Visual Argument

Recursive Peircean Product

One Peircean triad generates the table twice: internally as three dependence roles, externally as four oriented profiles, and synthetically as twelve warranted intersections.

3 internal roles4 external profiles12 mediated cells

Follow the Argument

The sequence moves from plain-language relations to formal notation. Each step establishes the vocabulary needed by the next.

Guided reconstruction

One Peircean grammar is applied recursively: as internal form ⟨1⟩, external differentiation ⟨2⟩, and mediated realization ⟨3⟩.

Step 1 of 8: Internal triad1 of 8

One grammar · recursively applied

The Peircean triad is the sole structural engine.

The same Firstness–Secondness–Thirdness grammar appears at three distinct levels. The labels name each level's structural function, not the cardinalities encountered there: 3, 4, and 12.

Meta-Firstness
Internal form
3 invariant roles
The complete role triad is held intact and repeated within every heading.
Meta-Secondness
External differentiation
4 directed profiles
Closed and indexed characters occupy two ordered boundary positions.
Meta-Thirdness
Mediated realization
12 classified cells
One realization law coordinates one role with one heading in every cell.

Meta-Firstness · internal form

The complete triad congeals into one invariant internal axis.

This axis contains all three dependence roles. Taken intact before any heading differentiates it, that complete triad functions as ⟨1⟩ at the higher architectural level and repeats within every heading.

Internal role axis ↓One continuous articulation, read through three dependence roles.Categorial continuity; no topology is asserted.

⟨1⟩P(x)

Presentation

Firstness

selected feature P

xpresented term

No distinct correlate is required.

At the selected level of analysis, the feature is specified without essential reference to a distinct correlate.

⟨2⟩R(x,y)

Discrimination

Secondness

xselected term

determined through

yexplicit correlate

At the selected level of analysis, the feature is specified through an explicit correlate, opposition, designation, reaction, position, or direction.

⟨3⟩M(x,y;z)

Mediated Integration

Thirdness

zmediator or law

xfirst term

ysecond term

One rule jointly organizes both terms.

At the selected level of analysis, a general rule or mediator jointly determines distinguishable terms as members of one organized structure.

⟨1⟩ retained wholeThe three moments remain one invariant form before the external heading axis differentiates them.

Primary-text warrant

Editorial synthesis

The triadic recurrence is explicit in both texts.

Kant identifies three moments within each class; Peirce explicitly reads his universal categories as corresponding to those recurrent triads.

Source 01Immanuel KantB110-111
A distinct third act
Direct quotation
“the third category in each class always arises from the combination of the second category with the first”
Editorial context · paraphrase
Kant immediately adds that the third is primitive because producing it requires a special act of understanding.
Use in this reconstruction
This anchors a three-position progression that preserves the difference between first, second, and genuinely synthetic third moments.
Source 02Charles S. Peirce12 Oct. 1904; CP 8.328-329
An explicit Kant-Peirce correspondence
Direct quotation
“They correspond to the three categories of each of the four triads of Kant's table.”
Editorial context · paraphrase
Peirce defines Firstness by non-reference, Secondness by relation to another, and Thirdness by mediation.
Use in this reconstruction
This licenses testing each Kantian triad by one Peircean dependence grammar rather than treating the comparison as a numerical coincidence.

Reconstructive limitThese passages warrant a comparative triadic grammar. They do not by themselves establish the paper's names for the three roles or its higher-level recursion.

Kant is cited by A/B pagination; Peirce by Collected Papers volume and paragraph. Linked public-domain transcriptions are supplied for inspection; the manuscript bibliography records the edition used by the paper.

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

Meta-Secondness · input and output

External differentiation begins with one ordered contrast.

Closed is Firstness-like relative to this boundary question; indexed is Secondness-like. Thirdness belongs to the passage between boundaries, not to a third boundary value.

Boundary: One of the two ordered positions in the passage being classified: its input or its output.

External axis ⟨2⟩One ordered passage has two distinct positions.

