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Another ancestor for the notion of function in this context, besides the
biological one, is then the notion of a mathematical function. The ˜˜func-
tion™™ we are talking about here would map given representations “
intuitions “ on to combinations of concepts in specific judgments.22
The ˜˜action™™ mentioned in the citation given above should not be
understood as a temporally determined psychological event.23 What
Kant is describing are universal modes of ordering our representations,
whatever the empirically determined processes by way of which those
orderings occur. They consist in subsuming individuals under concepts,
and subordinating lower (less general) concepts under higher (more
general) concepts. These subsumptions and subordinations are them-
selves structured in determinate ways, and each specific way in which
they are structured constitutes a specification of the ˜˜function™™ defined
above. Interestingly, introducing the term ˜˜function™™ in section one of

22
For a fascinating historical survey of the term ˜˜function,™™ its twofold meaning (biological
and mathematical) for Leibniz, for Kant™s immediate predecessors, and finally for Kant
himself, see Peter Schulthess, Relation und Funktion. Eine systematische und entwicklungs-
geschichtliche Untersuchung zur theoretischen Philosophie Kants (Berlin: De Gruyter, 1981),
pp. 217“47.
23
Michael Wolff maintains that according to Kant, the functions are not temporal, but the
actions (Handlungen) are (see Vollstandigkeit, p. 22). I do not think that is correct. To say
¨
that the actions by way of which representations are unified are temporal would be to say
that they are events in time. But surely this is not what Kant means. When he talks of
actions of the understanding what he means to point out is that the unity of representa-
tions is not given with them but depends on the thinking subject™s spontaneity. What
particular events and states of affairs in time might be the empirical manifestations of that
spontaneity are not questions he is concerned with. I would add that the actions in
question are no more noumenal than they are phenomenal: the concept ˜˜action™™ here
does not describe a property or relation of things, but only the status we can grant to the
unity of our representations: the latter is not ˜˜given™™ but ˜˜made™™ or it is the contribution of
the representing subject to the structuring of the contents of her representations.
TRANSCENDENTAL ANALYTIC
94

the Leitfaden chapter to describe the logical employment of the under-
standing is already making space for what will be the core argument of
the metaphysical deduction of the categories:
The same function, that gives unity to different representations in a
judgment, also gives unity to the mere synthesis of different representa-
tions in an intuition, which, expressed universally, is called the pure
concept of the understanding. (A79/B104“5)

I will return to this point in a moment
The ˜˜function™™ in question is from the outset characterized as a func-
tion of judging. This is because we can make no other use of concepts
than subsuming individuals under them, or subordinating lower con-
cepts under higher concepts, namely forming (thinking) judgments.
This being so, the ˜˜unity of the action™™ or function by way of which we
acquire concepts results in judgments that have a determinate form
(a determinate way of combining the concepts they unite).
There is thus an exact correspondence between the functions (˜˜unity
of the action of ordering different representations™™) the understanding
exercises in judging, and the forms of the judgments that result from the
functions. Unlike the functions, the forms are manifest in the linguistic
expression of the judgments.24
In section one of the ˜˜Leading Thread,™™ Kant makes use of two
examples of actual judgments to further elucidate the function of judg-
ing. The first is ˜˜All bodies are divisible.™™ He insists that in this example,
the concept of ˜˜divisible™™ is related to the concept of ˜˜body™™ (or the latter
is subordinated to the former) and by way of this relation, the concept
˜˜divisible™™ is related to all objects thought under the concept ˜˜body™™ (or
all objects thought under the concept ˜˜body™™ are subsumed under the
concept ˜˜divisible™™). A similar point is made again later in the paragraph,

24
Both Michael Wolff and Reinhart Brandt have drawn attention to the fact that for Kant,
there is no thought without language (see Wolff, Vollstandigkeit, pp. 23“4; Brandt,
¨
Urteilstafel, pp. 42, 110. In the Jasche Logic, Kant opposes the distinction that is usual in
¨
logic textbooks of his time, between judgments and propositions, according to which
judgments are mere thoughts whereas propositions are thoughts expressed in language.
Such a distinction is wrong, he says, for without words ˜˜one simply could not judge at all™™
(AAix, p. 109). Instead he distinguishes judgment and proposition as problematic versus
assertoric judgment (ibid.). But in fact, with a few exceptions Kant uses the term ˜˜judg-
ment™™ to refer to all three kinds of modally qualified judgments (problematic, assertoric,
apodeictic). Note also that in his usage, ˜˜judgment™™ refers on the one hand to the act of
judging, on the other hand to the content of the act (what we would call the proposition).
This is consistent with the fact that the function of judging finds expression in a form of
judgment (inseparably belonging to thought and language).
METAPHYSICAL DEDUCTION OF THE CATEGORIES 95

