Logic from a rhetorical point of view 9783110136838, 311013683X, 9783110886627, 3110886626

Citation preview

Grundlagen der Kommunikation und Kognition Foundations of Communication and Cognition Herausgeber/Editors Roland Posner, Georg Meggle

Witold Marciszewski

Logic from a Rhetorical Point of View

w DE

G_ Walter de Gruyter · Berlin · New York 1994

© Gedruckt auf säurefreiem Papier, das die US-ANSI-Norm über Haltbarkeit erfüllt. Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability

Die Deutsche Bibliothek — CI Ρ-Einheitsaufnahme Marciszewski, Witold: Logic from a rhetorical point of view / Witold Marciszewski. — Berlin ; New York : de Gruyter, 1994 (Grundlagen der Kommunikation und Kognition) ISBN 3-11-013683-X

© Copyright 1993 by Walter de Gruyter & Co., D-10785 Berlin Dieses Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Printed in Germany Druck: Werner Hildebrand, Berlin Buchbinderische Verarbeitung: Lüderitz & Bauer-GmbH, Berlin

To Haiina

Preface The plot of this essay is based upon the counterpoint of two ideas which could be put forward as mottos at the front of this book. According to St. Augustine and Descartes, both following Plato, the genuine t r u t h lives inside each of us, in interiore homine habitat ipsa Veritas, hence it is man's mind in which the t r u t h is to be intuitively searched for. According to Leibniz, t h a t pursuit is to be aided by a u t o m a t a able to symbolically depict t r u t h s and produce proofs as printouts, ut Veritas quasi picta, machine ope impresso, deprehendatur, hence assistance is to be found in the physical world of machines. Although seeming to oppose each other, these ideas tended to be synthesized by Leibniz. In his vision, the human mind-and-body is like a machine whose power and dignity consists in its being infinitely complex, in the literal mathematical sense of infinity, all t h a t complexity being wonderfully commanded by a centre. T h a t infinity and that command is what distinguishes the machines created by Nature or God from human-made machines, yet what they have in common makes it possible to model each kind on the other and to interact with one another. What has all that to do with rhetoric? Let me resort to the thought-provoking title of the editorial series in which this book appears, namely Foundations of Communication and Cognition. Previously this series' name was limited to its present first member; its extension, one may guess, is due to the realization t h a t any theory of communication has to be rooted in a theory of cognition. T h a t corresponds to what is recently presupposed by cognitive science, namely t h a t any interaction between an organism or a like system (say, a speaker) and its environment (say, his audience) is influenced by this system's cognitive map through which use responses are constructed to achieve intended outcomes. The intended outcome dealt with by rhetoric is the change of a certain

Vlll

Preface

cognitive state of an addressee effected by a cognitive state of an addresser with the use of a spoken or written text. T h i s definition is enough to show the input to cognitive science to be expected from rhetoric. T h o u g h the term 'cognitive science' is rather new, the problems raised in this theory go back to the beginnings of modern science in the seventeenth century when they appeared in the Cartesian and the Leibnizian theories of mind and matter (hence the comprehensive handling of these events in Chapter Three). Rhetoric in the version designed in this essay as cognitive rhetoric is t h a t theory of communicative interaction whose core involves the issues of rational argument; hence its foundations are to be seen in logic as the theory of the validity of arguments. Once upon a time there had been a strong link between logic and rhetoric but that link was almost broken because of the disappearance of rhetoric (for reasons discussed in C h a p t e r One) just at that time in which logic experienced a new phase of flourishing. However, when the end of this century witnesses the rapid development of communication studies, pervading semiotics, linguistics, psychology, computer science, etc., the old respectable name of rhetoric should be revived as the convenient term to denote a significant part of those studies. One more question should be settled before we form an alliance between that new rhetoric, as part of communication studies, and t h a t new logic which succeeds in providing mathematics with due foundations. C a n the same logic provide foundations for rhetoric? T h e answer is found in the relation between mental acts and what they produce. Their products include mathematical statements, proofs and theories, and these constitute the domain of abstract objects investigated by mathematical logic. However, the knowledge of products is very close to the knowledge of the methods of production. Therefore, while respecting the difference in ontological categories of the physical, the mental and the abstract, we should advance interactions among respective disciplines. To preserve the postulated categorial purity, one should distinguish between what I here call mind-philosophical logic (see Chapter T w o ) and general logic with its mathematical ramifications.

Preface

ix

Yet, a theory dealing with those mental acts which produce arguments and theories is important enough that it should take as much advantage as necessary of general logic dealing with those products. This is what the present essay tries to do in Chapters Four, Five and Six, presenting the theory of names (as related to the calculus of classes), truth-functional logic, and predicate logic, respectively. This exposition of logic is very elementary, indeed, so elementary that even those readers who are not familiar with theories of logic will by reading it be able to acquire the fundamental concepts referred to in the discussion of rhetorical and mind-philosophical ideas. Such a mode of exposition might prove boring to readers expert in theoretical logic. I hope, though, that my linguistic, historical and philosophical comments as well as rhetorical applications, all melted into the discussion of basic logical concepts, should appear not quite uninspiring even to those expert readers. The rest of the book, comprising the Chapters Seven to Ten, hinges on the assumption that there are two main manifestations of intelligence, namely reasoning and conceptualizing (conceptformation and -intepretation), and that the extent to which they can be simulated by artificial intelligence is different for each of them; reasoning processes can be formalized and automatized more easily and more efficiently than those of conceptualization. Moreover, it is claimed that conceptualization is more fundamental as it is presupposed by generalization in the sense which appears in the rules for the universal quantifier. Generalization is discussed with special attention in the Chapters mentioned above, because the serious error mostly committed by reasoners is that of hasty, premature generalizations which result in false universal statements. The problem of general ideas is approached at various angles. There is an extensive historical discussion of Kant's and his predecessors' views on mathematical general objects; there is also a detailed investigation, assisted by applications of a computerized proof-checker, on how generalizations may occur in a problem-solving process carried out by a chimpanzee. The emphasis put on the issues of generalization, as crucial for rhetoric-oriented mind-philosophical logic, belongs to the main features of this essay; it is handled in Chapters Seven

χ

Preface

a n d Nine ( t h e latter s u p p o r t i n g the former by s o m e anecdotical case studies). A n o t h e r characteristic issue, a p p e a r i n g in C h a p t e r s Eight and Ten ( a g a i n , case studies in the latter), involves the insistence on the role of implicit definitions being presented a g a i n s t the background of the o r t h o d o x logical theory of normal definitions (i.e. those in the form of equivalence or equality, meeting the eliminability criterion). T h e model of implicit defining, s u p p o s e d to be the main m e t h o d of instinctive conceptualization, is found in ostensive procedures a n d , on the other side, in axiom s y s t e m s of deductive theories. T h e whole discussion of reasoning is p e r m e a t e d by the d i l e m m a expressed in the m a x i m s quoted at the s t a r t . Should logic control the f o r m a t i o n of intuited o b j e c t s (conceptualization) a n d their truth-preserving t r a n s f o r m a t i o n ( o b j e c t u a l reasoning), or should it rather handle inference as a mechanical processing of tokens (symbolic r e a s o n i n g ) ? T h e former option is C a r t e s i a n (also K a n t i a n , intuitionistic, etc.) while the latter is Leibnizian (now adhered t o by the A I theorists). T h e vocation of mind-philosophical logic consists in e x a m i n i n g b o t h options a n d using each of them for a better u n d e r s t a n d i n g of the other. From a rhetorical point of view, logic is t o focus on o b j e c t u a l reasoning as being usual in our arguments; its theoretical u n d e r s t a n d i n g should increase efficiency of h u m a n communication. However, against the background of symbolic formalized reasoning, a s studied by symbolic logic, we win a deeper insight into the n a t u r e of o b j e c t u a l intuitive reasoning. In this way symbolic logic contributes to mind-philosophical logic a n d its rhetorical applications. *

T h e r e are more rhetorical points from which logic can be viewed, a n d an especially i m p o r t a n t one is t h a t concerned with rules of dialogue. A n advanced logical s y s t e m to shed light on t h a t rhetorical issue has been called dialogical logic, initiated by Paul Lorenzen in collaboration with K u n o Lorenz (Lorenzen a n d Lorenz [1978], Lorenz (ed.) [1979]). Its elaboration t o w a r d s p r a g m a t i c a l rules of a r g u m e n t a t i v e dialogue is d u e t o the s t u d y by Ε . M. B a r t h a n d E . C . W . K r a b b e [1982] From

Axiom

to Dialogue:

A

philosophical

Preface

xi

study of logics and argumentation. The system closer to practice of argument t h a n classical logic is S. Jaskowski's discussive logic, being applicable to arguments whose consistency is not proved (see Kotas [1975]). There are also linguistic approaches to dialogue rules. Let me mention two examples of such theories as making use of logical devices, and so hinting at those parts of logic which are relevant to rhetorical questions. One of them is Roland Posner's Theorie des Kommentierens [1972] which uses predicate logic to introduce a syntactic system for the description of relevance distribution in certain kinds of dialogues. T h e other example is the rich o u t p u t of the Prague Group of Algebraic Linguistics which uses predicate logic and some of its extensions to account for the relation between what is already known and what appears as new at a given stage of dialogue. These are just examples of the variety of logical approaches in communication theories. Let this brief listing suffice to express the present a u t h o r ' s awareness of how limited is his own approach to logically viewed rhetoric. However, these limitations afford opportunities to a more detailed study of selected topics, a study in t h a t historical perspective which is particularly pertinent t o t h e opposition of intuitive and formalized arguments. J a n u a r y 1993

Witold Marciszewski

Acknowledgements It is the a u t h o r ' s privilege and pleasure to express his acknowledgements and thanks. Those writers to whom I owe ideas developed in this essay are mentioned in appropriate references throughout the book. I would much enjoy telling about those debts in more personal terms, but then the size of this text would exceed acceptable limits. Let me only mention the name of Kazimierz Ajdukiewicz. I owe to him not only the knowledge and the kind of training which is thankfully remembered by his pupils, but also the strong encouragement t o approach logic in the pragmatic manner which, I hope, is manifested in this essay. As this book was to be written in English, I needed advice from someone whom I regarded as more competent in English than myself. I found him in the person of Olgierd Wojtasiewicz, an experienced translator of Polish philosophical and scientific literature into English; let me express my thanks for his reading the manuscript and helpful comments. Moreover, the manuscript has been read by the native speaker John Orman to whom it owes very advantageous corrections and refinements. Needless to say the responsibility for the final form of the text rests with myself. *

I owe more t h a n thanks to my wife Halina. She has been as much a part of writing this book as of my whole philosophizing and my joyful experiencing of the world of humans and of human-friendly machines.

Contents Preface

vii

Acknowledgements

xii

Chapter One: On the Rhetorical Point of View 1. Why rhetoric declined, and what remained of it 2. Descartes, Leibniz and Pascal facing a crisis in logic

....

1

...

1 6

Chapter Two: Mind-Philosophical Logic as a Theory of Intelligence

13

1. A terminological introduction 2. A case study and methodological comments 3. Conceptual potential and conceptual engineering

13 18 23

Chapter Three: Formalized versus Intuitive A r g u m e n t s T h e Historical B a c k g r o u n d 31 1. On how geometry and algebra influenced logic 2. The Renaissance reformism and intuitionism in logic . . . 3. Leibniz on the mechanization of arguments

31 44 57

Chapter Four: Towards the Logic of General N a m e s

68

1. From syllogistic to the calculus of classes 2. The existential import of general names 3. What names stand for: an exercise in Plato

68 75 79

xiv

Contents

C h a p t e r Five: T h e T r u t h - F u n c t i o n a l C a l c u l u s a n d t h e Ordinary U s e of C o n n e c t i v e s 1. 2. 3. 4.

The The The The

89

functional approach to logic 89 truth-functional analysis of denial and conjunction . 93 truth-functional analysis of disjunction 100 truth-functional analysis of conditionals 104

C h a p t e r Six: T h e P r e d i c a t e Calculus 1. 2. 3.

Subject, predicate, quantifiers Quantification rules, interpretation, formal systems Predicate logic compared with natural logic

112 112 . . . . 120 126

C h a p t e r Seven: R e a s o n i n g , Logic, a n d I n t e l l i g e n c e

. 141

1. Does a logical theory improve natural intelligence? . . . . 2. The internal logical code in human bodies 3. The problem of generalization in the internal code 4. W h a t intelligent generalization depends on 5. The role of a theory for intelligent generalization 6. Logic and geography of mind: mental kinds of reasoning . 7. Formal ('blind') reasoning and artificial intelligence . . . .

141 145 152 160 165 170 175

C h a p t e r Eight: D e f i n i n g , Logic, and I n t e l l i g e n c e

. . . 183

1. 2. 3. 4.

. . . 183 188 197 . . . 203

The ostension procedure as a paradigm of definition Normal definitions of predicates and names The holistic doctrine of definition Implicit definitions and conclusive conceptualization

C h a p t e r N i n e : S y m b o l i c Logic and Objectual Reasoning. Case Studies

220

1. 2. 3. 4.

220 224 231 239

On the case study method Cicero's reasoning in the light of symbolic logic Martha's objectual reasoning matched by symbolic logic . Aspasia's argument confronted with predicate logic . . . .

Contents

Chapter Ten: Implicit Definitions and C o n c e p t u a l N e t w o r k s . Case Studies 1. 2. 3. 4.

xv

257

A connectivist approach 257 The contrastive background: a definition for computers . 261 The case of a definition in the food market 265 The case of nonexistent Geist and similar cases 269

T h e Postscript as a B o o k - N e t w o r k Interface Material versus Formal A r g u m e n t s

278

References

288

I n d e x of N a m e s

297

I n d e x of Subjects

300

E x t e n d e d Table of C o n t e n t s

305

CHAPTER ONE

On the Rhetorical Point of View 1. W h y r h e t o r i c d e c l i n e d , a n d w h a t r e m a i n e d o f it 1.1. Once upon a time rhetoric was a vast and influential branch of learning, closely tied to Grammar and to Logic within the famous mediaeval Trivium. Nowadays it does not appear in research programmes nor in curricula, and historical studies alone mention it as a venerable monument of the past. On the other hand, the career of its sister disciplines Grammar and Logic has been a real success story. Grammar evolved into an immense field of linguistic studies, both theoretically pregnant and fruitful in applications, related to logic and mathematics, to empirical areas such as psychology and sociology, as well as to neurophysiology and computer science. Logic has proved a still greater success. After having been built on algebraic principles (which Aristotle himself did not dream of), it essentially contributed to building modern unified mathematics, opened up new prospects to philosophy, and paved the way to the idea of computers. Why did rhetoric fail to match the advances of its relatives? There is no simple answer to such a question. However, in order to define my 'rhetorical point of view' as held in this essay, I should attempt to suggest a sketchy answer. A more detailed account would require thorough research, for r h e t o r i c was deeply involved in the course of cultural and political history. It is why to substantiate any hypothesis regarding this issue would mean engaging in a comprehensive historical study. 1 1

Worthy of mention here is the monumental eight-volume study by the Polish historian of mediaeval philosophy (and my university professor and master)

2

One: On the Rhetorical Point of View

1.2. Rhetoric more t h a n Logic or G r a m m a r was involved in political, social and cultural circumstances of past periods which no longer exist in our times. For example, a politician's image is nowadays more shaped by his T V appearance t h a n by his ability to form long decorative phrases in his speeches. There was a time when rhetoric served t h a t special form of democracy which was characteristic of city-states or free cities in Antiquity, and in mediaeval and Renaissance Europe. However, modern democracy requires other means of influence. One should also take into account the increasing differentiation of political and cultural life. In view of t h a t process, as opposed to the situation in former periods, no universal methods of persuasion can nowadays be recommended to speakers and writers. Such methods, as codified by Aristotle, Quintilian, Cicero et al., were unanimously believed throughout centuries and millennia to be unquestionably valid. Now we are perfectly aware t h a t one must argue in quite a different manner, e.g., when addressing an EEC committee, and when negotiating with Muslim fundamentalists. This new awareness must have dawned in the 17th century when travellers and missionaries discovered cultures and mentalities so much differring from ours; certainly, Cicero's rules of persuasion proved of no particular use when faced with a Chinese or a Guarani audience. Both of these audiences exemplify the Jesuits' successful art of arguing, far from orthodox rhetorical prescriptions. In China, Jesuit missionaries had considerable success in convincing rulers and mandarins of the high scientific and technical performances of Western civilization, thus arousing respect for Christian ideas. In Paraguay, Jesuits managed to transform Guarani Indians Stefan Swiezawski Dzieje Filozofit Europejskiej XV Wieku [The History of European Philosophy in the 15th Century]; the first five volumes were published by The Catholic Theological Academy in Warsaw, 1974-1980. In spite of being written in Polish, the work may prove useful to a non-Polish reader due to abundant references as well as quotations in main European languages. There are partial translations of this work into French and English; a synthetic version appeared in one volume in French, see Swiezawski [1990], Other versions, corresponding to particular volumes of the Polish edition, are to appear. This work is especially useful for studying the history of rhetoric since the 15th century, as combining the mediaeval heritage with revived ideas of Antiquity provides us with a wonderful image of rhetoric's former power and glory.

1. Why rhetoric declined, and what remained of it

3

into docile a n d devoted Christian converts who obeyed orders t o f a r m t h e land, build churches, and p e r f o r m some a d m i n i s t r a t i v e tasks for their community. Certainly, t h e Fathers did not consult either Aristotle or Cicero. In spite of t h e flourishing s t a t e of rhetoric in t h e 17th a n d t h e next century, t h e widening of t h e world beyond t h e b o u n d a r i e s of t h e Graeco-Roman culture must have intiated t h e process of decline. F u r t h e r m o r e , in t h e 19th century t h e rise of new branches of learning, as e t h n o g r a p h y , sociology a n d psychology, c o n t r i b u t e d t o t h e art of dealing with people by t a k i n g into account their social a n d individual peculiarities. B u t even if people no longer expect a u t h o r a t i v e answers from t h e old rhetoric, there does remain t h e rhetorical problem of how to convince someone of my ideas. T h u s there remains a rhetorical point of view. W h a t nowadays can be seen from t h a t point is far from t h e ancient or mediaeval rethorical landscape. Nevertheless, it is t h a t old rhetoric t o which we owe our present ability t o see a n d t o s t a t e problems — e.g., t h a t of relations between logic and rhetoric; in this example one may see how new answers are d u e t o our inheriting some old questions. A n d where problems arise, we can investigate t h e m in a m a n n e r t h a t would suit our present ends a n d interests. 1.3. T h e same 17th century witnessed a n o t h e r t r e n d which undermined t h e logical side of rhetoric, namely a new situation in logic which accompanied t h e decline of t h e old paradigm for science. A look at this process should make t h e point of this c h a p t e r clear. Aristotelian logic reached its climax between t h e 12th and t i e 15th century. T h e n it marched in t h e vanguard of mediaeval rationalism which looked for its place within t h e limits — not too vast, indeed — of theological orthodoxy. T h e Christian faith was declared to accord with n a t u r a l reason, a n d t h e latter was exercised mainly by developing and applying logic. W h e n theology was dominant among t h e branches of learning, intellectual achievements in t h a t field m a t t e r e d more t h a n in any o t h e r one. A success in theology could have been measured only by two criteria: t h a t of accordance with o r t h o d o x teachings and t h a t of accordance with logic. T h e r e was no way of falsifying a theological conjecture t h r o u g h empirical reality, b u t it might have been refuted either as disagreeing

4

One: On the Rhetorical Point of View

with a theological axiom or as disobeying a logical rule; the latter would have referred to the mode of its being derived from formerly accepted propositions. However, due to the speeding up of the progress of science since the 16th century, people started to realize that the reputedly intelligent thinkers might have been fairly ignorant of the subtleties of Aristotelian logic. That observation was backed up through the critical examination of logic itself. It was hard to find in its procedures anything that could have aided Galileo's or Copernicus' discoveries, Gutenberg's involved technology (owed to a unique combination of craftsmen' skills), or Columbus' ideas. This is why the existing logic was so vehemently accused of uselessness and sterility both by Francis Bacon, who spoke on behalf of natural science, and by Rene Descartes, inspired by his own success in mathematics. Thus logic itself, being an important ally of rhetoric, started to suffer losses in the postmediaeval period. Temporarily it could even have improved the chances of rhetoric as an art of live speech and writing opposed to pedantic logic unable to move human souls. Yet, when tracing the history of rhetoric, especially its theory with Aristotle and Cicero as the greatest authorities, one must agree that its roots went back to logic. A figurative style as developed in the extra-logical branch of rhetoric was appreciated as a means to more powerfully influence an audience, but the main force of arguments was looked for in the rules of logic. Thus the decrease of the authority of logic must in the long run have diminished the position of rhetoric, too. That historical experience gives rise to the question concerning the chances of rhetoric after the revival and dramatic development of logic in our times. Is a new flourishing of rhetoric possible? We should not expect the answer in the affirmative as history is like a stream, and one can never enter the same stream twice. However, the rise of new theories and practical skills which deal with problems of efficient communication does leave much room for a new form of rhetoric. Firstly, there is the task of applying some achievements of modern logic to the art of successful communication, especially in regard to argumentation.

1. Why rhetoric declined, and what remained of it

5

That limited but important task should be attributed to what I suggest we call cognitive rhetoric, as a discipline which is (i) supported by a theory of natural logic taking advantage of symbolic logic (see Chapter Six, Section 3), and (ii) being developed for theorizing about arguments to be addressed to intelligent and benevolent audiences. This essay is meant as a reconnaissance of that nascent field of inquiry. The inquiry should take over the tasks of the old retired rhetoric, to carry it on in a new way, adjusted to the new times. The term 'cognitive rhetoric' is to play a significant role in the further discussion; the adjective contributes to explaining why rhetoric should take advantage of logic when logic is seen as fundamental for the study of cognition. However, this is not to mean that a new academic discipline is being planned. The distance between a fully formed discipline and other theoretical activities, even if these are distinguished by special names, can be measured with the help of a set of lucid distinctions devised by Posner [1988]. According to that essay, an academic discipline must include the following five components: (1) the domain as a set of objects, (2) the subject matter as a set of relevant properties (of these objects) referred to by suitable predicates, ( 3 ) the methods as a set of rules, (4) the body of knowledge as a set of asserted propositions, (5) the presentation as a set of means of expression (natural language, technical terminology, symbolism, diagrams, etc.). Cognitive rhetoric shares the domain with semiotics but differs from semiotics in item 2 as it involves predicates to express instructions and evaluations — in accordance with the old tradition of rhetoric as concerned with the art of an efficient activity (therefore, in Posner's terminology, its results have the status of a doctrine but not that of a theory). In the methodological aspect, it is characteristic for cognitive rhetoric, e.g., that it treats formal logic as a ladder to be mounted and then dispensed with (to use Wittgenstein's parable) in order to try a next approximation in rendering the nature of arguments. Therefore it possesses its own body of knowledge, e.g. the propositions comparing formal and material arguments. Thus, there are sufficient reasons to treat cognitive rhetoric as a special field of study, though not in so extensive and so advanced a way that it could be regarded as an academic discipline.

6

One: On the Rhetorical Point of View

2. D e s c a r t e s , Leibniz and Pascal facing a crisis in logic 2.1. Thus, in the 17th century began a crisis in logic which also undermined rhetoric. What can we learn from that story when attempting to build rhetoric again on a logical basis? According to the view so outspokenly stated by Thomas Kuhn, any crisis in science, as in politics, has to bring about a significant turn, or even a scientific revolution. This view sounds convincing to anyone familiar with what happened in science in the 17th century. Kuhn's [1962] views happen to be criticized for some claims belonging more to philosophy of science than to its history. Whether scientific revolutions must appear with cyclic regularity, is an issue as debatable as the analogous question in political philosophy. But that revolutions do happen is no risky contention, and that a most dramatic revolution within science did occur in the 17th century is common knowledge. 2 People concerned with logic were not isolated in their feeling that the old foundations proved to be wrong and that new foundations needed to be constructed. The feeling of crisis was overwhelming. It had a dramatic effect upon the Church and resulted in the emergence of Protestantism. It was also apparent in philosophy, and in views on human knowledge as well as views on physical reality. Revolutionary movement in that area resulted in the new paradigm which was self-consciously and, so to speak, enthusiastically mechanistic. Before we attempt to ponder the impact of that new framework upon logic and rhetoric, let us explain the stress being laid upon events of the 17th century, though the process in question must have started earlier. A brilliant justification of such a historical approach is given by Butterfield [1958] in chapter ten entitled "The place of the scientific revolution in the history of western civilization" (p. 180). Though everything comes by antecedents and mediations—and these may always be traced farther and farther back without the mind ever 2

E.g., in the entry 'modern' in Webster's Third New International Dictionary a typical use of the term in question is exemplified with Josiah Royce's saying "modern thought is a very recent affair, dating back only to the seventeenth century".

2. Descartes, Leibniz and Pascal facing a crisis in logic

7

coming to rest—still, we can speak of certain epochs of crucial transition, when the subterranean movements come above ground, and new things are palpably born, and the very face of the earth can be seen to be changing. On this view we may say that in regard not merely to the history of science but to civilization and society as a whole the transformation becomes obvious, and the changes become congested, in the latter part of the seventeenth century. We may take the line that here, for practical purposes, our modern civilization is coming out in a perceptible manner into the daylight. To a great extent our modern civilization is due to the mechanistic paradigm of science. 3 The crisis which endangered science at the end of Middle Ages was overcome with the emergence and successful development of t h a t paradigm. However, logic did not participate in that process. Its rules proved useless in making discoveries, and its content did not fit into the mechanistic framework. The latter point, though not the fault of logic itself, might have left it beyond the main current of thought. Thus two combined issues challenged philosophers: (i) how to make logic assist the creative thinking t h a t would lead to discoveries, and (ii) how to build it into the general mechanistic outlook. Three solutions have been suggested for this set of problems, each retaining its validity u p to our times, and each being pertinent to a rhetorical approach to logic. Let they be named after the most eminent philosophers in the 17th century, namely Descartes, Leibniz and Pascal. Descartes and Leibniz initiated new approaches to logic, competing with one another. Pascal was more concerned with an analysis of the mind than with creating a logical system, nevertheless he gave logic two strong impulses, one towards the theory of definition, and one towards probabilistic reasoning (to a certain extent, they came to be treated in a systematic way, too, namely in comments found in Port Royal Logic). W h a t will be most taken into account in the present discussion is Pascal's analysis of two kinds of intelligence which contributes much to mind-philosophical logic (in the sense explained in the next chapter). 3

How this paradigm has brought about the dramatic progress in natural science can be learnt from the quoted book by Butterfield, also from the very instructive The Evolution of Physics [1947] by Albert Einstein and Leopold Infeld.

8

One: Oil the Rhetorical Point of View

All of them were busy with the then fashionable trend to create a logic of discovery — logica inventionis, and all dealt with the problem of how to relate logic to the mechanistic worldview. While Descartes and Leibniz tried to elaborate a system answering these questions, Pascal focussed on describing the innate power of human intelligence without even trying to state a system of rules to guide this faculty; but it is the description of the mind given by him from which we can profit most when striving for a logical theory of intelligence. Let us first compare Descartes and Leibniz. When faced with the above-stated questions, they came to different conclusions. 2.2. Rene Descartes' solution fitted into his radical mind-body dualism concerning the relation between mind and matter. There are — he claimed — two independent substances, each of them existing in world of its own, even if they interact in a way with each other, namely body, existing in space (res extenso), and mind, living, so to say, in another dimension (res cogitans). Faced with such a split in reality, one must have asked where logic belonged. Descartes did not bother with Aristotelian logic which he regarded as useless for solving real problems. He created, instead, his own logic which he called rules to guide the mind (regulae ad directionem ingenii — to quote the title of one of his essays); he did not use the term 'logic' as such which would have called to mind the scholastic tradition. Later, however, when his ideas started to compete with traditional ones, his followers did not avoid the term, hence the denomination Cartesian logic. The phrase itself 'rules to guide the mind' hints at the Cartesian solution. Logic belongs to the realm of mind. Its rules prescribe how the mind should behave in order to tell what is true from what is false; for instance, that one should never bother about what other people have said regarding the matter in question, one should instead concentrate on what can be grasped by a clear and manifest intuition (rule iii). If one imagines logical rules as being as close as possible to an algorithm for problem solving, one has to feel disappointed with such advice. However, Descartes believed himself (not without some justification) to be addressing his rules to intelligent minds, not to machines; in this case he could expect

2. Descartes, Leibniz and Pascal facing a crisis in logic

9

an understanding of his ideas, even if not fully expressed in their literal formulation. He hoped that the method defined by such rules, which were extracted from his own, much successful, experiences in mathematical thinking, would help people in training and improving their minds. Thus he believed he had replaced the old and useless logic by a theory of scientific methods that would match the challenges of the new age. He also believed himself to have taught how to build a mechanistic theory of the material world with a method due to a deeper insight into the non-mechanical nature of the mind. 2 . 3 . Leibniz's reaction to the crisis of logic is found at the opposite pole. He was a monist who did not deny reality either of mind or of matter, but saw the mental as organizing the material at its very heart. The mind is no 'ghost in the machine' (according to Ryle's [1949] famous saying) but rather an engineer's idea that is to organize the machine's functioning. In this context logical rules appear well suited to control behaviour of a reasoning machine as arithmetical rules control the behaviour of a computing machine. Leibniz not only stated the programme of building a reasoning machine but also made some theoretical arrangements which we now see as necessary preparatory steps. Namely, he strenuously attempted to create a logical calculus which could be performed by a machine (Veritas machinae ope impresso). Eventually, he succeeded in something that roughly resembled the later Boolean algebra in which he managed, inter alia, to express the four types of Aristotelian categorical propositions. 4 Now we know that the algebraization of logic provides us with an algorithm to mechanically check the validity of any formula of the propositional calculus, while for the rest of logic (predicate calculus) there are ingenious methods to reduce it in a way to propositional logic. In this way Leibniz tried to put logic into the mechanistic framework. As for the demands of the mechanistic approach, his solution 4 Unfortunately, his discovery remained unknown up to the end of the 19th century, and was made independently by more authors from among whom George Boole has proved most successful. More on this subject is found in Chapter Three.

10

One: On the Rhetorical Point of View

appears to be much more successful t h a n t h a t of Descartes. But what about the other part of the programme of reforming logic, t h a t called logica inventionis, which intended to make logic of use in making discoveries, in looking for new truths? Leibniz believed t h a t a logical machine could be fit enough to perform such a demanding task. Were he right, then the logica inventionis project would remain in beautiful accord with the mechanistic framework. However, his point is not likely to be interpreted in a brief and lucid manner. Only to mention a possible course of interpretation, let it be recalled t h a t Leibniz put the demarcation line between mind and matter in a peculiar, so to say, 'set-theoretical' manner. 5 Namely, true to his juvenile insights expressed in the dissertation De arte combinatoria, he measured the distance between live creatures and inanimate m a t t e r by degree of complexity, namely infinite in the former case, finite in the latter. Then one might say t h a t a machine can approximate the mind's performance in a degree proportional to its being complicated, the limit of such a progression laying in infinity. The infinite complexity of life and mind would be actual while t h a t of a machine only potential, and thus one would save both the idea of the insuperability of the mind and the idea of the increasing possibility of its being replaced by a machine (both nicely combined with Leibniz's infinitistic framework, and sounding reasonable to present users of ELSI computers, i.e., those of E x t r a Large-Scale Integration). 2 . 4 . Pascal's contribution to logical and psychological foundations of rhetoric mainly consists in his famous distinction between esprit de geometrie and esprit de finesse. Their comparison from the logical point of view can be made in terms of premises and inferences, as did Pascal when comparing these mental faculties. Here are Pascal's own words on the esprit de finesse as characteristic of practical men being opposed to mathematicians. 6 5 As to the role of set-theoretical insights in Leibniz's thought, see the inspiring paper by Friedman [1975]. 6

The Thoughts of Blaise Pascal transl. C. Kegan Paul, London 1895, George Bell L· Sons, see section 'Various Thoughts', p. 310. The phrase itself does not appear in the quoted text; it appears earlier in a passage which is continued by the one here cited.

