In Jaśkowski [1], some properties of the consequences of formulas of the sentential calculus with no more than two occurrences of any variable have been settled. Analogous problems may be raised concerning the consequences of equalities in which the same restriction relates to individual variables. In the present paper, problems of this kind are solved and more general theorems are proved about the logical consequences of some formulas, called here multiconditionals.

I. (Bull's system OIC, defined in [1] as positive implication plus ССССpqqrСССpqrr, can be axiomatised by adding this to a weaker base.)

1. СpСqp

2. ССpqССqrСpr

3. ССССpqqrСССpqrr

4. ССССqrСprsССpqs = D22

5. ССrСpqСССpqСrqСrq = D43

6. СССССpqССrqСrqsССrСpqs = D25

7. ССpСqrССpqСpr = D64

In this paper we follow up our work in [2] on standard classes of recursively enumerable sets, and it will be supposed that the reader is familiar with [2]. One of the main problems left open in [2], that of determining whether or not every standard class has a least member is resolved by the construction of a standard class all of whose members are non-empty, and two of whose members are disjoint. This shows that there is a standard class which is not p.r. in the sense of [2] and we now prefer the adjective sequential for those standard classes which were called p.r. in [2]. Otherwise our terminology will be the same as in [2]. We shall also prove the theorem only stated in [2] that any standard class all of whose members have cardinality < 3 is sequential. Further, we give an example of a standard class which is not sequential and all of whose members have cardinality < 4.

By ΡL we shall mean the first order predicate logic based on S4. More explicitly: Let Ρ0 stand for the first order predicate calculus. The formalisation of Ρ0 used in the present paper will be given later. ΡL is obtained from Ρ0 by adding the rules the propositional constant □ and

Modal logic received its modern impetus from the work of Lewis and Langford [10]. In recent years, however, their axiomatic approach, aided by somewhat ad hoc matrices for distinguishing different modal systems, has been supplemented by other techniques. Two of the most profound of these were, first, the algebraic methods employed by McKinsey and Tarski (see [11] and [12]) and, second, the semantic method of Kripke (see [5] and [6]); and there have been others. The aim of the present series of papers is to afford a synthesis of these methods. Thus, though new results are given, the interest lies rather in revealing interconnexions between familiar results and in providing a general framework for future research. In general, we show that semantic completeness results of the Kripke kind can be deduced from the algebraic results by means of one central theorem (Theorem 21).

In [1], p. 171, Sacks asks (question (Q5)) whether there is a recursively enumerable degree of unsolvability d such that for all n ≧ 0. Sacks points out that the set of conditions which d must satisfy is not arithmetical. For this reason he suggests that a proof of (Q5) might require some new combinatorial device. The purpose of this note is to show how (Q5) may be proved simply by extending the methods of [l].2

In [2], Fischer has shown the existence of a bounded-truth-table degree of unsolvability which contains a collection of many-one degrees which has, under many-one reducibility, the order type, ω, of the positive integers. In [1; pg. 124], Dekker and Myhill give a construction which shows that the many-one degree of any simple set contains a collection of one-one degrees of simple sets which has, under one-one reducibility, the order type (ω + ω*) ·ω. A basic result of recursive function theory (due to Myhill) is that the creative sets form a many-one degree which consists of a single one-one degree (i.e. a single recursive isomorphism type).

It is well known that the recursive sets form three many-one degrees. Two of these, the degrees of the empty set and of the set of all integers, consist of single one-one degrees.

Markov's [2] notion of Normal Algorithm (NA), with which we assume familiarity, is the simplest of the better known ways of expressing the concept of algorithmicity. An NA consists of an ordered finite list of replacement formulae

The dots in parentheses indicate that a given replacement formula may be either simple (P → Q), or terminal (P → .Q). No restrictions are imposed on the lengths of the Pi and Qi in the replacement formulae, and the only rule of order of application is a priority rule to the effect that of several replacement formulae applicable to a word W at a given step, the earliest one is used.