An infinite set A is said to be hyperhyperimmune (h.h.i.) if, for any collection of disjoint simultaneously recursively enumerable (r.e.) finite sets, A must fail to intersect with one of those sets. Thus the elements of an h.h.i. set are, in a sense, very elusive. D. A. Martin [3] showed that the degrees of h.h.i. sets with r.e. complements are exactly the r.e. degrees with jump 0″. More generally, C. G. Jockusch [2] found a′ ≥ 0″ to be a sufficient condition for a to be the degree of an h.h.i. set and found a′ ≥ 0′ to be necessary. However, it was also shown that in the degrees as a whole neither condition gave a characterization of the h.h.i. degrees. The purpose of this note is to prove that a′ = 0″does characterize the h.h.i. degrees below 0′.

Theorem. The degrees below 0′ containing h.h.i. sets are exactly those degrees below 0′ with jump 0″.

Proof. From [2], if a′ ≥ 0″, then a contains an h.h.i. set.

Conversely, let A ∈ a where a′ < 0″ and a < 0′. Let {As ∣ s ≥ 0} be a recursive sequence of finite sets such that for each x, lims, Ax(x) exists and equals A(x).

For a set B, let B[m] denote B ∩ [0, m], and (if B is finite) let ∣B∣ denote the cardinality of B.

In what follows there is presented a unified semantic treatment of certain “paradox-free” systems of entailment, including Church's weak theory of implication (Church [7]) and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6]). We shall refer to these systems generally as relevant logics.

The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice—interpreted informally as the semilattice of possible pieces of information. Completeness theorems can be given relative to these semantics for the implicational fragments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems—in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the “paradoxes of material implication”.

The systems discussed are formulated in fragments of a first-order language with → (entailment), &, ⋁, ¬,(x) and (∃x) primitive, omitting identity but including a denumerable list of propositional variables (p, q, r, p1,…etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, A1,… are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The semantics of the systems are given in informal terms; it is an easy matter to turn the informal descriptions into formal set-theoretical definitions.