On Cognitive Value and Logical Consequence: A Reply to Dickie

Kent W. Erickson

Abstract: Imogen Dickie (2008) argues that in Begriffsschrift and On Sense and

Reference, respectively, Frege is concerned with two different puzzles corresponding to two

different kinds of ‘informativeness’ (i.e., ‘cognitive value’) regarding true identity

statements of the form ‘a = b’. I argue here, to the contrary, that in both places Frege was

in fact concerned with the same puzzle, and thus conclude that as an interpretive project

Dickie’s proposal fails, in particular, to faithfully represent the true motivation behind

Frege’s Bg account of identity statements.

1

1. Introduction

In a fairly recent article, Imogen Dickie (2008) identifies what she describes as an

‘interpretive difficulty’ with the ‘standard view’ of the transition from Frege’s so called

‘metalinguistic account’ of identity statements in Begriffsschrift (Bg hereafter) to his new ‘sense

view’ in On Sense and Reference (OSR hereafter). Frege’s appeal to senses in OSR is explicitly

motivated by broadly epistemological concerns regarding differences in cognitive value

[Erkenntniswert] between non-trivially true instances of identity statements of the form ‘a = b’

and trivially true instances of ‘a = a’.1 And in light of Frege’s rejection of his earlier Bg account

as against this backdrop, one is led to believe that in both places Frege was grappling with the

same basic question: Roughly, in virtue of what are true instances of ‘a = b’ potentially informative

in a way that trivially true instances of ‘a = a’ are not? However, Dickie argues as against this

interpretation that in each place Frege is in fact concerned with two different puzzles corresponding

to two different kinds of ‘informativeness’ (i.e., cognitive value). In Bg, according to her, Frege is

expressly occupied with what she calls the question of evolutionary informativeness, whereas in

OSR his reported target is the question of rational informativeness.

I will explain these distinctions below. But in short, I argue here, to the contrary, that Frege

was in fact concerned with the same puzzle throughout corresponding roughly to Dickie’s notion

of rational informativeness. Specifically, I submit that in both Bg and OSR Frege was primarily

concerned to explain which logico-semantic property of true ‘a = b’ formally justifies other

potentially knowledge-advancing inferences (viz., those of the form Fa Fb). For while Frege

clearly valued the knowledge gained from individual scientific discoveries, including ‘recognizing

1 As Dummett (1973, 123) put it, “Frege's notion of sense is a cognitive one: difference in cognitive value is

precisely what requires difference in sense to explain it.”

2

something as the same again’, what he valued more, throughout his career, are logical discoveries

regarding valid patterns of deductive reasoning—i.e., what follows from what and according to

which logical laws. As such, I contend, more generally, that for Frege the ‘real’ cognitive value of

any logical statement,2 including both non-trivial identities and ‘fruitful’ definitions, is constituted

by the deductive consequences of its logical form; true statements that support non-trivial

inferences are for Frege logically valuable, and hence cognitively/epistemically valuable, whereas

those that do not, are not. 3 If correct, I conclude that as an interpretive project (as opposed to a

rational reconstruction) Dickie’s proposal thus fails, in particular, to accurately represent the true

motivation behind Frege’s Bg account of identity statements.4

Toward this end, I begin in Section 2 with a more detailed review of the reported difficulty

along with Dickie’s proposed solution to it. Section 3 represents my take on Frege’s Bg account

of identity statements, and thus why, in general, I think Dickie’s account fails as an interpretive

project. Section 4 addresses the reported consequences of Dickie’s proposal and demonstrates why

they too are either misguided or otherwise non-tendentious. Section 5 closes with a few concluding

remarks.

2. Dickie’s Proposal

Frege’s opening discussion in OSR (see Frege, 1892, 151-2) is widely regarded a corrective

to his earlier Bg account of identity statements of the form ‘a = b’ as expressing a relation between

2 What he sometimes distinguishes as its “wirklich/eigentlich Erkenntniswert.” 3 This claim is not entirely new, and versions of it have been defended in various ways by others including William

Taschek, Peter Strawson, Michael Kremer, Michael Potter, Jamie Tappenden, and to the extent indicated below by

Dickie herself. 4 This indictment is perhaps too strong. For while Dickie initially represents herself as engaged in an interpretive

project, there are several points where she seems to slide toward a contemporary reconstruction of Frege. But quite

frankly I cannot ascertain precisely which of these two projects she is engaged in, if not perhaps some combination

of both. But I will continue to treat it here as an interpretive enterprise.

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the names ‘a’ and ‘b’ as opposed to the objects they designate. To refresh memories, he begins by

posing the following dilemma:

Equality [Gleichheit]5 gives rise to challenging questions which are not altogether easy to

answer. Is it a relation? A relation between objects, or between names or signs of objects?

In my Begriffsschrift I assumed the latter.

As Dickie rightly notes, this sharp dichotomy of choices appears to have been forced by Frege's

recognition in Bg of just two kinds of content—either purely metalinguistic or purely objectual.

However, in the target passage in OSR Frege proceeds to reject both options, beginning with an

explanation of why a purely objectual account of identity (or ‘equality’) won’t work:

Now if we were to regard equality as a relation between that which the names ‘a’ and ‘b’

designate, it would seem that a = b could not differ from a = a; i.e. provided a = b is true.

A relation would thereby be expressed of a thing to itself, and indeed one in which each

thing stands to itself but to no other thing.

The worry here is familiar, and while not explicit amounts to a reductio refutation of Mill’s theory

of proper names as logical constants that merely ‘stand for’ [bedeuten] their objects/referents. For

if understanding a name is simply to know what it refers to, then anyone who understands ‘a’ and

‘b’ should know straightaway that they corefer. Yet this requirement arguably places intolerable

demands on a speaker’s semantic competence. Furthermore, it would seem that under so called

‘objectual identity’ ‘a = a’ and ‘a = b’ should express the same trivial thought—viz., that the object

referred to twice over is self-identical—and which should then also be knowable a priori by

anyone who understands such statements. However, as Frege famously observes:

5 In a related footnote (ibid.) Frege mentions that he is using the term ‘Gleichheit’ in the sense of ‘Identität’ which

straightforwardly translates to English as ‘identity’.

4

[…] a = a and a = b are obviously statements of differing cognitive value; a = a holds a

priori and, according to Kant, is to be labeled analytic, while statements of the form a = b

often contain very valuable extensions of our knowledge and cannot always be established

a priori.

