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1. A WORD ABOUT PRESUPPOSITIONS This book is addressed to philosophers, and not necessarily to those philosophers whose interests and competence are largely mathematical or logical in the formal sense. It deals for the most part with problems in the theory of partial judgment. These problems are naturally formulated in numerical and logical terms, and it is often not easy to formulate them precisely otherwise. Indeed, the involvement of arithmetical and logical concepts seems essential to the philosophies of mind and action at just the point where they become concerned with partial judgment and" belief. I have tried throughout to use no mathematics that is not quite elementary, for the most part no more than ordinary arithmetic and algebra. There is some rudimentary and philosophically important employment of limits, but no use is made of integrals or differentials. Mathematical induction is rarely and inessentially employed in the text, but is more frequent and important in the apP'endix on set theory and Boolean algebra. • As far as logic is concerned, the book assumes a fair acquaintance with predicate logic and its techniques. The concepts of compactness and maximal consistency turn out to have important employment, which I have tried to keep self-contained, so that extensive knowledge of meta logical topics is not assumed. In a word, the book presupposes no more logical facility than is customary among working philosophers and graduate students, though it may call for unaccustomed vigor in its application.
Though with considerable delay, most of the writings of Polish logicians of the inter-war period are now available in English. This is not yet true of Polish philosophy. In the present volume English-speaking readers will fmd, for the first time, a sizeable collection of the articles of one of the most original and distinguished of Poland's philosophers of the present century, Kazimierz Ajdukiewicz (1890-1963). To be sure, Ajdukiewicz was a philosopher-logician from the beginning of his career. His first work of some importance, a monograph entitled From the Methodology of the Deductive Sciences (1921 post-dated; two abstracts published in 1919/20) exhibited two features which were to become charac teristic of the style of his later philosophy: On the one hand the monograph was the result of Ajdukiewicz's deep interest in the systems of modern logic, the foundations of mathematics, in the properties of deductive systems and their relevance to philosophy; on the other hand the monograph was an attempt at developing an 'understanding methodology' (in the sense of Gennan 'Verstehende Methodologie') of deductive sciences, i. e. a pragmatic study of axiomatic systems which would supplement purely formal investiga tions of those systems. The fonner made him a close ally oflogical empiricists; the latter was rooted in the henneneutic tradition of the second half of the 19th century (Dilthey) which spilled over into the 20th century (Spranger) and which was not cherished at all by logical empiricists.
The first section of this chapter describes the major goals of this investiga tion and the general strategy of my presentation. The remaining three sections review some requisite background material and introduce some terminology and notation used in the book. Section B contains a brief review of some of the ideas and notation of elementary logic and set theory. Section C contains an introductory discussion of kinds and at tributes. Section D presents some basic ideas about laws and law sentences. A. GENERAL PLAN OF THE BOOK Basic scientific research is directed towards the goals of increasing our knowledge of the wor1d and our understanding of the wor1d. Knowledge increases through the discovery and confirmation of facts and laws. Understanding results from the explanation of known facts and laws, and through the formulation of general, systematic theories. Other things being equal, we tend to feeI that our understanding of a c1ass of phenomena increases as we develop increasingly general and intuitively unified theories of that c1ass of phenomena. It is therefore natural to consider the possibility of one very general, unified theory which, at least in principle, governs all known phenomena. The dream of obtaining such a theory, and the understanding that it would provide, has motivated an enormous amount of research by both scientists and philosophers.
This book is intended to be a survey of the most important results in mathematical logic for philosophers. It is a survey of results which have philosophical significance and it is intended to be accessible to philosophers. I have assumed the mathematical sophistication acquired· in an introductory logic course or in reading a basic logic text. In addition to proving the most philosophically significant results in mathematical logic, I have attempted to illustrate various methods of proof. For example, the completeness of quantification theory is proved both constructively and non-constructively and relative ad vantages of each type of proof are discussed. Similarly, constructive and non-constructive versions of Godel's first incompleteness theorem are given. I hope that the reader· will develop facility with the methods of proof and also be caused by reflect on their differences. I assume familiarity with quantification theory both in under standing the notations and in finding object language proofs. Strictly speaking the presentation is self-contained, but it would be very difficult for someone without background in the subject to follow the material from the beginning. This is necessary if the notes are to be accessible to readers who have had diverse backgrounds at a more elementary level. However, to make them accessible to readers with no background would require writing yet another introductory logic text. Numerous exercises have been included and many of these are integral parts of the proofs.