Input

0closed1indexed

Output

0closed1indexed

The same two characters are available at each endpoint; their order is retained when a passage is classified.

Closed boundary0 closed

No free context parameter

The determination is stated without a free contextual parameter.

Indexed boundary1 indexed

Relative to an explicit context

The determination is relative to a subject, part, state, time, or relatum.

Primary-text warrant

Editorial synthesis

Receptivity and combination are textually distinct.

Kant separates what sensibility gives from the spontaneous act that combines it; Peirce separates positive character from determination through another.

Source 01Immanuel KantB129-130
The manifold is given; combination is enacted
Direct quotation
“combination of a manifold in general can never come to us through the senses”
Editorial context · paraphrase
Sensibility is receptive, whereas synthesis is an act of the understanding's spontaneity and must be unitary across combinations.
Use in this reconstruction
The passage motivates distinguishing an available determination from its directed articulation through an additional contextual term.
Source 02Charles S. Peirce12 Oct. 1904; CP 8.328
Reference changes the categorial role
Attributed source account · 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.
Use in this reconstruction
That contrast supplies the relative dependence test encoded by the two boundary characters.

Reconstructive limitNeither author uses the paper's terms closed and indexed. Those are explicit reconstructive tests applied to the input and output of a completed determination.

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

Two characters · two ordered positions

The external axis has four headings because direction matters.

Each input and each output can be closed or indexed. The four ordered combinations are therefore 00, 10, 01, and 11; swapping positions distinguishes Quantity from Modality.

Profile: The ordered pair formed by the input character and output character. Thus 10 and 01 are different profiles.

input2 charactersoutput2 charactersordered result4 profiles

Directed profile

Quality

one → one

Input0 closed

heading operationformal label Φ_Q

Output0 closed

Local determination within a value structure.

Directed profile

Quantity

many → one

Input1 indexed

heading operationformal label Φ_N

Output0 closed

Read an indexed manifold as one closed determination.

Directed profile

Modality

one placed in a field

Input0 closed

heading operationformal label Φ_M

Output1 indexed

Give one content a standing across indexed states or conditions.

Directed profile

Relation

many ↔ many

Input1 indexed

heading operationformal label Φ_R

Output1 indexed

Organize dependence among indexed relata under a law or rule.

Primary-text warrant

Editorial synthesis

Kant links four logical heads to the synthesis of a manifold.

The fourfold organization is not imposed on an unrelated theory of cognition: Kant presents the headings as functions of unity and then carries those functions into intuition.

Source 01Immanuel KantA68-70/B93-95
Four forms of unifying representation
Direct quotation
“the function of thought in judgment can be brought under four heads, each of which contains three moments”
Editorial context · paraphrase
Kant first characterizes judgments as functions of unity, then names Quantity, Quality, Relation, and Modality as the four heads.
Use in this reconstruction
The headings are therefore four differentiated modes of a common unifying activity, not four topics gathered after the fact.
Source 02Immanuel KantA77-80/B103-105
One function crosses judgment and intuition
Attributed source account · 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.
Use in this reconstruction
This is the strongest textual hinge for reading the four logical headings as externally differentiated ways of organizing a manifold.

Reconstructive limitKant gives the four headings, but he does not derive them from a two-by-two boundary square. The directed profiles are the paper's proposed explanation of that fourfoldness.

Portrait: Becker, 1768. Public domain.

Exchange input and output

Reversal fixes the symmetric profiles and exchanges the mixed profiles.

The mixed profiles 10 (Quantity) and 01 (Modality) are the only profiles that change under reversal: they have the same boundary count but opposite orientations.

Reversal: The operation that exchanges input and output: 10 becomes 01, and 01 becomes 10.
Reversal class: All profiles connected by that exchange: {00}, {10, 01}, and {11}.

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

Fixed under reversal
00Quality00Quality
Closed input and closed output remain the same when exchanged.
Converse pair
10Quantity01Modality
Both have one indexed boundary, but in opposite directions.
Fixed under reversal
11Relation11Relation
Indexed input and indexed output remain the same when exchanged.

