Relevance Logic > The Logic S (Stanford Encyclopedia of Philosophy)

Processing math: 100%

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).

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

Supplement to Relevance Logic

The Logic S

Browse

About

Support SEP

Mirror Sites