Visual Argument · Immanuel Kant and Charles Sanders Peirce

Kant's Table, Step by Step

Immanuel Kant distinguishes a manifold given in sensible intuition, the unity of apperception to which it must belong, and synthesis as the act that combines it. Charles Sanders Peirce distinguishes what is considered in itself, what is related to another, and what mediates between terms. This sequence tests a structural comparison between them across Kant's twelve categories.

3 internal roles4 external profiles12 proposed positions

Follow the Argument

The sequence begins with the philosophical comparison, builds the model one step at a time, and marks which conclusions are mathematical and which remain interpretive.

How the argument works

Kant and Peirce both ask how a manifold can count as one, although they ask that question in different philosophical settings. This argument proposes a correspondence between their three-part structures and then tests it against Kant's table.

Step 1 of 8: Kantian ground1 of 8

What the texts say
Kant separates a given manifold, the unity to which it must belong, and the synthesis that combines it. Peirce separates what is in itself, what refers to another, and what mediates.
What this work proposes
Read the manifold as presentation ⟨1⟩, apperceptive unity as the second term to which it must answer ⟨2⟩, and synthesis as mediation ⟨3⟩.
What would count as success
The comparison must explain more than the number three. It must make sense of the three categories under each heading, the four headings, and their twelve crossings.

Road map

Three roles and four heading profiles produce twelve places in the table.

The arithmetic is simple. The argument lies in how the roles and profiles are defined, and whether each category really fits the position assigned to it.

First question
What happens within a heading?
3 proposed roles
Each heading contains three categories, read here as presentation, discrimination, and mediation.
Second question
How do the headings differ?
4 directed profiles
Each heading is modeled by whether its input and output depend on an explicit context.
Final question
How do the two axes combine?
12 table positions
Every position in the table combines one proposed role with one heading profile.

Step 1 · the proposed correspondence

Kant separates what is given, the unity it must belong to, and the act that combines it.

A manifold is given in sensible intuition. Its representations must be capable of belonging to one self-consciousness, which Kant calls the unity of apperception. Synthesis is the act that combines the manifold under that unity. This work compares those three functions with Peirce's Firstness, Secondness, and Thirdness.

Why unity is second: The numbers name roles, not quantities. Unity is placed second because it is the distinct pole to which the given manifold must belong, not because unity somehow means two.
Why synthesis is third: Synthesis does not merely add another item. It is the act that connects the manifold with the unity required for one experience.

Correspondence ↓Three functions in Kant, compared with three kinds of relation in Peirce.This is the central philosophical proposal being tested.

⟨1⟩P(x)

Kantian termManifold of sensibilityA77/B102; B129-130

Structural reading

Presentation

Peircean role: Firstness

selected feature P

xpresented term

No distinct correlate is required.

The feature is considered as given, before it is referred to a distinct second term.

⟨2⟩R(x,y)

Kantian termUnity of apperceptionB131-135

Structural reading

Discrimination

Peircean role: Secondness

xselected term

determined through

yexplicit correlate

The feature is determined through a distinct second term: an opposite, selected item, state, position, or relatum.

⟨3⟩M(x,y;z)

Kantian termSynthesisA77/B102; B129-135

Structural reading

Mediated Integration

Peircean role: Thirdness

zmediator or law

xfirst term

ysecond term

One rule jointly organizes both terms.

A rule or mediating term connects distinguishable terms within one organized whole.

Names and proposed rolesKant's twelve categories retain their names and places under the four headings. The new role labels describe the job each category is proposed to perform within its heading.

Textual basis

What the sources show

Immanuel Kant and Charles Sanders Peirce both ask how many impressions can become one experience.

Kant distinguishes what is given to the senses, the single consciousness to which it must belong, and the act that combines it. Peirce also begins with a manifold of impressions and asks what is required to hold them together as one.

