewable conceptual content.

3. In B egriffsschrift , immediately after introducing the logical strokes,

Frege turns in §8 to the discussion of “identity of content,” before his

attention is drawn, in §9, to “The Function.” It is telling that the discus-

sion is juxtaposed in this way, given the janus-faced logical/semantic

nature of statements of identity to which we have already alluded.

Frege’s notation at least makes it clear where he stands, for the identity

symbol is not found among the strokes, but among the symbols, in with

12 The italicized clause is from the Geach translation, p. 12.

13 There is an exegetical issue here. Does Frege intend the substitution clause to be

part of the “meaning” of judgements of identity of content, or does he take it to be a

parenthetical remark about the logical role of judgements of identity of content within

8

those things that have conceptual content. This placement immediately

raises two issues:

(i) What is the logical form of identity statements such

that their content can be viewed?

(ii) How do identity statements play their logical role,

if the identity symbol is a “symbol,” and hence

non-logical?

To understand Frege’s answers to these questions consider how he ana-

lyzes the logical form of identity statements in Begriffsschrift .

F rege opens the section entitled “Identity of Content” with the

following remark: “Identity of content differs from conditionality and

negation by relating to names, not to contents.” ( B egriffsschrift , p.

124.) He makes this more explicit, at the close of the section, as

follows: 12

Now, let

|––— ( A / B )

mean: the symbol A and the symbol B have the same

conceptu al content, so that A can always be replaced

by B and conversely.

The triple-bar “identity of content” symbol that we see in the judgement

(type) displayed, which we may paraphrase as the relation “— has the

same co nceptual content as —,” has the blanks filled in not with sym-

bols for objects, but with symbols for symbols; judgements of such form

are judgements of identity of content. From this definition, we can see

right off that the answer s to our questions will be interconnected in a

very fundamental way, for what Frege maintains here is that i dentity

statements can play the logical role they do just because the conceptual

content of the symbols to be substituted one for another is the same. 13

the overall logical system? While the former is perhaps indicated by Frege’s

typographical use of italics, the latter interpretation is perhaps more plausible, given

the redundancy that woul d be introduced between the definition and the axiomatic

governance of substitution that Frege specifies with proposition (52) of Begriffsschrift .

Regardless, what is important here is the apparent causal connection Frege sees

between identity of content and substit ution; one substantial change we find in

Grundgesetze , where Frege assumes objectual id entity, is that this connection is

broken. Nothing about an identity statement, in and of itself, implies anything about

substitution of symbols.

9

Early on in Begriffsschrift Frege remarks that “the only thing

considered in a judgement is that which influences its po ssible

consequences . Everything necessary for a correct inference is fully

expressed; but what is not necessary usually is not indicated”

( Begriffsschrift , p. 113.) Given this, it must have been that Frege

thought that judgements of identity of content expressed the information

necessary in order for inference to proceed properly, and moreover that

this information must be metalinguistic. Why would he have thought

that? One of the things upon which Frege was most insistent was the

formal nature of proof in a logistic system. What Frege meant by this is

that wh ile inferential relations hold between thoughts, we can only

determine whether one thought follows from another with respect to the

form by which that thought is expressed. Proofs proceed, if you will, in

terms of what expressions (simple or complex) look like. Thus, because

of the nature of the notational system, proofs could be carried out by

proceeding from step to step via formal matching of symbols; h ow one

can proceed in a proof in virtue of such matching is stipulated by rules

o f inference. Usually, these stipulations are made externally to the

system; think of m odus ponens , which gives license to detach the

consequent from the antecedent of a conditi onal. But what if a pattern

of inference were licensed by some sort of proposition that occurs

internall y to the system; what sort of information would it need to

carry? Presumably, it too would have to carry the pertinent formal

information about the symbols involved needed to proceed from one step

in a proof to another; that is, it would have to carry metalinguistic

information. In the case of substitution, if it is to be licensed by a

statement of ide ntity, then that statement must carry the pertinent

information about the symbols such that they can be substituted one for

another. If we have a judgement that says of symbols that they have the

14 Although submitted to three different journals, the paper never saw print; see

Bynum’s remark on p. 21 of “On the Life and Work of Gottlob Frege.”

15 “Boole’s logical Calculus, p. 29.

10

same content, e.g. “ c / d ,” then we can move in a proof from the judge-

ment:

|––— P ( c )

to the judgement:

|––— P ( d ).

Frege characterizes such inferences by proposition (52):

|––— ( c / d ) e (f ( c ) e f ( d )),

glossed with the remark that it “says that we may replace c everywhere

by d , if c / d ” ( Begriffsschrift , p. 162).

If capturing information relevant for i nference is indeed the

reason fo r the metalinguistic identity of content, then the following

thought occurs: couldn’t this information be exported and stated as a

rule of inference? Couldn’t we just transpose the basic proposition into

a rule of inference in an informationally neutral way? According to the

following remark, from “Boole’s logical Calculus and the Concept-

script,” dated 1880/81, 1 4 Frege contemplated doing just this not long

after finishing Begriffsschrift : 15

In the preface of my Begriffsschrift I already said that

the restriction to single rule of inference w hich I there

laid down was to be dropped i n later developments.

This is achieved by converting what was expressed as a

judgement in a formula into a rule of inference. I do

this with formulae (52) and ( 53) [i.e. the law of self-

identity] of my Begriffsschrift , whose content I render

by the rule: in any judgement you may replace one

symbol by another, if you add as a co ndition the

equation between the two.

11

Observe that Frege describes the proposed r ule of inference in words

cast in a manner very similar to those used to describe basic proposition

(52); he speaks here as well me talinguistically of the relation of sym-

bols. So it seems that as long as the information relevant to proof re-

mains constant, basic propositions can be transformed into rules of in-

ference; these are two ways of saying the same thing, one way internal,

the other external, to the system.

While Frege’s remarks illuminate how he understood the logical

aspects of judgements of identity of content, he unf ortunately is not

entirely clear in elucidating what advantage he saw in stating

substitution under identity (of content) as a rule of inference. We may,

however, conjecture along the follow ing lines. Stating substitution

under identity as a basic proposition internal to the system forced upon

Frege a bifurcation of symbols. Although the symbols that occur in the

judgements that are inferentially related are, as Frege puts it,

“representatives of their contents,” wha t are substituted in the

conceptual notation are symbols, so we must be able to recognize that it

is the symbol “ b ” that is being substituted for the symbol “ a .” “Thus,”

Frege says, “with the introduction of a symbol for identity of content, a

bifurcation is necessarily introduced into the meaning of every symbol,

the same symbols stan ding at times for their contents, at times for

themselves.” ( Begriffsschrift , p. 124). If our concern is just with the

thoughts that partake of the inference, then the symbols stand for their

content; but if our concern is with the proof of that inference, then we

must be able to see the symbols as stand ing for themselves. Symbols

shimmer between these two ways of being seen. Note, however, that

Frege apparently did not feel any compunction to adopt such a bifur-

cation by assuming modus ponens as a rule of inference, and for good

reason. In a proof, we proceed from step to step by recognizing that

there is formal matching of symbols, and in order to state which

matchings are legitimate we need symbols that stand for symbols (cf. the

contemporary usage of sc h ematic letters and corner quotes). The

symbols that are so recognized, however, are tho se that are u sed in the

statements that are inferentiall y related; mention of the symbols to be

recognized in statements that may be inferentially related does not

require that in the forms so related that the symbols themselves be

menti oned . In the case of m odus ponens , validating that “ b ” follows

from “ a ” and “ a e b ” does not requ ire that any of the symbols be

mentio ned in the object language . Thus, while stating m odus ponens

16 Notice that if our conjecture is along the right lines, Frege would not have seen a

use/mention confusion in the Begriffsschrift theory at this point in his thin king,

although he may have come to see it that way later on. Church, for one, thought so:

“If use and mention are not to be confused, the idea of identity as a relation between

names renders a formal treatment of the logic of identity all but impossible. Solution

of this difficulty is made the central theme of über sinn und Bedeutung and is actually

a prerequisite to Frege’s treatment of identity in Grundgesetze der Arithmetik .” ( i bid ,

p. 3)

12

has a metalinguistic character, qua rule of inference, it would only be to

confuse use and mention to bifurcate the symbols of the language. In a

comparable way, there would be no call for bifurcation from

substitution if it were characterized as a rule of inference, so that the

artifice of taking the symbols of the conceptual notation as standing for

anything other than their contents could be abandoned, for as Frege

points out, it is only judgem ents of identity of content that require this.

No longer would symbols need to stand both for their contents and

themselves, a considerable simplification of the con ceptual notation.

But notice that if symbols are no long er bifurcated, then there is no

longer any place for an identity of content symbol; identity will have to

be otherwise defined. Frege recognized this in comparing his system to

Boole’s: “The first thing one notices is that Boole uses a greater

number of signs. Indeed I too have an identity sign, but I use it between

contents of possible judgement almost exclusively to stipulate the sense

of a new designation. Furthermore I now no longer regard it as a

primitive sign but would define it by means of others.”(“Boole’s logical

Calculus,” p. 35-6.) Although he says nothing more on the matter, his

making it a virtue that his system has fewer signs than Boole’s indicates

that Frege’s initial rationale for moving a w ay from the Begriffsschrift

theory was that it simplified the conceptual notation. But aside from

this, Frege most likely would not have seen any problem in returning to

the prior approach; however, considerations were to shortly weigh in,

starting with Grundlagen , that would move Frege t o a rather different

view. 16

Returning to the Begriffsschrift account, we can place our

finger on the re ason that Frege adopted the metalinguistic slant of

judgements of identity of content in that he thought that in this way the

symboliza tion of identity could play its logical role, allowing proper

movement from step to step in proofs. We must be able to see in a

17What awareness Frege had of the issue put this way at the time of writing

Begriffsschrift is open to debate. Hans Sluga, in G ottlob Frege (London: Routledge

and Kegan Paul, 1980) and “Semantic Content and Cognitive Sense” in L. Haaparanta

and J. Hintikka, eds., Frege Synthesized: Essays on the Philosophical and

Foundational Work of Gottlob Frege (Dordrecht: D. Reidel, 1986), points out that there

is recognition of the problem, in the context of Kantian views, by Lotze in his Logic of

1874, and that he developed a position not unlike Frege’s. This is one of the bases for

Sluga’s argument that Lotze was a significant influence on Frege, a view that has been

extensively challenged by Michael Dumment, in The Interpre tation of Frege’s

Philosophy (Cambridge: Harvard University Press, 1981) and in a number of articles

collected in Frege and Other Philosophers (Oxford: Clarendon Press, 1991). Dummett

does make a case that Frege had read Lotze prior to 1879, but believes him not to have

held Lotze in particularly high esteem, never mentioning him by name in his writings.

But regardless, it is clear that the significance of the problem d id not come home to

roost for Frege until much later, and then arising from a rather different source. See

discussion in §7, especially footnote 43.

13

judgement of identity that symbols have identical contents, for otherwise

such judgements would not fully express “everyth ing necessary for a

correct inference.” Although there is nothing in principle barring the

introduction of a symbol for objectual identity into the conceptual

notation, it should be clear, given Frege’s logical literalism in B egriff-

sschrift, that statements of objectual identity would not suffice for the

job at hand, for they would not contain all the needed information. All

that would be expres sed in such a judgement would be that there is a

unitary content, and this would at best beg the question of how

substitution of an object for itself could lead from one distinct form to

another. (Since the terms of an objectual identity are just “repre-

sentatives of their contents,” there is one and only one content in

question.) Insofar as Frege would have seen a problem with statements

of objectual identity in Begriffsschrift it was that they would not provide

the right information for inference, and would not play the logical role

Frege demanded of them. Note that this conclusion arises orthogonally

to the semantic relation of “ a = b ” to “ a = a .” In fact, Frege nowhere in

Begriffsschrift mentions the semantical problem that has come to be

known as “Fre ge’s Puzzle”- that “ a = a ” and true “ a = b ” express the

same proposition (have the same content) - and there seems to be no

reason that he would, for the problem for Frege would h ave been with

“ a = b ” itself, and that problem would be a logical problem. 17

4. In §8 of B egriffsschrift , aft e r informally introducing the notion of

identity of content, Frege turns immediately to a possible objection. We

18 Wittgenstein thought so; see §5.53 ff of the Tractatus .

19 I find that the translation by Geach of this passage from Begriffsschrift (p. 11) reads

better than Bynum’s: “what we are dealing with pertains merely to the expression , and

not to the thought .” ( Begriffsschrift , p. 124.)

14

may put it as follows. Why, we ask, if “ A ” and “ B ” have the same con-

tent, do we need a symbol for identity of content? An initial answer is

that in order to connect the ex pressions so as to form a judgement that

says they have the same content we must recognize the difference be-

tween there being identity of content and expressin g identity of content

in the conceptual notation; we want to be able to express identity of con-

tent because this is what warrants substitutions of formally unlike sym-

bols. This may appear, however, a rather hollow justification for if the

reason for having an identity symbol is just that we have unlike symbols

with the same content, doesn’t this just show bad design of the system? 18

Isn’t the identity symbol just an artefact, needed only because of an

assumption about symbols, and eliminable by requiring a biunique rela-

tion between symbols and content? Yes, Frege tells us, but o nly if “it

were here a matter of something pertaining only to expression , not to

thought .” 19 That is, the criticism would be appropriate if a judgement

of identity of content was no more than an assertion of coreference. But

this Frege denies holds of his theory; in it the matter rather pertains both

to expressions and to thought, and it is this that ultimately justifies hav-

ing an identity symbol within the conceptual notation.