Primary-text warrant

Editorial synthesis

Quantity and Modality point in opposite directions.

Kant characterizes Quantity by composition from parts and Modality by a standing assigned to an otherwise unchanged content.

Source 01Immanuel KantA162/B203
Quantity gathers a manifold into a whole
Attributed source account · paraphrase
An extensive magnitude is cognized through the successive synthesis of homogeneous parts into one representation.
Use in this reconstruction
This supports the Quantity profile's movement from an indexed manifold toward one completed determination.
Source 02Immanuel KantA74/B99-100; A219/B266
Modality assigns standing without adding content
Attributed source account · 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.
Use in this reconstruction
This supports the converse movement: one content receives a standing across explicit conditions or states.

Reconstructive limitThe source texts support the directional contrast. The exact reversal operation exchanging 10 and 01 is the reconstruction's formalization of it.

Portrait: Becker, 1768. Public domain.

Forget orientation · recover the triad

The heading tetrad is an oriented refinement of the same triad.

Once direction is forgotten, 00 (Quality) realizes ⟨1⟩; 10 (Quantity) and 01 (Modality) jointly realize ⟨2⟩; and 11 (Relation) realizes ⟨3⟩. The middle term splits in two only while orientation is retained.

Quotient: The coarser three-part classification obtained by treating profiles in the same reversal class as equivalent. It forgets direction while retaining zero, one, or two indexed positions.

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.

121

⟨1⟩
First-relative · class size 1
No indexed boundaries
00 Quality
closed → closed
⟨2⟩
Second-relative · class size 2
One indexed boundary
10 Quantity · 01 Modality
indexed → closed or closed → indexed
⟨3⟩
Third-relative · class size 1
Two indexed boundaries
11 Relation
indexed → indexed

Primary-text warrant

Editorial synthesis

Direction may vary while the dyadic role remains one.

Peirce treats one-sided action as Secondness and also permits Firstness and Secondness to recur within more complex categorial formations.

Source 01Charles S. Peirce12 Oct. 1904; CP 8.328-329
One-sided determination remains Secondness
Attributed source account · paraphrase
Peirce distinguishes the direction of action from the more general fact that a dyadic determination is involved, and places mediation beyond that dyad.
Use in this reconstruction
This supports treating Quantity and Modality as opposed orientations of one second-relative class.
Source 02Charles S. PeirceCP 1.530-543
The categories recur within one another
Attributed source account · paraphrase
Peirce distinguishes the Firstness of Secondness, the Firstness and Secondness of Thirdness, and genuine from degenerate mediation.
Use in this reconstruction
That recursive practice permits the four heading profiles to be classified again at a higher level without erasing their internal complexity.

Reconstructive limitPeirce does not state the orbit set {00}, {10, 01}, {11}. The quotient is exact mathematics; identifying its three classes with Peircean roles is a philosophical proposal.

Portrait: Sarony, 1891. Public domain.

Meta-Thirdness · mediated realization

One realization law synthesizes the internal and external axes.

Multiplication counts the cells; it does not explain the table. The structural claim is that one law μ coordinates each role with each heading, so every cell satisfies both warrants at once.

⟨1⟩ Internal continuity

One retained form

The three moments articulate one whole from within.

Categorial continuity; no topology is asserted.

⟨2⟩ External combinatorics

Two boundary characters occupy two ordered positions.

Input

0closed
1indexed

Output

0closed
1indexed

4directed profiles

one realization lawat every crossing

00Quality

10Quantity

01Modality

11Relation

⟨1⟩PresentationFirstness

⟨2⟩DiscriminationSecondness

⟨3⟩Mediated IntegrationThirdness

⟨3⟩ Mediated whole3 × 4 = 12classified intersections

Blue row-strands preserve internal role; red column-strands preserve external profile. One uniform map realizes their green intersections.

