**On the provenience and meaning of the concept
“exponent” in Kant’s Critique of pure reason***

*Sobre la proveniencia y el significado del concepto
“exponente” en la Crítica de la razón pura, de Kant*

André Rodrigues Ferreira Perez·

University of São Paulo,
Brazil

**Abstract**

In this text I shall explore the meaning of the
concept “exponent” in the *first Critique *by resorting to its
provenience. Beginning with a brief analysis of the two meanings Kant ascribes
to it the *Critique*, the exponent of a series and the exponent of a rule,
I intend to point out that by means of Kant’s concept of analogy, intimately
linked with proportion, we can find a route into some of the mathematics
textbooks of the 18^{th} century, which shed great light in the matter.
Thereafter, as a transition for returning to the *Critique*, we shall see
how, in the *Duisburgscher Nachlass*, the exponent plays a central role
for Kant as he thinks the emergency and necessity of rules in Philosophy, in
comparison to Mathematics. In this way I
hope to show how the “exponent” is taken up by Kant and made fruitful,
especially for the *Analogies of experience*.

**Keywords**

analogy, exponent, judgment, proportion, rule

**Resumen**

En este texto exploraré el significado del concepto
"exponente" en la *primera Crítica* recurriendo a su proveniencia.
Partiendo de un breve análisis de los dos significados que Kant le atribuye en la
*Crítica*, exponente de un serie y exponente de una regla, pretendo
señalar que, mediante el concepto kantiano de analogía, íntimamente ligado a la
proporción, podemos encontrar una ruta a algunos de los manuales de matemáticas
del siglo XVIII, que lanzan gran luz al este respecto. A partir de esto, como
transición para volver a la Crítica, veremos cómo, en el *Duisburgscher
Nachlass*, el exponente desempeña un papel central para Kant al pensar la
emergencia y la necesidad de las reglas en Filosofía, en comparación con la
Matemática. De esta manera espero mostrar cómo Kant incorpora el “exponente” y
lo hace fecundo, especialmente para las Analogías de la experiencia.

**Palabras clave**

analogía, exponente, juicio, proporción, regla

**Introduction**

The
concept of exponent is mentioned very few times in the *Critique of pure
reason*. This fact, at least according to a quantitative criterion, seems to
attest in favor of its irrelevance. It literally occurs in four passages: two
belonging to *Transcendental Analytic* and two, to *Dialectic*. In
the former case, the first mention occurs when systematically representing the
principles of pure understanding and, the second, in a conclusive excerpt from
the *Third Analogy*; in the latter case, we find it in the section *On
transcendental ideas* and, for the last time, in the first section of the *Antinomies*,
the *System of cosmological ideas*. Located sparsely, in different registers
of the work, this attestation indicates nothing about an alleged unit of signification
or an alleged explanatory capacity of the concept in question; on the contrary,
it could be said that the exponent is a pre-critical residue of a book
empirically composed in different periods, and nothing more.

However,
it is worth noting that three of the four occurrences explicitly share at least
one aspect, namely: the notion of series, in view of the moments of relation;
in particular, regarding the pure concept of causality. Thus, if in the *Analogies*
we have interconnection of phenomena subordinated to certain exponents[1], in the first book of *Dialectic* the exponent makes
possible the progression of the series of mediated inferences in general[2], and, in the *System of cosmological ideas*, it
is attributed to the pure concept of causality the exponent of a series, as a
series of causes (as conditions) to a given effect (as conditioned)[3]. In contrast, the first occurrence attributes the
exponent of a rule in general to the principles of pure understanding, as a
condition under which, given the concrete case through experience, the
particular laws of nature, subject to these principles, would be possible[4].

In
this sense, it is a question of how the relationship between rule and series in
general works (be it causes and effects, reasoning etc.) and, more importantly,
what the exponent means for, or what it can clarify about, this relationship.
The serial aspect of the concept in question would point to its mathematical
provenience and its formulation in judicative terms would be indicative of the
appropriation carried out by Kant. Thus, in order to avoid a genetic path that
would have as a starting point other texts than that of our author (what, therefore,
would make the presentation of the argument somewhat arbitrary) and some notes
by Kant himself in loose sheets or in marginalia of other works (in which case
the bet on the cohesion of the material would be very uncertain), we shall
start our route by considering the judicative element to establish a framework
from which I will carry out a regressive analysis of some mathematical sources,
in order to approach with greater certainty the serial character. I hope to
make clear how, by means of this regressive analysis, the exponent plays a
relevant role in Kant’s thinking, especially in establishing what would become
the *Analogies of experience*.

**I**

As
to the four passages referred to above, the one that offers more elements for
understanding the judicative meaning of the exponent is the one that mobilizes it
to explain the generation of *ratiocinatio polysyllogistica*. In the register
of a strictly logical use of our superior cognitive capacity, a syllogism or
inference of reason is explained as follows: in the major we have a universal
rule; in the minor, the subordination of the condition here expressed to that
of the major; finally, the conclusion expresses the assertion of the predicate
of the universal rule to the subsumed case. Schematically, according to KrV,
A304/ B360-1 and A330/B386:

[All] S is P - universal rule that expresses the assertion
of the predicate (P) to the condition (all S)

[All] Q is S - subordination of the condition of the
minor (Q) to the condition of the major (all S)

[All] Q is P – assertion of the predicate (P) of the
universal rule (all S is P) to the subsumed case (Q)

For
the sake of clarity, let us take the judgment “Socrates is mortal”. Such a
judgment can also be acquired through experience, insofar as I recognize
mortality as a ground of cognition, a *ratio cognoscendi*, of the intuitions
of Socrates, Plato et al. Unlike the context of the experience, however, two
aspects need to be highlighted: i) in the register of logical use, on the one
hand, it does not matter the transcendental reference of the concept-subject to
intuition. The singularity of intuition, which accounts for the immediacy in
which it represents the object, implies that it has no extension (otherwise it
would be an infimal concept[5]). However, precisely because it has no extension,
what is asserted of Socrates could never refer solely to part of what we
represent therein. And ii), since intuition and concept do not differ by the
object represented or by what the representation is referred to, but only by
the mode or form of the representation[6] (singular, universal; immediate,
mediate), it is perfectly licit that, in the syllogism, I represent the
singular Socrates *as if* it had common validity, as if it had an
extension, so that it can, thus, integrate the purely logical consideration
proper to the formal register with which we now deal - something like “all Socrates
”, as its total extension, is mortal, where the predicate subordinates, without
exception, everything that is under the concept-subject. In this sense, Kant
says that “an inference of reason is itself a judgment that is determined, a
priori, in the entire extent of its condition” (KrV, A322/B378)

Given
the judgment, it is required to find a concept that contains the condition by
means of which the assertion (of the predicate mortality) befits Socrates - in
this case, it is the concept of man, that is, the condition of mortality; i.e, Socrates,
taken in all its extension, expressed in the concept of humanity. Thus, the
representation Socrates (condition of the minor) is entirely subordinate to the
medium-term humanity (condition of the major, and assertion in the minor), from
which we have the minor premise: Socrates is man. Finally, the universal rule
that subordinates humanity and provides its predication to man is sought: all
men are mortal. In short: the rule “all men are mortal” establishes the
predication of something universal (mortality) to Socrates, under the condition
of humanity.

This
condition-conditioned relation explains the predication exemplified under two
aspects: the subordination of Socrates-man-mortal, in this order, each included
under the sphere of the other, as partial representations of the entire
extension of the concept, each time, superior; on the other hand, the inclusion
of mortal-man-Socrates, in this order, in the intension of the other, insofar
as they play the role of ground of cognition[7]. For this reason, the production of polysilogisms can
take place in two complementary directions. Reason intends either to find the
entirely determined concept (the infimal species) of an individual (therefore,
the one subordinated to the extension of all the others that are included in
its intension), or the supreme concept or superior ground of cognition (the one
included in the intension of all the concepts that are part of its extension).

