\============================================================ CAPUT ARS BREVIS C — DE FIGURIS / QUATUOR FIGURAE Ars Generalis Applied — Knowledge Base Layer Version: 1.2.0-CAPUT-AB-C-ROSETTA-INTEGRATED Status: STABILIZED / INTERPRETATIVE \+ OPERATIONAL Scope: Ars Brevis — Chapter 2 (The Four Figures) Authority: ARS BREVIS / AEGIS (non-mutating reference) Mutation Policy: VERSION-CONTROLLED ONLY Class: AEGIS / CAPUT \============================================================ PURPOSE \------------------------------------------------------------ This CAPUT formalizes the four figures of Ars Brevis as an ordered combinatory system. It preserves: • scholastic exposition of the figures • formal structure of principial and relational operation • pairwise and triadic combinatorics • the role of the medium in inference • Rosetta anchor for integrated Latin / English figural parsing This CAPUT serves as: • interpretative key for Chapter 2 • operator-semantic bridge for figural logic • anti-drift reference for the use of figures \------------------------------------------------------------ CAPUT PRINCIPLE \------------------------------------------------------------ The figures (figurae) are not merely illustrations. They are: • ordered strata of operation • devices for finding media • combinatory engines of the Art Latin anchor: Figurae non sunt mere illustrationes; sunt strata operationis ordinata, instrumenta ad invenienda media, machinae combinatoriae Artis. \============================================================ I. FIGURE A (FIRST FIGURE) \============================================================ L1 — Scholastic Form \------------------------------------------------------------ The first figure is called A. (Latin: Prima figura per elementum A significatur.) It contains nine principles and nine letters. It is circular because subject becomes predicate and predicate becomes subject. (Latin: Circularis est quia subiectum fit praedicatum et praedicatum fit subiectum.) In this figure, the artist seeks the naturally proportionate connection between subject and predicate so as to find media for conclusions. (Latin: Artifex naturalem proportionatam connexionem inter subiectum et praedicatum quaerit ut medium ad conclusionem inveniat.) Each principle is wholly general in itself. When contracted to another principle, it becomes subalternate. (Latin: Principium ad aliud contractum fit subalternatum.) When contracted to something singular, it becomes wholly specific. Thus the intellect ascends and descends by a ladder. (Latin: Intellectus scala ascendit et descendit.) The principles of this figure implicitly contain everything that exists. (Latin: Principia huius figurae omnia existentia implicite continent.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ Figure\_A := circular principial field over Σ\_A Where: Σ\_A := {B, C, D, E, F, G, H, I, K} resolve(l, A) → principial meaning Formal note: full pairwise connectivity may be used as a modern descriptive reconstruction, but it is not asserted here as the chapter’s own doctrinal formulation. \------------------------------------------------------------ Binding \------------------------------------------------------------ “circular” ⇔ principial convertibility field “subject becomes predicate and vice versa” ⇔ bidirectional predication “contracts to another principle” ⇔ subalternating contraction “ladder of ascent and descent” ⇔ reversible abstraction traversal “contains everything in existence” ⇔ universal reduction to A \------------------------------------------------------------ A. CONVERTIBILITY LAW \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ The principles in Figure A are mutually convertible. (Latin: Principia in Figura A inter se convertuntur.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ ∀ a\_i, a\_j ∈ A: predicable(a\_i, a\_j) ⇔ predicable(a\_j, a\_i) \------------------------------------------------------------ Binding \------------------------------------------------------------ “convertible” ⇔ symmetric principial predication within the pure field of Figure A \------------------------------------------------------------ Structural Note \------------------------------------------------------------ This CAPUT states convertibility at figural level. It does not determine by itself the later status of contracted, subject-conditioned, or mixed expressions. \------------------------------------------------------------ B. PREDICATION STRUCTURE \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ A principle may appear: • in itself as general • in another as subalternate • in a singular as fully specific (Latin: Principium potest esse per se generale, in alio subalternatum, in singulari specificum.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ predication(a\_i, a\_j) Types: • general • subalternate • singular (subject-conditioned) \------------------------------------------------------------ Binding \------------------------------------------------------------ “general” ⇔ uncontracted principle “subalternate” ⇔ contracted to another principle “specific” ⇔ instantiated in subject \------------------------------------------------------------ C. CONTRACTION LAW \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ Contraction reduces principial generality without destroying principial identity. (Latin: Contractio generalitatem principalem minuit sine identitatis detrimento.