SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIBNCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE,

PRAGMATIC LOGIC

SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIBNCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIORAL SCIENCES Editors: DONALD DA VIDSON, The Rockefeller University and Princeton University JAAKKO HINTIKKA, Academy of Finland and Stanford University GABRIEL NUCHELMANS, University of Leyden WESLEY C. SALMON, University of Arizona VOLUME 62

KAZIMIERZ AJDUKIEWICZ PRAGMATIC LOGIC Translated from the Polish by OLGIERD WOJTASIEWICZ D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND IBOSTON - U.S.A. PWN - POLISH SCIENTIFIC PUBLISHERS WARSAW-POLAND

Translated from the original Polish Logika Pragmatyczna, Warszawa 1965 Library of Congress Catalog Card Number 72-95887 ISBN-13: 978-94-010-2111-1 eisbn-13: 978-94-010-2109-8 DOl: 10.1007/978-94-010-2109-8 Distributors for the U.S.A., Canada and Mexico D. REIDEL PUBLISHING COMPANY, INC. 306 Dartmouth Street, Boston, Mass. 02116, U.S.A. Distributors for Albania, Bulgaria, Chinese People's Republic, Czechoslovakia, Cuba, German Democratic Republic, Hungary, Korean People's Democratic Republic, Mongolia, Poland, Rumania, Democratic Republic of Vietnam, the U.S.S.R. and Yugoslavia ARS POLONA - RUCH Krakowskie PrzedrnieScie, 7, 00-068 Warszawa 1, Poland Distributors for all remaining countries D. REIDEL PUBLISHING COMPANY P.O. Box 17, Dordrecht, Holland Copyright by PWN - Polish Scientific Publishers - Warszawa 1974 Softcover reprint ofthe hardcover lst edition 1974 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher

FOREWORD When asked in 1962 on what he was working Kazimierz Ajdukiewicz replied: Several years ago Polish Scientific Publishers suggested that I prepare a new edition of The Logical Foundations of Teaching, which I wrote before 1939 as a contribution to The Encyclopaedia of Education. 1 It was a small booklet covering elementary information about logical semantics and scientific methodology, information which in my opinion was necessary as a foundation of teaching and as an element of the education of any teacher. When I recently set to preparing the new edition, I rewrote practically everything, and a booklet of some 100 pages swelled into a bulky volume almost five times bigger. The issues have remained practically the same, but they are now analysed much more thoroughly and the threshold of difficulty is much higher now. The main stress has been laid on the methods used in the empirical sciences, and within that field, on the theory of measurement and the methods of statistical inference. I am now working on the last chapter of the book, concerned with explanation procedures and theory construction in the empirical sciences. When that book, which I intend to entitle Pragmatic Logic, is completed I intend to prepare for the press Vol. 2 of my minor writings, Language and Cognition, 2 which will cover some of my post-war papers. I will also co-operate in preparing a similar publication which is to appear in English in the United States. I also plan parallelly to continue my studies in logical semantics and scientific methodology. First of all, I would like to finish my outline of my papers Proposition as the Connotation of Sentence and Intensional Expressions. 3 I Both in Polish. 2 When Kazimierz Ajdukiewicz died in 1963, Polish Scientific Publishers continued work on Vol. 2 of LAnguage and Cognition (in Polish). Vol. 2, prepared for the press by Prof. Klemens Szaniawski and Dr Halina Mortimer, appeared in 1965. (Ed.) 3 Published posthumously in Studio Logica, Vol. XX, 1967 (in English). v

