In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is

(I) strong enough to express interesting properties not expressible by the classical language, but

(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

A. N. Prior in [9] has suggested an approach towards a second order logic of existence where, following medieval logicians, we distinguish “between predicates (like ‘is red’, ‘is hard’, etc.) which entail existence, and predicates (like ‘is thought to be red’, ‘is thought of’, etc.) which do not.”2 Let us refer to attributes (including relational attributes) which are designated by the former kind of predicate as existence attributes, or for brevity, e-attributes.

In this paper we prove infinitary analogues of model-theoretic results known for finitary logic. The infinitary language we deal with is Lω1ω which is roughly described by saying that, in addition to the usual formation rules of the lower predicate calculus with identity, also the formation of the conjunction and disjunction of countably many formulas is allowed.

In this note we sketch a decision procedure for S3.51 based on reduction to conjunctive normal form. Using the following theorem of S3.5: and its dual for M over a conjunction, any formula can be reduced by standard methods (as in S52) to a conjunction of disjunctions of the form where Í is (p ⊃ p), 0 is ∼(p ⊃ p) and α — λ are all PC-wffs (i.e. they contain no modal operators).

By γ, we mean the rule, “From ├ A and ├ Ā V B, infer ├ B”.1 This rule has played an important and a controversial role in a set of relevant logics free of certain well-known paradoxes of implication, like AĀ-→B and A-→(B-→B). Among these logics we count the pioneering systems of strenge Implikation presented by Ackermann in [1],2 as well as the Anderson-Belnap systems E of entail-ment and R of relevant implication.3

In this paper we continue our investigation of the Dedekind cardinals which was initiated in [2]. Those results are summarized below. Let ω be the finite cardinals and Δ the Dedekind cardinals. In [7] Myhill defined a class of functions ƒ: Χκω→ω, which he called the combinatorial functions, and which he applied to the study of recursive equivalence types.

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

Dummett's LC [1] is a system which characterizes all formulas of the propositional calculus which are valid in every chain (for definitions and notation see the first section of [2]). An L-algebra is a Heyting algebra in which (x → y) + (y → x) = 1 for all x, y. L-algebras bear the same relation to LC as Boolean algebras to the classical propositional calculus and Heyting algebras to the intuitionist propositional calculus.

The question, “What do initial segments of the degrees of unsolvability look like?” has interested recursive function theorists for several years. Sacks [4] hypothesized that Sis a finite initial segment of degrees if and only if S is order-isomorphic to a finite initial segment of some upper semilattice with a least element. Lachlan [2] suggested the generalization, S is an initial segment of degrees if S is order-isomorphic to some countable upper semilattice with both least and greatest elements.

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

In 1956 W. Ackermann proposed a new axiomatic set theory, that has received some attention in recent years. (See references [3], [4], [6], [7], and [9].) This theory distinguishes between sets and classes. In this paper we study mainly the natural models of this theory. We show, among other results, that the set-theoretical fragment of these models are also models of Zermelo-Fraenkel set theory. This result gives a partial answer to the question, raised by A. Levy, of the relative strength of Ackermann's set theory with respect of Zermelo-Fraenkel's.2

In this paper we introduce a system S5U, formed by adding to the modal system S5 a new connective U, Up being read “probably”. A few theorems are derived in S5U, and the system is provided with a decision procedure. Several decidable extensions of S5U are discussed, and probability logic is related to plurality quantification.

In [5, Corollary 3.1] D. A. Martin characterized the degrees of the hyperhypersimple sets as those r.e. degrees a satisfying a′ = 0″. In the present paper we investigate the degrees of the hyperhyperimmune (h.h.i.) sets. We do not achieve a characterization of these degrees, but we do show that the condition a′ ≥ 0″ is sufficient, and the condition a′ ≥ 0′ is necessary, for a degree a to contain an h.h.i. set.