when Kant explains that the concept ˜˜body™™ means something, for
instance ˜˜metal,™™ which thus can be known by way of the concept
˜˜body.™™ In other words, in saying ˜˜Metal is a body™™ I express some
knowledge about what it is to be a metal, and thus also a knowledge
about everything that falls under the concept ˜˜metal.™™ The two examples
jointly show that whatever position a concept occupies in a judgment
(the position of subject or the position of predicate, in a judgment of the
general form ˜˜S is P™™), in its use in judging a concept is always, ultimately,
a predicate of individual objects falling under the subject-concept of the
judgment. This in turn makes every judgment the major premise of an
implicit syllogistic inference whose conclusion asserts the subsumption,
under the predicate-concept, of some object falling under the subject-
concept (e.g. the judgment ˜˜all bodies are divisible™™ is the implicit pre-
mise of a syllogistic inference such as: ˜˜all bodies are divisible; this X is a
body; so, this X is divisible.™™ Or again: ˜˜All bodies are divisible; metal is a
body; so, metal is divisible; now, this is metal; so, this is a body; so, this is
divisible.™™ And so on). If it is true to say that we make use of concepts only
in judgments, it is equally true to say that the function of syllogistic
inference is already present in any judgment by virtue of its form. For
asserting a predicate of a subject is also asserting it of every object falling
under the subject-concept.
This is why, as Kant maintains in what is undoubtedly the decisive
thesis of this section, and perhaps of the whole Leitfaden chapter:
We can, however, trace all acts of the understanding back to judgments,
so that the understanding in general can be represented as a capacity to
judge [ein Vermogen zu urteilen]. (A69/B94)
¨

By ˜˜understanding™™ he means here the intellectual capacity as a whole,
what he has described as spontaneity as opposed to the receptivity or
passivity of sensibility. In agreement with a quite standard presentation
of the structure of intellect in early modern logic textbooks, Kant divides
the understanding into the capacity to form concepts (or understanding
in the narrow sense), the capacity to subsume objects under concepts
and subordinate lower concepts to higher concepts (the power of judg-
ment, Urteilskraft) and the capacity to form inferences (reason, Vernunft).
He is now telling us that all of these come down to one capacity, the
capacity to judge. The latter is not the same as the power of judgment
(Urteilskraft). One way to present the relation between the two would be
to say that the Urteilskraft is an actualization of the Vermogen zu urteilen.
¨
But for that matter, so are the two other components of understanding.
TRANSCENDENTAL ANALYTIC
96

So the Vermogen zu urteilen is that structured, spontaneous, self-regulating
¨
capacity characteristic of human minds, that makes them capable of
making use of concepts in judgments, of deriving judgments from
other judgments in syllogistic inferences, and of systematically unifying
all of these judgments and inferences in one system of thought.25
This explains why Kant concludes section one with this sentence: ˜˜The
functions of the understanding can therefore all be found if we can
completely present the functions of unity in judgments™™ (A69/B94). If
the understanding as a whole is nothing but a Vermogen zu urteilen, then
¨
identifying the totality of functions (˜˜unities of the act™™) of the under-
standing amounts to nothing more and nothing less than identifying the
totality of functions present in judging, which in turn are manifest by
way of linguistically explicit forms of judgments. Kant adds: ˜˜That this
can easily be accomplished will be shown in the next section.™™ The ˜˜next
section™™ is the section that expounds (as its title indicates) ˜˜the logical
function of understanding in judgments,™™ by laying out a table of logical
forms of judgments.
But of course, even if we grant Kant that he has justified his statement
that ˜˜the understanding as a whole is a capacity to judge,™™ this by itself
does not suffice to justify the table he presents. How is the table itself
justified?
Kant™s explanation of the function of judging decisively illuminates
the table he then goes on to set up. First, if the canonical form of
judgment is a subordination of concepts (as in the two examples ana-
lyzed above) then this subordination can be such that either all or part of
the extension of the subject-concept is included in the extension of the
predicate-concept: this gives us the quantity of judgments, specified as
universal or particular. Moreover, the extension of the subject can be
included in or excluded from the extension of the predicate-concept.
This gives us the title of quality, specified as affirmative or negative
judgment. The combination of these two titles and their specifications
provides the classical Aristotelian ˜˜square of opposites™™: universal affir-
mative, universal negative, particular affirmative, particular negative
judgments.
Within each of these first two titles, however, Kant adds a third
specification, which does not belong in the Aristotelian square of