Source 01Immanuel KantB129-130
The manifold is given; synthesis combines
Quoted text
“combination of a manifold in general can never come to us through the senses”
Source in context · paraphrase
Kant distinguishes the manifold supplied to intuition from the spontaneous act by which understanding combines it.
How it is used here
This gives the argument its first and third terms: a manifold is given, and an act combines it.
Source 02Immanuel KantB131-135
Combination answers to one apperception
Quoted text
“The I think must be able to accompany all my representations”
Source in context · paraphrase
Kant requires the manifold to be combinable in one self-consciousness and grounds combination in the original synthetic unity of apperception.
How it is used here
The proposed second role is this required unity: the manifold must belong to one consciousness if it is to count as one experience.
Source 03Charles Sanders Peirce1867, §§1, 6, 10-12
Peirce begins from the passage from many to one
Quoted text
“reduce the manifold of sensuous impressions to unity”
Source in context · paraphrase
Peirce seeks the elementary conceptions required between a manifold of impressions and unity, culminating in representation through an interpretant.
How it is used here
Kant and Peirce can therefore be compared on a precise question: how can a plurality be held together as one?
Source 04Charles Sanders Peirce12 Oct. 1904; CP 8.328-329
An explicit Kant-Peirce correspondence
Quoted text
“They correspond to the three categories of each of the four triads of Kant's table.”
Source in context · paraphrase
Peirce defines Firstness by non-reference, Secondness by relation to another, and Thirdness by mediation.
How it is used here
Peirce himself connects his three categories with the three places in each of Kant's four groups. That makes the comparison historically grounded, though it does not settle the exact pairing argued for here.

Limit of the claim The texts establish a real point of comparison, but they do not establish the exact pairing proposed here. Reading Kant's manifold, unity, and synthesis through Peirce's Firstness, Secondness, and Thirdness is this work's central interpretive claim. Kant's category names are not being replaced.

Immanuel Kant is cited by A/B pagination; Charles Sanders Peirce by Collected Papers volume and paragraph. The links open public-domain transcriptions. The manuscript bibliography records the editions used by the paper.

Portraits: Becker, 1768 · Sarony, 1891. Public domain.

Step 2 · input and output

Each heading is modeled as a passage with a starting point and a result.

At either end, 0 means closed: the determination can be stated on its own. 1 means indexed: it must be stated relative to something else, such as a subject, set, state, time, or relatum. The first digit describes the input; the second describes the output.

Build one profileChoose how the input and output depend on context, then read from left to right.

First digitInput boundaryChoose one

0closedstated on its own1indexedrelative to context

Second digitOutput boundaryChoose one

0closedstated on its own1indexedrelative to context

The order matters. 10 means indexed input → closed output; 01 means closed input → indexed output.

Boundary: One endpoint of the passage being classified: its input or its output.
Relation to Peirce: A closed end resembles Firstness because it is considered on its own. An indexed end resembles Secondness because it depends on another. The status of the whole passage is considered later.

Textual basis

What the sources show

The model begins with a simple contrast: stated on its own or stated relative to something else.

Kant distinguishes what the senses give from the act that combines it. Peirce distinguishes a character considered in itself from a determination that depends on another term. The model turns this contrast into a two-part test.

Source 01Immanuel KantB129-130
The manifold is given; combination is enacted
Quoted text
“combination of a manifold in general can never come to us through the senses”
Source in context · paraphrase
Sensibility is receptive, whereas synthesis is an act of the understanding's spontaneity and must be unitary across combinations.
How it is used here
This supports distinguishing what is simply given from what is determined through an added relation or context.
Source 02Charles Sanders Peirce12 Oct. 1904; CP 8.328
Reference changes the categorial role
Source summary · paraphrase
Peirce's First is what it is without reference to another; his Second is what it is with respect to a second, before any mediating third is considered.
How it is used here
The model records this difference at each end of a passage: 0 means stated on its own; 1 means stated relative to something else.

