Formal Logic
"Logic", one of the central words in Western intellectual history, compre hends in its meaning such diverse things as the Aristotelian syllogistic, the scholastic art of disputation, the transcendental logic of the Kantian critique, the dialectical logic
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SYNTHESE LIBRARY A SERIES OF MONOGRAPHS ON THE RECENT DEVELOPMENT OF SYMBOLIC LOGIC, SIGNIFICS, SOCIOLOGY OF LANGUAGE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, STATISTICS OF LANGUAGE AND RELATED FIELDS
Editors: B. H. KAZEMIER / D. VUYSIE
PAUL LORENZEN
FORMAL LOGIC TRANSLATED FROM THE GERMAN BY FREDERICK J. CROSSON
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
ISBN 978-90-481-8330-2 ISBN 978-94-017-1582-9 (eBook) DOI 10.1007/978-94-017-1582-9 1965
All rights reserved No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means without permission from the publisher
CONTENTS
Introduction
VII
1 1 6
CHAPTER I
Syllogistic 1 Fundamental Linguistic Concepts 2 The Syllogistic Moods
CHAPTER II
Classical Logic of Junctors 3 Conjunction and Negation 4 Adjunction 5 The System of Junctors
19 19 25 30
CHAPTER III
The Calculi of the Logic of Junctors 6 Calculization 7 Consistency and Completeness
40 40 44
CHAPTER IV
Effective Logic of Junctors 8 Affirmative Logic 9 Negation
48 48 61
CHAPTER
v Logic of Quantors 10 Some-Quantor and All-Quantor 11 Completeness and Undecidability
CHAPTER VI
73 73 88
Logic of Identity 12 Descriptions 13 Abstraction, Relations and Functions 14 Identity Calculus
99 99 105 111
Bibliography Table of Logical Signs Index of Names Index of Subjects
117 119 120 121 V
INTRODUCTION
"Logic", one of the central words in Western intellectual history, comprehends in its meaning such diverse things as the Aristotelian syllogistic, the scholastic art of disputation, the transcendental logic of the Kantian critique, the dialectical logic of Hegel, and the mathematical logic of the Principia Mathematica of Whitehead and Russell. The term "Formal Logic", following Kant is generally used to distinguish formal logical reasonings, precisely as formal, from the remaining universal truths based on reason. (Cf. SCHOLZ, 1931). A text-book example of a formal-logical inference which from "Some men are philosophers" and "All philosophers are wise" concludes that "Some men are wise" is called formal, because the validity of this inference depends only on the form ofthe given sentences - in particular it does not depend on the truth or falsity of these sentences. (On the dependence of logic on natural language, English, for example, compare Section 1 and 8). The form of a sentence like "Some men are philosophers", is that which remains preserved when the given predicates, here "men" and "philosophers" are replaced by arbitrary ones. The form itself can thus be represented by replacing the given predicates by variables. Variables are signs devoid of meaning, which may serve merely to indicate the place where meaningful constants (here the predicates) are to be inserted. As variables we shall use - as did Aristotle - letters, say P, Q and R, as variables for predicates. Our text-book example then derives from the forms "Some Pare Q" and "All Q are R" the form "Some Pare R". The inference from "if it rains or snows, then he does not come" and "it rains", to "he does
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