Frege puts matters in the following way:

. . . the same content can be fully determined in

different ways; but, that the same content , in a

particular ca se, is actually given by t wo different

modes of determination is the content of a judgement .

Before this can be made, we must supply two different

names, corresponding to the two [different] modes of

determination for the thi ng thus determined. But the

judgement requires for its expression a symbol for

identity of content to combine the two names. It

follows from this that different names fo r the same

content are not always merely an indifferent matter of

form; but rather, if they are associated with different

20 Begriffsschrift, pp. 125 - 6.

21 On Frege’s way of putting things in Begriffsschrift , it is only assumed that if there

are distinct modes of determination, they will be associated with distinct symbols; what

is not clear is whether Frege allows a single mode to be associated with distinct

symbols. If this were allowed, however, there would not be a need for a judgement of

identity of content, for that would be already determined by the very fact that they are

associated with the same mode of determination. The issue would then only be one of

notation. Suppose that the mode of determination associated with “3” and that

associated with “III” are the same; it would then be trivial to judge identity of content,

just as it would be to judge identity of content of repetitions of “3.” Thus, this case is

irrelevant to the matter at hand, although it may not be irrelevant to say that they are

translations ; i.e. that “3” and “III” are the symbols in distinct notational systems that

are associated with the mode of determination in question . (This is in essence the

argument Frege ma kes in “On Sense and Reference,” in which he explicitly allows

distinct symbols to have the same sense.)

15

modes of determination, they concern the very heart of

the matter. 20

The question facing Frege is why would we want to have unlike expres-

sions of the same content; what justifies this sort of multiplicity of sym-

bols? It is justified, Frege tells us, because expressions may be related

to contents in more than one way. The relation of expressions and con-

tent is not simplex or direct, but rather is mediated, by a mode of deter-

mination ( Bestimmungsweise ). Contents may be “given” in more than

one way, and if anything is biunique, it is the relation between expres-

sions and modes of determination of their content. Now we need to keep

things straight about this sort of relation in a way that can be symboli-

cally encoded in the conceptual notation; we do this by ass o ciating dif-

ferent labels - symbols - with distinct modes of determination. But then

all we can conclude from the occurrence of distinct symbols is that they

have distinct modes of determination, but n ot that they have distinct

contents. Whether they have the same or different content is left open

by the notation. It is closed by a judgement of identity of content; i.e. a

logical form in which the sym bols fall on either side of the identity of

content sign. If there are two different symbols in such a judgement, it

follows that the content is given in that judgement by two different

modes of determination, (for otherwise there would not be distinct sym-

bols). 21

Modes of determination, Frege thus tells us, are wha t justify

22 Grundlagen , p. 3

16

judgements of identity of conte nt; but more than that, it is through this

justification that we can touc h the thought, for this justification will

provide information fixing the judgements within the categories of

thought. In the G rundlagen , Frege asks, “Whence do we derive the

justification for [a content’s] assertion?” He answers as follows: 22

Now these dist inctions between a priori and a

posteriori, synthetic an d analytic, concern, as I see it,

not the content of the judgement but the justification for

making the judgement. Where there is no such

justification for making the judgement, the possibility

of drawing the distinction vanishes.

The justification to which Frege refers, we must understand, is not itself

part of the content of the judgement per se, and is thus not represented in

the judgement; it is rather extra -notational. Modes of determination are

th us in no way r epresented in judgement above and beyond that which

follows about them in virtue of the occurrence of (distinct) symbols.

What is expressed by a judgement of identity of content via its represen-

tation in the conceptual notation is solely that the symbols have identical

content; the justification for such a judgement, and hence its connection

to “thought,” is not so represented in such logical forms. This is not to

say, however, that modes of deter mination cannot themselves be ex-

pressed as contents of judgements, and in fact that they can be is of im-

portance in determining the category of judgements of identity of con-

tent.

Frege remarks that “If in carrying out [the proof of a

proposition], we come on ly on general logical laws and on definitions,

then the truth is an analytic one. . . . If, however, it is impossible to give

the proof without making use of truths which are not of a general logical

nature, but belong to the sphere of some special science, then the

proposition is a synthet ic one.”( G rundlagen , p. 4.) Thus, if we are to

categorize a judgement we must see what sort of premisses would justify

its truth; what sort of premisses would be needed for its proof. In

Begriffsschrift , in way of illustration, Frege asks us to c onsider two

points on a circle, “ A ” and “ B ,” where A is a fixed point on the

23 My thinking in this section has been influenced by remarks of Mark Kalderon.

17

circumference, “given through perception,” and B is computed as the

(rotating) point of intersection of a line from A to the circumference.

Frege then asks “What point corresponds to the position of the straight

line when it is perpendicular to the diameter?” The answer is A ; A and

B are the same point. “Thus, in this case,” Frege now tells us, “the

name B has the same content as the name A ; and yet we could not have

used only one name from the beginning since the justification for doing

so is first given by our answer. The same point is determined in two

ways”. ( Begriffsschrift , p. 125.) What Frege has illustrated here is a

“proof” of the judgement that “ A ” and “ B ” have the same content; the

role of the modes of determination is that they stand as premisses of this

proof. These modes of determination, however, are synthetic, one being

geometrical, the other perceptual. Thus, because of the way in which

this conclusion was reached, in this case, Frege puts it, “the judgement

as to identity of content is, in Kant’s sense, synthetic.”

Judged by conceptual content, introducing a new atomic symbol

into the conceptual notation with the same content as some other atomic

symbol, does not, in and of itself, change the expressiveness of the sys-

tem, unless b y that introduction there is some judgement that can be

expressed that otherwise could not be. This obtains in the case at hand,

for a synthetic judgement has become possible, established by a proof of

that judgement, where the critical premisses in the proof that fix its

status are the modes of determination. Without such modes of

determination there would be no way to establish that distinct

expressions, atomic or not, play distinct roles in proofs, even though

they have the same content. 23

Before continuing, let us quickly survey the territ ory we have

covered. In order to have a logistic system sufficiently general so as to

serve as a general system of reasoning, Frege needed a symbol for an

identity relation in the conceptual notation. Frege’s initial thought was

that in order for it to play its logical role, it had to be identity of content;

i .e. metalinguistic. This does not, however, detach judgements of

identity from thought, which Frege took to mean from the Kantian

categories of thought. One might think this because such judgements

are ostensibly about coreference of expressions. But t hey are about

something more, for the expressions related are associated with modes

18

of determination, and from these we can deduce the connection to

thought. In 1879, we thus see two themes in Frege’s thinking that strike

us as perhaps somewhat antiquated. One is his logical literalism, from

which the need to mention the terms of the identity in order to explicate

their logical role follows. The other is his adherence to the enduring

influence of fundamental Kantian notions. In contrast, his concern

about identity is strikingly modern, for in Begriffsschrift we see clearly

illuminated for the first time the tension associated with identity

statements, between their logical role and their sta tus as expressing

contentful propositions, which may be true of false.



5 . In introducing modes of determination, Frege’s concern was with

how to justify having a symbol of identity of content within the concep-

tual notation in the face of an objection that it would be otiose, for we

could just forbid in the first place symbols with the same conceptual

content. But such an objection, at least under one interpretation, is pa-

tently absurd, if it is to mean that there could not be both a simple and a

complex term (or multiple complex terms) for the same content, that we

could not, for instance, name and describe things. Rather, the objection

is only sensible if the issue is whether there can be more than one simple

atomic term for any given content. Given his goal of having a generally

applicab le system for reasoning regardless of the content of proposi-

tions, Frege must allow for this, as we commonly come across in the

sciences more than one simple term for a single thing. Showing by em-

pirical proof, (i.e. proof from empirical assumptions), that atomic terms

apply to the same thing can be the essence of a scientific discovery. If it

is justified to have more than one atomic term for a given object, we will

then need an identity of content sign, if we are to allow for the assertion

that they are terms for the same thing. Such assertions are not just

about the expressions, but also pertain to the thought because each

atomic term is itself justified by being associated with distinct modes of

determination. Such distinct modes of determination thus give good

reason for multiple atomic terms for a given object.

Seen in this light, Frege’s choice of a geom e tric example to

illustrate becom es understandable, although at first it might seem odd.

For although the system of Begriffsschrift is intended as a general

system, applicable to any domain of reasoning, Frege’s particular

concern is with the na ture of arithmetical judgements; the latter part of

the book establishes notions, in particular that of ancestral of a relation,

24 A number may be designated by more than one numeral if they are in different bases.

But this would be a case of how a number is designated in, so to speak, different

languages , and would be an issue of translation, rightl y a metalinguistic matter.

Equating numerals in different bases would be no different than equating Arabic and

Roman numerals; it would be to say no more than that “3” and “III” are different ways

of representing or insc ribing the same numeral (number-name). This relation,

however, is not identity of content as Frege saw matters, for it does not correspond to

a difference in mode of determination; the number 3 - that object - is determined in the

same way regardless of whether we write its numeral “3,” “III,” or “three.”

25 One place where Frege gives a judgement with equality is when introducing the

material conditional in §5 - he uses “3×7 = 21” as an example.

19

that were to play a vital role in Frege’s forthcoming logicism. Yet Frege

chooses not an arithmetical example, but a geome trical one, and one

might be curious why this is. The reason he avoids an arithmetical

example is that arithmetic is one domain whe re we get along with a

primitive biunique symbol/content relation - there is one and only one

atomic symbol for each number, its numeral, and hence there are no

simple expr essions of which we need to say that they have the same

content. 24 Thus, one way of seeing the problem with arithmetic is that it

just doesn’t have the right level of generality to serv e the point Frege

wishes to make; the trivial “2 / 2” wouldn’t do the trick. But with

regard to this aspect of arithmetic, there is also a more subtle issue at

play, which, when understood, sheds light on the path Frege took to his

“mature” view of identity, that of Grun dgesetze and “On Sense and

Reference.”

There is a curious aspect of Frege’s pr esentation in

Begriffsschrift and in the articles he wrote contemporaneous with it in

explication of his system. While Frege introduced the identity of content

sign in order to insure the general applicability of the conceptual

notation to all domains of inquiry, there is one crucial pl ace where he

does not employ this symbol. This is in judgements of arithmetic

equality. In stating sums, for instance, Frege nowhere writes them as

judgements of identity of content; he never writes an equation as “2+3 /

5,” but rather as“2+3 = 5.” Although it is not altogether apparent in

Begriffsschrift proper, 25 he makes this usage clear in a lecture entitled

“Applications of the ‘Conceptual Notation’,” presented in late January

1879, just six weeks after Frege completed B egriffsschrift , and in

“Boole’s logical Calculus and the Concept-scr ipt,” composed in 1880.

26 Including Frege’s own; see his usage in the mathematical work he had undertaken

by the time of Begriffsschrift, in particular “Methods of Calculation” (1874).

20

In both of these essays, Frege sought to show h ow the conceptual

notation could be employed to express certain complex arithmetic

propositions, providing in course a range of illustrations. To take but

one example, in “Applica t ions” in discussing the theorem that every

positive whole number can be “represented” as the sum of four squares,

Frege considers the following equation:

30 = a2 + d2 + e2 + g2.

Thus, the practice we observe is that Frege uniformly employed an

equality symbol, symbolized as “=”, in arithmetic propositions; this is in

addition to his assumption of the identity of content symbol (“ /”).

Given that F rege employed two different symbols, it is

reasonable to assume that he thought that they had two diff erent

meanings; it would not have been Frege’s style to have two distinct

symbols with the very same meaning. However, of these two symbols,

Frege only specifies the meaning of one of them, the identity of content

sign; no specification is given of the equality sign. But why should there

have been? Frege would have no more have perceived a need to define

the arithmetic equality sign as he wo uld have thought it necessary to

define the plus-sign. He would have thought the meaning to be obvious;

in accordance with mathematical practice, 26 the intuitive equation of an

arithmetical operation with either some number, or with some distinct

arithmetical operation. The equality relation as we find it in “2+4 = 6”

equates the sum-operation applied to 2 and 4 with the number 6; in

“2+4 = 2×3” with the product-operation applied to 2 and 3. These are

judgemen ts, as Frege puts it, “which treat of the equality of numbers

which have been generated in different ways.” (“Boole’s logical

Calculus,” p. 13)

The equality symb ol, as Frege would have construed it at this

time, is thus part of the vocabulary of arithmetic, and just as other areas

of mathematics may also sport their own specific equivalence relations,

e.g. congruence in geometry, equality represents an explicitly arithmetic

concept. In introducing the identity of content sign as the symbol of

identity in Begriffsschrift , Frege viewed it as a novel symbol within the

27 Bear in mind that in Begriffsschrift Frege did no t have a generalized notion of

denotation, such that “2+4” would denote its value, i.e. the number 6. This was only

fully realized considerably later, at the time of Grundgesetze .