Putting
aside the question of legitimacy, in the first case, of this claim of reason,
it is interesting to point out that the continuation of the chain, according to
its strictly logical operation, depends on the acquisition of the ground
according to which condition and assertion (S and is P, in our example) will be
bound in the major premise of each syllogism, in one word, the fundament of the
rule. According to this, there will be three types of fundament or ground: the
bond in a subject, the bond of the chained members and the bond of the parts in
a whole. Through them, reason seeks, respectively, the subject who is not
predicate, the presupposition that does not presuppose anything else, and the
aggregate of divided members for whose completion no more division is required[8].
Precisely these grounds of the universal rule are called exponents: “Now, every
series whose exponent (of the categorical or hypothetical judgment) is given
can be continued; therefore, the very same activity of reason leads to *ratiocinatio
polysyllogistica*” (KrV, A331/B387). We therefore have three modes of
syllogism, a typification that elicits its diversity from the universal rule
expressed in the major premise.

In
these terms, the concept in question is also present on in some *Reflexionen*
on logic[9]. For example, we find, in the well-known R3202, the
following: “a rule is an assertion under a universal condition. The relation of
the condition to the assertion, as the latter is under the former, is the
exponent of the rule” (AA XVI, 710 _{8-10}, 1790s). Also, in a set that
deals strictly with judgments, we read:

“The relation of concepts (exponent):

The subject to predicate

The ground
– consequence form of judgements

Whole – part

Categorical, hypothetical,
disjunctive”. (AA XVI, R3063, 636 _{4-11} (ca.1776-1779?, [1773-1775?],
1780-1783??).

Indeed, the essential
of the operation made possible by the given exponent resides in the logical bond
between condition and assertion, whose judicative expression constitutes a
universal rule. To the extent that, in this register, one does not go beyond
formal consideration, the exponents that provide the chain of the inferences
mentioned are not used only in syllogisms. They also apply to judgments that,
regardless of the origin of the expressed knowledge or cognition, can play the
role of universal rule (or law), i.e., they can be used as a premise in a
potential reasoning.

Thus
understood, all judgments, as universal propositions, apt to be used as a major
premise, are called principles. Note, however, that fundamental or superior
propositions of different origin, such as mathematical axioms (e.g.: between
two points there can be only one straight line), principles of pure
understanding (e.g.: everything that happens has a cause), or even propositions
acquired from experience via induction (e.g.: all bodies are heavy), can be
used as principles. If one considers them as to their origin, this denomination
would be, strictly speaking, inappropriate, since a principle, in an absolute
sense, should provide synthetic knowledge of the particular in the universal
through concepts alone[10]. In the case of axioms it is not possible for me to
know such a property of the line by concepts alone, but only in pure intuition;
as for the principles of pure understanding, although they originate from
concepts, they need a confluence from the sensible conditions of a possible
experience (one cannot, for example, derive from the mere concept of happening
that everything that happens has a cause); in the case of universal
propositions acquired by induction and formed by empirical concepts, sensation
is required. Regarding, however, the cases that can be subsumed in each of
these domains, it is correct to call them, comparatively, principles; in their
specific domain of application they configure superior knowledge[11]. For this reason, the meaning of principle carries
within it equivocation. According to our author: “the expression of a principle
is equivocal, and commonly it means only a knowledge that can be used as a
principle, even if, in itself and according to its own origin, it is not a
principium at all” (KrV, A300/ B356)[12]. Although the author resorts to this common meaning,
there is a preference for characterizing a principle in relation to the scope
in which this name can be taken in the strict sense:

Therefore, I would call knowledge by principles that
in which I know the particular in the universal by concepts. Thus, every
syllogism is a form of deriving a knowledge from a principle. For the major
premise always offers a concept which makes that everything that is subsumed
under its condition be known by means of it according to a principle. Now,
since all universal knowledge can serve as a major premise in a syllogism, and
understanding offers such universal propositions *a priori*, then these
can be called principles, in view of their possible use. (KrV, A300 / B357)

In
a transcendental register, which includes the question of origin, understanding
does not yield principles, since its superior propositions are not based on
mere thinking, that is, they do not offer knowledge by concepts alone, but by
concepts and intuition. Nevertheless, the consideration of the use abstracts
from the transcendental conditions of acquisition of these propositions and
attends only to what is essential in a rule, in general. Although the *raison
d'être* of the principles of pure understanding in the argumentative economy
of the text is linked to the reference to intuition (to the form of possible
experience), considered *a priori* synthetic judgments, the logical-formal
character is inseparable from what they have of fundamental, the logical aspect
of the judicative form, rooted in the domain of thinking, of which knowing is
only a case. We mean to point out that there is a non-excluding character in
what concerns the relation between the registers of general logic and that of
transcendental logic, as, for example, the objective validity of concepts, as
bound in judgments, which is sought out by the latter must still and
necessarily bear the form of thought, with which the first deals.

Now,
on the one hand, taken as judgments in general, the superior propositions
presented in the *System of principles*, as laws, must bear, in their judicative
“structure”, the relation between condition and assertion, thought in the
exponent of the rule (for the form of thinking); on the other hand, taken as *a
priori *synthetic judgments, a characterization relevant only in a
transcendentalized logic, there must be a determined reference to sensibility
(for the real, not just logical, possibility of the conceptual bond). We must
then grasp how can we, is this case, attribute objective reality to the logical
relation between concepts.

**II**

When
it comes to investigating the concept of exponent, this cleavage between the
strictly logical relation of concepts and their possible reference to something
= x is especially significant in the *Analogies of experience*. The three superior
propositions (which, as such, are laws that have the exponent of a rule) of
pure understanding found in this section of the *System of Principles*, establish
a domain called nature (in empirical sense), by which it is understood “the
interconnection of the phenomena according to their existence, according to
necessary rules, that is to say, according to laws” (KrV, A216/ B263). These fundamental
laws, the synthetic *a priori* judgments of relation, taken together,
exhibit the result of the *Analogies*: “our
analogies expound, properly, the unity of nature in the interconnection of all
phenomena under certain exponents that express nothing more than the relation
of time -[…]- to the unity of apperception” (KrV, A216/B263). It is not
clear, however, what specific contribution is required from exponents so that,
subordinating the phenomena, these can be known as interconnected, as to
existence. More than that, it would be hasty to assume the meaning of exponent
in this excerpt is the same as that found in the *Dialectic*, that is:
that the principles of relation, as judgments, can be interpreted according to
the relation of condition and assertion found by the exponent does not seem, at
this point, too problematic; but that the interconnection of the phenomena
occurs under exponents, this seems to require a different meaning of the term
than that linked to the judicative form.

At
this point, I would like to suggest that attention to the concept of analogy
may indicate a promising way to undo the knot. In the context in which this
concept comes to the fore, in an introductory section of the *Analogies*,
Kant is busy indicating why the principles of dynamical use differ from those of
mathematical use. One of the focal points of this difference lies in the
constitutive character of the latter, as opposed to the regulatory of the
former. This character is linked with a difference in the kind of synthesis
(whose superior concept is that of "bond": *conjunctio*, *Verbindung*),
specific to each variant. Avoiding entering into all the intricacies of these
notions, it should be said that the principles of mathematical use operate
through the kind of bond called composition (*compositio*, *Zusammensetzung*),
whereas those in dynamic use do it through connection (*nexus*, *Verknüpfung*)[13].

The
distinctive feature of the composition (homonymous of the mathematical
operation[14]) is that the bound elements are homogeneous with each
other and, therefore, it is licit to consider them mathematically; however,
they are not necessarily mutually implicated. Both in the case of composition
by aggregation and by coalition, homogeneity resides in that all composite
members are acquired by the construction act itself, since it is a limitation
of the parts of space, or of time. Thus, precisely in virtue of this
homogeneity, these constructed elements do not belong to each other because the
construction procedure is, at the same time, arbitrary: in the aggregation of a
square by the conjunction of the base of two triangles, or in the coalition of
an intensive magnitude by composition from a gradation of infinite intermediate
points between 0 and 1, none of the composite members is required to conceive
another; that is: unlike an accident in relation to a substance or an effect in
relation to a cause (whose bond involves some contingency, that of existence),
the constructed elements are, concerning their conception, independent of each
other. In this sense, the principles of mathematical use are characterized as
constitutive, since homogeneity and reciprocal independence allow me, so to
say, to acquire all the members of the composition (leastwise virtually) in the
aggregation or coalition procedure itself; in a word, the constructed concept
engenders the represented object itself.

