- [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.

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

© Copyright 2006 by John Halleck, All Rights Reserved. This page is http://www.cc.utah.edu/~nahaj/logic/structures/systems/s.rubin.html This page was last modified on December 17th 2006.