21

conceptual notation, not bound to any specific domain, and n ot as a

replacement for a well-known sign of arithmetic. Indeed, as far as

arithmetic is concerned, equa lity and identity are complementary

notions; those judgement s containing the equality sign could not be

replaced by judgements with the identity of content sign standing in its

stead, nor vice versa, wi th identity of content replaced by equality.

Unlike identity of content, equality is not a met alinguistic relation; the

symbols that occur in judgements of equality stand for their contents.

Thus, in holding between the sum operation as applied to the numbers 2

and 4 on the one hand, and the number 6 on the other, equality holds

between distinct conceptual contents. But then we could not replace

arithmetical equality with identity of content. For “2+4 / 6” to be true,

“2+4” and “6” must have the same conceptual content; if “2+4 = 6” is

true, “2+4 / 6” will be false. Equality, in what Frege would have taken

as its common arithmetic sense, although an equivalence relation, is not

identity. 27

Notice that aside from not being metalinguistic, judgements of

equality differ from judgements of identity of content in another

important way. It is characteristic of the latter judgements that

significant information relevant to conceptual content is secreted away,

revealed as modes of determination. But with equalities, the

mathematically relevant information is dis played in plain sight; the

mathematically relevant information expressed by “2+4 = 6” is just that

the sum of 2 and 4 is 6. There is no need of modes of determination to

reveal this information; once we have explicated the conceptual content

of the judgement, we will have elucidated all the rele v ant information

that it expresses. The circumstances with judgements of equality are no

diff erent in this regard than with a judgement such as “John left.”

Where modes of dete rmination are called for is just where there is a

relation that does not reveal the relevant information, in particular where

we say of two atomic names that they have the same conceptual content.

But, as noted, this is not a circumstance that we find in arithmetic, given

that there is only one numeral per number, and hence for which there

would be no place for judgements of identity of content.

28 Frege also uses identity of content to form statements that express the stipulation of

specific values for variables in a proof; cf. his usage in the derivation in Begriffsschrift

of proposition (68), in which he g ives “( ?a f ( a)) / b ” to state the value of a

propositional variable.

29 Begriffsschrift , p. 168. The final parenthetical emendation of the quotation appears

in the quoted text, placed there by the translator.

22

If these remarks are along the right track, it indicates that Frege

saw the extent of judgements of identity of content as quite

circumscribed. The operant notion here is j udgement , for although in

Begriffsschrift Frege never clearly specifies how identity of content is to

be deployed, he does distinguish the judgement (assertion) of identity of

content, and the stipulat ion of such identity. The primary case that

Frege remarks upon of the latter use is definition. 28 Distinguishing these

cases, Frege remarks t h at a definition “does not say “The right side of

the equation has the same content as the left side”” - this is what a

judgement would say - “but, “They are to have the same content.” [A

definition] is therefore not a judgement . . .The only aim of such de -

finitions is to bring about an extrinsic simplification by the

establishment of an abbreviation” ( Begriffsschrift , p. 167-8), so as to

formally simplify proofs and make them more comprehensible. There

are two characteristics o f this stipulative use for definition to remark

upon. First, while definitions are not judgements, and are distinctly

indicated in the conceptual notation by the use of a special symbol, once

the definition is laid down it may be employed as a judgement, and play

the logical role of such judgements: 29

Although originally [a definition] is not a judgement,

still it is readily converted into one; for once the

meaning o f the new symbols is specified, it remains

fixed from then on; an d therefore [a definition] holds

also as a judgement , but as an analytic one, since we

can only get out what was put into the new symbols [in

the first place].

A definition qua judgement is analytic because it is trivial, being merely

abbreviatory; there are no distinct modes of presentation to call upon for

the terms of the judgement, for we would then be getting more out of the

30 At least this is how it is rendered in the German original; in the English translation

by Bynum it is rendered with identity of content: “? ( d / a )”. If Bynum has correctly

brought the text into line with Frege’s intended usage, then it indicates that Frege

would have held a stronger thesis, that the identity of content sign would have to occur

between simple atomic symbols , presumably as a stipulation of the syntax of the

conceptual notation. I have deferred, however, to the usage in the original; that it is the

intended usage is supported by the formula Frege gives in “Boole’s logical Calculus”

(p. 23) to express that 13 is prime, which is identical to the formula given in

“Applications,” save app ropriate substitutions. It is given with the equality, not the

identity of content, sign. In conversations with Prof. Bynum, he was unfortunately

unable to recollect why he had made the noted change from the German original, aside

from observing that it would have been reasonable to assume that so shortly after

finishing Begriffsschrift , in a lecture devoted to explicating the system of that book,

that Frege would have employed the full range of notions introduced there. This

assumes, of course, that Frege inadvertently or mistakenly used the equality sign in this

particular place, as opposed to his proper usage of this sign in other places, such as

stating sums. I would like to thank Ignacio Angelelli and Christian Thiel, as well as

Terry Bynum, for discussion of this point.

23

new symbols than we put in . Second, in the statement of a definition

there will always be a complex expression, the one that is being abbrevi-

ated. In contrast, Frege’s practice indicates his intention that we take

his use of the singular definite article seriously when he says in charac-

terizing non-stipulative judgements of identity of content, those to which

mo d es of presentation are pertinent, that “ das Zeichen A und das

Zeichen B ” are related by identity of content. Apparently only single,

atomic symbols can stand to the sides of the identity of content sign.

Notice that nothing in what we have said proscribes “2+4 / 6”;

it is just that it would only be construed as defining the numeral “6” by

stipulating that it h as the same conceptual content. as “2+4.” There is

one place in arithmeti c, however, where we could imaginably employ

identity of content; observe that not only is “? (2 = 3)” true, so too is

“? (2 / 3).” Frege actual usage is to employ the equality sign. In

“Applications,” when characterizing prime numbers, Freg e gives the

clause “? ( d = a )” as pa rt of expressing that for any number a , it is

indivisible by any positive whole number d in the sequence beginning

with 2, such that d is different from a . 30 (On Frege’s use of variables,

roman characters are b o und by an implicit maximally wide scope

universal quantifier, while gothic c haracters are bound by narrower

scope quantifiers that are explicitly indicated.) This bolsters our initial

conclusion, that Frege only employed judgements of identity of content

31 In “Applications,” Frege gives the following formula, where “ ?”stands for

congruence:

?U ( CD ?C U e ( BD ? B U e D / U)).

“This is the case,” Frege says, “when and only when D lies on the straight li n e

determined by B and C .” (p. 204) Unlike in arithmetic, however, Frege has no other

option but to use the identity of content sign in this formula, since points are not the

sort of things that are congruent. (Congruence, as Frege specifies, holds between two

pairs of points.) Recall that Frege established by example in B egriffsschrift that points

may be multiply designated, with each designation being associated with a differe n t

mode of determination.

32 In fact, Frege addressed the issue of how to derive arithmetic formulas from one

another via substitution in work prior to B egriffsschrift , specifically in Methods of

Calculation , his thesis of 1874; cf. pp. 61 - 64.

24

where modes of determinati on are involved. Since we are dealing with

non- identities, we do not have two names for the same number, but

rather names of different numbers, and hence there is no call for modes

of determination. “? (2 = 3)” by itself is sufficient to express that the

two numbers 2 and 3 are distinct. 31

At the time of Begriffss chrift , it appears clear that Frege does

not fold mathemat ical equality into identity of content, and that he had

two symbols, one arithmetic, the other, whil e not outright barred from

arithmetic, since it is general to the logi cal of all domains, of no utility

there. But Frege no doubt would have become aware of a substantial,

and glaring, logical problem in this view. The problem is that in

Begriffssc hrift , substitution is characterized only under identity of

content, by proposition (52):

|––— ( c / d ) e (f ( c ) e f ( d )).

What is not to be found in Begriffsschrift is a characterization of substi-

tution under equality; equations do not fall under the proposition above.

Although Freg e may have thought that since equality is an arithmetic

notion, the substitution of equals for equa ls would be specified within

arith metic, 32 and not by a basic law of logic, it is still hard to imagine

that it would not have irked him that substitution was characterized dis-

tinctly with respect to the two notions, but in ways that would be com-

pletely parallel logically. This redundancy would surely have indicated

that something was amiss. The natural response, of course, would be to

33 Grundgesetze , p. 6.

34 Fifty years later, Tarski, in his Introduction to Logic (New York: Oxford University

Press, 1951), still felt the need to remark on the s e matters, and devoted an entire

section (§19) to discussion of “mathematicians who - as opposed to the standpoint

adopted here - do not identify the symbol “=” occurring in arithmetic with the symbol

of logical identity.” (p. 61) For them, equality is a “specifically arithmetical concept.”

The problem Tarski sees with this view is that there is a breakdown in the generality

25

seek a unification, one general notion applicab le in all cases. This is

precisely what Frege does, beginning with Grundlagen , in 1884.

6. Given that we have two overlapping notions, an initial strategy in

seeking to unify them would be to see if one could be reduced to the

other. This is not, however, the strategy that Frege takes; rather he opts

for a replacement strategy, introducing a new notion that subsumes the

old ones. This notion is objectual identity; any statement that validates

substitution is now to be ana lyzed as an i dentity statement . This in-

cludes not only what were formerly jud gements of identity of content,

but also mathematical equalities. Thus, in G rundlagen , Frege speaks of

“the identity 1 + 1 = 2”, and says that “identities are, of all forms of

proposition, the most typical of arithmetic.” (p. 69) In the Introduction

to Grundgesetze , Frege explicitly acknowledges this move: 33

Instead of three paralle l lines I have adopted the

ordinary sign of equality, since I have persuaded myself

that it has in a rithmetic precisely the meaning that I

wish to symbolize . That is, I use the word “equal” to

mean the same as “coinciding with” or “identical with”;

and the sign of equality is actually used in arithmetic in

this way.

As Frege now sees matters, with a rather revisionist ring given his own

prior view, the notion that mathem a ticians (including himself) actually

had in mind when they employed the equality symbol was identity, and it

is this “ordinary sign” that subsumes identity of content. But although

like equality it stands for an objectual relation, it is not a symbol pecu-

liar to mathematics, as is equality; rather it is a general symbol, applica-

ble to propositions about all sorts of things, as is identity of content. 34

of substitution; for mathematics “it becomes necessary to give a special proof that this

replacement is permissible in each particular case it is applied.” (p. 61)

26

As we have portrayed matters, the primary motivation for Frege

to seek unification arises from general considerations of logic that arise

narrowly with respect to the formal system of the Begriffsschrift . With

Grundlagen , however, the picture considerably widens, with the

emergence of Frege’s logicism, th e mathematical agenda that was to

inform all the remainder of Frege’s work. In Grundlagen , Frege took

on the task of showing something substantive about logic, t hat abetted

with appropriate logical definitions, arithmetic could be reduced to it.

Moreover, this reduction would be sufficient to establish that arithmetic

truths are analytic truths, since they could be proven solely from logical

laws and the definitions. Carrying through this program, however,

required clarifications of th e logic; things had to be made clear which

prior considerations had not shined sufficient light upon. In particular,

given the manner in which Frege undertook to define (cardinal) number,

identity statements became essential. The reason for this Frege lays out

in one of the most famous passages of the Grundlagen , opening §62:

How, then, are numbers given to us, if we cannot have

an y ideas or intuitions of them? Since it is only in the

context of a proposition that words have any meaning,

our problem becomes this: To define the sense of a

propo sition in which a number word occurs. That,

obviously, leaves us still a very wide choice. But we

have already settled that number words are to be

understood as standing for self-subsistent objects. And

that is enough to give us a c lass of propositions which

must have a sense, namely those which express our

recognition of a number as the same again. If we are to

use the symbol a to signify an object, we must have a

criterion for deciding in all cases whether b is the same

as a , even if it is not always in our power to apply this

criterion.

Frege’s reasoning in this remark begins with what he takes to have been

shown to this point in Grundlagen , that numbers are logical objects, and

35 More precisely, he examines what he takes to be a comp arable condition for the

direction operator.

27

he makes a query with respect to it. His answer is to invoke the context

principle - access to such obj ects is only through the truth of proposi-

tions about them - and to single out a certain sort of proposition - iden-

tity statements - as central, for in order to know what sort of obje ct a

number is, we must know the identity co nditions that obtain for them;

that is, the conditions u nder which identity statements about numbers

are true. These Frege gives by Hume’s Principle :

The number of F s = the number of G s iff F and G

are equinumerous.

It is necessary in order for something to be a numbe r that it satisfy the

criterion of identity given by Hume’s Principle.

But is it also sufficient? In the ensuing sections, through §69,

that form much of the heart of the G rundlagen , Frege seeks to ans wer

this question by exploring whether Hume’s Pr inciple can stand as a

contextual definition of the numerical operator. 3 5 The answer he

ultimately gives is that it cannot ; the reason is what is known as the

“Julius Caesar” problem. The problem is as follows: If a criterion of

iden tity is to serve as a contextual definition, it must obtain in a ny

identity statement in which the numerical operator occurs. However, as

given by Hume’s Principle, the criterion o nly applies if a number is

given in just this way; consequently, it is undefined in “The number of

F s = Julius Caesar.” But lacking a way of knowing whether this is true

or false, we are left short of the generality required of definition.

Frege’s response is to limit the cases by giving an explic it definition of

the numerical operator that entails only the relevant identity statements;

i.e. that entails Hume’s Principle. So although Frege backs away from

construin g Hume’s Principle as a contextual definition, he still holds

that it states a condition that must be met as part of the characterization

of number.

