DRAFT: What follows is the text of a talk presented at the Third Annual Early Analytic Conference, Purdue
University, April 9-10, 2004 Comments welcome; please do not quote or circulate.
Truth, Assertion, and the Horizontal: Frege on “the essence of logic”
William W. Taschek
The Ohio State University
Taschek.1@osu.edu
Abstract: In the opening to his late essay, “Der Gedanke”, Frege asserts without qualification that the word “true”
“points the way for logic.” But in a short piece from his Nachlass entitled, “My Basic Logical Insights”, Frege
writes that the word “true” makes “an unsuccessful attempt to point to the essence of logic,” asserting instead that
“what really pertains to logic lies not in the word ‘true’ but in the assertoric force with which the sentence is
uttered.” Properly understanding what Frege takes to be at issue here is crucial for understanding his conception of
logic, and in particular its normative status. In this paper, I focus my attention on clarifying the latter claim and
Frege’s motivations for making it. Along the way, I will address some peculiarities about the role of the horizontal
in Frege’s mature Begriffsschrift.
Anyone more than superficially acquainted with the writings of Gottlob Frege will be
familiar with a number of passages in which Frege identifies logic as a science essentially and
distinctively concerned with truth. They range from fairly early in his career to quite late.
Consider first a passage from the earlier of two posthumously published manuscripts that Frege
labeled “Logic”. The editors of his Nachlass date this earlier fragment from between 1879 and
1891. Frege begins—not surprisingly from the author of the recently published Begriffsschrift—
by stating that the task of logic is to establish laws of valid inference (2-3). This claim
immediately leads him to contrast the subject matter of logic with that of psychology, at which
point he writes:
…there is a sharp divide between these disciplines, and it is marked by the word ‘true’.
Psychology is concerned with truth in the way every science is, in that its goal is to
extend the domain of truths; but in the field it investigates it does not study the property
‘true’ as, in its field, physics focuses on the properties ‘heavy’, ‘warm’, etc. This is what
logic does. It would not perhaps be besides the mark to say that the laws of logic are
nothing other than an unfolding of the content of the word ‘true’. Anyone who failed to
grasp the meaning of this word—what marks it off from others—cannot attain to a clear
idea of what the task of logic is. (3)
Consider next a passage from the important Preface to Volume I of the Grudgesetze, which was
published in1893. Here, Frege puts the matter this way:
What is crucial to the treatment of the science of logic is the conception of logical laws,
and this in turn is connected with how the word ‘true’ is understood…. I understand by
logical laws not psychological laws of holding true, but laws of being true [Gesetzes des
Wahrseins].
What follows is, of course, one of Frege’s most extended and challenging critiques of
psychologism.
Four years later (1897), in the partial summary that precedes the principal text of the
second and longer of the two posthumous “Logic” manuscripts, Frege writes:
The word ‘true’ specifies the goal. Logic is concerned with the predicate ‘true’ in a
special way. The word ‘true’ characterizes logic. ([126])
In the corresponding passage from the main text, he fills this out as follows:
When entering upon the study of a science, we need to have some idea, if only a
provisional one, of its nature. We want to have in sight a goal to strive towards; we want
some point to aim at that will guide our steps in the right direction. The word ‘true’ can
be used to indicate the goal for logic just as ‘good’ for ethics and ‘beautiful’ for
aesthetics. Of course all sciences have truth as their goal. But logic is concerned with the
predicate ‘true’ in a quite special way, namely in a way analogous to that in which
physics has to do with the predicates ‘heavy’ and ‘warm’ or chemistry with the predicates
‘acid’ and ‘alkaline’. There is however the difference that … there is no one property by
which their essence [Wesen] is so completely characterized as logic is by the word ‘true’.
…Consequently, we can also say: logic is the science of the most general laws of
truth [of being true (Wahrseins)]. (227-228)
Then, in 1918, in the opening passage of his essay, “Der Gedanke”, Frege offers us the following
familiar remarks:
Just as ‘beautiful’ points the way for aesthetics and ‘good’ for ethics, so do words like
‘true’ for logic. All sciences have truth as their goal; but logic is also concerned with it in
a quite different way: logic has much the same relation to truth as physics has to weight
or heat. To discover truths is the task of all sciences; it falls to logic to discern the laws
of truth [of being true]. (325)
In order to avoid misunderstanding and prevent a blurring of the boundary between
psychology and logic, I assign to logic the task of discovering the laws of truth [being
true], not the laws of taking things to be true or of thinking. It is in the laws of truth that
the meaning of the word ‘true’ is unfolded [entwickelt]. (326)
2
Finally, in a 1919 fragment that the editors of his Nachlass have labeled “Notes for Ludwig
Darmsteadter”, Frege writes:
What is distinctive about my conception of logic is that I begin by giving pride of place
to the content of the word ‘true’, and then immediately go on to introduce a thought as
that to which the question ‘Is it true?’ is in principle applicable.
Given the steady supply of passages like these, it would hardly seem risky to conclude
that for Frege, the essence of logic is in some distinctive way bound up with—characterized or
indicated by—the notion of truth. Logic, it would seem, is concerned with truth just as physics
is concerned with weight or heat. So it would seem that just as the laws of physics are laws of
weight and heat, and whatever other properties belong to the distinctive subject matter of
physics, so the laws of logic just are the laws of truth or of being true. Indeed, it is noteworthy
that in nearly every passage in which Frege makes a point of characterizing the aim or essence of
logic by appeal to the notion of truth, he is also concerned to distinguish the subject matter of
logic from anything that could possibly be the subject matter of psychology. So, we find Frege
repeatedly insisting that it is precisely by its failure to do justice to the central and distinctive role
that the notion of truth plays in logic that psychologism with respect to logic is exposed as
bankrupt. As Frege points out in “Der Gedanke”, for example, when logical laws, the laws of
being true, are assimilated to psychological laws of thinking, of taking to be true, “truth has not
been given its proper place” (325).