2. Descartes, Leibniz and Pascal facing a crisis in logic

11

The reason that mathematicians are not practical is that they do not see what is before them, and that, accustomed to the precise and distinct statements of mathematics and not reasoning till they have well examined and arranged their premises, they are lost in practical life wherein the premises do not admit of such arrangement, being scarcely seen, indeed they are felt rather than seen, and there is great difficulty in causing them to be felt by those who do not of themselves perceive them. They are so nice and so numerous, that a very delicate and very clear sense is needed to apprehend them, and to judge rightly and justly when they are apprehended, without a rule being able to demonstrate them in an orderly way as in mathematics; because the premises are not before us in the same way, and because it would be an infinite matter to undertake. We must see them at once, at one glance, and not by a process of reasoning, at least up to a certain degree. In a n o t h e r passage (p. 311), Pascal uses t h e p h r a s e p e n e t r a t i v e intellect to n a m e t h e s a m e faculty. T h i s description is w o r t h q u o t i n g because of its use of t h e concept of premises which belongs t o t h e terminology of logic. Some are able to draw conclusions well from a few premises, and this shows a penetrative intellect. Others draw conclusions well where there are many premises. For instance, the first easily understand the laws of hydrostatics, where premises are few, but the conclusions so nice, that only greatest penetration can reach them. And those persons would perhaps not necessarily be great mathematicians, because mathematics embrace a great number of premises, and perhaps a mind may be so formed that it searches with ease a few premises to the bottom, yet cannot at all comprehend those matters in which there are many premises. These are two kinds of mind, the one able to penetrate vigorously and deeply into the conclusions of certain premises, and these are minds true and just. The other able to comprehend a great number of premises without confusion, and these are the minds for mathematics. The one kind has force and exactness, the other capacity. Now the one quality can exist without the other, a mind may be vigorous and narrow, or it may have great range and no strength. A f t e r more t h a n t h r e e centuries, t h e s e observations display new vitality — owing t o o u r familiarity with c o m p u t e r s , a n d t o o u r knowledge, even if m o d e s t , of t h e f u n c t i o n i n g of t h e brain. Certainly it is n a t u r a l for a c o m p u t e r t o i m i t a t e m a t h e m a t i c a l m i n d s d u e t o t h e e n o r m o u s m e m o r y capacities being "able t o c o m p r e h e n d a great n u m b e r of premises w i t h o u t c o n f u s i o n " . It is why c o m p u t ers are good at deducing d a t a f r o m explicitly e n u m e r a t e d , even if

12

One: On the Rhetorical Point of View

gigantic, sets of premises. As for the brain mechanism underlaying the penetrative mind, it may be considered as approaching this model. An advantage of the brain over the computer consists in an astronomical number of connections between neural cells so t h a t innumerable d a t a from many centres and levels can be combined and synthetized at an appropriately high level in order to yield a conclusion. The conclusion follows, so to say, from premises written nowhere, for no unit in itself records a whole premise, it may result from a combination and interplay of unimaginably numerous partial d a t a converging towards a whole to be integrated by a central unit. Such a connectivist model accounts for the feature which Pascal perceived as the ability to deal with enormous complexity of details; hence the name 'finesse' to suggest t h a t minuteness, and the adjective 'penetrative' to suggest the necessity of penetrating deep layers of a vast network (to be called a conceptual network in Chapter Eight). Obviously, a great part of t h a t process must occur at the subconscious level, so t h a t often a penetrative mind perceives only t h e result without being able to account for either the premises or the ways of conceptualizing and reasoning; this lack of awareness of our own mental processes is the price to be paid for their enormous efficiency. In spite of t h e denied access to t h a t immense fabric of mentalneural activity, we shall try to find out a factor at the conscious level owing to which the penetrative esprit, being a prerequisite of rhetorical acumen, could significantly improve its performances. T h e penetrative mind is defined as one suitably endowed with what I suggest we call 'conceptual potential' and 'conceptual engineering' as principal constituents of intelligence. They constitute the main subject-matter of mind-philosophical logic which is to provide rhetoric with a solid cognitive foundation. 7

7

Such a cognitive foundation requires an adequate terminology. To find a suitable English counterpart for the Pascalian esprit de finesse without coining new and unavoidably artificial terms, I suggest the word 'acumen' as a short and natural translation. In its original Latin meaning it denotes a top in acute form, and later, by extension, high intelligence, acuteness, wit.

CHAPTER TWO

Mind-Philosophical Logic as a Theory of Intelligence 1. A t e r m i n o l o g i c a l i n t r o d u c t i o n 1.1. To see the impact of Pascalian ideas upon rhetoric, specifically those discussed in the preceding chapter, it should be noted t h a t there are three rhetorical activities: (i) influencing motivations, (ii) influencing convictions, (iii) influencing a system of concepts. T h e first should be handled with the combined means of psychology and decision theory, while the remaining ones correspond to t h e two main chapters of logic, viz. those concerning reasoning and definition, respectively. T h e contention of this essay is to the effect t h a t the core of rhetorical acumen consists in intelligent defining concepts for the sake of intended inferences. This is why this book deals so extensively with definitions (Chapters Eight and Ten). However, before a systematic treatment of definitions can be considered, one needs a certain concept both to prepare t h e discourse on definitions and, for the time being, to have a substitute (of a more formal definition theory) for some tasks to be pursued in the chapters to follow, namely the concept of an argumental construct. Its introduction and exemplification should advance our understanding of the n a t u r e of rhetorical acumen. When an object is constructed in one's mind in such a way t h a t the assumption of its existence makes it possible to infer new propositions, i.e., such t h a t their inferring would be impossible without this assumption, then — throughout the present book — the object in question is said to be an argumental construct. Let us start by examining t h e ordinary use of the noun belonging to

14

Two: Miiid-Philosophical Logic as a Theory of Intelligence

this phrase. In a popular dictionary of English t h e t e r m is defined as

construct

an idea formed in the mind by combining pieces of information (Longman).

A more t h o r o u g h explanation runs as follows: something that is constructed esp. by a process of mental synthesis: as a: an object of thought constituted by the ordering or systematic uniting of experiential elements (as percepts and sense data) and of terms and relations b: an intellectual or logical construction c: an operational concept; d: the result of such a construction or concept. (Webster

III)

A typical context of this t e r m is provided by such phrases as ' t h e constructs of science', or 'theoretical constructs'. An i m p o r t a n t step in this terminological a r r a n g e m e n t consists in adding t h e adjective ' a r g u m e n t a l ' t o t h e noun ' c o n s t r u c t ' . T h u s we obtain t h e notion of an argumental construct as a construct created for the sake of argument. It is t h e main tenet of this book t h a t t h e m e t h o d of consciously creating a r g u m e n t a l constructs is w h a t we really practise in arguing, while t h e formal-logical inference rules in arguing act almost imperceptibly, as on a subconscious level. Nevertheless, t h e logical theory providing us with inference rules is necessary t o theoretically u n d e r s t a n d t h e n a t u r e of a r g u m e n t , a n d this u n d e r s t a n d i n g , in t u r n , should improve t h e art of a r g u m e n t as combining inferences with definitions. T h e notion of argumental constructs is rooted in t h e concept of implicit definition (as discussed in C h a p t e r Eight, Section 4). Such a definition possesses t h e f e a t u r e of creativity, t h a t is t o say, it introduces a new term as involved in t h e context of a new assertion, t h u s increasing t h e deductive power of t h e theory in question. 1 A n o t h e r source of this idea goes back to Leibniz who summed u p t h e main principle of his logic and his metaphysics in the famous maxim praedicatum inest subjecto. We should not bother a b o u t the 1

The idea of creative definition is due to the Polish logician S. Lesniewski (see Suppes [1957]). Lesniewski introduced this notion in order to express the requirement of non-creativity which was to be met by deductive systems constructed according to his methods. However, the logical work of the mind is unlike that postulated by Lesniewski for his systems.

1. A terminological introduction

15

grammatical doctrine presupposed by this maxim, namely t h a t every s t a t e m e n t can be reduced t o t h e subject-predicate s t r u c t u r e ; as a rule, implicit definitions do not comply with t h a t doctrine. W h a t counts, t h o u g h , in Leibniz's view, is t h e idea t h a t genuine knowledge consists of s t a t e m e n t s which are either definitions or logically follow from definitions. It is obviously t r u e of t h e axiomatized m a t h e m a t i c a l theories, for any set of axioms is identical with a set of implicit definitions, a n d t h e whole (infinite) rest of theorems is deducible from t h e axioms. As for empirical sciences, one observes t h e p a t t e r n t h a t t h e more theoretically advanced a science is, as is t h e case with theoretical physics, t h e more it is liable t o become axiomatized. As for our everyday knowledge, t h e r e are m a n y more definitions in it t h a n philosophers dreamt of, because any creative thinking consists in discovering new objects, properties, relations, etc., and in defining t h e m , as a rule, according t o t h e m e t h o d of implicit definitions. 1 . 2 . Let t h e following example help to s u b s t a n t i a t e t h e conception of t h e predominance of a definitional component in our knowledge. 2 In this chapter I hereby i n t r o d u c e t h e term 'mind-philosophical logic'. It is t o express t h e idea which occurred t o me in t h e m o m e n t of finishing this book (so t h a t I had t o r e t u r n to this C h a p t e r a n d revise it). This exemplifies t h e fact t h a t a new idea results f r o m encountering a new object which was, in this case, t h e finished book (earlier, when t h e object was not ready yet, this idea was t o o vague to be expressed in t h e first version of this c h a p t e r ) . T h e conceptual environment in which this new concept had grown was t h a t expressed in t h e t e r m s philosophical logic and philosophy of mind. If there are, I t h o u g h t , special philosophical logics p e r t i n e n t t o particular philosophical theories, e.g., deontic logic related t o ethics and t h e philosophy of law, t e m p o r a l logic related t o t h e philosophy of time, etc., t h e n there should be the mind-philosophical logic t o be related t o t h e philosophy of mind. 2 The paragraph which follows is to examplify the conception stated above, and to introduce the concept of philosophical logic as being crucial for this essay. It involves an autobiographical digression to add a feature of live experience to the discussion concerning the role of a definitional component in developing a theory.

16

Two: Mind-Philosophical Logic as a Theory of Intelligence

Obviously, such a wordplay is not enough to create a new dom a i n , b u t it is hoped t h a t t h e t h e discussion which is t o follow will explain t h e full meaning of t h e so-invented t e r m . It has proved t h a t a new field must be explored by logic if it is to make a serious contribution to cognitive rhetoric. Any theory of rational communication presupposes a theory of rational cognition, and cognition consists in acts of reasoning, defining, classifying, ordering, etc., all of t h e m being t h e s u b j e c t - m a t t e r of logic. However, if these constit u t e t h e firmly established subject of logic qua logic, why should we introduce t h e special adjective 'mind-philosophical'? T h e answer goes back t o t h e beginnings of t h e modern philosophy of logic as developed by E d m u n d Husserl and G o t t l o b Frege in particular (their ideas being earnestly followed by t h e Warsaw School). T h e rise of modern logic was accompanied by t h e strong anti-psychological claim regarding b o t h logic and m a t h e m a t i c s . C o n t r a r y to former views, logic has been seen as a theory which does not deal with any mental acts; instead, it deals with a domain of abstract objects, as are truth-values, sets, functions, concepts, propositions, theories, etc.; it is also Karl P o p p e r ' s claim t h a t logic is concerned with w h a t he calls ' t h e third world'. This t u r n coincided with t h e s t a r t of t h e brilliant success story of modern logic; nevertheless after a lapse of time it was observed, esp. by Kazimierz Ajdukiewicz, t h a t we also need a normative theory of cognitive activities of t h e mind. Ajdukiewicz [1974] suggested t h e term pragmatic logic as referring t o h u m a n activity (πρμα-γα) to distinguish t h a t branch of w h a t he called apragmatic logic as concerned with a b s t r a c t entities; such entities may be interpreted either as certain things in themselves (according t o Frege) or as p r o d u c t s of mental acts (the latter would comply with Kazimierz Twardowski's distinction of mental acts and mental p r o d u c t s ) . My present proposal follows Ajdukiewicz's claim, but in a new terminological guise — hopefully one which is b e t t e r a d j u s t e d to t h e current s t a t e of affairs in logic, philosophy, a n d their environment such as cognitive science or artificial intelligence theories. Ajdukiewicz himself did not yet consider biological interpretations of logic which brought a b o u t a new approach t o t h e old philosophical m i n d - b o d y problem. Nowadays those mental activities which,

1. A terminological introduction

17

according t o Ajdukiewicz, should have been t h e s u b j e c t - m a t t e r of pragmatic logic obtain an intriguing philosophical background inspired by neurophysiological discoveries, by c o m p u t e r models, etc. Philosophy of t h e mind is now needed t o integrate t h e results of biology, cognitive psychology, c o m p u t e r science, etc., hence logic which makes use of such a philosophical reflexion a n d , at t h e same time, is capable of c o n t r i b u t i n g t o it, is worthy of being called mind-philosophical logic. My decision t o introduce t h e phrase 'mind-philosophical logic' may appear to be a terminological fiat, b u t is s u b s t a n t i a t e d by t h e vision of a new o b j e c t , namely t h e vision of a field of s t u d y combining logic with biology, c o m p u t e r science, etc. This vision includes t h e idea t h a t t h e cognitively crucial process of implicit definitions can be best explained by a feedback in conceptual networks (as discussed in C h a p t e r Eight, Subsections 4.4 a n d 4.5). For example, t h e t e r m ' m e n t a l activity' is connected (in this a u t h o r ' s own conceptual network) with t h e t e r m ' p r a g m a t i c logic' ( t h r o u g h t h e verbal association activity - p r a g m a ) , t h e n Ajdukiewicz's definition of pragmatic logic is subsumed u n d e r a theory dealing with t h e mind, t h e n comes t h e realization t h a t t h e theory of t h e mind should refer t o some scientific results, a n d t h a t a certain integration of these results is a j o b for t h e philosophy of mind, hence logic's taking a d v a n t a g e of such studies a n d also c o n t r i b u t i n g t o t h e m deserves t o be called philosophical logic. T h e r e is in such a discussion a wealth of knowledge concerning facts b u t , nevertheless, it is more definitional t h a n factual knowledge because t h e a u t h o r in question makes t h e decision t o imbue t h e meanings of terms with certain facts known to him, so t h a t this knowledge cannot be refuted without changing t h e corresponding meanings (if, for example, t h e r e are no logical operations in t h e central nervous system of h u m a n s , then t h e term 'mind-philosophical logic' as suggested above has t o change its meaning, for its definition involves t h e task of integrating logical and biological d a t a ) . T h u s knowledge can be contained in definitions r a t h e r t h a n in mere records of facts, according to t h e quoted view of Leibniz. Were t h e above discussion systematically s t a t e d as an axiomatized theory, t h e n t h e above-mentioned s t a t e m e n t s as implicitly defining t h e primitive concepts would function as axioms of t h a t theory.

18

Two: Mind-Philosophical Logic as a Theory of Intelligence

T h e next Section is t o exemplify some mind-philosophical issues involved in t h e notion of argumental constructs. These examples will be commented in terms of t h e distinction between objectual a n d symbolic a r g u m e n t , involving an anticipation of some mindphilosophical concepts t o be extensively discussed later. 2. A c a s e s t u d y a n d m e t h o d o l o g i c a l c o m m e n t s 2 . 1 . T h e m e t h o d of case studies is deliberately applied in this book. It is theoretically motivated by t h e view t h a t an efficient concept f o r m a t i o n , or conceptualization, consists in making use of a suitably chosen context of use instead of giving complete definitions (technically called normal definitions - see C h a p t e r Eight, Sec. 2). This context may consist of axioms, or axiom-like statements, forming an implicit definition (see C h a p t e r Eight, Sec. 4), it may also consist of purposefuly selected examples, i.e., specimens of t h e class denoted by t h e concept t o be defined. If such examples are systematically examined, then we o b t a i n a case study. T h e predominance of the case-study m e t h o d in this essay results both f r o m t h e d o u b t concerning chances of finding complete definitions a n d from t h e belief in m a n k i n d ' s instinctive ability t o generalize (see C h a p t e r Seven, Sections 3, 4, 5). T h e present case s t u d y is to exemplify one of t h e methods of forming a r g u m e n t a l constructs, while t h a t m e t h o d , in t u r n , should as pars pro toto exemplify t h e general notion of argumental cons t r u c t . This particular m e t h o d deserves a n a m e of its own; let it be called t h e adjective shift. It creates a new argumental construct t h r o u g h prefixing a noun, say N, with an adjective A which till now did not accompany Ν b u t preceded some other n o u n , say Μ, and in t h a t previous context AM t h e adjective has acquired a meaning now t o be shifted t o t h e new context AN. Owing to such a move, properties expressed by A as a t t a c h i n g t o t h e object of Μ become transferred to t h e object of Ν, and this allows asserting new propositions a b o u t t h e latter. Here is a case t h a t not only exemplifies t h e adjective shift but also stresses its rhetorical relevance, as t h e d e b a t e in which it appeared is very characteristic of our times. It is t h e d e b a t e on t h e decreasing role of t h e Christian churches in present-day German

2. A case study and methodological comments

19

society, reported by Der Spiegel (15 Juni 1992) in the context of a poll concerning the faith of German people ( Was glauben die Deutschen?). Its results are commented in a story of the German theologian Eugen Drewermann — a penetrative critic of the Catholic Church who in one of his lectures introduced the phrase ein real existierender Katholizismus thus referring to the disreputable conception of real socialism. When discussing this conceptual construction, let me take advantage of those personal reminiscences which I owe to the cognitively privileged status of having lived in a country of real socialism (benefiting from 'knowledge by acquaintance' as surpassing 'knowledge by description'). Citizens of such a country remember the particular impression made on them by the phrase 'real socialism'. There was a distinct and deliberate tone of threat whenever this phrase appeared in a speech: "There are people who are impudent enough to blame socialism for not changing the earth into a heaven, as some idealists had dreamt of. We who firmly stick to socialism shall not engage into any discussion with those malcontents. Let them do what we do, namely take facts as they are without any hair-splitting, otherwise they will be taught their lesson." And so on, and so on, the speech might have continued without giving any reasons, any diagnosis, why the socialist policies proved a failure and what might have been done to improve affairs. The adjective 'real' or 'realistic' in that specific context perfectly indicates a state of decadence and helplessness so far advanced t h a t no theory, not even a demagogical one (as was usual in earlier stages of socialism), is able to explain reasons for the decay or advise a solution; its most extreme stage is t h a t marked by Fidel Castro's maxim "socialism or death". A linguistic invention which sounds best in German, namely 'Realsozialismus', patterned on 'Realpolitik', should have settled questions in a quasi-magical way, as if suggesting a sacred historical necessity which does not allow of any impious corrections (here one might quote the verse by Horacius quid corrigere nefas est)·, at the same time, any revision of the doctrine or policies seemed unthinkable, as if offending the same sacred necessity. As we shall need a brief term to designate t h a t complex of properties falling under the schema of 'real [...]ism' let the phrase decline syndrome assume t h a t role.

20

Two: Mind-Philosophical Logic as a Theory of Intelligence

2.2. Now let us discuss some logical effects of that witty adjective shift trick as applied by Drewermann. The whole complex of properties denoted by the adjective in the previous context becomes transferred to the new subject — Catholicism. Moreover, when one takes into account the most recent historical experience of the decline of socialism, this provides the sarcastic adjective 'real' with a new content, allowing fairly precise predictions. It should be noted that the poll reported in the same Der Spiegel issue hints that the predicted decline has indeed begun, as the title of the reportage reads "only one in four still a Christian" ( N u r noch jeder vierte ein Christ). Thus the adjective shift in question proves equivalent to proposing the assertion Catholicism is affected by the decline syndrome. This assertion, due to its being equipped with the content of historical experience as described above, has a considerable deductive power; it yields both explanations and predictions, as well as some reformatory ideas. All of this is valid, provided that one accepts the existential sentence "there exists real Catholicism". It is not necessary that it be accepted as a firm assertion, it may be assumed as a working hypothesis; in either case it is an apt basis to make inferences. It is why the object called real catholicism deserves the name of argumental construct. Here are some examples of issuing inferences. Real socialism collapsed because of its obstinate refusal to examine the causes of its weakness (it is exactly what some authors called 'unreformability of socialism'). The same endangers Catholicism if it does not change its policy of hiding its head in the sand. This perilous strategy involves an attitude toward more or less firm adherents that does not take into account their doubts, but instead merely blames them or even anathematizes for deviations. Intolerance was always inherent in Catholicism as in all totalitarian ideologies but it was formerly derived from a strength of conviction, not undermined by empirical refutations which were yet to come; now it issues from intellectual weakness, of the kind belonging to the decline syndrome, when one declines to learn from accumulated objections. The above reasoning is no literal report on Drewermann's speech or writing. Only the very notion of 'real Catholicism' with its sarcastic hint due to the adjective shift was taken over in order to

2. A case study and methodological comments

21

devise a specimen of argumental construct, that is a concept constructed as a basis for argument. This example, though, does not disagree with the author's original intention, and therefore it is a realistic example of possible debates using argumental constructs. The argumental construct we discuss also proves fertile with new questions, not only explanations and predictions. The most basic and urgent question is that of reformability. Would there have been any chance for socialism, provided it had abandoned the adjective 'real' (as declaring the will of preserving status quo) and had it reformed itself, or was it doomed to collapse anyway in due historical time? Does the Church have a similar chance of becoming a reformable institution, provided it gets rid of its 'realism' (i.e., conservatism)? Answering such questions would exceed the limits of the present case study, but raising them should exemplify the role of an agumental construct not only as a source of premises but also as a source of new issues for a debate. 2 . 3 . The above case study was to show a useful method of creating argumental constructs, namely the method of analogy operating with adjectival shift. It is recommendable in those cases in which we lack a ready theoretical context while we have, instead, a rich experiential material to suggest an ad hoc theory. It is why this method may be of interest for practitioners of persuasion. However, a typical environment of argumental constructs is found in deliberately devised theories — mathematical, scientific, etc. Some argumental constructs are called theoretical constructs; the latter term appears in the current methodology of science, while the phrase 'argumental construct' is suggested by the present author for purposes of rhetorical approach to logic to express a concept which is more extensive than the concept of a theoretical construct. In what sense it is more extensive? People create theoretical constructs as objects whose existence is not ascertained in any observation statements; instead, it is supposed in the hypothetical assumptions of a theory in order to explain and predict certain observable phenomena. Thus all concepts of empirical sciences, apart from those taken from mathematics and those reporting observations, refer to objects belonging to the category of theoretical constructs. Obviously, all of them

22

Two: Mind-Philosophical Logic as a Theory of Intelligence

are argumental constructs as well, for they are created just to be used for explanations and predictions, t h a t is a kind of inferences. There are, though, argumental constructs which do not fall within t h e category of theoretical constructs as defined above. Objects of mathematics, e.g., geometrical figures, share with theoretical constructs both the feature of being constructed and the feature of possessing an inferential power; for instance, from the fact t h a t a figure is a triangle we can infer properties of this figure itself and of certain other objects. On t h e other h a n d , they differ from the theoretical constructs of empirical science due to their being involved not in hypothetical assumptions but in firmly established theorems. Hence, the class of argumental constructs is more extensive t h a n t h a t of theoretical constructs of empirical sciences. I should yet remove a possible objection t h a t the argumental construct class has been defined too broadly, so t h a t it may prove coextensive with the class of any objects whatever; were it so, the new concept would be unnecessary, as being replaceable by the already existing notion of object. Let me reply with an argument ex concesso.3 I shall not fight the supposition of coextensivity, but shall only explain the usefulness of the concept of argumental construct, even if it shares extension with the concept of object. Both the noun 'construct' and the adjective 'argumental' hint at some features which are lacking in the concept of object. In these terms we are able to pose particularly logical questions, where the noun is concerned with proper methods of construction, while the adjective, with the question of how to endow a construct with possibly greatest argumental power. This is why the concept of argumental construct is a convenient tool for theorizing about arguments. Furthermore, the concept of argumental construct has a notable impact on viewing rhetoric from the logical angle. It is this notion which makes it possible to distinguish the logic which people use practically from theoretical logic which they tend to ignore but which proves necessary to account for their logical practice. This divide between logical theory and the practice of reasoning, as well as their interplay, can be fully grasped only when one is familiar 3

This would mean a concession to philosophical idealism regarding all objects as mental constructions.

3. Conceptual potential and conceptual engineering

23

with t h e former and carefully inquires into t h e latter. This t y p e of inquiry is entered u p o n in t h e next section, together with t h e first step towards presenting theoretical logic (also called formal or symbolic logic). 3. Conceptual potential and conceptual engineering 3 . 1 . T h e example of adjective shift discussed in t h e preceding section should make us aware of how much t h e deductive power of t h e mind, and hence t h e ability t o create a r g u m e n t a l c o n s t r u c t s , d e p e n d s on two factors which I suggest we call conceptual potential and conceptual engineering. H a d not Mr. D r e w e r m a n n h a d t h e concept of real socialism in t h e set of notions developed d u r i n g his mental life, had he not h a d t h e notion of Realpolitik to enrich t h e content of t h e former (e.g., t o heighten its harshness), t h e n his conceptual potential would have been less t h a n it actually proved t o be; then he would not be so able t o notice a n d t o discuss t h e quality in question as characteristic of Catholicism. Were he not ingenious enough t o discover t h e analogy and t o use t h e trick of adjective shift t o form t h e concept of real Catholicism, his ability of conceptual engineering would not m a t c h t h a t actually possessed by him. T h e conceptual potential of a mind d e p e n d s not only on t h e n u m b e r of concepts one can, consciously or unconsciously, make use of, b u t also on their consistent s t r u c t u r i n g , their relevance t o t h e problem in question, a n d (last but not least) their a d e q u a c y with regard t o reality. Conceptual engineering consists in t h e ability t o process t h e conceptual potential, especially in (i) forming new concepts out of existing ones and out of observed facts, and (ii) using these concepts in a r g u m e n t s t o reach a conclusion pertinent t o t h e considered problem. These two capacities, necessary t o p r o d u c e argumental constructs, have their c o u n t e r p a r t s in t h e two main theories of mind-philosophical logic, namely t h e theory of definition and conceptualization, and t h e theory of reasoning, respectively. These two theories of mind-philosophical logic, when t a k e n together, are fit t o grasp t h e phenomenon of intelligence, i.e., t h e virtue which manifests itself in efficient problem-solving; indeed,

24

Two: Mind-Philosophical Logic as a Theory of Intelligence

conceptual potential and conceptual engineering are those factors which substantially contribute t o problem-solving processes. In t h e heroic times of antipsychologism in logic such a s t a t e m e n t would sound like heresy b u t we are lucky enough t o live in times when computers constitute a bridge between logic a n d psychology. C o m p u t e r s were created by engineers who implemented ideas of logicians; no psychologist participated in t h a t process. However, once created, computers proved very interesting for t h e psychology of thinking, both as devices t o assist t h e activities of t h e mind a n d as models of information-processing capabilities. T h e idea has emerged of artificial intelligence as t h a t being able to imitate hum a n intelligence. Obviously, there must have also a p p e a r e d t h e notion of natural intelligence, t h o u g h previously it had m a d e no sense t o prefix t h e term 'intelligence' with this adjective. Now, as logic is partly responsible for t h e rise of artificial intelligence, leading in t u r n t o t h e generalized notion of intelligence, i.e., comprising b o t h its artificial and its n a t u r a l ramification, logic can no longer extricate itself from dealing with problems of intelligence. A n d , if t h e r e are people who wish t o preserve t h e primitive innocence of m o d e r n symbolic logic, which did not c o n t a m i n a t e itself by having a liason with psychology, t h e honourable solution will consist in establishing a new discipline to be called mind-philosophical logic, as suggested in this essay. It will comply with t h e spirit of t h e previously discussed t h o u g h t s of Pascal who was fascinated b o t h by t h e h u m a n mind, so mysteriously endowed with esprit de finesse, and by t h e idea of an a u t o m a t o n . 4 T h e two constituents of intelligence listed above may be partly d e p e n d e n t u p o n each other: as an example, t h e greater t h e ability of conceptual engineering, t h e more notions it produces to increase t h e conceptual potential. Nevertheless, there are reasons t o distinguish one from the other; e.g., there may be two persons who at a particular moment have similar conceptual potential owing to 4

Mind-philosophical logic can be conceived of as a normative theory of intelligence. However, the adjective 'normative' becomes redundant when the concept of intelligence itself involves the feature of being normative in the sense that intelligence is something of great value, and therefore ought to be aimed at. As for the Pascalian esprit de finesse, its description resembles the present description of combined faculties of conceptual potential and conceptual engineering.

3. Conceptual potential and conceptual engineering

25

similar education, b u t one of t h e m can process information more quickly, and this involves more efficient conceptual engineering; if, besides, t h e same person proves more t r a i n e d , say, in t h e critical evaluation of hypotheses, he scores still more points for t h e conceptual engineering certificate. T h u s conceptual potential has more t o do with erudition while conceptual engineering more with t h e creativeness which can be called logical — in a broad sense of this adjective, viz. t h e sense pertaining t o efficient information-processing. W h e n inquiring into conceptual engineering from t h e angle of mind-philosophical logic, we come t o t h e crucial distinction which will be worded in t h e phrases objectual reasoning a n d symbolic reasoning. T h e following example should explain their m u t u a l relation. Let us imagine, first, t h a t Sherlock Holmes is carrying out some deductions in his mind, which Doctor W a t s o n t h e n describes in his diary. Holmes's deductions result in certain a r g u m e n t a l constructs, usually concerning h u m a n characters involved in t h e case in question. From some traits observed at t h e s t a r t Holmes deduces other features of persons and situations, a n d so develops those constructs u p t o t h e point where he comes t o know t h e main agent's motives, abilities, m e t h o d s , etc. Now he is able t o infer from such a construct by w h o m , where, when, and for w h a t purpose the crime u n d e r investigation was c o m m i t t e d . All t h e time, t h e building u p of a construct consists in t h e processing of t h e object being t h o u g h t of, and not in processing sentences which might describe t h a t object. Owing t o t h a t , Holmes is capable of a s m o o t h transition from imagined t o actual facts, a n d can check t h e former by t h e latter. For instance, he imagines a m a n with a wooden leg t o be s t a n d i n g at a definite place, and t h e n he finds t h a t t h e r e is indeed an imprint of t h e wooden leg at t h e place where he expected to find it. T h u s there is a kind of continuity between w h a t he thinks and w h a t proves t o be t h e case. This is t h e sort of reasoning which I propose t o t e r m objectual. It deserves this adjective since t h e reasoner's t h o u g h t directly refers t o objects, and it t r a n s f o r m s t h e m by enriching a n d modifying their properties, without a need t o resort t o symbolic descriptions. Doctor Watson had to proceed otherwise. He a t t e m p t e d at a verbal rendering of Holmes's deductions t o impress readers with their

26

Two: Mind-Philosophical Logic its a Theory of Intelligence

perfection. Therefore he must have carefuly selected words a n d arranged t h e m in a sequence which was to m a p Holmes's intellect u a l processes. True, this example is far from being perfect since W a t s o n ' s texts do not constitute exact demonstrations, b u t when compared with Holmes's tacit reasonings, it provides us with a useful approximation to w h a t I call symbolic reasoning. Symbolic reasoning may be also called formal, for it constitutes t h e subject m a t t e r of symbolic logic which is also called formal logic. On account of t h e opposition in meaning between 'formal' and 'material', t h e kind of inference opposite t o t h e formal one deserves t o be termed material reasoning. B o t h pairs of t e r m s are equally justified, but t h e pair ' o b j e c t u a l ' - ' s y m b o l i c ' will be preferred in this essay. 5 3 . 2 . T h e notion of argumental constructs is a philosophical category, hence it should not lack a philosophical discussion to s u p p o r t t h e present mind-philosophical enterprise. T h e discussion should create an instructive context for t h e idea of objectual inference. This context is provided by t h e perennial controversy between rationalism and empiricism. Epistemological rationalism, as opposed to empiricism, claims t h a t t h e r e are notions which a h u m a n mind owes to its own inborn e q u i p m e n t , as has been most convincingly shown by Socrates in Phaedo with respect to m a t h e m a t i c a l concepts. However, even in Phaedo it is not supposed t h a t all m a t h e m a t i c a l concepts are as ready for use as they are, say, in t h e head of a brilliant student of m a t h e m a t i c s who is a b o u t t o take his exam. They should be somehow developed from something which is only potentially given. Now, there is t h e question of w h a t such a development consists in. Is it like recalling a forgotten piece of perfect knowledge, as Socrates a n d P l a t o claimed t o be t h e case, or r a t h e r like t h e inventive constructing of an object out of some given pieces, with results which may be at variance with those of other constructors? T h e latter option can be called constructivist rationalism, i.e., t h e view t h a t h u m a n reason creates a r g u m e n t a l constructs of already 5

For stylistic reasons I use both the term 'reasoning' and the term 'inference', treating them as equivalent.

