By G. B Keene

This textual content unites the logical and philosophical facets of set thought in a way intelligible either to mathematicians with no education in formal common sense and to logicians with out a mathematical heritage. It combines an uncomplicated point of remedy with the top attainable measure of logical rigor and precision. 1961 version.

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**Example text**

It is the class defined by: “… is not identical with itself” and is represented by the symbol: ∧. Thus we use the notation, ∧, for any class whose defining condition is such that nothing satisfies it. Self-augment of a Class We shall use Quine’s term “self-augment” (see: Quine, Mathematical Logic, p. 247), to mean the class whose members are either a given class itself or the members of that class. The self-augment of a class C is, therefore, the class defined by: “… is identical with C or is a member of C”.

Pairs, Ordered Pairs and k-Tuplets IN the Bernays system a distinction is drawn between two senses of the word “class”. The distinction is expressed by using the word “class” itself in one particular sense and by using, in addition, the term “set”. The point of the distinction will be explained later (see II. 2, p. 38). Now the theory of pairs, ordered pairs and k-tuplets which is a fundamental part of the system, concerns operations which are confined to sets. The nature of these operations is, however, quite independent of the domain of discourse to which they are applied.

All that is required is an understanding of the meaning and interrelation of the following symbols: ∼, , v, ⊃, ≡, (x)φ, (∃x)φ, ε All of them translate straightforwardly into a familiar word or phrase of ordinary speech. But they are not mere shorthand symbols. For each differs from its counterpart in ordinary speech in virtue of the fact that it is precisely defined; whereas the words of ordinary speech are not. We shall formulate, explain and illustrate the definition of each in turn. In doing so we shall make use of the two groups of letters: x,y,z,w, and: P,Q,R,S, which we call variables.