Rudolf Carnap

Studies in Inductive Logic and Probability

A bottle of especial utility is disclosed. The bottle may be made of synthetic material as polyvinyl chloride. The bottle is formed by the rotation of a faired generating line made up from a straight linear segment and arcs of selected radii and centers about a central longitudinal axis. The linear segment of the generating line is parallel with the axis and the arcs are tangent at their points of juncture with each other and with the linear segment. The generating line thus rotated defines a form from which a mold is readily and inexpensively made to produce a bottle having adequate strength. The bottle may be used to contain, in particular, carbonated and like beverages or liquids. The bottom of the bottle is interiorly convex reentrant shape and has an exterior integral diametral reinforcing rib.

Modalities and Quantification

The purpose of this article is to give a survey of some results I have found in investigations concerning logical modalities. The results refer: (1) to semantical systems, i.e., symbolic language systems for which semantical rules of interpretation are laid down; (2) to corresponding calculi, i.e., syntactical systems with primitive sentences and a rule of inference; (3) to relations between a semantical system and the corresponding calculus. The semantical systems to be dealt with are the following: pro positional logic (PL), functional logic (FL), and the corresponding modal systems, viz. modal propositional logic (MPL) and modal functional logic (MFL).

The Aim of Inductive Logic

Publisher Summary This chapter discusses the customary schema of decision theory involving the concepts of utility and probability. By inductive logic, the chapter indicates a theory of logical probability providing rules for inductive thinking. The chapter also explains the nature and purpose of inductive logic by showing the way it can be used in determining rational decisions. It mentions the term “probability” not in the objective sense but in the subjective sense —for example, as the degree of belief. This is a psychological concept in empirical decision theory, referring to the actual beliefs of actual human beings. The main conceptions of probability are often divided into two kinds, objectivistic and subjectivistic conceptions. These are not two incompatible doctrines concerning the same concept, but two theories that concern two different probability concepts, both of them legitimate and useful. The concept of objective (or statistical) probability is connected with relative frequencies in mass phenomena. It plays an important role in mathematical statistics, and it occurs in the laws of various branches of empirical science, especially physics.

Testability and Meaning--Continued

17. The Problem of a Criterion of Meaning...................... 2 I8. The Construction of a Language-System L ................... 4 19. Atomic Sentences; Primitive Predicates...................... 6 20. The Choice of a Psychological or Physical Basis .............. 9 21. Introduced Atomic Predicates.............................. 14 22. Molecular Sentences ..................................... I5 23. M olecular Languages...................................... 7 24. The Critical Problem: Universal and Existential Sentences.... 2o 25. The Scale of Languages ................................... 24 26. Incompletely Confirmable Hypotheses in Physics............. 28 27. The Principle of Empiricism ............................... 33 28. Confirmability of Predictions............................... 35

Inductive Logic and Inductive Intuition

Publisher Summary This chapter focuses on inductive logic and inductive intuition. Although in choosing axioms for inductive logic one has to rely on our inductive intuition, the intuitive plausibility of an axiom may be more or less strong; and in the course of the development of a system, there may be progress by an increase in plausibility. Inductive logic is the study of inconclusive but plausible (if preferred, reasonable but nondemonstrative) inference. The various kinds of inductive reasoning (chiefly generalization from observed cases and generalization from low level inductions) constitute just a subset of the class of nondeductive inference patterns. According to the analysis of inductive intuition, one may learn something about the objective probability from the observation of frequencies.