25
Above I have translated Vermogen zu urteilen as capacity to judge. Guyer and Wood have
¨
translated it as faculty of judging. On this difference, see ch. 1, n. 3, p. 18. See also KCJ,
pp. 7“8. On judgments and inferences, see ibid., pp. 90“3.
METAPHYSICAL DEDUCTION OF THE CATEGORIES 97

opposites: singular judgment under the title of quantity, ˜˜infinite™™ judg-
ment under the title of quality. In both cases he explains that these
additions would not belong in a ˜˜general pure logic™™ strictly speaking.
For as far as the forms of judgment relevant to forms of syllogistic
inference are concerned, a singular judgment can be treated as a uni-
versal judgment, where the totality of the extension of the subject-
concept is included in the extension of the predicate-concept. Similarly,
an infinite judgment (in Kant™s sense: a judgment in which the predicate
is prefixed by a negation) is from the logical point of view an affirmative
judgment (there is no negation appended to the copula). But those two
forms do belong in a table geared toward laying out the ways in which
our understanding comes up with knowledge of objects. In this context
there is all the difference in the world between a judgment by way of
which we assert knowledge of just one thing (singular judgment) and a
judgment by way of which we assert knowledge of a complete set of
things (universal judgment). Similarly, there is all the difference in the
world between including the extension of a subject-concept in that of a
determinate predicate-concept, and locating the extension of a subject-
concept in the indeterminate sphere which is outside the limited sphere
of a given predicate (see A72“3/B97“8, where Kant distinguishes the
infinite judgments from both the affirmative and the negative judg-
ments). Now it is significant that Kant should thus add, for the benefit
of his transcendental inquiry, the two forms of singular and ˜˜infinite™™
judgment to the forms making up the classical square of opposites. It
shows that if the logical forms serve as a ˜˜leading thread™™ for the table of
categories, conversely the goal of coming up with a table of categories
determines the shape of the table of logical forms.
This is even more apparent, I suggest, if we consider the third title,
that of relation. It should first be noted that this title does not exist in any
of the lists of judgments presented in the logic textbooks Kant was
familiar with.26 On the other hand, the three kinds of relation in judg-
ments (relation between a predicate and a subject in a categorical

26
Early modern logicians typically distinguish between simple and composite propositions,
and their list of composite propositions includes many more besides Kant™s hypothetical
and disjunctive judgments. More importantly, the distinction between ˜˜simple™™ and
˜˜composite™™ propositions puts Kant™s categorical judgment on one side, and Kant™s
hypothetical and disjunctive judgments on the other side of the divide. Only Kant includes
categorical, hypothetical, disjunctive judgments under one and the same title, that of
relation. For more details about early modern lists of propositions see KCJ, p. 98, n. 44.
Note that Kant mostly uses the term ˜˜judgment™™ to refer to the content of the act
of judging (an act which is also called ˜˜judgment™™) but he sometimes insists that when
TRANSCENDENTAL ANALYTIC
98

judgment, relation between a consequent and an antecedent in a
hypothetical judgment, relation between the mutually exclusive specifi-
cations of a concept and that concept in a disjunctive judgment) deter-
mine the three main kinds of inferences, from a categorical, a
hypothetical, or a disjunctive major premise. This is in keeping with
what emerged as the most important thesis of section one: the under-
standing as a whole was characterized as a Vermogen zu urteilen because in
¨
the function of judging as such were contained the other two functions of
the understanding: acquiring and using concepts, and forming infer-
ences. This being so, it is natural to include in a table of logical forms of
judgment meant to expound the features of the function of judging the
three forms of relation that govern the three main forms of syllogistic
inference.
Still, as many commentators have noted, it is somewhat surprising to
see Kant include as equally representative of forms of judgment that
govern forms of inference, the categorical form that is the almost exclu-
sive concern of Aristotelian syllogistic, and the hypothetical and disjunc-
tive forms that find prominence only with the Stoics. Does this not
contradict Kant™s (admittedly shocking) statement that logic ˜˜has been
unable to make a single step forward™™ since Aristotle (Bviii)?
I think there are two answers to this question. The first is historical: the
forms of hypothetical and disjunctive inference (modus ponens and tollens,
modus ponendo tollens and tollendo ponens) are actually briefly mentioned
by Aristotle, developed by his followers (especially Galen and Alexander
of Aphrodisias), and present in the Aristotelian tradition as Kant knows
it.27 The second answer is systematic: it takes us back to the remark I
made earlier. Kant™s table is not just a table of logical forms. It is a table of
logical forms motivated by the initial analysis of the function of judging
and by the goal of laying out which aspects of the ˜˜unity of the act™™ (the
function) are relevant to our eventually coming up with knowledge of
objects. In this regard it is certainly striking that Kant should have
developed the view that in the ˜˜mediate knowledge of an object™™ that is
judgment, we not only predicate a concept of another concept and thus
of all objects falling under the latter (categorical judgment), but we also
predicate a concept of another concept and thus of all objects falling
under the latter, under the added condition that some other predication
be satisfied (hypothetical judgment); and we think both categorical and
the judgment is assertoric, it should be called a proposition. See Logic, xx30“3, AAix,
pp. 109, 604“5.
27
See Wolff, Vollstandigkeit, p. 232.
¨
METAPHYSICAL DEDUCTION OF THE CATEGORIES 99