But as safe as the claim might at first appear that, for Frege, the essence of logic is in
some distinctive way bound up with the notion truth might, Frege’s Nachlass contains an
important set of remarks that casts serious doubt on whether, for Frege, the notion of truth does
or, indeed, can play such a central role in characterizing the essence of logic. The remarks I have
3
in mind occur in a manuscript to which the editors of the Nachlass assign the date 1915. Frege
himself entitled it, “My Basic Logical Insights”. And beneath this title he wrote, “The following
may be of some use as a key to the understanding of my results.” Over ten years had passed
since Frege had received the letter from Russell pointing out the contradiction in the logical
framework of the Grudgesetze. And it is overwhelmingly likely that by this time Frege was
willing to concede the failure of his logicist program. It was doubtless a period during which
Frege felt obliged to take stock of his life’s work and to articulate what he took to be the lasting
achievements of that work. Indeed, we find in his Nachlass from this same period other short
manuscripts with such titles as “A Brief Survey of My Logical Doctrines” and “What May I
Regard as the Result of My Work”. “My Basic Logical Insights” clearly belongs to this project
of reassessment.
The manuscript is striking as a whole; and we shall eventually have a chance to consider
virtually every line in it. But for the time being I want to draw your attention to a central
passage. After arguing that attaching the words “is true” to a sentential clause as predicate adds
nothing to the thought expressed by the original sentence, Frege writes:
This may lead us to think that the word ‘true’ has no sense at all. But in this case a
sentence in which ‘true’ occurred as a predicate would have no sense either. All one can
say is: the word ‘true’ has a sense that contributes nothing to the sense of the whole
sentence in which it occurs as predicate.
But it is precisely for this reason that this word seems fitted to indicate the
essence of logic. Because of the particular sense that it carried, any other adjective would
be less suitable for this purpose. So the word ‘true’ seems to make the impossible
possible: namely, to allow what corresponds to the assertoric force to assume the form of
a contribution to the thought. And although this attempt miscarries, or rather through the
very fact that it miscarries, it indicates what is distinctive about logic. And this,
accordingly, seems something essentially different from what is distinctive of aesthetics
and ethics. For there is no doubt that the word ‘beautiful’ actually does indicate the
essence of aesthetics, as does ‘good’ that of ethics, whereas ‘true’ only makes an abortive
attempt to indicate the essence of logic, since what logic is really concerned with is not
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contained in the word ‘true’ at all but in the assertoric force with which a sentence is
uttered. (322-24)
If we are to take these remarks seriously, it would seem—contrary to what we were led to
believe by the various passages noted earlier—that the word ‘true’ does not after all indicate the
essence of logic. Logic is not, in its essence concerned with truth, but with the assertoric force
with which a sentence is uttered.
If these remarks do not explicitly contradict Frege’s earlier claims, then, at the very least,
they raise serious interpretive difficulties about how we are to reconcile them with the various
truth oriented pronouncement concerning the goal, the task, the aim, and essence of logic that
Frege makes both before and after these remarks were written. How, for example, are we to
understand Frege’s repeated assertion that the laws of logic are the laws of truth [being true]—an
assertion he repeats as late as in “Der Gedanke”, which he published at least three years after
these notes were written? And what sense are we to make of his repeated analogy between the
way in which logic is distinctively concerned with truth and physics is distinctively concerned
with such properties as heat and weight? Finally, if truth does not in fact indicate the essence of
logic, then how are we now to assess his charge that psychologism’s failure lies precisely in its
failing to respect the central and distinctive role that truth plays in logic?
Of course, we always have the option of dismissing Frege’s remarks in “My Basic
Logical Insights” as an aberration. Frege was always struggling against what he viewed as the
logical imperfections of ordinary language to find a way to articulate his views about logic. And
perhaps, in the end, Frege found the strategy he adopted here misguided. After all—as we have
already noted—we don’t find Frege, in the later “Der Gedanke”, insisting that it is not truth but
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assertoric force that best indicates the essence of logic. There, rather, he appears to revert to his
truth oriented way of characterizing that essence. And the same goes for his even later
comments in the “Notes for Ludwig Darmstaedter”.
Nevertheless, I think it would be a serious mistake to dismiss these important remarks.
First of all, as Frege proceeds to say a bit further on in the “My Basic Logical Insights”
manuscript, it is precisely due to the logical imperfection of ordinary language that we seem to
find ourselves obliged to deploy the word ‘true’. While it may be that its use cannot precisely
capture the distinctive concern of logic, it would seem that it is the best tool we currently have to
gesture toward what really does indicate the essence of logic—the assertoric force with which a
sentence is uttered. Frege writes:
How is it then that this word ‘true’, though it seems devoid of content, cannot be
dispensed with? Would it not be possible, at least in laying the foundations of logic, to
avoid this word altogether, when it can only create confusion? That we cannot do so is
due to the imperfection of language. If our language were logically more perfect, we
would perhaps have no further need for logic, or we might read it off from the language.
But we are far from being in such a position. Work in logic just is, to a large extent, a
struggle against the logical defects of language, and yet language remains for us an
indispensable tool. Only after our logical work has been completed shall we possess a
more perfect instrument. (324)
It would seem, then, that Frege felt that he had little choice but to use and to emphasize the
centrality of the word ‘true’ for logic. And so it is no surprise that he does so again in “Der
Gedanke”—even if the remarks in “My Basic Logical Insights” express what Frege took to be a
deep insight about the nature of logic. But why didn’t he there just talk about assertoric force as
he did in “My Basic Logical Insights”? My own surmise is that Frege felt that he had not yet
found a way to say that logic’s principal concern is with assertoric force that he felt would not be
6
misunderstood as introducing an element of psychologism. The rhetoric of truth was in this
connection more compelling.
