An individual or whole we understand to be whatever is represented in any given discourse by signs belonging to the lowest logical type of which that discourse makes use. What is conceived as an individual and what as a class is thus relative to the discourse within which the conception occurs. One task of applied logic is to determine which entities are to be construed as individuals and which as classes when the purpose is the development of a comprehensive systematic discourse.The concept of an individual and that of a class may be regarded as different devices for distinguishing one segment of the total universe from all that remains. In both cases, the differentiated segment is potentially divisible, and may even be physically discontinuous. The difference in the concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to what these subdivisions, if any, must be, whereas to conceive a segment as a class imposes a definite scheme of subdivision—into subclasses and members.The relations of segments of the universe are treated in traditional logistic at two places, first in its theorems concerning the identity and diversity of individuals, and second in its calculus of membership and class-inclusion. But further relations of segments and of classes frequently demand consideration. For example, what is the relation of the class of windows to the class of buildings? No member of either class is a member of the other, nor are any of the segments isolated by the one concept identical with segments isolated by the other. Yet the classes themselves have a very definite relation in that each window is a part of some building. We cannot express this fact in the language of a logistic which lacks a part-whole relation between individuals unless, by making use of some special physical theory, we raise the logical type of each window and each building to the level of a class—say a class of atoms—such that any class of atoms that is a window will be included (class-inclusion) in some class that is a building. Such an unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or even upon the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Furthermore, a raising of type like that illustrated above is often precluded in a constructional system by other considerations governing the choice of primitive ideas.

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We shall consider finite groups of order of g which satisfy the following condition:(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).

This paper aims to establish three major theses: (1) Not only declarative sentences, but also interrogatives and imperatives, may be classified as true or as false. (2) Declarative, imperative, and interrogative utterances may also be classified as honest or as dishonest. (3) Whether an utterance is honest or dishonest is logically independent of whether it is true or is false.The establishment of the above theses follows upon the adoption of a principle for identifying what is meant by any sentence, declarative, interrogative, or imperative. The analysis aims to show that meaning is to be attributed to the uttered or written sentence-token, rather to the thereby exhibited sentence-type. Further, the meaning of the sentential token is to be identified with a purpose of the speaker, that the speaker would reveal to the addressee by uttering the sentence.The to be revealed purpose is analysed into two components: an ultimate concern (that the addressee stand in such and such a relation—e.g., of believing, or informing the speaker about, or making it true that) and an ultimate topic of concern (the state of affairs, i.e., proposition, relative to which the speaker would have the addressee stand in the specified relation).Sentential utterances “signify” different purposes by “expressing” different ultimate concerns and “indicating ”different ultimate topics of concern. Variations in expressed concern are correlated with variations in sentential form, such as declarative, interrogative and imperative. Variations in indicated topic of concern are correlated with variations in the subject and predicate of the uttered sentence. Thus, for example, utterances of “Johnny will jump in the lake,” “Will Johnny jump in the lake?” and “Johnny, go jump in the lake! ” all indicate one and the same ultimate topic of concern but express different ultimate concerns with this topic.A sentential utterance is true or false according as its indicated topic of concern is true or false. Hence, declaratives, interrogatives and imperatives may all be classified as true or as false. But honesty or dishonesty is a function (explained in the paper) of the expressed concern, rather than of the topic of concern. Hence, although utterances of all sentential forms are honest or dishonest, their honesty or dishonesty is logically independent of their truth or falsity.

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