With
the bond by connection, it happens differently. This is due precisely to the
fact that the link of existences takes place (the perceptions of effective
objects among themselves, in which case the connection is physical, or those
perceptions with the superior cognitive capacity, in which case the connection
is metaphysical). Regarding the principles of dynamic use, but above all those
of the *Analogies*, the regulatory character concerns the fact that it is
not possible to indicate the members connected in a determined way *a priori*.
Indeed, the contingency of the way in which perceptions meet with each other is
an index that it is not possible to establish a priori the perceptual data
itself, which always involves existence. According to the text:

Here, it is not to think of axioms, nor anticipations;
but, if a perception in the temporal relation with others (even if indeterminates)
is given to us, it is not possible to say *a priori*: *which* other
perception or *how great*, but only how it is bound necessarily with the
other in what concerns existence, in this modo of time (KrV, A179/B222)

As can be seen,
since the phenomenon cannot be constructed as to existence, the physical
connection will not exactly refer to the terms bound, but only to the relation
between them; that is why these principles are called regulatory. Thus, even if
they represent the real connection in an experience, to the *Analogie*s
can only be attributed a priori knowledge of existence in a comparative way[15]. The proper reference, e.g., of the principle of
causality, would not reside in denoting the necessity of the *terms* bound
in a causal connection; rather, its referent would be the form without which
such a relation could not be thought necessarily, insofar as by means of it we
can advance the series of existences through the relation of possible
perceptions, since we know the *form* in which this relation must take
place. In one word, by means the principle stated in the *Second Analogy* one
does not know effectivity itself, but the form according to which possible
effectivity must take place as to be thought in a determine matter.

Now, the impossibility of referring,
*a priori*, to the terms bound in physical connection is part of the core
of the concept of analogy in Philosophy, as opposed to its mathematical
meaning. In the introductory section to the *Analogies* we read that:

In Philosophy the analogies mean something quite
different from that which they represent in Mathematics.

In Mathematics

-[…]- they are formulae that enunciate the equality of
two quantitative relations, and [they are, AP] always *constitutive*, in
such a manner that, if two members of the proportion are given, with it the
third is also given [AA: three members… the fourth[16]], that is,
can be constructed.

In Philosophy

-[…]- analogy is not the equality of two *quantitative*
relations, but *qualitative*, in which from three given members I can know
and give *a priori* the *relation* to a fourth, but not *this*
fourth *member* itself;

And, in Philosophy, the way in which the other members
are found:

however I have indeed a rule to search for it in
experience, and a characteristic sign [*Merkmal*] to find it therein (KrV,
A179-180/B222).

The
approximation, albeit by means of difference, with the mathematical analogy
holds rich elements to understand philosophical analogy. Before looking more
closely at the similarities and dissimilarities between the two models, I will
try to go a little deeper into the notions of mathematical analogy and proportion
to better support my argumentative proposal. For now, one should note only that
here we have an indication that the way in which the exponent of the rule
(relation between condition and assertion in a judgement) and the exponent of
the series (interconnection of the phenomena under the exponents) correlate
must be linked, in some way, to proportion.

**III**

As
K. Reich has pointed out[17],
the concept of proportion appears in eighteenth-century mathematics manuals
linked to the concept of relation, in the wake of Euclid's book V of the
Elements. In an exemplary way, G. S. Klügel (1739-1812) explains, in his Mathematics
Dictionary (5 vols.) The notion of relation as follows:

Relation (ratio, λόγος) of two homogeneous quantities
to each other is, according to Euclid (V. explan. 3), the mutual reference in
which both these quantities are to each other, regarding their quantities, so
that, thus, the concept of a relation in general arises from the comparison of
two homogeneous quantities with each other. (KLÜGEL, 1831, T.5, B.2, 728).

A mutual correlation that takes
place between two homogeneous quantities (a condition without which they could
not be compared) does not, strictly speaking, constitute a relation if the
quantities are equal. More precisely, a relation takes place by the inequality
of the compared quantities; comparison by which the relation is established.
Thus, it can be said that the question that poses the terms in which the
relation will take place is a question concerning the *how* of mutual
reference between quantities; question related to *how much* or *how
many times*. As the comparison is linked to one or the other, the answer to
the question determines an arithmetical or a geometrical relation[18] (although this denomination has already been considered
inappropriate[19]).

The relation is understood as
inequality between numbers, and when considering its members (as correlated
quantities by comparison), one investigates how one quantity can emerge from
the other; in which case, of a known quantity, it is possible to discover the
other, at first unknown. In an arithmetic relation, this inequality is called
difference (See EULER, 1911 [^{1}1770], §§381-3, 146; also KLÜGEL,
1831, T.5, B.2, 729): it is the quantity that must be added to the antecedent
to arrive at the consequent, since it is obtained by subtracting the smallest
from the largest, such as in the relation between *a* and *b*, in
which *b* is greater and *a*, smaller, the difference, *d*, is
obtained by *b* – *a*. On the other hand, when it comes to the
geometrical relation, the inequality between the members is obtained when, in
relation *a *: *b* , the consequent is divided by the antecedent (or
on the contrary, since the order of the members is indifferent), so that if *a*
is known, *b *is reached by multiplying the antecedent by the *inequality*
(for antecedent 4 and consequent 2, we have inequality 1/2). Euler calls this
inequality, which is expressed in a fraction whose numerator is the consequent
and the denominator the antecedent, the name or denomination (*Benennung*)
(EULER, 1911 [^{1}1770], §§441-445, 164-166). Klügel indicates, in
another way, that the inequality can also be called exponent: “The number *e*,
by which to multiply A, to obtain B is called exponent or name of A: B”
(KLÜGEL, 1831, T. 5, B.2, 729). So does A. G. Kästner, “in whose hands”, says
Kant, “everything becomes precise, understandable and pleasant” (AA II, NG,
170)[20]. According to the famous professor: "The
exponent or name (exponens sive nomen) of a relation is the number that
indicates how many times the antecedent member is contained in the
consequent" (KÄSTNER, 1786 [1758], V, 134)[21].

Our interest, therefore, is the
geometrical relation. It, as we see, has three elements: antecedent, consequent
and exponent or name, so that it is possible to express the relation between *a*
and *b* as *a *: *ae*, where* e* is the exponent. Note,
moreover, that it is also possible to establish a relation of equality between
two relations (therefore, four members are required) whose internal members are
unequal (which is true for both arithmetical and geometrical). This occurs to
the extent, therefore, that there is an equality of inequalities, that is, that
the relations have, if geometrical, the same exponent. In this case, the mutual
reference between the relations constitutes a proportion.

Geometrical relations are *equal *to each other
when their exponents are equal to each other, so thus its evident that in both
relations the second member emerges in a single way from the first, and two
equal geometrical relations form a *geometrical proportion*. (proportio, ἀναλογία). (KLÜGEL,
1831, T.5, B.2, 732)[22].

Here we have a
fundamental property of geometric proportion, synonymous with analogy[23]. In two equal relations, A: B and C: D, not only is
the division of the first and second member equal to that of the third and
fourth, but also the product of the external terms (first and fourth) must be
equal to that of the middle terms (second and third) - multiplying both
fractions of the ratio A / B = C / D by B, we have A = BC / D; multiply once
more by D, then we have AD = BC. From this criterion or characteristic
property, the relation of analogy A: B :: C: D can be modulated (A: C = B: D;
D: B = C: A; D: C = B: A), and the proportion remains the same as long as the
product of middle terms remains the same as that of external terms (AD = BC).
Thus, if three terms are given (A: B :: C: ...), we find the proportional
fourth = BC / A, by Regula de tri or Regel detri (EULER, 1911 [1770], §§471,
478, pp.174 and 176; also KLÜGEL, 1831, T.5, B.2, 747). The continued
application of the three-term rule allows the subsequent acquisition of *n*
proportional relations, by the successive search for the proportional fourth in
each case, all under the same exponent (if the rule used is composed, it is
called, according to the number of given relations, e.g.: regula quinque,
septem etc.).