A second reason for not dismissing Frege’s remarks in “My Basic Logical Insights”, is
that it turns out that this is not the only place in which he suggests that the essence of logic is
most centrally bound up with assertoric force. With considerably less fanfare, he comes very
close to making the same claim in the later (1897) “Logic” manuscript, where he writes:
If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3
make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, if
I assert that it agrees with reality. Therefore, it is really by using the form of an assertoric
sentence that we assert the truth, and to do this we do not need the word ‘true.’ Indeed,
we can say that even where we use the form of expression ‘it is true that…’ the essential
thing is really the assertoric form of the sentence. (228-29)
“Essential”, that is, so far as logic is concerned. The move here from reflection on the
redundancy property of the truth predicate to a focus on the importance and fundamental status
of assertion clearly anticipates the later discussion.1
Thirdly, anyone acquainted with Frege’s reflections on logic and its axiomatic
formulation will have been struck by his unwavering insistance from the Begriffsschrift on that,
insofar as it is the goal of logic to articulate the laws of valid inference, logic must concern itself
essentially with judged or asserted contents. To be a well-formed Begriffsschriftsatz—and
therefore a potential premise or conclusion in any inference—requires being prefaced with a
1
While it is true that in this passage Frege mentions the assertoric form of the sentence and not the assertoric force
with which it is uttered, and while it is also true that in his later writings Frege emphasized that assertoric form does
not suffice for assertion, he nevertheless often wrote as if the assertoric form of an uttered sentence “contained”
assertoric force unless this was somehow cancelled. Cf. for example “On Sinn and Bedeutung” p. 158. This was
more common earlier than later in his career.
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judgment stroke.2 This suggests that from the very outset of his career Frege took the twin
notions of judgment and assertion to play a central role in his understanding of that with which
logic is essentially concerned. It is also noteworthy that in the Begriffsschrift, and the several
early essays in which he explains and defends the aims and significance of that work, the notion
of truth is barely mentioned.
And finally, despite Frege’s repeated appeal to the notion of truth in characterizing the
essence of logic, it is noteworthy that from 1897 on, but probably as early as 1891, Frege began
to view the predicate “is true” as in various ways anomalous, if not downright problematic. For
example, we find Frege arguing, among other things, (i) that attaching the words “is true” to a
sentential clause as predicate—as, for example, in “It is true that seawater is salty”—adds
nothing to the thought expressed by the sentence—“Seawater is salty”—alone, (ii) that the
relation of truth to a thought is that of a Bedeutung to a sense, (iii) that the word ‘true’ is sui
generis and indefinable, and indeed, (iv) that it is a mistake to think that truth is a property of
sentences or thoughts at all (cf. “Logic in Mathematics”, 234). Each of these claims raises
delicate issues, and the story about how they are all supposed to fit together is quite complicated.
In what follows, I will have the opportunity to discuss some of these claims , but I am afraid,
only in a fairly superficial way. Nevertheless, my present point is simply that, given the
problematic status of the word “true” for Frege, one should be extremely cautious in assessing
the import of the various truth-oriented characterizations of the essence of logic that Frege offers.
2
Of course logic is also obliged to recognize and concern itself with unasserted contents, but this is only because
this is required to do justice to the logical assessments we make of our judgments, assertions and inferences.
8
All in all, then, it seems to me that we have compelling reasons to take the remarks in
“My Basic Logical Insights” very seriously. Indeed, I am inclined to give them a certain pride of
place when it comes to understanding Frege’s distinctive understanding of the nature and status
of logic. What remains to be seen is whether, or to what extent, we can reconcile these remarks
with his other, truth-oriented pronouncements about the essence of logic. My own view, as you
will see, is that while one can go quite a ways toward reconciling them, in the end these prima
facie conflicting pronouncements are symptomatic of a deep tension in Frege’s overall
conception of logic.
To see what I have in mind, we need to remind ourselves of another prima facie
dichotomy in the way that Frege characterizes the nature and status of logic. On the one hand, as
is now well known, Frege views logic as a substantive descriptive science, though, to be sure the
maximally general science. As Frege says in the opening paragraph of “Der Gedanke”,
The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean
prescriptions which ought to be obeyed but with which actual occurrences are not always
in conformity. Laws of nature [on the other hand] are general features of what happens
in nature, and occurrences in nature are always in accordance with them. It is rather in
this sense that I speak of laws of truth. Here of course, it is not a matter of what happens
but of what is. From the laws of truth there follow prescriptions about asserting,
thinking, judging, inferring. (325)
As with any science, the aim of logic is to identify a finite set of basic laws—substantive
generalizations—appropriate to its subject matter, and to derive from these any other laws
applying to its domain. The axioms of Frege’s Begriffsschrift are these basic laws. What
distinguishes logic from other sciences is not, of course, that it has as its aim the establishment of
such general truths—for that is the aim of any science whatsoever. Nor are the laws of logic
distinguished from other laws by containing some specialized or proprietary vocabulary that
9
segregates its subject matter from that of other sciences; for the only (ineliminable) vocabulary
needed to express logical laws is a vocabulary that will be required to express the laws of any
science whatever3. Rather, what distinguishes logic from other sciences is the fact that its laws
are in no significant way restricted with respect to subject matter. The laws of logic are laws that
hold of all reality, all of “what is”. There is no subject matter—there is, so to speak, no truth—
that is not subsumed under the laws of logic; and it is in this sense that they count, for Frege, as
the most general laws of truth or, better, of being true [Wahrseins]. The finer details of and
motivations behind this “universalist” conception of logic have been well described and
documented by others; so I will not pursue this further here. I do, however, want to point out
that Frege’s tendency to characterize the laws of logic as the laws of truth invariably occurs in
contexts in which Frege seems intent on emphasizing his commitment to this substantive,
descriptivist conception of logic, which are, in turn, contexts in which Frege is largely focused
on criticizing psychologism.
