Kant's Table as a Recursive Peircean Product: A Structural Reconstruction of the Three Moments, Four Headings, and Infinite Judgment

Ian Scott Knight

Working paper · Immanuel Kant and Charles Sanders Peirce

Kant's Table as a Recursive Peircean Product

A Structural Reconstruction of the Three Moments, Four Headings, and Infinite Judgment

Author Ian Scott Knight
Version manuscript v4
Date September 2026
Status working paper

Central thesis

From many impressions to one experience

Immanuel Kant calls what is given through the senses the manifold of sensibility. These many representations can belong to one experience only under the unity of apperception: one self-consciousness able to accompany them. Synthesis is the act that combines them. Charles Sanders Peirce begins from a related question: what must happen for a manifold of impressions to be held together as one?

Kantmanifoldcombined underunitybysynthesis

Peircemanybrought toonethroughmediation

The proposed correspondence concerns three roles: what is given, the unity to which it must belong, and the act that combines it.

⟨1⟩Presentation
Firstness
Manifold of sensibility
The multiplicity given in sensible intuition, considered apart from the act that combines it.
Kant, Critique of Pure Reason, A77/B102; B129-130
⟨2⟩Discrimination
Secondness
Unity of apperception
The one self-consciousness under which the manifold must be combinable as mine.
Kant, Critique of Pure Reason, B131-135
⟨3⟩Mediated Integration
Thirdness
Synthesis
The spontaneous act that combines a given manifold and brings it under apperceptive unity.
Kant, Critique of Pure Reason, A77/B102; B129-135

Terms and sources. Kant's terms are manifold, unity of apperception, and synthesis. Firstness, Secondness, and Thirdness are Peirce's terms. The argument compares their functions; it does not attribute Peirce's vocabulary or notation to Kant. Analytic of Concepts B-Deduction Charles Sanders Peirce's New List

Abstract

Immanuel Kant presents twelve categories as three moments under each of four headings. The table is systematic, but its printed form does not explain why one axis has three places, why the other has four, or why they belong to one structure. This paper begins with Kant's account of experience: a manifold is given in sensibility, its representations must be able to belong to one unity of apperception, and synthesis combines them under that unity. It compares those three functions with Charles Sanders Peirce's Firstness, Secondness, and Thirdness, then tests the comparison against the table.

The proposed roles are presentation $⟨ 1 ⟩$ , discrimination $⟨ 2 ⟩$ , and mediated integration $⟨ 3 ⟩$ . The category names remain Kant's; the role labels describe what each category is taken to do within its heading. A completed determination may be closed at a boundary or indexed to an explicit subject, part, state, time, or relatum. Applying that distinction to input and output gives four profiles: closed–closed, indexed–closed, closed–indexed, and indexed–indexed. These are assigned to Quality, Quantity, Modality, and Relation. Reversal exchanges Quantity with Modality while fixing Quality and Relation; ignoring direction then gives a $1 : 2 : 1$ pattern. Formally,

K H ρ_{H} ≅ R \times H, = {c, i}^{{in, out}}, = λ \circ ω : H \to R

where the isomorphism preserves each category's heading, $ω$ counts indexed boundaries, $λ$ matches the ordered counts $0 < 1 < 2$ with the three roles, and $ρ_{H}$ is unchanged by input–output reversal $τ$ . In Quantity, the proposed order is Particular–Plurality– $⟨ 1 ⟩$ , Singular–Unity– $⟨ 2 ⟩$ , and Universal–Totality– $⟨ 3 ⟩$ .

The proposal explains the $3 \times 4$ shape, the $1 : 2 : 1$ division of the headings, the mirror between Quantity and Modality, the mathematical–dynamical division, and the different forms taken by the third moment. Its main exact result concerns infinite judgment. In a complete De Morgan lattice, negation after singular evaluation agrees with predicate complement before evaluation. On a non-singleton domain, however, the predicate-level factorization retains more structure, and pointwise complement is the unique substitution-natural extensional inner factor in the stated universal-from-existential construction. The formalism is a modern reconstruction. It is not attributed to Kant or Peirce, and it does not replace Kant's transcendental deduction.

