Logical Empiricism at Its Peak

Schlick, Carnap, and Neurath

Maria Neurath, Sahotra Sarkar, Moritz Schlick, Rudolf Carnap, Sahotra Sarkar

• Publisher: Routledge
• Year: 2021
• Print ISBN: 9781032165769
• eBook ISBN: 9781000526547
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

First Published in 1996. This volume reprints pieces from the Vienna Circle period between the manifesto and the adoption of semantics, as well as two commentaries. During this period, the logical empiricists were the most ambitious and the most confident about the success of their enterprise. The first section consists of four ideological classics, The second section reprints three papers on physicalism. The third section consists of three papers on logic and the fourth on reprints three papers on truth, induction, and confirmation.

The New Logic

DOI: 10.4324/9781003249573-10

BY KARL MENGER(Translated by H. B. Gottlieb and J. K. Senior*)

*‘Translator’s Note: This paper is a revised version of a lecture delivered in 1932 and published in “Krise and Neuaufbau in den exakten Wissenschaften.” (Deuticke, Vienna, 1933.)

THE rapid development of physics, the result of observations made and ideas introduced within the last few decades, has brought about a change in the whole system of physical concepts. This fact is common knowledge, and has already attracted the attention of philosophers. It is less well known that geometry too has had its crises, and undergone a reconstruction. For centuries, so-called “geometrical intuition” was used as a method of proof. In geometrical demonstrations, certain steps were allowed because they were “self-evident,” because the correctness of the conclusion was “shown by the adjoined diagram,” etc. A crisis occurred in geometry because such intuition proved to be untrustworthy. Many of the propositions regarded as self-evident or based upon the consideration of diagrams turned out to be false. And so Euclidean geometry was reconstructed by methods free from all intuitive elements and strictly logical in nature. Moreover, for more than a century, various other geometries have been devised purely as logical constructions. Since they start with assumptions different from those of Euclid, and lead to conclusions partly in contradiction with his theorems, they are called “non-Euclidean.” Nevertheless, each one of these geometries is a closed system of propositions exempt from contradiction. Recently, some of them have even found application in physics.

As far as experience and intuition are, concerned, anyone, even though he be unfamiliar with the details of physics and geometry, can well understand that new empirical discoveries may overthrow the most venerable of ancient theories, and that intuition, if it ventures too far afield, may be forced to retire from positions erroneously occupied. One subject, however, is generally supposed to be unchanging and unshakable. That subject is logic. Hence, to anyone who is not an adept in the field, the recent active discussion of crisis and reconstruction within the realm of logic may seem not only strange but incomprehensible.

As a matter of fact, for two thousand years, logic has been the most conservative of all the sciences. Aristotle, who is considered the father of the subject, assumes as a starting point that every proposition ascribes a predicate to some subject. The propositions are classified on the one hand as affirmative or negative, on the other hand as universal or particular. “All cats are mammals” is universal and affirmative; “Some mammals are cats” is particular and affirmative; “Some mammals are not cats” is particular and negative; “No cat is a fish” is universal and negative. According to Aristotle, all inference consists in deducing a third proposition from two propositions of the given form. For example, from the premises that all cats are mammals and all mammals are vertebrates, it follows that all cats are vertebrates. From the premises that all cats are mammals and no mammal is a fish, it follows that no cat is a fish. All the inferences which Aristotle considered possible he arranged in three figures divided into altogether fourteen varieties. In the Middle Ages, these modes of inference were expanded to four figures including all together nineteen varieties, and were designated by the names Barbara, Celarent, etc. If, from the premises that to every (resp. no) M belongs the predicate P, and that to every S belongs the predicate M, it is inferred that to every (resp. no) S belongs the predicate P, then the inference is said to be according to the mode Barbara (resp. Celarent). Moreover, the three principles of identity, contradiction and excluded middle, which were later called the fundamental principles of logic, were also formulated by Aristotle, curiously enough not in his logic but in his metaphysics. The kernel of his system, the above mentioned logic of subject-predicate propositions, was essentially all that was as logic for two thousand years thereafter.

In the Middle Ages, it is true, the scholastics undertook a number of important logical investigations. But more recently, and particularly during the period of the Enlightenment, it became customary to regard the Labors of the scholars in the Middle Ages as hair-splitting; their work in logic was thus forgotten and was replaced by less fundamental considerations. Similarly, Leibniz’s views on logic, which were far in advance of their time, also remained without direct effect. It was perfectly clear to Leibniz that a mere treatment of subject-predicate propositions was insufficient, and must be supplemented by a logic of relations. Furthermore, he treated the logical principles and their mutual relations more systematically than his predecessors, and devised the project of a “lingua characteristica,” which should permit all scientific propositions to be stated in precise form, and of a “calculus ratiocinator,” which should contain and treat mathematically all the methods of inference. But the clearest evidence that Leibniz’s views, like the works of the scholastics, found no echo, is the famous dictum of Kant in the introduction to the second edition of the “Critique of Pure Reason.” “That logic has trodden this sure path since the earliest times, can be seen from the fact that, since Aristotle, it has not been obliged to take a single backward step.—A further noteworthy fact is that until the present it has been able to take no step forward, and so seems apparently to be finished and complete.” And in Kant’s “Logic” occurs the statement “The logic of to-day is derived from Aristotle’s Analysis.—Moreover, since Aristotle’s time, logic has not gained much in content, and from its very nature it cannot. For Aristotle omitted no item of reason.”

