Entailment of ambiguity.

Area of Science:

• * Theoretical Biology
• * Systems Biology
• * Philosophy of Science

Background:

• * Rosen's principle of 'closure to efficient cause' is central to understanding complexity and organisms.
• * Existing interpretations of this principle have led to confusion among scholars.
• * The (M,R)-system provides a formal framework to elucidate this concept.

Purpose of the Study:

• * To clarify the principle of 'closure to efficient cause' using the (M,R)-system.
• * To demonstrate how this principle underlies the characteristics of complex systems.
• * To identify and explain an inherent property of this closure, namely ambiguity.

Main Methods:

• * Formalization of Rosen's closure using a set of three algebraic maps, termed the (M,R)-system.
• * Analysis of the structural properties of the (M,R)-system, revealing a circular hierarchy of containment.
• * Examination of the implications of this structure for the nature of efficient cause and complexity.

Main Results:

• * The (M,R)-system clarifies the structure of closure to efficient cause as a circular hierarchy.
• * Each map in the (M,R)-system represents an efficient cause, defined by morphology and laws of nature.
• * A critical, observable property of this closure is its inherent ambiguity, previously unnoticed.
• * This ambiguity provides a foundation for understanding properties of complexity such as non-computability and undecidability.

Conclusions:

• * The (M,R)-system offers a clear and formal understanding of Rosen's closure to efficient cause.
• * The inherent ambiguity of this closure is a fundamental aspect explaining the nature of complexity.
• * This framework potentially simplifies proofs for complex system properties compared to Rosen's original work.