Über die Aufgabe der Physik

Nachdem lange Zeit hindurch die Frage nach den Quellen der physikalischen Erkenntnis heftig umstritten worden ist, darf heute vielleicht schon gesagt werden, daß der reine Empirismus seine Herrschaft verloren hat. Daß der Aufbau der Physik sich nicht aut die Versuchsergebnisse allein stützen kann, sondern auch nichterfahrungsmäßige Grundsätze verwenden muß, ist Ja von der Philosophie schon längst verkündet worden. Doch erst nachdem Vertreter der exakten Wissenschaften die Eigenart der physikalischen Methode zu untersuchen begonnen hatten und dabei zu einer nicht-empiristischen Auffassung gelangt waren, ergaben sich Lösungen, die den Physiker selbst befriedigen konnten. Hier sind vor allem Poincaré und Dingler zu wichtigen Ergebnissen gelangt. Wir gehen hier von ihren Grundsätzen aus, wenden sie aber allgemeiner an, als es bisher geschehen ist. Dadurch finden wir eine Antwort auf die Frage nach der Leistung der Physik, die uns erkennen läßt, welche logischen Beziehungen zwischen einander scheinbar widersprechenden physikalischen Theorien bestehen, und unter welchen Voraussetzungen eine Entscheidung zwischen diesen Theorien getroffen werden kann.

The Task of the Logic of Science

The works of the Vienna Circle and related groups have as their object to investigate science either as a whole or in its individual branches: they analyse the concepts, sentences, proofs, and theories that appear in different scientific fields, and they do this not so much from the point of view of the historical development of science as from the logical point of view. This field of work, for which there exists as yet no commonly recognized general term, may perhaps be called the theory of science and more precisely the logic of science. By “science” is to be understood here the totality of accepted sentences; and this includes not only the assertions of the scientists but also those of everyday life; for there is no sharp boundary between these two fields.

International Encyclopedia of Unified Science.

One can say that, from the point of view of scientific empiricism, it is not the notion of ’system’, but that of ‘encyclopedia’ that offers us the true model of science taken as a whole. In the spirit of scientific empiricism, the Mundaneum Institute in The Hague is preparing an International Encyclopedia of Unified Science destined to serve as a complement to existing encyclopedias.

Dreidimensionalität des Raumes und Kausalität

Zusammenfassung der ErgebnisseI.In der Erfahrung sind zwei Stufen zu unterscheiden: Die primäre Welt besteht aus den noch nicht dinglich gedeuteten Sinnesempfindungen in ihrer einfachsten Ordnung nach Zeit-, Raum-und Qualitätsunterschieden. Alle Ordnung und Verarbeitung der Erfahrung von solcher Art, daß sie auch weggelassen werden kann, rechnet zur zweiten Stufe. Ihr Inhalt ist die sekundäre Welt; Beispiele: die gewöhnliche Welt des täglichen Lebens und die physikalische Welt.II.Der Begriff der Dimensionszahl (DZ) wird festgelegt. Während die sekundäre (die gewöhnliche und die physikalische) Welt die DZ (3+1) hat (d. h. 3 Raum- und 1 Zeitdimension), ergibt die Untersuchung für die primäre Welt (den Bereich der ungedeuteten Sinnesempfindungen) nur die DZ (2+1). Die Konstruktion der sekundären Welt schließt also eine Erhöhung der DZ um 1 ein.III.Nach einer Bestimmung des Begriffs der determinierenden und der beschränkenden Gesetze zeigt sich: In der sekundären (physikalischen) Welt gelten determinierende Gesetze bestimmter Art (ersten Grades). In der primären Welt bestehen weder determinierende noch auch bloß beschränkende Gesetze. Der Aufbau der sekundären Welt führt also die Determiniertheit neu ein.IV.Die beiden in die sekundäre Welt eingebauten Fiktionen: Dreidimensionalität des Raumes (gleichbedeutend mit Vierdimensionalität des Weltgeschehens) und Determiniertheit oder physikalische Kausalität stehen in logischem Abhängigkeits-verhältnis zueinander. Die erste ist durch die zweite bedingt.

Value concepts (1958)

Carnap wrote a continuation of his reply to Kaplan (§32 of Carnap’s replies in the 1963 Schilpp volume), which would, however, have made that reply, already by far the longest in the book, too long. So he set aside his projected notes for a continuation to serve as the basis for a separate paper, which he never got around to writing. It is transcribed here from his shorthand and translated into English, with some introductory notes to provide a little context.