Reality / Affirmative = Presentation × Quality

⟨1⟩Internal role axis3 positions

⟨1⟩⟨2⟩⟨3⟩

⟨2⟩External heading axis4 directed positions

00100111

μsynthesizes

⟨3⟩Mediated table3 × 4 = 12 cells

Role repeated within each heading

00Qualityclosed → closed

10Quantityindexed → closed

01Modalityclosed → indexed

11Relationindexed → indexed

Role 1PresentationFirstness

RealityAffirmative

PluralityParticular

PossibilityProblematic

SubstanceCategorical

Role 2DiscriminationSecondness

NegationNegative

UnitySingular

ActualityAssertoric

CausalityHypothetical

Role 3Mediated IntegrationThirdness

LimitationInfinite

TotalityUniversal

NecessityApodictic

CommunityDisjunctive

Primary-text warrant

Editorial synthesis

Given manifold, unifying act, attained unity.

Kant supplies the cognitive progression; Peirce supplies an explicit categorial reading of its poles. The realization law makes the relation between the axes inspectable.

Source 01Immanuel KantB129-135
Apperceptive unity presupposes synthesis
Attributed source account · paraphrase
A manifold is given in sensible intuition, but its combination is an act of spontaneity. Kant argues that the identity of apperception across that manifold is possible only through consciousness of this synthesis.
Use in this reconstruction
At the architectural level, receptive diversity and self-conscious unity are not static inventories: a synthesizing act relates them. This is the strongest Kantian warrant for an externally articulated axis.
Source 02Charles S. PeirceCP 1.302
Peirce reads the Kantian poles categorially
Attributed source account · paraphrase
Peirce places Kant's manifold of sense under predominant Firstness and Kant's achieved synthetic unity under predominant Thirdness, which he associates with Totality.
Use in this reconstruction
This independently supports the passage from unselected multiplicity to law-governed integration while leaving room for the mediating work between them.
Source 03Charles S. Peirce1867, §§6-12
Mediation is what carries a manifold to unity
Attributed source account · paraphrase
Peirce searches for elementary conceptions between manifold and unity, introduces discrimination through a correlate, and requires an interpretant to hold differentiated impressions together.
Use in this reconstruction
This supplies the relational logic behind μ: distinguishable coordinates become one classified cell only through a mediating rule.

Reconstructive limitThe texts motivate the architecture but do not contain the product decomposition or the map μ. Those remain modern, testable parts of the reconstruction.

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

Worked column · Quantity

Quantity realizes the three roles as open support, selection, and exhaustive integration.

The ordering below is the paper’s stated normalization. Each formal witness follows the corresponding role criterion.

Internal role axis ↓One heading retained through all three role positions.Column 10 · Quantity · indexed input → closed output

Role 1Presentation
Plurality / Particular
$\exists_{C} P = ⋁_{x \in C} P (x)$
Particular scope presents open support without designating which witness supplies it.
Role 2Discrimination
Unity / Singular
$ev_{a} (P) = P (a)$
Singular judgment designates this member of the manifold rather than leaving the witness open or ranging exhaustively.
Role 3Mediated Integration
Totality / Universal
$\forall_{C} P = ⋀_{x \in C} P (x)$
Universal scope integrates every coordinate under one exhaustive condition.

One vertical columnThe external profile remains fixed while the internal role advances from presentation through discrimination to mediated integration.

Primary-text warrant

Editorial synthesis

The Quantity normalization has independent textual pressure.

Kant distinguishes singular designation from universal scope and describes Totality as Plurality unified; Peirce independently aligns manifoldness and attained unity with Firstness and Thirdness, respectively.

Source 01Immanuel KantA71-72/B96-97
Singular designation is not universal range
Attributed source account · 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.
Use in this reconstruction
This supports assigning singular judgment a distinct middle role of designation rather than treating it as disguised universal quantification.
Source 02Immanuel KantB110-111
Totality is plurality taken as unity
Attributed source account · paraphrase
Kant explicitly presents Totality as Plurality considered as Unity and insists that this requires a special act of understanding.
Use in this reconstruction
This directly supports the third role as exhaustive, attained integration rather than bare cardinality.
Source 03Charles S. PeirceCP 1.302
Multiplicity and attained unity occupy different roles
Attributed source account · paraphrase
Peirce associates measureless variety and multiplicity with predominant Firstness, and Kantian synthetic unity or Totality with predominant Thirdness.
Use in this reconstruction
This supplies an independent Peircean constraint on the outer terms of the proposed Quantity progression.