3. Conceptual potential and conceptual engineering

27

existing d a t a and then tests them according to certain criteria, e.g., t h a t of conformity with predictions based upon experience. Constructivist rationalism assumes the existence of elementary pieces of knowledge, supposedly certain potential concepts, inborn t o all human minds (perhaps, even to all possible minds). W h a t kind of whole results from them may depend on varying historical, cultural, and biological circumstances as well as, presumably, free choices of individuals; e.g., one can treat cardinal numbers either as basic d a t a or as constructions made of equinumerous sets. This philosophical claim deserves t h e name 'rationalism' as it does acknowledge the role of inborn ideas and of reason's activity, in contradistinction t o the doctrine of empiricism; at the same time, the adjective 'constructivist' should distance this point from something like Platonic rationalism which admits neither of constructional activity nor of any kind of relativism. Hence this claim opens up a vast field for conceptual engineering, much larger t h a n t h a t subjected to the constraints either of Platonism or of empiricism. Constructivist rationalism with its pragmatical a t t i t u d e is significant in rhetoric as t h e art of argument. W h e n sticking to it, one should not commit t h e following error of Socrates. W h e n responding to the court, he thought t h a t he could have convinced the judges of his innocence, had he only had one day more to carry on his defense. He believed so because the process of recalling knowledge requires less time t h a n t h a t of creating argumental constructs, which must be accompanied and supported by various testing procedures. It was impossible in one day to implant Socrates's conceptual world in the heads of those obtuse and angry Athenian citizens who happened to be his judges. It was impossible not only for the lack of time but also for the number of judges, five hundreds persons selected to the office by drawing lots. In such a case one needs a variety of arguments, suited to different minds, according to different notions held by them, and according to various degrees of ability to follow the speaker's ideas. Obviously, such a variety of approaches would be dispensable if arguments were to appeal to certain emotional stereotypes of the Athenian judges, as such stereotypes are shared by most community members; just t h a t was the strong chance of Socrates' prosecutors. Yet, his own

28

Two: Mind-Philosophical Logic as a Theory of Intelligence

rhetoric was of a different kind. W h a t he wished to practise was cognitive rhetoric, as we shall call it, t h a t is when just the rational faculties of an audience are addressed. 3 . 3 . Once in possession of an argumental construct, a person carries out reasoning through processing the construction according to those particular rules which are proper to the domain in question, as can be seen in geometrical demonstrations. Let the following example, which I have kept as simple as possible, illustrate the inferential force of an argumental construct. J o h n approaches a door which can be opened with either pulling or pushing. This kind of functioning is known to John from his experience which enables him to recognize such a structure when he encounters it. This ability to recognize a pattern means t h a t John has the concept of a door of this kind, even if he happens not to know any name to distinguish this one from other kinds; if he does not, then a remembered picture, more or less vague, would suffice for John's communication with himself. Hence his argumental construct involves an outline of the door shape and, the disjunctive property related to its functioning, namely that it can be opened either through pulling or through pushing^the italicized phrase defines t h e argumental construct in question). This argumental construct can be processed into another construction, e.g., such t h a t the above disjunctive property is transformed into a non-disjunctive one, say being pulled in order to be opened. This can result from pushing the door with the negative experience; John then acquires additional information which enables him to apply a more effective construction, i.e., one involving the last mentioned (i.e., non-disjunctive) property. It is worth commenting t h a t a person who performs such a transformation is entirely involved with transforming the current argumental construct, and not with remembering and applying logical rules. Rules of formal logic, though active in such a process, remain unnoticed (so to say, transparent), acting in an automatic manner, as if from behind stage. The story told above is to exemplify what has been formerly called objectual inference (or objectual reasoning) in contradistinction t o symbolic inference (or symbolic reasoning). Now suppose t h a t a logician follows Doctor Watson as the chronicler of

3. Conceptual potential and conceptual engineering

29

somebody else's deduction, and, moreover, he wishes to show its logical validity. A logician in t h a t role may be compared to a literary critic who is to judge whether a text meets certain criteria, in this case criteria of logical validity. Hence, he must present the examined reasoning (which might have been a private mental process dealing with argumental constructs) in t h e public form of a text, t h a t is as a sequence of symbols. From this point on our chronicler has two options. One of them consists in using the produced text as a device to help the reader to empathize with t h e presented reasoning, so t h a t t h e reader can relive t h e hero's experience, and thus t o innerly experience its validity; this was Doctor Watson's method, adopted also by the present writer in the above description of J o h n ' s reasoning. 6 The second option consists in producing a symbolic inference, t h a t is one which does not deal with extralinguistic objects, dealing instead with sequences of symbols being presented to t h e reader, and, so to speak, being transformed before his eyes. T h e question which is posed in such a procedure is: do t h e transformations which lead from the premises to the conclusion comply with relevant rules of processing symbols? Here is the symbolic reconstruction of John's inference as reported above. "This door opens by pulling or by pushing.

It does not open by pushing.

Hence it opens by pulling." When looking at this text as a symbolic inference, we entirely disregard which objects it refers to, a door or anything else. We heed only to its structure, that is a configuration of symbols. Here the structure is as follows. The first premiss has the disjunctive form either ρ or q (the letters ρ and q standing for respective sentences), while the second premiss has the form not-q, thereby being the denial of the second constituent of the disjunction. Now we apply t h e rule of symbolic logic, bearing t h e traditional Latin name modus tollendo ponens(i.e., asserting through denying) which allows us to assert any sentence 'p' equiform with the first constituent of an asserted disjunction 'either ρ or q', if one asserts 6

Those who may be expected to share this attitude are adherents of intuitionistic logic as a theory of proof where a proof is understood as a mental construction, in contradistinction to a formalized proof being a sequence of symbols. Cf., e.g., the article 'Intuitionistic logic' by S. Krajewski in Logic [1981], The idea of objectual reasoning (though not the term itself) is convincingly vindicated by I. Kant, as discussed in Chapter Seven, Subsec. 3.2.

30

Two: Mind-PhilosophicaJ Logic as a Theory of Intelligence

the sentence in the form of denial of the second member, i.e. 'not-q' (there is a parallel rule regarding t h e denial of t h e first constituent). This example should help us t o realize a f u n d a m e n t a l problem concerning relations between rhetoric and symbolic logic. Arguments which we actually make in discussions, debates, etc. are usually objectual inferences, in t h e sense exemplified above, while symbolic logic deals with symbolic inferences alone. If so, w h a t can symbolic logic contribute to t h e enhancing of objectual arg u m e n t s ? T h e contribution t o be expected is not a direct one, nevertheless it is very significant. We shall see it later (in C h a p t e r Six, its last Section). A step towards this, m a d e in t h e next chapter, consists in a deeper investigation into t h e difference between o b j e c t u a l and symbolic inference. T h o u g h these terms in their literal form do not occur in t h e next c h a p t e r , t h a t C h a p t e r is mainly devoted t o t h e same distinction. It deals with t h e programs for logic stated in t h e 17th century. As briefly mentioned in t h e preceding C h a p t e r , t h e Cartesian program was decidedly anti-formalist, claiming t h e necessity of the mind's focussing u p o n t h e investigated object in order t o properly deduce pertinent facts. This is a kind of program to be addressed t o natural h u m a n intelligence. T h e Leibnizian program was consciously formalist, being oriented towards t h e construction of a reasoning machine, hence towards artificial intelligence. A machine is not expected t o deal with mental objects; instead, it has t o be fed with sequences of symbols and rules t o process such sequences. T h u s t h e m e t h o d s of reasoning discussed in t h e context of t h e Cartesian program should be seen as exemplifying objectual reasoning, while those appearing in t h e Leibnizian context illustrate t h e notion of symbolic reasoning. At t h e same time this historical discussion is t o contribute to the realization of t h e sense in which logic can become a mind-philosophical study. It will be seen how deeply t h e philosophy of mind is involved in philosophy of logic.

CHAPTER THREE

Formalized versus Intuitive Arguments The Historical Background 1. O n h o w g e o m e t r y a n d a l g e b r a i n f l u e n c e d l o g i c 1.1. In C h a p t e r One we discussed t h e 17th century crisis in t h e history of logic as bearing on t h e relations between logic a n d rhetoric. In this chapter t h e heritage of t h a t century will be more closely examined t o b e t t e r u n d e r s t a n d t h e lines of development leading to modern logic. T h e distinction between o b j e c t u a l (material) a n d symbolic (formal) reasoning suggested in C h a p t e r T w o needs a historical background in regard to its bearing on rhetoric. This is mainly t h e background of t h e 17th century, whose significance was pointed t o in C h a p t e r One. However, t h e u n d e r s t a n d i n g of this period requires a glimpse into t h e age-old process of interaction between logic and m a t h e m a t i c s which s t a r t e d in t h e Greek antiquity. T h e t u r n of t h e 5th century B. C. witnessed one of those awakenings of h u m a n intellect which proved t h e t u r n i n g points in t h e history of m a n k i n d . We see such an awakening in t h e remotest past in the development of speech, later t h e birth of t h e concept of number and of systems of counting, and j u s t three centuries ago t h e formulation of t h e first mathematical theory of t h e universe in t h e form of Newtonian mechanics. T h e discovery of logical consequence a n d t h e resulting reliability of proofs resembling t h e reliability of c o m p u t a t i o n s was an event of t h e same dimension. Such was t h e birth of logic, codified for t h e first time in Aristotle's Analytics (ca. 350 B.C.) as t h e theory of syllogism. Twenty centuries later, in t h e 17th century, Leibniz,

3 2 Three: Formalized versus Intuitive Arguments - the Historical Background

congenial with Aristotle, assessed t h a t discovery in his Nouveaux essais sur I'entendement humain (Book IV, 17) as follows: The invention of syllogism is one of the most important and finest inventions of the human mind, and it is that kind of mathematics whose significance is not sufficiently known; and it may be said that it contains the art of infallibility if we only know how and are able to make use of it. Leibniz pointed to t h e m a t h e m a t i c a l n a t u r e of logic, which was possible in his times owing to t h e advances in algebra. T h e times of Aristotle himself saw a different kind of connection between logic and m a t h e m a t i c s , namely t h e m u t u a l methodological inspiration between logic and geometry. Historians of Greek science pronounce t h e names of these two disciplines in one b r e a t h when they, for instance, refer t o the climate of logic and geometry, which we have known so well since Euclid a n d the impetus imparted to Greek geometry and logic (by t h e m a t h e m a t i c a l m e t h o d s of t h e Babylonians). 1 T h e methodological feedback between logic and geometry consisted in t h e fact t h a t t h e ideas worked out in logical theory used t o pass t o t h e praxis of geometricians, who in t u r n provided logicians with t h e best p a t t e r n s of proofs. Aristotle, when writing his Analytics, drew from t h e interpretations of geometry t h a t were known to him t h e p a t t e r n of necessary knowledge a n d reliable inference, while Euclid, when writing his Elements half a century later, availed himself of t h e methodological concept of c o m m o n axioms ( τ α κοινά), t h a t is, those which are not specific, say, to geometry, b u t are drawn from some more general theory; note in this connection t h a t t h e Aristotelian example of such an axiom — "if equals be taken from equals t h e remainders are equal" — occurs on t h e list of axioms in t h e first book of Euclid's Elements. 1 . 2 . T h e second encounter between logic and m a t h e m a t i c s was due t o algebra and took place for t h e first time in t h e 17th century. Algebra itself as the science of t h e solution of equations, t h a t is finding t h e unknowns in equations, originated in Babylonia in the third millennium B.C., and found special application in the comp u t a t i o n of shares in inheritance cases. Shortly afterwards it came 1

See De Solla Price [1961],

1. On how geometry and algebra influenced logic

33

to be known in Egypt, which is testified by papyri dating from ca. 1800 B.C. W h e n arithmetic and algebra, born in Babylonia, met in the Hellenistic period with Greek geometry and logic, they gave rise to that gigantic 'tree of knowledge' of which our civilization is the fruit (cf. de Solla Price [1961]). A r a b scholars made great achievements in the development of algebra and in transferring it to Europe. This applies in particular to Al-Kwarizmi (9th century A . D . ) , who also contributed to the spreading of Hindu numerals, later called Arabic numerals. T h e very word 'algebra' originated from the title of his work that begins with the words 'al gabr' (the whole title meant 'the transfer of a constituent from one side of an equation to the other'). The decisive advances in algebra were made possible by the development of symbolic language, which was the contribution of the Europeans from the 13th century onwards (the Arabs, like the Babylonians before them, used in their computations phrases drawn from natural language). Symbols of arithmetical operations were coming successively into use, and the breakthrough was marked by the introduction of letters as symbols of numbers ( F . Viete, 1591); thus the idea of a variable entered mathematics. T h e realization of that breakthrough manifested itself in the birth of the term analytica speciosa to single out the mathematics using such notation that one symbol does not correspond to a single object (e.g., the figure '2' to the number 2), but to a class, or species, of some numbers (the Latin adjective speciosa means also 'beautiful', 'magnificent', etc., hence it reflected the fascination with that invention). It was Descartes who became the coryphaeus of that analytics when in 1637 he published his Geometry, to which Discours de la methode was the annex; in this work he gave a synthesis of geometry with algebra or analytics (hence the term 'analytic geometry'). Owing to the maximal generality which the notation using letters gave to algebra, Leibniz was in a position to realize that a letter need not refer to classes of numbers, but can refer to any classes of objects of any kind. T h e use of letters was invented for logic already by Aristotle, but only the successes of algebra could give birth to the idea of an algebraic treatment. T h a t was one of the greatest of Leibniz's ideas concerned with logic. It was partly

3 4 Three: Formalized versus Intuitive Arguments - the Historical Background

materialized by himself, but it was first published in print two hundred years later, after t h e same discovery had been m a d e in t h e m e a n t i m e by other authors. Yet t h e p e r m a n e n t imprint of algebra u p o n t h e mentality of people living in t h e 17th century consisted in their experience of how an a p p r o p r i a t e language renders thinking more efficient a n d signally contributes to t h e solution of problems. This fact played its role in t h e development of a movement for improving t h e whole language of science. 1 . 3 . T h e second discovery of t h e algebra of logic took place in England in t h e mid-19th century, and was due to a Pleiad of prominent algebraicians, of whom G. Boole (1815-1864) rendered the greatest services t o logic. He was helped in t h a t respect by t h e then nascent comprehension of t h e a b s t r a c t n a t u r e of algebra, which is to say t h a t an algebraic theory does not refer to any specified domain (which was particularly emphasized by G. Peacock in Treatise on Algebra, [1830]). Algebra can, on t h e contrary, find application or, more precisely, interpretation, in a class of structurally similar domains t h a t can be described with t h e means provided by a given algebraic theory. In this way algebra, after having developed from t h e old science of solving equations, has become t h e most general theory of structures, t h a t is systems of objects for which certain operations are defined. Such operations are described from t h e point of view of their properties, e.g., c o m m u t a t i v i t y or its lack, or t h e performability of operations with a neutral element (such as zero for addition in a certain algebra having an interpretation in t h e arithmetic of n a t u r a l numbers), a n d t h e like. One of t h e int e r p r e t a t i o n s of this calculus called Boolean algebra (since Boole was its main promoter) corresponds to t h a t part of logic which is known as t h e t r u t h - f u n c t i o n a l calculus. T h e same algebra has another interpretation in traditional syllogistic, and it was just t h a t interpretation which was so penetratingly anticipated by Leibniz. In order fully to grasp t h e meaning of t h a t interpretation we have t o refer t o Boolean algebra in its contemporary form. It has at least two operations (the others can be introduced by definitions), one of t h e m called complement and symbolized by and t h e other called multiplication and symbolized by 'o'. Moreover two objects of those operations are considered; they are singled

1. On how geometry and algebra influenced logic

35

out from the set of all those objects with which a given theory may be concerned, and are denoted by the symbols Ί ' and Ό ' (other notations are also used, but we choose t h e one which is most convenient typographically). Multiplication satisfies the conditions of commutativity and associativity, and such as: XoO = 0, X o l = X, while complement is defined by the conditions: ~ 1 = 0, ~ 0 = 1. Boolean algebra can be interpreted arithmetically, t h e domain of natural numbers being limited to zero and unity (in such a case all the laws o f t h a t algebra are satisfied). It can also be interpreted in many other ways, but two interpretations, already mentioned above, one in the domain of sentences, and the other in t h a t of sets, are f u n d a m e n t a l for logic. In the domain of sentences, multiplication is interpreted as t h e linking of sentences by ' a n d ' (conjunction); complement is interpreted as the negation of a given sentence, 1 as t r u t h , and 0 as falsehood. If a formula consisting of those symbols and variable symbols standing for sentences is always true, t h a t is, if it is t r u e regardless of whether t r u t h or falsehood is assigned to t h e variables which occur in t h a t formula, then it is a law of logic and belongs to the logical theory termed truth-functional, or sentential, calculus. By way of example let us mention the law of contradiction, which reflects the t r u t h of the idea t h a t it is not so t h a t something is and is not the case, in symbols: ->(p Λ ρ) = 1 for all substitutions, t h a t is for both ρ = 1 and ρ = 0. By making use of the symbols of conjunction ( a ) and negation (->) we can express all the laws of t h e sentential calculus, but in order to bring it near to t h e reasonings found in science and everyday life it is convenient to introduce other connectives which as a rule occur in reasoning, such as 'if p, then q\ which we define by the formula ~>(pA^q). It states t h a t , in accordance with our understanding of the t r u t h of the conditional sentence, it is not so t h a t t h e antecedent holds (i.e., is true) while its consequent does not hold. We can likewise define t h e sentential structure 'p or q' by using t h e formula ->(->p A ->q). 1 . 4 . When interpreting Boolean algebra in the domain of sets (which traditionally was termed the extensions of names) we can in a natural way express t h e four traditional kinds of general — or categorical — sentences, which form the building material of

3 6 Three: Formalized versus Intuitive Arguments - the Historical Background

syllogisms. 2 The result of the operation of multiplication, i.e., A B , is this time interpreted as the common part of the two sets, t h a t is as the set of those objects which are in both A and B. The complement of the set A, i.e, — A, is interpreted as the set consisting of all objects (in the domain under consideration) which are not in A. Further, the symbol Ί ' now means the set of all objects in a given domain, and Ό', the empty set, i.e., the set which has no element. And here are the interpretations of the categorical sentences in Boolean algebra. Every A is B: A • —B = 0; for instance, t h a t every lion (Λ) is a predatory animal (B) means the same as t h a t the set of those lions which are not predatory animals is empty, which is to say that there are no such lions; this condition performs its definitional function under the reservation (made already by Aristotle) t h a t t h e set A is not empty, which is to say t h a t Α φ 0. No A is Β: A Β = 0; e.g., the fact t h a t no lion is a hare means the same as t h a t there are no lions which are hares (the reservation of the nonemptiness of A is valid as above). Some Λ is ß (in other formulations: Some A are Β; There is at least one A which is Β): Α · Β φ 0; e.g., the fact t h a t a certain lion is tame means the same as t h a t the set of t a m e lions is not empty, i.e., t h a t there are elements of our universe of discourse (for instance, the domain of animals) which are lions and at the same time are tame. Some A is not Β (other formulations, as above): Α —Β φ 0; e.g. t h e fact t h a t a certain lion is not tame means the same as that the set of lions which are not tame is not empty, in other words, t h a t there are lions which are not tame. Such a translation of traditional logic into Boolean algebra enables us to activate a strong deductive a p p a r a t u s of algebra for obtaining economical and elegant proofs of the theorems of traditional logic. Such was also the intention of G. Boole [1847] himself, which can be seen in the title of his work: The Mathematical Analysis of Logic, Being an Essay toward a Calculus of Deductive Reasoning (Cambridge). 2

This subject is treated more comprehensively in Chapter Four. Its present brief account is only intended to facilitate the understanding of Leibniz's pioneering ideas as presented in the sequel of the present Section.

1. On how geometry and algebra influenced logic

37

The anticipation of t h a t interpretation, to be found, in Leibniz's treatise of 1686 (first published in 1903) entitled Generates inquisitiones de analyst notionum et veritatum (inquiry into a general theory of concepts and judgements), differs from the modern form mainly by the fact t h a t in place of the symbols '^έ 0' it has the Latin phrase 'est ens', i.e., 'is an entity' (or, briefly, 'est', i.e., 'exists'), while the symbols ' = 0' have the analogue in 'non est ens', i.e., 'is not an entity' (or 'non est', 'does not exist'). These phrases can be interpreted in at least two ways: as stating the existence of objects which are the extension of a certain concept, i.e., a certain set (extensional interpretation), or as stating the existence or non-existence of combinations of properties, i.e. the intension of a certain concept (intensional interpretation). Leibniz himself was open to both interpretations; he valued t h e extensional one as technically efficient, but believed the intensional one to be better because of certain philosophical considerations. T h e idea of the translation of a sentence consisting of subject, copula, and subjective complement, such as "every man is intelligent" into an existential sentence of the kind "non-intelligent manhood does not exist", was of scholastic provenance, to which Leibniz clearly referred. The brilliance of his own idea consisted in noticing an analogy between such sentences and equations of the algebra he had formulated (which included the operation of linking concepts and negating a concept). In those equations on the one side we find such a combination of concepts (one of which may be negated), and on the other, one of the two 'magnitudes' expressed by the terms 'ens' and 'non-ens'. The above stating how algebra came to link logic to mathematics finely illustrates two fundamental laws in the history of human thought, one of which we may name (in accordance with Leibniz) the law of continuity (lex continui) and the other the law of the logic of development. Both of them motivate certain rules of historical research, which are as significant as the rule of the study of the past from the position of the present. Continuity consists in the fact that historical reality does not make leaps (natura non facit saltus, as Leibniz used to say), even though t h a t property of the process of history is hardly perceptible in view of our irresistible, and otherwise legitimate, inclination to divide human history into

3 8 Three: Formalized versus Intuitive Arguments - the Historical Background

periods (which suggests demarcation lines between epochs), and also in view of the recently fashionable tendency to stress revolutions in the history of science. And yet every idea moves forward by making steps so small t h a t they can be treated as being close to zero. T h e impression of leaps which we have is due to defects in our knowledge (caused by gaps in the sources or by insufficient assiduity in research) and also to abbreviations and simplifications t h a t are inevitable in view of the boundlessness of the continuum of history. T h a t law of continuity results in the research directive stating t h a t for every process t h a t shows gaps we have to seek sources and facts which would fill those gaps with intermediary links. Guided by this directive, we should see the algebra of Leibniz's logic in connection with the uninterrupted development of algebra t h a t was going on for some time (especially when it comes to algebraic notation), and also to the continuation by Leibniz of t h e ideas of scholastic logic, a continuation which consisted above all in grafting them onto the tree of algebra. The case of the algebra of logic also illustrates the second of the regularities mentioned above, namely the law of the inner logic of the development of t h o u g h t . Owing to a historical accident, Leibniz's texts were not published early enough to influence the development of the algebra of logic, but even though t h a t accident could have delayed the process of development, it could not stop it for good. T h a t was so because the very content of traditional logic included the nuclei of its further development toward the computational approach (e.g., the use of variables, and the fact t h a t the relation between the subject and the subjective complement lent itself to the extensional interpretation). And if an idea contains an important element, then t h a t fact has such a gravitational influence upon human minds t h a t finally someone notices that element and articulates the consequences inherent in it. This again leads us to t h e methodological conclusion t h a t in reconstructing the lines of development of a given idea, we have to follow them from the inner content of t h a t idea which will lead us to the appropriate sources, and on obtaining new source d a t a we have to compare them with the results of earlier deduction. T h a t idea guided the present author when, convinced of the computational nature of syllogistic, noticeable from the point of view of contemporary logic,

1. On how geometry and algebra influenced logic

39

he inscribed in the genealogical tree of logic not only Leibniz and Boole, but also the numerous algebraicians living in the 16th, 17th, 18th and 19th centuries, and even those Schoolmen who toiled to express sentences with the subject-and-subjective-complement structure in the form of sentences on existence or non-existence. 1.5. What lesson can we draw from the above story of the algebra of logic? Note that the historian's task is to trace the paths of thought in order to distinguish the highways from side roads, to watch turns and forks, and to find those paths which lead us farthest into the future. B u t to trace the direction in which human thought moves one must have — as in the case of tracing the movement of a body in space — at least two points. B y having one such point in the 17th century, and after having found the second in the 19th century, one could state that the destiny of logic led it toward the algebraic interpretation. T h a t observation gave rise to new questions: How far could logic have advanced by going in that direction? Did that direction determine the future of logic for good, or did it merely define one of the stages? If the latter was true, what was the next stage to be, and why did it take that form rather than another? We have to answer these questions briefly before we proceed to discuss the other factors of development which had their sources in the 17th century. T h e founders of the algebra of logic, that is G. W. Leibniz and G. Boole, and also A. De Morgan (1806-1871), E. Schröder (1841-1902), C. S. Peirce (1839-1914), and others, were convinced that the whole of logic can be contained in algebraic calculus. They were also aware of the fact that traditional logic, even after its algebraic reconstruction, lacked the means required to describe relations (for instance, it was impossible to render in its language even such a simple relational sentence as "for every number there is a number greater than it"). T h a t was why the next stage was to consist in adding the algebra of relations to the already existing algebra of sets (i.e., the analogue of the traditional theory). These new researches, carried out mainly by De Morgan, Schröder, and Peirce, gave rise to the theory of relations, which became an important and indispensable discipline in the border area between logic and set theory, but its language (combined with that of the algebra of sets) did not suffice to express mathematics in its entirety, either.

4 0 Three: Formalized versus Intuitive Arguments - the Historical Background

At this point one could ask the question: Why should the language of logic perform such an important function? The intention of 19th century algebraicians was merely to improve traditional logic from the mathematical point of view. The reply is that the late 19th century, regardless of the intentions of the various authors, witnessed the need, and at the same time the possibility, of constructing a universal symbolic language of mathematics in which well defined symbols and precise syntactic rules would replace the vocabulary and syntax drawn from natural language. A great step forward in that direction was made in the period from the 15th to the 17th century, which saw the spreading of the symbols of arithmetical operations and relations. But at least one more step remained to be made, namely the rendering of the logical form of sentences by means of an appropriate symbolism and related specialized logical syntax. Such a step was vigorously planned in the 17th century, when the idea was advanced of constructing a rigorous and universal language of science, ideographical in character, called characteristica rationis (reasoned or conceptual writing), characteristica universalis etc. But at that time linguistic and logical means (and perhaps also the appropriate philosophical approach) were not available. On the other hand, those algebraicians who were active in the mid-19th century, even though they worked out some such means, neither planned such a great work nor were in a position to do it, because that required going beyond Aristotelian logic and the algebraic schema. At the turn of the 19th century logic entered a new path by providing mathematics with an universal and rigorous symbolic language, which had also far-reaching (though not universal) applications in other disciplines and in everyday discourse. For that to happen the 17th century programme of an universal language had to be revived, which did take place owing to the publication for the first time, in 1840 (by J. E. Erdmann) of some logical writings of Leibniz which formulated that programme (e.g., De scientia universali seu calculo philosophico, De natura et usu scientiae generalis, Fundamenta calculi ratiocinatoria, Non inelegans specimen demonstrandi in abstractis). There were three founders of contemporary logic, independent of one another when it comes to ideas

1. On how geometry and algebra influenced logic

41

even though they had intensive contacts with one another: Gottlob Frege (1848-1925), Giuseppe Peano (1858-1932), and Bertrand Russell (1872-1970). All of them referred to Leibniz's ideas, and two of them believed themselves outright to be the executors of his testament by carrying out his program of universal language. One of them was Frege, who undoubtedly has first claim to this title. In 1879 he published a work which presented the whole of contemporary logic (i.e., the sentential calculus and the predicate calculus) under the title Begrijfsschrifl, meaning conceptual writing, and can in Latin be pointedly rendered by Leibniz's term 'characteristica rationis'. His reference to Leibniz's terminology was intentional, which is proved by the fact that two years later he wrote a paper entitled Booles rechnende Logik und die Begriffsschrift (which remained unpublished after having been rejected by editors of three leading periodicals), where he explicitly referred to his intention to carry out Leibniz's program. 1.6. We have discussed so far three approaches of logic to mathematics, each of them bringing new links between the two. In Subsec. 1.1 reference was made to two such links observable in the past. The contact with geometry consisted in the fact that a similar conception of deductive science was propagated by logical theory (Aristotle's Posterior Analytics), and put into practice by geometry. Either trend somehow influenced the other from the 4th century B.C. until the 17th century A . D . and beyond. The influence of logic upon geometry found a strong reflection in the oft-recurrent problem of the logical independence of Euclid's fifth axiom from the remaining ones, which reached an epoch-making peak in the birth of non-Euclidean geometries. The 17th century saw a new connection because, on the one hand, variable symbols came to be intentionally used in algebra, which in logic had been in practice since Aristotle, and on the other, the similarity which was borne out in this way promoted the algebraic interpretation of logic, achieved by Leibniz. Even though Leibniz's endeavours at first fell into oblivion, the fate of the idea itself proved independent from historical accidents, and the idea materialized again in the works of Boole and other 19th century algebraicians.

4 2 Three: Formalized versus Intuitive Arguments - the Historical Background

But not all of Leibniz's ideas were revived in t h e work of those later algebraicians, because the notion of a universal language of science, in which the algebraic calculus would be built in as an instrument of deduction, remained as it were hidden in shadow. It was brought to daylight again only by Frege and Peano, congenial readers of Leibniz's texts, but with the essential correction t h a t the idea of a universal language was to materialize not in the form of an algebraic calculus but in the form of the much more powerful predicate calculus. T h e whole story has been narrated so far on two temporal planes, t h a t of the 17th century and t h a t of our times, with the additional incursion, made in a perspectivic shortening, into antiquity, in accordance with the principle of explaining the past by t h e present and also by its own past. In moving further in that direction we have to mention one more, the most recent, rapprochement between logic and mathematics. This new connection is totally outside the perspective of the 17th century, which is why it should also be mentioned here, in order to show not only the proximity of t h a t epoch relative to ours but also the distance between the two. When new logical calculi developed at the t u r n of the 19th century, namely the predicate calculus (which overcame the limitations of algebra) and the sentential calculus (which may be treated as a certain interpretation of Boolean algebra), new vistas were opened to the logicians. On the one hand, it was now possible to develop and improve the calculi themselves by formulating ever new versions and by constructing other calculi t h a t could be superstructured upon the former (such as modal logics). On the other, since those calculi abounded in problems to be investigated, it was also possible to pose questions about the consistency of each of them, their completeness, the purposes they could serve, the relations among the axioms and concepts of a given system (the problem of independence), and finally the relations among the various systems (the interpretability of one of them in terms of another, t h e consistency of one of them on the assumption of the consistency of another, etc.). Precisely the same questions can be posed about mathematical theories. Among these, the arithmetic of natural numbers is of particular importance from the logical point of view (it was first axiomatized by Peano, and superstructured on

1. On how geometry and algebra influenced logic

43

logic and set theory by Frege and Principia Mathematica), because the remaining branches of mathematics can in a way be reduced to arithmetic: the 19th century witnessed the reduction to the latter of the arithmetic of real numbers, to which geometry had been reduced earlier owing to Descartes. 3 Thus, when at the world congress of mathematicians in 1900 D. Hilbert presented to the mathematical community the task of proving the consistency of mathematics (in view of the antinomies discovered at that time), it was known that it would suffice to concentrate on the problem of the consistency of the arithmetic of natural numbers using for that purpose the logical apparatus, both that which was already known at that time and that which was still to be created (in that field Hilbert himself together with his famous school proved incomparable). We also owe to Hilbert the term 'metamathematics', used to denote such studies. They were carried on intensively in the 1920's and 1930's, and the pride of place, next to Hilbert's school in Göttingen, went to the Austrian logician K . Gödel (especially when it comes to his most revealing proof of the nonprovability of the thesis on the consistency of arithmetic) and the Polish logician A. Tarski (noted especially for founding logical semantics, which opened to logic new and unexpected prospects). As metamathematics developed it was more and more penetrated by mathematical concepts and methods as indispensable instruments of research. They were in particular drawn from arithmetic (e.g., the use of recursive functions initiated by Gödel), algebra (e.g., Tarski's algebraic approach to non-classical logics), topology and set theory (which from its very inception was connected with logic). We also have to record the interpenetration of logic with mathematical linguistics and a u t o m a t a theory. In this way m e t a m a t h e m a t i c s , which was originated as the logical theory of mathematics, itself underwent mathematization. T h a t mathematical orientation of logic, which dates back to some important trends of the 17th century, should be considered in the present context as a contrastive background for the rhetorical orientation of logic. There is a challenging problem of how that 3 A reader interested in Peano's axioms will find them in Chapter Eight, Subsec. 4.2. as an example of the definitional role of axioms.