But, now, it is also noteworthy that in virtually every context in which Frege lays out this
picture of logic as the maximally universal descriptive science, we find him also insisting on the
fundamentally normative character of logic. The penultimate sentence of the first of the two
passages from “Der Gedanke” quoted immediately above is characteristic: “From the laws of
truth there follow prescriptions about asserting, thinking, judging, inferring.” Here is how Frege
puts it in the second “Logic” manuscript:
Like ethics, logic can also be characterized as a normative science. How must I think in
order to reach the goal, truth? We expect logic to give us the answer to this question, but
we do not demand that it should go into what is particular to each branch of knowledge
and its subject matter. On the contrary, the task we assign logic is only that of saying
what holds with the utmost generality for all thinking, whatever its subject matter. We
3
There is a noteworthy exception here. It is not the case that every science requires names of courses-of-values.
10
must assume the rules for our thinking and for our holding something true are prescribed
by the laws of truth. The former are given along with the latter. (228)
Of course, to claim that the laws of logic articulate “what holds with the utmost generality for all
thinking, whatever its subject matter [my emphasis]” is, as Frege fully realizes, potentially quite
misleading. Unless one keeps squarely in mind that what is at issue here is what holds
prescriptively of all thinking, one is likely to take the laws of logic to be descriptive laws of
thinking and, thereby, fall into the morass of psychologism. “So,” as Frege points out later in
this same text,
If we call [the laws of logic] laws of thought, or, better, laws of judgment, we must not
forget we are concerned here with laws which, like principles of morals or laws of the
state, prescribe how we are to act, and do not, like laws of nature, define the actual course
of events. Thinking, as it actually takes place, is not always in agreement with the laws
of logic any more than people’s actual behavior is always in agreement with the moral
law. (246-47)
Notice the interesting shift in emphasis here from characterizing the laws of logic as prescriptive
laws of thinking to prescriptive laws of judgment. This is noteworthy because it suggests a
connection—a connection I will be emphasizing later—between Frege’s commitment to viewing
logic as essentially prescriptive and his claim in “My Basic Logical Insights” that the essence of
logic is best indicated not by the word “true” but by reflecting on the assertoric force with which
sentences are uttered. Since assertion just is, for Frege, the public manifestation of judgment,
and since logic, understood now in its normative guise, articulates prescriptive laws governing
judgment—and so, presumably, assertion—Frege’s suggestion that the essence of logic is
located in assertoric force can be understood as, in effect, an inchoate acknowledgement that the
essence of logic is to be found in its normative role vis-à-vis judgment and assertion and not, or
at least not exclusively, in its descriptive role. But I am getting ahead of myself here.
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There are two intimately related issues about the normative status of logic that we need to
get clearer about. On the one hand, we need to understand the sense in which Frege thought that
the relevant prescriptions “follow from”, as he sometimes puts it, the laws of logic qua
descriptive generalizations. On the other hand, we need a clearer sense of precisely how Frege
understands the kind of normative force that the laws of logic have over judgment and inference.
At first glance it can look as if Frege thought that there is nothing distinctively
prescriptive about logical laws as such, as opposed, say, to the laws of any other science. For
example, Frege says in a passage from the second “Logic” manuscript, almost immediately
following the last quote, that
We could with equal justice think of laws of geometry and laws of physics as laws of
thought or laws of judgment, namely as prescriptions to which our judgments must
conform in a different domain if they are to remain in agreement with the truth. (247)
This suggests that Frege held that from any law, any general truth, there follows a prescription
that one ought to think in accordance with it. In this case, logical laws are not distinctive in
virtue of their issuing in prescriptions about how one ought to judge, but are distinctive only in
virtue of the maximal generality of their subject matter and, so, only in the scope of their
prescriptive influence. A similar suggestion seems to be implicit in the following important
passage from the Preface to the Grundgesetze:
That the logical laws should be guiding principles for thought in the attainment of truth is
generally admitted at the outset; but it is easily forgotten. The ambiguity of the word
‘law’ is fatal here. In one sense it states what is, in the other it prescribes what should be.
Only in the latter sense can the logical laws be called laws of thought, in laying down
how one should think. Any law that states what is can be conceived as prescribing that
one should think in accordance with it, and is therefore in that sense a law of thought.
This holds for geometrical and physical laws no less than for logical ones. The latter then
only deserve the name ‘law of thought’ with more right if it should be meant by this that
they are the most general laws, which prescribe universally how one should think if one
is to think at all. (202)
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A bit later he adds:
Anyone who has once recognized a law of truth has thereby also recognized a law that
prescribes how judgments should be made, wherever, whenever, and by whomever they
may be.
It would seem, from these passages, that Frege is suggesting that the normative status of
the laws of logic, their normative authority over our thinking—or, better, judging—is
immediately consequent upon their status for us as general truths. In this respect, the normative
authority that the laws of logic have over our thinking is not different in kind from that which
derives from our acceptance of any law, whether in physics, geometry, or psychology. To be
sure, there is a difference in scope, but not a difference in kind. If this is right, then the laws of
logic cannot be distinguished from the laws of physics on the grounds that the former are in
some distinctive sense prescriptive while the latter are not.
