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

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

Abstract

Kant presents twelve categories under four headings with three moments in each, but the printed table does not itself explain why one axis is triadic, why the other is fourfold, or why their crossing should form one system. This paper gives a downstream structural reconstruction in which all three features instantiate one recursive Peircean grammar.

Within every heading, the moments realize presentation $⟨ 1 ⟩$ , discrimination $⟨ 2 ⟩$ , and mediated integration $⟨ 3 ⟩$ . The heading axis is generated differently but not independently of that grammar. A completed determination may have a closed boundary, stated without a free contextual parameter, or an indexed boundary, stated relative to an explicit subject, part, state, time, or relatum. Assigning those two boundary characters independently at input and output yields four directed profiles: closed–closed, indexed–closed, closed–indexed, and indexed–indexed. Their orientation-forgetting quotient has exactly three classes: no indexed boundary, one indexed boundary in either direction, and two indexed boundaries. The four headings are therefore an oriented refinement of a Peircean triad: Quality, the two opposed directions Quantity and Modality, and Relation. The role axis is the invariant internal form repeated within every heading; the heading axis is the external differentiation of that form; and the uniform realization map joining them is their mediation. Formally,

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

where the first isomorphism preserves projection to the heading base, $ω$ counts indexed boundaries, $λ$ identifies the ordered weights $0 < 1 < 2$ with the ordered roles, and $ρ_{H}$ is invariant under input–output reversal $τ$ . In Quantity the reconstruction deliberately asserts Particular–Plurality– $⟨ 1 ⟩$ , Singular–Unity– $⟨ 2 ⟩$ , and Universal–Totality– $⟨ 3 ⟩$ .

This recursive architecture explains not merely the number twelve, but why the table has one threefold axis and one fourfold axis, why the four divide as $1 : 2 : 1$ , why Quantity and Modality are converse mixed cases, why Kant's mathematical–dynamical division tracks output boundary, and why one third-moment role can receive four different realizations. Its exact mathematical core concerns infinite judgment. In a complete De Morgan lattice, outer value-negation and inner predicate-complement coincide after singular evaluation. On every non-singleton domain, however, the predicate-level factorization strictly refines the value-level factorization, and pointwise complement is the unique substitution-natural extensional inner factor in universal-from-existential aggregation. The result is a modern, conditional, and falsifiable reconstruction, not a historical attribution of this formalism to Kant or Peirce and not a substitute for Kant's transcendental deduction.

What the Paper Claims

Moment structure. Each of Kant's four headings can be role-normalized by one invariant triad: presentation $⟨ 1 ⟩$ , discrimination $⟨ 2 ⟩$ , and mediated integration $⟨ 3 ⟩$ .
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 companion computational audit exhaustively checks large finite families of models and searches for counterexamples to the fixed finite claims. It is corroboration and regression testing, not a substitute for the general proofs in the manuscript.

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.

Visual Thesis

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

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.

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

Computational Audit Summary

3roles
4headings
12classified cells

Reversal classes: {00} · {10, 01} · {11}.
Plurality / Particular: $\exists_{C} P = ⋁_{x \in C} P (x)$ .
Unity / Singular: $ev_{a} (P) = P (a)$ .
Totality / Universal: $\forall_{C} P = ⋀_{x \in C} P (x)$ .

Read the audit page.

What Remains Interpretive

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

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.

Downloadable Artifacts

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Computational audit
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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.