Limit of the claim Kant and Peirce do not use the terms closed, indexed, input, or output in this way. Those are tools introduced here to make the proposed comparison testable.

Portraits: Becker, 1768 · Sarony, 1891. Public domain.

Step 3 · four possible profiles

Two possible statuses at two ordered positions produce four profiles.

The four possibilities are 00, 10, 01, and 11. Because input comes first, 10 and 01 say different things. The digits record whether each boundary is closed or indexed; they do not count objects. This work proposes the following assignment: Quality 00, Quantity 10, Modality 01, and Relation 11.

Profile: A two-digit summary of contextual dependence at input and output: 0 means closed and 1 means indexed. This is not a count of objects; the one/many contrast belongs within Quantity, as Unity and Plurality. Thus 10 means indexed input and closed output, while 01 means the reverse.
Heading assignment: The four codes follow from the two choices. Pairing a code with a Kantian heading is a philosophical proposal that needs its own argument.

input2 statusesoutput2 statusesordered result4 profiles

Proposed profile

Quality

Example · determination retained locallyclosed → closed

Input0closedOutput0closed

The determination has no free contextual parameter at either boundary.

Proposed profile

Quantity

Example · indexed items gathered into a resultindexed → closed

Input1indexedOutput0closed

An indexed input is gathered into a closed determination.

Proposed profile

Modality

Example · content assessed across conditionsclosed → indexed

Input0closedOutput1indexed

A closed content is given standing relative to indexed conditions.

Proposed profile

Relation

Example · indexed relata organized by a ruleindexed → indexed

Input1indexedOutput1indexed

Both boundaries remain indexed to relata organized by a relation or rule.

Textual basis

What the sources show

Kant organizes judgment under four headings and connects them with the synthesis of intuition.

Quantity, Quality, Relation, and Modality are four ways in which a judgment can have unity. Kant then says that the same functions also give unity to the synthesis of intuition.

Source 01Immanuel KantA68-70/B93-95
Four forms of unifying representation
Quoted text
“the function of thought in judgment can be brought under four heads, each of which contains three moments”
Source in context · paraphrase
Kant first characterizes judgments as functions of unity, then names Quantity, Quality, Relation, and Modality as the four heads.
How it is used here
This establishes the four headings that the model must explain.
Source 02Immanuel KantA77-80/B103-105
One function crosses judgment and intuition
Source summary · paraphrase
A manifold is given, imagination synthesizes it, and concepts of understanding give that synthesis unity. Kant then attributes unity in judgment and unity in intuition to the same function.
How it is used here
This makes it reasonable to ask whether the four headings describe four different ways of organizing a manifold.

Limit of the claim Kant supplies the four headings and their connection with unity. He does not derive them from the four 00, 10, 01, and 11 profiles. That derivation is proposed by this work.

Portrait: Becker, 1768. Public domain.

Step 4 · reverse the passage

Swapping input and output leaves 00 and 11 unchanged but exchanges 10 and 01.

The reversal rule is r(a,b) = (b,a). Quality (00) and Relation (11) have the same status at both ends, so reversing them changes nothing. Quantity (10) and Modality (01) become each other. This makes them the only nontrivial mirror pair. The comparison below asks whether that reversal is accompanied by a parallel operator sequence in their two modern models.

Reversal: The operation r(a,b) = (b,a), which exchanges input and output. Thus 10 becomes 01, and 01 becomes 10.
Reversal class: A group of profiles that become one another under reversal: {00}, {10, 01}, and {11}.
Boolean case: With only true and false as values, join is OR and meet is AND. NOT exchanges them by De Morgan's laws. Evaluation is different: it selects one member or state and reads the value there.

Reversal preserves the symmetric profiles 00 and 11 and exchanges only the mixed profiles 10 and 01.

Local operator mirror · 10 ↔ 01

The reversed profiles carry a parallel operator sequence.

