 A characteristic matrix for a system L is a matrix that satisfies all and only the theorems of L. A matrix is finite if its set K of truth-values is finite. A finite characteristic matrix yields a decision procedure, where a system is decidable if every formula of the system that is not a theorem is falsified by some finite matrix (this is the finite model property). Yet Dugundji (1940) shows that none of S1–S5 has a finite characteristic matrix. Hence, none of these systems can be viewed as an \(n\)-valued logic for a finite \(n\). Later, Scroggs (1951) will prove that every proper extension of S5 that preserves detachment for material implication and is closed under substitution has a finite characteristic matrix.