The interest to us of Hume’s Principle is tha t qua identity

criterion it states conditions on the truth of i dentity statements .

Acco rding to the context principle, in order to gain access to logical

objects such as numbers, we must be able to form propositions that are

36 Of course this would be fine if numerals were numbers; that is, if one adopted the

formalist perspective on number. Frege, however, was consumed with scorn for this

view; cf. discussion below.

37 Grundlagen , p. 74

28

about such objects. If these are to be propositions about their identity,

then the iden tity relation must hold of the objects; it m ust be objectual

identity. What will not suffice are propositions about two expressions,

that they refer to the same object. Even though “ a / b ” is true if and

only if “ a = b ” is, the former could only be used t o give a c riterion of

identity for numerals, not for numbers; the point is to give criteria of

identity for objects, not a criteria for coreference of the ways of

designating objects. 36 Objectual identity was thus the only notion of

identity that would do for purposes Frege now has in mind, the objectual

characterization of number.

In the c entral sections of G rundlagen , before arriving at his

negative c onclusion, Frege defends the proposed contextual definition

specifically with respect to that part of it stated as an identity statement.

In doing so he explicates the identity notion he has in mind in two

crucial ways. First he specifies the content of such statements: 37

Our aim is to construct the c o ntent of a judgement

which can be taken as an identity such that each side of

it is a number. We are therefore proposing not to

define identity specially for this case, but to use the

concept of identity, taken as already known, as a means

for arriving at that which is to be regarded as identical.

Since numbers ar e objects for Frege, identity here is o bjectual identity

in the most general sense, the relation that any object whatsoever bears

to itself. Second, Frege elucidates that statements of objectual identity

play the logical role expected of ident ities; they do so because Frege

takes this logical role to be the defining characteristi c of identity. So

Frege writes in Grundlagen , §65 that he adopts Leibniz’s “definition of

identity” in terms of substitution salva veritate - “Thing s are the same

as each other, of which one can be substituted for the other without loss

of truth” - elaborating that “in universal substitutability all the laws of

identity are contai ned.” (p. 77). What we thus observe with these re-

38 “Truth-conditions” are not meant here in the Tars kian sense; Frege does not have

truth-conditions in this sense. Rather what is meant are the conditions under which a

sentence denotes the True.

39 What we give is actually what Frege labels Basic Law IIIa in §50, which he derives

from the generalized Basic Law III by instantiation of quantifiers. In this section, Frege

proves a number of consequences of Basic Law III, including the law of self identity,

i.e.:

|––— ( a = a ).

40 Grundgesetze , p. 71.

29

marks is th e emergence of Frege’s “mature” view that he would hold

consistently throughout his subsequent work: identity statements are

statements of objectual identity, such that in their presence in proof sub-

stitution is validated.

The way Frege puts matters in the Grundlagen is of course

quite informal, with vestiges, in the way he talks of substitution, of the

logical literalism that so strongly colored his presentation in Begriff-

sschrift . Frege shakes this off, however, when he comes to giving his

formal presentation in the firs t volume of G rundgesetze , nine years

later; there, for the first time, an identity theory is stated in a

recognizably modern fashion. Unlike in B egriffsschrift , in which

iden tity of content was introduced by definition, the identity symbol in

Grundgesetze is an unde fined term; what are specified are the truth-

conditions of statements in which this symbol occurs. 3 8 These Frege

gives in §7: ““ '' = )” shall denote the true”, he writes, “if '' is the same

as ), in all other cases it shall denote the false.” Also unlike

Begriffsschrift , Frege no longer speaks of the re p lacement of symbols,

when in §20 he specifies the logical role of identity statements, via Basic

Law III: 39

|––— ( a = b ) e (f ( a ) e f ( b )).

Frege describes Basic Law III in the following way: 40

If '' = ) is the true, the n [ ?f ( f ( )) e f ( ''))] is also the

True; i.e., if '' is the same as ), then '' falls under

every concept under which ) falls; or, as we may also

41 Frege’s letter to Peano is undated, but the editors of Frege’s P hilosophical and

Mathematical Correspondence place it in the period between 1894 and 1896.

42 Note that in describing an inference there is n o bar to r eading its premisses

metalinguistically; interpreting the identity sign in the object lan guage (“=”) as

objectual identity does not preclude saying me talinguistically that the symbols on either

side have the same content. Thus, just as it would be appropriate to describe the impli-

cational statement, which could stand a s a premise of a modus ponens inference, by

saying: “if a sentence ‘ a ’ implies a sentence ‘ b ’,” so too would it be appropriate to

descr ibe the identity statement, which could stand as a premise of a substitution

inference, by saying: “if ‘ a ’ and ‘ b ’ have the same cont ent.” In places, as in the

quotation in the text, Frege avails himself of this way of speaking; other examples are

found in Grundgesetze §105: “We ourselves use the equality sign to express that the

30

say: then every statement that holds for ) holds also for

''.

And finally, unlike in B egriffsschrift , Frege explicitly assumes in

Grundgesetze a generalized notion of denotation, allowing mathematical

equality, along with identity of content, to be subsumed under

(objectual) identity, for now a n umber can be denoted by its numeral

and by other complex expressions that can be stated in the conceptual

notation. Not only is the numeral “5” the (atomic) name of a number,

but so is the complex name “2+3”: “2+3 = 5” is a true i dentity state-

ment because both “2+3” and “5” denote the number five.

From a purely logical perspective, all is now in place as far as

identity is concerned, unified with respect to objectual identity. But yet

there still remains for Frege a substantial semantic issue about identity

statements that he must address if he is to defend his view of number

against its detractors.

7. As noted, in G rundgesetze , Frege specifies his unders tanding of

equalities as objectual identity statements; to repeat the remark from the

Introduction to G rundgesetze cited above, he says: “I use the word

“equal” to mean the same as “coinciding with” or “identical with”; and

the sign of equality is actually used in arithmetic in this way.” In a con-

temporaneous letter to Peano, Frege reasserts this view: “I take iden-

tity,” he says, “to be the meaning of the equals sign.” (p 126.) 4 1 How-

ever, in the continuation of the remark from Grundgesetze , Frege notes

an objection to his view that he curtly dismisses: 42

reference of the group of signs on the left-hand side coincides with the reference of the

group of signs on the right,” and in “Func tion and Concept” (p. 22): “What is

expressed in the equation ‘2.2 3 + 2 = 18’ is that the right-hand complex of signs has the

same reference as th e left-hand one.” But while these comments are in language

somewhat reminiscent of Begriffsschrift , they are not offered up as analysis, at pain of

confusing use and mention.

43 As far as can be discerned from Frege’s writing, it is from Thomae that Frege became

explicitly aware of the significance of this issue for his theory of number; there is some

issue as to when he became aware. §138 is in Volume II of G rundges e tze , dated by

Frege October 1902, and published in 1903. Although it is well-known that the

publication of Grundgesetze in mul tiple volumes was forced upon Frege by his

publisher, given the lukewarm receptions his previous publications had received, (see

the remarks by Bynum, ibid , p. 34 ff ), we can definitively place the drafting of §138 as

no earlier than 1898, the date of publication of t h e second edition of Thomae’s

Elementare Theorie der analytischen Functionen einer complexen Ver?nderlichen , from

which Frege drew the passage quoted, which appears on the second page of Thomae’s

31

The opposition that may arise against this will very

likely rest on an inadequa te distinction between sign

and thing signified. Of course in the equation “2 2 = 2 +

2” the sign on the left is different from that on the right;

but both designate or denote the same number

But what is this objection to taking equality as identity that rests on such

a fundamental confusion, and why is Frege so quick to dismiss it?

The objection is as follows: If equals is identity, then “ a = b ”

collapses into “ a = a, ” and all arithm etical equations would be trivial.

In the letter to Peano, Frege puts it this way:

What stands on the way of a general acceptance of this

view is frequently the following objection: it is thought

that the whole content of arithmetic would then reduce

to the principle of identity, a = a , and that there would

b e nothing more than boring instances of this boring

principle. If this were true, mathematics would indeed

have a very strange content.

Frege states the objection again in Grundgesetze §138, but this time he

gives an explicit reference, quoting a pas sage from Thomae, his col-

league at Jena: 43

book. Interestingly, this passage does not appear in the first edition of Thomae’s book,

published considerably earlier, in 1880. Rather, on the page cited, we find the

following remark: “Equating two numbers n = m means either something trivial,

namely n = n means that the number n is the number n , or n and m are dif ferent in

some respect, and only after abstraction do these numbers attain equality from this

difference.” (p. 2) Although the wording has changed, the content of this passage

appears to match, for Frege’s concerns, that which he quotes. (Michael Dummett is

thus not quite correct when in his Frege : Philosop h y of Mathematics (Cambridge:

Harvard Universi ty Press, 1991) he says that the first edition of Thomae’s book

contains all the pas sages Frege cites from the second edition; cf. footnote 1 on page

241.) Given that Frege cites the first edition of Thomae’s book in §28 of G rundlagen ,

published in 1884, we can assume that he was fa miliar with the issue early on. It is

thus plausible that §138 was drafted earlier, contemporaneous with the wri ting of

Volume I., but updated with the newer passage as Freg e prepared Volume II for

publication. (The only other reference in the section is to a publication of Dedekind’s

from 1892.) Thomae, incidently, cites the Grundlagen on the first page of the 1898

edition of his book, (along with works of Dedekind), only to remark that discussion of

them would take him too far afield. Given that both the first volume of G rundgesetze

and “On Sense and Refe rence” among other relevant works appeared in the period

intervening between the two editions of Thomae’s book, and that as colleagues at Jena

they apparently had fairly extensive personal discussions, Frege must no doubt have

been greatly annoyed at his colleague’s simply repeating his argument, without taking

account of Frege’s response. By 1906, Frege can no longer contain his ire at being

ignored, publishing an acidic “Reply to Mr. Thomae’s Holiday Causerie ,” in which,

among other things, he directly takes on Thomae’s views on equality and abstraction

(pp. 344-5), and remarks that he is “convinc ed that with my critique of Thomae’s

formal arithmetic, I have destroyed it once and for all.” This is followed up with an

equally acerbic essay in 1908 “Renewed Proof of the Impossibility of Mr. Thomae’s

Formal Arithmetic.” In these essays, Frege’s hostility towards the views of his

academically more successful colleague is vividly palpable, a reflection of how totally

their relationshi p had soured. While Thomae in 1896 had strongly supported an

unsuccessful attempt to advance Frege’s career, in 1906 he writes the following about

Frege to the university administration: “After all, we only have Colleague Frege left.

To my regret I cannot keep secret that his effectiveness has diminished in recent times.

The reasons for that cannot be established with certainty. May be one should look for

them in Frege''s hypercritical tendencies,” lea d ing the administration to report that “The

Honorarprofessor Hofrat Dr. Frege has probably never been a good docent.” At this

point, any hope Frege may have had for academic advancement were dashed. We, of

course, are quite c e rtain of the problem that affected Frege at this time - Russell’s

paradox. Thomae apparently did not recognize the cause; it is likely that he had not

read Grundgesetze , and hence did not comprehend the impact of the paradox, much,

one would conjecture, to Frege’s annoyance. For a very interesting discussion of the

relations of Frege and Thomae, from which the quotations have been drawn, see Uwe

Dathe, “Gottlob Frege und Johannes Thomae: Zum Verh?ltnis zw eier Jenaer

Mathematiker,” in Gottfried Gabriel and Wolfgang Kienzler, eds . , Frege in Jena :

32

Beitr?ge zur Spurensicherung (Würzburg: K?nigshausen & Neuman: 1997). (I would

like to thank my colleague Christian Werner for helpful discussion and translations.)

44 Having co mplimented Dedekind, Frege finishes the section by lambasting him for

praising the work of the mathematical formalist Heine.

45 The formalists were not the only enemies Frege saw to his view of number; casual

inspection of the initial sections of the G rundlagen reveal empiricism and

psychologism as also among their number. My thesis is just that as far as identity

statements are concerned, it is the formalists who take center stage. If Frege’s goal was

to establish that numbers are logical objects, the impediment placed by the formalists

was to their objectivity. The only place where Frege directs his remarks on equality

to anyone other than the formalists is in his review of Husserl’s P hilosophie der

Arithmetik I of 1894, where he remarks th at “psychological logicians . . . lack any

und erstanding of identity. This relation cannot but remain perfectly mysterious to

them; for if words designated ideas throughout, one could never say ‘ a is the same as

b ’; for to be able to say this, on e would first have to distinguish a f rom b , and they

would then just be different ideas.” (p. 200.) Frege continues this passage by stating

his agreement with Husserl that Leibniz’s principle of substitution “does not deserve

to be called a definition, even if my reasons are different than his. Since any definition

33

. . . Thomae remarks: ‘Now if equality or the equality

sign = were only to stand for identity, then we would be

left with trivial knowledge, or if one prefers, the

conceptual necessity a is a ( a = a )’.

