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 → (q → q) is an RM theorem with no shared variable. 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
RM	—	R-mingle	R plus the mingle axiom
RM3	—	The three-valued matrix	The Sugihara chain −1 < 0 < +1
→	U+2192	Sugihara conditional	Not ¬a ∨ b
LP	—	Logic of Paradox	RM3 minus the conditional
0	—	Both	The designated glut value

Metatheory. RM3's interest is that it is where the two paraconsistent traditions meet and disagree: the many-valued route arrives at it by asking for a three-valued paraconsistent logic with a decent conditional, and the relevant route arrives at it by adding an axiom that destroys the property relevance logic exists for. Dunn's completeness theorem — RM is complete for the Sugihara algebras, and RM3 is the three-element one — is what ties them, and the failure of variable sharing is what keeps them apart. It also sits underneath Strict-Tolerant logic's construction: ST's TT relation is LP, and adding the contrapositive to the ideal three-valued paraconsistent logic gives RM3, which is the same matrix seen from the other side.

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 living in the relevant family's literature — p → (q → q) is a theorem and there is no relevance in it. The mingle axiom A → (A → A) has no independent motivation: it was added to see what happens, and what happens is a well-behaved matrix and a lost property. Three values with a designated glut inherits LP's interpretive problem: nothing says what 0 means beyond its behaviour in the tables.