However, it is interesting to note
that Kästner, on the other hand, also uses the Regeldetri in another context[24], that of progression, in which there is properly a
series, given the non-indifference of the *order* of proportions. This is
evidenced above all by placing the use of this operation not for the search of
the proportional fourth, for which three known terms are required, but based on
two given terms, for which the third is sought, in a serial relation. According
to the author: “a geometric series is given when either its first and second members
are given or, instead, its exponent is given; for all the following members are
found through the three-term rule” (KÄSTNER, 1786 [1758], VI, 148). In this
case, in order to configure a series based on the relations of the same
exponent, they must have equal middle terms, such as a: b :: b: c :: c: d.
Otherwise, it could not be said that in the series, as a progression of
proportions, the terms that follow or precede have the same exponent. Thus, see
the series:

a^{4}/b^{3} :: a^{3}/b^{2 } :: a^{2}/b :: a :: b :: b^{2}/a
:: b^{3}/a^{2} :: b^{4}/a^{3 }...b^{n}/a^{n-1}

The first thing to
note, for the sake of clarity, is that the term "a" can be read, in
its complete "expression" as a^{1}/b^{0} and, the
term "b" as b^{1}/a^{0}. Secondly, less important, is
that in the form A:B :: B:C, the progression would be written a^{4}/b^{3}
: a^{3}/b^{2 }:: a^{3}/b^{2 } : a^{2}/b etc., which makes it
unnecessary to repeat, regarding the expression, the middle term. In any case,
it is understood that the exponent between any two proportions of this
progression is always the relation b/a, since the consequent term, divided by
the antecedent (t_{n+1}/t_{n}) is always equal to b/a.

Now,
when it comes to the acquisition of the chain or series, in the quote above
Kästner had indicated that there are two ways in which this is possible: either
at least two members are given, or the exponent is given. In the first case,
from a specific and localized relation, I can, by means of a comparison
procedure, find the exponent, the universal relation between the terms (the *terminus
generalis*); in the second, in possession of the latter, I can specify it,
and determine the singular relations between the members, provided that at
least two members are given. However, there is an important difference between
the two procedures, in which, it is true, I manage to construct the chain ad
infinitum, but in a different way. By comparing two given relations, I find the
exponent b/a and, in possession of this exponent, by the three-term rule, form
the series. When comparing the terms 1/8 :: 1/4, I find ratio 2, and I get 1/8
:: 1/4 :: 1/2 :: 1 :: 2 :: 4 :: 8 etc. I therefore discover how to find a term *c*,
since *c*_{n}* *= (c_{n-1})2. On the other hand, when
the exponent is given, it is not necessary that the constructed series is equal
to the series in the first case. Here it is much more about building a family
of series or possible series[25]:
both 1 :: 2 :: 4 :: 8 and 3/2 :: 3 :: 6 :: 12, in such a way that it is
possible to obtain the law of progressions in general (not just those of
exponent = 2, as in the example), namely, that c_{n }= e^{n-1},
where *e *is the exponent, since a term of position *n* is the
exponent multiplied by itself n-1 times, whatever the exponent. To put in another
way, in the first case I have the exponent for the construction of particular
relations; in the second, the universalized exponent, for which the
construction of serial relations between proportions is, firstly, possible.

**IV**

We
have seen that the use by Kant of the exponent is aimed, on the one hand, to
explain the strict logical relation between condition and assertion in a
judgment in general (the exponent of the rule), and, on the other, to explain
relation by means of which we think a series of causes-effects (the exponent of
the series). Now, in the *Lose Blätter aus dem Duisburg’schen Nachlass*
(henceforth DN), we find a conception of the exponent that takes place a
sensibility-oriented register, that is to say, not solely in a logical-formal
sense, but *also *in the logical-transcendental sense. As we shall see, it
will be possible to gain some terrain as to clarify both the meaning of
condition and assertion, and to deepen our understanding concerning the sense
in which the philosophical use of the exponent differs from the
mathematical.

Thus,
in these reflections of the mid-1770s, we find the concept of rule worked out in
relation to the exponent. In DN Kant names what is required for the emergence
of a rule. According to the author:

For the emergence of a rule three parts are required:
1. *x* as the datum for a rule (object of sensibility or, better, real
sensible representation). 2. *a*, the *aptitudo* for a rule or the
condition by means of which it [the real sensible representation, AP] is, in
general, referred to a rule. 3. *b*, the exponent of the rule (AA XVII,
4676, 656 _{8-12 }[1773-1775]).

The insertion of a
datum for a rule in the form of *something *(*Etwas*)* x*, is an
additional element in relation to the strictly formal conception of "assertion
under a universal condition", as we saw above. Understanding *x *as
the datum for the rule, as an indeterminate *sensible something *that is
virtually determinable when represented as subordinate to the rule, implies
conceiving or anticipating the object of a possible judgment, whose
consummation is set in terms of a determined reference of the relation between
condition and assertion. Here, in the DN, however, the pressing issue is more
that of acquiring the rule, than that of the convenience of the rule to the
case. Thus, there is a demand for the satisfaction of the condition, designated
by the symbol *a*, by which *x *(at first refractory to conceptual
universality) may befit the rule (at first, refractory to the individuality of
the sensible data). Properly, *a* is what offers determinability to *x*,
that is to say, it is that by virtue of which *x*, as something intuited,
can be converted into something thought; thus, *a* is at the same time a condition
*sine qua non* of the thinkability of *x* and of the concrete
reference of the rule: *a *designates *x* as a *thought* object.
In this condition, *a* is defined as a concept (among others: “-[…]- the
phenomenon *x*, of which *a* is a concept” [AA XVII, 4680, 665 _{5-6},
1773-1775]; “*x* always means the object of the concept *a* [AA XVII,
4674, 644 _{27-28}, 1773-1775]).

It would now be tempting to assign
to *b*, the exponent of the rule, the role of concept-predicate. In some *Reflexionen*
of the κ phase (1769), the symbols *x*, *a* and *b* were
mobilized to explain the insertion of an object, while something indeterminate,
in the judicative structure[26]. From the usual model of the categorical form (or
predicative, in this case) a judgment whose referent is potentially something =
x takes place “-[…]- when something *x* that I know through the representation
*a *is compared with another concept (^{g }b), in order to include
or exclude it ”(AA XVII, 3920, 344 _{24-26} [1769]). Aside from the
fact that the judicative form is limited to affirmation or denial, it is
necessary to note that the relevant change at the time of DN resides in that *b*
is no longer a mere predicate and, as in the R4676 mentioned above, responds to
a function of the relation between concepts in a judgment. Although in the same
DN we find some passages that seem to take *b* only as the
concept-predicate in a categorical judgment[27], here the concept of rule, by means of the exponent (*b*
as a relation between concepts, and not just as a concept-predicate), must
apply (or be universalized) to all relations between concepts in a judgment,
therefore, also to hypothetical and disjunctive forms[28].

Now,
in the DN Kant recognizes, according to the relations between concepts in a
judgment, three exponents: “there are, in this, three exponents: 1. of the
relation to the subject, 2. of the relation of consequence between them, 3. of gathering-together
[*Zusammennehmung*] ”(AA XVII, 4674, 647 _{17-19} [1773-1775]);
and according to the three exponents, three possible relations of *a* and*
b* in a judgment whose referent is something sensible = x: “In judgments,
however, there is a relation of *a* : *b*, which both refer to *x*.
*a* and *b *in *x*, *x* by means of *a *: *b*,
finally *a* + *b* = *x*.” (AA XVII, 4676, 657 _{8-10}
[1773-1775]). Although the categorical exponent, “of the relation to the
subject”, when expressed as “a and b in x” may suggest the predication of the
subject-concept and the predicate-concept to some sensible object, the
attribution of the relation between condition and assertion thought of in the
exponent cannot be identified with a judicative “formula” (such as subject,
verb and predicate), whose expression is a proposition; rather, the
aforementioned relations should be understood acts of thinking, as it is
highlighted in the sequence of R4674:

There are, in this, three exponents: -[…]-. The
determination of *a *in these *momentis* of apperception is the
subsumption under one of these *actibus* of thinking; one knows the
concept *a* (^{g} as in itself determinable and, therefore,
objectively) when one subsumes it under one of these universal acts of
thinking, through which it ends up under a rule. (AA XVII, 4674, 647 _{16-24}
[1773-1775])

We
see here that *b*, the exponent, is a determination of *a*. This
determination is understood in terms of subsuming the condition of the rule under
one of the three universal actions or acts of the mind. This determination, in
turn, allows the formation of the rule or, to put it another way: in what
appears to be an “ascendant” movement, will permit the intellectualization of *x
*as the exponent, *b*, determines the way in which *a *is to be
thought in regard with *x*; that is to say, *b* determines *a*
in *x*. In this way, is not just the case that the exponent is determinant
in relation to the concept *a *by means of which we think *x* (as
thought of *as *independent of its singularity[29]),
but also, in some way, *x *determines *a* (insofar as *x *specifies
the concept, or concepts, under which it is thought). Thus, in this context, to
say that the determination of *a *resides in its subsumption under one of
the three universal acts of thinking, is to say that the exponent provides us
with the way in which the representation of *x *as something thought (*a*)
must, for its turn, be thought or conceptualized. By itself, unregarded its
expression in a concrete judgment, the exponent, strictly speaking, is not
itself a determination *of* something, as a predicate of a *rule *in
a real relation would be; but it is the *manner in which* this
determination ought to take place by means of the virtually acquired rule[30].
In these terms, we can say that the exponent *b *can be characterized as
the *function of the rule*. According to R4680: “In the appearance *x*,
whereof *a *is a concept, there must be contained, in addition to what is
thought by *a*, conditions of its specification, which make a rule
necessary[31],
whose function is expressed by *b*” (AA XVII, 4680, 665 _{5-8},
1773-1775).