Quantity ranges over members of a domain; Modality ranges over admissible states. In these two modern models, their three positions are realized as join, evaluation, and meet.

10 · indexed → closedQuantitymembers of a domain

Corresponding roles

01 · closed → indexedModalityadmissible states

Plurality

⋁ / \exists

some member

⟨1⟩ in this pairJoin

Possibility

◊

some state

Unity

ev_{a}

this member

⟨2⟩ in this pairEvaluation

Existence

ev_{d}

the designated actual state

Totality

⋀ / \forall

every member

⟨3⟩ in this pairMeet

Necessity

□

every state

Cross-heading abstraction

The two orders give the same result.

$H$ = Quantity or Modality

Aggregate, then negate

⟨1⟩ · HSome / possible

⟨2⟩ · QualityNegation

Negate, then aggregate

⟨2⟩ · QualityNegation

⟨3⟩ · HEvery / necessary

In the formula, the right-hand operation acts first. On the left, negation acts on the aggregate. On the right, it acts on each indexed value before aggregation. Both routes say that the content holds at no member or admissible state.

Quantity

Modality

Boolean dualityBoolean negation exchanges the mirror's outer operators.In the Boolean case, “not some” becomes “every not.”

Middle positionThe designated index stays fixed. Negation changes the value found there, not which member or state was selected.

Historical precedent

The operator analogy is established; its category assignment is not.

Across the category rows. In 1904, Charles Sanders Peirce explicitly aligned Firstness, Secondness, and Thirdness with the three positions in each of Immanuel Kant's four category triads. That establishes a precedent for comparing the rows across all four headings.

The explicit operator analogy. In 1946, Rudolf Carnap gave the earliest explicit modern statement documented here. He wrote that necessity N is “analogous to a universal quantifier” and possibility ◇ is “analogous to an existential quantifier.” His state descriptions make the shared pattern semantic: necessity ranges over every state-description, while possibility holds in at least one. Gottfried Wilhelm Leibniz supplied the possible-world background; Saul Kripke later generalized the semantics by relating states through accessibility.

What this work adds. Carnap's analogy establishes the precedent for pairing some with possible and every with necessary. It does not formulate the typed role equivalence above, derive it from Kant's table, establish the designated middle position, or justify the particular pairings of Plurality with Possibility, Unity with Existence, and Totality with Necessity. Nor does it show that Quantity and Modality occupy the reversed profiles 10 and 01. Those are the claims tested here.

Gottfried Wilhelm Leibniz, Theodicy (1710)Charles Sanders Peirce, letter to Victoria Lady Welby (1904)Rudolf Carnap, “Modalities and Quantification” (1946)Saul Kripke, “A Completeness Theorem in Modal Logic” (1959)

Textual basis

What the sources show

Quantity gathers parts into one whole; Modality places one content under conditions.

In Kant's account of magnitude, Quantity proceeds from parts toward a whole. Modality does not add content; its categories are Possibility, Existence, and Necessity. The proposed profiles represent these as one content placed under different conditions.

Source 01Immanuel KantA162/B203
Quantity gathers a manifold into a whole
Source summary · paraphrase
An extensive magnitude is cognized through the successive synthesis of homogeneous parts into one representation.
How it is used here
The model therefore reads Quantity as a movement from context-dependent parts toward one completed whole.
Source 02Immanuel KantA74/B99-100; A219/B266
Modality assigns standing without adding content
Source summary · paraphrase
Modal form does not enlarge the judgment's content; it concerns how that content stands in relation to thought, experience, and the faculties of knowledge.
How it is used here
The model reads Modality in the other direction: one content is considered relative to a condition or state.

Limit of the claim The texts support the contrast between composition and modal standing. They do not encode it as 10 and 01. That formal reversal belongs to the model.

Portrait: Becker, 1768. Public domain.

Step 5 · ignore direction

If direction is ignored, the four profiles fall into three groups.