Perhaps this is indeed Frege’s view. I am inclined to think, however, that this cannot be
quite right. For as the concluding remarks in the first of the two passages from the Grundgesetze
just quoted reveal, Frege clearly does think that the laws of logic have a proprietary claim to the
title “laws of thought” conceived of normatively. The question, then, is whether Frege believed
that this proprietary status is to be (or even can be) accounted for simply on the basis of what we
might call logic’s maximal descriptive scope. What the concluding remark of that passage says,
recall, is that the laws of logic deserve to be called ‘laws of thought’ (in the normative sense)
with more right than, say, the laws of physics, only if they are taken to “prescribe universally
how one should think if one is to think at all [my emphasis].” The clear implication here is, first,
that Frege did indeed think the laws of logic deserved to be called the laws of thought with more
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right than the laws of physics, but, additionally, that what is distinctive about logical laws, in
virtue of which they earn this right, is that they are, in an important sense, constitutive norms of
thinking as such, where to assume that they are constitutive norms of thinking as such is to
assume that to count as a thinker at all, one must acknowledge the normative claim that logic has
over ones practice of judging, asserting, and inferring.
It is implausible, however, to view any prescriptions issuing from the laws of physics or
geometry as constitutive of thinking in any analogous sense. One’s status as a thinker is in no
way compromised by failing to acknowledge the truth of a law of physics or even geometry. If
we were to confront someone who failed to acknowledge the truth of some law of physics that
we ourselves accepted, there is no temptation—nor is it plausible to suppose Frege would be
tempted—to suggest that “here we have a hitherto unknown type of madness” (203). But in the
Grundgesetze, this is precisely how Frege proposes we should react if we were to confront
anyone who appeared not to acknowledge the categorical normative demand that issued from a
law of logic.
To be sure, if I do acknowledge the truth of a law of physics—or, for that matter, of any
other special science—I do thereby find myself under an obligation to “think in accordance with
it”. But what can this mean except that I am under a logical obligation to have my other
judgments be consistent with this one and, in particular, to be willing to acknowledge as true all
instances of the general claim? This, though, is an essentially logical obligation; it does not in
any sense derive independently of logic from the law of physics itself. I suggest, then, that Frege
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did take the normative status of the laws of logic to be distinct in kind from the normative status
of laws in the specialized sciences—and not merely distinct in scope.
These last reflections show that there is something quite peculiar, if not incoherent, in the
idea that a constitutive prescription about how one ought to judge can be immediately grounded
in our acceptance of any true generalization—independently, that is, of our already
acknowledging the normative demands logic places on our thinking. The problem for Frege is
that this would seem to be just as true for those generalizations that he took to express logical
laws, as it is for the laws of physics. On Frege’s universalist conception of logic, when I accept a
logical law, I acknowledge as true a substantive generalization. To be sure, it is maximally
general with respect to subject matter. But so what? Here, as before, by accepting the law as
true, I find myself under an obligation to “think in accordance with it.” But again, there seems to
be no way to understand this obligation except as grounded in a prior logical obligation to have
all my other judgments be consistent with this one and, in particular, to accept all instances of
this general truth. What, if anything, grounds this prior obligation? We cannot allow that it is
itself grounded in my acceptance of some other substantive logical law on pain of the sort of
regress exposed by Lewis Carroll’s parable of Achilles and the Tortoise.4 Eventually, we must
find ourselves acknowledging a commitment to a logical norm concerning judgments—in effect,
a rule of inference5—where this commitment cannot itself consist in (or be grounded upon) our
accepting as true some further substantive law. If this is right, then the prescriptive force of any
substantive logical law must itself be viewed as grounded in our prior commitment to some
Carroll, Lewis. “What the Tortoise Said to Achilles” Mind 4 (1895) 278-80.
It is noteworthy that Frege’s rules of inference are always couched in terms of assertions. The permissions they
encode are explicitly restricted to expressions prefaced by the judgment stroke.
4
5
15
fundamentally non-descriptive normative logical principles, which themselves are not grounded
in our acceptance of any more fundamental substantive claim.
Is this a problem for Frege? Well, yes and no. On the one hand, Frege was quite clear to
distinguish rules of inference from logical axioms in his formal system. And it appears that he
was also fairly clear about the ineliminability of rules of inference—though, so far as I can tell,
there is no place where he explicitly addresses this issue. To this extent, then, he seems to have
acknowledged and taken into account an ineliminable normative element in logic.
Indeed, on further reflection, I think it is fair to say that for Frege, the essence of logic is
revealed, in the first instance, by reflecting on its constitutive normative status vis-à-vis our
practice of judgment assertion, and inference. This is most readily seen in the context of the
specific critique of psychologism that Frege mounts in the Preface to the Grundgesetze. In
general, Frege’s criticisms of psychologism consist in his pointing out how, in one way or
another, a psychologisitic approach to logic fails to capture something that belongs to the essence
of logic. In the Grundgesetze, he clearly takes himself to have shown that psychologism is
committed to a conception of logical laws that is incapable of doing justice to the constitutive
prescriptive role that the laws of logic play vis-à-vis thinking as such. In particular,
psychologism cannot support a conception of judgment or assertion according to which they are
constitutively subject to normative governance by the laws of logic in the way Frege conceived
them to be. And it is precisely in this connection that the claim he makes about assertoric force
in “My Basic Logical Insights” comes into its own. If the laws of logic just are those laws that
issue in constitutive norms governing judgment and assertion, then if we are to understand their
16
distinctive status, their essence, we need to fully appreciate the aim that is constitutive of our
practice of assertion. And it is just this that is revealed in the assertoric force with which a
sentence is uttered.