4 4 Three: Formalized versus Intuitive Arguments - the Historical Background

mathematically oriented logic could be used for rhetorical applications. 2. T h e R e n a i s s a n c e r e f o r m i s m a n d i n t u i t i o n i s m in logic 2 . 1 . The development of all ideas, and hence also that of logic in the 17th century, is immersed in the melting pot of general civilizational development, in the cultural ferment of the period. The present Section will be dedicated to that problem. The century which will be described here brings out with particular clarity the fact that the masterpieces of intellect grow from the roots of tradition but also from a lively dialogue with the milieu that is contemporaneous with their authors. And that milieu means not only congenial individuals, but also the audience consisting of readers, disciples, opponents, snobs, etc., in a word, the entire enlightened public. What that public believes, to what it aspires, what concepts it uses to grasp the real world, what it reads and what it discusses - all these are factors of vital significance for intellectual enquiries. Moreover, one has to bear in mind the relationships between science and the totality of culture, on the one hand, and the economic conditions of a given society, on the other. The flourishing of the arts and sciences coincides as a rule with a flourishing economy, as is indicated by the history of Miletus, the golden age of Athens, the Italian cities in the late Middle Ages, the Netherlands, England and France in the 17th century. Hence if the ordinary man participates in the creation of economic well-being, then he contributes to the creation of spiritual values as well. It is worth while in this connection to quote the following description of the epoch now under consideration: 4 Then, in the late sixteenth and early seventeenth centuries, had come the wave of commercial and financial expansion - companies, colonies, capitalism in textiles, capitalism in mining, capitalism in finance - on the crest of which the English commercial classes, in Calvin's day still held in leading-strings by conservative statesmen, had climbed to a position of dignity and affluence. 4

See Tawney [1937], p. 182.

2. The Renaissance reformism and intuitionism in logic

45

Economic expansion favoured advances in science among other things by the demand for t h e services of mathematicians: Simon Stevin (1548-1620) worked out a notation for decimal fractions, intended to facilitate the computation of bank interest in t h a t capital of finance which the Netherlands were at t h a t time for Europe. Pascal constructed a calculating machine for t h e needs of his f a t h e r who was a tax collector. T h e same expansion gave rise t o large numbers of skilled technicians, leading Leibniz, when leaving for Paris in 1672, to expect t h a t he would find a mechanic there who would help him construct t h e prototype of a calculating machine. The demarcation line between scientists and craftsmen was liquid at t h a t time, which is illustrated by t h e large number of craftsmen in the London Royal Society, the greatest collective authority in science in t h e 17th century. Another interesting confirmation of t h a t fact can be found in t h e correspondence between Leibniz and the young Christian Wolff. When the latter offered Leibniz his dissertation on cog wheels, Leibniz'in his kind reply suggested t h a t Wolff should consult German craftsman on t h a t m a t t e r . In this way a blacksmith or a locksmith as potential partner in t h e discussion with Wolff participated in the formation of scientific culture of their times. One more fact from t h a t period is worth mentioning in this connection: the deliberate coining of scientific terms in national languages so t h a t navigators and clockmakers, not versed in Latin, could s t u d y mathematics, astronomy, etc. It were the Dutch who excelled in t h a t : to this day they have terms dating from t h a t time which in no way resemble their Latin equivalents. The advances in mechanical and optic crafts, in printing and in other arts and their impact upon the advances in science (especially in the construction of research instruments) is a topic which deserves lengthier discussion, but I have to confine myself here to recommending it to t h e reader's attention. Let us look now at t h e other direction in t h a t relationship: how the spiritual climate of the period shaped t h e development of finances, commerce, travels, and inventions. T h e spirit of expansion had its religious motivation in the P u r i t a n doctrine as well as a secular motivation in the ideology of 'the kingdom of m a n ' , propagated as early as in the 15th century. 5 The finest advocates of 5

See Tawney [1937], Swiezawski [1974], p.149.

4 6 Three: Formalized versus Intuitive Arguments - the Historical Background

the latter included Francis Bacon (1564 - 1616) and also William Shakespeare (1561 - 1626) as the author of the dramatic fairy tale The Tempest, in which the wizard Prospero proves able to control the elements. In Book I of aphorisms, concerned with the explication of nature and the kingdom of man (included in Novum Organon, 1620), Bacon wrote emphatically that human reason must be fully freed and purified so that the same road should lead to the kingdom of man, based on sciences, as that which leads to the kingdom of Heaven, which no-one can enter unless one becomes like a little child (i.e., free from prejudices like a child; aphorism 68). The mastery of Nature, lost as a result of the original sin, is to be regained by technical inventions: no authority, no sect, no star has influenced humanity more than mechanical inventions have (aphorism 129). The author of these words was not only a well-known philosopher but also the Chancellor of the Kingdom of England, and hence a London craftsman could feel fully appreciated and encouraged to develop his ingenuity, the more so as from the stage of Shakespeare's theatre The Globe he could hear flattering ideas e.g., that an inventor resembles a wizard who controls Nature. There was one more heritage of the Renaissance which came of age fully in the 17th century, namely Pythagorean and Platonic philosophy, from its earliest beginnings most closely intertwined with mathematics (which was allegedly announced at the entrance to Plato's Academy). The Platonic trend was always present in European thought, especially from the time when it was reinforced by the ideas of Plotinus (204-270) and in that new form, called NeoPlatonism by historians, established contacts with young and vigorous Christendom, in both its orthodox version and that marked by the influence of gnosis. A significant example of that synthesis can be seen in the person of Proclos of Constantinople (410-485), one of the widest known commentators of Euclid and a Platonizing theologian, to whose ideas Kepler used to refer. The first Christian writers and Church fathers were as a rule Neo-Platonians. This applies also to the greatest of them, St. Augustine of Hippo (354430), whose ideas were truly reincarnated in the 17th century in the work of Descartes, Pascal, and the milieu of Port Royal, and in the late 19th century came to the rescue of G. Cantor, the author

2. The Renaissance reformism and intuitionism in logic

47

of the theory of infinite sets, in his clash with the philosophy of mathematics t h a t followed the Aristotelian approach to infinity. The Pythagorean and Platonic trend particularly penetrated the consciousness of the enlightened public from the 15th century, which witnessed the renaissance of Platonism in Italy, especially in Florence, until t h e 17th century, when t h a t trend reached Cambridge, and in t h a t way influenced Newton. T h e spread of Platonism at t h a t time can be traced in two ways. One can follow its more popular version, manifested in the interest in the mysticism of numbers, the cabala, astrology, and t h e accompanying emergence of associations concerned with those matters. Another form of t h a t popular version can be seen in the presence of certain Platonic elements in the teaching of mathematics as the methodological pattern of all science. By following t h a t path we reach the mass substratum of an intellectual current t h a t gives t h a t current its material strength and spread. The other p a t h on which we can trace the reach of a given intellectual current is t h e number of outstanding authors who represent it. Here are some prominent personalities of t h a t time connected with the propagation of the Platonic vision of the world: Marsilio Ficino (1433-1499), Nicolaus of Cusa (1401-1464), Leonardo da Vinci (1452-1519), Nicolaus Copernicus (1473-1543), Galileo Galilei (1564-1642), Johannes Kepler (1571-1630). This wave of Platonism had its share in preparing the next one, which came in the 17th century. The latter less frequently referred to Plato himself, but it was nevertheless permeated by Platonic views on the possibility of and need for the mathematization of all knowledge and on the a priori nature of cognition. We thus see in the 17th century two vast currents, one of them linked to technology and economics and ideologically based on the slogan of ' t h e kingdom of man', and the other permeated by Platonic metaphysical speculation. Both influenced logic by postulating in agreement t h a t logic should give t h e human mind an instrument whereby it could arrive at the t r u t h . While they had one and the same goal in view they disagreed as to the method of reaching the t r u t h : the former staked on induction, whereas the latter, preoccupied with mathematics in the Platonic manner,

4 8 Three: Formalized versus Intuitive Arguments - the Historical Background

postulated a purely deductive method (more geometrico). Subsections 2.2 and 2.3 will be dedicated to the successive presentation of the former and the latter. 2 . 2 . T h e advances in logic in the 17th century, the authority which t h a t discipline enjoyed and the hopes it had aroused originated from the faith in the power of reason, and also the faith t h a t this power can be increased. The expansion of logic was to be achieved by its development and popularization after the appropriate reform which it had to undergo: the logic inherited from Aristotle and the Middle Ages, the advocates of the reform claimed, would not prove equal to the task. Before we tell the story of the reformers we have to note, in order to obtain a proper background, t h a t the teaching of logic was marked by the conservative trend observable in some university towns (Helmstedt, Wittenberg, Glessen, Königsberg). T h a t trend developed in the 16th century mainly owing to Philip Melanchton (1497-1560); being associated with this co-founder of Protestantism it was called a variety of Protestant logic. (There was also a reformist trend, which originated from Peter Ramus, who enjoyed great authority among the Protestants as the martyr who lost his life on St. Bartholomaeus Night.) The conservative trend reverted to Aristotle, and hence to the source, in the spirit of the Renaissance and the Reformation, disregarding the novelties contributed by scholastic logic. But this was not trend which dominated in the 17th century. If it must be mentioned, then only as a party to the polemics and as the contrastive background against which we can more clearly see the reformist tendencies. 6 The second contrastive background must be seen in the preceding period. The idea of a reform of logic developed in the 16th century; the 15th century was still busy with commenting the classics, namely Summulae logicales of Petrus Hispanus (1220-1277; as 6

A detailed factographical discussion of logic in the period under consideration is given by Arndt [1965]. His narrative illustrates very well the style (mentioned at the beginning of this essay) of pursuing the history of science by abstracting from the present-day state of a given discipline (this may, perhaps, be explained by the function of his text, which is the editorial introduction to Wolff's Logic).

2. The Renaissance reformism and intuitionism in logic

49

Pope he was known as John XXI), and later achievements, such as those dating from t h e 14th century. A special merit for t h a t continuation went to Paul of Venice (d. 1429), whose Logica Parva (1428) continued to be t a u g h t at many universities for two centuries to come. A certain logical idea of Paul of Venice in his Logica Magna found its way to the writings of Leibniz and came to play a certain role in the development of extensional logic. 7 The history of the reform of logic begins with Peter Ramus (Pierre de la Ramee, 1515-1572), remembered as a vehement opponent of Aristotle whom he blamed for being u n n a t u r a l in his approach to logic. Nevertheless, he took over a great deal from Aristotle, namely the theory of definitions, the theory of judgement, and the theory of inference (which is perfectly natural because a new logic is not created in a day, and in this way even t h e revolutionaries confirm the law of continuity). Two things made him differ essentially from Aristotle, but one of them was in t h e sphere of programs (in which one can relatively easily become an innovator), and the other has its source in age-old tradition, but not the Aristotelian. In his program Ramus included t h e requirement t h a t logic be adjusted to natural h u m a n thinking, without t h a t artificial abstractedness for which he blamed Aristotle. His followers took up those slogans, and for the two centuries t h a t followed we find in the title of numerous logical compendia various psychological terms connected with the conception of logic as the science of living human thinking and not of any abstract entities. T h e best known titles of this kind include La logique ou l'art de penser by A. Arnould and P. Nicole (first published in 1662, it had had ten French and as many Latin impressions by 1736); Medicina mentis sive artis inveniendi praecepta generalia by E. W . von Tschirnhaus, 1687 (several other courses in logic had t h e same title presenting logic as the way of healing one's mind); Introductio ad philosophiam aulicam, seu ... libri de prudentia cogitandi et ratiocinandi by Ch. Thomasius, 1688 ('philosophia aulica' meant court philosophy, which required — as claimed the second part of the title — prudence in thinking and reasoning). T h e same may be said about the title of the 7

See Nuchelmans [1983], Sec. 11. 3. 1.

5 0 Three: Formalized versus Intuitive Arguments - the Historical Background

work written by Ch. Wolff (1679-1754), a philosopher who at first was an a d h e r e n t of t h e Cartesian school in logic, and hence an opp o n e n t of syllogistic, b u t later, under Leibniz's influence, became an advocate of syllogistic. He was also t h e principal c o m m e n t a t o r of Leibniz's philosophy in Germany. And here is t h e characteristic title of his book: Vernünftige Gedanken von den Kräften des menschlichen Verstandes und ihrem richtigen Gebrauche in Erkenntnis der Wahrheit, 1713 (Reasonable T h o u g h t s a b o u t the Powers of H u m a n U n d e r s t a n d i n g a n d their P r o p e r Use in t h e Cognition of t h e T r u t h ) ; t h e book had had 14 German impressions by 1754, and also four impressions of a Latin version and two impressions of a French version. Even t h o u g h in its content Wolff's work did not belong t o t h e psychological t r e n d , it was a d j u s t e d to it by its title, which shows t h e strength of t h e t r e n d represented by R a m u s and t e r m i n a t e d only in t h e late 19th century by t h e concentrated a t t a c k against psychologism in logic from the position of t h e mathematically-oriented philosophy of logic (G. Frege, E. Husserl, and others). A n o t h e r novelty relative to Aristotle's ideas consisted in t h e division of logic into the science of j u d g e m e n t s a n d inference and t h e science of making inventions, in which much space was dedicated by R a m u s t o definitions (the connection between t h e method of making inventions and t h e formulation of definitions will be fully seen in Leibniz). It was not an absolute novelty, because such a division was introduced by t h e Stoics more t h a n a dozen centuries earlier and was t r a n s m i t t e d t o t h e Middle Ages by Boethius (480524), b u t t u r n i n g it into t h e principal slogan which fitted with the aspirations of t h e epoch was t h e work of R a m u s . W h e n it comes to Francis Bacon, not only one half of logic but t h e whole of it was to consist, in his opinion, in t h e art of making discoveries which would pave t h e way for t h e kingdom of m a n . Bacon designed a logic of induction, which was totally to replace t h e existing logic of deduction. T h e characteristic feature of deduction is t h a t t h e conclusion does not contribute any information t h a t would be not contained in t h e premisses: it has as much information as t h e premisses have or has less information t h a n t h e premisses convey. He was not t h e only one t o attack logic in this way, because a whole chorus of critics, among whom t h e voice of

2. The Renaissance reformism and intuitionism in logic

51

Descartes sounded loudest, blamed syllogistic — t h a t is t h e theory of deduction t h a t was universally accepted at t h a t time — for not contributing a n y t h i n g new t o our knowledge. B u t while Descartes and his followers wanted to replace syllogism by a n o t h e r deduction, modelled on t h e experience of m a t h e m a t i c a l thinking, whose creative character would not be questionable, B a c o n , guided by an empiricist ideology, failed t o perceive such an alternative solution. Hence, if logic was directly t o serve t h e expansion of h u m a n knowledge, he had to stake everything on t h e inductive m e t h o d . B u t at t h a t infantile stage of t h e logic of induction people failed t o realize t h a t for an increase of information one has to pay with a reduction of certainty, or, to p u t it more precisely, a reduction of probability which has certainty as its u p p e r limit. This is so because a general law is intended t o be a conclusion d r a w n f r o m observations t o infinitely m a n y possible cases and therefore tells us more t h a n t h e observations reflected in t h e premisses, which always cover finitely many cases. B u t it is uncertainty which is t h e price paid for such an increment of knowledge, because while N a t u r e may to a certain moment confirm t h a t general conclusion, there are no reasons t o be sure t h a t it will confirm it in each subsequent case. If this is so, then t h e logic of induction is not any alternative solution to t h e logic of deduction but can at most be its complement (which, by t h e way, t o this day is still in a stage of planning, rather t h a n in t h a t of achievement). Bacon's ideas, even t h o u g h they finely expressed t h e spirit of t h e epoch, did not pave any new ways for logic despite t h e intention of their a u t h o r ( t h a t f o u n d reflection even in t h e title of his work, Novum Organon, which referred t o Organon, t h e title of Aristotle's logical writings). T h e greatness which logic was destined to a t t a i n a n d which it did attain in our century, was reached by t h e old Aristotelian road when it merged with t h e p a t h along which m a t h e m a t i c s was developing. Those two p a t h s came closer t o one a n o t h e r for t h e first time in t h e 17th century, and t h e process was d u e to Leibniz. T h u s it was he and not Bacon who became t h e forerunner of f u t u r e logic. A n d those thinkers who, while underestimating Aristotle, did not commit the mistake of u n d e r e s t i m a t i n g deduction, also proved t o have come closer to w h a t was a h e a d . T h a t process will be discussed in t h e next subsection.

5 2 Three: Formalized versus Intuitive Arguments - the Historical Background

2 . 3 . T h e problem of how logic should be is not merely a m a t t e r of logical theory. It deeply p e n e t r a t e s t h e philosophical vision of t h e world a n d t h e mind. Bacon's design for a reform of logic originated from t h e empiricist conception of cognition which ascribes to t h e h u m a n mind t h e role of a passive receiver: it is like a screen on which N a t u r e casts t h e image of itself t h r o u g h t h e objective of m a n ' s sense organs, and if t h e mind is active in any respect then only by posing questions to N a t u r e . Such empiricism is opposed by apriorism, also termed rationalism (in one of t h e meanings o f t h a t word), which states t h a t there are t r u t h s which man comes to know before all sensory experience; they are given to him as it were in advance (Latin a priori). In t h a t sense, reason (Latin ratio) is independent of experience. According t o t h e rationalists, t h e essence of cognition consists in t h e realization by t h e h u m a n mind of those t r u t h s , given it a priori, which are general and necessary, and in deducing from t h e m answers t o questions which interest us. Premises drawn from sensory experience and experimental d a t a may be helpful in t h a t process of deduction, but they never suffice to lay t h e foundations of our knowledge. Hence logic as an i n s t r u m e n t of cognition, interpreted in t h e rationalist sense, must be a logic of deduction which not only serves t h e presentation and ordering of t h e t r u t h s t h a t are already known (in t h e 17th century, scholastic logic was blamed for being confined to t h a t role), b u t it can also lead one to discoveries a n d reveal t r u t h s t h a t had been unknown earlier. This is best proved by t h e role of deductive reasoning in m a t h e m a t i c s , which offers irrefutable examples of t h e existence of a priori t r u t h s . T h e fertility of deduction in t h e sphere of m a t h e m a t i c s fascinated o u t s t a n d i n g minds in t h e 17th century, for it was imposing as compared with t h e sterile, it was believed, p e d a n t r y t h a t marked t h e theory and praxis of deduction in its scholastic version. T h a t was why t h e new programme of deductive logic (the adjective 'deductive' will henceforth be d r o p p e d ) postulated t h e study of those m e t h o d s of thinking which had been intuitively used by mathematicians of all times; those m e t h o d s were t o be formulated in plain u n d clear rules and would t h u s yield a truly useful logic. It was on t h a t p a t h , it was expected, t h a t t h e art of creative thinking consisting in arriving at new t r u t h s ( a r s inveniendi) would be propagated.

2. The Renaissance reformism and intuitionism in logic

53

Before we examine the implementation of that programme we have to pay attention to the peculiar difficulty that is inherent in it. Deduction consists in transferring to the conclusion the content, that is information, already contained in the premises, hence in the conclusion there can be at most as much information as there is in the premisses but no more. On the contrary, a discovery consists in arriving at new information one did not have previously. This is why (deductive) logic as an instrument for making discoveries may appear unthinkable, something which is self-contradictory. But whoever raised such an objection against the concept of creative deduction would be guided by a conception of the mind that differed from that which marked the 17th century reformers of logic. The opinion characteristic of those times had its roots in Platonism, which dominated enlightened minds in the 16th and the 17th century from Florence to Cambridge (two famous strongholds of Platonism). In the Platonic interpretation, apriorism assumed a certain conception of what we would today call subconsciousness. In Plato himself that conception was associated with the theory of anamnesis, which meant cognition as reminiscence of truth originating from the pre-existential phase of the mind. Hence a discovery meant nothing else than bringing to daylight a truth that was formerly concealed. In other words, it could metaphorically be described as the shaping of that truth as it were from a substance which was given earlier, in the similar way as a bud and a fruit are shaped from the substance of a given plant. That theory was linked to, and brought in fuller relief by, the characteristically Platonic conception of the teacher as someone who helps his disciples to bring to light what was in the shade or a penumbra. Socrates as described in Platonic dialogues defined that activity by metaphorically comparing it to obstetric art, and St. Augustine coined the term illuminatio to name that process whereby human thought comes to light. This gave rise to the conception of dialogue as illuminating interactions, and also the conception of dialectic as the theory and art of dialogue which has deduction as its main instrument (which can clearly be seen in Plato's dialogues). Dialectic later came to be termed logic. Thus from the very inception of Platonism logic was to serve the

5 4 Three: Formalized versus Intuitive Arguments - the Historical Background

process of causing the truths that are innate to the mind to pass from the state of latency to that of clear presence. Thus it is not new contents that we owe to logical deduction but a new way in which they exist in human minds. And the fundamental question of the new program of logic was as to what rules are to be used in decoding the truths encoded in the back of our minds. 2.4. The classical answer to this question was given by Rene Descartes (1596-1650), followed by a group of authors collectively called the Cartesian school. It included Baruch Spinoza ( 1 6 3 2 emendatione (on 1677) with an opuscle Tractatus de intellectus the improvement of one's mind), E. W . von Tschirnhaus ( 1 6 5 1 1708) as the author of Medicina mentis (see 2.2 above), and also the authors of the Logic of Port Royal, whose popularity greatly contributed to the spreading of the Cartesian methodology. Descartes's famous method is expounded in his two works whose very titles express the idea of the art of making inventions, ars inveniendi. One of them is the Discours de la methode pour bien conduire sa raison et chercher les verites dans les sciences, published in Leyden in 1637 (the climate in the Netherlands was more favourable to new ideas than the vicinity of the Sorbonne). That good guidance of one's mind, mentioned in the title of the Discours, was presented in even greater detail in the treatise Regulae ad directionem ingenii (rules for the guidance of one's mind), published posthumously in Amsterdam in 1701. The latter work offers a more detailed insight into the Cartesian method (it has 21 rules while the Discours has merely four), but the former was the text which continued to influence the community of scholars throughout the first half of the 17th century, and this is why we shall concentrate on it. Here are rules informing the readers how to guide one's mind to arrive at the truth. The first of them states the famous principle of Cartesian doubt. Let these rules be distinguished by numbers, each number preceded by ' D M ' for Discours de la methode. D M - 1 . Not to accept anything as true before it comes to be known as such with all self-evidence; this is to say, to avoid haste and prejudice and not to cover by one's judgement anything except for that which presents itself to the mind so clearly and explicitly that there is no reason to doubt it.

2. The Renaissance reformism and intuitionism in logic

55

DM-2. To divide every problem u n d e r consideration into as m a n y particles as possible a n d is required for its b e t t e r solution. DM-3. To guide one's t h o u g h t from t h e simplest and easiest objects in order t o rise, later on, slowly t o t h e cognition of more complex ones; in doing so one has to assume regular connections even among those which do not form a n a t u r a l series. DM-4. To make everywhere precise specifications and general reviews so as t o be sure t h a t nothing has been o m i t t e d . Characteristically enough, t h e Discours in its first edition appeared not as a separate item b u t as t h e annex t o t h e opus whose role in t h e history of m a t h e m a t i c s and philosophy is c o m p a r a b l e to t h a t of Euclid's Elements, namely Geometrie. A n a l y t i c g e o m e t r y , expounded there for t h e first time, combined two branches of m a t h ematics which had previously been developed separately, namely geometry a n d t h e arithmetic of real n u m b e r s with accompanying algebra. T h a t work used t h e concept of function in an advanced way and essentially contributed to t h e invention by Leibniz a n d by Newton (independently of one a n o t h e r ) of m a t h e m a t i c a l analysis (differential and integral calculus), at t h a t time called infinitesimal calculus. T h a t gigantic step toward t h e integration of m a t h e m a t ics (the next comparable one was only t h e origin of set theory in t h e late 19th century) aroused such an a d m i r a t i o n and such hopes in t h e contemporaries t h a t t h e second half of t h e 17th century was animated by t h e vision of a f u t u r e science, t e r m e d mathesis universalis, which would cover t h e whole of knowledge, philosophy included, in t h e form of a single m a t h e m a t i z e d theory. W h a t was t h e a d v a n t a g e of t h e combination in one publication of an exposition of geometry and reflection on m e t h o d ? Now t h e exposition of geometry itself could be used, from t h e m e t h o d ological point of view, as t h e model for t h e deduction of theorems from initial self-evident facts formulated in axioms, but it did not provide any advice u p o n how one is t o arrive at t h e self-evident s t a t e m e n t s needed as premises of proofs. T h a t required a description of t h e a p p r o p r i a t e operations performed by t h e h u m a n mind, and just such a description was provided by t h e Discours de la methode. T h e problem of finding self-evident premises needed to prove theorems e x p a n d e d in t h e Cartesian school a n d its milieu t o a

5 6 Three: Formalized versus Intuitive Arguments - the Historical Background

vast set of problems pertaining to the distinction between the deduction of consequences from theorems already known, which was then called the synthetic m e t h o d , and the search for premises to support a given theorem, called the analytic m e t h o d . Here is the statement of t h a t distinction in the Logic of Port Royal, whose authors refer in a note to a manuscript by Descartes, to which they were given access. Method can be described generally as the art of a good arrangement of a series of thoughts in arguments used either for the discovery of truth, if it is not known, or for its demonstration to others, if it is already known. There are thus two methods. One of them is the method of discovering the truth, termed analysis or the method of solution; it might also be termed the method of invention. The other is the method of making the truth which has already been found accessible to others. It is termed synthesis or the method of composition, and it might also be termed the method of exposition. (Book IX, Chap. 2). T h e terms m e t h o d of solution and m e t h o d of composition which occur in t h a t description were to render a similar distinction existing already in scholastic logic and earlier in Euclid and Pappus. T h e theory of the analytic method, t h a t is of the ars inveniendi, on which the efforts of that generation of philosophers were focussed, evolved in two opposite directions, the psychological and the formalistic. T h e psychological trend, which more and more concentrated on recommendations pertaining to the attitudes and behaviour of the mind (with the accompanying moralizing tendency) was particularly strongly voiced in the writings of Christian Thomasius ( 1 6 6 6 - 1 7 2 8 ) and in those of his adherents, grouped in the then recently ( 1 6 4 9 ) founded university in Halle. Thomasius, who followed Ramus in that respect, shared with the latter the (so to speak) practical orientation in logic. His disciples and adherents were strongly influenced by Protestant pietism, which at that time had its centre precisely in Halle, and were thus influenced by the sentimentalist movement with its inclinations to irrationalism. No wonder, therefore, that ultimately the idea of the methodological identity of mathematics and philosophy, and hence the idea of mathesis universalis, was abandoned by them. This ends, in the late 17th century, one of the ramifications in t h e evolution of logic. B u t the 17th century also witnessed the emergence of an antipsychological trend, which announced the birth of mathematical

3. Leibniz on the mechanization of arguments

57

logic and made first steps in t h a t direction. T h a t trend, in distinction from the former, will here be termed formalistic, even though t h a t name is not fully suitable. T h a t trend, too, raised the problem of the art of discovery, but freed it from psychologism, owing to the the idea of algorithm. Those plans and ideas, even though they did not emerge from a historical vacuum and, as usual, marked t h e fruition of the work of numerous generations, are — in historical transmission — associated with the name of Leibniz. 3. Leibniz o n t h e m e c h a n i z a t i o n o f a r g u m e n t s 3.1. In the present section we shall discuss, even more clearly than so far, how logic was in the past in the light of what it is today. This perspective does not predetermine the choice of the instruments of interpretation. From the sets of concepts which we have at our disposal we shall select t h a t which includes the concept of algorithm as a recipe for the mechanical performance of tasks of a given kind, this being the class of concepts which will best serve us in looking at the future of logic in its civilizational context. Before we proceed to discuss t h a t concept (see Subsec. 3.2 below) we have to concern ourselves with t h a t of logical form as the one which played an essential role in the development of logical algorithms. The concept of logical form has developed from the observation t h a t certain sentences, e.g., in English, are accepted as true solely on the basis of their structure, with disregard of empirical and any other extralinguistic data. Such are, for instance, the following sentences. It rains or it is not the case that it rains. If it thunders, then it thunders. It is not the case t h a t it (both) dawns and does not dawn. If it dawns, then the cock crows, and hence if the cock does not crow, then it does not dawn. If a certain shoemaker is a lover of music, then a certain lover of music is a shoemaker. 2 = 2.

The law is the law. It is said about such sentences t h a t they are accepted solely on the strength of their logical form, and they are accordingly termed

5 8 Three: Formalized versus Intuitive Arguments - the Historical Background

logical t r u t h s . In order t o define t h e concept of logical t r u t h in a general m a n n e r , a n d hence also t h e concept of logical f o r m , one makes use of t h e p a r t i t i o n all of expressions of a given language i n t o two classes: one of t h e m includes t h e expressions called logical terms, such as ' a n d ' , 'or', ' n o t ' , 'if .. . t h e n . . . ' , 'some', 'every', ' = ' , a n d t h e like, while t h e o t h e r includes all t h e remaining expressions, called extralogical. Owing to this distinction, t h e intuition inherent in t h e formulation t h a t logical t r u t h s are accepted solely on t h e s t r e n g t h of their s t r u c t u r e , can b e m a d e more precise in t h e following way. A sentence is called a logical t r u t h if it remains t r u e a f t e r t h e replacement in it of extralogical expressions by any expressions ( d r a w n f r o m t h e same g r a m m a t i c a l category) with t h e proviso t h a t such a replacement is m a d e consistently, which is to say t h a t on replacing A by Β in one place we d o so in all those (and only those) places in which A occurs in t h a t sentence. T h u s t h e sentence 'It rains or it is not t r u e t h a t it rains' is a logical t r u t h because we o b t a i n a t r u e sentence if we replace in it t h e sentence 'it rains' by a n y o t h e r sentence, for instance ' T h e world was created in six d a y s ' . Such replaceability is of great significance for t h e recording of logical t r u t h s . Since for their t r u t h it is quite indifferent w h a t extralogical expressions occur in t h e m , it is b o t h possible a n d very useful t o replace these or o t h e r extralogical expressions by letters which in a sense indicate t h e blanks t o be filled. T h o s e letters should a t t h e same time indicate t h e g r a m m a t i c a l category of expression which may be used to fill t h e place occupied by a given letter in t h e schema. T h i s is why special conventions are a d o p t e d in t h a t m a t t e r ; for instance, t h e letters i x \ 'y', 'z' indicate t h e place of p r o p e r names, t h e letters 'ρ', 'ς', ' r ' , t h e places of sentences, etc. In this way we o b t a i n s c h e m a t a of sentences which are always t r u e , for instance t h e following. ρ or it is n o t t r u e t h a t p. If q, t h e n q\ χ = x\

If p, t h e n g' hence 'if n o t - q , t h e n not-p' (where ' n o t ' is synonym o u s with 'it is not t r u e t h a t ' ) .

