Career induction stops here (and here = 2)

Abstract

Now that Martin in [3] has solved the PW problem, the stock of ancient open problems for the relevant logics has been reduced to one. (Fresh problems are ever upon us, to be sure, such as the question of semantical completeness for relevant quantification theories raised in [4]. But I speak here only of problems of sufficient age and venerability to have defeated a generation of scholars a sufficient but not necessary condition is that the problem should have been mentioned in [5] and the relevant logics have provided but one such.) The Last Problem, for the relevant logics, is the question of their decidability. It is, to borrow a phrase used by J. Ehrlichman in [6] (in another problematic situation) the Big Enchilada. While the honors of Old Age are due the problem, the reader who elects to try it is to be warned that it remains in the best of health, and it is not likely soon to join PW among those problems that have gone to their Reward. (The Reward is to have its solution published somewhere, with all honor, glory, and, in extreme cases, tenure due the champion that has vanquished it.) Moreover, Old Age has made the decidability problem for R, E, and their closest relatives even more crotchety. Beware, dear Reader. This problem has defeated better men than you, as your humble author will be the first to testify. Most of those who have dragged themselves away from hand-tohand combat with the Big Enchilada (and contented themselves with fresh battles on easier terrain, such as Quantum Mechanics and the General Theory of Relativity) have left in complete defeat, all their ideas having been busted and with nothing left to try. (Your bedraggled author will testify to that, too.) One Warrior, however, has had a plan. Aside from the brief but telling encounter with the decision question recounted by Kripke in [7], it is the only plan to have produced a significant advance in the status of the decidability question. Ever. The scholar with the plan is Belnap. He calls his plan 'career induction' (whence the title of this essay). The plan is to whittle away at the formulas of R, a degree at a time. Since the 0-degree formulas