Mingle Logic

Origin. Anderson, Belnap (1960s). R plus mingle. Weakened relevance. Intermediate system. Between R and classical.

Models. Relevant logic with mingle axiom. Weakened variable sharing. Denseness condition. Ternary semantics variant.

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. Connexive tendency. More classical.

Semantics: Ternary relation with density. Rabc and a ≤ c implies Raac. Dense accessibility. Modified Routley-Meyer.

Variable sharing: Still required for theorems. A → B theorem implies shared variable. But weaker than R. More implications valid.

Relation to R: R ⊂ RM ⊂ Classical. Proper intermediate. Shares R's rejection of irrelevance. Adds mingle.

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

Symbols.

Metatheory. Relevant + mingle. Density. Intermediate strength. Variable sharing preserved.

Applies to. Relevant logic. Intermediate systems. Connexivity. Classical recapture.

Limitations. Less pure relevance. Odd theorems. Historical mainly. Less studied than R.