3. Leibniz on the mechanization of arguments

59

Logical truths expressed in such a schematic form are called laws of logic; they owe their general validity, which is typical of laws, precisely to their schematic character. The fact that logical truths are truths solely owing to their form paves the way, as we shall see, for the construction of algorithms of two kinds: those which give us a mechanical method of deciding whether a sentence is a logical truth, and those which give us a mechanical method of deciding whether a sequence of sentences claimed to be a so-called formalized proof is in fact a proof of that kind. The invention of algorithms which could serve those purposes was one of the great plans of Gottfried Wilhelm Leibniz ( 1 6 4 6 1716), who expected that the invention would result in an epochmaking progress of human knowledge. His expectations came true only in our century, and that was accompanied by the discovery of the fact that they can come true only to a certain extent, because it is not possible to algorithmize the whole of mathematics ( K . Gödel's famous result proving the incompleteness of arithmetic), and hence that it is a fortiori impossible to algorithmize the whole of our knowledge. But even those partial solutions are important enough to treat the concept of logical form (which paved the way for them) as one of the greatest attainments of the human mind. Here are several data from the history of that discovery, or rather the process of discovering, because here, too, we deal with a process which continued for centuries. The first flash of the intuition concerned with logical form must be seen in indirect reasoning, that is reasoning by reduction to absurdity, which is to be found in the enquiries of early Greek mathematicians, and which is particularly frequent in Plato's dialogues (where as a rule it is used by Socrates, who treated it as main tool of his dialectic). An indirect proof starts from the assumption that the statement to be proved, say C (for conclusion) is false. If this assumption (of the falsity of C ) results in a contradiction, this means that it is false, than the proposition C itself has to be true. The very fact of making use of such proofs points to the sensing of logical form, because in an indirect proof two desirable properties of our thinking are separated clearly from one another as if in a prism: the factual correctness, that is truth, of the premisses

6 0 Three: Formalized versus Intuitive Arguments - the Historical Background

a n d conclusions, and t h e formal correctness, t h a t is agreement of inference with t h e laws of logic. Aristotle of Stagira (384-322 B.C.), t o whom goes t h e unquestionable credit of creating t h e first system of formal logic, often m a d e skilful use of indirect proofs, but w h a t m a d e him t h e founder of logic was t h e fact t h a t in formulating the laws of inference he used for t h a t purpose letters in t h e role of n a m e variables. This is not t o say t h a t he already had a clearly shaped concept of logical form, b u t it is beyond d o u b t t h a t his epoch-making invention in t h e sphere of notation guided t h e t h o u g h t s of all his followers toward t h e idea of t h a t form. Logicians from the philosophical school of t h e Stoics (active for five centuries, beginning with t h e 5th century B.C.) were undoubtedly aware of t h e formal n a t u r e of logic obtained as a result of t h e use of variable symbols. They used numbers as sentential variables a n d t h u s created t h e nucleus of t h e logical theory which we now call t h e sentential calculus, while Aristotle's syllogistic was a calculus of names. T h e wealth of ancient logic, including t h e teaching of the Stoics, was t r a n s m i t t e d to later generations by Boethius (480-524 or 526), a R o m a n statesman from t h e times of Theodoricus. Logicians active in t h e late Middle Ages m a d e a successive i m p o r t a n t step forward by linking t h e logical form to t h e functioning of those expressions which were termed syncategorematic a n d analysed by t h e m in detail. T h e realization of t h e fact t h a t syncategorematic expressions determine the logical form of a sentence is particularly clear in t h e case of J o h n Buridan (rect o r of Paris university in t h e first half of t h e 14th century). We find in B u r i d a n a conception of logical form which resembles t h e c o n t e m p o r a r y one and includes t h e distinction between categorem a t i c terms a n d syncategorematic ones, t h e latter being those which determine the logical form of a given sentence. He also took into consideration t h e impact of t h e categorematic t e r m s upon t h e logical form; for instance, t h e form of a sentence lA is A' differs from t h e form iA is B\ Such a concept of logical form was taken over from Schoolmen by Leibniz, who t u r n e d it into t h e focal point of his conception of logic. Yet he remained isolated in t h a t respect, p e r h a p s in view of t h e very strong antiformalistic and antischolastic tendencies which e m a n a t e d from t h e milieu of b o t h R a m u s and

3. Leibniz on the mechanization of arguments

61

Descartes. T h a t concept was rediscovered only in t h e mid-19th century by A u g u s t u s De Morgan, one of t h e f o u n d e r s of t h e algeb r a of logic. 8 T h u s Leibniz had t o play t h e historical role of being t h e last Schoolman a n d at t h e same time t h e first logician of our epoch. 3.2. Cartesian methodology called for a direct contact of t h e mind with t h e object of cognition. Leibniz's methodology pointed t o t h e indispensability of a n o t h e r way of thinking, in which t h e contact of t h e mind with t h e o b j e c t of cognition is not direct, b u t takes place t h r o u g h t h e intermediary of signs as t h o s e i n s t r u m e n t s of thinking which represent t h e object assigned t o t h e m . O n e of t h e ways of using signs as i n s t r u m e n t s consists in o u r grasping with our t h o u g h t t h e sign instead of t h e object (its d e s i g n a t u m ) . T h a t intermediate way of thinking a b o u t things, in which t h e t h i n g itself is not present in t h e sphere of our a t t e n t i o n , was called blind thinking (caeca cogitatio) by Leibniz. 9 O p e r a t i o n s on large numbers are its simple example. As long as we remain in t h e sphere of small numbers, for instance when multiplying t h r e e by two, we still can be guided by some image of t h e o b j e c t itself; e.g., we imagine an a r b i t r a r y b u t fixed triple of things a n d join to it one triple more. Such an operation can be performed physically or mentally even if we do not have at our disposal symbols of t h e numerical system. It is otherwise when we have t o multiply numbers of a dozen or so digits each. In such a case we are deprived of t h a t visual contact with t h e m and have to rely on sequences of figures which represent t h e m . Such sequences are physical objects assigned to a b s t r a c t o b j e c t s , namely n u m b e r s , a n d operations on figures are unambiguously assigned t o t h e corresponding operations on numbers. For instance, j u x t a p o s i t i o n , t h a t is writing one after a n o t h e r t h e symbols '2', '·', '3', is an operation on signs, and t h e corresponding operation on n u m b e r s consists in multiplying two by three. 8

See Bochenski 1956, Sec. 42.01. This notion is here presented in the 17th century environment. Its topicality in the context of Artificial Intelligence is discussed in Chapter Seven, Section 7.2. 9

6 2 Three: Formalized versus Intuitive Arguments - the Historical Background

If we add to this description of blind thinking one element more, t h e n we obtain t h e concept of algorithm, which played t h e key role in Leibniz's logic. T h e term 'algorithm' itself was not used by him in its present-day meaning. T h a t role was played by t h e term calculus, i.e., c o m p u t a t i o n , or by metaphorical expressions like caeca cogitatio and filum cogitationis (thread of thinking). T h e latter t e r m referred to Ariadna's t h r e a d , t h a t is to t h e tool which enabled Thesaeus to move a b o u t in t h e L a b y r i n t h and to find his way o u t , t h a t is t o solve his problem without any thinking, and w i t h o u t t h e knowledge of t h e Labyrinth itself. T h e description of t h e algorithm is to be completed by t h e s t a t e m e n t t h a t it is a procedure consisting of a series of steps such t h a t every step unambiguously determines t h e next one in accordance with precisely defined rules. Those rules determine t h e successive steps in t h e procedure under consideration by referring solely to physical properties (such as size and s h a p e ) of objects, e.g., t h e shape of figures, and t h e positions they occupy. One can carry out t h e algorithm of addition in columns without having t h e slightest idea of n u m b e r s a n d availing oneself only of t h e rule which states t h a t if t h e sign '2' is written under '3', then one has t o write '5' in t h e next row in t h e same column, etc., and of t h e rule t h a t one has to s t a r t such operations from t h e rightmost column and after filling it t o pass to t h e next column on t h e left until t h e very end ( t h a t is t o say, until one reaches t h e leftmost symbol, which is repeated in t h e last row if there are e m p t y places above or under it). On joining t h e self-evident condition t h a t t h e n u m b e r of t h e steps must be finite we o b t a i n t h e full description of algorithm, which can be s u m m e d u p as follows. An algorithm is a recipe for a procedure pertaining to a definite class of tasks (e.g., addition of sequences of figures), which in each case guarantees t h e obtaining of t h e correct result after t h e performance of a finite n u m b e r of operations, and t h a t owing t o t h e fact t h a t the objects of such operations (e.g., sequences of figures) are reliably identifiable owing t o their perceivable physical properties (e.g., shape, position). W h e n t h e idea of algorithm came to be mentally worked out by Leibniz, it had had a picturesque past which is a good example of t h e harmonious work of history and t h e cooperation of

3. Leibniz on the mechanization of arguments

63

the various civilizations. The idea and the term are associated with Al-Khwarizmi, the Arabian mathematician in the 9th century, who systematized computational methods in his manual by means of sequences of instructions. When those methods became popularized at first in Italy (Leonardo Pisano, 1180-1240) and later in Germany (Adam Riese, 1492-1559) and in other countries, people started speaking about 'computations in Al-Khwarizmi's sense', that is, in accordance with such detailed instructions. AlKhwarizmi's manual was based on a Hindu astronomical work dating from the 7th century A.D., which a certain Hindu brought to Baghdad in 773. It included the decimal system, invented in India ca. A.D. 400, which offered an opportunity for such mechanical computations as those referred to above (which could be subsumed under 'blind thinking'). 3.3. In t h a t melting pot of ideas which was Leibniz's mind, the idea of mechanical computations merged with his looking at logical form through the prism of algebra as it was known at t h a t time, and also with two other ideas, which he had received from the past but transformed and combined in his own way. One of them, propagated especially by Joachim Jungius of Hamburg (1587-1657), consisted in the opinion t h a t the essence of thinking consists in the analysis of concepts, t h a t is in decomposing compound concepts into simpler ones, until one arrives at the simplest ones possible. Those simple and non-decomposable elements form, as Leibniz used to say, the alphabet of human thoughts. He saw the model of such a procedure in the decomposition of a number into factors which yields a product of prime numbers. Another idea, very popular in the 17th century, was that of a universal ideographic language, whose single symbols would be assigned not to sounds but, as in Chinese, directly to concepts. T h a t planned language was then called philosophical language, conceptual script, universal script, etc. Here are, by way of example, two titles of works on t h a t subject t h a t were popular in t h a t time: Gregor Delgarno published in London in 1661 his treatise Ars signorum vulgo character universalis et lingua philosophica (The art of signs or universal script with a philosophical language), and then in 1668, also in London, An Essay Toward a Real Character and a Philosophical Language was published by John Wilkins.

64 Three: Formalized versus Intuitive Arguments - the Historical Background A c c o r d i n g t o Leibniz, t h i s c o n c e p t u a l l a n g u a g e s h o u l d be cons t r u c t e d in c o m p l i a n c e w i t h J u n g i u s ' s idea, which is t o say t h a t simple s y m b o l s , e l e m e n t s of t h e a l p h a b e t , s h o u l d r e p r e s e n t t h e simplest n o n - d e c o m p o s a b l e c o n c e p t s , while c o m b i n a t i o n s of symbols, analogically t o algebraic o p e r a t i o n s , s h o u l d express a p p r o p r i a t e c o m p o u n d c o n c e p t s . T h e rules p e r t a i n i n g t o such o p e r a t i o n s on s y m b o l s were t o f o r m a logical calculus t h a t would b e a l g o r i t h m i c in n a t u r e a n d would g u a r a n t e e error-free r e a s o n i n g : Ut errare ne possimus quidam si velimus, et ut Veritas quasi picta, velut Machine ope in charta expressa deprehendatur. (So that one could not err even if one wished to, and would find the truth given as it were ad oculos, rendered in writing by the machine). T h i s f o r m u l a t i o n , t o be f o u n d in Leibniz's l e t t e r t o O l d e n b u r g ( d a t e d O c t o b e r 28, 1675), looks like a p r o p h e t i c description of t h e c o m p u t e r w h i c h , c o n t a i n i n g in itself an a l g o r i t h m of p r o v i n g theor e m s or of verifying proofs carried o u t by m a n , p r o d u c e s t h e result o n t h e screen or on a sheet of p a p e r c o m i n g o u t f r o m t h e p r i n t e r . I t is also said in t h a t l e t t e r t h a t t h e t r u t h would be delivered in a visible m a n n e r a n d i r r e f u t a b l e owing t o an a p p r o p r i a t e mecha n i s m ( m e c h a n i c a ratione). His similar i d e a was t h a t o n e could easily avoid e r r o r s in r e a s o n i n g if t h e o r e m s were given in a physically t a n g i b l e m a n n e r (si tradentur modo quodam palpabili), so t h a t o n e could reason as easily as o n e c o u n t s (ut non sit difficilius ratiocinari quam numerare). T h e s u b s t a n t i a t i o n of t h a t p r o g r a m m e is t o b e f o u n d in Leibniz's l e t t e r t o T s c h i r n h a u s ( M a y 1678): Nihil enim aliud est Calculus, quam operatio per characteres, quae non solum in quantitate, sed et in omni ratiocinatione locum habet. (Computation is nothing else than operation on graphic symbols, which takes place not only in counting but in all reasoning as well). S u c h c o m p u t a t i o n was s o m e t i m e s called by Leibniz Calculus Ratiocinator. O n o t h e r occasions he used t h e t e r m Characteristica Rationis, as was t h e case in his l e t t e r t o R ö d e c h e n of 1708: Characteristicam quandam Rationis cujus ope veritates rationis velut calculo quodam, ut in Arithmetica Algebraque, ita in omni alia materia quatenus ratiocinationi subjecta est, consequi licet. (A conceptual symbolism, by means of which one could arrive at truths of reason as it were by computation, like in arithmetic and in algebra, and in every other matter if it can be subjected to reasoning.)

3. Leibniz on the mechanization of arguments

65

Such a strong call for the computationalization of any domain of human thinking resembles the contemporary program of strong Artificial Intelligence. 10 A s argued in this chapter, it is a requirement which has foundations in the age-old philosophical and civilizational development. Thus Leibniz's name appears as a symbolic bridge between this splendid tradition and the nascent civilization in which automated information-processing was to become a decisive factor for further development. 3 . 4 . When concluding this chapter and hinting at its relations to both preceding and following ones, I take advantage of the phrase 'symbolic bridge' as referring to Leibniz above. Leibniz's ideas deserve to be extensively commented. However, the problem of their historical influence is a matter that must be handled in a sophisticated manner. Only his program for mathematical logic and his idea of the universal formalized language were known to posteriority, while his results themselves were not discovered until the end of the 19th century. 1 1 Yet the very fact that Leibniz's programme was being realized without Leibniz, up to the present stage of Artificial Intelligence, is instructive and thought-provoking. The intellectual forces acting on behalf of this development proved so strong that even the loss of Leibniz's so significant contribution, caused by the delay over two-centuries of its being published, did not hamper the progress. Leibniz was the herald of the coming epoch, but one whose voice had not been heard until the time in which his visions started to be materialized, and his results anew achieved by others — with limitations whose unavoidability has in the meantime been recognized. This is a precarious position, indeed; the assessement of Leibniz's position therefore requires a sophistication which does not attribute too much to him, and at the same time is able to acknowledge the great role he nevertheless played — as a symbol of epoch-making ideas and of the civilizational turn. This is why my 10

See Chapter Seven, Subsec. 7.3.

11

More d a t a on this subject are found in Chapter Four, Subsections 1.1 and

1.2.

6 6 Three: Formalized versus Intuitive Arguments - the Historical Background

reporting on Leibniz's ideas provides me with a most convenient background t o present t h e current situation in mind-philosophical logic. B o t h t h e strength and t h e weakness of these ideas throw much light on our present problems. 1 2 T h e r e is a wonderful counterpoint between t h e Leibnizian and t h e Cartesian approaches to mind and logic. Even Leibniz's drawbacks contribute to our u n d e r s t a n d i n g of t h e mind when confronted with t h e views and approaches of Descartes and Pascal. W h e n his belief in t h e power of mechanized t h o u g h t proves unrealistic, we are able to answer t h e question 'why?' with t h e help of other protagonists. Descartes' and Pascal's preoccupation with t h e m i n d ' s inner life counterbalances Leibniz's tendency toward replacing mental acts by operations on symbols which could be formalized and mechanized. In t h e Cartesian-Pascalian perspective we see t h e mind as dealing with objects and relations which are so numerous, subtle and involved t h a t in many cases a verbalization or symbolization must prove inadequate. T h e above historical counterpoint is parallel t o t h e opposition of o b j e c t u a l and symbolic reasoning as outlined in the preceding C h a p t e r . It is argued there t h a t — as a rule — an efficient reasoning, as dealing with objects appearing to t h e mind, eludes a t t e m p t s at verbalization, and a fortiori a t t e m p t s at formalization in a symbolic language. Only in some cases can it succeed, a n d even then it produces texts which are cumbersome for intelligent h u m a n s , hence for w h a t we call n a t u r a l intelligence. On t h e other h a n d , if we wish t o be helped by artificial intelligence, e.g., in some dull jobs, we must endow it with a power of reasoning dealing with symbols alone. W h y ? Because it is possible, having a two-letter alphabet (e.g., electric pulse and its lack) to devise a symbolic code corresponding to t h e resources of a h u m a n language. On t h e other h a n d , no one has so far succeeded in m a p p i n g t h e immense resources of h u m a n t h o u g h t onto such a code. T h u s a reflection concerning rhetoric should be focussed on the modes 12

There is still another reason to motivate the interest in Leibniz's logicotechnical projects. They are rooted in his metaphysics so that the technical and the metaphysical side of his thought shed light on each other. This, however, is a separate problem which should be noted as the subject for a further study but cannot be combined with the subject-matter of this essay.

3. Leibniz on the mechanization of arguments

67

and laws of objectual reasoning, as apt to create forcible argumental constructs, while symbolic logic provides us with a useful contrastive background, and with devices (to be applied creatively, not mechanically) for some partial solutions. In this chapter, certain programs and ideas, which do not prove to have had any bearing on the present situation in mind-philosophical logic and its surroundings, have been dealt with at some length. W h a t Ramus, Melanchton, and even Bacon, designed and postulated for logic in their time died away without any impact on the further progress of logic. Nevertheless, the story of their endeavours and dreams undoubtedly deserves t o be told and heard, for we can learn from it which intellectual forces are fit enough to shape h u m a n thought and civilization. And against such a background, the main protagonists of the process to be studied can be seen and appreciated all the better.

CHAPTER FOUR

Towards the Logic of General Names 1. From syllogistic t o t h e calculus of classes 1.1. T h e preceding c h a p t e r tells of p r o g r a m m e s of reforming logic, all of t h e m being concerned with w h a t we now call traditional logic. By t h e end of t h e 19th c e n t u r y t h e r e a p p e a r s a new kind of logic. It was so new t h a t people needed t o call it by a new n a m e . T h e t e r m logistic d a t e d from t h e I n t e r n a t i o n a l Congress of Philosophy of 1904, where it was suggested i n d e p e n d e n t l y by Itelson, Lalande, a n d C o u t u r a t . Louis C o u t u r a t , t h e renowned discoverer of Leibniz's logic, might have followed Leibniz w h o employed eit h e r t h e n a m e calculus ratiocinator, or logica mathematical or else logistica for his logical systems. T h a t expression, a f t e r a fairly wide reception in the thirties, later went o u t of use. T h e designation m a t h e m a t i c a l logic, also following Leibniz, proved more lasting b u t n o w a d a y s it denotes only a p a r t of w h a t was originally m e a n t , viz., t h e logical s t u d y of m a t h e m a t i c a l systems, being distinguished f r o m t h e logical s t u d y of philosophical systems; at present, t h e l a t t e r h a p p e n s to be called philosophical logic. M a n y a u t h o r s a n d institutions prefer t h e t e r m symbolic logic as sufficiently general a n d rendering t h e concern with t h e symbolic aspect of proof procedures, while in some historical contexts, it is t h e expression m o d e r n logic which proves more convenient. 1 As far as t h e difference between t h e old a n d t h e modern logic is concerned, t h e opinions of historians a n d philosophers of logic may vary f r o m stressing their radical incompatibility t o t h e claim t h a t t r a d i t i o n a l logic forms just a tiny p a r t of t h e m o d e r n version. 1

See, e.g., Logic [1981], "Logic, modern, history of" by W . Marciszewski.

1. From syllogistic to the calculus of classes

69

Apart from philosophical motivations, such differences may depend on views concerning the nature of historical development. Those who see the development of ideas as a dramatic process, involving sudden turns and revolutions, are more inclined to stress the novelty of modern logic, while more conservative minds perceive continuity. It is the contention of this Chapter that syllogistic paved the way for the calculus of classes and that this, in turn, contributed to what can be called the c o m p u t a t i o n a l f o r m a l i z a t i o n of logic. This stage of formalization has become possible due to the algebraization of logic claimed since the 17th century (as extensively reported in Chapter Three) which was a continuous process; it reached its mature stage in Boolean algebra in the middle of the 19th century. This algebra has become the paradigm of modern computational logic (Boole's rechnende Logik, as Frege called it); its theories duly deserve the name of calculi. As for the feature of formal restructuring, that is a new conception of logical form, the change was more rapid and unexpected, due to Frege's modelling of the structure of logical formulas on the mathematical language instead of imitating the syntax of natural languages. Both points mentioned above are of consequence for the main problem of this essay, that is the problem of what modern logic can contribute to cognitive rhetoric (cf. Chapter One, Subsec. 1.3). It is worth noting that whenever rhetoric flourished in the past, it had close links with traditional logic. Is it possible to create comparably strong links between rhetoric and modern logic? To work out premises for answering this question, in the subsections following this one I shall briefly discuss the issues of computational formalization and formal restructuring. 1.2. The program of computational formalization of logic was much in vogue in the 17th century (as reported in the preceding chapter). It started with Thomas Hobbes who compared reasoning to computing, and culminated with Leibniz, who in his numerous calculi tried to reduce reasoning to counting (his famous calculemus as a recipe for problem-solving). In order to yield the idea of computational formalization, this trend towards computing in logic must have met with another one, that going back to the

70

Four: Towards the Logic of General Names

Schoolmen, whom Leibniz appreciated for their notion of logical form (cf. Three, Subsec. 3.1). That amounted to developing logic as a theory of particular physical objects, such as shapes of expressions, contrary to the Cartesian conception of logic as dealing with the behaviour of the mind. This should be construed as the processing of some physical objects as representatives of abstract logical objects as are, for example, truths. It is the very core of formalization that results which hold in an abstract domain are produced in an indirect way which consists in processing physical entities, suitably coordinated with abstract ones. These two features, computization and formalization, can be combined in a natural way, yet they are independent of each other. One can develop mathematics in a non-formalistic way, as do contemporary intuitionists, and one can adopt formalization procedures which are not computational, e.g., formalized religious rites. That stream in the history of logic which united these features has been finally crowned with the mechanization of reasoning — desired, planned, and envisaged by Leibniz (cf. Chapter Three, Subsec. 3.3). Mechanization is a special kind of formalization, namely such that linguistic forms are no longer figures produced with something like a pencil, but are configurations of physical impulses which belong to the functioning of a machine, and at the same time can be interpreted by humans; for instance, an electric pulse is interpreted as number one, its lack as number 0, while their sequences produce all numbers rendered in binary arithmetical notation. Owing to such correspondence between numbers as abstract entities and electric pulses as physical entities, machines can be commissioned to compute and to reason. The latter proved possible owing to an ingenious reduction of reasoning to computing. Such a reduction was first attempted and accomplished within the field of traditional logic at some advanced stage of its development. The feasibility of such transformation depends on the way of interpreting the four forms of general statement recognized by Aristotle, and regarded by him and his followers as the complete classification of syntactic forms to be employed in logic. They are general in the sense that they consist of general names, namely two

1. From syllogistic to the calculus of classes

71

names combined by the copula. Thus no sentence concerning individuals, t h a t is containing an individual name as its grammatical subject, belongs to t h a t syntactic pattern of traditional logic. In this logic, no logical rules deal with inferences which involve names referring to individuals as individuals (there were only a t t e m p t s to refer them to one-element classes). This feature should catch due attention when compared with the fact t h a t modern logic takes just the opposite approach; namely, t h e basic syntactic p a t t e r n is t h a t of a sentence consisting of an individual name and a predicate, i.e., an atomic sentence. This sheds light on t h e scale of the formal (i.e., syntactic) restructuring having been brought by the transition to modern logic. However, the failure in completing the repertoire of logically relevant syntactic forms proved advantageous at a certain stage of development; it gave logic t h a t simplicity which encouraged a computational approach with those tools alone which were at hand in t h a t phase. Such an approach became possible after t h e mediaeval logicians developed the theory of distributio terminorum which provided a device to check the validity of syllogistic reasonings. It was the beginning of t h e extensional interpretation of general sentences, i.e., t h a t in which both subject and predicate are taken as names of classes. 2 Thus in the 17th century, when both scholastic logic and algebraic methods were perfectly assimilated by many philosophers, 2

T h e theory of term distribution, to a great extent due to William of Shyreswood (d. 1249), defined the ways in which a general term may be taken, i.e., whether in its full extension or partial extension. T h u s it belonged to the theory of suppositio, i.e., the examining of ways in which a term may be taken (as an individual, a class, etc.), but in another respect it contributed to the extensional conception of logic, admired by Leibniz for its technical advantages. Within this framework even the delicate problem of individual terms was solved, namely in the way suggested by William of Ockham (ca. 1300-1350) w h o treated an individual term as taken in its full extension, hence behaving as a general term; this paved the way for the set-theoretical concept of unit class (cf. Bochenski [1956], item 34.02). Aristotle himself rather ignored the problem of semantic interpretation of general statements, having had enough reasons to be satisfied with his technical achievement. A s to possible semantical interpretations of Aristotle's general statements, see Kneale and Kneale [1962], II, 5; as to theories of distribution and supposition, see ibid., passim.

72

Four: Towards the Logic of General Names

the time was ripe to attempt at an algebraic computization of logic. This was first done by Leibniz, but his results did not influence later authors since they remained unknown up to the end of the 19th century; only then were Leibniz's logical manuscripts discovered by Louis Couturat. 3 More luck was had by Johann Heinrich Lambert (1728-1777), who admired Leibniz's logical genius but did not know anything about his algebraization of logic and worked it out by himself to publish it in his many logical volumes (at least the seven ones which are available at present), especially one under the much telling title Neues Org anon oder Gedanken über die Erforschung und Bezeichnung des Wahren, published in Leipzig 1764 (cf. Lambert [1782], the chapter Versuch einer Zeichenkunst in der Vernunftlehre). The terms Bezeichnung und Zeichenkunst hint at the tendency towards formalization, while the phrase Novum Organon indicates the historical role of the work, seen as a counterpart to the Organon of Aristotle. In the same century there were more authors who tried an algebraic computization of logic, but it is a long story which should be told at another place. 4 Lambert deserves to be mentioned as a convincing evidence t h a t similar results may be obtained in the same historical period independently of each other, as if they were governed by an objective law of development holding in a realm of abstract ideas. 1.3. In the 19th century the algebraical calculus of logical objects reached its maturity, again with many authors acting simultaneously. The most lucky among them proved to be the British mathematician George Boole [1847], [1854]. He created an advanced and viable theory which entered the history of logic under his name. Boolean algebra has become the first complete paradigm of computational logic and an indispensable tool in other branches of science and technology including computer science and the study of brain activity. 3 Ail instructive exposition of Leibniz's results is found in Kneale and Kneale [1962], and in the editor's comments to the critical edition, accompanied by a German translation of Leibniz's Generates inquisitiones de analyst notionum et veritatum, Schupp [1982]. 4 Many important data on this subject can be found in Risse [1964] and in Styazhkin [1969].

1. From syllogistic to the calculus of classes

73

In Boole's algebra, likewise in the earlier a t t e m p t s of giving logic a calculative form, the Aristotelian syllogistic dealing with extensionally interpreted general statements proved a suitable laboratory for the computational treatment of logic. First, however, one needs to have cleverly invented those values which may appear in equations as results of operations performed on classes. Such values can be clearly defined due to the concept of the universe of discourse. For a while let us focus our attention upon that concept as introduced by Augustus de Morgan (1806-1871), another English pioneer of algebraical logic. 5 A universe of discourse is what contains all the entities to be discussed in a given discourse, investigation or theory. These entities can be grouped into classes included in (i.e., being subclasses o f ) the universe in question. Now let us distinguish two subclasses, the greatest and the least, from among those included in the universe. Obviously, the greatest equals the universe itself while the least is one having zero elements; let them be called the universal class and the e m p t y class, respectively. In principle, it is of no consequence what symbols we choose to denote these limiting cases, but in practice it proved convenient to use the symbol Ό' for the empty class and Ί ' for the universal class. For some partial analogies with arithmetical operations help to interpret logico-algebraical equations in which these values appear as results of operations upon classes. 6 To render a general statement in algebraic form we still need the equation sign and signs for two operations called complementation and intersection, both perfectly compatible with our handling of classes in everyday language. 7 De Morgan's calculus in his Formal Logic of 1847 is essentially similar to that of Boole but not so fully worked out, esp. in the notation and applications. 5

Leibniz vaguely anticipated these two values. In his terminology the word ens would have corresponded to 1, and the word non-ens to 0. 6

7 This is worth noting from the rhetorical point of view as confronting logical calculi with everyday arguments. In this context 'everyday language' should mean at least the Indoeuropean languages. T h e outstanding Polish logician Roman Suszko (d. 1979) used to repeat that mathematics, as based on the idea of class, could only have developed so enormously in the environment of those ethnic languages which involve the notion of class at the base of their syntactic structures, as do Greek, Latin, etc.

74

Four: Towards the Logic of General Names

T h e complementation of a class, say A, is relative to the universe in question, say U; namely the complementation of A, symbolically, —A, is the class of all those and only those elements of U which are not elements of A, t h a t is all the rest which remains after 'removing' A from U. The intersection of classes, say A and B, symbolically A B, is the class of all those and only those entities which are elements of both A and B. The Aristotelian general statements take the following forms. 1. universal affirmative: Every A is B, e.g., every masculinist is a male. 2. universal negative: No A is B, e.g., no masculinist is a male. 3. particular affirmative: Some A i s e . g . , some masculinists are male. 4. particular negative: Some A is not B, e.g., some masculinists are not male. 8 Here are the corresponding Boolean equations: (1) (2) (3) (4)

A -B = 0 A B =0 Α -Β φ 0 Α - Β φ 0. Now we can check the validity of some logical theorems in a purely computational way. For instance, there are theorems t h a t (1) is equivalent to the denial of (4), and (2) is equivalent to the denial of (3); t h a t two statements are equivalent means t h a t they have the same truth-value, i.e., either both are true or both are false. Obviously, to deny (1) means to replace the equality sign by the sign of inequality, and then one obtains (4). The same transformation holds for (2) and (3). Two statements such that 8

I am not sure whether the term 'masculinist' exists in English. This uncertainty is deliberately left as this gives opportunity to show that the understanding of terms substituted for schematic letters is irrelevant for understanding the logical relations between considered sentences. However, those who feel better when understanding till the constituents of the statement in question may fancy the definition that a masculinist is a member of the Men's Liberation Movement.

2. The existential import of general names

75

one of them is equivalent t o the denial of t h e other are said t o be cont r a d i c t o r y . Using transformations of this and similar kind one can demonstrate the validity of Aristotelian syllogisms, e.g. t h a t famous one (as opening the list of syllogistic forms in all expositions): every masculinist is a male; every male is fearless; hence, every masculinist is fearless. The application of the class calculus to proving the validity of such a form of syllogistic argument requires a more systematic introduction which would lead us away from t h e course of the present discussion. 9 In order to perform some other calculations we need the principle that the empty class has no members in common with any other class; in other words, the intersection of the empty class with any class A equals the empty class, in symbols: 0 · A — 0. One can hardly find anything more obvious t h a n this principle: t h e empty class as having no members at all cannot share any members with another class. This obviousness is worth emphasis from the rhetorical point of view, it should makes us aware of how close the calculus of classes is to our everyday thinking. It is equally obvious t h a t any statement of the form 1 tells us t h a t there are no .4s not being Bs (i.e., their intersection is e m p t y ) , e.g., every cuckoo is a bird means t h a t there is no cuckoo being a non-bird, i.e., the class of cuckoos not being birds is empty. However, these obvious assertions have a consequence which may seem shocking to everyday language users. For instance, those who do not believe in witches would be compelled to agree t h a t , say, every witch is an accomplice of the devil. Indeed, if the class of witches is empty, then its intersection with any class whatever is empty: if there are no witches at all, then there are no witches plotting with the devil. T h u s the statement in question proves true (under the weak interpretation of a universal s t a t e m e n t ) .

2. The existential import of general names 2 . 1 . The conclusion of the preceding section is thought-provoking. The fact t h a t such obvious and natural principles so easily lead to 9

A reader interested in this issue may consult an instructive exposition in Copi [1954], Appendix A.

76

Four: Towards the Logic of General Names

consequences which seem artificial or even false (e.g., t h a t every witch is the devil's accomplice) sheds much light on relations between natural language arguments and logical calculi. The crux of t h e matter is the existential import of names, i.e., the presupposition t h a t the use of a name implies reference to an existing thing. W h a t does it mean? Our language develops in the environment of existing things; whenever a name is being used, it is implicit in the very fact of its use t h a t this name stands for something. This is no accident and no arbitrary convention. This is a necessity deeply built in our lives. T h e alternative strategy would be something like t h a t : first to presuppose t h a t the name in question stands for nothing, i.e. its extension equals the empty class, and next to check if, by chance, not the opposite is the case. In other words, non-existence would be constantly presupposed while existence would require a proof. Imagine a hunter who addresses his companion with the warning "a bear is about to attack us". The existential import of the name ' b e a r ' implies t h a t the companion accepts the consequence "there is a bear (close to us)". Should he not accept t h a t and demand a proof, the bear would quickly devour both hunters. Survival and development are due to collaboration among humans, and collaboration means taking for granted t h a t our communication does not consist in talking about nothing. This rule of the existential import of names is so universal t h a t it controls both everyday speech about empirically given things and the mathematical discourse concerning abstract constructions. There is just t h e difference, by no means minor, t h a t in mathematics it is existence t h a t should be proved in each case by providing a suitable construction; in everyday communication the existence of what it refers to is presupposed, and non-existence should be proved, if necessary (therefore we do not trust liars but, at the same time, we do not assume from the start t h a t everyone is a liar). In such a framework, the above quoted statement about witches should be regarded as false since it would imply t h a t witches do exist — in the virtue of the existential import of the name 'witch'. And if a statement implies anything false, it has to be false itself. On the other hand, the proof of its t r u t h is flawless on the basis of some obviously true premises.