This monograph is designed to provide an introduction to the principal areas of tense logic. Many of the developments in this ever-growing field have been intentionally excluded to fulfill this aim. Length also dictated a choice between the alternative notations of A. N. Prior and Nicholas Rescher - two pioneers of the subject. I choose Prior's because of the syntactical parallels with the language it symbolizes and its close ties with other branches of logi cal theory, especially modal logic. The first chapter presents a wider view of the material than later chapters. Several lines of development are consequently not followed through the remainder of the book, most notably metric systems. Although it is import ant to recognize that the unadorned Prior-symbolism can be enriched in vari ous ways it is an advanced subject as to how to actually carry off these enrichments. Readers desiring more information are referred to the appropri ate literature. Specialists will notice that only the first of several quantifi cational versions of tense logic is proven complete in the final chapter. Again constraints of space are partly to blame. The proof for the 'star' systems is wildly complex and at the time of this writing is not yet ready for publi cation.
The present study which I have subtitled A Study in Law and Logic was prompted by the question of whether an investigation into law and legal systems could lead to the discovery of unrevealed fundamental patterns common to all such systems. This question was further stimulated by two interrelated problems. Firstly, could an inquiry be rooted in specifically legal matters, as distinct from the more usual writings on deontic logic? Secondly, could such inquiry yield a theory which would nevertheless embrace a strict and simple logical structure, permitting substantive conclusions in legal matters to be deduced from simple rules governing some basic concepts? Before the development of deontic logic, W. N. Hohfeld devoted his efforts to this question at the beginning of this century. However, with this exception, few jurists have studied the interrelation between law and logic projected in this way. Nevertheless, two great names are to be found, Gottfried Wilhelm Leibniz and Jeremy Bentham-both philo sophers with legal as weIl as logical training. Bentham's investigations of logical patterns in law have only recently attracted attention; and as for Leibniz, his achievements are still almost totally unexplored (his most important writings on law and logic have not even been translated from Latin). My initial interest in the question was evoked by Professor Stig Kanger. Although primarily a logician and philosopher, Stig Kanger has been interested also in the fundamentals of legal theory.
This anthology consists of a collection of papers on the nature of dis positions and the role of disposition concepts in scientific theories. I have tried to make the collection as representative as possible, except that problems specifically connected with dispositions in various special sciences are relatively little discussed. Most of these articles have been previously published. The papers by Mackie, Essler and Trapp, Fetzer (in Section 11), Levi, and Tuomela appear here for the first time, and are simultaneously published in Synthese 34, No. 4, which is a special issue on dispositions. Of the previously published material it should be emphasized that the papers by Hempel and Fisk have been extensively revised specially for this anthology. The papers are grouped in four sections, partlyon the basis of their content. However, due to the complexity of the issues involved, there is considerable overlap in content between the different sections, especially between Sections land 11. I wish to thank Professors James Fetzer and Carl G. Hempel for helpful advicc in compiling this anthology.
This book grew out of previously published papers of mine composed over a period of years; they have been reworked (sometimes beyond recognition) so as to form a reasonably coherent whole. Part One treats of informative inference. I argue (Chapter 2) that the traditional principle of induction in its clearest formulation (that laws are confirmed by their positive cases) is clearly false. Other formulations in terms of the 'uniformity of nature' or the 'resemblance of the future to the past' seem to me hopelessly unclear. From a Bayesian point of view, 'learning from experience' goes by conditionalization (Bayes' rule). The traditional stum bling block for Bayesians has been to fmd objective probability inputs to conditionalize upon. Subjective Bayesians allow any probability inputs that do not violate the usual axioms of probability. Many subjectivists grant that this liberality seems prodigal but own themselves unable to think of additional constraints that might plausibly be imposed. To be sure, if we could agree on the correct probabilistic representation of 'ignorance' (or absence of pertinent data), then all probabilities obtained by applying Bayes' rule to an 'informationless' prior would be objective. But familiar contra dictions, like the Bertrand paradox, are thought to vitiate all attempts to objectify 'ignorance'. BuUding on the earlier work of Sir Harold Jeffreys, E. T. Jaynes, and the more recent work ofG. E. P. Box and G. E. Tiao, I have elected to bite this bullet. In Chapter 3, I develop and defend an objectivist Bayesian approach.
This book presents a unified and systematic philosophical account of human actions and their explanation, and it does it in the spirit of scientific realism. In addition, various other related topics, such as psychological concept formation and the nature of mental events and states, are dis cussed. This is due to the fact that the key problems in the philosophy of psychology are interconnected to a high degree. This interwovenness has affected the discussion of these problems in that often the same topic is discussed in several contexts in the book. I hope the reader does not find this too frustrating. The theory of action developed in this book, especially in its latter half, is a causalist one. In a sense it can be regarded as an explication and refin~ment of a typical common sense view of actions and the mental episodes causally responsible for them. It has, of course, not been possible to discuss all the relevant philosophical problems in great detail, even if I have regarded it as necessary to give a brief treatment of relatively many problems. Rather, I have concentrated on some key issues and hope that future research will help to clarify the rest.
