substance

hypermath | formal-universality | dictionary

What it is

substance is the second vertex of the triangle: what a form carries — its content, its matter, the clay it is made of. substance(x) :: Prop asserts that x has definite content. It is the middle vertex because syntax generates it and it generates semantics. The ax-sub-nec axiom establishes that substance necessitates semantics.

substance :: Form -> Prop

Semantically closed in L1 Section IV: substance(x) := congruent-class(x) at =~ level.

What it is not

• Not value in the sense of evaluation. No computation happens to extract substance. Substance is structurally assigned, not computed.
• Not meaning. Meaning is semantics. Substance is what a form carries before it is related to other forms. Once substance is embedded into relations, it becomes semantics.
• Not content in the information-theory sense. No bits, no measure. substance is a Prop — an assertion that content exists and is definite in this form.

What it clarifies

Clarifies the definition of =~ (congruent): x =~ y iff substance(x) = substance(y). The opaque predicate substance :: Form -> Prop is exactly what congruence tests.

Clarifies the triangle: syntax is the entry, substance is what syntax fills, semantics is what substance projects into the relational context.

Concrete example

The L0 Section VIII executive opaques each carry substance. syntax-opaque carries the substance shape assignment; derives-opaque carries directed approach. These are opaque substance assignments — their content is not further derived within L0. The closes in L1 Section IV give each opaque its semantic content.

Visualization

Clay on a potter's wheel after it has been given a shape but before it has been fired. The shape is there (syntax). The material is there (substance). It has not yet been placed in relation to other finished vessels (semantics). It is mid-formation.

Speculation

Whether substance at L2 becomes derivable from the derivation-matrix entries — whether the substance of a form is its row in the D matrix — is OPEN. If D[x] (the outgoing row) encodes substance, then substance dissolves into the derivation matrix architecture and no longer needs to be a primitive opaque. This would compose substance with congruent and simulation into a single unified account via D.