2. The existentiell import of general names

77

T h e alleged paradox dissolves if we consider t h e role of logical calculus in t h e analysis a n d evaluation of a r g u m e n t s . A calculus should supply means t o resolve a c o m p o u n d expression into constituents which are relevant t o t h e validity of t h e a r g u m e n t in question. T h e calculus of classes makes it possible t o distinguish t h e negative existential constituent there is no A not being Β in t h e universal affirmative s t a t e m e n t s . W h e t h e r such a sentence does or does not include t h e positive existential constituent there is an A should be decided for each concrete context t o which one applies one's logical analysis. For t h e purpose of rendering t r a d i t i o n a l syllogistic in a m a t h e m a t i c a l form it was advisable t o a d m i t t h e option without t h e positive existential c o m p o n e n t . In this way t h e class of syllogistic theorems has been divided into those which are provable under this weak interpretation of universal s t a t e m e n t s and those which require a strong interpretation, t h a t is including t h e assertion t h a t t h e g r a m m a t i c a l s u b j e c t is not empty, i.e., having t h e existential i m p o r t . Whenever t h e weak i n t e r p r e t a t i o n does not suit our u n d e r s t a n d i n g of a concrete context, we are free t o choose t h e strong one. Aristotle himself preferred t h e latter route, and his followers c o n t r i b u t e d to a b e t t e r u n d e r s t a n d i n g of his intentions owing t o t h e analysis outlined above. 2 . 2 . Let us consider other examples of applying t h e calculus of classes to t h e analysis of those a r g u m e n t s which involve general statements. T w o particular s t a t e m e n t s have been said to be subaltern to t h e respective universal s t a t e m e n t s and sub-contrary t o each other while two universal s t a t e m e n t s are said t o be contrary to each other; there is also t h e relation of being contradictory t o each other discussed earlier (see above 1.2 in this c h a p t e r ) . A subaltern s t a t e m e n t is logically entailed by t h e respective universal s t a t e m e n t only u n d e r t h e strong i n t e r p r e t a t i o n of t h e latter. Suppose t h a t a naive reasoner is not aware of t h e difference between t h e two interpretations. T h e n he might be puzzled by an a r g u m e n t resorting t o t h e subalternation rule which claims: from any true universal you should infer the respective particular statement. O u r reasoner is liable to (recklessly) acknowledge t h e t r u t h of (a) Every witch is a witch.

78

Four: Towards the Logic of General Names

For any thing is identical with itself, and then on the basis of subalternation he is bound to acknowledge t h a t (b) There is a witch which is a witch, and so to ascertain the existence of at least one witch. Even worse, from the universal negative statement (c) No witch is a real entity he should infer (d) There is a witch which is not a real entity. Such puzzles can easily be solved after translating general statements into their class-theoretical counterparts. It is enough to observe t h a t (a) is true only under the weak interpretation, t h a t is one to the effect A • —A = 0; in fact, there are no members shared by the classes A and its complement —A, also in the case when A is empty. However, the t r u t h due to the emptiness of A gives us no reason to infer the non-emptiness of A. If, on the other hand, we take a universal statement in the strong interpretation, then the existence of an object belonging to t h e class in question is granted from the very start. We deal then with the compound assertion (1*) Α φ 0 and A • —B = 0 as the basis of a reasoning which leads to (3) A • Β φ 0, t h a t is to the corresponding particular statement (cf. the list of statements in Subsection 1.2). If A is not empty, then (one has to conclude) also Β is non-empty; were it empty, its complement mentioned in (1*) would be non-empty, and then together with t h e non-empty A it could not yield the empty class referred to in equation (1*). A rhetorical moral which follows from this discussion runs as follows: before we use a universal statement in an argument, we should check if there are reasons to claim the non-emptiness of the subject. If the existence of the entities referred to is neither postulated nor proved in the theory in question, then the reasoner should apply a suitable procedure to prove existence. An everlasting paradigm of such procedures is found in Euclid. They deserve careful examining but before it is made we should enlarge our repertoire of means of expression in a formalized language. In t h e above proof of the law of subalternation there appear logical

3. What names stand for: an exercise in Plato

79

terms which do not belong to the language of the theory of classes, as 'and', 'not' and 'if...then'; they are borrowed from ordinary English as is also 'either...or' being usual in mathematical and other arguments. Their formalization in an exact logical language should grant more rigour to arguments; at the same time it sheds much light upon the structure of arguments in everyday languages. It is why the issue requires a systematic treatment which will be t h e subject of the next chapter. However, before we resume the discussion of those calculi which are particularly suitable for analysing arguments, it is in order to point out some significant limitations of the calculus approach. This is meant not to decrease the importance of logical calculi but rather to use them in the role of a filter which should distinguish computable and non-computable constituents of extramathematical arguments. 3. W h a t n a m e s s t a n d for: a n e x e r c i s e in P l a t o 3 . 1 . In the preceding section we dared a risky logical enterprise of coordinating some English contexts, such as those involving 'every' and 'some', with certain operations of the calculus of classes. The risk, which we share with the inventors of this calculus themselves, consists in choosing each of these words from among many with similar functions: do we consider each of them to represent a whole category of English words which are interchangeable with each other (e.g., 'some' might be replaced by 'a(n)', 'at least one', etc.), or do we give them a specific meaning of their own? In the latter option we restrict the applicability of logical calculus under consideration to those words which have been coordinated with operations of t h a t calculus, and so the calculus proves of little use to render the wealth of argument forms in natural languages. On the other hand, when following the former option, we have to examine that wealth and so face all the vagueness of natural-language expressions and the resulting arbitrariness of interpretation; however, it is that option which has to be chosen if logic is to serve any rhetorical purpose. In particular, it is the articles a and the which deserve most careful examination as being the most frequent function words in

80

Four: Towards the Logic of General Names

t h e use of English and similar languages. An intriguing problem t h a t must be attacked at t h e very start results from t h e fact t h a t t h e same indefinite article ' a ' in some contexts is used as equivalent t o 'every' while in others as equivalent to 'some'. Peculiarities of t h e definite article are still more thought-provoking, for sometimes it is used t o express individuality and sometimes t o express generality, and t h e latter even in a specially strong sense which a m o u n t s to claiming t h e existence of general objects, or universale. A full spectrum of uses, including t h e last mentioned extreme, can be best found in P l a t o , and it is why some texts taken from his Republic will be used for our logico-linguistic exercises. 10 T h e interpretation suggested in this discussion does not pret e n d to render Plato's original t h o u g h t , for such a study should take into account t h e original text as obeying specific rules for t h e Greek articles. It is rather to be regarded as t h e discussion of an English text which, owing to its relationship to P l a t o , provides us with an ample spectrum of uses of articles. In order to tell each of these uses from t h e others, I suggest some t e r m s invented ad hoc instead of distinguishing t h e m by numbers (as is usual in dictionaries of English). These terms allude to t h e mediaeval theory of suppositions, which is particularly suitable for such discussion since t h e logicians using Latin, a language without articles, could not even have had t h e possibility of resorting t o t h e systematization provided by dictionaries. It is why they have invented t h e technical t e r m suppositio— not in t h e sense of a conjecture, guess, etc. b u t in t h e sense derived from t h e context: (nomen) supponit pro (aliquo), i.e., a n a m e stands for a thing. It is this sense in which t h e English c o u n t e r p a r t supposition is to be employed in t h e present section. As for t h e adjectives distinguishing kinds of supposition, my terminological inventions are somehow inspired by mediaeval Latin terminology b u t do not tend to follow it; they are devised purely for t h e p u r p o s e of t h e present discussion. 10

All the texts are taken from the English translation by B. Jovett, The Dialogues of Plato, vol. 3, The Republic, 3rd edition, Clarendon Press, Oxford 1892. The numbers in parentheses indicate the book and the passage, the latter according to a standard numbering (Ed. Steph.) applied by Jovett.

3. What names stand for: an exercise in Plato

81

I shall distinguish two suppositions which can be expressed by the indefinite article, namely member supposition and class supposition, and two expressed by the definite article, namely individual supposition and eidetic supposition, all that being discussed with regard to the problem of the existential import of names as raised in the preceding section. 3 . 2 . Member supposition corresponds to that use of the indefinite article which consists in using it as a function word before nouns in the singular form (apart from proper names) and mass nouns when the individual in question is undetermined, unidentified, or unspecified, esp., when the individual is being first mentioned or called to notice; for example, 'there was a tree in the field', or 'a man walked past him'. This grammatical characterization should be complemented by a condition which we owe to the calculus of classes. Let 'some' in the sentence 'some dogs are philosophers' be replaced by the indefinite article, thus yielding 'a dog is a philosopher'. Is this new sentence equivalent to the former, or not? If one answers in the affirmative, then one takes 'a dog' in member supposition, i.e., one has in mind an individual member (or more of them) of the class of dogs. If one answers in the negative, then one resorts to another supposition (to be discussed later). To prove this statement with regard to the member supposition, it is enough to prove the existence of at least one dog being a philosopher, while the other supposition will require more than that; hence in arguing one should be aware of which supposition comes in question. The following passage of Republic suggests a criterion to recognize an occurrence of member supposition. Socrates intends to prove that the watchmen in the State should be wise men because watching involves distinguishing between the friendly and the hostile individuals, and that requires a knowledge characteristic of philosophers; the discussion starts with a familiar experience concerning dogs recalled by Socrates (the first speaker in the dialogue quoted below) and successively being confirmed by Adeimantus (ii, 376). — A dog, whenever he sees a stranger, is angry; when an acquaintance, he welcomes him, although the one has never done any harm to him, nor the other any good. Did this never strike you as curious?

82

Four: Towards the Logic of General Names

— The matter never struck me before; but I quite recognize the truth of your remark. — And surely this instinct of the dog is very charming; - your dog is a true philosopher. — Why? — Why, because he distinguishes the face of a friend and of an enemy only by the criterion of knowing and not knowing. And must not AN animal be A lover of learning who determines what he likes and dislikes by the test of knowledge and ignorance? — Most assuredly. T h e r e are at least two criteria to ascertain that it is the member supposition which comes into play in the above passage. T h e first of them appears at the very start. It is the temporal setting of the argument, namely the dog in question 'sees a stranger', hence it must be a concrete individual since only the objects of that category can perform such activities. T h e other criterion consists in the validity of replacing a by the in a suitable moment. Such a moment comes when the listener is already acquainted with the object referred to which was unknown to him when the story started. In our story this occurs in the statement ending with the conclusion 'your dog is a true philosopher'. In this context it is clear that people talk about a concrete individual dog, that owned by Adeimantus, hence that use of the (which is to be called individual supposition); whenever we deal with such a sequence of using first the indefinite and next the definite article (in individual supposition), the former occurs in the member supposition. T h e term 'member' in the role of adjective should indicate that the individual in question is not considered as an individual but as a representative of that class to which the general name, as 'dog' refers to. T h e statement of the form 'a dog is a philosopher' is true only if there exists a dog, hence the name being its subject must have the existential import. Were there no dogs in the world, this would have made this statement false (obviously, there is another condition of truth, namely that a member of the class of dogs be a member of the class of philosophers; this, however, is irrelevant to the present problem). 3 . 3 . Class supposition corresponds to w h a t can also be expressed with the help of the words like any and each preceding a general

3. What names stand for: an exercise in Plato

83

name which is followed by a restrictive modifier, for instance 'a man guilty of kidnapping wins scant sympathy', or 'a man who is sick cannot work well'. This definition, due to grammarians, rightly hints at a restrictive modifier as characteristic of such suppositions, e.g., the modifier 'who is guilty of kidnapping' and 'who is sick' in the above examples. However, a logician has a reason to ask whether the general name following the indefinite article might not be regarded as a restrictive modifier in some constructions like those discussed above. Consider the following statement made by Plato: Ά S t a t e was thought by us to be just when the three classes in the S t a t e severally did their own business' (iv, 435). Disregarding unnecessary elements, let us reduce this statement to the following form: A State is just when the three classes severally do their business. There is no restrictive modifier to follow the subject 'a state', nevertheless the statement should be interpreted as universal since it is thought as listing a necessary condition in the definition of a just state (as is obvious with the context of this sentence). T h e perverse idea suggested by this example is to the effect that the subject may prove to be its own restrictive modifier. To see that let us paraphrase the considered definition as follows: A system, being a perfect State is one in which the three classes severally do their business (viii, 546). T h e trick consists in finding a dummy subject, here 'system', which is always possible (in any case it may be a universal dummy subject like 'thing') so that the old subject becomes a restrictive modifier. Therefore, for some contexts it may be difficult to decide whether we deal with a construction characteristic of member supposition or with a construction admitting a hidden modifier. T h e latter can be termed a class supposition on account of its natural translatability into a statement concerning classes, as in those previously listed examples: 'the class of guilty men is included in the class of those who win scant sympathy', and 'the class of sick men is included in the class of those who cannot work well'. T h e example of Republic would now run: the class of just states is included in that of systems in which the three categories of citizens severally do their business.

84

Four: Towards the Logic of General Names

The suppositions under consideration contrast with one another with regard to existential import. Member supposition grants existential import to a name while class supposition does not. Imagine t h a t there is nowhere in the world a State designed by Plato. This does not refute the t r u t h of his definition; it just means t h a t there are no perfect states deprived of those three categories (i.e., philosophers as rulers, and guardians, and artisans), each of them flawlessly doing its own business. Even if this ideal does not materialize, the Platonian idea of State, as expressed in the above statement, may remain true. The distinction between member supposition and class supposition starts to vanish when we enter a realm of entities which are indistinguishable from one another as are, for instance, geometrical points. A dog which sees a stranger (to make use of the quoted Socrates' story) is different from a dog which in the moment does not see any stranger, and this temporal difference makes the said dogs different from one another. Hence in such a context the indefinite article makes the name following it stand for individual members of a class, and not for t h a t class itself. However, what about members of a class which are not liable to bear any individual features? Then it is indifferent whether one speaks of one member or of all members; whatever is being said about one of them is true about all the rest. This case is worth a special study which may result in a revision of the grammatical doctrine t h a t the indefinite article marks class supposition only when there appears a restrictive quantifier (to hint at a common property which, so to speak, cancels individual differences). To suggest a starting point for such a possible study, let me quote a formulation of general theorems in t h e example of Euclid's Theorem 14 of Book I. It runs as follows: If, at a point in a straight line, two other straight lines, on the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. Other indefinite articles of the same function are hidden in the plural forms as '(any) two right angles', etc. The above feature of such a mixed (so to speak) member-class supposition proves especially interesting from the logico-rhetorical point of view, namely t h a t the error in arguing which consists

3. What names stand for: an exercise in Plato

85

in jumping to a hasty generalization may stem from a wrongly presupposed indiscernibility of members of a certain class. For instance, one jumps to the conclusion 'a person who is Jewish has enormous mathematical abilities' — where the article a produces a class supposition (to refer to the class of Jewish persons), and 'who is Jewish' is a restrictive modifier — on the basis of the premise ' a person who is Jewish has enormous mathematical abilities', where a is taken in member supposition to mean 'there is a person who is Jewish with enormous mathematical abilities'. If there acts in the given mind an unconscious assumption that all Jews are alike in mental qualities (i.e., indiscernible in some respect), it should strengthen the tendency to mistake one of these suppositions for the other. Supposedly, such errors are more likely to be committed by those persons in whose brains there are relatively weak connections between the zone of verbal activities and the zone in which representations of real things are recorded; such brains are more liable to be misled by purely verbal similarities or identities, belonging to what Francis Bacon called idola fori. 3 . 4 . There is a remarkable parallelism between two suppositions corresponding to the indefinite article and those corresponding to the definite article. Two functions of the definite article which are logically significant are the following. T h e definite article is used: (i) for mentioning a particular thing either because we already know which one is being talked about or because only one exists; (ii) before a noun in the singular to make it general, e.g. 'the lion is a wild animal' ( = l i o n s are wild animals), 'the computer has revolutionized office work'. T h e difference between (i) and member supposition consists in the kind of knowledge possessed by a speaker: whether he knows the thing in its individuality, as in case (i), or only as a member of a definite class; yet, these different cognitive situations involve an individual. Let this feature be rendered by the term individual supposition; in member supposition the elements of a class are not recognized in their individual traits, yet they are referred to as 'schematic' individuals, so to speak, and in this sense the suggested terminological choice proves justified. Individual supposition occurs in the English version of Republic in the way conforming to general rules which hold for English; there

86

Four: Towards the Logic of General Names

is nothing specific to distinguish it from other t e x t s or a u t h o r s . New issues brought up by Republic are related to u s a g e (ii), being a n a l o g o u s t o class supposition b e c a u s e of the feature of generality. Let it be called e i d e t i c s u p p o s i t i o n in the sense of the term t o be explained later in a context of P l a t o ' s t h o u g h t . T h e r e is a p a s s a g e a b o u t the perfect shoemaker t o prepare discussion a b o u t the perfect g u a r d i a n . The shoemaker was not allowed by us to be a weaver, or a builder — in order that we might have our shoes well made; but to him and to every other worker was assigned one work for which he was by nature fitted. ( B o o k ii, 374). Now let us imagine a botcher who is neither by n a t u r e fitted nor sufficiently trained to m a k e g o o d shoes. Does he fulfil the notion expressed in the general supposition ''the s h o e m a k e r ' ? T h e r e are no such d o u b t s as far as either m e m b e r supposition or class supposition is concerned. It makes sense if one says t h a t he knows a shoemaker who is a botcher ( m e m b e r s u p p o s i t i o n ) , or if one says something like this: a shoemaker should pay t a x e s as do all c r a f t s m e n . In the latter context, the term in question a p p e a r s in class supposition, a s the p h r a s e refers t o each one belonging to the class of shoemakers. T h e s a m e supposition, i.e., t h a t in which a n a m e s t a n d s for a class, a t t a c h e s t o the n a m e ' a shoemaker' in the following statement: as any other person, may prove unfit for his [CSJ A shoemaker, occupation. Such s y m p a t h e t i c u n d e r s t a n d i n g of h u m a n weakness ranges over the class of all people, hence all shoemakers, too. T h i s universality is d u e t o the m o d a l word may: whatever actually h a p p e n s t o s a m e persons of s o m e class may h a p p e n to other members of the s a m e class. Would this j u s t i f y inference from class supposition t o general s u p p o s i t i o n ? Let us listen to the 'tone' of the following s t a t e m e n t which results from C S by replacing a with the: [ES] The shoemaker, as any other person, may prove unfit for his occupation. T h e person talked a b o u t in E S is no P l a t o n i a n shoemaker, for the latter ex definitione, hence necessarily, has occupational fitness which does not necessarily a t t a c h to the former. T h e 'tone' in which one utters sentences like E S is t h a t of praising perfection. It

3. What names stand for: an exercise in Plato

87

is the tone penetrating the whole text of Republic as dealing with the perfect State and its perfect members. 3.5. The supposition produced by the definite article in its generalizing function deserves to be called e i d e t i c s u p p o s i t i o n since it deals with an ideal entity; the Greek terms είδος and t6ea are synonymous, and the former is more fitting to create a new technical term since the latter has already too many senses in ordinary and philosophical English. Have we, one may ask, to follow Plato in this respect, and thus to commit ourselves to his controversial philosophical assumptions? It suffices to answer that deeds should be judged by their fruits. In the present logico-rhetorical framework, the chapters concerning definitions should prove how fruitful the adopted Platonian approach is. Here are some further examples of eidetic supposition. In the human soul there is a better and also a worse principle; and when the better has the worse under control, then a man is said to be master of himself, (iv, 431). In this short passage there appears member supposition with conspicuous existential import ('there is a better principle', etc.), anaphoric use of the definite article ('the better', 'the worse'), class supposition ( ' a man in which the better has the worse under control'), and against this background the clearly distinguishable eidetic supposition the human soul. To hint at such generality as is attached to this principle, Plato needs more than class supposition; he must resort to eidetic supposition because the said properties of soul are not only general but also essential and pertaining to the ideal of soul. And so of the individual; we may assume that he has the same three principles in his soul which are found in the State, (iv, 435). T h e same thought is developed in the following passage. Such is the good and true City or State, and the good and true man is of the same pattern; and if this is right, every other is wrong; and the evil is one which affects not only the ordering of the State, but also the regulation of the individual soul, ( v , 449). Here Plato seems to have a feeling that he touches upon the very essence both of the soul and the state, that is the structure consisting of three elements which should act in harmony if the soul or the state in question is to be true. Thus eidetic supposition has nothing to do

88

Four: Towards the Logic of General Names

with appearances; in the sphere of appearances generality may be explained through class supposition, but in the realm of genuine things only eidetic supposition does justice to genuine generality (i.e., not accidental but founded in general objects). Obviously, it is another way of saying t h a t eidetic supposition deals with ideals since — in the Platonian perspective — only ideal things are real things. Plato's answer to the question of how to conceive generality may be questioned but cannot be disregarded. It will contribute to t h e discussion of generality in chapters t o follow, one dealing with reasoning (Seven) and one dealing with defining (Eight). The way to this discussion leads through the presentation of those logical theories which were lucky to become the s t a n d a r d of modern logic (while the theory of general names is found at its margin), namely t h e truth-functional calculus and the predicate calculus. As I end this exercise in Plato, let me address the contemporary philosophers from the noble family of Minimalists (as are empiricists, nominalists, behaviorists, etc.). They are very serious persons who take any Platonic inclinations as a sign of mental frivolity or even deviation. They should be asked to compare the world of physical solids with the world of geometrical objects, and to account for semantic differences between statements describing these two realms. First let them answer if they recognize such differences, and if do, let them next try t o account for them in a logico-linguistic theory free of any Platonic bias. This should be a genuine contribution to the theory of definition. This theory heavily draws on fundamental logical calculi, and it is why we need first to discuss them. In will be done in the natural order, first truth-functional logic and then predicate logic.

CHAPTER FIVE

The Truth-Functional Calculus and the Ordinary Use of Connectives 1. T h e f u n c t i o n a l a p p r o a c h t o logic 1.1. It was mentioned in Chapter Four (Subsec. 2.2) t h a t in demonstrations of class-theoretical theorems we need some logical connectives, which do not belong to the language of the theory of classes, namely such words as 'not', 'and', 'either ... or', 'if ... then', etc. T h a t they appear in every reasoning, and t h a t they form the subject-matter of a logical theory (to be discussed in this chapter) is now a fact which is obvious to every educated person. However, it was far from obvious for many centuries, and even less obvious was the idea t h a t syllogistic, e.g., in its class-theoretical version, should belong to a uniform logical system together with the theory of logical connectives. The latter was somehow anticipated by Stoic logic and by the mediaeval treatises De consequentiis but an attempt was never made at unifying these two branches of logic, until the turning point made by the German mathematician, Gottlob Frege (1848-1925). The unifying idea, one t h a t pervades the whole of mathematics and the whole of modern logic, is the concept of function. Though practically used in arithmetic, algebra, analysis, etc., from the very beginnings of these disciplines, its first theoretical formulation did not appear until 1749 when Leonard Euler (1707-1783) explained a function as a variable quantity that is dependent upon another variable quantity. For many purposes such a roughly stated definition was sufficient, but the further development of mathematics

90

Five: T h e Truth-Functioned Calculus

brought a more general and abstract concept of function which makes use of class-theoretical notions. 1 This abstract notion is not restricted to dependence of quantities, e.g., numbers, but it comprises any fact of the correspondence between elements of a set, say A, and elements of another set, say V. Let it be so that to each element of A by the correspondence ν = φ(α) there is assigned exactly (not more, and not less than) one element of V. Then that element Vj of V which is assigned to (one or more) elements a i , a2,...,a n of A is said to be the value of the function in question for the arguments ai, a2,...,a n . In other words, the function φ is defined on the set A, called the set of its arguments, or its domain of definition, and ranges over the set V of its values, or its range. For example, the relation of having a father assigns exactly one individual to one or more individuals; in this example the domain of definition and the range are identical (e.g., the class of humans). These may also happen to be different; for instance, exactly one ticket corresponds to a group of participants (e.g., a family) of a performance (thus the domain is a class of humans, and the range, a class of documents). 1 . 2 . In some contexts the term 'operation' proves more convenient than 'function', esp. when referring to those functions which one used to call operations from the very beginning of one's school instruction. Such are arithmetical operations which perspicuously exemplify the concept of function as a correspondence: in the operation of division, for instance, to each two elements of the domain of natural numbers there is coordinated exactly one element of the range of rational numbers. The unification of logic on the basis of the concept function has been possible owing to Frege's idea that there are functions for which both the domain and the range equal the two-element class 1

Instead of the term 'class-theoretical' one often uses the term 'set-theoretical'

in contexts like the present one. T h e theory of classes is regarded as the elementary and basic part of set theory, the latter dealing with the issues of infinite sets.

Hence whatever is class-theoretical is also set-theoretical, but for

the present purposes I prefer the former term as connected with the historical development narrated in the previous chapter.

91

1. The functional approach to logic

of truth-values. T h e t e r m truth-value is t o denote two a b s t r a c t objects, one of t h e m called t r u t h , t h e other f a l s e h o o d . T h e next bold step was t o look at sentential connectives as connecting j u s t truth-values, a n d at t h e same time to disregard t h e o t h e r aspects of a sentence, as its meaning, its form, etc. This step was t o some extent prepared by t h e development of t h e calculus of classes with which Frege was fully familiar. 2 To notice t h e continuity (which does not yet diminish t h e surprising novelty of Frege's insights) let us recall t h e notions of t h e e m p t y class a n d t h e universal class, a n d of operations on classes, as discussed in t h e previous c h a p t e r (subsec. 1.3); t h e operations mentioned t h e r e are complementation and intersection. T h e complement of t h e e m p t y class a m o u n t s to t h e universal class, a n d vice versa, which is a trivial common-sense observation, as t h e e m p t y class and t h e universal class form together t h e universal class. In symbols: - 0 = 1 a n d - 1 = 0; e.g., if one takes t h e class of plants as universal ( l ) , then (in a universe of discourse so defined) t h e class of non-plants (—1) equals 0. T h e intersection operation as applied t o these two distinguished classes is defined by t h e following equations: 1 1 = 1, 1 - 0 = 0, 0 - 1 = 0, 0 - 0 = 0. Obviously, t h e elements which t h e universal class has in common with itself again form t h e universal class; there are no elements which would be common t o t h e universal class a n d t h e e m p t y class, i.e. t h e operation results in t h e e m p t y class, etc. It may be asked why one should dwell on logical facts so element a r y t h a t they belong t o t h e basics of general education. However, they deserve mention to s u p p o r t t h e claim t h a t such basic notions of elementary logic comply with t h e common mental conduct of people who s t a t e a r g u m e n t s in ordinary language. T h u s t h e functional approach which results in those logical notions proves t o be fruitful from t h e rhetorical point of view as well as well. This determines t h e strategy of subsequent discussion. T h e r e are at least two t r u t h - f u n c t i o n a l operations which u n d o u b t e d l y accord with 2

See G. Frege, Booles rechnende

p.180.

Logik und die Begriffsschrift

in Frege [1973],

92

Five: The Truth-Functional Calculus

some uses of ordinary connectives; this encourages us to define other connectives in terms of those basic ones, and so to examine relations between them and between their ordinary-language counterparts. 3 1 . 3 . Teachers of formal logic used to pay much attention to the socalled paradox of implication, which consists in a certain discrepancy between the inference rules for the ordinary-language conditional and their truth-functional counterpart. The problem is regarded by some scholars as serious; logical calculi, one claims, should help to check the validity of arguments, and this is possible only if inference rules are identical in ordinary language arguments and in the calculus applied to check them. There is also a problem, though less frequently raised, of the ordinary use of either ... or and the meaning of its formal counterpart, called disjunction. There are in fact some problems, but they result rather from too superficial a philosophy of logic than from an actual conflict between logic and ordinary language. This philosophy is related to what might be called folk logic; let this term be patterned on the nowadays fashionable phrase 'folk psychology'. 4 Folk logic involves the principle pertaining to logic itself that logic should rule the whole of human cognitive conduct or, at least, the conduct of a philosopher, a scholar, etc. This principle is associated with the development of logic from its very beginnings, from Aristotle's Organon up to viewing logic as Medicina mentis in the way characteristic of the Cartesian School (cf. Chapter Three, Subsec. 2.2). In modern logic this principle is hardly traceable, but there are traces of it in folk logic inherited from older logical theories. No wonder that they have a hold on some minds, namely those happily 3

This claim may seem so evident that it should not require any defence. There is, though, the great, even if recently decreasing, influence of "ordinary language philosophy" (which emerged after the Second World War and was centred in Oxford University) which makes it fashionable to disregard formal logic and to base a theory of argument by attending to the structure of ordinary usage alone. This was an understandable reaction to the excessive formalism of the prewar neopositivistic manifesto, but the reaction itself proved also exaggerated, and now the time is ripe for a more balanced approach. 4

The latter is discussed in Sterelny [1990], the book which is here of interest for some other reasons, too.

2. The truth-functional analysis of denial and conjunction

93

rooted in good old traditions, and so liable to overlook something new. Even if logic taken in the broadest sense, hence without any restricting adjective, should obey the above principle, this certainly does not pertain to formal logic, i.e., t h a t consisting of logical calculi. Due to awareness of this, the present discussion distances itself from those who first fall into a t r a p carefully devised by themselves, and then display a wonderful ingenuity to get out of it. This is why the exposition starts from two logical connectives which combine the feature of being basic, i.e., being such t h a t all the other connectives are definable in terms of them, with the feature of being closest to the meanings of their counterparts in truth-functional logic. Owing to t h a t definability relation, the closeness in meanings should be inherited by the connectives defined in terms of those chosen as basic. This forms the subject matter of the next section. 2. T h e t r u t h - f u n c t i o n a l a n a l y s i s o f denial a n d c o n j u n c t i o n 2 . 1 . For a moment we should forget logic and its applications and deal only with an abstract calculus of two objects, one of them being denoted by Ί ' , the other one by Ό' (any resemblance of these shapes to symbols known earlier is only accidental). Let the two-element set including 0 and 1 be both the domain of definition and the range of functions we are to define. We are free to create functions of as many arguments as we like, t h a t is one-, two- , three-argument functions, etc. For instance, one can define the three-argument function ψ by which to each ordered triple of O's and l's as arguments (the number of such orderings being 2 3 for any triple) there corresponds 0 unless the sequence consists of O's alone; in this case let the value of the function be 1, as expressed in the formula ^(0,0,0) = 1. As one can define functions of arbitrarily many arguments, there are infinitely many such functions. For reasons to become apparent later, we take into account only one-argument and two-argument functions. For the former category there are four (2 2 ) functions to be defined, for the latter — sixteen (4 2 ), as seen in the following tables; the symbol at the top of a column (a letter in T l , a numerical symbol in T2) can be used to name the function in question.

94

Five: The Truth-Functional Calculus X

A

Β

c

D

1

1

1

0

0

0

1

0

1

0

Before displaying the next table, let us take advantage of this one t o discuss the transition from the abstract calculus to its interpretation in logic. T h e interpretation consists in making t h e symbols Ί ' and Ό' stand for abstract objects called truth-values, namely the True and the False, respectively, and then making the function symbols (A, B, C, D) stand for operations on truth-values. Thus function C becomes the operation of denial, called also negation, which performed on the True yields the False, and performed on the False yields the True. It is not a priori granted t h a t each of such combinatorily obtained functions has an intuitive logical interpretation, or t h a t all interpretations are equally intuitive. In any case, the interpretation of Β as assertion — in a similar sense as saying 'ditto', i.e., confirming something having been already said — is fully intuitive. It is more difficult to find an intuitive realization for A and D, but these functions may have some technical uses. As for the functions Β and C, it is the latter, i.e., the denial, which has enormous importance for formulating and scrutinizing arguments. Its behaviour is exactly like the behaviour of the ordinary expressions which are attached to a single statement, either prefixed to the whole statement, as in the phrases 'it is not true t h a t ' , 'it is not the case t h a t ' , etc., or prefixed to the predicate, as in the word 'not'. In each case the True is transformed into the False, and vice versa. Thus the construction of Table T l , though made entirely a priori and listing mere possibilities, supplies us with a logical means which is omnipresent in actual arguments. A similar experience awaits us in the case of two-argument functions. 2 . 2 . Each argument in a pair takes either the entity 0 or t h e entity 1 as its value, and this yields four combinations, viz., (1,1), (1,0), (0,1), (0,0). For each pair of argument values there is exactly one value of the given function, t h a t is either 1 or 0. Four pairs require four such coordinations to be made in 16 possible ways, as shown in the following table.

