#### 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 |

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
(*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).