Reconstructive limitThe resulting Particular-Plurality, Singular-Unity, Universal-Totality alignment is intentionally revisionary with respect to Kant's printed row order.

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

Complete table · twelve intersections

Inspect each category as a role-heading pair.

Rows preserve the three role tests; columns preserve the four directed passages. Selecting a cell reveals both warrants and its formal witness.

⟨1⟩
Read downA row fixes the internal role.
⟨2⟩
Read acrossA column fixes the directed heading.
⟨3⟩
Select a cellThe intersection must satisfy both.

Left edgerow role: blue ⟨1⟩, red ⟨2⟩, green ⟨3⟩
Top edgeheading quotient class: blue ⟨1⟩, red ⟨2⟩, green ⟨3⟩
Header endpointscolumn-wide input → output: closed or indexed
Selectiongold marks the current cell; dark header ticks locate its row and column

Role ↓ Heading →

00Qualityclosed → closed

10Quantityindexed → closed

01Modalityclosed → indexed

11Relationindexed → indexed

Role 1PresentationFirstness

Role 2DiscriminationSecondness

Role 3Mediated IntegrationThirdness

Selected intersection

Column · external heading →

Row · internal role ↓

μ · selected cell · ⟨3⟩row and column jointly satisfied

Category witness

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 ⊆ X × X

Rows ↓ source xᵢColumns → target xⱼ

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

Dependency graph

Reciprocal directed cycles

Every dependency has a directed return, integrating the relata reciprocally.

Example matrix

Symmetric across the diagonal

R* ∩ converse(R*)

Rows ↓ source xᵢColumns → target xⱼ

Every off-diagonal pair is accompanied by its transpose.

positive determination

Why it satisfies the role
Why it satisfies the heading

Classification test

Why this cell rather than its neighbors?

Peircean role anchorAttributed paraphrase
Kantian category anchorAttributed paraphrase
Formal witness
Epistemic status

Primary-text warrant

Editorial synthesis

The table claims systematic unity, not merely completeness by enumeration.

Kant demands a common principle and Peirce requires mediation wherever differentiated elements are held together as one intelligible structure.

Source 01Immanuel KantA80-81/B106-107
A system must arise from a common principle
Attributed source account · paraphrase
Kant contrasts his derivation with a rhapsodic search: the categories are to form a complete, ordered system generated from the faculty of judgment.
Use in this reconstruction
The matrix must therefore display one repeatable rule of classification, not just preserve the familiar list of twelve names.
Source 02Immanuel KantA182/B224; A189/B232; A211/B256
Relation advances from persistence to direction to reciprocity
Attributed source account · paraphrase
The Analogies articulate Substance as permanence, Causality as succession under a rule, and Community as coexistence through reciprocal interaction.
Use in this reconstruction
That progression supplies a stringent visible test for the Relation column's three graph-theoretic witnesses.
Source 03Charles S. Peirce1867, §§9-12
A genuine third is not a heap of pairwise links
Attributed source account · 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.
Use in this reconstruction
The selected cell must therefore exhibit a rule that coordinates both axes, not merely sit at their geometric crossing.

Reconstructive limitNo cited passage proves that the twelve cells have this exact product form. The inspector exposes the cell-by-cell burden so the reconstruction can be challenged locally.

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

Recursive Peircean Product

Follow the Argument

Internal form

External differentiation

Mediated realization

Presentation

Discrimination

Mediated Integration

No indexed boundaries

One indexed boundary

Two indexed boundaries

One retained form

Plurality / Particular

Unity / Singular

Totality / Universal