FOREWORD This statement explains the origin of Pragmatic Logic and its destination: it was to be a deeper-reaching and modernized version of what had been said in The Logical Foundations of Teaching. To carry out that task its author had to transform a contribution to The Encyclopaedia of Education into an advanced university handbook. Kazimierz Ajdukiewicz used to point out on various occasions that such a handbook was needed. He emphasized that while in the Polish literature of the subject there were monographs on formal logic (with two modern and excellent items recently added to that list4 ) there was no book on the basic concepts of semantics and scientific methodology that would satisfy present-day requirements. That gap would best be filled by a comprehensive book by many authors, which he had in mind. Pragmatic Logic was to be a provisional solution. The title of the book pointed to close links with the practice of scientific research and reflected the author's opinion that the methodologist's task is above all to codify those procedures which are in fact used in research and to substantiate them with respect to the goals of science. This view of the tasks of scientific methodology explains the structure of the book. A significant illustration is offered by the chapters on probabilistic laws and on rules of statistical inference. These problems, merely indicated in The Logical Foundations of Teaching, have in Pragmatic Logic been expanded to the extent unprecedented in Polish works on scientific methodology. The decision to treat probabilistic methods so extensively and so penetratingly was certainly dictated by the comprehension of their present-day importance in the empirical sciences. It must also be said in this connection that Kazimierz Ajdukiewicz's opinion on the importance of statistical inference had been formed even at the time when the general theory of statistical inference was still little developed. Works in that field by Polish logicians were largely inspired by him. Pragmatic Logic appears as a text which was intended by its author to serve as a handbook for the students of philosophy. But it is also a book which, like all that which was written by Kazimierz Ajdukiewicz, bears the mark of his intellectual personality. It is not my task A. Grzegorczyk, An Outline of Mathematical Logic (an English-language version in preparation); I. Slupecki & L. Borkowski, Elements of Mathematical Logic and Set Theory, Oxford, 1967. VI

FOREWORD here to list all those parts of the book which are original contributions to science (some of which are being here published for the first time). I will mention only, by way of example, some characteristic issues: the classification of types of inference; the concept of the conclusive nature of inference; the theory of measurement; the concept of a law of science. It is legitimate, I think, to apply to Pragmatic Logic that formulation which Kazimierz Ajdukiewicz himself used when writing about Tadeusz Kotarbinski's Gnosiology: "(... ) textbooks usually are compilations, expositions of things already accepted, and as a rule do not contribute much new to science. In the present case, although the work is called a textbook, it is not merely that". 5 Finally, I have to state that which is the most difficult thing to write: Pragmatic Logic is an unfinished work. The author's death in 1963 prevented him from completing the last chapter of the book and from giving final formulation to those already written, especially those which were written last and which Kazimierz Ajdukiewicz was unable to polish up. It may not be said about any part of the present book that it has been given its final form. This is obvious to all those who had known Kazimierz Ajdukiewicz in his work as a man who would always strive for better solutions and would never rest satisfied with what he has achieved. Klemens Szaniawski, cr. T. Kotarbinski, Gnosio!ogy, Oxford, 1966, p. SIS. vii

CONTENTS Foreword v Introduction 1. Logic as a Foundation of Teaching PART I WORDS, THOUGHTS AND OBJECfS Chapter I Expressions and Their Meanings 2. Understanding of Expressions. 3. Meaning of Expressions... 4. Language and Meaning 5. Speech as a Way of Communicating Thoughts 7 8 12 13 Chapter II Statements and Their Parts 6. Proposition and Sentence........ 16 7. Parts of Statements. Syntactical Categories 18 8. Complex Statements.......... 22 9. Simple Statements........... 25 10. Statement Schemata and Statements Derived from Them. 27 Chapter III Objective Counterparts of Expressions A. Extension of Terms II. Designating and Designata 31 31 ix

CONTENTS 12. Denotation and Extension 13. Relations between Extensions 14. Unions and Intersections of Sets 15. Logical Partition B. Intension of Terms 16. Complete and Characteristic Intension 17. Linguistic Intension 32 34 39 40 43 43 45 Chapter IV Ambiguity of Expressions and Defects of Meanings 18. Ambiguity....... 19. Vagueness....... 20. Incomplete Formulations. 48 52 56 Chapter V Definitions 21. Two Ways of Understanding the Term "Definition" 57 22. The Concept of Nominal Definition... 58 23. Definitions by Abstraction and Inductive Definitions. 62 24. Errors in Defining................ 68 25. Stipulating and Reporting Definitions....... 70 26. Definitions by Postulates and Pseudo-definitions by Postulates............ 77 27. The Concept of Real Definition 81 Chapter VI Questions and Interrogative Sentences 28. The Structure of Interrogative Sentences 29. Decision Questions and Complementation Questions. 30. Assumptions of Questions. Suggestive Questions 31. Improper Answers 85 87 88 89 x