To utter a sentence with assertoric force is just to express a judgment. For Frege, the
notion of judgment is fundamental. Throughout his career, Frege consistently explicates the
notion of judgment as the acknowledgment of the truth of a thought. Frege insists, however, that
this explication is not any sort of definition. It would be a mistake—precisely the sort of mistake
that Frege’s insistence that this is not a definition was intended to forestall—if, for example, one
supposed that understanding the nature of judgment required a prior and independent grasp of the
notion of truth or, for that matter, of thought. And in particular, it would be a mistake if we were
to allow this explication to seduce us into understanding truth as if it were a substantive property
of thoughts. For this might in turn lead us to assimilate judgment to predicating truth of a
thought. But Frege repeatedly inveighs against making just this mistake.
That this is a mistake, Frege took to be clearly exposed by reflecting on the “redundancy”
feature of the ordinary truth predicate when applied to thoughts. The following passages from
the 1914 “Logic in Mathematics” manuscript is characteristic. He begins by allowing that,
Of course treating truth as a property of sentences or of thoughts is in accordance with
linguistic usage. If we say ‘The sentence “3>2” is true’ the corresponding thing holds of
the thought. Still the predicate true is quite different from other predicates such as green,
salty, rational, for what we mean by the sentence, ‘The thought that 3>2 is true’ can more
simply be said by the sentence ‘3 is greater than 2’. Thus we do not need the word ‘true’
at all to say this. And we see that really nothing at all is added to the sense by this
predicate. In order to put something forward as true, we do not need a special predicate:
we only need the assertoric force with which the sentence is uttered. [233]
He then adds,
17
If a man says something with assertoric force that he knows to be false, then he is lying.
This is not so with an actor on the stage, when he says something false. He is not lying,
because assertoric force is lacking. And if an actor on the stage says ‘it is true that 3 is
greater than 2’ he is no more making an assertion than if he says ‘3 is greater than 2’.
Whether an assertion is being made, therefore, has nothing at all to do with the word
‘true’; it is solely a matter of the assertoric force with which the sentence is uttered. [234]
The word “true” is neither necessary to express a thought, nor is it sufficient, when attached as a
predicate to a thought or sentence, to effect an assertion. But now it will hardly be surprising to
find that it is this very line of reasoning that leads Frege to his conclusion, in “My Basic Logical
Insights”, that it is not the word “true” but the assertoric force with which a sentence is uttered
that correctly indicates the essence of logic. After offering his standard elucidatory explication
of judgment as the acknowledgement of a thought as true, Frege proceeds as follows:
The word ‘true’ is not an adjective in the ordinary sense. If I attach the word ‘salty’ to
the word ‘seawater’ as predicate, I form a sentence that expresses a thought. To make it
clearer that we have only the expression of a thought, but that nothing is meant to be
asserted, I put the sentence in the dependent form ‘that seawater is salty’. Instead of
doing this I could have spoken as an actor on the stage as part of his role, for we know
that in playing this part an actor only seems to speak with assertoric force. Knowledge of
the sense of the words ‘is salty’ is required for an understanding of the sentence, since it
makes an essential contribution to the thought—in the mere word ‘seawater’ we should
of course not have a sentence at all, nor an expression of a thought. With the word ‘true’
the matter is quite different. If I attach this to the words ‘that seawater is salty’ as a
predicate, I likewise form a sentence that expresses a thought. For the same reason as
before I put this also in the dependent form ‘that it is true that seawater is salty’. The
thought expressed by these words coincides with the sense of the sentence ‘that seawater
is salty’. So the sense of the word ‘true’ is such that it does not make any essential
contribution to the thought. If I assert ‘it is true that seawater is salty’, I assert the same
thing as if I assert ‘seawater is salty’. This enables us to recognize that the assertion is
not to be found in the word ‘true’, but in the assertoric force with which the sentence is
uttered. (322-23)
It is precisely this insight, he goes on to claim, that leads us to see that it is not the word ‘true’
that indicates the essence of logic but the assertoric force with which a sentence is uttered.
18
To be sure—as Frege’s elucidatory explication of judgment reveals—it is fundamental to
our understanding of judgment and assertion that we appreciate that their aim is truth.
Judgments are by their nature subject to assessment as correct or incorrect; and the notion of
correctness constitutive of judgment is inextricably bound up with the notion of truth. But, for
Frege, we have no independent grip on the notion of truth apart from our appreciation of it as the
constitutive aim of—the constitutive norm of correctness for—judgment. And, for Frege, our
appreciation of truth as a norm of correctness for judgment is ultimately bound up with our
appreciation of judgments as constitutively subject to normative governance by the laws of logic.
It is in just this sense, then, that for Frege—as he puts it in “Der Gedanke”—“It is in the laws of
truth that the meaning of the word ‘true’ is unfolded [entwickelt]” (326). Or, as we saw Frege
write already in the earlier of the two “Logic” manuscripts, “It would not perhaps be besides the
mark to say that the laws of logic are nothing other than an unfolding of the content of the word
‘true’” [3].
While it cannot be denied that Frege did indeed acknowledge an ineliminable normative
element in logic (which he sought to accommodate in his formal system through his inference
rules together, of course, with his insistence that logic is concerned, in the first instance only
with judged contents), it also cannot be denied that Frege felt committed to an ineliminable
descriptive element in logic as well. This shows itself in his unwavering insistence that an ideal
formulation of logic will be axiomatic and include, as substantive general axioms, logical laws—
precisely those laws he calls laws of truth.
