The Operator System (Δ–Ψ)
Introduction
Praxeological Meta-Structure Theory (PMS) is defined by a closed set of eleven operators, denoted Δ–Ψ.
Each operator represents a structural action, not a meaning, state, or object. The operators are formally irreducible: none can be derived from a composition of the others without loss of structural specificity.
PMS does not require metaphorical, psychological, or semantic interpretation. The operator system is fully defined by its operators, their layers, and their dependency constraints.
The PMS Operators (Δ–Ψ)
The table below lists the complete operator set. Descriptions are intentionally minimal and structural.
| Symbol | Name | Structural Function |
|---|---|---|
| Δ | Difference | Introduces a distinction within an otherwise undifferentiated structure. |
| ∇ | Impulse | Initiates activation or directional change without specifying form. |
| □ | Frame | Establishes a bounded context or scope for subsequent operations. |
| Λ | Non-Event | Marks unrealised or excluded possibilities within a structure. |
| Α | Attractor | Introduces a tendency toward convergence or amplification. |
| Ω | Asymmetry | Creates directional or role-based differentiation. |
| Θ | Temporal Iteration | Produces ordered repetition or sequential structure. |
| Φ | Reframing | Transforms an existing frame without destroying structure. |
| Χ | Distancing | Separates, isolates, or insulates domains or processes. |
| Σ | Integration | Commits and stabilises prior structural transformations. |
| Ψ | Self-Binding | Establishes invariants that constrain future operations. |
No operator presupposes semantics, intention, or domain-specific meaning. Interpretation occurs outside the operator system.
Layer Structure
The eleven operators are organised into four structural layers. Layers reflect increasing degrees of structural consolidation.
Layer 1 — Differentiation & Activation
Operators: Δ, ∇
- Δ introduces distinctions.
- ∇ initiates movement or activation.
This layer defines the minimal conditions for structured action to begin.
Layer 2 — Framing & Directionality
Operators: □, Λ, Α
- □ establishes bounded contexts.
- Λ records unrealised or excluded alternatives.
- Α introduces convergence tendencies within a structure.
This layer governs environmental formation and directional bias.
Layer 3 — Transformation & Control
Operators: Ω, Θ, Φ
- Ω creates asymmetry or role differentiation.
- Θ produces temporal or iterative structure.
- Φ transforms existing frames without collapsing them.
This layer enables controlled transformation within structured contexts.
Layer 4 — Consolidation & Constraint
Operators: Χ, Σ, Ψ
- Χ isolates or separates domains.
- Σ integrates and commits transformations.
- Ψ binds structures through invariants.
This layer governs stability, persistence, and long-term constraint.
Dependency Rules
Operators in PMS cannot be composed arbitrarily. The system defines structural preconditions for valid composition.
Core Dependency Principles
-
Asymmetry requires structure
Ω may only follow a distinction (Δ) or an established frame (□). -
Iteration requires prior structure
Θ operates only on already differentiated or framed configurations. -
Reframing requires a frame
Φ can only act on an existing □. -
Attractors require differentiability
Α presupposes a Δ-field. -
Integration requires content
Σ may only occur after at least one prior operator. -
Self-binding requires consolidation or separation
Ψ may only follow Σ or Χ.
These rules form a partial order, not a linear sequence. Multiple valid compositions exist, but structurally invalid sequences are excluded.
Consequences of the Operator System
Structural Consequences
- No operator may be omitted or merged
- No operator may be redefined semantically
- Composition is primary
- Invalid sequences are formally excluded
Methodological Consequences
- PMS is not explanatory psychology or phenomenology
- PMS does not compete with domain theories
- Analysis precedes interpretation
Practical Consequences
- Systems can be audited structurally
- Constraints can be enforced mechanically
- Derived systems must remain operator-faithful
Epistemic Consequences
- No worldview is implied
- Disagreement occurs outside the system
- Stability replaces persuasion
Formal Status
The operator system is:
- closed (no additional operators allowed),
- minimal (no operator is redundant),
- substrate-independent,
- non-semantic by design.
The canonical specification of operators, layers, and dependency rules is defined in:
PMS.yaml
https://github.com/tz-dev/Praxeological-Meta-Structure-Theory
All derived systems (PMS-STACK, PMS-QC, MIPractice) inherit this operator system without modification.
From Operators to Grammar
This page defines what exists in PMS: the complete operator set, its layers, and its non-negotiable dependency rules.
What it does not yet define is how this system is made formal, auditable, and machine-readable.
The operator system becomes operational only once it is expressed as a normative grammar.
→ Continue to:
Formal Grammar & PMS.yaml
(The canonical specification of PMS as a machine-readable standard)