The groups are {00}, {10, 01}, and {11}, with sizes 1 : 2 : 1. They contain zero, one, and two indexed ends, respectively. That grouping is exact. Comparing it with Peirce's three categories is a further philosophical proposal.

Quotient: The mathematical name for the simpler classification obtained when profiles that differ only by reversal are treated as equivalent.

Forgetting direction merges 10 with 01 and leaves the symmetric profiles 00 and 11 fixed, producing class sizes 1 : 2 : 1 for {00}, {10, 01}, and {11}, respectively.

Proposed Peircean readingWith no indexed ends, a passage can be considered on its own. With one, it depends on another term. With two, both ends depend on context.Two indexed ends do not by themselves establish Thirdness. The model must also show that a rule coordinates them. Relation is the crucial test of that claim.

⟨1⟩
C0 · Compared with Firstness · class size 1
No indexed boundaries
00 Quality
self-contained at both ends
⟨2⟩
C1 · Compared with Secondness · class size 2
One indexed boundary
10 Quantity · 01 Modality
context on one side of the passage
⟨3⟩
C2 · Compared with Thirdness · class size 1
Two indexed boundaries
11 Relation
two context-bound sides coordinated together

Textual basis

What the sources show

Reversing a one-sided profile changes its direction, not the fact that it has two unequal sides.

The profiles 10 and 01 are different because their directions are opposite. Yet both contain one self-contained end and one context-dependent end. The model therefore groups them together when direction is ignored.

Source 01Charles Sanders Peirce12 Oct. 1904; CP 8.328-329
One-sided determination remains Secondness
Source summary · paraphrase
Peirce distinguishes the direction of action from the more general fact that a dyadic determination is involved, and places mediation beyond that dyad.
How it is used here
This helps explain why Quantity and Modality may face opposite ways while belonging to the same one-sided group.
Source 02Charles Sanders PeirceCP 1.530-543
The categories recur within one another
Source summary · paraphrase
Peirce distinguishes the Firstness of Secondness, the Firstness and Secondness of Thirdness, and genuine from degenerate mediation.
How it is used here
Peirce's own practice shows that his categories can describe patterns within more complex structures. It does not by itself prove this particular grouping.

Limit of the claim The three groups {00}, {10, 01}, and {11} follow exactly from the reversal rule. Calling them Firstness-like, Secondness-like, and Thirdness-like is a further philosophical interpretation, not a mathematical result.

Portrait: Sarony, 1891. Public domain.

Step 6 · cross rows and columns

Three role positions crossed with four heading profiles give twelve places.

The multiplication is straightforward: each of the three roles can appear under each of the four headings, so 3 × 4 = 12. The symbol μ names the proposed assignment of a Kantian category to each place. The grid proves the count; it does not prove that the assignments are correct.

Cartesian product: The set of all twelve possible pairs made from three roles and four headings.
Assignment rule μ: The claim that every role-heading pair has one corresponding category in Kant's table.

Formal count3 role positions × 4 directed profiles = 12 coordinates

Proposed assignmentAt each crossing, μ names the category assigned to that role and heading.

Internal role ↓μ(r,h)External heading →

00Qualityclosed → closed

10Quantityindexed → closed

01Modalityclosed → indexed

11Relationindexed → indexed

⟨1⟩PresentationFirstness

⟨2⟩DiscriminationSecondness

⟨3⟩Mediated IntegrationThirdness

Selected crossingReality / Affirmative = Presentation × Quality

The blue line marks the selected role. The red line marks the selected heading. Their green crossing is the category proposed to satisfy both descriptions.

3 × 4 = 12

Textual basis

What the sources show

Three proposed roles combined with four proposed profiles give twelve positions.

Kant's table has three categories under each of four headings. This work interprets the rows as three roles and the columns as four profiles. Pairing every row with every column produces the table's twelve positions.