19
But now given the ways in which Frege has exposed the problematic nature of the
ordinary language predicate “true”, and given his suggestion that a more adequate conception of
the notion of truth involves appreciating its status as a constitutive norm of judgment, how are
we to understand Frege’s repeated characterization of these laws of logic as substantive laws of
truth. In particular, how, if at all, can we rescue Frege’s repeated analogy between the way in
which the laws of physics concern themselves with such fundamental properties as heat and
weight and the way in which the laws of logic concern themselves with truth? The laws of
physics will clearly contain predicates whose sense will oblige us to recognize such fundamental
properties and heat and mass. But in what sense, if any, do the laws of logic contain predicates
that oblige us to recognize a property of truth? If truth were a property of thoughts, we might
expect the laws of logic to be laws concerning when this property holds of, or what follows from
its holding of, various thoughts. But we know already that Frege was keen to deny that the word
‘true’ picked out any such property—and that language is misleading precisely to the extent to
that it seduces us into understanding truth in this way. Moreover, it is plain from Frege’s mature
Begriffsschrift, as presented and deployed in the Grundgesetze, that Frege’s logic at no point
finds itself obliged to deploy a predicate whose extension is the set of all true thoughts or all true
sentences. In this sense, then, it is quite clear that Frege does not deploy a truth predicate in
anything like the modern, broadly semantic, sense. So how does Frege want us to think of the
laws of logic, in their descriptive guise, as substantive laws of truth?
Well, while it certainly is true that Frege’s axiomatic formulation of logic contains no—
nor, in its presentation, is he obliged to appeal to any—truth predicate in the modern sense, is it
right to conclude that his logic is not concerned in a substantive way with truth? Or, to put this
20
another way, is it right to conclude that his logic contains nothing at all worth calling a truth
predicate? What about Frege’s horizontal? From a logical point of view, Frege’s horizontal is,
after all, a predicate—for it is a first-level concept name. Moreover, the only object in its
extension is the True—that is, the object Frege identifies with the Bedeutung of true sentences.
In this quite specific sense, then, the horizontal is a kind of truth predicate: it is a predicate true
of the True and of no other thing. Moreover, the horizontal is an absolutely essential feature of
Frege’s mature Begriffsschrift. The negation stroke, the condition stroke, and Frege’s concavity
can operate only on expressions prefaced by the horizontal. And most importantly of all, in
Frege’s logic it makes no sense to attach the judgment stroke to any expression that is not
prefaced by the horizontal—even in those cases where that expression clearly does express a
thought or otherwise names the True. For Frege, then, the horizontal is a predicate that occurs
essentially in [the content of] every judgment. This being the case, there is a straightforward
sense in which the laws of logic concern its Bedeutung in a way analogous to the way the laws of
physics will concern the Bedeutungen of the heat and weight invoking predicates that occur in
them.
Don’t get me wrong. I am not suggesting that Frege was motivated to introduce the
horizontal into his Begriffsschrift simply to preserve the analogy between the concern that the
laws of logic have with truth and the concern that the laws of physics have with heat and weight.
Frege’s motivations for introducing the horizontal were considerably more complicated than that,
bound up as they clearly were with his motivations for viewing sentences as proper names of
truth values. And in this latter regard, his motivations were surely a fairly complex (and, by my
lights, problematic) admixture of philosophical and systematic concerns, centering both on
21
Frege’s commitment to the function/argument analysis of semantic complexity at the level of
Bedeutung and his fundamental commitment to viewing truth as the constitutive aim of
judgment. Nevertheless I do believe that the role of the horizontal in his Begriffsschrift does
give a fairly clear and perspicuous expression to the sense in which Frege held that the laws of
logic exhibit a substantive, and not merely normative, concern with truth—the True.
Now while I think that reflection on Frege’s use of the horizontal in his Begriffsschrift
does help us to appreciate the sense in which he took logical laws to concern themselves with
truth in a substantive way, I also believe that his commitment to an ineliminable descriptive
element in the essence of logic remains in fundamental tension with his conception of logic as
providing constitutive norms for thinking as such. Moreover, I think that Frege’s understanding
and deployment of the horizontal in his mature logic provides us with an interesting
manifestation of the difficulties I have in mind and have already gestured at. As I already noted,
Frege’s motivations for introducing the horizontal were bound up with his motivations for
assimilating sentences into the same logical type as that of ordinary singular terms—the logical
type to which Frege gave the general label “proper name”. And though I have no space here to
spell out what I take to be the motivations for this feature of his logico-syntactic taxonomy, it
should at least be evident that Frege’s conception of the role of the horizontal in his
Begriffsschrift presupposes the assimilation of sentences to proper names.
This assimilation is, of course, thought by many to be deeply problematic. Michael
Dummett’s characterization of it as “an almost unmitigated disaster” is well known.6 In recent
6
Dummett, Michael. Frege: The Philosophy of Language, (Harvard University Press, Cambridge, Mass.; 1973) pp.
644.
22
years, however, a number of philosophers have made significant efforts to mitigate Dummett’s
criticism—some of whom do an excellent job of showing how, given Frege’s universalist
conception of logic, Frege’s commitment to the function/argument analysis of logical structure
and the understanding of quantificational generality that goes with it, provided him with a deep
internal motivation for this assimilation.7 And I would agree that so long as one focuses on
Frege’s universalist conception of logic, these philosophers offer a compelling response to
Dummett’s critique. Nevertheless, I continue to think that Dummett’s worry has some merit.
For as soon as one turns ones attention to Frege’s commitment to the constitutive normative
status of logic, vis-à-vis thinking as such, the assimilation of sentences to proper names does
indeed prove problematic. Indeed, I would argue that Frege’s universalist conception of logic,
given its commitment to a substantive conception of logical laws, and his normative conception
of logic, according to which the principles of logic introduce constitutive prescriptions on
thinking as such, are not easily reconcilable.
Here, at least, is the beginning of a story about how I think this tension is manifested in
some difficulties connected with Frege’s assimilation of sentences to proper names and his
deployment of the horizontal. Frege’s segregation of symbols in his Begriffsschrift into different
logical types was intended to encode fundamental differences in their inferential role and, so,
logical significance (especially in the context of inferences involving generalization). To assume
that two distinct expression are of the same logical type is to assume that—apart from a possible
difference in their specific Bedeutung, which will be marked by their being distinct symbols of
the same (broad) syntactic type—there is no inferentially relevant difference between them.
