A sequent calculus P, for a negationless first-order predicate logic is given. The completeness of the system with respect to a truth definition in the style of Kripke may be established classically. The system is equivalent to P2 in [7]. (To be precise, it is equivalent to P2 without the symbol % and without the axiom -> 9". These were introduced in [7], merely by way of typographical economy. Obviously a similar com? pleteness proof could be given for a system containing these.) Naturally P satisfies the same readability definition as that given for P2, and is negationless in a semantic sense, both with respect to the readability definition and to the Kripke-type definition. A syntactical non-nullity theorem is also available. From the viewpoint of pure predicate calculus, the system gives a satisfactory cha? racterization of negationless reasoning. However a system in the style of Px in [7] is generally more appropriate for the development of a negationless theory. For a particular first-order negationless theory, one naturally has in mind an in? tended interpretation if he is thinking constructively. Even if the system should be characterizable by a (finite) set of axioms or axiom schemata, it is not clear that the representation of negationless reasoning in P is entirely satisfactory. Consider r the set of (closed) axioms of the theory. Then a natural analogue to intuitionistic or classical usage would be to identify theorems of the theory with formulas A such that \-p T -> A. However not all of the sequents in the proof of r -> A in P need be negationlessly interpretable under the intended interpretation. (There exist subsystems of P which have this property.) The situation is, of course, further complicated by the fact that in negationless systems, rules are more likely to be required than axioms or axiom schemata. The development of arithmetic in A1 or A2 is an example [7]. This was the reason for proposing the more complicated Plt For the system P, the sequent r -> O is understood to mean that for any negation? less interpretation for which all the formulas of r are true, at least one of those of ? is true also; and furthermore that there exists some negationless interpretation for which all the formulas of Y are simultaneously satisfiable and for each formula of 0, one for which it is satisfiable. The fact that a concept of falsity is used in the counter-example does not of neces? sity make the counter-example negationlessly meaningless. The general argument as it is outlined here is of course, non-constructive but it is easy in some cases to think