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Universal Syntax

The Line sibling — how marks connect. The rules that determine which compositions are valid and the algorithms for parsing them.

The Four Algebraic Sorts

Every symbol in Universal Language belongs to exactly one of four sorts. Sort determines which operations can be applied.

e

Entity

Things that exist — points, embedded assertions, results of modification

•○{•}modify_entity(m, e)
r

Relation

Connections between entities — lines, curves, directed or undirected

——→⌒compose(r₁, r₂)
m

Modifier

Transformations that alter — angles, extracted properties

∠θabstract(e)m₉₀
a

Assertion

Complete statements — predications, conjoined/disjoined frames

predicate(e, r, e)conjoin(a₁, a₂)negate(a)

Six Well-Formedness Rules

A construction is syntactically valid if and only if it satisfies all six rules. These are necessary and sufficient.

Rule 1: Sort Matching

Every operation input must match its declared sort. predicate takes (e, r, e), not (r, e, e).

✓ Does each argument have the right sort?

Rule 2: Enclosure Closure

Every enclosure boundary must close — the Jordan Curve Theorem requires it. No open boundaries.

✓ Is every boundary a complete closed curve?

Rule 3: Connection Completeness

Every relation must connect exactly two entities (its endpoints). No floating lines.

✓ Does every line/curve touch exactly two entity-points?

Rule 4: Single Root Frame

A well-formed sentence has exactly one outermost assertion frame. Everything is inside one boundary.

✓ Is there exactly one top-level frame?

Rule 5: No Orphans

Every element in a sentence must be reachable from the root frame via containment or connection.

✓ Can you reach every symbol from the outermost frame?

Rule 6: Modifier Attachment

Every modifier must be attached to exactly one entity or one relation. No free-floating modifiers.

✓ Is every angle/modifier applied to something specific?

Sentence Types

Atomic

A single predication: subject + relation + object in one frame

•→• (entity relates to entity)→predicate(e₁, r, e₂) → a

Compound

Two or more frames combined by conjunction, disjunction, or embedding

Frame₁ ∩ Frame₂ (conjunction)→conjoin(a₁, a₂) → a

Recursive

A sentence frame embedded inside another sentence as an entity

○{[•→•]} (the fact that this relates to that)→embed(a) → e, then use e in outer predicate

The 5-Pass Reading Procedure

  1. 1

    Pass 1: Enclosures

    Identify all enclosures — sentence frames and concept boundaries. Determine containment hierarchy.

  2. 2

    Pass 2: Connections

    Identify all lines and curves between entities. Map the connection graph.

  3. 3

    Pass 3: Angles

    Read angles at each connection point. These encode qualitative relationships.

  4. 4

    Pass 4: Points

    Identify all points (entities) and their positions within frames and connections.

  5. 5

    Pass 5: Curvature

    Read curvature of connections. Straight = static relation. Curved = process/change.

Interactive Syntax Tools

Expressionvalid
predicate(•, →, •)Notation: •→•
Rule Validation (0/6)
○Rule 1: Sort Matching
○Rule 2: Enclosure Closure
○Rule 3: Connection Completeness
○Rule 4: Single Root Frame
○Rule 5: No Orphans
○Rule 6: Modifier Attachment

Relationship to Siblings

Syntax defines validity: can this construction be composed? Grammar explains meaning: why does this composition mean what it does? The Lexicon records the resulting definitions, and the Thesaurus maps paths between them.

The sort classification (Entity, Relation, Modifier, Assertion) used here and the symmetry classification (Noun, Verb, Adjective, Determiner) used by Grammar classify different aspects of the same symbols. The atoms are defined in Symbology.