The first of the crises in logic was caused by defects in this Aristotelion system which were brought to light by mathematics. When confidence in geometrical intuition had been shaken, there came a purely logical reconstruction of geometry. This consisted of a complete enumeration of all the hypotheses (axioms) concerning fundamental concepts not explicitly defined, and a deduction of the whole system of theorems from these axioms by strictly logical methods without any appeal to experience, diagrams, self-evidence or intuition. An analogous purely logical construction of arithmetic followed. This trend in mathematics naturally carried within a search for clarity in regard to all the principles of inference used in the deduction of mathematical propositions from axioms. And in the course of this search, it became evident that the old logic was utterly insufficient for the demands of modern mathematics in respect to both precision and completeness. It was therefore chiefly mathematicians who undertook and carried out the desired reconstruction of logic. A short sketch of the chief stages in this reconstruction (which is called logistics) will next be given; but it should be understood that this logistics, although much more far-reaching than the Aristotelian logic, is to-day counted as part of the classical content of logic. It will appear that the real problems of the new logic only begin where logistics stops.

The first step in the reconstruction was the development of the so-called calculus of classes (also called algebra of logic) particularly by Boole, Pierce and Schröder in the second half of the last century.¹ The Aristotelian logic, as stated, deals mostly with the question: If the relations of two classes to a third class are known, what can be said about the relations of the two classes to each other? For example, if the relations of the class of all cats and the class of all vertebrates to the class of all mammals are known, what can be said of the relations of the class of all cats to the class of all vertebrates? The calculus of classes is more extensive; for it is a systematic theory of the relations of any number of classes. Aristotle only investigates inclusions. He asks whether of two classes one is entirely or partly contained in the other, or whether one is entirely or partly outside the other. But the calculus of classes investigates many other relations between classes besides inclusion, and undertakes many other operations with classes. Among other things, it treats systematically of the “join” and “intersection” of two classes A and B—that is the class of all elements in either A or B as well as the class of those in both A and B. In contradistinction to Aristotle, the null class (containing no member) is also considered. For instance, the intersection of the class of all cats and the class of all fish is the null class. The calculus of classes, starting from a certain few propositions, gives a systematic treatment of the relations of all classes. Among its theorems, there are nineteen which correspond to the Aristotelian scholastic modes of inference.²

¹See Boole, “The Mathematical Analysis of Logic,” Cambridge (1847), and “An Investigation of the Laws of Thought,” London (18$4) and Chicago (1916); Schrader, “Vorlesungen über die Algebra der Logik,” 3 vols. Teubner, Leipzig (1890-1910) and “Abriss der Logik,”ibid. (1909-1910). See also the first chapters of Lewis and Langford “Symbolic Logic,” The Century Company, New York. ²According to Ladd Franklin, these can moreover be combined into a single theorem, namely the theorem that for the three classes A, B and C, it cannot be simultaneously true that the intersection of A and B is the null class, the intersection of not-B and C is the null class, and the intersection of A and C is not the null class. In the case of the mode Barbara, the premises are: All M is P and all S is M. That is, the intersection of the class M with the class not-P is the null class, and the intersection of the class S with the class not-M is the null class. Then, according to the Ladd Franklin formula, it is impossible that the intersection of the classes S and not-P should not be the null class. In other words, the intersection of the class S and the class not-P is the null class. That is, all S is P, and the assertion of the mode Barbara is thereby proved.

But in the calculus of classes, these nineteen theorems are not the only ones; moreover they occupy in it no distinguished position. They are not among its initial propositions (in the first place because the whole of this calculus can be based on much fewer than nineteen propositions), and so they are not necessary for the founding of a systematic calculus of classes. But even if it were desired to assume so needlessly large a number of initial propositions, the particular nineteen Aristotelian ones would not be sufficient for the construction of the whole calculus of classes. This calculus is therefore a definite advance over the old logic of class inclusion.

A second step leads further. The calculus of classes is a theory deduced from a few initial propositions concerning the relations of classes, just as Euclidean geometry is deduced from a few initial propositions (axioms) concerning the relations of points, lines, and planes.³ The calculus of classes is thus a special mathematical theory; it is, however, far from containing the whole of logic, for logic does not confine itself to the consideration of classes. If further propositions are deduced from any set of propositions, this deduction is called logical inference. The subject matter of this inference is, however, not classes but propositions. And yet logic is expected to deal with the rules of inference. The second step in the expansion of logistics, which goes back principally to Pierce and Schröder, was therefore the development of a calculus of propositions. This calculus teaches how propo...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Table Of Contents
• Series Introduction
• Volume Introduction
• Ideology
• Physicalism
• Logic and the Philosophy of Mathematics
• The New Logic
• Ethics
• Unity of Science
• Commentaries
• Acknowledgments