What the Reconstruction Claims

Central correspondence.Kant's account of combination differentiates a manifold given in sensible intuition, the unity of apperception under which it must be combinable, and synthesis as the act that brings it under that unity. The reconstruction assigns these functions to $⟨ 1 ⟩$ , $⟨ 2 ⟩$ , and $⟨ 3 ⟩$ , respectively. Peirce's own passage from a sensuous manifold to unity through mediation makes the comparison structural rather than merely numerical.
Moment structure.Each of Kant's four headings can be role-normalized by one invariant triad: presentation $⟨ 1 ⟩$ , discrimination $⟨ 2 ⟩$ , and mediated integration $⟨ 3 ⟩$ . This changes the categories' comparative order, not their identities or names: the table remains Kant's twelve categories.
Operator structure.Quantity and Modality instantiate an exact join–designation–meet pattern; in Boolean models, the outer operations become OR and AND and are exchanged by negation through De Morgan duality. Quality realizes positive determination, opposition, and limitation; Relation realizes persistence, direction, and reciprocity. The invariant is the functional order shared by these heading-specific operations.
Heading structure.Closed versus context-indexed determination at input and output yields four directed profiles: $00$ , $10$ , $01$ , and $11$ . Input-output reversal fixes $00$ and $11$ while exchanging $10$ and $01$ , so the heading tetrad is an oriented $1 : 2 : 1$ refinement of a triad.
Table structure.The normalized table is a role-trivialized family over the four headings, represented by $K ≅ R \times H$ at the level of classified cells.
Quantity correspondence.The role normalization intentionally pairs Particular–Plurality– $⟨ 1 ⟩$ , Singular–Unity– $⟨ 2 ⟩$ , and Universal–Totality– $⟨ 3 ⟩$ .
Infinite judgment.Negative and infinite singular judgments may coincide extensionally while differing in compositional structure; on non-singleton domains, the predicate-level infinite factorization strictly refines the value-level negative factorization.

What Is Mathematically Proved

the exact reversal-orbit structure of the four boundary profiles and its $1 : 2 : 1$ quotient;
the product-over-the-heading-base statement conditional on the role-structure realizations;
indexed adjunctions for existential and universal aggregation;
indexed De Morgan duality;
strict factorization refinement for negative versus infinite judgment;
singleton collapse of indexed versus local readouts;
Yoneda characterization of reindexing-natural indexed operations;
substitution-natural extensional uniqueness of pointwise complement in the universal-from-existential factorization;
the stated modal De Morgan dualities within the selected semantics;
the elementary relation-theoretic properties of the Community surrogate.

Supplementary scripts reproduce finite examples and check the implementation of selected definitions. They neither establish the general results nor provide evidence for the philosophical interpretation. Download the machine-check source bundle.

Visual Thesis

The correspondence begins before the table: a manifold is given, it must belong to one apperception, and synthesis combines it under that unity. The same triadic grammar then recurs as three internal roles, four oriented profiles, and twelve mediated intersections.

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.

The count 3 × 4 = 12 records the result. The reconstruction concerns the single rule that makes every crossing answer to both axes.

Follow the eight-stage argument

What Remains Interpretive

The mathematics does not by itself prove that Kant's table is this structure. The substantive philosophical burden remains in:

defending the architectural correspondence manifold– $⟨ 1 ⟩$ , apperceptive unity– $⟨ 2 ⟩$ , and synthesis– $⟨ 3 ⟩$ without presenting it as Kant's own notation;
identifying closed and indexed determination as the relevant endpoint contrast;
assigning Quality, Quantity, Modality, and Relation to $00$ , $10$ , $01$ , and $11$ , respectively, at one semantic level;
interpreting the reversal quotient as Peircean at the heading level;
establishing that the same presentation–discrimination–mediation grammar is genuinely realized in all four headings;
defending the revisionary Quantity correspondence;
showing that the modern formal reconstruction illuminates Kant's systematic ambitions without being attributed historically to Kant.

Formal Model

Mathematical surrogates that make the claimed structure inspectable.

Historical-Systematic Argument

The philosophical claim that this formal structure captures the texts well.

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Citation

@unpublished{Knight2026RecursivePeirceanProduct,
  author = {Knight, Ian Scott},
  title = {Kant's Table as a Recursive Peircean Product: A Structural Reconstruction of the Three Moments, Four Headings, and Infinite Judgment},
  year = {2026},
  month = sep,
  note = {Working paper, manuscript version v4},
  url = {https://ianscottknight.com/research/kant-table/}
}

Keywords

Kant; table of categories; Peirce; recursive structure; boundary profile; order theory; categorical logic; infinite judgment; De Morgan duality.

Manifold of sensibility

Unity of apperception

Synthesis