Taking this observation for granted, Frege then goes on to explain why a purely metalinguistic

solution fares no better:

Nobody can be forbidden to use an arbitrarily producible event or objects as a sign for

something. In that case the sentence a = b would no longer be concerned with the subject

matter, but only with its mode of designation; we would express no proper knowledge by

its means. But in many cases this is just what we want to do.

As Frege notes here, the assignment of names to their objects is arbitrary, which is to say a matter

of linguistic convention. As such, it is then equally arbitrary whether any two names designate the

same object. Thus, to assert ‘a = b’ under Frege’s Bg account of identity is to merely report a fact

about language use rather than a substantive semantic fact about the relevant object(s) of reference.

And so in this respect true instances of ‘a = b’ are according to Frege no more informative than

trivially true instances of ‘a = a’. Moreover, Frege suggests here that when we make identity claims

we do not normally (or anyhow ‘in many cases’) take ourselves to be talking about names but

rather the objects referred to. Hence the dilemma.

The conclusion of this discussion is likewise familiar: There must be more to the semantic

content of an expression than its meaning/reference [Bedeutung]. Specifically, since ‘a = a’ and

‘a = b’ differ only in the substitution of type-distinct yet coreferential names, but that qua signs a

mere orthographic difference can have no bearing on differences in cognitive value, Frege reasons

that:

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A difference [in cognitive value] can arise only if the difference between the signs

corresponds to a difference in the mode of presentation of the thing designated.

Frege goes on to qualify that differences in ‘modes of presentation’ correspond to a difference in

sense [Sinn], where Fregean senses are posited as abstract logico-semantic entities that speakers

associate with expressions and that in turn determine their reference.

The relevant point is again that Frege’s invocation of sense in OSR is clearly motivated by

broadly epistemological considerations related to differences in cognitive value between two

otherwise truth-conditionally equivalent sentences. And in view of Frege’s rejection of his earlier

Bg account as against this backdrop, one is led to believe that Frege was in fact concerned to

explain the same puzzle in both places: Roughly, in virtue of what are true instances of ‘a = b’

potentially informative in a way that trivially true instances of ‘a = a’ are not? But if the answer

so obviously owes not to a difference in signs, then one wants to know how someone of Frege’s

acumen might have ever thought it did? Or as Dickie states the worry, in OSR Frege represents his

Bg account ‘as not even a contender as a solution to the puzzle of informative identities’, which

she adds ‘is hard to understand’ and indeed ‘just seems wrong’. For depending on what one means

by ‘cognitive value’, Dickie notes that ‘it may be informative to be told that a sentence’s truth

value remains invariant under replacement of one name by another’, and which is reportedly what

one stands to learn from true ‘a = b’ under Frege’s Bg account.

As way of reconciling this apparent tension, Dickie submits, again as against the standard

view, that in each place Frege was in fact concerned with two different puzzles corresponding to

two different kinds of ‘informativeness’. In Bg, according to her, Frege is specifically concerned

with what she calls the question of evolutionary informativeness, whereas his appeal to senses in

OSR is designed to address what Dickie calls the question of rational informativeness.

6

In the first instance, Dickie characterizes the question of evolutionary informativeness (with

respect to identity statements) as follows:

THE QUESTION ABOUT EVOLUTIONARILY INFORMATIVE IDENTITIES—What

account of what understanding a name involves must we give in order to explain the fact that a

subject may, without rational inconsistency, understand two co-referential names without

knowing that they co-refer, and may then acquire knowledge that the names are in fact names

for the same thing?

As Dickie clarifies, this question is so called because ‘it is a puzzle about how a specific kind of

situation and a specific kind of transition in the evolution of the epistemic life of an individual are

possible’. Specifically, she adds, it involves a transition ‘to the situation in which the subject knows

that the names co-refer’, which is again reportedly what one stands to learn from true ‘a = b’ under

Frege’s Bg account of identity.6 In short, Dickie suggests that since the resources available to Frege

in Bg are only equipped to explain evolutionary informativeness, we must assume that this was his

target explanadum. The suggestion, in other words (or as I understand it), is that Frege’s Bg

account was designed to explain precisely what it is capable of explaining.

In OSR, by contrast, Dickie contends that Frege is concerned with a deeper notion of

cognitive value—the kind that really matters to logical appraisal:

[…] Frege is no longer concerned with evolutionary informativeness and differences in

evolutionary cognitive value. He is concerned with the possibility of what I shall call

‘rational informativeness’. This type of informativeness is bound up with Frege’s

conception of how a deductive proof can provide justification for moving from affirmation

of its premisses to affirmation of its conclusion.

6 I dispute this finding in Sections 3 and 4 below.

7

According to Dickie the rational informativeness of true ‘a = b’ is determined by the rational

distance between the names ‘a’ and ‘b’ within ‘the network of logically self-evident rational

relations between statements [in which these names occur]’. I will return to Dickie’s notion of

'rational distance' momentarily. But so far as I can tell, what she means by a ‘logically self-evident

chain of inference’ is what Frege considered a gapless deductive proof wherein each step is

formally justified by the general rules of pure logic, often together with the aid of certain axioms,

theorems, and/or ‘fruitful’ definitions. Thus, in the traditional Fregean vernacular 'logically self-

evident' here just means analytically or deductively provable.7 By contrast, a sequence of

judgments that is not logically self-evident is by Fregean standards formally invalid.

With this background in mind, Dickie argues that:

Frege’s picture of proofs as built up out of logically self-evident steps, and of a rational order of

logically self-evident relations between thoughts, carries with it the possibility of a distinctive

kind of difference in cognitive value [my emphasis] between extensionally equivalent

expressions. This is the possibility that two expressions might stand for the same object (if they

are names) or the same function (if they are predicates) but that a chain of inference constructed

using one expression might be logically self-evident [i.e., valid/deductively provable] while the

parallel chain of inference constructed using the other is not.

By way of illustration, Dickie employs a variant of the following proof in Peano Arithmetic that 1

+ 2 = 3, where ‘S0’ stands for ‘the successor of 0’, ‘SS0’ stands for ‘the successor of the successor

of 0’, and so on.8 We begin here by stating Peano’s two axioms of addition,

(A1) x [x + 0 = x]

7 In my view Dickie’s use of the term ‘self-evident’ here is somewhat awkward, as Frege scholars typically use this

term to mean something more like ‘trivial’ in the sense of not requiring special cognitive effort. Yet Frege clearly

believed that a proof could be ‘self-evident’ in Dickie’s sense while also requiring substantial cognitive effort. 8 For sake of clarity, my reformulation of Dickie’s example here is a bit more explicit, yet its content is preserved.

