. . . postulating the validity resp. satisfiability of a logical state- ment is equivalent to a statement about the size of the domain.
In the most general sense one can say the decision problem is solved if one has a procedure that determines for each logical expression for which domains it is valid resp. satisfiable.
is valid for all domains
Examples of formulas which are valid in every domain are those derived from the predicate calculus. Since one suspects that this system gives all such [valid] formulas, one would move closer to the solution of the decision problem with a characterization of the formulas provable in the system.
A general solution of the decision problem, whether in the first
or second formulation, has not appeared till now. Special of the decision problem · · · have been attacked and solved
cases by P.
2One finds Schro¨der alluding to this fact in Vol. III of Al ebra der Lo ik, but he does not try to justify his remarks.