Sound Proof System is Consistent
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Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\mathscr P$ be a proof system for $\LL$.
Suppose that $\mathscr P$ is sound for $\mathscr M$.
Then $\mathscr P$ is consistent.
By assumption, some logical formula $\phi$ is not an $\mathscr M$-tautology.
Since $\mathscr P$ is sound for $\mathscr M$, $\phi$ is also not a $\mathscr P$-theorem.
But then by definition $\mathscr P$ is consistent.
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.4$: Conditions for an Axiom System: Theorem $1$