hypothetical predications in the context of a unified and, as much as
possible, specified conceptual space (expressed in a disjunctive judg-
ment). These added conditions for predication (and thus for knowing
objects under concepts) find their full import when related to the corres-
ponding categories, as we shall see in a moment.
The fourth title in the table is that of modality. Kant explains that this
title ˜˜contributes nothing to the content of the judgment (for besides
quantity, quality and relation there is nothing more that constitutes the
content of a judgment), but rather concerns only the value of the copula
in relation to thinking in general™™ (A74/B100). The formulation is some-
what surprising, since after all none of the other titles was supposed to
have anything to do with content either: they were supposed merely to
characterize the form of judgments, or the ways in which concepts were
combined in judgments, whatever the contents of these concepts. But
what Kant probably means here is that modality does not characterize
anything further even with respect to that form. Once the form of a
judgment is completely specified as to its quantity, quality, relation, the
judgment can still be specified as to its modality. But this specification
concerns not the judgment individually, but rather its relation to other
judgments, within the systematic unity of ˜˜thinking in general.™™ Thus a
judgment is problematic if it belongs, as antecedent or consequent, in a
hypothetical judgment; or if it expresses one of the divisions of a concept
in a disjunctive judgment. It is assertoric if it functions as the minor
premise in a hypothetical or disjunctive inference. It is apodeictic (but
only conditionally so) as the conclusion of a hypothetical or disjunctive
inference. Such a characterization of modality is strikingly anti-
Leibnizian, since for Leibniz the modality of a judgment would have
entirely depended on the content of the judgment itself: whether its
predicate is asserted of its subject by virtue of a finite or an infinite
analysis of the latter. Note, therefore, that Kant™s characterization of
modality from the standpoint of ˜˜general pure™™ logic confirms that the
latter is concerned only with the form of thought, not with the particular
content of any judgment or inference.
So the table, in the end, is fairly simple: it is a table of forms of concept
subordination (quantity and quality) where, to the classical distinctions
(universal and particular, affirmative and negative), is added under each
title a form that allows special consideration of individual objects
(singular judgment) and their relation to a conceptual space that is
indefinitely determinable (infinite judgment). And it is a table where
judgments are taken to be possible premises for inferences (relation)
TRANSCENDENTAL ANALYTIC
100

and are taken to derive their modality from their relation to other
judgments or their place in inferences (modality).
Kant™s claim that the table is systematic and complete is not supported
by any explicit argument. Efforts have been made by recent commenta-
tors to extract such an argument from the first section of the Leitfaden
chapter, the most systematic effort being Michael Wolff™s. Even he,
however, recognizes that the full justification of Kant™s table of logical
forms comes only with the transcendental deduction.28 Indeed, in its
details the table can have emerged only from Kant™s painstaking reflec-
tions about the relation between the forms according to which we relate
concepts to other concepts, and thus to objects (forms of judgment), and
the forms according to which we combine manifolds in intuition so that
they fall under these concepts. It is a striking fact that the first mature
version of Kant™s table of logical forms appeared not in his reflections on
logic, but in his reflections on metaphysics. This seems to indicate that
the search for a systematic list of the categories and a justification of their
relation to objects determined the establishment of the table of logical
forms of judgment just as much as the latter served as a leading thread
for the former.29
I now turn to the culminating point of this whole argument: Kant™s
argument for the relation between logical forms of judgment and cate-
gories, and his table of the categories.