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(A2) x ∀y [x + Sy = S(x + y)]

along with the following rule that I will call Distribution (D):

(D) x [S(x) = Sx, S(Sx) = SSx, S(SSx) = SSSx…]

For example, (D) merely allows us to rewrite ‘S(S0)’ as ‘SS0’, or ‘S(SS0)’ as ‘SSS0’, and so on.

Lastly, we take for granted the rules of Universal Instantiation (UI) and Substitution of Identicals

(SI), which together with (A1), (A2), and (D) support the following chain of inference:

1. S0 + 0 = S0 (A1, UI: x = S0)

2. ∀y (S0 + Sy = S(S0 + y)) (A2, UI: x = S0)

3. S0 + S0 = S(S0 + 0) (2, UI: y = 0)

4. S0 + S0 = S(S0) (3, A1)

5. S0 + S0 = SS0 (4, D)

6. S0 + S(S0) = S(S0 + S0) (2, UI: y = S0)

7. S0 + SS0 = S(S0 + S0) (6, D: S(S0) = SS0)

8. S0 + SS0 = S(SS0) (5, 7, SI: S0 + S0 = SS0)

9. S0 + SS0 = SSS0 [i.e., 1 + 2 = 3] (8, D)

This proof is formally valid—its conclusion follows logically from stated axioms and rules of

inference. Or as Dickie says, it constitutes ‘a series of logically self-evident steps’. However,

Dickie invites us to consider the result of substituting occurrences of ‘SS0’ (highlighted above)

with the coreferential expression ‘S0 × SS0’ to yield the following sequence:

1. S0 + 0 = S0 (A1, UI: x = S0)

2. ∀y (S0 + Sy = S(S0 + y)) (A2, UI: x = S0)

3. S0 + S0 = S(S0 + 0) (2, UI: y = 0)

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4. S0 + S0 = S(S0) (3, A1)

5*. S0 + S0 = S0 × SS0 (SI: ‘S0 × SS0’ for ‘SS0’)

6. S0 + S(S0) = S(S0 + S0) (2, UI: y = S0)

7*. S0 + S0 × SS0 = S(S0 + S0) (SI: ‘S0 × SS0’ for ‘SS0’)

8*. S0 + S0 × SS0 = S(S0 × SS0) (SI: ‘S0 × SS0’ for ‘SS0’)

9*. S0 + S0 × SS0 = SSS0 (SI: ‘S0 × SS0’ for ‘SS0’)

As Dickie rightly notes, while steps 5*, 7*, 8*, and 9* are all true ‘they no longer follow from

logically self-evident steps […]; the replacement of ‘SS0’ with the co-referring ‘S0 × SS0’ has

turned a proof into a non-proof’, which is again to say one that is formally invalid. Or as stated in

purely logico-semantic terms, Dickie observes that while truth-preserving substituting ‘S0 × SS0’

for ‘SS0’ ‘does not preserve the location of the statement in which the substitution occurs in the

network of logically self-evident rational relations between statements’.

Now, according to Dickie the reason that such substitutions are not location-preserving owes

to a difference in what she calls the rational distance between ‘SS0’ and ‘S0 × SS0’. Of course as

introduced ‘SS0’ and ‘S0 × SS0’ are not proper names, per se, but rather abbreviations of definite

descriptions. This difference will be relevant in later discussion. But assuming for the moment, as

Frege did, that definite descriptions count as proper names, Dickie defines the notion of ‘rational

distance’ in these terms as follows:

There is ‘rational distance’ between names ‘a’ and ‘b’ if and only if, for some γ1, …, γn, where

γ1, …, γn is a chain of inference which contains occurrences of ‘a’ but not of ‘b’, replacing all

occurrences of ‘a’ with occurrences of ‘b’ transforms γ1, …, γn from a chain of inference which

is logically self-evident to one which is not, or from a chain of inference which is not logically

self-evident into one which is.

10

The leading idea here, again as I understand it, is that each Fregean thought occupies a certain

(presumably eternal) location in the rational order of logically self-evident relations between

thoughts. And since on Frege’s mature view the thought expressed by a declarative sentence

(including identity statements) is compositionally determined by the senses of its constituent parts,

the location of that thought within the network of logically self-evident relations is ultimately

determined by the respective locations of its constituent senses. Dickie puts the point this way:

Frege intends an expression’s sense to be its contribution to determining where the thoughts

expressed by sentences containing it lie in the natural order of thoughts: the pattern of thoughts

determined by logically self-evident relations between them.

And to this she adds that ‘all right deductive reasoning owes its justification to this pattern of

logically self-evident relations’.

Hence, under the Fregean assumption that type-distinct yet coreferential names typically

(always?) differ in sense, their substitution alters the location of the thought expressed with respect

to its inferential relations to other thoughts. The resulting ‘distance’ between substitution instances,

again according to Dickie, is best explained in terms of the rational distance between the

substituted names (or rather I think what she means to say is the distance between their respective

senses). In regard to the example above, the differences in sense between ‘SS0’ and ‘S0 × SS0’ is

manifest in differences in the inferential properties of the thoughts expressed by sentences in which

these expressions occur, and which in turn explains the resulting substitution failures (in this case,

why the substitution of coreferential expressions fails to preserve validity).

So characterized, Dickie goes on to define the notion of ‘rational informativeness’ (again

specifically with respect to identity statements) in terms of differences in rational distance between

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coreferential names (and more generally between any two formally distinct yet extensionally

equivalent expressions):

An identity statement between co-referring names ‘a’ and ‘b’ is ‘rationally informative’ if and

only if there is rational distance between ‘a’ and ‘b’.

Thus, true statements of the form ‘a = b’ are rationally informative according to Dickie because

on Frege’s OSR account there exists rational distance between [the respective senses of] the names

‘a’ and ‘b’. By contrast, since tokens of type-identical expressions necessarily have the same sense

there can be no such distance between them, thereby explaining why statements of the form ‘a =

a’ are not rationally informative and hence why ‘a = a’ and ‘a = b’ differ in cognitive value under

Frege’s OSR account.

