Mingle Logic

Origin. The matrices are Sugihara's (1955); Anderson and Belnap named and studied RM = R + mingle; Dunn (1970) proved completeness for the Sugihara algebras. Intermediate system: R ⊊ RM ⊊ classical.

Models. Relevant logic with the mingle axiom — which costs it relevance. Variable sharing fails outright. Semantics is algebraic: totally ordered De Morgan monoids (Sugihara algebras).

Formalism.

Mingle axiom: M: A → (A → A). Self-implication chains collapse. Idempotence of implication. Strengthens R.

System RM: R + M. Relevant logic with mingle. Stronger than R. Weaker than classical.

Characteristic: (A → B) ∨ (B → A) invalid in R. Valid in RM — linearity, which is why the algebras are chains. Not connexive: Aristotle's thesis ¬(A → ¬A) fails, as A = −1 in RM3 shows.

Semantics: Dunn (1970): RM is complete for the Sugihara algebras — De Morgan monoids that are totally ordered. RM is the intersection of all finite Sugihara matrices, and RM3 is the three-element one. This algebraic route, not a Routley–Meyer frame condition, is the standard semantics for RM.

Variable sharing — fails: (A ∧ ¬A) → (B ∨ ¬B) is an RM theorem and shares no variable. So RM is not a relevant logic, despite being R plus one axiom. Dunn's weak relevance result is what survives: if ⊢ A → B without shared variables, then ⊢ ¬A and ⊢ B.

Relation to R: R ⊂ RM ⊂ Classical. Proper intermediate. Keeps R's rejection of explosion — RM is paraconsistent. Loses R's rejection of irrelevance, which is the whole cost of mingle.

Negation: Standard relevant negation. De Morgan. Contraposition holds.

Symbols.

Symbol	Unicode	Meaning
M	—	mingle axiom
RM	—	R + Mingle
→	U+2192	relevant implication
R	—	relevance logic

Metatheory. RM is decidable, unlike R — the Sugihara matrices give a decision procedure, and this is the practical payoff of an otherwise unmotivated axiom. Completeness for Sugihara algebras (Dunn 1970). Intermediate strength. Variable sharing is lost.

Applies to. Relevant logic. Intermediate systems. Semilinear substructural logics. Classical recapture. Three-valued paraconsistency, via RM3.

Limitations. Not relevant, despite the name and the neighbourhood. Odd theorems. Mingle has no independent motivation. Less studied than R.