CONTENTS 32. Thoughts Expressed by an Interrogative Sentence 33. Didactic Questions 91 93 PART II INFERENCE Chapter I Formal Logic and the Consequence Relation 34. Formal Logic........... 35. Logical Consequence 36. The Relationship between the Truth of the Reason and the Truth of the Consequence. 37. Enthymematic Consequence 97 98 101 104 Chapter II Inference and the Conditions of Its Correctness 38. The Concept of Inference...... 39. Conditions of Correctness of Inference 106 107 Chapter III Subjectively Certain Inference 40. The Conclusiveness of Subjectively Certain Inference. 109 41. The Conclusiveness of Subjectively Certain Inference in the Light of the Knowledge of the Person Involved 115 42. Deductive Inference 117 43. Deducing..... 119 Chapter IV Subjectively Uncertain Inference 44. The Conclusiveness of Subjectively Uncertain Inference 120 45. Logical Probability versus Mathematical Probability 122 46. Statistical Probability 123 47. Reductive Inference.............. 130 xi

CONTENTS 48. Induction by Enumeration 49. Inference by Analogy.. 50. Induction by Elimination. 138 156 160 PARTlII METHODOLOGICAL TYPES OF SCIENCES Chapter I The Division of Sciences into Deductive and Inductive 51. The Methodology of Sciences.. 52. Deductive and Inductive Sciences 185 190 Chapter II Deductive Sciences 53. Deductive Sciences at the Pre-axiomatic Intuitive Stage. 194 54. Deductive Sciences at the Axiomatic Intuitive Stage.. 195 55. The Philosophical Controversy over the Substantiation of Primitive Theorems in Intuitively Approached Deductive Sciences................... 198 56. Deductive Sciences at the Abstract Axiomatic Stage 202 57. Formalized Deductive Theories 207 (a) Rules of Defining 207 (b) Rules of Inference.... 211 (c) Construction of Formalized Theories 218 58. Deductive Theories from the Apragmatic Point of View 222 (a) Consistency of Theories...... 222 (b) Independence of Axioms..... 228 (c) Deductive Completeness of Theories 231 (d) Completeness of Deductive Systems 234 Chapter III The Inductive Sciences A. The Empirical Foundations 238 xii

CONTENTS 59. Irrevocable Assumptions and Theorems in the Inductive Sciences 238 60. Protocol Statements.............. 241 61. The Method of Direct Experience as Subjective and Unrepeatable 246 62. Observation and Experiment 249 B. Counting and Measurement 254 63. Selected Concepts in the Theory of Relations. Numbers and Counting.... (a) Preliminary Remarks (b) The Concept of Relation (c) Properties of Equivalence Relations. Types of Equi- 254 254 255 valence Properties... 260 (d) Many-one, One-many Relations. The Concept of Equipotence of Sets.. 263 (e) Numbers 265 (f) Counting 64. Selected Concepts in the Relations Theory (ctd.). Ordering Relations. Isomorphism and Homomorphism of Rela- tions........ 271 (a) Preliminary Remarks... 271 (b) Ordering Relations.... 271 269 (c) Isomorphism of Relations. 274 (d) Homomorphism of Relations 276 (e) Abstraction Relations... 278 65. Magnitudes and Scaling.... 280 (a) Primary and Secondary Properties of Abstraction 280 (b) Ordered Families of Abstraction Properties. Magnitudes.... 66. Additive Magnitudes........ xiii 281 286

CONTENTS 67. Examples of Definitions of a Physical Sum of Magnitudes... 68. The Measurement Function.............. 69. Measurement Proper.... 70. Measurement without a Unit of Measurement and Measurement without a Zero Point 294 297 303 310 Chapter IV Inductive Sciences and Scientific Laws A. General Laws 71. General Laws which State Relationships between Constant and Variable Properties 72. General Laws which State Relationships between Variable Properties. Functional Laws.. B. Statistical Laws Laws and Parametric 73. Statistical Laws which State the Degree of Association of Constant Properties.... 74. Laws of Statistical Distributions.... (a) Laws of Distribution of Probabilities of Discrete Variables (b) Laws of Probability Density Distribution for Continuous Variables (c) Binomial Distribution (d) Normal Distribution 75. Laws on Correlation of Variable Properties 316 316 318 324 324 331 331 334 341 345 355 Chapter V Statistical Reasoning 76. Introductory Remarks 377 xiv