Source 01Immanuel KantA68-70/B93-95
Kant supplies four headings with three moments each
Source summary · paraphrase
Kant brings the functions of thought in judgment under Quantity, Quality, Relation, and Modality, with three moments under each head.
How it is used here
This supplies the table's visible 3-by-4 arrangement.
Source 02Charles Sanders PeirceCP 1.530-543
The categories can recur at distinct levels
Source summary · paraphrase
Peirce distinguishes Firstness and Secondness within more complex categorial formations and differentiates genuine from degenerate Thirdness.
How it is used here
This permits the rows to be described with Peircean terms while the four Kantian headings remain a separate axis.

Limit of the claim The arithmetic is exact once those axes are accepted. The difficult claim is that each Kantian category belongs at its proposed row and column for one consistent reason.

Portraits: Becker, 1768 · Sarony, 1891. Public domain.

Step 7 · test the formal witnesses

The three roles take a different form under each heading.

No single operator sequence governs the whole table. Quality requires a scale of degree; Quantity, a domain of members; Modality, a range of admissible states; and Relation, a structure of connected terms. The atlas declares those four settings before testing all twelve categories, so each formal symbol is read only within the model that gives it meaning.

Modern model: A precise example of how a category could play its proposed role. The formulas belong to this reconstruction, while the category names and source passages remain Kant's.

Typed formal atlasTwelve categories require four kinds of model.Kant's names; explicitly modern witnesses.

Before reading the symbols

Each column declares its own formal setting.

These settings keep values, operations, quantifiers, and relations from being mistaken for one another. Assume the joins and meets displayed below exist; when V has only true and false, they become ordinary OR and AND.

00 · Qualitybounded degree scale

(V, \leq, ⊥, ⊤)

The scale is normalized between complete absence and a positive reference; limited degrees lie between them.

10 · Quantitydomain valuation

P : C \to V, a \in C

A predicate assigns values to members of a domain; the required joins and meets are assumed to exist.

01 · Modalitystate valuation

u : S \to V, d \in S

One content is evaluated over admissible states, one of which is designated as actual.

11 · Relationdependence relation

R \subseteq (X \times X) ∖ 1_{X}

An irreflexive relation records dependence among distinct relata; its one-way and reciprocal parts are derived below.

Internal role ↓ Heading →

00Qualitydegrees of determination

10Quantitysome / this / every

01Modalitypossibility / existence / necessity

11Relationpersistence / direction / reciprocity

⟨1⟩ · Role 1Presentationpositive · some · possible · persistent

Quality · degrees of determination

RealityAffirmative

reference pole

r = ⊤

normalized positive reference

Quantity · some / this / every

PluralityParticular

join / existential

⋁_{x \in C} P (x)

some support

Modality · possibility / existence / necessity

PossibilityProblematic

state join

◊_{S}

some admissible state

Relation · persistence / direction / reciprocity

SubstanceCategorical

identity surrogate

1_{X}

identity as a persistence surrogate

⟨2⟩ · Role 2Discriminationopposition · selection · direction

Quality · degrees of determination

NegationNegative

zero value

r = ⊥

complete negation

Quantity · some / this / every

UnitySingular

evaluation

ev_{a}

this member

Modality · possibility / existence / necessity

ExistenceAssertoric

state evaluation

ev_{d}

the designated actual state

Relation · persistence / direction / reciprocity

CausalityHypothetical

asymmetric part

R_{\to}

one-way dependence

⟨3⟩ · Role 3Mediated Integrationlimitation · all · necessary · reciprocal

Quality · degrees of determination

LimitationInfinite

interval condition

⊥ < r < ⊤

bounded positive degree

Quantity · some / this / every

TotalityUniversal

meet / universal

⋀_{x \in C} P (x)

exhaustive support

Modality · possibility / existence / necessity

NecessityApodictic

state meet

□_{S}

every admissible state

Relation · persistence / direction / reciprocity

CommunityDisjunctive

symmetric core

R \cap R^{op}

direct reciprocal dependence

Quality · a normalized degree scale

Reality supplies the positive reference; Negation supplies the zero; Limitation lies between them.