To sum up, Dickie argues that in Bg Frege addresses himself to the question of evolutionary

informativeness as characterized by the transition from not knowing that two names corefer to

knowing that they do. However, according to her by the time of OSR Frege has turned his sights

to the question of rational informativeness:

THE QUESTION ABOUT RATIONALLY INFORMATIVE IDENTITIES—What account of

what a subject’s understanding of a name involves must we give in order to allow for the fact

that there might be rational distance between ‘a’ and ‘b’ even though ‘a’ and ‘b’ co-refer?

In short, Dickie concludes that between Frege’s metalinguistic view of identity statements in Bg

and his subsequent sense view in OSR, only the latter is capable of explaining differences in

rational informativeness. Specifically, since Bg again recognizes just two kinds of content—either

purely objectual or purely metalinguistic—it lacks the theoretical resources to explain the rational

distance between coreferential expressions, and hence the rational informativeness of the identity

statements in which appear; for this, Fregean senses are needed. Moreover, if we assume that in

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Bg Frege was not concerned to explain the rational informativeness of true ‘a = b’, then one might

feel compelled to conclude, with Dickie, that in Bg and OSR, respectively, Frege was in fact

occupied with two different questions corresponding to two different notions of informativeness.

Dickie goes on to argue that her proposal has consequences regarding (i) an adequate

criterion of difference for senses, (ii) the special role/significance of definitions in Frege’s logic,

and (iii) the relationship between Frege’s notions of modes of determination in Bg and his modes

of presentation in OSR. I return to these consequences just briefly in Section 4 below and why I

believe they, too, are either misguided or otherwise non-tendentious. But in service of this goal the

next Section attempts to show why, more generally, Dickie’s proposal fails as an interpretive

project, and in particular because it fails to accurately identify the real motivation behind Frege’s

Bg view of identity statements.

3. My Take on Frege’s Bg Account of Identity Statements

At issue here is again the real (or ultimate) motivation behind Frege’s Bg account of identity

statements, which while not stated in precisely these terms strikes me as fairly straightforward. At

bottom, Frege needs a way to formally justify potentially knowledge-advancing substitution

inferences of the form ‘Fa Fb’, which while synthetic in Kant's sense are intuitively truth-

preserving on condition that the names ‘a’ and ‘b’ corefer, or I shall say when they have the same

objectual content.9 Now, since in Bg Frege limits himself to just one rule of inference—modus

ponens—the only way to license the transition from Fa to Fb is through a chain of reasoning that

has ‘Fa Fb’ as the conclusion of an instance of modus ponens. Working backwards, in other

words, the question is with what to replace the ‘?’ in the following sequence of Fregean judgments:

9 Recall that for Frege a judgment is justified to the extent that inferences drawn from it under the general rules of

logic do not lead to contradiction.

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(1) |— ? (Fa Fb)

(2) |— ?

(3) |— (Fa Fb)

The obvious candidate is an identity statement of the form ‘a = b’. But then one must define what

such statements mean, and again in a way that formally warrants the inference from Fa to Fb. The

issue here, however, is that at the time of Bg the symbol ‘=’ was customarily associated with

arithmetical equality which by definition expresses a relation between numbers. Yet if, as Frege

believed, arithmetic is reducible to logic, then what he needs is a notion of identity that is

maximally general—one that applies to any sort of objects whatsoever.

One option here is to just stipulate that the symbol (‘=’) applies not only to numbers but to

any objects whatever. Yet while fully general, a purely objectual definition of identity immediately

raises the issue noted by Frege in OSR regarding how to distinguish true ‘a = b’ from trivially true

instances of ‘a = a’. The difficulty again arises, in particular, under the Millian assumption that

names merely stand for their objects. For given a logical language that permits the introduction of

content-identical names—i.e., names with the same Bedeutung—those names should, in

consequence of Leibniz’s Law, be everywhere substitutable, salva veritate. Implicit in Leibniz’s

principle of Substitution, however, is also a strict principle of Semantic Compositionality

according to which substitutions of content-identical names should not only be truth-preserving

but also content-preserving.

Importantly, however, in his Preface to Bg Frege commits himself to the view that judgments

with different logical consequences must also differ in content. Specifically, he writes:

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[…] the contents of two judgements can differ in two ways: either the conclusions that can be drawn

from one when combined with certain others also always follow from the second when combined

with the same judgements, or else this is not the case.

The observation here is trivial: Two judgments either differ in their deductive consequences or

they do not. Its implications are important, however. For what Frege seems to be offering here is

a general criterion of content individuation; the relevant criterion being that whatever judgeable

contents are, two judgments have (or express) the same content if and only if they have the same

deductive consequences. By parity of reasoning, one assumes that two judgments differ in content

for Frege just in case they differ in their deductive consequences. Now, since there is no avoiding

the conclusion that ‘a = a’ and ‘a = b’ differ in their logical consequences, then by this criterion

they must also differ in content.10 Thus, to adopt a purely objectual account of identity leaves Frege

no way to distinguish the respective contents of ‘a = a‘ and ‘a = b’ on the basis of differences in

their deductive consequences (or at least not without abandoning either Compositionality or

Millianism, or perhaps both).

Thus, Frege appears to be stuck with just one other option, which is to distinguish the

respective contents of ‘a = a’ and ‘a = b’ on the basis of differences between the names ‘a’ and

‘b’. And indeed to this end Frege introduces a new symbol, the triple bar ‘≡’, to stand for what he

calls identity of content [Inhaltsgleichheit]. For instance, a well-formed judgment of the

Begriffsschrift is given by (2'):

(2') |— (a ≡ b)

10 Unfortunately, Frege is not entirely clear in Bg what, at the sentential level, he takes the content of a judgment to

be. But the answer to this question needn’t detain us here.