In
light of what was said, it is possible to better understand the meaning of the
relation *a *:* b*, in which both symbols refer to *x*. It
should first be noted that there are three cases wherein this can occur:
mathematical construction, exposition, and observation. As Kant says,
respectively: “In the first case the relation *a *: *b *follows from
the construction of *a *= *x*. In the second, it is drawn from the
sensible condition of the intellection of *a*; in the third, from the
observation. The first two synthesis are *a priori* (all three objective)”
(AA XVII, 4676, 655 _{6-10}, 1773-1775). The difference of these modes
of reference to *x *is grounded in the kind of sensible condition that is
taken up to form a rule. As far as we are concerned only with the first two
modes[32],
it should be said that in the construction, or synthesis through composition,
what matters is the condition by means of which the object is *given*,
that is to say, the conditions of *sensible intuition*. E.g.: in the pure
form of space, I can, solely by the possession of the concept “triangle”,
compose a figure whose concept necessarily contains the procedure for its
construction, that is, three angles whose sum equals 180 degrees; or, to
further exemplify, in the pure form of time, I can, by means of the concept of
a geometrical[33]
series in which the procedure of construction, c_{n }= e^{n-1}
is contained, compose all possible series in which the progressions consists in
multiplying the *ratio *for itself *n* times. The strict universality
of this kind of concept resides in the fact that the concept and, at the same
time, the procedure for the construction of the *x* it implies, are valid
for any and all triangles (be isosceles, scalenus etc.) or for any and all
geometrical progressions (be the *ratio *2, 1/3 etc.). To say then that in
this case *a *: *b *follows from the construction of *a *= *x*
means that the sensible condition for *a*, and by consequence, the
condition for a rule,* *is given by the arbitrary construction of *x*,
in which case, so to say, there is no gap between object and concept..

Now, although in the case of exposition the
model of necessity seems to be closely inspired by mathematical construction,
these two ways of determining *a *in *x* can, by no means, be
identified. In R4684 Kant says that “Therefore, we represent us the object by
means of an analogon of construction” (AA XVII, 4684, 670 _{20-21},
1773-1775)[34].
Here, he has in mind the concept of triangle, which gives us a rule of
composition that is valid for any and all triangles; the same kind universal
validity of the rule should somehow apply to those cases in which, e.g.,
something that occurs always follows from something that precedes. This
representation, as a universal act for determination of appearances, must yield
a rule *as if *the connection therein represented was constructed in the
inner sense. The problem here is that this kind of connection cannot, on the
one hand, be constructed, nor is given by (which is absent in mathematics)
sensation alone. The reason thereof is that the kind of object that in this
case must be thought under *a* is always linked to *existence*[35],
which, on the one hand, is only possible *a posteriori* and, on the other,
does not present itself and by itself to us as *connected* with other
existences (therefore necessarily), albeit conjoined (just generally). The main
question for Kant is then what, if anything, can we know *a priori* and
necessarily about this existences or, to rephrase it in terms of the DN: how,
from this contingent existences, can emerge an universal and necessary rule or
law[36]?

Whilst
construction required nothing more than pure forms of intuition, exposition, in
dealing with existence, requires *also* perception, understood here as the
way in which existence is posited in the inner sense, e.g.: “perceptions are
not solely appearances, i.e., representations of appearances, but of their
existence. –[…]– The perception is the position in the inner sense in general”
(XVII, 4677, 659 _{14-18}, 1773-1775).
The requirement of perception can be understood *grosso modo* by
the fact that *our* mind cannot generate objects in regard to their
existence (as it would be the case with intellectual intuition), although it *can
think* them in a determining manner. In our present context it should
suffice to say that “existence” indicates, in opposition to arbitrarily
generated mathematical objects in pure intuition, that something effective (and
therefore in itself radically independent from our intellectual capacities) is *given*
to us via reality in sensation (as matter of perception), that is, is posited
to our receptivity. In this sense, *a *must represent the condition
whereupon our perceptions have the unity required to form a rule, that is, so
that the rule may refer to perceptions in general. In a sense, *a *ought
to be both sensible, as to be able to represent the datum *in concreto*,
and intellectual, for it must also represent something universal in *x*.
As Kant puts it in R4684:

If a concept is also sensible, but universal, then it
must be considered in its *concreto*, for example, triangle in its
construction. If the concept does not signify pure, but empirical intuition,
i.e., experience, then the *x* contains the condition of relative position
(*a*) in space and time; i.e., the condition of determining universally
something therein.

Moreover, appearances are determined through time, but
in the *synthesi* the time [is determined, AP] through an appearance,
e.g., of that which exists or occurs or coexists. These [relations, AP] are the
most universal in appearance, whereof reality is the matter. (XVII, 4684, 671 _{10-19}, 1773-1775).

In
the case of Mathematics, it could be said that the conditions for conceiving *a
in concreto* would be met by the concept of triangle since it both
represents triangle *in general* and the very procedure for generating any
particular triangle[37].
With empirical intuition, however, we have already seen that it is not possible
for human intellect to absolutely posit effectivity itself, that is, to
generate the perceptual data. What we can relatively posit is the data, not
made but given elsewhere, in temporal relation. That is also to say that
appearances are given to us, due to the forms of our sensibility, in space and
time (appearances determined through time), but the act of positing in inner
sense *temporalizes *the given, i.e., institutes *order *in time
(time determined through appearances). Therefore, the notions, according to the
above-mentioned excerpt, of appearances of that which exists in time, that
which occurs in time and that which coexists in time will provide us with the
sensible, but universal concepts; or, more precisely, concepts *under *sensible
conditions[38] (ad
1781 called Schemata), that account for the ways in which time is to be
determined. Just as *a* represents the universal in perception, or that
which is universal in positing in time, *b *can now be further specified
as “-[…]- the exponent of the relations of perceptions, hence of determining
their place according to a rule” (XVII, 4676, 655 _{19-21}, 1773-1775).

As we have seen, *a *was the “*aptitudo
*for a rule or the condition by means of which it [the real sensible
representation] is, in general, referred to a rule”; this reference to the rule
ought to arise through the exponent that gives us the manner in which *a *is
to be determined. Let us take the concept *a *in the case in which the
position in time takes place as something that occurs, that is, in the case in
which existence is posited in the mode of succession. Through the exponent of
relation of consequence, the particular succession is apt to be thought as a
member of a series of possible perceptions, which are chained by means of we
relating them to each other in terms of ground and consequence. So, when
something *d* occurs, we determine its place in time by positing for it
something as its ground (*c*), that is, something from which it came to
be. In the series of possible perceptions, the same relation must apply to,
say, something *f* that occurs. We would say that just as I must posit
something *c *as to determine *d *in its time-relation, I must posit
something *e*, so the same may occur with *f*, that is, c : d :: e :
f. This series, therefore, would be ordered by means of the exponent of ground
and consequence, as the manner to determine the positing in time, *a*.

