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	Name	Meaning
→	U+2192	Rightwards arrow	Relevant implication of RM
¬	U+00AC	Not sign / Negation	De Morgan negation, contraposable
∧	U+2227	Logical and	Extensional conjunction
∨	U+2228	Logical or	Extensional disjunction
⊢	U+22A2	Right tack / Turnstile	Theoremhood
⊂	U+2282	Subset of	Inclusion of logics, R ⊂ RM ⊂ CL
⊊	U+228A	Subset of with not equal to	Proper inclusion, R ⊊ RM ⊊ CL
−	U+2212	Minus sign	Sign of the negative element −1 in a Sugihara chain
M	—	Mingle	The axiom A → (A → A)
RM	—	R-Mingle	R together with M
RM3	—	Three-element Sugihara matrix	Values −1, 0, 1
R	—	Relevance logic R	The base RM extends
CL	—	Classical logic	The upper bound of the chain

Metatheory. RM is R together with mingle, A → (A → A), and sits properly between R and classical logic, R ⊊ RM ⊊ CL. Dunn (1970) proved RM complete for the Sugihara algebras — De Morgan monoids that are totally ordered — and RM is exactly the intersection of the finite Sugihara matrices, of which RM3 is the three-element one. Because the finite matrices determine the logic, RM is decidable, in contrast to R, E and T, all of which Urquhart (1984) proved undecidable. Linearity is what mingle adds algebraically: (A → B) ∨ (B → A) is an RM theorem though not an R theorem, which is why the algebras are chains and why RM belongs with the semilinear substructural logics. Variable sharing fails, since (A ∧ ¬A) → (B ∨ ¬B) is a theorem with no shared variable, but Dunn's weak relevance result survives: if ⊢ A → B with no variable in common, then ⊢ ¬A and ⊢ B.

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

Limitations. RM is not a relevance logic: variable sharing fails outright, (A ∧ ¬A) → (B ∨ ¬B) is a theorem, and the criterion that motivates the family is abandoned by the one axiom that defines the system. What replaces it is only Dunn's weak relevance condition, which permits irrelevant theorems whenever the antecedent is refutable or the consequent provable. Mingle has no independent philosophical motivation comparable to relevance itself; it is adopted because the resulting algebras are chains, and the decidability is a consequence of that choice rather than of any account of implication. RM is not connexive either — Aristotle's thesis ¬(A → ¬A) fails, as the value −1 in RM3 shows. The totally ordered semantics forces a linear comparability of propositions that has no evident reading in terms of information or relevance.