CONTENTS 77. Estimation of Parameters 78. Levelling of Errors of Measurement as Example of Estimation of Parameters............. 79. Verification of Hypotheses and Statistical Tests Supplement: Proving and Explaining Subject Index 379 395 411 440 451 xv

INTRODUCTION I. WGIC AS A FOUNDATION OF TEACHING The task of the school is not only to convey to the pupils information in various fields, but also to develop in them the ability of correctly carrying out cognitive operations. These two tasks, which in practice are being carried out parallelly, have been termed, respectively, the material and the formal goal of teaching. But to be able to train his pupils in a correct performance of cognitive operations the teacher must himself provide the standard of correct thinking. This, however, does not suffice: the pupils must carry out the operations themselves, and the teacher must see to it that they do that properly. Hence, if a pupil, when proving a theorem, or explaining a phenomenon, or defining a concept, commits an error, the teacher must draw his attention to the fact. But it does not suffice if he tells the pupil that he committed an error; he also ought to point out where the error is and in what it consist. The practical ability to think correctly, which every teacher should have, will alone suffice for him to notice that the pupil made an error; it will probably also suffice for him to explain to the pupil where the error is. But it need not necessarily suffice for him to be able to tell in what that error consists and what its nature is. To be able to do so he must know those concepts and terms which make it possible to discuss cognitive operations and their types, properties, etc. He also must have a theoretical knowledge of the conditions which the various cognitive operations must satisfy in order to be correct. The concepts and terms concerned with cognitive operations, their types, properties, elements, etc., have been worked out by that branch of logic which is termed methodology. (Hence these terms and concepts are called methodological.) The same branch cf logic provides information about the conditions of correct (i.e., purpose-serving) performance of the various cognitive operations. By becoming familiar with logical methodology the teacher assimilates those concepts and terms which 1

INTRODUCTION are required when discussing the pupil's mental work; he also thus acquires the knowledge of the theorems which explicitly lay down those conditions which are necessary and sufficient for correct performance of various cognitive operations. Thus by familiarizing himself with elements of logical methodology the teacher has at his disposal the concepts, terms, and theorems which he will need to carry out his duties properly. But his knowledge of the said terms and concepts will help him not only to attach right labels to the errors committed by his pupils: he will find it of assistance whenever, for any reasons, he wants to discuss science and research. Suppose that the physics teacher, in order to inform his pupils about the law of gravitation and its substantiation, decides to tell them how Newton arrived at the fonnulation of that law. When doing so he will perhaps begin by telling the pupils that the said law was born in Newton's mind as a hypothesis, from which he succeeded to deduce the law which states how the Moon revolves round the Earth and how the planets revolve round the Sun, the law which agrees with observations within the margin of error. That agreement between the consequences of the said hypothesis with empirical data is its confirmation, which Newton thought to be sufficient to accept that hypothesis as a general law. If he teaches them in this way, he not only informs his pupils about the law of gravitation, but also explains to them in what that law consists. But to do so he must use such terms as "hypothesis", "deduction", "verification of hypotheses", etc., and all these terms are methodological ones. It is true that they are in current usage and hence are known to every teacher, including those who have never studied logic, but it is also true that the concepts which in the minds of such teachers stand behind those terms lack in clarity and precision. The study of logic will impart those tenns clear and unique meanings. It is only when the teacher comes to use such methodological terms properly, when he does not confuse, for instance, a theorem verified with a theorem proved, and deduction with just any kind of inference, that his methodological analyses and his appraisal of the proof value of a given method of substantiating a theorem will be valid. The logical foundations of teaching are that amount of the knowledge of logic which a teacher should have to be properly prepared to 2