He gives this passage in contrast to Dedekind’s view, which embodies

three points that Frege “exactly agrees” with:

(1) the sharp distinction between sign and its reference

(2) the definition of the equ ality sign as the identity

sign,

(3) the conceptions of numbers as the reference of

number signs, not as the signs themselves

Acceptance of these tenets place Dedekind’s - and Frege’s - “view in the

starkest contrast to every formalist theory, which regards sign s or fig-

ures as the real objects of arithm etic.” 44 The enemy has now been lo-

cated and the issue joined, and it could not be more fundamental - the

nature of number. 45

is an identification, identity itself cannot be defined,” and concludes with the remark

that “The author’s explanation , ‘We simply say to any two contents that they are

identical if there is an identity between . . . the characteristic marks which happen to

be our centre of our interest’ . . . is not to my taste.” It is the scent of formalism in this

remark that Frege surely found distasteful.

46 Frege quotes this passage, from the second edition of Thomae’s Elementare Theorie

der anakytischen Functionen einer complexen Ver?nderlichen, in Grundgesetze , §88,

in critique of Thomae’s likening numbers to chess pieces, defined by the role they play

in a mathematical game; cf. Thomae’s remarks in the section of his Elementare Theorie

that Frege cites. For discussion of Frege’s critique of formalist mathematics found the

second volume of Grundgesetze , §§88 - 137, see Michael Dummett, F rege: Philosophy

of Mathematics (Cambridge: Harvard University Pres, 1991) , ch. 20, and Michael

Resnik, Frege and The Philosophy of Mathematics (Ithaca: Cornell University Press,

34

For mathematical formalists ( such as Thomae), the conclusion

to be reached from the argument cited is that equals is not identity. But

what then is an equality? The answer they give is that it is a relation be-

tween expressions; given their conflation of distinct numerical

expressions with distinct numbers, equality is to be understood as an

equivalence relation between distinct things. What “2 + 3 = 5” says is

that two numbers are numerically equivale nt, their equivalence

established by abstracting away from where they differ. Given that it

also involves an equivalence between unlike terms, one might be

tempted here to analogize to Frege’s n otion of identity of content in

Begriffsschrift . Doing so however would be to neglect not only that

Frege did not take arithmetic al equalities as judgements of identity of

content, but also a view that persevered throughout Frege’s writi ngs,

that arithmetical statements, no differently than any other statements,

are meaningful statements. In arithmetic, this meaningfulness arises

from the reference relation of numerals to numbers, this being an

instance of the more general relation holding between signs and objects.

Arithmetic, of course, also has its formal side, in the proofs of theorems

(about the numbers) from the axioms. But for proof, we need only

attend to signs in the formal mode, qua symbols, to warrant taking each

step from meaningful statement to meaningful statement; what the

symbol stands for is not germane for this. Frege is adamant, however,

that although proofs are to be un derstood as symbol manipulations,

mathematical concepts are not given by such manipulations. But this is

what obtains on the formalists’ view of number, as Thomae remarks: 46

1980), ch. 2.

47 Letter to Peano, p. 127.

35

It does not ask what numbers are and what they do, but

rather what is demanded of them in arithmetic. For the

formalist arithmetic is a game with signs that are called

empty. This means that they have no other content (in

the calcu lating game) than that they are assigned by

their behavior with respec t to certain rules of

combination (rules of the game).

Thus, that Frege found the fo rmalists’ de-trivialization strategy com-

pletely unacceptable is none too surprising, as it is ultimately b a sed, in

his view, on an incoherent notion of number. Numbers cannot be j ust

formal marks, to be manipulated by r ule; this would result, as Frege

remarks to Peano, in a “chaos of numbers”: 47

There would not be a single number which was the first

prime number after 5, but infinitely many: 7, 8 - 1, (8 +

6): 2, etc. We would not speak of ‘the sum of 7 and 5’

with the definite article, but of ‘a sum’ or ‘all sums’,

‘some sums’, etc.; and hence we would n ot say ‘the

sum of 7 and 5 is divisible by 3’.

Abstract ion is no patch, for the notion of numerical equivalence onto

which it is to converge must presuppose a notion of number such that

distinct numbers, given by distinct numerical symbols, can be the same

number. This is to be the accomplishment of abstraction, by which,

Frege says, “things are supposed to become identical by being equated.”

But how can abstraction, a “capability of the human mind” according to

Thomae, turn two things into one? - “if the human mind can equate any

objects whatever, [abstraction] is especially meaningless, and the mean-

ing of equating will also remain obscure.” Frege asks: “What do [the

formalists] want to achieve by abstracting? They want - well, what they

4 8 This and the two previous quotations are from “Reply to Mr. Thomae’s Holiday

Causerie ,” pp. 344-5, in which Frege, apparently incensed by Thomae, remarks that

“After reading what [Thomae] said about abstraction, I vented my feelings in this verse:

Abstraction’s might a boon is found

While man does keep it tamed and bound;

Awful its heav’nly powers become

When that its stops and stays are gone

49 This passage is from Frege’s unpublished “Logic in Mathematics” (p. 224), in which

the argument first given in the letter to Peano appears again, although, dated (1914),

it was written many years later.

36

really want is identity.” 48 To obtain this, however, we do not want

formalist obscurity rooted in their confusion of numeral and number;

rather, we need, according to Frege, to carefully distinguish symbols

from what they stand for; we need to attend to their s emantic relation.

If there can be distinct symbols that are signs for the same thing, it can

be said that the “signs ‘2 + 3’, ‘3 + 2’, ‘1 + 4’, ‘5’ do designate the

same number.” 49 Equality can be construed as identity; the chaos

evanesces.

Of course, rejectin g the formalist conception of number and

accepting Frege’s points (1), (2) and (3), does not make the p r oblem

raised in Thomae’s remark go away in and of itself. What does make it

go away is the recognition, all too clear to Frege, that the premise of the

argument is simply mistaken - identity statements of the form ja = b k do

have “grea ter cognitive content than an instance of the principle of

identity.” But whatever this greater cognitive content is to consist in,

such that it separates “ a = b ” from “ a = a ,” it cannot be such that it

would “prevent us from taking the equals sign . . . as a sign of identity.”

(Letter to Peano, p. 126.) But what is it that satisfies these dual criteria?

It is precisely at this point of the discussion that the notion of s ense

makes its appearance.

8. Let’s consider the issue facing Frege. If we harken back to

Begriffsschrift , where there are two notions, equality and identity of con-

te nt, Frege only perceived the need to associate modes of presentation

with judgements of the la tter sort. Frege did this in order to introduce

inform ation that was not otherwise specified by the judgement itself as

part of conceptual content; modes of determination are called for with

50 Geach, ibid , p.144.

37

atomic names because all that is indicated as part of conceptual content

by their representation in the conc eptual notation is their denotation.

Things are different in the case of mathematical equality, however; they

make no call for modes of determination. This is because all the mathe-

matically significant information is al ready manifest intranotationally;

what else could be relevant to the thought expressed by “2+3 = 5” than

just that 5 is the sum 2 and 3? This information, however, is exactly

what is obscured if “2+3 = 5” is to be read as an identity statement, for

all that would be relevant to its truth would be coreference to the number

5. All we would have would be d enotational information; what is

masked is th e mathematical information that the number denoted is the

sum of 2 and 3. The tension we see here is nicely put in the following

remark by Geach: 50

. . . what is con veyed by mathematical equations is the

strict identity of what is mentioned on either side of the

equation; thus 6:3 = 1+1 because 6:3 is the number (not

a number) which when multiplied by 3 yields the result

6, and 1+1 is that very num ber. . . What makes the

equation informative is that though the same number is

mentioned on both sides, it i s presented as the value of

two different fun ctions - the quotient function and the

sum function.

The problem facing Frege is how to capture both of these sorts of infor-

mation. On the B egriffsschrift notion of content, Frege is denied the

wherewithal to analyze arithmetic equalities as identities; a general no-

tion of denotation can provide such wherewithal, but then we lose our

grip on the mathematical information expressed. But this grip must be

maintained, if the criticism of the formalists is to be met.

How then is Frege to capture both sorts of mathematical

information ? He achieves this by introducing the more fine-grained

notion of content embodied in his theory of thoughts, as this is chiefly

developed in the early in 1890’s , primarily in the seminal essays,

“Function and Concept” and “On Sense and Referen ce,” and in

Grundgesetze . This theory is elegant in its simplicity: thoughts are

51 The former quotation is from an unpublished manuscript entitled “Logic” of 1897,

p. 139; the latter from “Notes to Ludwig Darmstaedter,” dated July 1919, p. 253.

38

complexes, the sum of constitue n t senses of which they are composed.

Senses are to be characterized functionally; they ha ve the property of

determining (or presenting) reference ( Bedeutung ). Grouped together to

form a thought, they are capable of determining a truth value as

referen ce; but in order to do so, they each themselves must be able to

determine a reference. (Their references may be either objects or n -level

concepts.) Thoughts are e xpressed by sentences (form s of the

conceptu al notation) in virtue of sentences being composed of s igns ,

symbols that express senses. Content, Frege now says in way of

contrasting his new view to that of Begriffsschrift , “has now split for me

into what I call ‘thought’ and ‘truth-value’, a s a consequence of

distinguishing between sense and denotation of a sign.” ( Grundgesetze ,

pp. 6 - 7)

On this scheme the components of content are r elated in two

distinct way. One is by the semantic relation of determination or

presentation of reference by sense; the other is by judgement , which is no

longer t o be understood as a property of an undifferentiated monolithic

content. Frege describes judgements in “On Sense and Reference” as

“advances from thoughts to truth value,” (p. 65), in G rundgesetze §5 he

says “by a judgment I un derstand the acknowledgment of the truth of a

thought ” a nd in a letter to Husserl he remarks that “Judgement in the

narrower sense could be characterized as a transition from a thought to a

truth value.” (p. 63). This transition, Frege makes clear is a cognitive

relation: “ W hen we inwardly recogniz e that a thought is true, we are

making a judgement ,” Fr ege writes, with the emphasis; he amplifies by

remarking that “Both grasping a thought and making a j udgement are

acts of a knowing subject, and are to be assigned to psychology. But

both acts involve something that does not belong to psychology, namely

the thought.” 51 It is with respect to the status of a thought with respect

to its role in judgement that Frege introduces the n otion of c ognitive

value - those aspects of judgement that ar e determinable from the

thought itse lf are the cognitive value of the thought. While Frege does

not ela borate very much on the notion of cognitive value, he does make

clear one pivotal aspect of the notion. This is that the cognitive value of

a thought reflects wh ether to judge it we must attend to the semantic

52 Gareth Evans, in discussing Frege in The Varieties of Reference (Oxford: Clarendon

Press, 1982), gives the following condition on cognitive value: “a sentence S has a

different cognitive value from a sentence S ’ just in case it is possible to understand S

and S ’ while taking different attitudes towards them.” (p. 19) This c o ndition is

undoubtedly false; Evans has confused here expressing different thoughts with having

different cognitive value. “Cicero is a Roman” and “Tully is a Roman” on Fregean

lights express different thoughts, and one can surely have different attitudes towards

them; they do not thereby differ in cognitive value. If two sentences express thoughts

that have different cognitive values, then they express different thoughts; the inverse,

however, does not hold. Two sentences may express different thoughts, but have the

same cognitive value.

39

connection of Sinn and Bedeutung . Any thought in which we must take

into account the particulars of the determination of reference by the

constituent senses in order to advance to truth will have greater cognitive

value than thoughts for which this is not required, and any thoughts that

so differ in cognitiv e value will consequently enter into distinct

judgements. 52

With these notions in hand, Frege is now in a position to make

his response to the formalist critique, giving an answer to what has come

to be known as Frege’s puzzle. In G rundgesetze §138, immediately

following the passage cited in the previous secti on, in which he quotes

Thomae’s remark, Frege says:

The knowledge that the Evening Star is the same as the

Morning Star is of far greater value than a mere

a pplication of the proposition ‘ a = a ’ - it is no mere

result of a conceptual necessity. The explanation lies in

the fact that the sense of signs or words (Evening Star,

Morning Star) with the same reference can be different,

and that it is pr e cisely the sense of the proposition -

beside its reference, its truth-value - that determines its

cognitive value.

The formalists (T homae in particular) have, according to Frege, just

made a “mistake.” It is the mistake that arises from not attending to the

thought expressed, from an “ inadequate distinction between sign and

thing signified,” the mistake noted in the Introduction to G rundgesetze .

Interpreting equality as identity does not make “ a = a ” and “ a = b ” have

the same cognitive value. It is trivial to judge the truth of the former

53 For Frege, while “ a = a ” is trivial, it remains sensible; it expresses a thought. This

was not held by everyone. Moore, for instance, remarks that “A sentence of the form

“ — is identical with —” never expresses a prop[o sition] unless the word or phrase

preceding “is identical with” is different from that which follows it.” (G. E. Moore.

“Identity” in Commonpl ace Book: 1919 - 1953 , edited by Casimir Lewy, (London:

George Allen and Unwin, 1962.)

40

given the thought it expresses, for mere inspection of its form shows it to

be composed in accordance with a logical law; it is an instance of the

law. Its truth is known a priori , regardless of the particulars of how the

constituent senses determine reference. 53 It is just such particulars,

however, that we need to regard in order to judge the trut h of “ a = b ”;

because “ a ” and “ b ” can have different senses, in order to make a judge-

ment we must ascertain whether they determine the same reference. And

because of this, “ a = b ” has greater cognitive value t han “ a = a ” and

must express a differ e nt thought; since the latter is a “mere result of a

conceptual necessity,” the latter, a fortiori , cannot be. With this insight,

Frege has answered the challenge of the formalists.