95

2. The truth-functional analysis of denial and conjunction

xy

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1 1

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

1 0

1

1

1

1

0

0

0

0

1

1

1

1

0

0

0

0

0 1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

0 0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

T2

When we t a k e the s y m b o l s Ί ' and Ό' a s s t a n d i n g for truthvalues, it proves t h a t some functions listed in the a b o v e t a b l e can be interpreted a s English connectives. In p a r t i c u l a r , function 8 renders the behaviour of ' a n d ' . When ' a n d ' connects two s t a t e m e n t s b o t h referring to the T r u e (i.e., simply, being t r u e ) the connection results in the True, otherwise it results in the False. T h i s function is termed c o n j u n c t i o n . T h e r e is a noteworthy f e a t u r e of the ordinary use of ' a n d ' which cannot be rendered in the a b s t r a c t theory of truth-values. I suggest that this f e a t u r e be discussed in t e r m s of p r a x e o l o g y , i.e., the general theory of purposeful human action. A m o n g the basic praxeological categories are t h o s e of efficiency a n d e c o n o m y of action (this e x a m p l e hints at a relation between p r a x e o l o g y a n d decision theory; the latter p r e s u p p o s e s as given s o m e f u n d a m e n t a l concepts worked out by the f o r m e r ) . 5 T h e concepts of praxeology are deliberately so general t h a n they can have i n s t a n t i a t i n g a p p l i c a t i o n s in various d o m a i n s of h u m a n action, e.g., in economic activities as studied by L u d w i g von Mises [1949]. Certainly, the d o m a i n of h u m a n communication is an i m p o r t a n t field t o be studied with the help of praxeological categories, such as p u r p o s e f u l n e s s , efficiency, economy, rationality, utility, etc. S o m e of these concepts are t o be a d o p t e d to e x a m i n e the use of ' a n d ' as well a s other connectives. The term praxeology was first used in 1890 by Alfred Espinas in his article 'Les origines de la technologie', Revue Philosophique 15, year 30, 114-115, and his book published in Paris 1897, with the same title. The idea of praxeology forms the basis of the economic theories of Ludwig von Mises as presented, e.g., in his Human Action [1949], The Polish philosopher and logician Tadeusz Kotarbinski (1886-1981) pioneered a philosophical theory of human action under the same term; see, e.g., his Praxiology [1965]. 5

96

Five: The Truth-Functional Calculus

Anyone u t t e r i n g a c o m p o u n d s t a t e m e n t of t h e form ρ and q ( t h e letters indicate t h e places t o be filled by constituent statements) does so for a certain purpose, and this implies t h a t b o t h constituents carry useful information and t h a t there should be a nexus between their meanings. T h e latter condition prevents a situation in which two messages, b o t h useful for t h e addressee but pertaining t o very different subjects, would be combined in one conjunctive sentence. C o m p a r e , for example, t h e following utterances: (a) 'Here s t a r t s a slippery section of t h e road; anyway, scape on its sides is very attractive';

the land-

(b) 'Here s t a r t s a slippery section of t h e road and t h e landscape on its sides is very attractive'. T h e u t t e r a n c e (a) may be so interpreted t h a t there is none or only a slight connection between t h e components s t a t e m e n t s . T h e speaker is aware t h a t in t h e second s t a t e m e n t he allows himself t o express his casual association or remembering, not necessarily justified by its usefulness for t h e listener, while t h e first s t a t e m e n t is a warning t o express a care; this awareness of t h e lack of a closer connection is hinted t h r o u g h t h e use of 'anyway'. T h e use of ' a n d ' in (b) suggests a stronger connection. It is likely t o be interpreted as follows. T h e r e is an additional reason to be cautious, namely t h e danger of being distracted by t h e b e a u t y of t h e landscape in circumstances requiring special a t t e n t i o n , i.e., when driving a car. Were it not so, were t h e second member of t h e conjunction added only as a report on t h e speaker's esthetic impressions, this construction could then be charged with committing t h e praxeological error of t h e lack of w h a t I shall call communicative relevance; this kind of relevance consists in combining two or more messages into one syntactic construction, e.g., conjunction, only if there is a nexus between their meanings. W h a t nexus is required, is defined by t h e set of conventions governing t h e ordinary usage of expressions. 6 6

I take the convenient term kommunikative Relevanz from Posner [1972], a book in which this concept plays an important role. The present use of this term does not match its elaborated definition in the said book, it is rather introduced ad hoc, to take advantage of its suggestive associations.

2. The truth-functional analysis of denial and conjunction

97

T h e conjunctive construction (b) in the above example lacks communicative relevance in the case of being so interpreted t h a t there is no nexus of content or a purpose between the messages conveyed by its components. Such a lack of nexus belongs t o t h e category of praxeological fallacies in communication (other fallacies of this kind are to be discussed in what follows). These should be distinguished from logical fallacies in arguments. T h e latter are definable on the basis of a logical calculus as is to be shown below. 2.3. To assess the validity of an argument, t h a t is its being free from any logical fallacy, we need t h e concept of logical truth which, when applied to a logical formula, can be roughly defined as a formula true by virtue of meanings of its logical terms alone (a more precise definition is to be given later with t h e help of t h e t r u t h functional calculus). Another name for logical t r u t h is tautology. Let it be noted t h a t the latter term has its origin in ancient rhetoric. In its Greek form ravroXojia it appeared in the R o m a n rhetorical writer Dionisios of Halikarnas (at the time of Augustus) to mean a pleonasm, i.e., a redundant phrase t h a t does not convey any new information with respect to what has been said previously. The assimilation of this term t o logic is due t o Ludwig Wittgenstein (1889-1951) who in his Tractatus defined tautology as a sentence t h a t admits of any possible state of affairs (lässt jede mögliche Sache zu, item 4.462), hence does not provide us with any information about the actual world. The trait of being a tautology is conspicuous, e.g., in t h e denial of contradiction, t h a t is in saying t h a t two contradictory statements cannot both be true. Truth-functional logic offers us a calculatory method of checking whether a formula does or does not belong to tautologies. Let it be shown in the mentioned example referred to as t h e law of (the denial of) contradiction (the part in the parentheses is usually omitted). In the formulas of truth-functional logic, the English words ' n o t ' and ' a n d ' are replaced by some artificial symbols; let us accept the symbolic notation in which denial is rendered by t h e sign and conjunction by the sign Λ. 7 To perform calculation we need 7

A review of logical notations (each of them having a heuristic merit) can be found in Logic [1981], article 'Logic, modern' by W. Marciszewski, item 3. 8.

98

Five: The Truth-Functional Calculus

pertinent parts of Tables T 1 and T 2 . Let these parts be singled out as T N for denial (negation) and T C for conjunction; because of their use in calculating the truth of tautologies such devices are called t r u t h - t a b l e s . pAq

1

9 1

Ρ

TN

1

Ρ

-.ρ

1

0

0

1

0

0

1

0

0

1

0

0

0

TC

With the help of these truth-tables we calculate the truth of the law of contradiction CL

(ρ Α -.p),

where ρ is a sentential v a r i a b l e to represent any statement whatever, in the following way. In the case of ρ = 1, after substituting 1 for p, we have the following equalities: -.(1Λ -.1) = -.(ΙΛΟ) =

-.0=1.

In the case of ρ = 0 there are the equalities: -.(0Λ-.0) = -.(OA 1) =

-.0=1.

T h u s CL proves true for any truth-value of a statement which would fill up the place marked by the letter p. T h e existence of such forever-true assertions, the truth of which is provable in a purely calculatory way, is a real philosophical discovery. It was anticipated by Leibniz's notion of a statement being true in all possible worlds. In fact, at the place of ρ in CL there may occur any sentence whatever, hence any sentence describing any possible world (e.g, "all humans have spherical bodies"), and the assertion like CL will always prove true. Let me further exemplify the notion of tautology by a version of the theorem termed the law of simplification. SL

->((p A q) Α -ιρ),

~·((ρ A q) A ->q)

A s there appear two sentential variables, each being substitutable either by 1 or by 0, there are four possible substitutions, namely:

99

2. The truth-functional analysis of denial and conjunction

Table of substitutions

Ρ

q

substit.

1

1

i

1

0

ii

0

1

iii

0

0

iv

For substitution i, one obtains -•((Ι Λ 1) Λ -il) = -.((1 Λ 1) Λ 0) = —.(1 Λ 0) = -.0 = 1. A simple check for substitutions ii, iii and iv again shows t h a t the truth-value of SL equals 1, thus SL proves true in every state of affairs ('in all possible worlds'), i.e., it proves to be a tautology. 2.4. The above result sheds light on the difference between logical validity and praxeological correctness in communication, as discussed above (Subsec. 2.2). The lack of communicative relevance (a praxeological fallacy) in example (b) does not do any harm to the logical validity of argument which might be based on SL. Let us recall the statement in question: (b) 'Here starts a slippery section of the road and the landscape on its sides is very attractive'. By virtue of SL it can rightly be inferred from (b), assumed as true, t h a t its first component as well as the second one has to be true. For SL, due to the denial sign prefixing t h e formula, rejects the case t h a t (b) would be true whilst its first constituent would be not-true; hence, provided the t r u t h of (b) one has to acknowledge the t r u t h of its first constituent; the same deduction applies to the second constituent. Thus the argument proves logically valid even if its first premise, viz., the conjunction, lacks communicative relevance. This clearly shows t h a t a critical examination of an argument should be split into two parts, one concerning the logical aspect of the argument, the other its praxeological aspect. However, the latter aspect happens to be regarded as logical by folk logic which tends to subsume some other problems of correctness under logic, including those termed above as praxeological. This distinction should remove headaches occurring to those teachers of logic who give examples like (b) and then face the objection that there is (in the sense of folk logic) something illogical

100

Five: The Truth-Functional Calculus

in them. This prophylactic measure taken to prevent misunderstandings and objections will prove even more useful when dealing with other logical connectives, those to be discussed in the next section. 3. T h e truth-functional analysis of disjunction 3.1. Paul is either a man of genius or is crazy (example 1). Such a disjunctive statement may be heard when someone's ideas go far beyond either common views or an average capacity of understanding (example 2). Let this opinion be examined as an example of the ordinary use of disjunction. Let us imagine that the connective either ... or has somehow disappeared from English. Should it mean that we no longer be able to express disjunctive thoughts (like those in examples 1 and 2 above)? Fortunately, the answer is optimistic. The apparently scant set of connectives reduced to denial and conjunction proves, in fact, able to express disjunction and some other operations, though at the cost of a more complex or cumbersome linguistic structures. Example 1 can be translated into that impoverished language as follows: It is not the case that Paul is no man of genius and that Paul is not crazy. In the language of the truth-functional calculus, when abbreviating the first component of the denied conjunction as G and the second as C, one renders the statement under consideration in the following form: Exl -i(->G Λ -.C). Now, using tables TN and TC we can calculate which truthvalues of the component statements make the whole statement (i.e., the denied conjunction of denials) true and which make it false. For instance, when G = 1 and C — 0, the calculation runs as follows: ->(-il Λ -.0) =

ι ( 0 Λ 1) =

->0 =

1.

After having exhausted all the four substitutions, we obtain the following table.

101

3. The truth-functioncil analysis of disjunction

G

c

->(->£? Λ -ιC

1

1

1

1

0

1

0

1

1

0

0

0

3.2. Does the above result agree with our instinctive understanding of the connective either ... or? Suppose t h a t Paul is not a man of genius and is not crazy, as listed in the last line of T D * . Then the statement discussed as an example, according to the ordinary understanding of either ... or, becomes false, as is stated in t h e table, too. T h e same ordinary understanding confirms the results of calculation in those lines in which one of t h e statements is t r u e while the other is false. There may be a problem about the first line in which both component statements are taken to be true. Suppose t h a t Paul is both a genius and is crazy, which would even accord with some theories of genius (e.g., t h a t developed by T h o m a s Mann in his Doctor Faustus). One might then ask if the speaker had a good reason to use t h e form of disjunction. However, this question is praxeological, and not logical. Here the rule of communicative behaviour seems to function, in prescribing t h e following: CRD (for Communication Rule for Disjunction): use the disjunction only in those cases in which you do not know which member of the disjunction is true, while you know that one of them is true. Should this rule be completed to the effect that the last clause would read: 'you know that at least one of them is true'? Example 2 (at the start of this section) is meant to shed light on this question. 8 When the present a u t h o r uses the statement referred to as example 2, he follows t h e rule in question in t h e version involving the at least proviso. For the sake of convenience of description, let the disjunctive property to be a man of genius or crazy be referred 8 Let the quoted text be repeated for the reader's convenience. It runs as follows. Paul is either a man of genius or is crazy (example 1). Such a disjunctive

s t a t e m e n t may be heard w h e n someone's ideas go far beyond views or an average capacity of understanding ( e x a m p l e 2).

either

common

102

Five: The Truth-Functional Calculus

t o as D. T h e present a u t h o r is not certain whether the average person a t t r i b u t e s D t o P a u l because of t h e enormous difference of content between Paul's own ideas and h i s / h e r own or because of difficulties in u n d e r s t a n d i n g Paul's ideas; I can, however, be certain t h a t at least one of these circumstances accounts for t h a t opinion concerning Paul, and this means t h a t his s t a t e m e n t will also be verified when b o t h circumstances prove t o be t h e case. Analogously, the view a t t r i b u t i n g D t o Paul would be verified in t h e case of Paul's being b o t h a m a n of genius and crazy, provided t h a t those who maintain this view do interpret 'either ... or' in this non-exclusive way, viz., a d m i t t i n g t h e occurrence of both alternatives, called inclusive disjunction. T h e cautious clause provided is used because of t h e awareness t h a t there is an ambiguity a b o u t this connective in English a n d its c o u n t e r p a r t s in some other languages. Acting as t h e a u t h o r of t h e statement referred to as example 2, I am able to report on my own interpretation of 'either ... or' occurring in this s t a t e m e n t and t h u s to use this s t a t e m e n t as an exemplification of t h e connective defined in T D * . However, if someone prefers t o use 'either ... or' in t h e way which does not admit of b o t h alternatives, i.e., as exclusive disjunction, t h a t int e r p r e t a t i o n can be rendered in t h e t r u t h - f u n c t i o n a l language as well, namely it corresponds to operation 10 in Table T2 (see Subsec. 2.2 above). In some languages there are two words to do t h e d u t y for t h e English 'either ... or'. In Latin vel a n d aut correspond t o inclusive and exclusive disjunction, respectively. 9 W h e n using the t e r m 'disjunction' without any adjective, I mean t h e inclusive sense of disjunction, corresponding to operation 2 of T 2 , a n d expressed in terms of denial a n d conjunction in Table T D * above. Instead of rendering t h e disjunction a d o p t e d in T D * , it is practical t o devise a special single symbol for it; let it be 'V' (to suggest t h e meaning of t h e Latin vel). Here is its t r u t h - t a b l e ( T D * rewritten as T D ) . 9

A reader more interested in this subject should consult The Language of Reason by T. J. Richard (Pergamon Press, 1978), Sec. 7.5.

3. The truth-functional analysis of disjunction

Ρ 1

9 1

pVq

1

0

1

0

1

1

0

0

0

103

1

3.3. T h e correspondence between t h e ordinary connective 'either ... or' and its t r u t h - f u n c t i o n a l c o u n t e r p a r t does not e x h a u s t its whole meaning. T h e r e are two praxeological features to a d d something to t h a t meaning. O n e of t h e m derives from t h e requirement of communicative relevance, as previously stated for conjunction (in 2.4 above). In t h e ordinary use of disjunction n o b o d y would say something like this: 'either Newton discovered gravitation or Bill Clinton will succeed in reducing t h e US d e b t ' . There is no point in uttering such a s t a t e m e n t , in spite of its being t r u e (due t o t h e first component s t a t i n g a historical fact), hence such an u t t e r a n c e would be a praxeological fallacy. In addition t o sharing this f e a t u r e with c o n j u n c t i o n , a disjunctive s t a t e m e n t has one f e a t u r e which is characteristic of it alone. This specific f e a t u r e is defined in the rule CRD (see Subsec. 3.2 above). According t o this rule, a disjunctional s t a t e m e n t claimed as t r u e should express t h e combined s t a t e of certainty as t o its t r u t h as a whole, and of uncertainty as to t h e t r u t h of either of its members. In t h e ordinary use of disjunction, nobody would say something like: ' B o n n is either t h e capital of G e r m a n y or t h e capital of C u b a ' — when everybody, including t h e speaker himself, knows t h a t t h e first c o m p o n e n t is t r u e and t h e second is false. W h a t a b o u t a situation in which t h e speaker alone knows which component is t r u e while his audience does not? S u p p o s e t h e P r i m e Minister of a S t a t e were asked a b o u t t h e d a t e of a national election, to be a n n o u n c e d by him, b u t for t h e time being wished to conceal the d a t e without telling a lie; therefore he mentions two dates in the disjunctive m a n n e r ('either this or t h a t ' ) , one of t h e m being t r u e according to his knowledge, t h e o t h e r one being wrong. T h e question belongs t o w h a t might be called t h e ethics of communication, related to praxeology of communication; this should

104

Five: The Truth-Functional Calculus

perhaps be settled by recourse to the precept of telling the whole t r u t h with admissible exceptions to this precept. Whatever ethical solution would be right, from the rhetorical point of view it is advisable to take advantage of logical structures which allow of speaking the t r u t h without imparting too much information. To state this idea in more general terms, it should be said that the conjunction serves to augment information while the disjunction serves to reduce it. Let us consider a single statement p. It follows from, e.g., ρ A q, and not vice versa, hence the conjunction tells us more than its single component. On the other hand, the disjunction ρ V q follows from the single ρ (as well as from q), and not vice versa, hence the single component tells us more than the disjunction as a whole; the more components it includes, the less information it conveys. These logical laws provide speakers and writers with an expedient method of dosing information according to their purposes, yet without damaging the t r u t h . 4. T h e truth-functional analysis of conditionals 4 . 1 . If one is endowed with instinctive logic, one observes the consequence rules (example 3). If I use this statement as a case to be studied, this is due to some inspiration drawn from St. Augustine (example 4). The statement is closely related to the main point of this essay, it might even be used as an introductory motto; hence, it deserves being examined for both its content and its form. And what about St. Augustine? In his work De doctrina Christiana (iv, 3) he made a witty comment on the rules of eloquence, t h a t is certain rhetorical and logical canons. It reads: it is the case t h a t people observe rules because they are eloquent, not t h a t they adopt them to become eloquent. In his own words: implent quippe regulae, quia sunt eloquentes, non adhibent ut sint eloquentes.10 Both Augustine's remark and the tenet of this book emphasize the import of the instinctive skill at arguing. This, in turn, implies t h a t we need a logical theory first in order to better understand our minds, and only then to improve them due to such understanding. To sum up, example 3 is so chosen t h a t its form should serve as an instance of conditional statements, while its content should help 10

See Migne Latinus, vol. 34, p. 91.

4. The truth-functional analysis of conditional

105

in studying some relations between form and content in conditional assertions. To start with, as in the preceding section, we imagine that the connective 'if ... then ...', (henceforth abbreviated to i f ) , has disappeared from the English language. This does not mean that we are unable to express conditional thoughts such as those in examples 3 and 4 above. Again we can use denial and conjunction to express the same thought, or at least that part of it which is crucial for the logical validity of an argument. Example 3 can be translated into such reduced language as follows: It is not the case that one is endowed with instinctive logic and does not observe the consequence rules. Example 4 might take the following form: It is not the case that I use this statement [...] and this is not due to some inspiration drawn from St. Augustine. Or, in a freer translation: I use this statement not without some inspiration drawn from Augustine. Let us scrutinize example 3 with the truth-functional calculus, abbreviating the first component of the denied conjunction as Ε (for 'Endowed') and the second as Β (for 'oBserves'). Then the statement takes the following form: Ex3 -.(£? Λ ->B). Now, using Tables TN and TC we can calculate which truthvalues of the component statements make the whole statement, i.e., the denied conjunction, true and which make it false. For instance, if Ε = 1 and Β — 0 (see the second row in TC) the result of calculation is as follows: -.(1 Λ -.0) = --(1 A 1) = -.1 = 0. Taking into account one by one the pairs of truth-values listed in the other rows, we ascertain that in all remaining cases the examined function takes 1 as its value. This result is displayed in the following table called TI, where the letter I hints at the term implication, which is another name for conditional.

106

Five: The Truth-Functional Calculus Ε

Β

->(E A ->B)

1

1

1

1

0

0

0

1

1

0

0

1

4 . 2 . Do these results agree with the intuitive understanding of the connective if as guided by our logical instinct? Suppose the one mentioned in example 3 is identical with someone called Smith who is known to be endowed with instinctive logic and to disobey logical consequence rules; in symbols, ( l ) Ε A B. This instance obviously denies the statement 'If Smith is endowed with instinctive logic, Smith observes the consequence rules', that is ( 2 ) if E, then B. Since 1 is the denial of 2, it follows that the denial of 1 amounts to 2; this means that the statements ->{E Α -ιB) and if E, then Β do state the same fact. Thus at this point our intuitive reasoning agrees with the truth-functional analysis. Suppose that Smith does not enjoy any instinctive logic, and he observes consequence rules (for communicative relevance 'but' would be better than 'and' in the present sentence, but logically these connectives are equivalent). T o express this symbolically: —Έ Α Β (see the third row of T I * ) . Is, then, the conditional intuitively true?

T h e answer is in the

affirmative. Once we have agreed that if E, then Β means the same as ->(E Α ~>B), we should assert that - Έ A Β verifies this conditional as it verifies its equivalent, viz. ->(E A ->B). A n d the latter is verified, since when ->E is the case, then the statement Ε must be false; then any conjunction containing Ε (as is Ε A ->5) has to be false, hence the denial of such a conjunction as ->(E A -p)

q

Ρ

q)

or, equivalently, ("'Ρ

which bears the name of the famous mediaeval logician Duns Scotus, and which caught the attention of such champions of logic and There is a tremendous wealth of literature devoted to the logic of relevance. Let me just mention the classical two-volume position by Anderson et al. [1975], [1992], whose first volume is dedicated to W. Ackermann as the founder of this branch of logic. The first attempts to make implication closer to natural reasoning are due to Lewis and Langford [1932]. A very useful overview of the current state of relevant logic, combined with the author's own ideas and results, is found in Read [1988]. 11

4. The truth-functional analysis of conditional

109

mathematics as Bertrand Russell and Henri Poincare. T h a t attention is recorded in some anecdotes. Poincare [1924] used to say t h a t a novice in mathematics has troubles in proving theorems only u p to the moment in which he unconsciously assumes a contradiction; the rest then goes very smoothly, as he is able to prove anything he wishes, even a t r u t h . Russell, when asked to give an example of how any statement whatever, say t h a t Russell (a renowned atheist) is the Pope, might follow from the self-contradictory statement 5 = 2 + 2 , suggested t h a t 3 be subtracted from both sides of this supposed equality; it follows t h a t 2 = 1 , thus two different persons, viz. Bertrand Russell and the Pope, form one person; hence Russell is the Pope. There is nothing about this law t h a t should startle people engaged in arguing, even if they are not so sophisticated in logic as were Russell and Poincare. If a partner commits a contradiction, then he is expected to demand the acknowledgment of any claim he makes. For, according to his own convenience, he may make use either of one or the other member of the contradiction which, consciously or unconsciously, he has made. It should be noted t h a t the requirement of communicative relevance in the use of a conditional, as stated in the above quotations related to the logic of relevance, was formulated in a stronger form t h a n was done above in the cases of conjunction and disjunction. For the conditional, it is required not only t h a t its components be concerned with the same subject or domain; there is a yet stronger requirement, viz., t h a t of 'a necessary connexion' or of 'a logical nexus'. This new feature, though, should also be construed in terms of communicative relevance. The use of a conditional is subjected to the rule t h a t it is communicatively relevant only if a logical nexus holds between the antecedent and the consequent. 4.4. The requirement of communicatve relevance with regard to a conditional seems so essential t h a t it may be asked why this problem is disregarded in expositions of truth-functional logic and in comments concerning its applications. It proves, however, t h a t when in arguing one is guided by some suitably selected inference rules concerning the use of conditional, then one is not bothered by any paradoxical consequences. In what follows I shall give an

110

Five: The Truth-Functional Calculus

example of such common-sense rules, granting t h e communicative relevance. So far, t h e phrase 'inference rule' was used in this essay in an intuitive way, assuming t h a t whoever u n d e r s t a n d s t h e notion of inference and t h a t of a rule should properly construe t h e phrase in question. A more systematic t r e a t m e n t , in t h e sense of listing and c o m m e n t i n g some set of inference rules, is t o be f o u n d in t h e next c h a p t e r . T h e most elementary introduction to t r u t h - f u n c t i o n a l logic can be m a d e without formulating inference rules, due t o t h e t r u t h - t a b l e m e t h o d which makes it possible to check validity of any formula without inferring it from other formulas. Hence it is not necessary for t r u t h - f u n c t i o n a l logic to be constructed and presented in t h e form of a deductive system, t h a t is to say, a set of theorems in which each theorem is either asserted as an axiom, i.e., without a proof, or is derived from t h e previously accepted theorems with t h e use of inference rules. It is possible for t r u t h - f u n c t i o n a l logic to be constructed as a deductive system, and for some methodological reasons many aut h o r s present it in t h a t form, b u t this is not necessary in the present exposition as focussed on t h e use of t r u t h - f u n c t i o n a l connectives in ordinary-language arguments. However, t h e discussion of conditionals and their praxeological aspects can profit from a comment on a chosen example of inference rules. T h e means of inference offered by symbolic logic function either as theorems stating t h a t such-and-such is t h e case (logical theorems s t a t e t h a t for all possible worlds) or as rules concerning operations. Hence in constructing a deductive system one has a range of free choice, limited only by some conditions t o be satisfied by t h e system (as consistency, independence of axioms, completeness, etc.). For logical systems it is possible to be constructed without theorems at all, only in t h e form of a set of inference rules. Nevertheless, it is useful to take into account b o t h mentioned forms. Forms with particular pertinence t o t h e use of conditionals are discussed in w h a t follows. Firstly, there is t h e theorem called m o d u s p o n e n s after its old Latin n a m e modus ponendo ponens to mean t h e mode of reasoning which s t a r t s from a position, in t h e sense of an affirmation, and thereby leads t o an affirmation (while other modes start from a

4. The truth-functional analysis of conditional

111

negation, or lead to a negation, etc.). Here is its truth-functional formulation: [PP]

((p-g)Ap)-?.

The rule which is its counterpart, called either t h e modus ponens rule or Detachment Rule (the latter name will be preferred), runs as follows: [DR] from φ —up and φ infer φ. For instance, a politician bases his calculations on asserting the statements of the form: (i) If the Government creates unemployment, then severe industrial unrest will follow. (ii) The Government creates unemployment. Then by the virtue of D R the politician predicts severe industrial unrest (which may help him in finding a mode of persuasion). W h a t additional light can this shed on the praxeological relevance of a conditional (apart from points discussed in 1.3 above)? The use of conditionals in the context of D R belongs to the most typical ones. We need conditional statements in order to make inferences guided by t h a t rule. For this purpose one must believe in the t r u t h both of the conditional and of its antecedent. A n d , as to the former, people (except in some artificially constructed cases) have no other grounds for such belief as their knowledge (or, at least, belief) in a physical, conceptual, or else logical, nexus between the antecedent and the consequent. Only t h a t makes them sure t h a t if the antecedent is the case, then the consequent must be the case, too. Even if people are aware t h a t truth-functional logic allows them to assert a conditional without knowing such a nexus exists, they never do t h a t for this simple reason t h a t the knowledge about a nexus is (as a rule) the sole source of knowledge a b o u t t h e t r u t h of the conditional. Authors who present symbolic logic (if not guided by rhetorical considerations) are not bound to concern themselves with the discrepancy between the truth-functional definition of a conditional and its actual applications. W h e t h e r they comment on it or not, people accept conditionals within the range of their cognitive possibilities, and these restrict application to those cases which involve a nexus between the constituents; this is specially evident in the use of the detachment rule.

CHAPTER SIX

The Predicate Calculus 1. Subject, predicate, quantifiers 1.1. In the beginning was the Word teaches us t h e Gospel according to St. J o h n . B u t which kind of word? T h e N a m e , destined to play t h e role of t h e subject in a sentence, or r a t h e r t h e Predicate? A n d , if we opt for t h e Name, should it be t h e P r o p e r Name, as in ' J o h n t h e Evangelist', or rather t h e General Name, as in ' M a n ' ? These grammatical distinctions are of consequence for logic, too. In discussing t h e m I shall take advantage of a grammatical theory suited for t h e language of modern logic. T h e dilemma as worded above in t h e biblical quotation can be expressed in t h e theory called categorial grammar. T h o u g h t h e adjective 'categorial' refers t o a f e a t u r e common t o all g r a m m a r s , as each of t h e m offers a categorization, in this case there is a special reason t o take advantage of this term. Namely, t h e theory in question is built on a distinction which makes it possible t o develop a calculus of categories. It is the distinction between basic categories and derived categories of expressions. This terminology implies t h a t some categories are derivable from other ones; at b o t t o m there are those which are not derivable themselves but provide t h e rest with t h e basis for derivation. T h e set of rules to define valid derivations, namely those which result in syntactically coherent (i.e., grammatical) expressions, forms t h e calculus characteristic of such a g r a m m a r . T h u s , when metaphorically asking what kind of word was at the beginning, we raise t h e question a b o u t categories to be acknowledged as basic. T h e answer given in terms of categorial g r a m m a r allows us to clearly observe t h e grammatical difference between t h e language of Aristotelian logic and t h a t of modern logic: in t h e

1. Subject, predicate, quantifiers

113

former there is the basic category of general names which does not occur at all in the latter, while in t h e latter there is t h e basic category of individual names which does not occur at all in the former. As for predicates, in the standard version of predicate logic called first-order logic, there is an infinite set of predicate categories, each of them being directly derivable from t h e category of individual names. 1 The set of predicate categories is infinite since a predicate can be derived, t h a t is to say, formed out of one (individual) name, or two, three, four names, and so on (theoretically) to infinity (in practice, though, we deal with a finite and rather a limited number). The names (which in this context are always construed as individual names) giving rise to a predicate are called its arguments. T h u s there are one-argument (one-place, unary), two-argument (two-place, binary), three-argument (three place, ternary), etc., predicates. Let these categories be exemplified by t h e following predicates: '... is a cat', '...is a descendant of ...', '... lies between ... and ...', respectively, where each string of dots is to be filled u p by a name, and the number of such strings corresponds to t h e arity (i.e., t h e number of arguments) of a predicate. An expression belonging to a derived category is called a functor by analogy to a mathematical function sign which is also accompanied by a sequence of arguments. A functor is seen as an 'active' element in forming a compound expression out of simpler ones, hence the classification of functors is made both according to the category of expression which a given functor makes up and the categories of expressions from which the compound is made. The syntactic description of predicate logic involves two basic categories, viz., t h a t of names and t h a t of sentences; let them be 1

In logics of higher orders (see 3.3 below) there are predicates whose derivation from the basic category of individual names is not direct; directly a predicate of order η derives from predicates of order η — 1, i.e. those of which η-order predicates can be predicated. A brief information about higher-order logics is found in 'Predicate logic' by W. Marciszewski in Logic [1981]; this author gives an introduction to categorial grammar in the same volume, while its more advanced discussion is found in Buszkowski, Marciszewski, van Benthem (eds.) [1988],

114

Six: The Predicate Calculus

symbolized by t h e indexes ' n ' a n d ' s ' , respectively. A u n a r y predicate forms a sentence o u t of one n a m e ; this fact is conveniently symbolized by t h e c o m p o u n d index 's:n', where t h e letter before t h e colon (sometimes a fraction line is used i n s t e a d ) hints at t h e category of t h e c o m p o u n d expression, a n d t h e letters following t h e colon (or, w r i t t e n under t h e line) hint at t h e a r g u m e n t s from which t h e c o m p o u n d is p r o d u c e d . Correspondingly, a binary predicate is indexed as s:nn, while t h e expression resulting f r o m a sentence (such as an a r g u m e n t ) t r a n s f o r m e d into a n a m e should be indexed as n:s. 2 T h e s y n t a c t i c fact t h a t predicates in m o d e r n logic do not belong t o basic categories does not u n d e r m i n e their i m p o r t in t h e semantic dimension, t h a t is t h e role of conveying i n f o r m a t i o n . This role, which in t r a d i t i o n a l logic is played by a general n a m e , f u n c t i o n i n g either as t h e g r a m m a t i c a l s u b j e c t or as t h e predicate of a sentence, is in m o d e r n logic taken over by predicates; t h e (individual) names act only as elements necessary for syntactic c o n s t r u c t i o n , while t h e i n f o r m a t i v e f u n c t i o n belongs wholly t o predicates. 1 . 2 . Let us more closely examine t h e fact t h a t t h e predicate alone, a n d not in collaboration with t h e g r a m m a t i c a l s u b j e c t , is t o furnish i n f o r m a t i o n a b o u t t h e s t a t e of affairs referred t o by t h e sentence in question. T h i s semantic difference entails a radical difference of t h e syntactic s t r u c t u r e s of sentences. To explain this issue, let us consider t h e following sentence (in which t h e s u b j e c t p h r a s e is underlined while t h e predicate is in t h e slant t y p e ) : a:

Every m a n who names me t r a i t o r is lying like a villain.

T h e above sentence is a p a r a p h r a s e ( u n f o r t u n a t e l y , a clumsy one) of t h e following exclamation f o u n d in Shakespeare: 2

The index n:s indicates, e.g., the category of the functor 'that', as brought forward by the following analysis. 'It rains' is a sentence, and 'that' transforms it into the name 'that it rains'. That the latter is a name follows from the fact that it can be used as an argument of a sentence-forming functor, as 'is bad', to result in the sentence 'That it rains is bad'. It is not the only syntactic interpretation of 'that'; this particle has more philosophically and logically interesting intepretations (this problem is extensively discussed by Marciszewski [1988]).