If,
say, the procedure of mathematical construction applied here, not only the
effective given would be unnecessary but we would also be able to acquire the
complete series, past, present and future, of the relation of consequence for
the totality of what occurs, since we would be able to construct all the terms
in the chain and arrive at a uncondicioned (*Unbedignt*), to use the
terminology of the *Antinomies*. Thus, it would be possible, e.g., from a
given effect, to arrive at the *first cause *in time, by means of a chain
of regressive conditions. To mention the same example of the last section, be
it the progression:

a^{4}/b^{3} :: a^{3}/b^{2 } :: a^{2}/b :: a :: b :: b^{2}/a
:: b^{3}/a^{2} :: b^{4}/a^{3 }...b^{n}/a^{n-1}

Here, any t_{n+1}/t_{n}
means that t_{n+1}_{
}is the effect of the cause t_{n}. By means of the exponent b/a
each *causal relation* could be linked by its middle term, so that each
consequent *relation* could be taken as an effect of a preceding one, and
each antecedent *relation*, as a cause of a consequent one: a^{4}/b^{3}
: a^{3}/b^{2 }:: a^{3}/b^{2 }: a^{2}/b
etc., so that the occurrence a^{3}/b^{2 }both causes a^{2}/b,
and is the effect of a^{4}/b^{3} (remembering that, as
mentioned above, the identity of the middle term makes it unnecessary to
explicitly refer to it every time).

But,
strictly speaking, mathematical construction is not justified in *a priori *knowledge
of experience *as such*. Here, by an equality of proportions, an analogy,
we cannot provide *a priori* the *terms *related themselves, as if
our minds posited existences. To refer to the introductory section of the *Analogies*,
in Philosophy: “analogy is not the equality of two *quantitative* relations,
but *qualitative*, in which from three given members I can know and give *a
priori* the *relation* to a fourth, but not *this* fourth *member*
itself” (KrV, A179-180/B222). According to its mathematical provenience in
geometrical proportions, also in Philosophy the exponent has a role to play in
determining a series. Here, however, given the impossibility of using our
representations in a constitutive manner, the possession of the exponent does *not
*allow us, *a priori*, to provide any *determine *causal relation
and, *a posteriori*, to provide the terms of the series for that which
there is no corresponding datum. What we nevertheless know is that *if *existence
as succession is posited to our receptivity, then it *must* be brought
under a rule by means of a given exponent (in this case, the relation of
consequence), as we can only know *a priori* the relation of possible
perceptions, never the perceptions themselves.

**V**

As
we have just seen, there is something like an anticipatory character linked to
the operation the exponent represents. However, this sort of *a priori*
anticipation of an aspect of the form of experience does not, for Kant, *determines*,
but as he puts it:

-[…]- it only says that something is, according to a
rule still to be found, determinable according to a certain given exponent. It
serves, therefore, to try this determination and to expose the appearance,
being the *principium *of judging [*Beurtheilung*] the appearance,
e.g., what occurs has in some precedent its ground. (XVII, 4677, 659 _{22-24},
660 _{1-3, }1773-1775).

Albeit the
exponent being the determination of *a*, e.g., the existence as succession
(*a*) thought as relation of consequence (*b*), it is not a
determining judgment as the universal rule presented, e.g., in *Second
Analogy*, which states that “all changes occur in accordance to the law of
the connection of cause and effect” (KrV, B232), or a particular rule, say,
that all changes in matter have an exterior cause[39].
Instead, the exponent provides determinability to the given, by means of which
the determination can be “tried”, that is, by means of which the given may be
recognize as a case of a (potential) rule. The procedure, so to say, as to make
the datum determinable, so it can be determinately thought under concepts, is
to expose its appearance. Accordingly, Kant names this procedure exposition,
which is an alternative to construction[40].

By
exposition of appearances it is understood “-[…]- the determination of the
ground whereupon the interconnection of sensations reside”, as the “ground of
all relation and of concatenation [*Verkettung*] of representations”
(XVII, 4674, 643 _{20-22, 10-11}, 1773-1775). By its turn,
concatenation is conceived as “-[…]- the representation of the inner action of
the mind to connect representations” (XVII, 4674, 643 _{14-15},
1773-1775); connection that can occur, as we have seen, in terms of a relation
to a subject, of a relation of consequence or of a relation of
gathering-together. Therefore, although the exponent is not itself a
determining judgment, it is the ground by means of which the determination is
made possible. These inner actions account for the very condition of unity of
our conscious representations, that is, that in connected or related
representations by means of this inner action “there is unity not in virtue of:
in which; but: through which the manifold is brought in one” (XVII, 4674, 643 _{17-18},
1773-1775). This “through which” is understood a little later in the same R4674
as “-[…]- the subjective representation (of the subject) itself, but made
universal, for I am the original of all objects. Therefore, it is conjugation
as function, which makes the exponent of a rule” (XVII, 4674, 646 _{12-14},
1773-1775).

*Cum
grano salis*, this bears
an interesting resemblance to what Kant calls, in the §15* *of
B-Deduction, *qualitative unity.* Here, he is discussing the concept of
bond, *Verbindung*, as representation of the synthetic unity of the
manifold:

The representation of this unity cannot emerge from
the bond; rather, as it is added to the representation of the manifold, it
makes possible, in the first place, the concept of bond. This unity, which
precedes *a priori* all concepts of bond, is not that category of unity
(§10); for all categories are based upon logical functions in judgments, but in
these bond is already thought and, thus, unity of given concepts. (…)
Therefore, we must search even higher for this unity (as qualitative, §12),
namely in that which contains in itself the ground of the unity of different
concepts in judgments and, thus, of the possibility of the understanding, even
in its logical use. (KrV, B130-1).

This qualitative
unity is that of the logical functions, as for this excerpt, a pre-categorial
condition of unity. Here I would like to suggest that though this unity is the
ground of conceptual unity in judgments it does not have to be, itself, a
conceptual unity; like the “I think”, as the continuation of the text (§16)
might suggest. Anyhow, the “I think” is
not original, say, in the sense that it precedes that inner action of the mind
as its condition. Rather: it is only by means of a *recognition* of a
common subjective ground in all binding activities that it is possible to think
something as a numerical identity therein, something *thought of* as that
which is the “transcendental actor”, as the condition of possibility, of these
acts. However, given my scope here, I do not intend to deal with the
difficulties involving the conditions for self-consciousness or self-ascribed
representations, nor with a sound characterization of the synthetic unity of
consciousness. For me, it suffices to frame this unity in terms of activity[41].

Looking
a little further into it, as for the reference to §12, Kant notes the following
concerning unity:

For in all knowledge of a object there is *unity*
of the concept, which one may call *qualitative unity*, insofar as by it
only the unity the comprehension [*Zusammenfassung*] of the manifold of
knowledge is thought, as, say, the unity of the theme in a drama, in a speech, in
a fable. (KrV, B114)

The qualitative
unity as mentioned here seems a much, so to say, looser one than that strictly
represented by concepts. It is analogous of that of a drama, a fable etc. It
seems to be the unity of conceiving in one, but not necessarily in a
conceptual, reflected manner. As will be said later in the B-Deduction, the *Zusammenfassung*
of the manifold according to the forms of sensibility yields formal intuition
(KrV, B161n). Although we read in the continuation that this representation is
the determination of sensibility by the understanding, the unity of this
intuition pertains to space and time and not to concepts of the understanding.
The seems very much alike the way in which, as seen in the DN, in the existence
as succession (which by itself has some form of unity, though not that of a
case of a given universal rule) we would arrive at an order in succession by
means of the relation of consequence; but not at the judgment that states that
everything that occurs presupposes a cause. In any case, the exponent operates
as that by means of which a given particular must be posited in time as to be
brought (or thought of as contained) under the universal even if, as the above
passage indicates, this universal is not currently given. As mere inner action
of mind to connect representations, it should not (though made universal, as
the subjective form of our concepts) be hastily equated with a determining
function in judgement. It seems to be much more the case to conceive it as mere
spontaneity of thinking, as we read, e.g. in a passage from the Paralogisms:
“Thinking, taken by itself, is merely the logical function, hence the sheer
spontaneity of the bond of a manifold of a merely possible intuition -[…]-”
(KrV, A428).

To
be sure, if my argument so far is, albeit not in every bit, leastwise sound, it
indicates how Kant could have made fruitful for Philosophy a concept
encountered in Math textbooks[42].
Although it does not seem to occupy a prominent place in the *Critique*
when taken literally, the analysis of the DN shows its importance for the work
in progression, as what there was still in a faltering manner called Analogy of
phenomena, Analogy of nature, Analogy of understanding, Analogy of experience[43]
would latter be embodied in the seemingly core section of the *Analytic of
Principles*. This embodiment is also indicated by the way Kant characterizes
retrospectively in the *Phaenomena and Noumena*, the superior rules of
pure understanding expounded in the *System of Principles*, by attributing
to them the same procedure encountered in DN, that of exposition: “Its
principles [of understanding] are merely principles of exposition of
appearances-[…]-” (KrV, A247/B303); and: “-[…]- what we have sustained so far:
that our pure cognitions of the understanding in general are nothing more than
principles of the exposition of appearances” (KrV, A250).