See, for example, Tyler Burge’s “Frege on Truth” in Frege Synthesized, ed. by L Haaparanta and J. Hintikka (D.
Reidel, Dortrecht; 1986) 97-154 and Thomas Rickett’s “Quantification, Sentences, and Truth-values”, unpublished
manuscript.
7
23
Thus, one would expect that the well-formedness of a Begriffsschrift expression will be
preserved under substitution of expressions of the same logical type. Now it is certainly the case
that wherever we have a well-formed expression containing a first-level function name, we can
always substitute the horizontal for that first-level functor, and the result will be a well-formed
expression of the same logical type as the expression with which we began. The reverse,
however, is not the case.
Consider the expression type (previously mentioned) that Frege calls a
“Begriffsschriftsatz” in the Grundgesetze. This, as Frege puts it there, is “a sign that consists of
a judgment stroke and a name of a truth-value prefixed by a horizontal” (219). As Frege writes
in “On Function and Concept”, such an expression “does not designate [bezeichnet] anything; it
asserts something” (142). This is a particularly important expression for Frege, for in the system
of the Begriffsschrift, as we have already noted, it is only Begriffsschriftsatze that can serve as
premises and conclusions in inferences. But now if one substitutes for a horizontal that occurs
immediately after the judgment stroke in a Begriffsschriftsatz any other first-level functor, the
result will not be a Begriffsschriftsatz—even if the expression following the judgment stroke that
results from this substitution is a proper name of the True.8 Indeed, the result will not be a wellformed Begriffsschrift expression at all. So not even every proper name of the True, yet alone
every proper name, can be substituted one for the other while preserving the inferential role of
the containing expression—at least not if the containing expression is a Begriffsscriftsatz.
8
Nor can the condition stroke, the negation stroke or the concavity be attached to a proper name, whether or not it is
a proper name of the True, unless that name is prefixed by the horizontal.
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The point is that Frege’s conception of the normative role of logic requires that we
recognize a fundamental logical difference between those contents that are judgeable and those
that are not and, therefore, between those expressions—sentences—that express judgeable
contents and those non-sentential singular terms that do not. It is only the sense of a sentence—a
thought—that can be judged or asserted. And it is judgments, and the inferences we make using
them, that are subject to the normative rule of logic. Since, for Frege, it is only judgments or
assertions that can play the role of premises and conclusions in inferences, then, from the
normative perspective, we are obliged to think of sentences and ordinary singular terms as
expressions of distinct logical types. But it is precisely this logically relevant distinction—which
the normative perspective obliges us to take note of—that Frege’s assimilation of sentences to
proper names obliterates.
Of course, having obliterated the distinction, Frege—with a touch of technical genius—
uses his horizontal to try, in effect, to recapture it. But now, as our reflections of a moment ago
show, this only works if the horizontal itself is, in effect, viewed as having an anomalous logical
status among first-level concept expressions. For as we saw, it is not the case that the result of
substituting any other first-level functor for the horizontal in a well-formed Begriffsschrift
expression will always result in a Begriffsschrift expression with the same inferential role. Once
the demand is made that only expressions prefaced by the horizontal can be judged or asserted—
or for that matter, negated or conditionalized, or quantified into—the horizontal has, in effect,
been assigned a logical role that distinguishes it sharply from all other first-level functors. The
normatively grounded distinction in logical type between those expressions that express thoughts
25
and those singular terms that do not has now been shifted to an unacknowledged difference in
logical type between the horizontal and all other first-level functors.
But insofar as Frege supposed that the introduction of the horizontal enabled him to
recapture the—from the normative perspective—essential distinction between expressions that
express thoughts and those that do not, he was quite mistaken to suppose that he had thereby
effected a successful reconciliation between what he took to be the normative and descriptive
demands of logic. For, as we saw previously, to the extent that it makes any sense to view a
prescription concerning how one ought to judge as issuing from our acceptance of a general
truth, whether a law of physics or a substantive law of logic, it does not make sense to view that
prescription as deriving immediately from that truth but, rather, as ultimately grounded in a prior
logical obligation to judge consistently with our acceptance of it. For Frege, this prior obligation
is just the constitutive obligation everyone has to judge in accordance with the norms of logic, if
he is to judge at all. But acknowledging this places serious pressure on Frege’s substantive
universalism, which is not relieved by his deployment of the horizontal in his logical laws. For if
the only prescriptions that “follow from” our acceptance of a substantive logical axiom are
ultimately grounded in a prior obligation to make sure ones judgments are consistent with
whatever else one acknowledges to be true, then how are we to understand the status of these
axioms or any other laws of logic? In particular, how are we to make sense of the idea that logic
involves an ineliminable descriptive element?
Frege’s acknowledgement of the need for rules of inference over and above his axioms
shows that he appreciates that there is an ineliminable normative element to logic. But the status
of the axioms—his assumption that they are likewise fundamental—is problematic. They are, of
26
course, epistemically fundamental for Frege in the sense that they are not themselves justified by
appeal to other propositions. How, though, from Frege’s point of view, are we to make sense of
our commitment to their truth—or, for that matter, our commitment to any logical truth—except
by saying something like this: I am obliged by the demands of logic to acknowledge them as
true. If this is right, if it would be a violation of the norms of logic to deny the truth of these
general thoughts, then these norms precede that acceptance and do not derive from it. The true
essence of logic would seem then to lie precisely in these norms—for Frege, constitutive norms
of judgment and assertion—and not in any truth or our acknowledgement of any truth. It is in
this sense then, I submit, that “what logic is really concerned with is not contained in the word
‘true’ at all but in the assertoric force with which a sentence is uttered.”
27