In the Introduction to the Polish-language version of the present book I expressed the hope that Polish studies in semiotics would before long be numerous enough to make possible another anthology on semiotics in Poland containing material published since 1970. That hope has in fact come true. The fact that semiotic research has been gaining momentum in this country is reflected in the growing interest in the discipline, in expanding international contacts, and in the steady increase in the number of publications. Thus, 1972 saw the setting up of the Department of Logical Semiotics, headed by the present writer, at Warsaw University Institute of Phi losophy. The seminar on semiotics, which I started in 1961, had met more than two hundred times by the end of 1976; since 1968, meetings have been held jointly with the Polish Semiotic Society. Another semi nar, confined to university staff and concerned with logical semiotics, which was inithted in 1970, had met more than fifty times by the end of 1976. The former seminar often plays host to foreign visiting pro fessors; so far scholars from Australia, Belgium, Britain, Canada, Czechoslovakia, France, the German Democratic Republic, Italy, the Netherlands, the Soviet Union, and the United States have attended.
This book is intended as a contribution to the foundations of the sciences of man, especially the social sciences. It has been argued with increasing frequency in recent years that the vocabulary of social science is to a large extent an action vocabulary and that any attempt to systematize concepts and establish bases for understanding in the field cannot, therefore, succeed unless it is firmly built on action theory. I think that these claims are sub stantially correct, but at the same time it seems to me that action theory, as it is relevant to social science, still awaits vital contributions from logic and philosophy. For example, it has often been said, rightly I believe, that situa tions in which two or more agents interact constitute the subject-matter of social science. But have we got an action theory which is rich enough or com prehensive enough to allow us to characterize the interaction situation? I think not. Once we have such a theory, however, we should be able to give an accurate account of central social phenomena and to articulate our concep tions about the nature of social reality. The conceptual scheme advanced in this book consists, in the first instance, of solutions to a number of characterization problems, i. e. problems which may be expressed by questions of the form "What is the nature of . . .
The Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference were held in JyvaskyIa, Finland, June 29-July 6, 1976. The Conferences were organized by a committee which consisted of the editors of the present volume. The Conferences were supported financially by the Ministry of Education of Finland, by the Academy of Finland, and by the Division of Logic, Methodology, and Philosophy of Science of the International Union of History of Science. The Philosophical Society of Finland and the Jyvaskyla Summer Festival gave valuable help in various practicalities. 35 papers by authors representing 10 countries were presented at the two meetings. Of those papers 24 appear here. THE EDITORS v TABLE OF CONTENTS PREFACE v PART 1/ PROOF THEORY GEORG KREISEL / Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory 3 DAG PRAWITZ / Proofs and the Meaning and Completeness of the Logical Constants 25 v. A. SMIRNOV / Theory of Quantification and tff-calculi 41 LARS SVENONIUS/Two Kinds of Extensions of Primitive Recursive Arithmetic 49 DIRK VAN DALEN and R. STATMAN / Equality in the Presence of Apartness 95 PART II / INFINITARY LANGUAGES VEIKKO RANTALA / Game-Theoretical Semantics and Back-and- Forth 119 MAARET KAR TTUNEN / Infinitary Languages N oo~.
3 in philosophy, and therefore in metaphilosophy, cannot be based on rules that avoid spending time on pseudo-problems. Of course, this implies that, if one succeeds in demonstrating convincingly the pseudo-character of a problem by giving its 'solution', the time spent on it need not be seen as wasted. We conclude this section with a brief statement of the criteria for concept explication as they have been formulated in several places by Carnap, Hempel and Stegmiiller. Hempel's account ([13J, Chapter 1) is still very adequate for a detailed introduction. The process of explication starts with the identification of one or more vague and, perhaps, ambiguous concepts, the so-called explicanda. Next, one tries to disentangle the ambiguities. This, however, need not be possible at once. Ultimately the explicanda are to be replaced (not necessarily one by one) by certain counterparts, the so-called explicata, which have to conform to four requirements. They have to be as precise as possible and as simple as possible. In addition, they have to be useful in the sense that they give rise to the formulation of theories and the solution of problems. The three requirements of preciseness, simplicity and usefulness. have of course to be pursued in all concept formation.
Jaakko Hintikka was born on January 12th, 1929. He received his doctorate from the University of Helsinki under the supervision of Professor G. H. von Wright at the age of 24 in 1953. Hintikka was appointed Professor of philosophy at the University of Helsinki in 1959. Since the late 50s, he has shared his time between Finland and the U.S.A. He was appointed Professor of philosophy at Stanford University in 1964. As from 1970 Hintikka has been permanent research professor of the Academy of Finland. He has published 13 books and about 200 articles, not to mention the various editorial and organizational activities he has played an active role in. The present collection of essays has been edited to honour Jaakko Hintikka on the occasion of his fiftieth birthday. By dedicating a Festschrift to Jaakko Hintikka, the contributors wish to pay homage to this remarkable man whom they see not only as a scholar of prodigious energy and insight, but as a friend, colleague and former teacher. The contributors hope the essays collected here will bring pleasure to the man they are intended to honour. All of the essays touch upon topics Hintikka has taken an direct or indirect interest in, ranging from technical problems of mathematical logic and applications of formal methods through philosophical logic, philosophy of language, epistemology and history of philosophy to philosophical aesthetics.