The argument Frege gives here would appear to be completely

general; whether a tho ught is trivial has to do with its composition, and

not with what sorts of things “ a ” and “ b ” stand for. But yet again, as in

Begriffsschrift , Frege turns in the mids t of a discussion of the “basic

laws of arithmetic” to a non-arithmetical illustration of this point. Such

examples are employed in support of his position not only in

Grundgesetze , but also in his other remarks on identity statements of this

period. In the undated letter to Peano, in his remarks “On Mr. Peano’s

Conceptual Notation and My Own” (1897), in an unsent letter to

Jourdain (1914), in “Function and Concept” (1891), and in “Logic and

Mathematics” (1914), perceptual examples are employed, whether along

the lines of the famous case of the “The Evening Star is the Morning

Star,” or of Astronomer X’s and astronomer Y’s comets that turn out to

be the same (in “On Mr. Peano’s Conce ptual Notation and My Own”),

or the case of explorers who see a mountain from different directions, the

“Afla” and “Ateb” of the le tter to Jourdain. But such examples are not

used to the exclusion of specifically arithmetical examples; these are

found in the letter to Peano, “On Mr . Peano’s Conceptual Notation and

My Own,” “Function and Concept,” in a letter to Russell (1904), and

54 “Logic and Mathematics,” p. 225

55 Frege, however, does not pursue the analysis of identity statements in “On Concept

and Object,” (the s ense/reference distinction is not introduced there). Rather, his

remarks are in the context of an admonition to distinguish this case from “is” when

used in predications that denote concepts, as in “Phosphorus is a planet.”

41

perhaps most clearly in “Logic and Mathematics”: 54

. . . one cannot fa il to recognize that the thought

expressed by ‘5 = 2 + 3’ is different than that expressed

by the sentence ‘5 = 5’, although the difference only

consists in the fact t hat in the second sentence ‘5’,

which designates the same number as ‘2 + 3’, takes the

place of ‘2 + 3''. So the two signs are not equivalent

from the point of view of the thought expressed,

although they designate the very same number. Hence I

say that the signs ‘5’ and ‘2 + 3’ do indeed design ate

the same thing, but do not express the same s ense . In

the same way ‘Cop ernicus’ and ‘the author of

heliocentric view of the planetary system’ designate the

same man, but have different senses; for the sentence

‘Coper nicus is Copernicus’ and ‘Copernicus is the

author of the heliocentric view of the planetary system’

do not express the same thought.

In this passage, Frege’s last known remark on identity statements, he

clearly indicates the intended generality of the method; it is to apply in a

uniform fashion to all cases, regardless of whether an identity statement

is arithmetic or not. Not only is “2 + 3 = 5" to be taken as a statement of

identity, so too is “Hesperus is Phosphorus” or “Copernicus is the author

of the heliocentric view of the planetary system”; in the latter examples,

“‘is’ is used like the ‘equals’ sign in arithmetic, to express an equation.”

(“On Concept and Object,” p. 44.) 55 By looking at the thought ex-

pressed, at the way that it is internally composed, the cognitive value of

an identity statement can be assessed independently of the subject matter

of the statement. Thus, if we were to take arithmetic statements as ana-

lytic, a s Frege does in G rundlagen , the difference in the relation of

thought to judgement bifurcates among them in the same way as it does

56 In “On Sense and Reference,” Frege remarks that “ a = a ” is “according to Kant, is

to be labelled analytic,”, but he refrains from labeling “ a = b ” synthetic, remarking only

that it “cannot always he established a priori .” (p. 56) Interestingly, it is Kant’s sense

of analytic he refers to here, by which Frege means logically trivial, and not his own

notion, which he had gone to great lengths to establish in Grundl agen . In fact, the

analytic/synthetic distinction plays little role, if any, in Frege’s think ing during the

period we are discussing; hence the tempered nature of the remark in the last sentence

of the text. Thus, in Grundgesetze Frege speaks of the goal of his project as showing

that arithmetical truths are logical, not analytic, truths. While there is much to be said

about this, briefly the reason for this is that given the primacy of thought, (as opposed

to its derived status in Begriffsschrift and Grundlagen ), at this point Frege can say little

more than t hat arithmetic truths are analytic because they are about logical objects;

while synthetic truths are about non-logical objects. This is all that could be discerned

from the information contained in the thought itself . But this would not be a very

robust notion of analytic and synthetic, and certainly would not satisfy anyone looking

for an explication of the Kantian notions. But of course the theory of thought otherwise

pays large dividends for Frege, sufficiently so that he could abandon an objective that

had previously so animated his thinking. (I am indebted to Aldo Antonelli for

discussion of this point.)

42

for non-arithmetical statements, even though for the latter the distinction

in cognitive value would cleave the same distinction as the Kantian dis-

tinction of analytic and synthetic. 56

Let us take stock at this point. The issue of the mathematical

information expressed by “2+3 = 5” now comes down to distinguishing

its sense and reference, and the relation thereof. In particular, the

thought this sentence expresses contains as a constituent sense the sense

of “2+3” - that the sum-operation is applied to 2 and 3 - that determines

5 as the reference of “2+3,” the same number that is the reference of “5.”

It is thus the thought expressed by “2+3 = 5” that carries the

mathematically significant information, not the reference. (Assuming the

latter is the mistake at the heart of the fo rmalists’ critique.) The w ay it

carries it, via two distinc t senses, determines its cognitive value as

greater than that of the thought expressed by “5 = 5.” To contrast with

Begriffsschrift , what Frege has done has taken the content of judgements

of equality and transformed it into the thought expressed by a

mathematical identity, including the structure discernable in the form of

the judgement, “distinguish[ing] parts in the thought corresponding to the

parts of a sentence, so that the structure of t he sentence serves as an

57 “Compound Thoughts,” p. 55. Here Frege clearly reveals a s yntactic view of

thoughts; he contin ues the remark above by saying “To be sure, we really talk

figuratively when we transfer the relation of whole and part to thoughts.” This raises

issues about the structure of thoughts; for example, are there parts of thoughts that

don’t express senses, that serve as “connecting tissue,” holding thoughts together as

wholes, (as we might think of syncategoremic expressions doing in syntax)? If there

are, can they have an effect on the thought expressed. This relates to the issue of

whether we can have equivalent ways of composing a given sense; cf. discussion of the

status of Basic Law V in §9. (Again, thanks to Aldo Antonelli for discussion of this

point.)

58 Note that in order to make the argument for greater complexity of thought we need

a language in which there are multiple atomic names with the same denotation; thus

again arithmetic falls short.

59 There are differences between the notions however that we must observe. In

Begriffsschrift , what was determined was conceptual content; modes of determination

are of an object and are associated with terms. Senses, on the other hand, are

themselves objects, albeit abstract and logical, that are expressed by terms; now the

determination itself, as a constituent part of sense, is part of the content. The

correspondence of Bestimmungsweise and Sinn was first noted by Ignacio Angelelli, in

Studies on Gottlob Frege and Traditional Philosophy , (Dordrecht: D. Reidel, 1967),

pp. 38 - 40, and subsequently has been remarked upon by Terrell Ward Bynum

“Editor’s Introduction,” in Gottlob Frege, Conceptual Notation and Related Articles ,

translated and edited by Terrell Ward Bynum, (Oxford: Clarendon Press, 1972), pp. 65

- 68; Richard Mendelsohn, “Frege’s B egriffsschrift Theory of Identity,” Journal of the

History of Philosophy , 22, 3, (1982), 279 - 99: Peter Simons, “The Next Best Thing

to Sense in Begriffsschrift ”, in J. Biro and P. Kotatko, eds., F rege: Sense and

Reference One Hundred Years Later , (Dordrecht: Kluwer, 199 5 ); and Michael Thau

and Ben Caplan, “What’s Puzzling Gottlob Frege,” ms, UCLA, 1999.

43

image of the structure of the thought.” 57 If this imaging is isomorphic, as

in “2+3 = 5,” the structure of the thought expressed will be no more

complex than that of the sentence itself. Isomorphism fails in one central

case, however; with atomic names the stru cture of the thought will be

more complex than that of the sentence. 58 The source of this complexity

is where Frege finds common ground with his earlier views; it is with the

modes of determination ( Bestimmungsweise ) of Begriffsschrift , now

wrapped up in the more general notion of sense. 59 Thus by parity, along

with the generalization of denotation from atomic to complex terms goes

the gen eralization of modes of determination; what had applied to a

narrower class of terms now applies inclusively to the larger class. But

this common germ should not occlude the ch anged role of this notion in

60 It is of interest to compare at this point remarks of Ramsey’s in “The Foundations of

Math ematics,” (in Frank P. Ramsey, T he Foundations of Mathematics and other

Logical Essays , (New York: The Humanities Press, 1950), pp. 16-17.) Observing that

“in ‘ a = b ’ either ‘ a ’, ‘ b ’ are names of the same thing, in which case the proposition

says nothing, or of different things, in which case it is absurd. In neither case is it the

assertion of a fact. . . . When ‘ a ’, ‘ b ’ are both names, the only significance which can

be placed on ‘ a = b ’ is that it indicates that we use ‘ a ’, ‘ b ’ as names of the same thing

or, more generally, as equivalent symbols.” Thus, Ramsey’s response is to entertain

a view reminiscent of that of Begriffsschrift , but unlike that view, it is strictly a relation

between symbols, and does not encompass the subtlety of Frege’s modes of

determination. Ramsey briefly explores whether such a construal of identity statements

could suffice for mathematics; concluding in the negative, he rejects the account, (as

well as Russell and Whitehead’s account in Principia Mathematica ), and endorses the

view of Wittgenstein in the Tractatus that dispenses with an identity symbol in favor

of the convention that each distinct symbol has a distinct meaning (reference). Cf. pp.

16-20, 29-32

44

the direction of explanation. In B egriffsschrift , Frege begins with a

notion of judgements as forms and appeals to modes of determination in

order to explain their connection to though t. In his mature work,

explanation goes the o t her way round. Frege begins with thoughts, and

explains by appeal to their constituent senses their connection to

judgement. What Frege believes he obtained from looking at matters this

way was a defense of his view of number. Numbers are logical objects,

and true identity statements about them are logical truths; but this does

not leave them bereft of mathematical conte nt, given the theory of

thought. And it was this that was the pay-off of his new perspective on

identity statements. 60

9. Central of the thesis we have been exploring is that in order to fully

comprehend why Fr e ge introduces the Sinn/Bedeutung distinction we

must comprehend the role it plays in d efense of his theory of number.

But it is important to bear in mind that the distinction itself is not signifi-

cant for the constructi on of that theory; the sense/reference distinction

plays no role in that. The construction does, however, exploit the theory

of identity as part of specifying what sort of logical objects numbers are;

for Frege in Grundgesetz e numbers are extensions of certain concepts,

and the theory of identity is needed to characterize the criteria of identity

for such extensions. As was discussed in §6, Frege’s approach is antici-

pated in Grundlagen , but it is in the Grundgesetze (an d in an introduc-

tory manner in “Function and Concept”) that Frege thought he had suffi-

61 In Grundgesetze, §10, Frege shows that truth-values can be reduced to certain

courses-of-values; so taken, Frege only introduces one novelty to his ontology, not two.

See footnote 64.

62 In “Boole’s logical Calculus” (1880/81) Frege makes clear that the material

biconditional is to be expressed by the conjunction of conditionals; unlike his identity

sign, “Boole’s identity sign,” Frege writes, “does the work of my conditional strokes:

B e A and A e B .” (p. 37) In Begriffsschrift §3, Frege says what it would be for two

judgem ents to have the same conceptual content - “the consequences which can be

derived from the first judgement combined with certain others can always be derived

from the second judgement combined with the same others”. Frege gives this condition

in the service of elucida ting what are irrelevant differences in representation with

45

ciently clarified his views so that he could properly address the issue. It

was central to Frege’s view of identity in this period that in conformance

with his insistence on a clear distinct ion between function and object, it

is only objects that fall under the identity relation, not functions. Identity

i s to be construed strictly as objectual identity. However, what is to be

taken as an object is considerably elaborated in Grundgesetze , to include

two sorts not introduced previously, truth-values and courses-of-val-

ues. 61 Taken in coordination with the articulation of levels of functions,

this reification provided, Frege thought, the tools for putting into place

the foundations of the not ion of number, through the statement of the

infamous Basic Law V. The sad story of Basic Law V is well-known,

and I will not rehearse here how it paved the road to paradox. It is, how-

ever, worth pausing to consider its formulation, for it will allow us to get

a complete picture of Frege’s mature theory of identity.

In Grundgesetze , Frege propounds the these s that (i) truth-

values are a type of logical object, and (ii) sentences are complex names,

having truth-values as their references. Given the generalized notion of

denotation in Grundgesetze , the syntax allows for id entity statements

such as (I):

Russell was English = Frege was German (I)

For an identity statement like (I) to be true, the complex terms that stand

to the sides of the identity sign must refer to the same thing, either both

to the True or both to the False, (although of course their senses differ).

