Supplement to Relevance Logic
The Logic R
Here is a Hilbert-style axiomatisation of the logic \(\mathbf{R}\).
Our language contains propositional variables, parentheses, negation, conjunction, and implication. In addition, we use the following defined connectives:
\[\begin{align} A\vee B &=_{df} \neg(\neg A \amp \neg B) \\ A \leftrightarrow B &=_{df} (A \rightarrow B) \amp(B \rightarrow A) \end{align}\]
Axiom Scheme
Axiom Name
1.
\(A \rightarrow A\)
Identity
2.
\((A \rightarrow B) \rightarrow((B \rightarrow C) \rightarrow(A \rightarrow C))\)
Suffixing
3.
\(A \rightarrow((A \rightarrow B) \rightarrow B)\)
Assertion
4.
\((A \rightarrow(A \rightarrow B)) \rightarrow(A \rightarrow B)\)
Contraction
5.
\((A \amp B) \rightarrow A,(A \amp B) \rightarrow B\)
& -Elimination
6.
\(A \rightarrow(A\vee B), B \rightarrow(A\vee B)\)
\(\vee\)-Introduction
7.
\(((A \rightarrow B) \amp(A \rightarrow C)) \rightarrow(A \rightarrow(B \amp C))\)
& -Introduction
8.
\(((A\vee B) \rightarrow C)\leftrightarrow((A \rightarrow C) \amp(B \rightarrow C))\)
\(\vee\)-Elimination
9.
\((A \amp(B\vee C)) \rightarrow((A \amp B)\vee(A \amp C))\)
Distribution
10.
\((A \rightarrow \neg B) \rightarrow(B \rightarrow \neg A)\)
Contraposition
11.
\(\neg \neg A \rightarrow A\)
Double Negation
Rule
Name
1.
\(A \rightarrow B, A\vdash B\)
Modus Ponens
2.
\(A, B\vdash A \amp B\)
Adjunction