Supplement to Relevance Logic
The Logic S
Here is a Hilbert-style axiomatisation of the logic \(\mathbf{S}\) (for “syllogism”).
Our language contains propositional variables, parentheses and one connective: implication.
| Axiom Scheme | Axiom Name | |
| 1. | \((B\rightarrow C) \rightarrow((A\rightarrow B) \rightarrow(A\rightarrow C))\) | Prefixing |
| 2. | \((A\rightarrow B) \rightarrow((B\rightarrow C) \rightarrow(A\rightarrow C))\) | Suffixing |
| Rule | Name | |
| 1. | \(A \rightarrow B, B \rightarrow C \vdash A \rightarrow C\) | Transitivity |
| 2. | \(A \rightarrow B \vdash(B \rightarrow C) \rightarrow(A \rightarrow C)\) | Rule Suffixing |
| 3. | \(B \rightarrow C \vdash(A \rightarrow B) \rightarrow(A \rightarrow C)\) | Rule Prefixing |
The logic T-W is \(\mathbf{S}\) with the addition of the identity axiom \((A\rightarrow A)\). Martin’s theorem is that no instance of the identity axiom is a theorem of \(\mathbf{S}\). It is a corollary of Martin’s theorem that in T-W if both \(A\rightarrow B\) and \(B\rightarrow A\) are provable, then \(A\) and \(B\) are the same formula (see Anderson, Belnap, and Dunn (1992) §66).