In this paper we show how a modification of results due to Simons ([6]) yields a set of independent axiom schemata for von Wright's M ([8], p. 85), with a single primitive rule of inference. We first describe a system M*, then show its equivalence with M, and finally show that our schemata are independent.1. Axiomatization ofM*. We adopt the notational conventions of McKinsey and Tarski ([4], p. 2), as amended by Simons ([6], p. 309), except that we take “(α ⊰ β)” as an abbreviation for “˜◇˜(α 0→ β)”, rather than for “˜◇(α ∧ ˜β)”. Our only rule of inference is:Rule. If ⊦ α and ⊦ (˜◇˜α → β). then ⊦ β.We have six axiom schemata:We require a number of theorems for the proof of equivalence of M* with M.Theorem 1. If ⊦ α and ⊦ (α → β), then ⊦ β.Theorem 2. If ⊦ α and ⊦ (α ⊰ β), then ⊦ β.Proof by hypothesis, A5, and Theorem 2 (twice).Theorem 3. If ⊦ (α ⊰ β), then (˜◇β → ˜ ◇α).Proof by hypothesis, A6, and Theorem 2.Theorem 4. If ⊦(α ⊰ β), then ⊦[˜˜(γ ∧ α) ⊰ ˜˜(β ∧ γ)].
A reduction in the number of independent axiom schemata for $S4$.
