## 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
 Rule Name 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 (AA). 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 AB and BA are provable, then A and B are the same formula (see Anderson, Belnap, and Dunn (1992) §66).

Open access to the SEP is made possible by a world-wide funding initiative.
Please Read How You Can Help Keep the Encyclopedia Free

The SEP would like to congratulate the National Endowment for the Humanities on its 50th anniversary and express our indebtedness for the five generous grants it awarded our project from 1997 to 2007. Readers who have benefited from the SEP are encouraged to examine the NEH’s anniversary page and, if inspired to do so, send a testimonial to neh50@neh.gov.