Draft:Archidifferon

The archidifferon is a formal construct in nonmonotonic logic, defined as the minimal logical–ontological basis of difference. Introduced in the context of Reiter's Default Logic, it serves as a formal mechanism that breaks monotonicity while preserving classical conservativity.

Definition

The archidifferon is represented as a default rule:

$\Delta =\langle M(x)\land \neg M(y):\neg M(y)/x<y\rangle$

• Prerequisite: $M(x)\land \neg M(y)$
• Justification: $\neg M(y)$
• Consequence: $x<y$

This rule applies only if $M(y)$ is not derivable, ensuring consistency.

Properties

• Nonmonotonicity: Adding $M(b)$ cancels previously inferred $a<b$.
• Minimality: Among all rules of the form $\langle P(x)\land \neg P(y):\neg P(y)/xRy\rangle$, this is minimal by a lexicographic order (predicate name, number of symbols, and normalized length K′).
• Skeptical inference: A formula is derived only if it appears in every extension.
• Conservativity: Classical conclusions are preserved under extensions (see Reiter 1980).
• Universality: Generalizes to any unary predicate $P$.

Metric and Normalization

Section 5 of the associated formalization defines a normalization process and the metric $K'$ used to compare candidate rules.

Definition

The Archidifferon is defined as the fundamental, non-derivable ontological structure that makes any distinction, structure, or sense possible. It is not a substance, not a being, and not subject to negation. Instead, it operates as the necessary precondition for the emergence of difference.

History and Context

The concept relates to:

• Heraclitus: Logos and opposition.
• Plotinus: The One and differentiation.
• Hegel: Phenomenology of Spirit.
• Heidegger: Sein und Zeit.
• A. Mou (2018): The Ontology of Difference.

See also

• Nonmonotonic logic
• Default logic
• Ontology (philosophy)