15

Informally, Frege says that (2') expresses the judgment that ‘the symbol a and the symbol b have

the same conceptual content, so that a can always be replaced by b and conversely’. Yet (2') alone

fails to express what is needed to formally justify the conclusion in (3) above. Rather, what is

needed, in addition, is an axiom, or definition, or rule of inference that codifies Leibniz’s Principle

of Substitution in the meaning of the identity sign. In Bg, this lacuna is filled by what Frege labels

basic proposition (52), which we can think of as a kind of contextual definition of the identity

relation as formulated in (1'):11

(1') |— (a ≡ b) (Fa Fb)

Echoing Frege, the judgment in (1') merely makes explicit that one may freely exchange the

symbol ‘b’ for ‘a’ in any formula of the general form ‘Fa’, salva veritate (and hence salva

validate), under the identity of their content—a judgment which itself must be asserted somewhere

in the body of the proof in the form of (2'). Having done so, what we wind up with, by modus

ponens, is the following sequence of formally valid judgments:12

(1') |— (a ≡ b) (Fa Fb)

(2') |— (a ≡ b)

(3) |— (Fa Fb)

This proposal of course has drawbacks; one being that it seemingly13 introduces a systematic

ambiguity into the meaning of each atomic name in the language—a situation that would

ultimately not sit well with Frege. For as he notes, in the context of a statement of identity of

content names at once come to stand for themselves, yet in all other contexts, such as (3), names

11 As reported by May (2000), in “Boole’s logical Calculus and the Concept-script,” published shortly after

Begriffsschrift in 1881, Frege considers exporting his basic proposition (52) as a rule of inference, yet which leads to

certain complications that are again largely orthogonal to my purposes here. 12 There are of course other ways to derive (3); e.g., by rules of biconditional exchange and conjunction elimination. 13 See Section 4 below for clarification of my ‘seemingly’ hedge here.

16

merely stand for their objects. As importantly, Frege anticipates the very objection he would later

raise in OSR, which is that his definition for identity of content lends the appearance that identity

statements pertain only to the expression and not the thought expressed. I assume this is familiar

territory, but in response to both worries Frege invokes the following example:14

<Figure 1 goes about here>

Figure 1. Fregean Modes of Determination

Frege elaborates (I’m paraphrasing) that as the line AB moves about point A in the direction of the

arrow so does point B such that when line AB is perpendicular to the diameter of the circle the

points A and B coincide, which is to say that the names ‘A’ and ‘B’ have the same content. This

result is geometrically provable. However, one can also just see that this is the case simply by

looking at the illustration. As Frege puts it, the same point is determined in two different ways: (i)

immediately through intuition, and (ii) as the point B when the line is perpendicular to the diameter.

There is much to be pondered about Frege’s rather brief and somewhat unclear discussion of

identity of content in §8 of Bg. But most relevant for purposes here he writes:

The need for a symbol for identity of content thus rests on the following: the same content can be

fully determined in different ways; but that, in a particular case, the same content is actually given

by two modes of determination is the content of a judgement. Before this judgement can be made,

two different names corresponding to the two modes of determination must be provided for that

that [the relevant content] is thereby 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

14 This illustration does not appear in Bg but borrows from Beaney (1997: 64).

17

for the same content are not always merely a trivial matter of formulation, but touch the very heart

of the matter if they are connected with different modes of determination. [Frege’s emphases]

Now to suggest here, as Frege does, that ‘the same content is actually given by two modes of

determination is the content of a judgement’ might appear to conflict with his earlier explication

of statements of identity of content as expressing the judgment that ‘the symbol a and the symbol

b have the same conceptual content, so that a can always be replaced by b and conversely’. To the

contrary, however, as Robert May (2001, 17) observes (correctly in my view), ‘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’.

May’s point, in other words (or as I understand it), is that each step in a Fregean proof,

including judgments of identity of content, requires independent justification. With respect to the

present example, since judgments of the form ‘A ≡ B‘ are again synthetic in Kant's sense, their

justification will typically take the form of an inductive proof. And so unless one merely stipulates

that the names ‘A’ and ‘B’ are content-identical then one’s judgment that ‘A ≡ B‘ is true requires

independent proof that ‘A’ and ‘B’ are in fact content-identical names. Part of this proof, according

to Frege, will include the judgment that the same point/object is determined in different ways, and

that these distinct ‘modes of determination’ (MODs) are conventionally linked to different names.

Thus, if one can establish (by whatever means) that ‘A’ and ‘B’ in fact correspond to different

MODs of the same object, one has thereby justified one’s judgment that ‘A ≡ B’ is true. Notice

that we are talking about two different judgments with different contents; the former involves a

judgment to the effect that the same object is given by two different MODs, and that these two

MODs are associated with the names 'A' and 'B', respectively. By contrast, the identity statement

18

‘A ≡ B’, according to Frege, expresses the judgment that the names ‘A’ and ‘B’ have the same

conceptual content and are therefore substitutable, salva veritate (and hence salva validate).

The upshot is this: Whatever MODs are for Frege, in one direction they are intimately related

to the objects they determine (presumably via some sort of determination relation), and in the other

direction MODs are associated with particular names. And so as I interpret the passage above, it

is in this way that different names for the same object ‘touch the very heart of the matter’ insofar

as they are connected with different MODs. In other words, judgments of identity of content are

for Frege about their objects, if only indirectly, in virtue of being justified by the connection

between coreferential names and their respective MODs, which is in turn what justifies the

use/introduction of such names in the context of a logical proof.

So characterized, one cannot help but notice the resemblance between MODs in Bg and

Frege’s modes of presentation (MOPs) in OSR. To wit; both notions correspond to ways of

determining a Bedeutung. The chief difference, near as I can tell, is that in Bg Frege has not yet

recognized the theoretical advantage of integrating MODs/MOPs directly into the content of a

judgment in terms of the thought [Gedanke] that it expresses. Indeed, pace Dickie I find the more

curious interpretive question of OSR to be why Frege there makes no mention whatever of his Bg

notion of MODs, if perhaps only to spare himself the embarrassment of not having recognized its

proper theoretical role sooner.15

But setting this curiosity aside, the relevant observation is this: In Bg true identity statements

of the form ‘a = b’ (or as it were ‘a ≡ b’) are problematic for Frege in that they betray a

fundamental tension between the demands of formal correctness (i.e., questions of validity and

logical consequence) and those of semantic evaluation (i.e., questions of truth and reference). For

15 Frege here only makes reference to a name’s “mode of designation” and by which I take him to mean its purely

formal/orthographic properties.

19

on the one hand, a Millian conception of names coupled with a strict principle of Semantic

Compositionality implies that the substitution of coreferential names should not only be truth-

preserving but also content-preserving. On the other hand, Frege wants to distinguish the

respective contents of ‘a = a’ and ‘a = b’ on grounds that they obviously differ in their logical

consequences. But the real conundrum for Frege, as I see it, is that in Bg he has not yet hit on, or

at least has not yet fully developed, the idea that the logically relevant content of a judgment—i.e.,

the thought/Gedanke that it expresses—is not only object-directed but also has internal logical

structure that determines (or otherwise ‘influences’) its deductive consequences.

