I may add the remark that my personal fortunes were of great help in the writing of this book. The migration of intellectuals which followed the disastrous political developments in Germany has greatly contributed to an exchange of the various standards of civilization; and I for one cannot but be grateful to a fate which led me into various countries, not as a traveler, but as a teacher and collaborator in the education of youth. I thus have taught logic and scientific methods in various countries and various languages; and I have studied the reaction of students of many nationalities to an instruction in a scientific logic. I have seen that the logistic approach to philosophy is not bound to a certain type of mind, or of milieu or of educational system, but represents a most successful clarification of ideas, by which all forms of scientific pursuit will profit. At the same time, the necessity of teaching in several languages led me to try to adapt the methods of symbolic logic to the study of conversational language; and I thus undertook an inquiry which turned out to be useful for the understanding both of logic and of language. The present book is the first systematic presentation of such dual use of the logistic symbolism.
The requirements of teaching and the desire to connect logic with the actual use of language have determined the structure of the book. Throughout, emphasis is laid on the applicability of the logistic symbolism to the meanings of conversational language. From the very beginning, the prepositional operations are introduced in two interpretations, distinguished as the adjunctive and the connective interpretation, and it is shown that the first, useful as it is for the calculus, cannot exhaust the meaning of the corresponding terms of conversational language until it is supplemented by another which can account for the feeling of connection that is associated, for instance, with the usual notion of implication. It is shown that such connective operations can be constructed from the adjunctive ones by the help of the metalanguage. Though mathematicians are inclined to disregard this problem as irrelevant for their purposes, a logic which claims to be the logic of conversational language and of scientific thought cannot overemphasize it. To prove that a satisfactory solution can be attained without abandoning the principles of what has been called an extensional logic constitutes one of the major objectives of the book.
This line of inquiry is continued by an analysis of grammatical categories aimed at the construction of a logistic grammar, at least the outlines of which are now traceable. The now generally accepted notion of prepositional function is used as the major tool of this analysis; to carry it through, however, a number of special investigations had to be made, the results of which, I hope, will be of interest to both the logician and the linguist. I should like to mention in this respect the analysis of event arguments, of token-reflexive symbols, of the tenses of verbs, of the nature of the adjective and the adverb, and of logical terms. I should be glad if the philologist could make use of these contributions by a logician who advances no claim to be an expert in philology, but who feels that the state of traditional grammar is hopelessly muddled by its two-millennial ties to a logic that cannot account even for the simplest linguistic forms.
The interest in practical applications cannot make an analysis of the foundations dispensable. In fact, whoever has taught knows that a clear insight into the meanings of fundamental notions and methods is a prerequisite for successful teaching. In an introductory chapter as well as in repeated nontechnical inserts I have tried to clarify the notions and methods used. I was thus led to an analysis of the nature of deductive methods and a discussion of the proofs of consistency, and I arrived at results that may interest the mathematician who feels disturbed by doubts about the value of deductive proof. It appears to me that, although logic and mathematics are nonempirical sciences, statements about the reliability of logical and mathematical statements belong in an empirical metalanguage; there is no need, therefore, to claim absolute certainty for them. Among the technical features of the book, what may be interesting for the logician is the extension of the truth-table method to the calculus of functions, the introduction of the higher calculus of functions by a method which permits the derivation of the formulas of this calculus from those of the simple calculus, and the definition of connective operations aforementioned, which is accompanied by a definition of the modalities. An exposition of the foundations of mathematics, on the other hand, is not included in the plan of this book. I trust, however, that the mathematician who has studied symbolic logic from my book will find it easy to follow the presentation in other books dealing with this subject.
The notation I use is a simplified version of Russell's symbolism, with the exception of the sign for the negation, for which a horizontal line on top of the formula offers great advantages. Furthermore, I Ihave abandoned Russell's system of dots and have used parentheses, since they are so much easier to read.
A few words may be added advising the student how to read this book. Those who wish to study symbolic logic thoroughly will do best to follow the presentation in the order given. Those, however, who are chiefly interested in the linguistic applications of symbolic logic may make a short cut by omitting chapters IV and VI, with the exception of §§ 39-40, which should be read. Such readers will be sufficiently prepared to understand, on the whole, the analysis of conversational language given in chapter VII; and they may even read chapter VIII successfully. All students, however, are invited to test their abilities with the exercises given in the appendix for all sections marked at the end by the symbol '(Ex.)'. I have found that an understanding of the methods and the value of symbolic logic will never be reached until a sufficient familiarity with the technique of the symbolization is attained. For the benefit of those who feel uncertain about their achievements, the solutions of all these little problems are given in a second appendix. Should the reader feel unable to do a certain group of problems, it might be helpful to consult the solution of the first one of the group. Teachers who use this book in classes will find it easy to construct, along the lines given, additional exercises of their own.
It goes without saying that the ideas gathered in a book of this kind represent a selection from the contributions of many writers, and that it is not always possible to trace an individual discovery back to its author. Among the logicians to whom I am indebted, however, two men stand out who have shaped modern logic in its essential lines. The first is Bertrand Russell, of whose work I will single out here only the theories of propositional functions and of descriptions, the use of what he calls material implication, and the elaboration of a practical notation including operators and bound variables. The second is David Hilbert, of whose contributions to logic I mention here, in particular, the program of a complete formalization of the object language and of a proof of consistency. I had the good fortune to learn from both men, not only by reading their publications but also by personal contact: from Hilbert when I was a student in the University of Göttingen; from Russell when, some decades later, he was my colleague in the University of California at Los Angeles, long after I had studied his books. Among those who made their way into logic at the same time as I did, I should like to mention my friends Rudolf Carnap, whose insistence on a separation of language and metalanguage I took over, using it to extend his theory of tautological implication to a general theory of nomological statements, and Charles Morris, whose general theory of signs I used for the exposition of the sign nature of language. I cannot give here an account of the contemporary development of symbolic logic, and so I may be excused if I do not name the many others whose opinions have influenced me through their writings or through personal discussions. The ideas of symbolic logic, in fact, constitute today an atmosphere common to a large group of philosophers for whom the name logical empiricists has been coined; without my personal participation in the activities of this group I would not have been able to write this book. Let me express the hope that the present book will help to solidify the conceptions adhered to by the group, and help to win further adherents for its ideas.
Of those who helped me in the writing of the manuscript I should like to name my former assistants in Los Angeles: Dr. Norman C. Dalkey, Dr. Abraham Kaplan, and Mrs. Elinor Resting Charney. Many detailed formulations were developed by the help of their criticisms, and a great many of the exercises were constructed by them. I thank, furthermore, Professor Rudolf Carnap and Professor Charles Morris for the suggestions of improvements they made after reading the manuscript. My gratitude also goes to my wife, Maria Reichenbach, for the various forms of help and advice she afforded me in the writing of this book.
My warmest thanks, however, I wish to express to all the students of my courses in logic, in Germany, in Turkey, and in the United States, for their active interest in the subject. The never ending flow of their questions for the meaning of the terms and methods used impelled me to make my ideas clearer and clearer, and to develop methods of exposition which I never would have found in the solitude of the philosopher's study. Let me hope that this book will show what can become of a philosophy when it is worked out and experienced in the close community of teacher and students, when it is the live subject of a common inquiry into the nature of thought.
HANS REICHENBACH
University of California
at Los Angeles
October 1946