Supplement to Relevance Logic
The Logic S
Here is a Hilbert-style axiomatisation of the logic S (for “syllogism”).
Our language contains propositional variables, parentheses and one connective: implication.
|Axiom Scheme||Axiom Name|
|1.||(B→C) →((A→B) →(A→C))||Prefixing|
|2.||(A→B) →((B→C) →(A→C))||Suffixing|
|A → B, B → C ⊢ A → C||Transitivity|
|A → B ⊢ (B → C) → (A → C)||Rule Suffixing|
|B → C ⊢ (A → B) → (A → C)||Rule Prefixing|
The logic T-W is S with the addition of the identity axiom (A→A). Martin's theorem is that no instance of the identity axiom is a theorem of S. It is a corollary of Martin's theorem that in T-W if both A→B and B→A are provable, then A and B are the same formula (see Anderson, Belnap, and Dunn (1992) §66).