$⊥$ complete negation $r$ limited degree $⊤$ positive reference

Limitation $⊥ < r < ⊤$

Kant treats reality in appearance as admitting degrees and complete negation as their zero point. To distinguish the three categorical positions, this model normalizes the full positive reference as ⊤ and complete absence as ⊥. Reality names the reference pole; Negation names its zero; Limitation is a positive determination bounded between them. The quantifier some belongs to Quantity's model of the first position; it does not define the first position itself. Quality's first position is a reference pole, not a merely existential condition over positive degrees. Here ⊤ is a formal normalization, not the claim that every Kantian reality has maximum empirical intensity.

The middle row

Exclusion can define a selection; evaluation reads it.

x1⊥
x2⊥
a⊤
x4⊥
x5⊥

$δ_{a} = (⊥, ⊥, ⊤, ⊥, ⊥)$ $ev_{a} (P) = P (a)$

In a Boolean encoding, the mask marks a true and every other coordinate false. That exclusion explains the intuitive connection with NOT. The resulting operations for Unity and Existence are nevertheless evaluations: P(a) for the selected member and u(d) for the designated actual state. Boolean complement changes a value; evaluation chooses where to read one.

Why Relation is different

Persistence, direction, reciprocity

1_{X}

R_{\to}

R \cap R^{op}

Relation concerns connections among things, not truth values spread across a collection, so the atlas uses relational surrogates. The identity relation ( $1_{X}$ ) models identity through change; the asymmetric part ( $R_{\to}$ ) retains the one-way pairs in the dependence relation; and the symmetric core ( $R \cap R^{op}$ ) retains exactly those pairs related in both directions. A merely indirect path back is not counted as direct reciprocal interaction.

What is sharedEach heading is asked the same three questions: what is presented, what introduces a distinct second term, and what joins the terms into one structure. The formal object answering those questions may be a value, a condition, an evaluation, a join or meet, or a relation. The shared order does not supply a shared operator or quantifier.

Textual basis

What the sources show

The same three questions take a different form under each Kantian heading.

Under Quantity, the issue is some, this one, or every. Under Modality, it is Possibility, Existence, or Necessity. Quality concerns affirmation, negation, and limitation; Relation concerns persistence, directed dependence, and reciprocity. The modern formulas make these differences explicit instead of pretending that every heading uses the same operation.

Source 01Immanuel KantB110-111
Each third category requires a distinct act of combination
Source summary · paraphrase
Kant describes Totality as Plurality considered as Unity, Limitation as Reality joined with Negation, Community as reciprocal causality, and Necessity as existence given through possibility.
How it is used here
This supports a common pattern with four different realizations: in each heading, the third category joins the first two by a special act.
Source 02Immanuel KantA71-72/B96-97
Singular judgment designates one
Source summary · paraphrase
Kant says a singular judgment may behave like a universal one in inference while remaining essentially different as cognition: it stands to the universal as unity to infinity.
How it is used here
The model treats Unity as the selection of one member: neither an open plurality nor the whole collection.
Source 03Immanuel KantA182/B224; A189/B232; A211/B256
Relation advances from persistence to direction to reciprocity
Source summary · paraphrase
The Analogies articulate Substance as permanence, Causality as succession under a rule, and Community as coexistence through reciprocal interaction.
How it is used here
The proposed Relation model follows that sequence: what persists, what depends on what, and what interacts reciprocally.

Limit of the claim The formulas are modern models, not formulas found in Kant. The Boolean reading applies directly to the proposed models of Quantity and Modality. The quantifier 'some' describes Quantity's first position, not the first position as such, and therefore does not define Reality. Quality and Relation require models of degree and relation instead. The usefulness of each model must be judged separately.

Portrait: Becker, 1768. Public domain.