Thus, under the assumptions (i) and (ii), Frege was able to utilize the

identity sign as a convenient way of expressing material equivalence. 62

respect to the conceptual notation. But what Frege does not provide in B egriffsschrift

is the syntactic wherewithal to state such identity; judg ements of identity of content

only hold with respect to terms that denote their conceptual contents. Thus, we don’t

have any counterpart to (I); i. e. we don’t have “Russell was English / Frege was

German.” But even if we did, it could not the work of (I), since it is false. “Russell

was English” plainly does not have the same conceptual content as “Frege was

German,” for one contains Russell, the other Frege.

63 “Comments on Sense and Meaning,” p. 120 - 1.

46

From the standpoint just elaborated , notice that what we would

say about (II) is just the same as (I); it too is true just in case what

stands on either side of the identity sign refer to the same truth value:

Max is a renate = Max is a cordate (II)

There is something more, however, that we would like to say about (II)

than what we also say about (I); (II), unlike (I), contains equivalent pred-

icates, for every renate is a cordate, and every cordate a renate, some-

thing we can state by (III):

Renates = cordates. (III)

(III), it wo uld appear, equates concepts; but if so, then according to

Frege, it is not well-formed, for an identity statement can only contain

names of complete (saturated) objects besides the identity symbol: 63

the relation of equality, by which I understand complete

coincidence, identity, can only be thought of as holding

for objects, not concepts. . . . we may not write M = O,

because here the letters M and O do n o t occur as

function -letters. But nor may we write M ( ) = O ( ),

because the argument-places have to be filled.

“An isolated function-letter without a place for an argumen t,” Frege

says, “is a monstrosity.” ( Grundgesetze , §147.) Thus, insofar as there is

a relevant parse of (III), it is as “( ) is a renate = ( ) is a cordate;” but

this could only be thought to be well-formed if one were to confuse con-

c epts and objects, what is unsaturated with what is saturated. We are

64 Frege recognized that the sufficiency of this approach may not be obvious, especially

in light of the Julius Caesar problem that had led him to reject the contextual definition

of number in Grundlagen , and he devotes §10 of Grundgesetze to this matter. Frege

thought that with his method of introducing extensions in Grundgesetze , which is not

by contextual definition, he c o uld overcome the problem. The problem as now con-

ceived is that to ask whether an arbitrary object p is a course of values is reduced to

whether “ |f ( g) = p ” is true. This would only result in the appropriate class of identity

statements because Frege thought that he could characterize all objects as courses-of-

values. This cannot work, however, for if it did, there would be a consistency proof for

the system of Grundgesetze . (Thanks to Aldo Antonelli for bringing this to my

attention.) For discussion of this section of Grundgesetze , one of the most complicated

47

not, however, at a complete loss here, for the equivalence of concepts

can be specified under a weaker condition, that the same objects fall un-

der each concept. Concepts that fall under the complex second-level re-

lation:

?a M( a) = Q( a),

will meet this condition. The resulting generalization:

?a f ( a) = g ( a),

is not itself an identity statement - the identi ty sign as it occurs here is

part of a complex second-level relation, under which first-level concepts

fall - although its truth depends upon identity statements like (II), whose

terms denote truth-values, being true.

The importance of being able to specify su ch a relation for

concepts is that it paved the way for Frege to introduce an important

novelty of Grundgesetze , the introduction of terms referring to courses-

of-values , of which e xtensions are the special case for concepts. Frege

was insistent on the view that extensions are logical objects; it is a claim

that stands at the heart of his logicism, for i n G rundgesetze , Frege

explic itly defines numbers as the extensions of certain concepts.

Extensions themselves (and courses-of-values in general), however, are

not defined, but rather are legitimately introduced by satisfying a criteria

of identity. Frege''s idea here was simply this: If we ask what sort of

objects extensions are, it is sufficient to answer that th ey are the sort of

objects that are the same just in case the same objects fall under the

corresponding co ncepts. 6 4 What constitutes a corresponding concept is

and difficult in the book, see Michael Dummett, Frege: Philosophy of Mathematics ,

p. 209 ff , Michael Resnik, F rege and The Philosophy of Mathematics , p. 208 ff and A.

W. Moore and Andr e w Rein, “ Grundgesetze , Section 10” in L. Haaparanta and J.

Hintikka, eds., Frege Synthesized: Essays ont he Philosophical and Foundational Work

of Gottlob Frege (Dordrecht: D. Reidel, 1986).

48

made explicit by the compositio n of complex names of extensions. As

Frege notates, “ |f ( g)” refers to an extension; then the first-level function

f ( >) that stands as the argument of the second-level function |R( g) is the

corresponding concept. Thus, two extensions are the same:

|f ( g) = ?g ( "),

if and only if the concepts to which they correspond are identical. Con-

cepts are identical if and only if they fall under the second-order relation

above; that is, if the same objects fall under them. Identity of extension

is therefore to be equated with t he extensional equivalence of the con-

cepts; the result is:

( |f ( g) = ?g ( ")) = ( ?a f ( a) = g ( a)).

Frege remarks: “We can transform the generality of an identity into an

identity o f courses-of-values and vice versa. This possibility must be

regarded as a law of logic, a law that is invariably employed, even if tac-

itly, whenever discourse is carried on about the extensions of concepts.”

( Grundgesetsze, §9.) This “law” of logic is Frege’s Basic Law V.

Frege thought at the time of the Grundgesetze that because Basic Law V

was a logical law, that extensions being logical objects was secured,

since they satisfy a logical criteria of identity, and thus that his theory of

number was justified. But, of course, Frege was wrong about this, as he

instantly understood upo n learning of Russell''s paradox, and the incon-

sistency of Basic Law V.

It should be clear that as we have descri bed the role of Basic

Law V in Frege’s mature theory of number, we have had no recourse to

the sense /reference distinction, even though it takes the form of an

identity statement. This is not particularly surprising, given that Frege''s

goal was to establish extensions as references. But we may ask a further

question about Basic law V, that arguably does implicate the sense/re-

ference distinction. Frege remarks at various points that in order for a

65 Cf. “A Brief Survey of My Logical Doctrines,” pp. 197 - 8.

49

statement to qualify as a logical law it must at least meet two standards,

that it be a logical truth, and that it be self-evident. Frege, however, is

somewhat foggy about what constitutes satisfying these conditions, but a

natural initial thought, given the notions at hand, would be to look at the

intensional structure of the statement of the law, and appeal to synonymy

of the terms. Indeed, this is where Frege initially glances; in “Function

and Concept” he identifies the thoughts expressed by the two statements

joined in Basic Law V. Each “expresses the same sense, but in a

different way,” he says. (p. 27.) No doubt Frege''s thinking here is that if

Basic Law V is a logical law then it must at least be distinguished from

a garden variety material equivalence like (I). What he wants to capture

is that, in essence, the identity of extensions is no different than the

identity of the correspondi ng concepts; t hey are one and the same

circumstance, (just as, for instance, that lines are parallel is the same as

their having the same direction). The theory of the G rundgesetze ,

however, only has identity of reference and identity of sense; it provides

no way of directly expressing this other sort of equivalence relation. It

is, however, entailed by identity of sense, and this is Frege initial

conjecture about Basic Law V.

If Basic Law V is justified as a logical law by its intensional

structure, then it would appear that the sense/reference distinction i s

implicated, at a very fundamental level in Frege''s theory, for if the

statements that stand as the terms of Basic Law V express the same

thought, then Basic Law V is analytic. But, if they do express the same

thought, it is not in a way that satisfies Frege''s other criterion for logical

laws, that they be self-evident. Frege allowed that there may be various

over t forms that may express a single thought. For example, active

sentences and their passive counterparts are formally disti nct, yet

arguabl y the transformation between them leaves untouched the senses

that compose the thought expressed. There is only one thought

expressed, composed of the same senses. Statements so related Frege

called equipollent ; as Frege observes, because equipollent stat e ments

express the identical thought, to recognize the thought expressed by one

statement is to recognize the thoug ht expressed by the other. 6 5 The

statements that stand as the terms of Basi c Law V, however, are not

equipollent. They contain expressions that ref er to distinct second-level

66 We should be careful here to distingui sh the issue whether two distinct complex

terms can express the same thought from abbreviatory definition, in which the sense

of non-complex term is equated with that of a complex term. Atomic terms may have

compositionally complex senses; this composition is just not transparently revealed as

it would be with non-atomic terms.

67 “Comments on Sense and Meaning,” p. 122. This paper is dated by the editors of

Frege’s Posthumous Writings as 1892 - 1895. While the editors adopt the convention

of translating Bedeutung as “meaning,” I aver to the more common “reference” in the

quotation.

68 Review of Husserl, p. 200.

50

functions, and given that distinctness of reference entails di s tinctness of

sense, the thoughts they express therefore must contain different senses.

If thoughts are to be defined as the composition of the ir constituent

senses, there is no guarantee that by recognizing a thought composed in

one way, that we can also recognize it composed in some other way. But

then there is also no guarantee that Basic Law V is self-evident. 66

The problem here is that there is really no wiggle room in

Frege’s theory to have thoughts be the same, yet composed of different

senses; no doubt it would have struck Fr ege that this does not square

with his other assumptions about sense and reference. What is

consistent with these views is that the terms of Basic Law V do not

express the same thought, a view buttressed by the observation that:

(( |f ( g) = ?g ( ")) = (( |f ( g) = ?g ( ")),

contrasts in cognitive value wit h Basic Law V just as “Hesperus is

Hesperus” contrasts w ith “Hesperus is Phosphorus,” or “5 = 5” with

“2+3 = 5,” implying that the terms express different thoughts. Frege’s

reaction, however , was somewhat different; he retreats to an

extensionalist position, which he maintains throughout Grundgesetze .

He tells us during this period that “reference and not the sense of words

[are] the essential thing for logic . . . the laws of logic are first and fore-

most laws in the realm of reference and only relate indirectly to sense.” 67

In his review of Husserl’s Philosophie der Arithmetik in 1894, Frege

elaborates: 68

This reveal s a split between psychologistic logicians

69 The position we have developed in the last paragraphs echoes th at of Michael

Dummett; see his Frege : Philosophy of Mathematics (Cambridge: Harvard University

Press, 1991), especially pp. 168 - 176, and The Interpretation of Frege’s Philosophy ,

ibid , especially pp. 529 - 537. In the latter book, and in F rege and Other Philosophers,

ibid , Dummett mounts an extensive critique of Hans Sluga, who is the chief proponent

of identity of sense view; see his Gottlob Frege (London: Routledge and Kegan Paul,

1980 ), especially pp. 149 - 157, and “Semantic Content and Cognitive Sense” in L.

Haaparanta and J. Hintikka, eds., Frege Synthesized: Essays on the Philosophical and

Foundational Work of Gottlob Frege (Dordrecht: D. Reidel, 1986). Sluga states (in the

latter article) that Frege “initially introduced the notion of sense in order t o explain

why Axiom V is not a synthetic truth.” (p. 61.) His argument for this view stems from

the claim that Frege’s Begri ffsschrift theory only applied to synthetic truths, so that

“Frege is no longer able to explain how any but the most trivial arithmetical equations

can turn out to be logical truths.” (p. 58.) To do this, Sluga argues, Frege needed the

sense/reference distinction in order to countenance analytic truths through identity of

sense. In maintaining that the terms of Basic Law V express the same thought, Sluga

appears to deny that they have different component senses when he remarks that “It is

only our subjective perception and our manner of speaking that distinguish the

statement about functions from that about value-ranges . . . a t hought concerning a

51

and mathematicians. What matters to the former is with

the sense of the words, as well as the ideas which they

fail to distinguish from the sense; whereas what matters

to the latter is the thing itself: the reference of the

words.

Thus, regardless of whether the two sides of Basic Law V have the same

sense or different, they determine the same reference, and that is all that

matters for logic. In particular it doesn’t matter to the characterization

of numbers as logical objects; all Frege needs to characterize the logical

role of Basic Law V is the weaker notion of material equivalence. But

although taking an agnostic view, Frege ap parently retained his qualms

about the issue left open. In a well-known remark, Frege, in commenting

upon Russell’s Paradox, says about Basic Law V that “I have never con-

cealed from myself its lack of the self-evidence which the others possess,

and which must properly be demanded of a law of logic.” ( G rundesetze ,

Appendix II.) But there is no reason to think that Frege thought that this

self-evidence would have been found through refl ection on the

sense/reference distinction; as Frege saw the issue it cut much deeper, to

the heart of “how numbers can be conceived as logical objects.” We

know now that it cut to a point where it is insuperable. 69

function is the same as one concerning a value-range.” ( Gottlob Frege , ibid, p. 157.)

It is difficult to see how to reconcile this with the fact that different references entail

different senses, unless it is maintained that the second-level concept-terms in Basic

Law V refer to the very same concept. For this and other reasons, Sluga’s rendering

strikes one as a most implausible reading of Frege.

70 This is well-chronicled in Bynum’s “On the Life and Work of Gottlob Frege,” ibid.

71 In unpublished “Comments on Sense and Reference,” Frege elaborates on the issue

discussed in the previous section, remarks that are missing, we might conjecture, from

“On Sense and Reference” at least in part because of the technical notation required

to make the point; indeed, he remarks that he will elaborate his view of concepts “for

52

10. While one can admire the elegance and coherence of Frege’s ac-

count, there is on e thing missing. There is no argument for the central

concept: sense. Why should we think that such things exist, with the

properties that Frege ascribes to them? The importance to Frege of pro-

viding such an argument is clear, for he does not wish to be accused of

introducing some unreal, mystical notion just in order to avoid a founda-

tional problem with his view of number. The remedying of this defect is

the task that Freg e takes on in his seminal essay on the topic of identity

statements, “On Sense and Reference.”