In
this way, if principles of pure understanding provide us with “the concept that
contains the condition and, so to say, the exponent of a rule in general -[…]-”
(KrV, A159/B198), we do not deal with a closed system, which does not permit
growth. It is much more the case that by means of including the exponents,
whereupon we set the fundament of all relation in appearances, in the intension
of phenomena we, therefore, acquire a ground by means of which we may encounter
recognizable features (as a *Merkmal *by means of which we recognize the
minds on activity in appearances[44])*
* that allow the particulars in nature
to be brought under the extension of rules, given or still to be found, in a
determine manner. In the same way, by the “-[…]- unity of nature in the
interconnection of all phenomena under certain exponents -[…]-” (KrV,
A216/B263) we could understand the positing of a common ground through which
appearances are yet not determined, but determinable in an unending process of
thinkable completion.

In
exposition we refer the ground of all relation and concatenation of our
representations to the sheer spontaneity of thinking. But, as such, the
universal acts of the mind or inner actions are by no means restricted to
working with sensible data. In the open field of mere thinking, the exponent
shows more clearly its mathematical provenience. Paying here the price of not
being able to denote given objects for what is thought or, as in Mathematics,
to compose the infinite chain of cases drawn from the procedure implied in the
given exponent, reason nonetheless possesses the same operating cog, albeit, if
salutary, only a regulatory one. This operative concept, as the exponent of a
rule, accounts for the very act under which Kant thinks our chaining of
judgments, in the *ratiocinatio polysyllogistica* in general, or in
thinking the unconditioned in the series of conditions.

**Bibliographical references **

CODATO, L. N. (2003), *Forma lógica na Crítica da
razão pura*, Tese de doutoramento no Departamento de Filosofia
da Universidade de São Paulo.

_____________. (2004), *Extensão e forma lógica na
Crítica da razão pura*. In.:
Revista Discurso [USP], n.34, 2004, pp.145-202

EULER, L. (1911 [^{1
}1770]), *Vollständiger Anleitung zur Algebra*. In.: Leonhardi Euleri Opera Omnia. Leipzig u. Berlin: B. G. Teubner.

KANT, I. (1998), *Kritik
der reinen Vernunft*. Hamburg: Felix Meiner.

*________. *(1924), *Kant’s handschriftlicher Nachlaß (Logik).*
In.: Kants gesammelte Schriften. Hg. von der Königlich Preuẞischen Akademie der
Wissenschaften, Band XVI. Berlin: Walter de Gruyter.

________. (1926), *Kant’s
handschriftlicher Nachlaß (Metaphysik, erster Theil).* In.: Kants gesammelte
Schriften. Hg. von der Königlich Preuẞischen Akademie der Wissenschaften, Band
XVII. Berlin: Walter de Gruyter.

KÄSTNER, G. (1786 [^{1
}1758]) *Anfangsgründe der Arithmetik, Geometrie, ebenen und sphärischen
Trigonometrie, und Perspectiv*. Der mathematischen Anfangsgründe 1ten Theils
erste Abtheilung. Vierte vermehrte Auflage. Göttingen: Vandenhoek und Ruprecht.

KLÜGEL, G. S. (vol.
1, 1803; vol. 2 1805; vol. 4, 1823; vol. 5, 1831), *Mathematisches Wörterbuch
oder Erklärung der Begriffe, Lehrsätze, Aufgaben und Methoden in der Mathematik*.
*Erste Abtheilung: Die reine Mathematik*. Leipzig: Schwickert.

REICH, K*.*
(2001),* Die Vollständigkeit der Kantischen Urteilstafel*. In.: Klaus
Reich Gesammelte Schriften. Hamburg: Felix Meiner.

SCHULTHESS, P. (1981), *Relation und Funktion*: *eine
systematische und entwicklungsgeschichtliche Untersuchung zur theoretischen
Philosophie Kants*. Berlin:
Walter de Gruyter.

* I would like to thank FAPESP, which
funded my master’s degree research by means of which this work was made
possible (process n. 2017/09999-1, Fundação de Amparo à pesquisa do Estado de
São Paulo). A first version of this
text I now present was discussed in the “Primeras Jornadas sobre la Dialéctica
Trascendental de la Crítica de la Razón Pura” (2019, Santiago, Chile). For the objections and suggestions
made on that occasion I am grateful to Prof. Luis Placencia, Javier Fuentes,
Nicolás A. Silva Sepúlveda, Felipe Cardoso Silva and Paulo Borges de Santana
Junior. I must say, however, that whilst their commentaries surely made this
work a lot better, I was not able to do justice to their valuable
contributions, so that all the errors one may eventually find here are
exclusively my fault.

·
Philosophy PhD student
in the Philosophy Department of the University of São Paulo, Brazil. Email
address: andre.perez@usp.br

[1] “Our analogies expound, properly,
the unity of nature [in empirical sense, AP] in the interconnection of all
phenomena under certain exponents that express nothing more than the relation
of time (insofar as it contains in itself all existence) to the unity of
apperception, which can only take place in the synthesis according do rules”
(KrV, A216/B263). I shall quote the works of Kant according to the AA (with
exception to the KrV, in which case, as usual, the excerpts are referred to by
A and B). All translations are mine unless otherwise stated.

[2] “Now, every series whose exponent
(of categorial or hypothetical judgment) is given, may be continued; therefore,
this very action of reason leads to the *ratiocinatio
polysyllogistica*, which is a series of inferences that may be continued
until undetermined distances, be it by the side of conditions (*per
prosyllogismos*), be it by that of the conditioned (*per episyllogismos*).”
(KrV, A331/B287-8).

[3] “The same is also valid for
substances in community, which are mere aggregates and have no exponent of a
series (…). In consequence, only the category of *causality remains*,
which offers a series of causes for a certain given effect, whereof we can add
that, from the later, as the conditioned, to the formers, as conditions -[…]-”
. (KrV, A414/B441-2).

[4] These [superior principles of the
understanding, AP] alone, however, provide the concept
that contains the conditions and, so to say, the exponent of a rule in general;
whilst experience provides the case that is under the rule” (KrV, A159/B198).

[5] Cf. See the specification principle
KrV, A655-6/ B683-4; compare with the elucidation of quantity in the table of
logical functions, KrV, A71/B98.

[6] See KrV, A320/B376-7.

[7] Concerning *Merkmal* as
partial representation and ground of cognition, see: AA XVI, R2283, 299
(ca.1780-1789? [1776-1778??]); R2285, 299 (ca. 1780-1783?
1776-1779??); R2286, 299-300 (ca.1780-1783). Concerning this subject see the
doctoral thesis of Luciano Nervo Codato (*Forma lógica na Crítica da razão
pura*, 2003), particularly pp.89-97. Part of the content was published in *Extensão
e forma lógica na Crítica da razão pura* (Revista Discurso [USP], n.34,
2004, pp.145-202), particularly pp. 189-194.

[8] See KrV, A323/B379-381

[9]
Mentioned and briefly analyzed by K. Reich (2001, 80-81 [66-67]).

[10] See. KrV, A300/B356-7.

[11] In the case of Mathematics this in
mentioned in A300/B356. Regarding principles of pure understanding, in KrV,
A148/B188 and A302/B358.

[12] The difference between principle
and principium as already mentioned in the *System of principles*. See
KrV, A160/B199.

[13] See KrV, B201-2, note

[14] Stricly speaking, one cannot
identify the procedure of composition in the case of principles of mathematical
use and in that of Mathematics. In a context in which it is a question of
differentiating Philosophy and Mathematics, Kant says that: “In fact, I
referred in the *Analytic*, in the table of principles of pure
understanding, to certain axioms of intuition; but the principle there
introduced was not itself an axiom, but served only to indicate the principium
of possibility of axioms in general and, itself, was only a principle from
concepts”(KrV, A733/B761). See also KrV, A732/B760 and KrV, A733/B761.

[15] See KrV, A225/B273 and A226/B279

[16] In the AA it was intended to
correct Kant's text. As we will see below, when investigating the mathematical
notion of (geometric) proportion, it is true that the acquisition of a fourth
unknown term requires three known ones; however, in a progression, from the
relationship of at least two members, as Kant says, it is possible to acquire
the ratio by which the others are found.