In 1936, G. Birkhoff and J. v. Neumann published an article with the title The logic of quantum mechanics'. In this paper, the authors demonstrated that in quantum mechanics the most simple observables which correspond to yes-no propositions about a quantum physical system constitute an algebraic structure, the most important proper ties of which are given by an orthocomplemented and quasimodular lattice Lq. Furthermore, this lattice of quantum mechanical proposi tions has, from a formal point of view, many similarities with a Boolean lattice L8 which is known to be the lattice of classical propositional logic. Therefore, one could conjecture that due to the algebraic structure of quantum mechanical observables a logical calculus Q of quantum mechanical propositions is established, which is slightly different from the calculus L of classical propositional logic but which is applicable to all quantum mechanical propositions (C. F. v. Weizsacker, 1955). This calculus has sometimes been called 'quan tum logic'. However, the statement that propositions about quantum physical systems are governed by the laws of quantum logic, which differ from ordinary classical logic and which are based on the empirically well-established quantum theory, is exposed to two serious objec tions: (a) Logic is a theory which deals with those relationships between various propositions that are valid independent of the content of the respective propositions. Thus, the validity of logical relationships is not restricted to a special type of proposition, e. g. to propositions about classical physical systems.
The subject of the following study is theories of memory. The first part is a study of one broad type of theory which is very widely adhered to at this time. It enjoys great popularity among neuro physiologists, neuropsychologists, and, more generally, among scientifically oriented people who have directed their attention to questions about memory. Further, this way of looking at the matter is not confined to scientific professionals. Indeed, we can find popularized versions of the view in magazines like Time and Reader's Digest. So in the first part of the book, I will give a presentation of the view in its general form. The theory will be presented in such a way as to reveal the features which make it tempting, which make it seem to be a very natural way to explain the phenomena of memory. (And, clearly, from the number of adherents the view has won, it is tempting, and it does seem to be to go about explaining memory. ) After setting forth a natural way this generalized version of the theory, I will next present material by various authors who hold this view. This will allow the reader to get some idea of the different forms which the theory (the 'memory trace' or 'engram' theory) takes. The last step is a critic ism of the theory. In the second part of the book, the attack on trace theory will be strengthened by a further criticism.
The goal of the present volume is to discuss the notion of a 'conceptual framework' or 'conceptual scheme', which has been dominating much work in the analysis and justification of knowledge in recent years. More specifi cally, this volume is designed to clarify the contrast between two competing approaches in the area of problems indicated by this notion: On the one hand, we have the conviction, underlying much present-day work in the philosophy of science, that the best we can hope for in the justifi cation of empirical knowledge is to reconstruct the conceptual means actually employed by science, and to develop suitable models for analyzing conceptual change involved in the progress of science. This view involves the assumption that we should stop taking foundational questions of epistemology seriously and discard once and for all the quest for uncontrovertible truth. The result ing program of justifying epistemic claims by subsequently describing patterns of inferentially connected concepts as they are at work in actual science is closely connected with the idea of naturalizing epistemology, with concep tual relativism, and with a pragmatic interpretation of knowledge. On the other hand, recent epistemology tends to claim that no subsequent reconstruction of actually employed conceptual frameworks is sufficient for providing epistemic justification for our beliefs about the world. This second claim tries to resist the naturalistic and pragmatic approach to epistemology and insists on taking the epistemological sceptic seriously.
11 original. Modifications which I introduced are radical and often far going. In my opinion the Polish text had two main drawbacks. It was overloaded with informal considerations and at the same time formal concepts included in some parts of the book were presented in a too complicated way. Of course one of the motives to revise it was also the fact that much time has passed since I finished writing the Polish version and obviously certain decisions and ideas contained in the first edition seem not quite relevant now. So it is not only the desire to make the exposition clearer but also the reasons of substantial nature which motivated writing a revised version. I do not think it desirable to bother the reader with a detailed discussion of all changes to which the Polish version was subjected and that is why I will confine myself to pointing out only the most significant ones. Explanations concerning logical and set-theoretical notions applied in the book have been shortened as much as possible, in the Polish version one whole chapter was devoted to the discussion of them.
This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under mine certain strong arguments against these positions precisely by invok ing G6del's and related work. I also began to realize that the Church Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor ganizing it in the direction of the present book.