The Algebra of Construal: 5 Formalising Construal—Towards an Algebra

The previous episode demonstrated how multi-perspectival reasoning emerges from the interaction of construals. We now turn to formalisation. Construal is not merely a conceptual principle; it is a systematic operator with rules, types, and relational interactions. Formalising it as an algebra provides a framework to study, compare, and propagate meanings systematically.

Construal as an Algebraic Operator

At the most abstract level, a construal can be treated as an operator $C$ acting on a system $S$ through a perspective $P$, producing an instance $I$:

$$C:S\times P\to I$$

From this, several algebraic notions emerge:

1. Composition ($\circ$) – Sequential application of construals. If $C_1$ and $C_2$ are construals, their composition generates a new instance:

$$I=(C_2\circ C_1)(S,P)$$

The order matters if construals are non-commutative, producing different emergent instances.

2. Interaction ($\otimes$) – Parallel or relational application, where construals influence one another:

$$I=C_1\otimes C_2(S,P)$$

Here, the emergent instance reflects both the combinatory and interfering effects of the construals.

3. Transformation – A construal may alter the system itself, generating new potentialities for future actualisations:

$$S^{\mathrm{\prime }}=C(S,P)\text{where }{S}^{\mathrm{\prime }}\mathrm{\ne }S$$

Rules of the Algebra

While the algebra is not fully numerical, several operational rules can be postulated:

• Associativity: The sequential application of multiple construals is associative if the emergent outcome is independent of internal grouping.

• Non-commutativity: Order of construals often matters; perspective shapes the emergent instance.

• Identity: There exists a “neutral” construal $C_0$ that maps a system to itself without altering the instance.

• Inverse or Reflexivity: Some construals may be partially reversible, creating “undo” operations in relational propagation.

Illustrative Example: Emergent Social Meaning

Consider a social system where multiple agents apply distinct construals to a shared event:

• $C_A$ emphasises relational context.

• $C_B$ emphasises temporal dynamics.

• $C_C$ emphasises potential outcomes.

The emergent social meaning is not reducible to any single construal. Using the algebraic approach:

$$I_{emergent}=(C_A\otimes C_B)\circ C_C(S,P)$$

This formalism allows systematic exploration of how construals combine, interfere, and propagate, illuminating patterns that were previously implicit.

Implications

By formalising construal as an algebra:

• We provide a language to study relational generation of meaning.

• Interactions among construals become predictable in structure, even if outcomes remain emergent and context-dependent.

• We lay the groundwork for computational or mathematical models of relational epistemology, multi-agent reasoning, and the propagation of possibility.

Looking Ahead

The final episode of this series will examine applications and horizons: how the algebra of construal can inform cognitive science, social systems, linguistics, AI, and philosophy, demonstrating that relational ontology is not only descriptive but generative—a platform for new modes of reasoning and insight.