RM3

Origin. The three-element Sugihara matrix, from Sugihara's work on mingle (1955); the logic R-mingle (RM) is Anderson and Belnap's (Entailment I, 1975, §29), and RM3 is its three-valued characteristic matrix, due to Dunn (1970), who proved RM's completeness for the Sugihara algebras and RM3's role among them.

Models. The three-valued paraconsistent logic that everyone reaches independently — add contraposition to the "ideal" three-valued paraconsistent matrix and this is what you get. It is LP with an implication: LP has three values and no detachable conditional, and RM3 adds one, at the cost of the variable-sharing property that made relevance logic relevant. It is the boundary between the relevant family and the many-valued one, and belongs to both.

Formalism.

Values: {−1, 0, +1}, designated: {0, +1}. Read: −1 false only, 0 both, +1 true only. The Sugihara matrix on the chain −1 < 0 < +1.

Connectives: ¬a = −a a ∧ b = min(a, b), a ∨ b = max(a, b) a → b = −a ∨ b if a ≤ b = min(−a, b) otherwise The conditional is the Sugihara one: it is not ¬a ∨ b.

What holds: Modus ponens: a, a → b ⊨ b. Detachment works, unlike in LP. Contraposition: (a → b) → (¬b → ¬a). No explosion: a ∧ ¬a ⊭ b, since 0 is designated and 0 ∧ −0 = 0.

What fails — the mingle axiom: RM = R + mingle: A → (A → A) Mingle is not a relevant principle; adding it to R gives RM, and RM3 is RM's three-valued matrix. RM fails variable sharing: (p ∧ ¬p) → (q ∨ ¬q) is an RM theorem with no shared variable. Check it in the matrix: p ∧ ¬p ∈ {−1, 0} and q ∨ ¬q ∈ {0, +1}, so the antecedent never exceeds the consequent and the Sugihara conditional returns −(p ∧ ¬p) ∨ (q ∨ ¬q), designated in every case. (The often-quoted p → (q → q) is not an RM theorem: take p = +1 and q = 0, so q → q = 0, and +1 → 0 = min(−1, 0) = −1, undesignated.) So RM is paraconsistent and not relevant, which is why Anderson and Belnap treat it as a curiosity and the many-valued literature treats it as a discovery.

Position: RM3 is the only three-element Sugihara algebra. Every RM theorem is RM3-valid; RM is the intersection of all finite Sugihara matrices. As a matrix logic: RM3 = LP + the Sugihara conditional.

Symbols.

Symbol	Unicode	Name	Meaning
→	U+2192	Rightwards arrow / Sugihara conditional	Not ¬a ∨ b; see the two-case table
¬	U+00AC	Not sign / Negation	Order-reversing involution, ¬a = −a
∧	U+2227	Logical and	Meet on the chain, min(a, b)
∨	U+2228	Logical or	Join on the chain, max(a, b)
⊨	U+22A8	True / Double turnstile	Matrix consequence, preservation of designation
⊭	U+22AD	Not true	Failure of matrix consequence
≤	U+2264	Less-than or equal to	The Sugihara chain order
<	U+003C	Less-than sign	Strict order, −1 < 0 < +1
−	U+2212	Minus sign	Negation on values, and the value −1
+	U+002B	Plus sign	Marks the value +1
∈	U+2208	Element of	Membership in the value set
0	U+0030	Digit zero	The designated glut value, "both"
1	U+0031	Digit one	Magnitude of the two classical values
RM	—	R-mingle	R plus the mingle axiom
RM3	—	The three-valued matrix	The Sugihara chain −1 < 0 < +1
LP	—	Logic of Paradox	RM3 minus the conditional
R	—	Relevance logic R	The base RM extends

Metatheory. RM3 is the unique three-element Sugihara algebra, and Dunn (1970) proved RM complete for the Sugihara algebras as a class, with RM the intersection of all the finite ones. Since RM3 is only one member of that family, the logic it determines properly extends RM: every RM theorem is RM3-valid, but RM3 is not a characteristic matrix for RM. Being a fixed finite matrix, RM3 is decidable by evaluation, and a countermodel is a single assignment checkable in polynomial time, so validity lies in coNP. Detachment holds — a, a → b ⊨ b — which is exactly what LP's material conditional fails to deliver, and the negation-conjunction-disjunction reduct of RM3 is LP itself. RM3 is paraconsistent: a ∧ ¬a ⊭ b, because the middle value is designated and closed under negation. That designation also distinguishes RM3 from three-valued Łukasiewicz logic, which designates only the top value and is therefore explosive. Contraction and mingle both hold, being inherited from R and from M.

Applies to. Paraconsistent logic with detachment. The Sugihara algebras and the semilattice semantics for R. The boundary between the relevant and many-valued families. Three-valued approaches to the paradoxes, where RM3's conditional is the one LP lacks.

Limitations. Variable sharing fails, so RM3 is not a relevant logic despite its place in that literature — (p ∧ ¬p) → (q ∨ ¬q) is a theorem and there is no relevance in it. The mingle axiom A → (A → A) has no independent motivation beyond producing a well-behaved matrix. Disjunctive syllogism fails as well: assign p the value 0 and q the value −1, and (p ∨ q) ∧ ¬p is designated while q is not, so the conditional RM3 supplies does not restore the classical inference LP lost. Contraction is present, so RM3 cannot base naive comprehension or a transparent truth predicate: Curry's derivation needs nothing more. As a single matrix it validates schemas RM does not prove, so it settles more than the logic it illustrates. And it inherits LP's interpretive problem: nothing says what 0 means beyond its behaviour in the tables.