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ contract(a\_i, a\_j) \------------------------------------------------------------ Binding \------------------------------------------------------------ “contracts to another” ⇔ specialization “without loss” ⇔ preserved identity \------------------------------------------------------------ Structural Note \------------------------------------------------------------ Stronger partialization doctrines belong to ALBUS-level object consolidation and are not imposed here. \------------------------------------------------------------ D. GENERAL ↔ SPECIFIC LADDER \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ The intellect descends from universal to subalternate to specific, and ascends by the same ladder. (Latin: Intellectus a universali ad subalternatum ad specificum descendit, et eadem scala ascendit.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ Descent / ascent across levels of abstraction \------------------------------------------------------------ Binding \------------------------------------------------------------ “ascend / descend” ⇔ reversible abstraction movement \------------------------------------------------------------ E. MEDIUM DISCOVERY FUNCTION \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ The artist seeks the proportionate connection between subject and predicate in order to find a medium. \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ find\_medium(a\_i, a\_j) \------------------------------------------------------------ Binding \------------------------------------------------------------ “medium” ⇔ inferential bridge \------------------------------------------------------------ F. UNIVERSAL CONTAINMENT LAW \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ All things in existence are reducible to the principles of Figure A. (Latin: Omnia existentia ad principia Figurae A reducuntur.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ ∀ x: reducible(x, A) \------------------------------------------------------------ Binding \------------------------------------------------------------ “reduce to principles” ⇔ principial intelligibility \------------------------------------------------------------ I. COMPRESSION NOTE \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ Because principles are mutually convertible, fewer can suffice for broader principial expression. (Latin: Quia principia plene convertuntur, pauciora sufficiunt ad omnia exprimenda.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ convertibility ⇒ compressibility \------------------------------------------------------------ Structural Note \------------------------------------------------------------ Specific minimal bases (e.g., triadic reductions) belong to object-level ALBUS reconstruction and are not fixed here. \============================================================ II. FIGURE T (SECOND FIGURE) \============================================================ L1 — Scholastic Form \------------------------------------------------------------ The second figure is called T. (Latin: Secunda figura per elementum T significatur; tres in ea implicatos complectitur triangulos, et quilibet est generalis ad omnia.) It contains three triangles. \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ Figure\_T := {T1, T2, T3} \------------------------------------------------------------ Binding \------------------------------------------------------------ “three triangles” ⇔ triadic relational operators \------------------------------------------------------------ A. TRIANGLE T1 \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ Everything exists in difference, concordance, or contrariety. (Latin: Omnia sunt in differentia, concordantia vel contrarietate, et nihil sine his potest exsistere.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ relation(x,y) ∈ T1 \------------------------------------------------------------ B. TRIANGLE T2 \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ All things have beginning, middle, and end. (Latin: Omnia habent principium, medium et finem.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ process(x) → {principium, medium, finis} \------------------------------------------------------------ C. TRIANGLE T3 \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ Everything exists in majority, equality, or minority. (Latin: Omnia sunt in maioritate, aequalitate vel minoritate.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ compare(x,y) ∈ T3 \------------------------------------------------------------ D. UNIVERSALITY LAW \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ Nothing relevant to figural relational articulation is treated outside T-structure. (Latin: Nihil extra structuram T-relationalem exsistit.