A previous paper proposed a finitary system of logic which was surmised to be basic in the sense that every finitary system of logic could be defined within it. In the present paper this anticipated result, or rather its syntactical counterpart, is definitely established as Theorem VI. The calculus K is a formalization of the basic logic. Its main features were described in the previous paper, acquaintance with which is presupposed here. Attention is called to the fact that from now on the class Y of atomic U-expressions will be assumed to be finite instead of infinite. This is a relatively minor change and does not affect the previously established results. The account of the nature of calculi, as given in the other paper, is to be made more precise by stating that a calculus is any class of expressions having as a Gödel representation a recursively enumerable class of integers. A system of logic is regarded as being finitary if it is capable of formalization by means of an appropriate calculus.