So where, then, does this leave us regarding the question of cognitive value? While Frege

does not explicitly appeal to differences in ‘cognitive value’ in Bg, he doubtlessly believes that

true instances of ‘a = b’ are epistemologically significant in a way that trivial instances of ‘a = a’

are not. Generally speaking, however, Frege is less concerned with what one stands to learn from

particular instances of true ‘a = b’ as with what else can be validly deduced from them. For recall

that among Frege’s lifelong goals, beginning with Bg, was to demonstrate how the resources of

pure logic alone stand to advance our knowledge in abstraction from ‘the particularity of things’,

which is to say independently of the particular subject matter to which it is applied, whether it be

arithmetic, geometry, the special sciences, or what have you.16 For instance, in a posthumous essay

Frege writes (see Frege, 1898/99, 57):

If you ask what constitutes the value of mathematical knowledge, the answer must be: not

so much what is known as how it is known, not so much its subject-matter as the degree to

which it is intellectually perspicuous and affords insight into its logical interrelations.

16 Frege writes: “I am convinced that my Begriffsschrift can be successfully applied wherever a special value has to

be placed on the validity of proof […].”

20

This sentiment is not peculiar to mathematical judgments, as in the Grundlagen (Frege, 1884: 92)

Frege repeats, more generally, that:

The aim of proof is not only to place the truth of a proposition beyond all doubt, but also to

afford insight into the dependence of truths on one another.

The upshot is this: While Frege naturally valued individual scientific discoveries, including

‘recognizing something as the same again’ (a kind of empirical recognition), what he valued more

are logical discoveries, and in particular recognizing the deductive consequences of a true

proposition as determined by its logical form and again as justified by the relevant logical laws,

axioms, and/or definitions by which those consequences follow from other pre-established truths.

The general picture that emerges from this discussion is this: True instances of ‘a = b’ are

non-trivial for Frege, which is to say cognitively valuable, not merely in virtue of the particular

facts they express but because they are needed to formally justify other potentially knowledge-

advancing inferences (viz., Fa Fb). In general, I think it fair to say that throughout his career

what Frege took to be the ‘real’ cognitive value of any logical statement is constituted by the

deductive consequences of its logical form: A given (true) judgment is logically valuable, and thus

cognitively/epistemically valuable, to the extent that its logical structure allows one to prove things

that would otherwise be unprovable. Or to use Dickie’s language, ‘This type of informativeness is

bound up with Frege’s conception of how a deductive proof can provide justification for moving

from affirmation of its premisses to affirmation of its conclusion’, just as it is in OSR. My claim

in other words, and again pace Dickie, is that Frege’s view about the logical significance of true

instances of ‘a = b’, and hence of their cognitive/epistemological significance, is essentially the

same in both Bg and OSR.

21

As best I can tell, this same principle applies to what Frege considered to be genuinely

‘fruitful’ definitions. The definition in (1') above, for example, illustrates how for Frege a good

definition shows its worth in carrying out a proof. Specifically, this example illustrates the way in

which ‘Fa Fb’ and ‘a = b’ (or again ‘a ≡ b’ as it were) are ‘organically’ related, as Frege might

put it. For while ‘Fa Fb’ and ‘a = b’ are by definition analytically (i.e., truth-conditionally)

equivalent, taken in isolation they are not logically equivalent in the sense of having the same

deductive consequences (and hence on Frege’s view in both Bg and OSR they are also not content-

identical). Rather, to repeat an earlier point it is only after the identity sign has been so defined

that one can validly assert ‘a = b’ (or again ‘a ≡ b’) in order to prove ‘Fa Fb’. On my reading

of Frege, it is precisely the role of a good definition to bridge such gaps.

Indeed these observations are manifest, in spades, in Dickie’s own example from earlier. For

just as one cannot prove that 1 + 2 = 3 in Peano arithmetic without Peano’s axioms of addition,

one cannot validly derive the same conclusion with substitutions of ‘S0 × SS0’ for ‘SS0’ without

also stating Peano’s axioms for multiplication. That is, the substituted expression ‘S0 × SS0’ in

Dickie’s second proof contains an undefined operator; viz., the multiplication operator ‘×’. Yet by

adding Peano’s axioms of multiplication, we can readily turn this ‘non-proof’ back into a valid

proof again—one that follows from ‘logically self-evident steps.’ For this, all we need is to prove

that ‘SS0’ and ‘S0 × SS0’ designate the same number, which repeating steps 1-5 from above, and

adding (M1) and (M2) below, is a fairly trivial exercise:

Peano’s Axioms of Addition:

(A1) x (x + 0 = x)

(A2) x ∀y (x + Sy = S(x + y))

Peano’s Axioms of Multiplication:

22

(M1) x (x × 0 = 0)

(M2) x y (x × Sy = x + (x × y))

Prove: SS0 = S0 × SS0

1. S0 + 0 = S0 (A1, UI: x = S0)

2. ∀y (S0 + Sy = S(S0 + y)) (A2, UI: x = S0)

3. S0 + S0 = S(S0 + 0) (2, UI: y = 0)

4. S0 + S0 = S(S0) (3, A1)

5. S0 + S0 = SS0 (4, D)

6. S0 × 0 = 0 (M1, UI: x = S0)

7. y (S0 × Sy = S0 + (S0 × y)) (M2, UI: x = S0)

8. S0 × S0 = S0 + (S0 × 0) (7, M2, UI: y = 0)

9. S0 × S0 = S0 + 0 (6, 8, SI: S0 × 0 = 0)

10. S0 × S0 = S0 (1, 9, SI: S0 + 0 = S0)

11. S0 × S(S0) = S0 + (S0 × S0) (7, M2, UI: y = S0)

12. S0 × SS0 = S0 + (S0 × S0) (11, D)

13. S0 × SS0 = S0 + S0 (10, 12, SI: S0 × S0 = S0)

14. S0 × SS0 = SS0 (5, 13, SI: S0 + S0 = SS0)

Notice that step 13 here is identical to the previously invalid substitution step 5* in Dickie’s second

proof from above. And of course with the current sequence in hand we could go on to complete

the original proof in terms of ‘S0 × SS0’ by replacing steps 7*-9* from above with 15-17 below:

15. S0 + S0 × SS0 = S(S0 + S0) (14, SI: ‘S0 × SS0’ for ‘SS0’)

16. S0 + S0 × SS0 = S(S0 × SS0) (14, SI: ‘S0 × SS0’ for ‘SS0’)

23

17. S0 + S0 × SS0 = SSS0 (14, SI: ‘S0 × SS0’ for ‘SS0’)

Now, while in my view Dickie’s example fails to establish the rational distance between ‘SS0’ and

‘S0 × SS0’ as intended, it also does not rule out this hypothesis. And indeed I think Dickie is

basically right about this much. For one could argue, more concretely, that the rational distance

between ‘SS0’ and ‘S0 × SS0’ is represented here by the number of logical steps needed to derive

the one from the other (roughly the 'distance' between steps 5 and 14 above).