Kant™s argument for the table of the categories
I said earlier that the fundamental thesis of section one of the Leitfaden
chapter is ˜˜Understanding as a whole is a capacity to judge.™™ I might now
add that the fundamental thesis of section three (˜˜On the pure concepts of
the understanding or categories™™) is that judgments presuppose synthesis.

28
Ibid., pp. 45“195, esp. p. 181.
29
The Logik Blomberg (1771) and the Logik Philippi (1772) give a presentation of judgments
that remains closer to Meier™s textbook, which Kant used for his lectures on logic, than to
the systematic presentation of the first Critique. See AAxxiv“1, pp. 273“9 and 461“5;
Logic Blomberg, in Lectures on Logic, pp. 220“5. For an occurrence of the two tables in
Lectures on Metaphysics of the late 1770s, see Metaphysik L1, AAxxviii“1, p. 187. But
see also Reflexion 3063 (1776“8), in Reflexionen zur Logik, AAxvi, pp. 636“8. For a more
complete account of the origins of Kant™s table, see Tonelli, ˜˜Die Voraussetzungen zur
Kantischen Urteilstafel in der Logik des 18. Jahrhunderts,™™ in Friedrich Kaulbach and
Joachim Ritter (eds.), Kritik und Metaphysik. Heinz Heimsoeth zum achtzigsten Geburtstag
(Berlin: De Gruyter, 1966). Also Schulthess, Relation und Funktion, pp. 11“12;
Longuenesse, KCJ, p. 77, n. 8; p. 98, n. 44.
METAPHYSICAL DEDUCTION OF THE CATEGORIES 101

In a way, this statement is a truism. After all, ˜˜synthesis™™ means noth-
ing more than ˜˜positing together™™ or ˜˜combination,™™ and it is obvious
that any judgment of the traditional Aristotelian form ˜˜S is P™™ is a
positing together or combination of concepts. Indeed Aristotle defined
it in just this way, and the Aristotelian tradition followed suit all the way
down to Kant, including Port-Royal™s logic of ideas.30 What is new,
however, in Kant™s notion of synthesis, is that it does not mean only or
even primarily a combination of concepts. As far as concepts of objects
given in sensibility are concerned, the combining (synthesis) of those
concepts in judgments can occur only under the condition that a com-
bining of parts and aspects of the objects given in sensibility and poten-
tially thought under concepts also occurs. The rules for these
combinings is what transcendental logic is concerned with.
But why should there be syntheses of parts and aspects of objects
presented to our sensibility? Why should it not be the case that empiri-
cally given objects just do present themselves as spatiotemporal, qualita-
tively determined wholes that have their own presented boundaries?
Kant does not really justify the point in section three of the Leitfaden
chapter. The furthest he goes in that direction is to explain that in order
for analysis of sensible intuitions into concepts to be possible, synthesis of
these same intuitions (or of the ˜˜manifold [of intuition], whether it be
given empirically or a priori™™ [A77/B102]) must have occurred. The
former operation, as we saw from section one of the Leitfaden chapter,
obeys the rules of the logical employment of the understanding. The
latter operation must present the sensible manifold in such a way that it
can be analyzed into concepts susceptible to being bound together in
judgments according to the rules of the logical employment of the
understanding.



30
See Aristotle, De interpretatione, 16a11; Arnauld and Nicole, Art of Thinking, part ii, ch. 3. As
we saw in the previous section, Kant nevertheless gives new meaning to the idea of
judgment as a combination of concepts, since in his view the activity of judging determines
the formation of concepts, so that the unity of judgment is strictly speaking prior to what it
unites, namely concepts. Note also that in the main text I write that ˜˜synthesis™™ means
positing together as well as combination. In saying this I would like to emphasize the fact
that as with all of Kant™s terms pertaining to representation, one should give ˜˜synthesis™™
the sense of the act of synthesizing as much as that of the result of the act. Similarly,
˜˜combination™™ means combining as much as the result thereof. Depending on the context,
it is sometimes helpful to use the term expressly connoting the action of the mind rather
than the term connoting the result or intentional correlate of the action. In any event, both
dimensions are always present for Kant.
TRANSCENDENTAL ANALYTIC
102

Here it will be useful to recall the problem laid out in the letter to Herz
mentioned in the first section of this chapter. Mathematical concepts
present their own objects by directing the synthesis of an a priori (spatial)
manifold according to rules provided by the relevant concept (e.g. a line,
a triangle, a circle). But we cannot do that in metaphysics, because there
the objects of our concepts are not just constructed in pure intuition.

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