Inasmuch as my discussion of the formal properties of “System L,” presented elsewhere, is substantially the same as that presented by Lewis in a previous issue of this Journal, I shall confine my remarks in this brief rejoinder to the problem of deducibility with respect to necessary and impossible propositions—since it is specifically the solution of this problem which will determine the adequacy of the calculus of logical implication as a canon and critique of deductive inference.

The problem, as originally presented, can be stated succinctly in the following inconsistent triad: (1) All necessarily true principles are logically dependent on or deducible from one another, i.e., are equivalent, (2) All principles of logic and mathematics are necessarily true, and (3) Some principles of logic and mathematics are not logically dependent on or deducible from one another, i.e., are independent. These three propositions are logically inconsistent in the sense that the conjunction of any two of them will logically imply the falsity of the third. Now if Lewis should affirm the first and third of these propositions, then he should concede the falsity of the second. This concession, it was believed, he was not likely to make, inasmuch as it would seem to involve a repudiation of a doctrine (the tautological character of every logical and mathematical truth) with which he has particularly identified himself. If, in view of this fact, he should accept, instead, the second and third of these propositions, then he should concede the falsity of the first.