Closure of Self-Determining System Based on Causal and Constitutive Relations

Abstract: A self-determining system is defined as one in which causes originating within the system influence the system itself. This definition raises the question of how to specify system boundaries. Although the concept of "closure" is commonly used for this purpose, defining boundaries solely in terms of causal relations introduce challenges, such as how to handle external causes and circular causality. To address this issue, we introduce two types of asymmetric relations: causal and constitutive. We propose that system boundaries can be defined as closures of loops formed by these relations, referred to as causal-constitutive loops. By constraining constitutive relations, the resulting system necessarily includes internal causes and thereby satisfies self-determination. Furthermore, to prevent reduction to supervenience, constitutive relations must involve at least two independent variables. This minimal requirement leads to two interdependent loops, which implies a dual-process organization.

• The paper defines CC-loops by integrating causal and constitutive relations to establish clear system boundaries.
• It avoids conceptual pitfalls such as circular causality and supervenience by enforcing strict structural rules on variable dependencies.
• The framework has practical implications for synthetic biology, cognitive science, and autonomous systems through its dual-process organizational model.

Closure of Self-Determining Systems: Formalization via Causal and Constitutive Relations

Conceptual Foundations

The paper establishes self-determination as a property of systems where internally originating causes influence the system itself. This reorientation from biological self-maintenance to self-determination demands a precise method for bounding such systems. Prior frameworks that rely solely on closure via causal relations (e.g., Rosen's closure of efficient causes or Maturana and Varela's operational closure) either risk excluding relevant external causes or succumb to conceptual difficulties like circular causality and the exclusion problem associated with supervenience.

The authors argue that unifying two distinct, asymmetric relations—causal (temporal precedence) and constitutive (mereological dependence)—enables robust system boundary definitions while retaining openness to external influence. Causal relations characterize the directed effect of variable changes on outcomes, while constitutive relations account for the simultaneous dependence between wholes (functions) and their parts (variables), without temporal ordering.

Formalization of Causal–Constitutive Loops (CC-Loops)

The study introduces CC-loops as closed structures formed by combining causal and constitutive relations. Causal relations are operationalized as $b := F(a)$, with $a$ (cause) and $b$ (effect) as variables and $F$ as the transmission mechanism. Constitutive relations are defined such that $F$ is constituted by a set of variables $\{c_j\}$ via $F :\sim \mathcal{G}(c_j)$, reflecting the mutual realization (mereology) of the whole by its parts.

To avoid collapse to supervenience—which would render the system unable to exert causal influence on its own constituents—the constitutive mechanism must involve at least two independent variables. In this construction, all variables and functions comprising the system are interconnected via CC-loops, but external influences (e.g., $a$) are explicitly allowed, provided they do not participate in CC-loops.

Structural Requirements and Prohibitions

The framework imposes critical structural prohibitions:

• Constitutive relations are restricted to causal transmission mechanisms, avoiding trivial variable decompositions.
• Loops formed solely by causal relations (circular causality) are proscribed, circumventing the "chicken-and-egg" dilemma.
• Constitutive relations must not reduce to supervenience (single variable dependence), which would invoke the exclusion problem from philosophy of mind: supervenient entities cannot causally influence their subvenient base.
• Effect variables (e.g., $b$) cannot constitute their own causal mechanism ($F$), preserving causal asymmetry.

Additionally, causal relations among constitutive variables (e.g., between $a$0 and $a$1 in $a$2) are disallowed to ensure proper loop formation and avoid destabilizing the constitutive organization.

Practical and Theoretical Implications

The presented formalism enables precise demarcation of system boundaries in artificial or biological systems, addressing the challenge of external causal openness versus internal causal closure. The minimal requirement of dual interdependent CC-loops implies that self-determining systems must be organized as dual-process structures, which could inform models in synthetic biology, distributed cognition, and autonomous robotic systems. The explicit exclusion of supervenience and circular causality resolves longstanding issues in system autonomy and agency.

Variational constraints on constitutive relations suggest directions for future research: mapping material interactions to constitutive dependencies and elucidating emergent organization in biological or cognitive architectures.

Conclusion

This work rigorously defines self-determining systems as entities whose boundaries are formed by the closure of causal and constitutive relations, embodied in CC-loops with at least two independently manipulable variables. The incorporation of external causes within this closure circumvents isolationism and establishes a minimal, formal requirement for self-determination. The framework's implications extend to theoretical models of autonomy and practical designs of artificial agents, proposing avenues for advancing organizational theory in artificial life and cognitive science.