Step 8 · check every placement

The final table lets each proposed assignment be checked separately.

Choose a category. Its row states the role it is supposed to play; its column states how its heading treats context at input and output. The inspector then gives the reason for each classification, a relevant source passage, and the modern model used here.

Read across a rowThe internal role stays fixed; the left edge carries its color.
Read down a columnThe directed heading stays fixed; its endpoints appear above.
Select an intersectionGold marks the category; dark ticks locate its row and column.

Role ↓ Heading →

00Qualityclosed → closed

10Quantityindexed → closed

01Modalityclosed → indexed

11Relationindexed → indexed

Role 1PresentationFirstness

Role 2DiscriminationSecondness

Role 3Mediated IntegrationThirdness

Selected category

RowColumn

Modern model

Reality

Column · external heading →00 · Quality

Row · internal role ↓Role 1 · Presentation

Dependency graph

No dependency arrows

Each relatum persists without dependence on a distinct relatum.

Example matrix

Identity only

1ₓ

Rows ↓ source xᵢColumns → target xⱼ

The diagonal records each relatum as itself, not an inter-relatum edge.

Dependency graph

Directed acyclic example

Dependence advances through the ordering; no arrow leads back.

Example matrix

Strictly upper-triangular

R→ = R ∖ Rᵒᵖ

Rows ↓ source xᵢColumns → target xⱼ

Every marked source-target pair points forward in the ordering.

Dependency graph

Direct reciprocal pairs

Every displayed dependency is paired with a dependency in the reverse direction.

Example matrix

Symmetric across the diagonal

R ∩ Rᵒᵖ

Rows ↓ source xᵢColumns → target xⱼ

Every off-diagonal pair is accompanied by its transpose.

normalized positive reference

Why it fits the row
Why it fits the column

Compare the alternatives

How this choice differs from its neighbors

Peircean sourceSource summarized
Kantian sourceSource summarized
Modern formula
Status of the claim

Textual basis

What the sources show

The final test is whether one rule explains all twelve placements.

Kant says the categories must form a system derived from a common principle rather than a list collected by trial and error. The matrix therefore has to do more than reach the number twelve: it must explain every placement by the same two questions about role and profile.

Source 01Immanuel KantA80-81/B106-107
A system must arise from a common principle
Source summary · paraphrase
Kant contrasts his derivation with a rhapsodic search: the categories are to form a complete, ordered system generated from the faculty of judgment.
How it is used here
A successful reconstruction must explain all twelve categories by one repeatable rule.
Source 02Immanuel KantA182/B224; A189/B232; A211/B256
Relation advances from persistence to direction to reciprocity
Source summary · paraphrase
The Analogies articulate Substance as permanence, Causality as succession under a rule, and Community as coexistence through reciprocal interaction.
How it is used here
The Relation column must therefore show a clear progression from persistence, through one-way dependence, to reciprocal interaction.
Source 03Charles Sanders Peirce1867, §§9-12
A genuine third unifies through mediation
Source summary · paraphrase
Peirce argues that comparison requires a mediating representation and that a manifold is unified only when its differentiated elements are held together through an interpretant.
How it is used here
Each selected cell must satisfy both its row and its column. The crossing in the diagram shows exactly which two claims are being combined.

Limit of the claim The 3-by-4 form is the hypothesis being tested. The cell inspector shows the reason for each assignment so that a weak placement can be challenged without accepting or rejecting the whole table at once.

Portraits: Becker, 1768 · Sarony, 1891. Public domain.

Kant's Table, Step by Step

Follow the Argument

What happens within a heading?

How do the headings differ?

How do the two axes combine?

Presentation

Discrimination

Mediated Integration

No indexed boundaries

One indexed boundary

Two indexed boundaries

Reality supplies the positive reference; Negation supplies the zero; Limitation lies between them.

Exclusion can define a selection; evaluation reads it.

Persistence, direction, reciprocity