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ Figure\_T provides universal relational articulation \------------------------------------------------------------ E. INTELLECTUAL LADDER \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ When the intellect ascends to general objects, it is general; when it descends to particulars, it is particular. (Latin: Cum intellectus ascendit ad generalia, generalis est; cum descendit ad particularia, particularis est.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ abstraction ladder \------------------------------------------------------------ F. INTERACTION WITH FIGURE A \------------------------------------------------------------ L1 — Scholastic Form \------------------------------------------------------------ Figure T serves Figure A by distinguishing principles. (Latin: Figura T servit Figurae A discernendo bonitatem ab alia bonitate, et bonitatem a magnitudine, etc.) \------------------------------------------------------------ \============================================================ III. FIGURE 3 (THIRD FIGURE) \============================================================ L1 — Scholastic Form \------------------------------------------------------------ Figure 3 contains 36 cameras, each made of two letters. (Latin: Figura Tertia 36 cameras continet, unamquamque ex duabus litteris constantem.) Each camera signifies subject and predicate. (Latin: Camera significat subiectum et praedicatum.) Each principle is attributed to all the others. (Latin: Unumquodque principium omnibus aliis attribuitur.) No camera may contradict another. (Latin: Nulla camera alii contradicat, et omnes cum conclusione debeant concordare.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ Figure\_3 := pairwise combinatory structure \------------------------------------------------------------ \============================================================ IV. FIGURE 4 (FOURTH FIGURE) \============================================================ L1 — Scholastic Form \------------------------------------------------------------ Figure 4 has three circles: (Latin: Quarta figura tribus constat circulis: exteriore immobili, medio mobili, interiore mobili.) By rotation, triadic cameras are generated. (Latin: Rotando circulos medium et interiorem sub circulo exteriore fixo, camerae triadicae generantur.) The middle letter functions as the medium. (Latin: Littera media fungitur ut medium.) By continued rotation, 252 cameras are produced. (Latin: Per continuam rotationem, 252 camerae producuntur.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ Figure\_4 := triadic rotational combinatory structure \------------------------------------------------------------ Structural Note \------------------------------------------------------------ In Figure 3 the medium is discovered. In Figure 4 the medium is positioned. (Latin: In Figura Tertia medium invenitur; in Figura Quarta medium ponitur.) \============================================================ V. FIGURAL HIERARCHY \============================================================ L1 — Scholastic Form \------------------------------------------------------------ The four figures form an ordered system. (Latin: Quatuor figurae systema ordinatum constituunt: Figura A principia, Figura T respectiva, Figura Tertia cameras binarias, Figura Quarta cameras triadas.) \------------------------------------------------------------ L2 — Operator Form \------------------------------------------------------------ A → T → 3 → 4 \============================================================ VI. GLOBAL MEDIUM LAW \============================================================ L1 — Scholastic Form \------------------------------------------------------------ The medium joins subject and predicate. (Latin: Medium est id per quod subiectum et praedicatum coniunguntur ut conclusiones trahantur.) \------------------------------------------------------------ \============================================================ VII. MEMORY CONDITION \============================================================ L1 — Scholastic Form \------------------------------------------------------------ The four figures must be known by heart. (Latin: Quatuor figurae corde tenendae sunt, sine quo artifex bonum usum practicum Artis facere non potest.) \------------------------------------------------------------ \============================================================ VIII. NON-COLLAPSE RULE \============================================================ This CAPUT must: • preserve L1 and L2 distinction • preserve figural sequencing • preserve CAPUT vs ALBUS separation • preserve CAPUT vs CARCER separation This CAPUT must not: • execute OPERA • redefine TENET • collapse figures into abstractions \============================================================ IX. FUNCTION \============================================================ CAPUT ARS BREVIS C governs: • the four figures as ordered system • principial, relational, binary and triadic structures • medium discovery and positioning • figural hierarchy • integrated Latin/English Rosetta stabilization \============================================================ END CAPUT ARS BREVIS C \============================================================