How is Frege to establish the reality of sense? He does this by

arguing, at length and with great force, that senses can be Bedeutung ,

that they can be references. Since senses can be referred to, they must

exist; given Frege’s realism about abstract objects, senses can b e

r eferences in just the same way that numbers can be. Looked at this

way, the central goal of “On Sense and Reference” is to give an

“existence proof” of senses. Frege develops this with great care,

choosing his words judiciously; since this essay appeared in a general,

non-technical philosophical journal, Frege does not want his audience to

view hi m as pursuing an arcane issue in mathematics. The difficulties

Frege faced with publication are well-known. 70 Frege, consistently stung

by the limited, yet highly critical reception of his work among his

contemporaries, appeared to feel that he had never achieved the proper

“voice” for expressing his views, and hence vacillated between more and

le ss technical presentations of his work. This was particularly acute

around the time of the publication of “On Sense and Reference,” in the

early 1890''s, for this is when Grundgesetsze was deemed too technical to

be published in o ne volume. 7 1 As with G rundlagen with respect to

the reader who is not frightened of the concept-script” (p. 120).

72 In the text, Frege asks whether equality is a relation; in a footnote, he says that he

“use[s] this word in the sense of identity.” (p. 56).

53

Begriffsschrift , “On Sense and Reference” stands to Grundgesetsze in

manner of presentation, except that in “On Sense and Reference,” Frege

disguises the mathematical implications. He is so successful in this that

almost serendipitously he lays much of the foundations, both

conceptually and tech nically, for a whole independent branch of

philosophy, the philosophy of language. Indeed, his central argument is

such a tour de force that it has stood the test of time a s an analysis of

oblique contexts in natural lan guage, independently of its original

purpose as an argument for the reality o f Sinn. B ut we should not be

misled by this achievement into mistaking what “On Sense and

Reference” is about. It is an essay about identity statements. But to be

about identity statements is to be about something m uch more

fundamental for Frege - the nature of number. Throughout Frege keeps

the enemy squarely in sight. Although they go unmentioned, it is the

usual suspects - the mathematical formalists and their incoherent theory

of number.

Frege begins his most famous essay by posing the question: is

identity a relation? 72 There is a lure, Frege admits, to viewing identity

as a relation between “names or signs of objects,” a view by his own ad-

mission he was tempted in Begriffsschrift . The lure is to be found in the

apparent avoidance of the problem, originally p osed by the formalists,

engendered by the opposite answer - identity as a relation holding of the

object named - of being unable to distinguish true “ a = b ” from “ a = a .”

So Frege asks us to suppose that identity is a relation between signs;

under what conditions would it truly hold? Whatever the conditions are,

they must involve what it is that the symbols have in common. For

Frege this is the “same designate d thing;” only if they have this in

common could the relation be said to truly hold of the signs for that

object. But nevertheless, even though a = b and a = a now appear to be

differen t relations, we are really no further along with the problem. An

object, Frege observes, may be symbolized in any number of ways; this

is a fact of life. But if this is all that identity statements are about, there

still will not be sufficient information to calcu late a difference in

73 “On Sense and Reference,” p. 57.

54

cognitive value. To assess a difference in cognitive value, we must

consider the reaso n for this multiplicity of symbols. If distinct signs

c orrespond to distinct “modes of presentation” of a reference, only then

is there good reason for distinct signs. A statement that an object has two

names does not deliver the information w hy it has two names; bu t

without such information an identity statement would be left b ereft of

cognitive value.

Frege is, of course, being a bit disingenuous when he poses this

view as his prior view of Begriffsschrift . That analysis could not even

be stated under the assumptions present at the tim e of “On Sense and

Reference ,” since the notion of conceptual content is no longer defined.

Content is now broken into two parts, sens e and reference, so if there

w ere to be an identity of content relation between expressions it would

have to be either identity of sense - “— has the same sense as —” or

identity of reference - “— has the same reference as —”. The

Begriffsschrift identity of content symb ol (“ /”) cannot, however, be

identified with either of these. Identity of sense, where we gloss “has” as

“express,” is of course too strong, and Frege in fac t nowhere defines

such a symbol; thus our concern is to be with identity of reference. But

identity of reference is not the same as identity of conceptual content.

This is because only the latter relation is one that holds between

expressions qua how their contri bution to propositional content is

determined by their a ssociated modes of determination. The former

no tion, the one Frege criticizes in the opening paragraph of “On Sense

and Reference,” is simply a coreference relation; it holds of expressions

detached from modes of presentation. The critical point is that if they

are detached from modes of presen tation, then they are also detached

from cognitive values, and hence no distincti ons can be drawn that turn

of difference of cognitive value. Where there is no such connection of

sense to reference there can be no differenc e in cognitive value. This is

the conclusion Frege reaches in the following passage: 73

Nobody can be forbidden to use any arbitrarily

producible event or object as a sign for something. In

that case the sen tence a = b would no longer refer to the

subject matter, b ut only to its mode of designation; we

74 Ibid . We cannot make of Frege’s choice here of a geometrical rather than arithmetic

example what we did in Begriffsschrift . In the context of examples employed in other

writings of this period, the ratio nale here appears to be more rhetorical than

substantive; Frege would not have wanted to have been seen as begging the question,

given that his goal was to independently motivate the key notion in an argument against

he formalists notio n of arithmetical equality. In the next paragraph he extends the

geometric example to that of “the evening star” and “the morning star,” but remains

mum throughout on arithmetic.

55

would express no proper knowledge by its means. But

in many cases this is just what we want to do. If the

sign ‘ a ’ is distinguished from the sign ‘ b ’ only as object

(here by means of its shape), not as sign (i.e. not by the

manner in which it designates something), the cognitive

value of a = a becomes essentially equal to that of a = b ,

provided that a = b is true. A difference can arise only if

the dif ference between the signs corresponds to a

difference in the mode of presentation of that which is

designated.

“‘ a ’ is coreferential with ‘ a ’” and “‘ a ’ is coreferential with ‘ b’” both

e xpress thoughts; the former is a priori , the latter a posteriori . Hence,

both “‘ a ’” and “‘ b ’” express senses. However, there is something miss-

ing from these senses: the mode of presentation part. We don’t have

modes of pr esentation of the things these sentences are about because

these th i ngs are already present; sense doesn’t determine reference be-

cause the reference, the sign itself, is already there. That is fine if our

purpose is to say something about the linguistic devices by which we go

about making statements about things. But what we require in the gen-

eral case are not statements about statements about things, but state-

ments about things themselves; with the former we are at a level too de-

tached to express proper knowledge of objects. The point in the interior

of a trian gle at which lines from the vertices to the mid-point of the op-

posite side intersect may be designated as the intersection of any two of

the th ree lines. “So,” Frege says in concluding the opening paragraph,

“we have different designations for the same point, and these names

(‘point of intersection of a and b ,’ ‘point of intersection of b and c ’)

likewise indicate the mode of presentation; and hence, the statement con-

tains actual knowledge.” 74

75 Russell’s “Grey’s Elegy” argument of “On Denoting” can be understood as attacking

Frege at this point; cf. Simon Blackburn and Alan Code, “The Power of Russell’s

Criticism of Frege: ‘On Denoting’ pp. 48-50.” Analysis , XXXVIII , 2 (1978): 65-77.

On Blackburn and Code’s int erpretation, the problem Russell directs towards Frege

(and his own earlier views) is that while the description in question denotes a sense,

there is no way to give a meaning that d etermines that sense as the denotation; to put

it in Fregean terms, there is no indirect sense to determine customary sense as indirect

reference.

76 Cf. Grundlagen , §§25-27

56

The view that Frege is criticizing in commencing “On Sense and

Reference” we can now see is not his own view of B egriffsschrift , but

rather is a criticism of the view that he criticizes there. Consequently his

remarks to this point in the essay are highly reminiscent of what he says

in Begriffsschrift in isolating the role of “modes of determination”

( Bestimmungsweise ). However, much has changed in the ensuing years

in Fr ege’s comprehension of the significance of this notion, for what he

has come to understand is that there is a way of capturing this notion in

the theory of thought that supports a notion of objectual identity, a

“relation between objects . . . in which each thing stands to itself but to

no other thing.” Rhetorically, Frege does not explicitly state that it is

firmly establishing this point, and definitively refuting the fo rmalists

argument against equal ity understood as identity, that is the goal of the

essay. Rather he begins with the second paragraph of the paper the

process of leading us to this conclusion by laying out a th eory of s igns ,

those symbols that express a sense ( S inn ), “wherein the mode of

presentation is conta ined,” and s tand for or designate a reference

( Bedeutung ), the object so presented. The moves Frege then makes over

subsequ ent paragraphs are well-known: First he establishes that senses

can be references, and then, since references are objects, that senses are

objects. His initial pass at showing this is with proper names; p rima

facie, their customary senses can be referred to, albeit indirectly, i n

reported speech with the locution ‘the sense of the expression “A”’. 7 5

The second premiss is then established by deflecting the only alternative,

that senses are ideas; ideas are undermined as references by their

inherent subjectivity. Senses can’t be ideas any more than numbers can

be; if they were, then they would not have the sorts of properties that we

expect references to have. 76 Frege now takes a second pass at the

argument, this time with much more force and detail, turning to natural

77 “On Sense and Reference,” p. 67.

78 “On Sense and Reference,” p. 78.

57

language. For declarative sentences, Frege argues, their sense is a

thought, their reference, a truth-value. This much is established by a

substitution argume nt; the latter, but not the former, remains constant

und er substitution of coreferential parts of a sentence. But there are

telling exceptions, contexts in which trut h-value does not remain

constant; the se exceptions can be isolated grammatically as sentences

that themselves contain subordinate clauses (oblique contexts). For the

remainder of the essay up to the closing paragraph, Frege explores these

exceptions in great depth and with considerable linguistic sophistication,

establishing that they fall under the following principle: 77

In such cases it is not permissible to rep lace one

expression in the subordinate clause by another having

the same customary refer ence, but only by one having

the same indirect reference , i.e. the same customary

sense.

For F rege, the central tenet is that anything that is B edeutung is objec-

tive; in oblique contexts, senses are Bedeutung , given the substitution

argument; hence, senses are objects.

Having put all the pieces into place, Frege is now ready to

deliver the punch line in the final paragraph of the paper: 78

When we found ‘ a = a ’ and ‘ a = b ’ to have different

cognitive values, the explanation is that for the purpose

of knowledge, the sense of the sentence, viz., the

thought exp ressed by it, is no less relevant than its

reference, i.e. its truth val ue. If now a = b , then indeed

the reference of ‘ b ’ is the same as that of ‘ a, ’ and hence

the truth value of ‘ a = b ’ is the same as ‘ a = a. ’ In spite

of this, the sense of ‘ b ’ may differ from that of ‘ a ’, and

thereby the thought expressed in ‘ a = b ’ differs from that

of ‘ a = a. ’ In that case the two sentences do not have the

same cognitive value. If we understand by ‘judgment’

79 Letter to Peano, p. 126.

58

the advance from the thought to its truth value, as in the

above paper, we can also say that the judgments are

different.

At the end of his most justifiably famous paper, Frege can give this argu-

me nt, with all its implications, with the firm conviction that he has ar-

gued as compellingly as possible to place the central notion on which it

turns - Sinn - on firm foundational footings. In his letter to Peano that

we cited above, Frege complains that: 79

As far as the equals sign is concerned, your remark that

diff erent authors have different opinions about its

meaning leads to co n siderations that very many

mathematical propositions present themselves as

equations and that others at least contain equations, and

if we place this against your remark, we get the result

that mathematicians agree indeed on the exte rnal form

of their propositions but not on the thoughts they attach

to th em, and these are surely what is essential. What

one mathematician proves is no t the same as what

another understand by the same sign. This is surely an

intolerable situation which must be ended as quickly as

possible.

There is only one possible meaning of equality, according to Frege, that

can re store the peace: identity. “On Sense and Reference” is the final

piece of the puzzle, presented in a precise and decisive, yet non-techni-

cal, way, for establishing the identity theor y of G rundgesetsze . While

that theory is distinct from that of Begriffsschrift , at its heart is an in-

sight with its origins in the earlier monograp h; what Frege finds in his

mature work of the late 19 th and early 20 th centuries is a way of express-

ing this insight semantically that allows him to dismiss outright what he

viewed, in contrast to his own views, as the incoherent and chaotic view

of number held by many of the most influential mathematicians of his

day. With the machinery of sense and reference, no doubt Frege’s most

important bequest to modern thought, the account of identity statements

59

he developed to accomplish this is so brilliant and full of both philosoph-

ical and linguistic insight, that it has outlasted the demise of both his

view of number and many of the assumptions that animated his philo-

sophical outlook. How we are to evaluate Frege’s account once removed

from th is context, (especially with respect to contemporary skepticism

about sense), is a matter we leave for the sequel.

University of California, Irvine

23 October 2000

60

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61

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62

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