INTRODUCTION teach well. What has been said above seems to justify the claim that the knowledge of terminology and precise methodological concepts, and also the knowledge of elementary methodological theorems which lay down the conditions of correctness of the principal types of cognitive operations must be included in the logical foundations of teaching. But the logical foundations of teaching are not confined to the knowledge of the principal methodological concepts and theorems. They also cover the knowledge of the principal concepts and theorems of another branch of logic, known as the logic of language or logical semiotic. The need to have a knowledge of that field is linked with a specific task with which secondary school teachers are faced. They ought to train their pupils in (I) formulating statements which have a tangible meaning, and are not just empty generalities that state nothing definite, (2) formulating statements in an unambiguous way, and (3) formulating statements so that they state what they are intended to state. In other words, the pupils should be trained to make statements that are matter-of-fact, unambiguous, and precise. The knowledge of formulating one's statements so is indispensable not only at school, but in everyday life as well. Nonobservance of these three requiremens may be tolerated in those cases where speech serves to express emotions or to arouse them, e.g., in poetry and in unscrupulous agitation, but never in those cases where cognition and/or rational (i.e., cognition-based) action are at stake. Hence it is evident that developing in the pupils the ability and the urge to make statements which are matterof-fact, unambiguous and precise is one of the principal tasks of school education. The logic of language, otherwise termed logical semiotic, analyses the functions of speech, and in particular is concerned with the mapping of facts in speech. In this way it prepares the set of concepts and the terminology which are indispensable for informing about all kinds of infringements of the principle that statements must be matter-of-fact, unambiguous and precise formulations; it further systematizes such infringements, and indicates the ways of preventing them. This is why it seems that the theory of teaching, at least in that section which is concerned with training the pupils in semantically correct speech, must be based on logical semiotic. But not only those who are working on the theory of teaching will find it useful to become familiar with that 3

INTRODUCTION branch of logic. The same applies to those who actually do the teaching. The study of logical semiotic will make them sensitive to infringements of semantic correctness in speech, i.e., will help them notice such infringements, but also to recognize their nature; it will also show them how to use language in a way that would be free from such shortcomings. It seems, accordingly, that the logical foundations of teaching, which cover those sections of logic the knowledge of which helps the teacher to teach better, should include, next to elements of logical methodology, elements of logical semiotic as well. The main core of elementary logic, i.e., logic in the narrower sense of the term as the discipline which lists and systematizes all the schemata of deductive inference (and the underlying logical tautologies). seems to be less important. for the teacher. This is so because in everyday thinking he encounters only those cases of inference which follow very simple schemata of deduction, and the wealth of other schemata, listed in formal logic, finds application but rarely. Hence it does not seem worth while to burden the teacher's memory with them. On the other hand, it is worth while to draw the teacher's attention, first, to the variety of meanings which in everyday speech many terms have, including those terms which occur in the simplest schemata of deductive inference, i.e., what are called logical constants, such as "or", "every", "some", etc., secondly, to the most frequent errors in deduction. What has been said above provides sufficient suggestions for the composition of the book to be entitled Pragmatic Logic. It will have three parts, to deal, respectively, with the logic of language, with most elementary information about formal logic, and with the principal elements of logical methodology. They will be discussed in that order, since in such an arrangement each part that comes later requires the knowledge of what has been said in earlier part or parts, but not conversely. 4

PART I Words, Thoughts and Objects

CHAPTER I-EXPRESSIONS AND THEIR MEANINGS 2. UNDERSTANDING OF EXPRESSIONS The rustle of leaves, the singing of birds, the noise of a passing motorcar we hear. The expressions of a language of which we have command we not only hear, but also understand. It is not easy to explain in what the understanding of an expression consists. The same kind of response to an expression heard is not always called the understanding of that expression. It is very often said that a person understood a given word when the hearing of that word intertwined in his mind with a thought about an object other than the word in question. For instance, a person who knows Latin thinks about the Earth on hearing the word "terra"; he thinks that the Earth is round on hearing the statement "terra est rotunda". But it is not always required that the hearing of a word should in a person's mind intertwine with a thought about an object other than the word in question when it is said that that person understood that word. It will be said, for instance, that we understand the word "whether", as it occurs, e.g., in "I do not know whether he will be here", even though on hearing that word we do not direct our thoughts toward any object other than the word in question. We would also, perhaps, say that a soldier understood an order if he did what he was told to, even if the order was formulated in a language which he does not understand in the first of the meanings mentioned above. (In this case we say that a person understands an order in the same sense as we might say that a dog understands those orders to which he has been trained to respond.) As can be seen from these explanations, the word "understand" is used in different senses. Without going here into any detailed analysis of these various meanings of the word "understand" we shall bear in mind, in the discussion that follows, the first meaning of that word, namely that a person understands an expression if on hearing it he directs his thoughts to an object other than the word in question. In 7