Supplement to Relevance Logic

The Logic NR

Here is a Hilbert-style axiomatisation of the logic NR.

Our language contains propositional variables, parentheses, necessity, negation, conjunction, and implication. In addition, we use the following defined connectives:

AB =df ¬(¬A & ¬B)
AB =df (AB) & (BA)
Axiom Scheme Axiom Name
1. AA Identity
2. (AB) → ((BC) → (AC)) Suffixing
3. A → ((AB) → B) Assertion
4. (A → (AB)) → (AB) Contraction
5. (A & B) → A,(A & B) → B & -Elimination
6. A → (AB), B → (AB) ∨-Introduction
7. ((AB) & (AC)) → (A → (B & C)) & -Introduction
8. ((AB) → C)↔((AC) & (BC)) ∨-Elimination
9. (A & (BC)) → ((A & B)∨(A & C)) Distribution
10. (A → ¬B) → (B → ¬A) Contraposition
11. ¬¬AA Double Negation
12. □(AB) → (□A → □B) K
13. (□A & □B) → □(A & B) K&
Rule Name
AB, AB Modus Ponens
A, BA & B Adjunction
A ⊢ □A Necessitation

Copyright © 2012 by
Edwin Mares <Edwin.Mares@vuw.ac.nz>

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