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

Copyright © 2020 by
Edwin Mares <>

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