At any rate, I suspect that Frege had something like this in mind with respect to putative

differences in cognitive value between, say, the identity statement in step 14 above and the trivially

true identity ‘SS0 = SS0’. But more generally I trust that Frege would agree that the real cognitive

value of (the thought expressed by) any true judgment, including identity judgments, is constituted

by the deductive consequences of its logical form (in whatever way logical forms get cashed out).

Moreover, I have attempted to demonstrate here that this is more or less Frege’s position in both

Bg and OSR. Before concluding, I want to just very briefly consider a few reported consequences

of Dickie’s proposal and indicate why, given the results of this Section, they too are either equally

misguided or otherwise non-tendentious.

4. Reported Consequences of Dickie’s Proposal

To begin, Dickie cites a related problem with a widely accepted criterion for determining

whether two expressions differ in sense, which she formulates as follows:17

17 While Dickie does not indicate as much, this particular formulation appears to derive from Gareth Evans in his

Varieties of Reference (Evans, 1982: 18), yet is traceable to a passage from On Function and Concept (Frege, 1891:

138) where Frege himself writes: “If we say 'the Evening Star is a planet with a shorter period of revolution than the

Earth,' the thought we express is other than in the sentence 'the Morning Star is a planet with a shorter period of

revolution than the Earth'; for somebody who does know that the Morning Star is the Evening Star might regard one

as true and the other as false.”

24

THE INTUITIVE CRITERION OF DIFFERENCE FOR SENSES—‘a’ and ‘b’ differ in sense

if and only if it is possible for a subject who understands ‘a’, ‘b’, and ‘F’ to affirm ‘Fa’ and deny

‘Fb’ without loss of rational coherence.

The problem with this criterion, according to Dickie, is that it’s vulnerable to counterexample.

Here she asks us to imagine reading up on philosophical, social, and political activity in early

Twentieth Century England. One repeatedly finds reference to someone called ‘Russell’ in some

contexts and in others someone known only as ‘Bertie’, yet without realizing that ‘Russell’ and

‘Bertie’ corefer. In consequence, one ends up forming contradictory beliefs about the actual

referent of these names, and thus potentially drawing false conclusions from them (or perhaps

failing to draw certain true conclusions).

In such circumstances, Dickie contends that one can understand ‘Bertie’ and ‘Russell’

without associating these names with particular Fregean senses ‘that endow them with distinctive

inferential properties’. In general, she notes that ‘there are good arguments for the conclusion that

ordinary proper names are not normally associated with any distinctive way of identifying its

bearer, except as the bearer of the name’.18 By contrast, Dickie allows that definite descriptions do

have senses that determine their inferential properties, which thereby explains the rational distance

between coreferential definite descriptions such as those abbreviated by ‘SS0’ and ‘S0 × SS0’, and

hence the rational informativeness of identity statements such as (14) from above.

Now if ordinary names lack senses, or anyhow that their senses are inferentially inert, then

any contradictory beliefs formed about their referents, and any false inferences drawn from them,

cannot be attributed to a difference in sense. Rather, according to Dickie the inferential properties

of ordinary proper names ‘are determined just by their bearers and their status as labels for their

18 The “good arguments” referred to here derive from the likes of Scott Soames and Saul Kripke.

25

bearers’. In other words, the beliefs we form using ordinary names, and interferences we draw

from them, are metalinguistic in nature. And so understood, Dickie concludes that all cases of

informative identity statements involving ordinary proper names are best explained in terms of

evolutionary informativeness, which is to say a kind of informativeness that is not generated by

differences in sense.

In short, what such putative counterexamples show, according to Dickie, is that the intuitive

criterion of difference for senses above is in need of revision. Specifically, this criterion is

inadequate by her lights because ‘it ignores the distinction between evolutionary and rational

informativeness’. What is needed, in particular, is a criterion ‘which aligns difference in sense with

rational distance’ along the following lines:

THE REVISED CRITERION OF DIFFERENCE FOR SENSES—‘a’ and ‘b’ differ in sense if

and only if, for some γ1, …, γn, where γ1, …, γn is a chain of inference which contains

occurrences of ‘a’ but not ‘b’, replacing all occurrences of ‘a’ with occurrences of ‘b’ transforms

γ1, …, γn from a chain of inference which is logically self-evident to one which is not, or from

a chain of inference which is not logically self-evident to one which is.

More plainly, by Dickie's criterion ‘expressions differ in sense if and only if substituting one

expression for another in a sentence changes the location of the thought it expresses’.

In brief reply to this proposal, there is again no avoiding the fact that ‘a = a’ and ‘a = b’ have

different inferential properties, even when ‘a’ and ‘b’ are ordinary proper names. Of course if, as

Dickie suggests, ordinary proper names lack inferentially relevant senses then the criterion above

simply does not apply. But then as Dickie also suggests this leaves us with only a metalinguistic

explanation of the logical difference between ‘a = a’ and ‘a = b’ when 'a' and 'b' are coreferential

names, yet which as previously noted Frege himself summarily rejects in OSR. Moreover, Frege

quite clearly believed that ordinary proper names do have senses, and that coreferential names

26

typically differ in sense. And so again positioned as an interpretive project Dickie’s proposal here

seemingly misrepresents Frege’s own view on the matter.19 But let’s bracket this issue. For if

Dickie instead intended this particular proposal to be a plausible contemporary reconstruction of

Frege' stated doctrines, then we can debate the relative merit of her proposal on these terms.

