S (Rubin)

• [Canty, 1965] "A Note on the Axiomatization of Rubin's system (S)"
• Original Source: J . E. Rubin, "Bi-modal logic, double-closure algebras, and Hilbert space", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, v. 8 (1962), 305-322.

Notes

Based on

• Definitions
  • p => q ==def L1(p>q)
  • M1 p ==def ~L1~p
  • M2 p ==def ~L2~p
• Rules
  • R1: Modus Ponens for => ( from |-p, and |- p => q, infer |- q)
  • R2: Adjunction for & (from |- p, and |- q, infer |- p & q)
  • R3: Uniform substitution (US)
• Axioms
  • A1: (α & β) => (β & α)
  • A2: (α & β) => α
  • A3: α => (α & &alpha)
  • A4: ((α & β) & γ) => (α & (β & γ))
  • A5: α => ~ ~ α
  • A6: ((α => β) & (β => γ)) => (α => γ)
  • A7: (α & (α => β)) => β
  • A8: L2 α => L2 L2 α
  • A12: L2 α => L1 α
  • Plus the result of replacing L2 with L1 (and M2 with M1) in the above axioms and rules.

[Canty, 1965, p190]

Canty's article is missing definitions, I added probably the right set -JH

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