We give here an account of a series of lessons in which a class of Juniors carried out some simple experiments with numbers. In these experiments the numbers, expressed in the familiar Hindu-Arabic notation, were subjected to certain sequences of operations from which there resulted fairly obvious digital patterns—reversals, interchanges, transpositions, rhythms and so on. These patterns were the focal points to which the attention of the class was directed.

The present age is witnessing the rapid growth of a wide variety of communication and control systems using electrical signals. Examples are radio and commercial communications, television and facsimile transmission, radar and aircraft control systems, computers and industrial control mechanisms. All these have in common the fact that information has to be transmitted and processed through a channel, and they can all be represented by a block schematic such as in fig. 1. The efficiency of these systems must be assessed by comparing their information handling capacity with their cost in time, transmission channel bandwidth, equipment complexity and money. It is therefore essential to define a quantitative measure of information and to derive relations between this quantity and the parameters of a given communication system and also to determine criteria for a maximum information flow under given conditions. Information theory concerns itself with these problems, based on simplified and somewhat idealized mathematical models of actual systems. The origins of information theory go back to the 1920’s, but consistent logical foundations for the theory were laid by Shannon in the late 1940’s and the modern theory dates from then. It might appear that information is too loosely defined as a concept to be submitted to quantitative analysis; and indeed, before we can do so, we must clearly differentiate between the amount of information in a message, its meaning (semantic aspects), and its value (philosophical aspects).

With his unaided faculties, a man is comparatively limited in performance. His vision, hearing and legs operate only over small distances, he cannot lift very heavy weights nor weave much cloth in an hour. But by making use of devices such as the aeroplane, the telephone, and the powered loom, he can exploit natural properties and forces, and thereby attain what is literally a superhuman standard of performance in all kinds of activities.

Since the end of the last century railway engineers have known that when a railway vehicle passes round a curve a very large force can exist between the outer leading flange and the rail, even under equilibrium conditions. However it was not until Mr. S. R. M. Porter’s paper appeared in 1934 that a satisfactory method of calculating this force was established. Mr. Porter included in his paper two problems of considerable complexity which would take many hours or even days to solve. This may be one of the reasons why his paper is not more widely known.

Boolean class algebra is a special case of lattice theory and it is well known that the propositional calculus is Boolean. It is also well known that there are sixteen and only sixteen truth functionally distinct relationships between two elementary propositions in the pronositional calculus.

Some time ago Professor Prior circulated some typewritten notes due to Mr. C. A. Meredith on logical questions. These were written in the Polish notation and Professor Prior kindly translated those on equational logic into a symbolism more familiar to us. How much of the present paper was actually in those notes, it would now be impossible to say, but they introduced us to the subject and inspired this paper which, without them, would not have been written.

There is a growing realization among mathematicians and logicians of the many-sided role played by the axiom of choice in various branches of mathematics. Many of them tend to accept the axiom of choice as a legitimate principle provided, of course, it is proved to be independent in a suitable axiom system. This tendency has been accelerated by Gödel’s proof of the compatibility of this axiom in a reasonably broad system of axioms [2]. Such a view seems to have been shared by Fraenkel and Bar-Hillel [1; pp. 44-80] in their excellent exposition of the function of the axiom of choice in the modern mathematics in general and the axiomatic set theory in particular.