[17] REICH, 2001, 80-82 [66-68]. Reich
also indicates, in these pages, the textbooks we shall investigate in the
sequence.

[18] As L. Euler puts it: “(...) when
one asks about inequality, it can occur in two ways; then, either it is asked
how much one [quantity] is greater than the other, or how many times one is
greater than the other. Both types of determination are called relation; the
first one is usually named arithmetical and the second, geometrical relation”
(EULER, 1911 [^{1}1770], §390, p.148). In the same way Kästner: “when
we investigate, through subtraction, how great one number is in comparison to
another, we consider both these numbers in their arithmetical relation (ratio
arithmetica). When, however, we investigate, through division, how many times
one number in contained in the other, or what sort of part it is from the
other, we consider them in their geometrical relation (ratio geometrica)”
(KÄSTNER, 1786 [^{1}1758], V, p.129).

[19] For Euler: “this denomination has
nothing in common with the subject matter itself, but was adopted arbitrarily”
(EULER, 1911 [1770], §390, 148); In the same way Klügel: “Both names
[arithmetical and geometrical] are inappropriate” (KLÜGEL, 1831, T.5, B.2,
729). Also, Kästner notes that: “according to the use of the ancients, the name
relation pertains only to the geometrical. The moderns also attributed it to
the arithmetical comparison between two numbers. Therefore, when this word is
put without one of its epithets, one must understand, every time, geometrical
relation” (KÄSTNER, 1786 [^{1}1758], V, 129).

[20] In the context of the compliment
Kant indicates the way, the clearest and most determined, with which Kästner
works with the concept of negative quantities, referring to the *Anfangsgründe
der Arithemtik*, cited above, on pp. 59-62 of Kant’s edition.

[21] For Euler, differently from Kästner
and Klügel, the exponent is the number that indicates the grad of a power, like
its current meaning: a^{100}, pronounced *a *elevated [*elivirt
oder erhaben*] to a hundred, expresses the hundredth power of *a*. The
number written on top, in our case 100, is usually named exponent” (EULER, 1911
[1770], §172, 64). Klügel, in his Dictionary, points out a cleavage in the term
in question: “Exponent is, firstly, the number that indicates the grad in a
power, and can be a whole, fractioned, rational or irrational, positive or
negative number. Secondly, it is the number by means of which one must multiply
the consequent member of a relation as to arise the antecedent member [as the
order in here indifferent, AP] (KLÜGEL, 1805, T.2, B.1, 170-171). We should
note that the difference in both meanings is more one of context than a
conceptual one. In the one case, a relation (a : b); in the other a series (a,
b, c, d…), such that if the initial relation were a : a.a, the series of powers
could be a, a^{2}, a^{3}, in which case the exponent is *a*.
As a grad of a power, expressed as a_{n}= R^{n-1}, the exponent
(n-1) indicates the position of a term in the series, the reason why Klügel
considers more adequate to name (n-1) as position-index or index-number [*Stellenzeiger*
oder *Stellenzahl*] (KLÜGEL, 1805, T.2., B.1, 171).

[22] The same applies for both Euler
(1911 [1770], §461. 171) and Kästner (1786 [^{1}1758], V, 132 and 134).

[23] “Analogie (analogia) is synonymous
with proportion. It is the greek word ἀναλογία, by which Euclid expresses the
equality of two relations, see proportion” (KLÜGEL, 1803, T.1, B.1, 77)

[24] As to the fist context, that of *proportion*,
see KÄSTNER, 1786 [1758], V, 137.

[25] I should thank my colleague, Paulo
Borges de Santana Junior, for bringing to my attention the case with the family
of possible series.

[26] See SCHULTHESS, P., 1981, pp.78-86.

[27] E.g..: “The proposition of identity contains
the comparison of two predicates, *a *and *b*, with *x* –[…]-“ (AA
XVII, R4676, 653 _{12-13},_{ }[1773-1775]).

[28] For a slightly different
interpretation see SCHULTHESS, 1981, 252-253.

[29] As Kant says in R4674: “-[…]- such
proposition [of subsumption of *a *under one of the universal acts of
thinking, AP] is a principle of the rule, therefore, of the knowledge of the
appearance by the understanding, [rule, AP] by means of which it [the
appearance, AP] is considered as something objective that is thought in itself
independently of the singularity of what was given” (XVII, 4674, 647 _{24-27},
1773-1775).

[30] I must say that given the character of text we
now have to deal with, it is by no means undisputed if some characterizations
of key concepts that occur in these *loose sheets *should be interpreted
in a univocal, restricted manner; it seems to be the case that conflicting
conceptions stay side by side here. E.g., the addition of the same period to
the R4674 “x is therefore the determinable (object) that I think through the
concept *a*, and *b *is its determination (^{g} or the manner
to determine it)” (AAXVII, 4674, 645 _{28-29}, 1773-1775). It seems to
me that the determination itself and the way it must occur, therefore,
something as its prescriptive character, should not be identified.

[31] That is not to say that the rule is
required or that we necessitate it (in the sense of nötig); but that it applies
unrestrictedly (notwendig).

[32] In any case, see R4678 (661 _{8-12},
1773-1775) for the exemplification of “observation”

[33] Although “geometrical” brings to
mind spatiality, we must remember that, as Euler, Kästner and also Klügel have
noted, this nomenclature is entirely arbitrary, and has no bearing on the
subject proper.

[34] In the same way, the principles of
exposition are “-[…]- analoga of axioms, which take place *a priori*, but
only as anticipations of all laws of nature in general” (XVII, 4675, 658 _{1-2},
1773-1775).

[35] Kant refers to it here in the DN as
the *real object*. As for the vocabulary of the Critique, it should be
understood as the *effective* object, in so far as the *real *of
sensation is there connect with quality and is subjective, although virtually
objective. In the *System of Principles*, the dynamical categories
(relation and modality) deal with effectivity, *Wirklichkeit*, and not
just *Realität*.

[36] To be sure, Kant also asks about de
conditions of acquisition of determinate rules or empirical laws of nature.
However, the most pressing question, as in the Critique, concerns the
principles of pure understanding. See, for example: “Hence determinate rules of
synthesis can be given to us only through experience, but their universal norm
[can be given] *a priori*” (XVII, 4679, 663 _{19-20}, 1773-1775).
See also the edition of 1781, where Kant says the following: “Now, the
representation of a universal condition according to which a certain manifold
(…) can be posited is called a rule and, when it must be so posited, it is called
a law” (KrV, A104).

[37] By now it should be clear that *a
*in the DN (although surely not in a univocal sense) represents what in the *Critique*
will take place in the Schematism. Unfortunately, we will not be able, in this
paper, to follow this very interesting route.

[38] “In a synthetical judgment it can
never be in relation 2 pure concepts of reason with each other, but one pure
concept of understanding with a concept under sensible conditions” (XVII, 4584,
671 _{24-26}, 1773-1775).

[39] See AA IV, MAdN, 543

[40] “We must expose [exponieren, AP]
concepts when we cannot construct them” (XVII, 4678, 660 26-27, 1773-1775).

[41] Here it is posed, so to say, the
limit of my argument. These last parts, namely, what concerns §15 of the
B-Deduction and §12 are not yet for me clearly connect to the DN. This possible
relation was, as I recollect, indicated by Prof. Placencia in last years
“Primeras Jornadas”, for which I am very grateful; as also was pointed out, as
a hypothesis, what seems to me now the logical continuation of the analysis, to
step in the *third Critique*, as the exponent appears there as means in
which we bring representations of imagination to concepts, as opposed to
demonstration, in which in expose our concepts in sensibility, or at least a
common *Merkmal *in them (see AA V, KdU, §57, Anm. I, 341-344).

[42] Note that it would not have been the first
time, as can be seen in the concept negative quantities; and, as at least
Schulthess points out in the course of his thesis, the same would have happened
with “function”.

[43] Respectively, in the *Reflexionen* 4675, 648 e 4682, 669; R4675,
652; R4681, 667 e R4684, 672; R4602, 606 (1773-1775)

[44] “I have indeed a rule to search for
it [the fourth member] in experience, and a characteristic sign [*Merkmal*]
to find it therein (KrV, A179-180/B222).

DOI: https://doi.org/10.5281/zenodo.5775953

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