Relatedly, Dickie argues that her revised criterion offers a better account of the special

significance that Frege assigns to definitions. I haven’t the space to review her argument in full

detail. But Dickie submits (again on Frege’s behalf) that the informativeness of a correct definition

is better characterized under her revised criterions of difference for senses in that:

An expression’s sense determines its place in the pattern of deeper inferential connections: this

is Frege’s ‘natural order’ of logically self-evident inferential relations between statements. And

all right deductive reasoning owes its justification to this pattern of logically self-evident

relations. So the most fundamental justification for the use of an expression is justification in

terms of its sense. But not all justification is fundamental justification. The view of the

explanatory role of the notion of sense suggested in §2 enables us to say that before recognizing

the correctness of the definition you are operating with expressions which you do understand

but for whose use you have only non-fundamental justification. Before coming to accept the

definition you are able to make some justified moves involving the expressions. These moves

are ‘justified’ in that they do not lead you into contradiction and you may have some inchoate

grasp of why they are the right moves to make. But before recognising the definition’s

correctness you do not have fundamental justifications for the moves you make using the defined

term. In looking for the definition you are looking for the fundamental justification which lies

beneath your non-fundamental justification: the position in the network of logically self-evident

19 As mentioned in note 17 above, for example, Frege himself endorses a version of what Dickie calls the “intuitive

criterion of difference for senses.”

27

inferential relations which will explain why the non-fundamental inferences you have been

engaged in so far work.

The relevant claim seems to be this: If one’s grasp of a definition only provides one with a non-

fundamental (i.e., metalinguistic) justification for drawing certain inferences, then such statements

are merely evolutionarily informative. On the other hand, to be fundamentally justified in accepting

the correctness of a definition is to have a clear grasp of both its sense and its logical consequences,

which according to Dickie’s criterion is to recognize the position of its defiens (its defining phrase)

within ‘the network of logically self-evident inferential relations’, and hence its inferential

relations to other thoughts. It is in this latter way, according to Dickie (on Frege’s behalf), that a

correct definition can be rationally informative.

Now, as a rational reconstruction of Frege I have no qualms, in principle, with Dickie’s

revised criterion of difference for senses. And indeed it shows a certain affinity with Frege’s Bg

criterion for the individuation of judgeable/conceptual content as sketched in Section 3. As an

interpretive project, however, Dickie’s account again fails to faithfully track Frege’s own view on

the topic. For I have argued that for Frege of both Bg and OSR the logical significance of any true

proposition (including correct definitions) is determined by the deductive consequences of its

logical form. And so if this is all that Dickie’s intends to claim with respect to definitions, then I

take her proposal to be largely non-controversial among most Frege scholars.

The third reported consequence of Dickie’s proposal is that it helps explain the difference in

role between Frege’s notion of modes of determination in Bg (MODs here) and his modes of

presentation in OSR (MOPs here)—although Dickie herself occasionally uses these terms

interchangeably. Given space limitations, let me again just very briefly summarize. Dickie seems

to think that in Bg Fregean MODs were posited to help explain the question of evolutionary

28

informativeness— what account of understanding a name explains the fact that a subject may,

without rational inconsistency, understand two co-referential names without knowing that they co-

refer, and may then acquire knowledge that the names are in fact names for the same thing?’ In

OSR, by contrast, MOPs are needed to explain rational informativness. In Dickie’s words:

In the Begriffsschrift the patterns of rational relations laid down by proofs are made up only of

expressions and their ordinary contents (their referents) and Frege appeals to modes of presentation

to explain why a specific kind of pattern will always arise. By the time of ‘On Sense and Reference’

he thinks that modes of presentation are the stuff from which the pattern itself is made.

I again question Dickie’s proposal here, or whether I even understand it. For if what I suggested

in Section 3 holds, then judgments of identity of content in Bg are themselves justified by the fact

that different names are associated with distinct MODs of the same object. Notice, however, that

to judge that two names are associated with different MODs of the same object presupposes

knowledge (or anyhow belief) that those names corefer. 20

The relevant point is this: Statements of identity of content for Frege do not merely express

the fact that two names corefer but rather that if they do then they are everywhere substitutable,

salva veritate (and hence salva validate). In terms of comprehension, we can say that to understand

an identity statement under Frege’s Bg account requires recognizing not only that the names ‘a’

and ‘b’ corefer (a judgment which again must be independently justified) but also the logical

implications of this fact; namely, that one is thereby justified in drawing substitution inferences of

the form ‘Fa Fb’. Notice further that this account is perfectly compatible with the Millian

20 It will be noticed that one cannot understand an identity statement without understanding the meaning of the

identity symbol. And if, as Frege suggests in Bg, to understand is in part to understand that two names are related by

their MODs, then one must know that for an identity statement to be true it must be the case that the names flanking

the identity symbol are associated with MODs of the same object. This is a judgment one must be prepared to accept

in order to be justified in accepting the identity judgment.

29

assumption that understanding a name is to understand nothing more than what that name refers

to. For when Frege suggests that in the context of a statement of identity of content names ‘at once

stand for themselves’, this could very well be (and indeed appears to be) a function of the semantics

of the identity symbol itself as opposed to the meanings of the names involved. But my claim, at

bottom, is that in Bg Frege is not expressly concerned to explain what Dickie characterizes as the

question of evolutionary informativeness because he is concerned with more than what explains

how subjects come to acquire the knowledge that two names corefer.

That said, Dickie is quite right to point out that by the time of OSR Frege thinks that MOPs

are ‘the stuff’ from which valid patterns of deductive reasoning are made. But to repeat an earlier

point, the chief difference between MODs and MOPs, as I see it, is that in Bg Frege has not yet

recognized the theoretical advantage of integrating MODs directly into the content of a judgment

in terms of the thought that it expresses and, moreover, that the thought expressed is not only

object-directed but also has internal logical structure that determines its deductive consequences.

5. Conclusion

In conclusion, I again find no firm indication in Bg that Frege is here expressly concerned

with what Dickie calls the question of evolutionary informativeness. Rather, as I read Bg Frege’s

chief concern there was with how to formally justify potentially knowledge-advancing substitution

inferences of the form ‘Fa Fb’. And given limited theoretical resources, his best/only option

was to define the identity relation in a way that encapsulates Leibniz’s principle of Substitution in

the meaning of the identity symbol. That is, Frege is concerned less with what one stands to learn

from recognizing the truth of particular instances of ‘a = b’ (or ‘a ≡ b‘) as with what else can be

validly deduced from them. For this and other reasons I again conclude, more generally, that for

30

Frege of both Bg and OSR the ‘real’ cognitive value any true thought/proposition/judgment,

including both identity thoughts and so called “fruitful” definitions, is constituted by the deductive

consequences of its logical form.

References

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