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Key Concepts

Material Bases

A material base B = consists of:

  • L_B: An atomic language — a set of atomic sentence strings (e.g., "rain", "wet_ground")
  • |~_B: A base consequence relation — a set of sequents (Gamma, Delta) where Gamma and Delta are sets of atomic sentences

The base encodes defeasible material inferences: reasoning patterns that hold in normal circumstances but can be overridden by additional information.

Containment Axiom

Every material base automatically satisfies Containment: if Gamma and Delta share any element (Gamma ∩ Delta ≠ ∅), then Gamma |~_B Delta. This is the analogue of the identity axiom in classical logic.

Exact Match (No Weakening)

Base consequences require exact syntactic match. If the base contains {A} |~ {B}, then {A, C} |~ {B} is not an axiom. This is what makes the system nonmonotonic — extra premises can defeat inferences.

Sequents

A sequent Gamma => Delta represents a reason relation: the sentences in Gamma (the antecedent) provide reason for at least one of the sentences in Delta (the succedent).

  • Multi-succedent: Delta can contain multiple sentences. Gamma => A, B means "Gamma provides reason for A-or-B."
  • Empty antecedent: => A means A is unconditionally assertable.
  • Empty succedent: A => means A is incoherent (leads to nothing).

Nonmonotonicity

In NMMS, adding premises can defeat inferences. If {rain} |~ {wet_ground} but the base has no consequence from {rain, covered}, then:

  • rain => wet_ground is derivable
  • rain, covered => wet_ground is not derivable

This models the everyday pattern: rain normally makes the ground wet, but if the ground is covered, the inference is defeated.

Nontransitivity

NMMS also lacks Mixed-Cut (the structural rule for transitivity). Even if A |~ B and B |~ C, it does not follow that A |~ C. Each inference step must be independently justified by the base.

Supraclassicality

Despite lacking Weakening and Mixed-Cut, NMMS is supraclassical: all classically valid sequents are derivable. The law of excluded middle (=> A | ~A), double negation elimination (~~A => A), and all classical tautologies hold.

Explicitation Conditions

The logical connectives "make explicit" reason relations through these biconditionals:

  • DD (Deduction-Detachment): Gamma |~ A -> B, Delta iff Gamma, A |~ B, Delta
  • II (Incoherence-Incompatibility): Gamma |~ ~A, Delta iff Gamma, A |~ Delta
  • AA (Antecedent-Adjunction): Gamma, A & B |~ Delta iff Gamma, A, B |~ Delta
  • SS (Succedent-Summation): Gamma |~ A | B, Delta iff Gamma |~ A, B, Delta