115

1. Subject, predicate, quantifiers

ß\ W h a t e v e r in t h e w o r l d he is t h a t n a m e s m e t r a i t o r , v i l l a i n - l i k e he lies. W h i l e α is a single s e n t e n c e , d u e t o t h e f a c t t h a t it i n v o l v e s o n l y o n e o c c u r r e n c e of t h e v e r b ' i s ' , β is c o m p o s e d of t w o s e n t e n c e s , as seen in t h e d o u b l e o c c u r r e n c e of 'is' a c c o m p a n i e d b y d o u b l e o c c u r rence of t h e s u b j e c t ' h e ' . S u c h a s u b j e c t , a s b e i n g o n l y a p r o n o u n w i t h o u t any specific content, does not convey a n y

information,

h e n c e t h e t a s k of c a r r y i n g t h e w h o l e i n f o r m a t i o n is p e r f o r m e d b y t h e p r e d i c a t e s ' n a m e s m e t r a i t o r ' a n d ' v i l l a i n - l i k e lies'. N o w let us r e p l a c e t h e p e r s o n a l p r o n o u n ' h e ' b y t h e v a r i a b l e χ r a n g i n g o v e r t h e e n t i r e u n i v e r s e (as e x p r e s s e d b y t h e p h r a s e ' w h a t ever in t h e w o r l d ' ) , a n d let us m a k e t h e c o n d i t i o n a l s t r u c t u r e o f β more explicit t h r o u g h introducing the connective 'if...then'.

Thus

we obtain the following (again, underlining the s u b j e c t and slanting the predicate): 7 : For a n y χ if χ names me traitor, t h e n χ villain-like

lies.

T h e f o r m 7 is t h e t y p i c a l t r a n s f o r m a t i o n of a u n i v e r s a l o n e - s u b j e c t s e n t e n c e , in t r a d i t i o n a l logic r e c k o n e d a m o n g t h e s o - c a l l e d general or categorical s t a t e m e n t s , i n t o a c o n d i t i o n a l s e n t e n c e in w h i c h t h e w h o l e i n f o r m a t i o n is c o n t a i n e d in p r e d i c a t e s . 3

English,

like

o t h e r n a t u r a l l a n g u a g e s , c a n use b o t h f o r m s , c o n d i t i o n a l as well as c a t e g o r i c a l , w h i l e in t h e logical l a n g u a g e s e x a m i n e d h e r e , o n l y o n e of t h e s e m e t h o d s is a d o p t e d , n a m e l y t h e c a t e g o r i c a l in t r a d i t i o n a l logic (cf. C h a p t e r F o u r , S u b s e c .

1.2, etc.)

α-form and the

c o n d i t i o n a l 7 - f o r m in m o d e r n logic. B o t h forms should be carefully examined from the rhetorical v i e w p o i n t , f o r t w o r e a s o n s at l e a s t .

First, because we

practise

r h e t o r i c s in n a t u r a l l a n g u a g e s , w h i c h e m p l o y b o t h s t r u c t u r e s ; seco n d , t h e r e a r e s e r i o u s p h i l o s o p h i c a l m o t i v a t i o n s b e h i n d e i t h e r of t h e s e logical s t r u c t u r e s . 3

T h o s e p h i l o s o p h i c a l issues, in t u r n , a r e

The terms 'sentence' and 'statement' are used interchangeably, and so are the terms 'general statement [sentence]' and 'categorical statement [sentence]'. As to the latter pair, its first member was introduced in Chapter Four, Subsec. 1.2, where the adjective 'general' was more convenient as hinting at the generality of the subject discussed in that context. In the present context it is the adjective 'categorical' which proves more convenient as opposing the adjective 'conditional'.

116

Six: The Predicate Calculus

concerned with cognitive processes of consequence for t h e a r t of argument. T h e r e is an ambiguity in t h e m e a n i n g of t h e t e r m ' p r e d i c a t e ' which should be removed before we proceed with our inquiry. T h e a m b i g u i t y a p p e a r s in t h e sentences m a d e o u t of two t e r m s a n d copula, such as "She is an a r t i s t " . T h e parsing of such a sentence can result either in she [is an artist] or in she is [an artist] (as above, t h e underlining m a r k s t h e s u b j e c t , while t h e slant, here combined with brackets, m a r k s t h e predicate). T h e first parsing agrees with a s t a n d a r d g r a m m a t i c a l rule, t h e second follows t h e usage of t r a d i t i o n a l logic in which t h e copula a p p e a r s between t h e t e r m s , t h e t e r m preceded by it being called 'predicate 1 ( L a t i n praedicatum). To avoid ambiguity, I reserve t h e expression predicate for t h e case of twofold p a r t i t i o n , t o comprise copula a n d t h e n a m e t h a t follows it, a n d a d o p t t h e expression predicate t e r m t o t h e case of t h e threefold p a r t i t i o n in which t h e verb 'is' is predicated of two entities, one of t h e m denoted by t h e s u b j e c t t e r m , t h e other one by t h e predicate t e r m . This way of speaking is usual in describing t h e sentence forms of t r a d i t i o n a l logic. 4 1.3. As said with reference t o t h e examples a a n d /?, when t h e s u b j e c t t e r m is a p r o n o u n or a variable, it is not able t o convey any i n f o r m a t i o n a b o u t t h e e n t i t y which t h e sentence in question refers to. In such cases, t h e whole task of carrying information is p e r f o r m e d by t h e predicate t e r m . To simplify t h e m a t t e r , let us consider such a simple sentence as 'he is a liar', called atomic by logicians. An a t o m i c sentence is one t h a t consists solely of a predicate (as 'is a liar') accompanied by t h e a p p r o p r i a t e n u m b e r of terms, i.e., individual expressions, either c o n s t a n t s or variables; in a n a t u r a l language t h e role of variables may be played by p r o n o u n s (e.g., ' h e ' ) . A c c o m p a n y i n g expressions are called a r g u m e n t s of t h e predicate in question. T h e a p p r o p r i a t e n u m b e r of a r g u m e n t s d e p e n d s on t h e meaning of t h e 4

See, e.g., Kneale and Kneale [1962], p. 65.

1. Subject, predicate, quantifiers

117

predicate accompanied by t h e m . T h e meanings of '... is a liar', '....is big', '... walks', etc., hint at one entity t o be predicated of, hence predicates of this kind are called one-place, or o n e - a r g u m e n t , or u n a r y predicates. T h e class of two-place (binary) predicates can be exemplified by expressions like '... '... '... '... '...

is t h e f a t h e r of ...', is a friend of ...', is bigger t h a n ...', dances with ...', precedes ...';

while t h e expression '... lies between ... and ...' is an example of a three-place ( t e r n a r y ) predicate. Theoretically, t h e number of places (as marked by blanks in our examples) is unlimited; practically, it conforms to t h e needs of communication a n d capacities of our minds. It should be noted t h a t t h e triples of dots used above as blanks, t o indicate t h e n u m b e r of a r g u m e n t places, perform t h e same role which is characteristic of variables; t h e latter also m a r k free places t o be filled in. If one prefers t h e blanks technique, t h e n in t h e case when t h e same o b j e c t is t o be referred to more t h a n once, t h e place for t h e expression referring t o it should be marked with identical blanks, say a d a s h '—' while blanks for o t h e r a r g u m e n t s should have a different shape. T h e n one would p u t '... precedes —' t o express t h e same as t h e expression lx precedes y' does, while '... precedes ...' would correspond to ' ζ precedes x\ Because of t h e practical inconvenience of blanks technique, we use r a t h e r letters of various shapes, called variables. However, t h e analogy with blanks should be remembered t o properly u n d e r s t a n d t h e use of letters as variables; in logic we use letter a r g u m e n t s also for o t h e r purposes, hence seeing variables as blanks helps t o avoid m i s u n d e r s t a n d i n g s . 1 . 4 . T h e discussion concerning variables as blanks was to shed light on t h e n a t u r e of predicates, and their partition according t o n u m b e r of a r g u m e n t s . Now it is in order t o discuss t h e functioning

118

Six: The Predicate Calculus

of a r g u m e n t s . T h e r e are two m e t h o d s t o fill a blank with an expression referring to an object; if all blanks are filled, t h e predicate becomes a sentence. One of t h e methods consists in using a proper name. Let t h e place marked with 'a;' in 'a: is a liar' (or, equivalently, 'he is a liar') be filled with t h e proper n a m e 'Epimenides' to result in t h e sentence 'Epimenides is a liar'. In this context we can see t h e double role of p r o n o u n s . They may be used either in t h e function of variables or blanks, as shown in t h e examples above, or as substitutes for proper names. If in t h e presence of Epimenides one hints at him and says 'he is a liar', then ' h e ' means 'Epimenides'. It can be said t h a t such a procedure t r a n s f o r m s t h e pronoun (or, variable) ' h e ' into a proper n a m e which is m a d e out of this pronoun and t h a t situation which involves t h e gesture of hinting and t h e object hinted at. T h e r e is an English verb which fittingly describes w h a t is going on in a case like t h a t of referring t o Epimenides t h r o u g h 'he'. It is t h e word ' t o bind'. In t h e hinting procedure t h e expression 'he' becomes bound t o a definite person, say, t h a t of Epimenides. Due t o t h e binding, 'he' changes its linguistic function, it is no longer a variable, in spite of preserving t h e same physical shape. T h e same holds when a letter, say 'a;', is used as a variable. This can be b e t t e r seen in a generalized form of binding which is t h e following. Let us consider t h e view of a pessimist to t h e effect t h a t 'eve r y b o d y is a liar', or (equivalently) 'all people are liars'. In a half-symbolic language it can be stated as follows: 6: for any χ holds: χ is a liar. Now 'a:' has a different meaning t h a n inside t h e predicate lx is a liar'. In 6, 'a:' refers to any entity, i.e., to whatever in t h e world (in accordance with Shakespeare's phrase in ß, in 1.2), while inside t h e predicate in question it has no reference at all, lacking any meaning in t h e same way as an e m p t y space. Hence t h e phrase 'for any χ holds' (or, shorter, 'for any a;') performs t h e role of binding the variable i.e., of t r a n s f o r m i n g it into a symbol which refers to something. To distinguish these two roles, t h a t of a blank and t h a t of a symbol having reference, logicians decided to employ t h e t e r m s a free variable or real variable, and a b o u n d variable or apparent variable, respectively.

1. Subject, predicate, quantifiers

119

T h e adjectives ' b o u n d ' a n d 'free' are currently used nowadays, while t h e other pair is going out of use. However, t h e old-fashioned t e r m s are desirably suggestive. T h e p h r a s e 'real variable' is to remind us t h a t only n o n - b o u n d symbols, i.e., those f u n c t i o n i n g as blanks, are genuine variables, while binding deprives t h e m of t h a t function, so t h a t they occur as variables only apparently, on account of having been left in t h e same s h a p e as before binding. 5 T h e r e is a n o t h e r way of binding a variable, viz. with t h e p h r a s e for some ..., or (equivalently) there is ... such that (i.e.., satisfying w h a t follows; instead of ' t h e r e is' one may say ' t h e r e exists' or 'exists'). W i t h this phrase we obtain sentences like t h a t : ζ:

There is χ such that: χ is a liar.

If our universe ('whatever in t h e world') is defined as t h e set of all people, t h a t is, it does not involve apes, angels, etc., then ζ simply means: 'some people are liars'. T h e use of t h e plural in this translation is only for stylistic reasons, as t h e same can be said with t h e sentence: ' a t least one m a n is a liar' (this phrasing should be a d d e d to those listed above, as being a n o t h e r stylistic variant). T h e phrases used in δ a n d in ζ (including their synonyms), a p a r t from their common task of binding variables, have a n o t h e r f e a t u r e in c o m m o n . In a vague b u t unquestionable way they deal with some quantities, namely numbers of objects. For any means 'as many entities as are in t h e universe in question', while for some means 'not less t h a n one'. Even if not very precise, they hint at certain quantities, a n d this is why these phrases and their symbolic c o u n t e r p a r t s in formulas have been called quantifiers. It has been shown in t h e above discussion how quantifiers interplay with predicates in forming sentences. First, variables are added to predicates as their a r g u m e n t s t o result in t h e s t r u c t u r e of a 5

David Hilbert used different letter forms to distinguish these kinds of symbols: the lower-case letters from the beginning of alphabet for free variables, and those from the end for bound variables. This recommendable precision has not been followed by other authors, who regarded that the context is sufficient to prevent ambiguity. Common practice prevails, and therefore I do not follow Hilbert's way here, but it is worth remembering that through this simplification a useful notational device is lost.

120

Six: The Predicate Calculus

sentential formula (called also 'propositional formula', or 'open sentence'), next come quantifiers to form a full sentence, which, if we need to clearly distinguish it from a formula, is called a closed sentence. Quantifiers belong to the category called variable-binding ope r a t o r s . T h e r e may be other quantifiers besides the two discussed above, for instance 'there is exactly one', 'there are not less than two', 'there are infinitely many' (for various kinds of infinity), and so on. 6 However, in its standard version logic is content with two variable-binding operators which form sentences, namely the universal quantifier, to which a variable owes its referring to the whole universe, and t h e existential quantifier — t h a t to which a variable owes its referring to at least one entity in the universe. 7 T h e occurrence of these two categories, predicates and quantifiers, accounts for the fact that there are two equivalent names for modern logic, viz., p r e d i c a t e logic and quantification logic. In this essay, t h e term 'predicate logic', or 'the predicate calculus', is preferred as one that directs our attention to some comparisons between modern and traditional logic, the latter being seen as a logic of names.

2. Quantification rules, interpretation, formal systems 2 . 1 . T h e question of how to use a word or a symbol amounts to the question of its meaning, while meaning is what is being brought in by a definition. This general comment is in order at the start of the present section, for it leads to a moral that is both logical and rhetorical. T h e moral is addressed in particular to those who in the name of scientific rigour are forever demanding 'precise' definitions, meaning by this statements in the lexicographical form 'j4 means so-and-so'. Such people used to react with a contemptuous Ί do not understand' when a speaker was unable to recite such an 'exact' 6

More on this subject in Logic [1981], "Quantifiers" by S. Krajewski.

Besides quantifiers, there are variable-binding operators which play another syntactic role: when binding a variable, they transform a predicate not into a sentence (as the quantifiers do) but into a name. The latter are important from the rhetorical angle as helping to make logic closer to every-day arguments, hence special attention is paid to them later (Subsec. 3.4). 7

2. Quantification rules, interpretation, formal systems

121

definition. However, m a n y i m p o r t a n t notions, e.g., those occurring in t h e axioms of a theory, cannot be so defined (unless one commits an infinite regress). In such cases, an efficient m e t h o d of defining consists in showing words in use, and this practice should also be followed in everyday a r g u m e n t s and discusions. A convenient t e r m for this kind of definition is implicit definitions. Logic a n d m a t h e m a t i c s supply us with paradigms of t h a t procedure, a n d t h e case of quantifiers is in this sense paradigmatic. T h e use of a quantifier consists in either a d d i n g it t o a formula or omitting it in t h e course of inference. If this is done in a truth-preserving way, t h a t is t o say, a t r u e formula remains t r u e after having been so t r a n s f o r m e d , then t h e inference in question is logically valid. T h e listing of truth-preserving uses of a quantifier a m o u n t s t o a definition explaining its meaning; such a list can be regarded as a concise introduction to predicate logic. W h e n reading a classical text consisting of proofs, say Euclid, one clearly sees t h a t such operations were ubiquituous in t h e practice of reasoning, even if not codified in any logical theory available at t h e time in question. Such codification was first accomplished in contemporary logic. Before s t a t i n g t h e rules of using quantifiers, let me i n t r o d u c e a convenient n o t a t i o n . 8 T h e variety of phrases used for wording t h e universal quantifier in a n a t u r a l language will be represented by t h e symbol 'V', followed by t h e letter t o indicate t h e variable (within t h e succeeding formula) being b o u n d by this quantifier. Let Φ(χ) be any formula containing V . 9 T h e universal quantifier forms a formula like this: ν*Φ(χ) 8 There are other notations for quantifiers (see Logic [1981]); the one chosen for present purposes has the merit of being suggestive inasmuch as it depicts the universal and the existentiell quantifiers as stylized abbreviations for 'All' and 'Exists', respectively. 9

As usual in mathematical practice, the letters from the middle of the Greek alphabet will be used to denote any formula whatever, without hinting at its content and structure; the only relevant information is to the effect that the formula includes the variable being bound by the quantifier prefixing that formula.

122

Six: The Predicate Calculus

Analogously, various ways of expressing that an object is suchand-such, will be unified with introducing the symbol of existential quantifier which forms the following formula:

For each quantifier there are rules on introducing it into a formula and eliminating it from a formula. In the rules there occur symbols acting as proper names of objects satisfying the formula in question. These will be, so to say, dummy names, inasmuch as we do not deal with a concrete domain from which definite things might be picked up, but rather with a schematic representation of any possible domain. T h e symbols which are to function as schematic, or indefinite, proper names are lower-case letters beginning the alphabet, viz., a, b, c, etc. 2.2. W e eliminate the universal quantifier when we transform a universal proposition into a singular one, that is into a proposition referring to one from among the instances of the universal formula. This means that together with dropping the quantifier we replace the bound variable by the proper name of an entity satisfying the formula in question. T h e corresponding transformation rule runs as follows: [EU]

from ν χ Φ ( χ )

infer

Φ(α),

where a, as explained above, is the name of an arbitrary object satisfying the formula Φ. Let the inference be illustrated by taking 'x = x ' for 'Φ(χ)'; then from i y x ( x = x ) ' there follows 'a = a'. If one takes into account a definite domain, say that of natural numbers, then the above universal identity results in ' 1 = 1 ' , ' 2 = 2 ' , etc. T h e abbreviation which is to function as the label of this rule stands for Elimination of the Universal quantifier. T h e next rule to be considered is that of Introduction of the Universal quantifier. If a formula is satisfied by every entity in the domain in question as is, e.g., 'χ = χ' in the domain of all things, then it is allowed to be transformed into a universal proposition, that is to be prefixed by the universal quantifier. T h e assertion of its being so universally satisfied is expressed by taking it as a premise of inference. W i t h this comment in view, the rule is to be stated as follows:

2. Quantification rules, interpretation, formal systems

[IU] from

Φ(ζ)

123

infer

Analogical operations hold for the other quantifier. Elimination of the Existential quantifier means, as in E U , dropping the quantifier and replacing the respective variable by a proper name, but with the proviso [p] that the same name was not earlier introduced with eliminating the existential quantifier in another formula. Should one ignore this restriction, then a false proposition might result from applying this rule, e.g., to the formulas '3 c (x is a Hary and ' 3 r ( x is a sainty after replacing 'x' by the same name in both formulas. [EE] from 3χΦ(χ)

infer

Φ (α), provided [ρ].

T h e last rule to be listed sheds light upon what one calls the existential import of names as discussed earlier (Chapter Four, Sec. 2) and in this Chapter (Subsec. 3.4). It is the rule of Introduction of the Existential quantifier that runs as follows: [IE] from Φ(α) infer

3χΦ(ζ).

2.3. The statement of rules given above is most general, covering all possible structures of the formulas prefixed with quantifiers. The Greek letter Φ ( ζ ) represents any formula whatever, if only that formula contains the free variable i x \ It may be the simplest sentential expression involving a one-place predicate, as iPx\ or one with a more-place predicate, as or else an expression with more predicates combined by connectives (i.e., 'and', 'or', 'if...then', etc., e.g., i\/x(Px and Ryzy), as well expressions containing more quantifiers (either before the formula or inside it), as, e.g., ' V r 3 y ( P x y and V^Q^)'; another way of making a structure more involved depends on denying (i.e., prefixing with a symbol meaning 'it is not the case t h a t ' ) either the whole formula or some of its components. This way of representing arbitrary formulas ensures the desirable generality of the inference rules presented above. A s a rule, the structure of a formula is determined by the interplay of word order and punctuation signs; only in the so-called Polish notation, devised by Jan Lukasiewicz, does the syntactic structure depend on word order alone, but this theoretical merit is achieved at the cost of perspicuity. Hence, in practice, we benefit from both means of structuring expressions, using parentheses as

124

Six: The Predicate Calculus

the punctuation signs (no other punctuation devices are needed, since parentheses prove sufficient to define the scope of the symbols involved). In particular, parentheses hint at the scope of a quantifier, and so tell us which variables are bound by the quantifier in question. For instance, the two following conditionals have different meanings because of structural differences resulting from the scope of the quantifier: [1] VxPx->Qx, [2] Ve(.P« - Q*). In [1] only the antecedent occurs in the scope of the quantifier, while in [2], as marked by the parentheses, the scope extends to the end of the formula. 2.4. At last, the very notion of a formula should be defined. This Subsection is to deal with the concept of a formula, with interpreting formulas by reference to the universe of discourse, and with distinguishing between formal and interpreted languages. The concept of a sentential formula of a definite language, in short, a formula, is an expansion of the concept of a sentence; in logic, the latter term usually stands for an expression which has the grammatical form of a sentence, and involves no free variables. Every sentence is a sentential formula, but an expression with free variables is no sentence; it can be transformed into a sentence either through binding all variables or through replacing all free variables by proper names. More systematically, the concept of formula is defined as follows. We start from defining the set of atomic formulas. For this purpose, we must have the list of predicates (let them be symbolized by some upper-case letters) as well as the list of individual variables and names (i.e., proper names) in the language in question. The elements of the latter category, comprising symbols for individuals, are briefly called terms. Atomic formulas are obtained through juxtaposing predicates and terms in the following order: first a predicate, then as many terms as result from that kind of predicate: one term follows a one-place predicate (e.g., Pa, Pix, Qy), a pair of terms — a two-place predicate, i.e., referring to a

2. Quantification rules, interpretation, formal systems

125

two-place relation (Rab, Rax, Sxy), a triple — a three-place predicate, i.e., referring to a three-place relation (e.g., Rixyz, as in 'x lies between y and z), a n d so on; often t h e t e r m s , as a r g u m e n t s of t h e predicate in question, are put in parentheses a n d separated by commas, e.g., R\(x, y, z), b u t no ambiguity arises if one a b a n d o n s such p u n c t u a t i o n marks. Once having obtained t h e set of atomic formulas, we define t h e set of c o m p o u n d ones, t h e composition being of two kinds. O n e of them consists in prefixing a formula (either atomic or earlier obtained from atomic ones) with a quantifier, while t h e other involves combining earlier existing formulas by means of sentential connectives, such as ' a n d ' , 'or', 'if...then', t h e latter set including also symbols which may precede formulas, such as a negation sign (to be read 'it is not t h e case t h a t ' ) . T h e n u m b e r of such connectives varies d e p e n d i n g on certain linguistic conventions; we are not b o u n d to make decisions now in this m a t t e r , it is enough to note t h a t a precise definition of a formula of a given language takes into account all such symbols accepted for t h e system in question. Having t h u s settled t h e syntactic question of producing atomic formulas from predicates and t e r m s , and producing more comp o u n d formulas from less c o m p o u n d ones, we can s t a t e a crucial semantic problem: w h a t do such formulas refer to? For example, what is t h e formula 'V x (r = x)' a b o u t ? T h e answer sheds light on t h e t u r n brought a b o u t by modern logic in our thinking a b o u t language. B o t h in n a t u r a l languages and in traditional logic, a proposition possesses a meaning w i t h o u t any reference t o t h e whole domain of t h o u g h t being presupposed in t h e discourse in question. In modern logic, t h o u g h , a formula does not receive any interpretation until one defines t h e set of o b j e c t s t o be referred t o by individual variables. Such a set is called t h e universe of discourse or, shorter, t h e universe; it is said t o provide t h e language with interpretation. W i t h o u t interpretation a language is only a formal system, t h a t is a set of rules concerning formation a n d t r a n s f o r m a t i o n s of certain strings of symbols which do not d e n o t e anything. Such a separation between syntactic a n d semantic c o m p o n e n t s of a language has far-reaching consequences. It makes it possible

126

Six: The Predicate Calculus

for a computer to effectively manipulate linguistic symbols without any need of simulating their understanding; after these manipulations are produced it is up to a human being to give them meaning. And, once having understood the nature of a computer, we are better able to make illuminating comparisons between machines and organisms, especially human bodies as producing texts and dialogues. The means for producing formal systems when combined with the concept of interpretation enable us to introduce the concept of a formalized system, or a formalized theory. Imagine a system in which theorems are proved in such way that each inference step is justified by formal inference rules, i.e., rules which take into account only the physical shape of formulas. Imagine that, at the same time, the system is given an interpretation, hence its terms refer to objects in a definite domain. For instance, the variables of Boolean algebra constructed as a formal system become interpreted by assigning to them objects which belong to the universe of classes, and the operation symbols are interpreted as operations on classes; then we have to do with a formalized theory, i.e., one sharing the inferential rigour with formal theories and at the same time being intepreted. Owing to this combination, a formalized theory is manageable both by humans and by computers, and at the same time it has a meaning and importance for humans. This is exactly what Leibniz dreamt of, with the limitation that there are theories, as important as, say, arithmetic, in which it is not possible to prove all their truths in such a purely mechanical way; thus their formalization can be only partial. 10 Nevertheless, formalization remains a tool which essentially contributes to the use of computers as devices assisting human intelligence. 3. P r e d i c a t e logic compared w i t h natural logic 3.1. The presentation of symbolic logic in this and in the preceding chapters provides us with sufficient material to introduce the 10

This famous limitative result is due to Kurt Gödel (1906-1978) (Gödel [1931]). More information on this and other limitative results of modern logic can be found in Logic [1981], articles by S. Krajewski: 'Completeness', 'Consistency', 'Decidability', 'Recursive functions' 'Truth'.

3. Predicate logic compared with natural logic

127

concept of natural logic. Its nearest conceptual environment on t h e side of logic itself is formed by t h e notion of symbolic logic, especially predicate logic, a n d t h a t of objectual reasoning as opposed to symbolic reasoning. On t h e other h a n d , this concept is closely related to t h a t of cognitive rhetoric as a theory f o u n d e d on logic. 11 T h e English term logic means not only t h e science of correct reasoning, defining, etc., b u t also a reasonable thinking, a good sense. Let t h e t e r m theoretical logic be applied t o t h e f o r m e r , while the latter, being a quality or a conduct of mind, deserves t h e name of natural logic. In fact, there is a m u l t i t u d e of n a t u r a l logics, as great as t h a t of various n a t u r a l languages, or even as t h a t of h u m a n individuals; b u t even if there are so m a n y of t h e m , they have much in c o m m o n , sufficiently much t o form a vast field to be called ' n a t u r a l logic' in t h e singular. T h e field so called has t o be subjected t o some laws which, being concerned with behaviour, can be s t a t e d as certain rules. There then arises t h e question of how such rules of n a t u r a l logic are related to t h e rules of theoretical logic, especially its s t a n d a r d symbolic version known as predicate logic. Are these sets of rules identical, or disjunct, or else overlapping? May n a t u r a l logic profit from relations with theoretical logic, a n d vice versa? To answer these questions, it should first be realized t h a t t h e r e are two component p a r t s of n a t u r a l logic, a biological c o n s t i t u e n t and a cultural c o n s t i t u e n t , t h e latter being mainly linguistic. 1 2 Owing to t h e biological constituent, each of us is capable of objectual (material) inferences. T h a t is t o say t h a t in t h e reasoning a b o u t an object one may come to t h e t r u e conclusion without verbalizing either t h e premises or t h e conclusion. T h e reasoning consists then in a sequence of mental transformations of t h e o b j e c t in question. It can be nicely shown for some geometrical o b j e c t s , as 11

The concepts of objectual and symbolic reasoning are first introduced in Chapter Two, Subsection 3.1, and are then discussed in the context of generalization procedure in Chapter Seven, Subsec. 3.2. The definition of cognitive rhetoric is found in Chapter One, Section 1.3 in fine. 12

The biological constituent of natural logic is discussed in Chapter Seven, Subsec. 2.2, in relation to von Neumann's ideas.

128

Six: The Predicate Calculus

did I. Kant in the example of the proof t h a t the sum of the angles in any triangle equals two right angles. 1 3 Once an objectual inference is verbalized as a sequence of sentences, it can be examined to search for logical rules to justify particular steps in t h a t mental processing of the object. As far as mathematical reasonings are concerned, predicate logic proves sufficient to account for their validity. This means t h a t , in mathematical domains, natural logic, as an innate skill at objectual reasoning, can handle a similar range of problems as t h a t for which symbolic logic in its predicate version has been suited. Certainly, in a case like t h a t scrutinized by Kant, i.e., belonging to geometry, the biological constituent of natural logic is essential, because the intuition of space (to use K a n t ' s category) is apt to guide not only human but also animal reasoning. On the other h a n d , we should not disregard the linguistic constituent such as t h e geometrical terminology; without it, the problem of the sum of the triangle angles could not have been raised at all. However, t h e linguistic constituent in geometrical inferences may lack logical terminology; it helps, but one may do without it. 1 4 In this sense, the logic of reasoning like t h a t mentioned above has been called natural — as one marking people from their birth, or acquired by them spontaneously without any effort to master it, without any study of logical theories. The question raised in the title of this Section is concerned with the natural logic so conceived. W h e n it comes to its comparison with predicate logic, there comes t h e question of their relation to each other to be tackled in what follows. 3 . 2 . There are two methods of extending predicate logic beyond t h a t set of means which involves the categories of expressions and inference rules introduced so far. One of these methods consists in adding new axioms or new rules which are not derivable from 13

This example is comprehensively discussed in Chapter Seven, Subsections

3.2, 3.3, 4.1. 14

This observation is confined to reasonings concerning individual mathematical objects. When a reasoning is concerned with a whole mathematical theory, e.g., when its consistency is examined, then theoretical logic is necessary ex definitione, as a theory dealing with cognitive values of other deductive theories.

129

3. Predicate logic compared with natural logic

the existing ones; the other consists in defining new categories of expressions, with corresponding axioms or rules, with the help of already existing means. The latter does not increase the deductive power of predicate logic but makes it much more operative. Both methods deserve a careful study from the rhetorical point of view to confront the inferential means of predicate logic and those which people owe to their natural logic. The first and most usual extension consists in adding axioms which allow new deductions, and at the same define the symbol of identity '=' as a new logical constant, the binary predicate added to the truth-functional connectives and the quantifiers. It can be defined either by appropriate rules or by a set of axioms. Since the latter is an instructive example of what is called implicit definition (to be discussed later), it is advisable to formulate it as a set of axioms. It consists of two axioms from which other propositions characterizing identity can be deduced. The axioms are as follows. χ —χ (x = y) —+ (.A(a;) —• j4(y))

reflexivity; extensionality.

The terms 'reflexivity' and 'extensionality' are names of properties defined by the respective formulas. Here are the other properties characteristic of the relation of identity, derivable from the above axioms. (x — y) —• (y = x) ((x = y) A(y = ζ)) —• (χ = ζ)

symmetry; transitivity.

The identity theory added to the predicate calculus makes it possible to introduce new individual names. The method consists in forming a name out of a predicate with the help of the quantifiers and the identity symbol. All propositions which can be expressed in this new form, i.e., involving names, are also capable of being stated in the old form, i.e., with the use of predicates alone. Although this extension does not advance the deductive power of predicate calculus, it is of utmost importance, because it enables the introduction of function symbols into the language of mathematics and thus ensures maximum efficiency in computing (see Subsec. 3.4 and 3.5 below).

130

Six: The Predicate Calculus

Obviously, t h e relation of identity or, as it may also be called, equality, is not alien to n a t u r a l logic. However, at t h e linguistic level of this logic the symbol ' = ' does not have an exact c o u n t e r p a r t either in English or in similar languages. Expressions such as ' t h e same', 'identical with', etc., are not so convenient, so operative, in t h e syntactic aspect, while t h e verb 'is', which syntactically is most similar t o ' = ' , is b u r d e n e d with an ambiguity; sometimes it m e a n s t h e same as ' = ' , for instance in t h e context 'two and two is f o u r ' , b u t is different in t h e sentence ' t h e r e is an even n u m b e r ' ; in still other contexts 'is' should be interpreted as a c o u n t e r p a r t of t h e inclusion sign of t h e theory of classes. T h e moral to this comparison is t h a t we deal here with a case in which theoretical logic helps n a t u r a l logic in clearing up an ambiguity, and so contributes t o its e n h a n c e m e n t . 3 . 3 . Predicate logic can be developed without t h e theory of identity. However, there would be no point in such abstention since t h e predicate ' = ' is necessary for practical reasons in t h e language of m a t h e m a t i c s , and there are no objections to be raised against it from a philosophical point of view. This is why t h e theory of identity is usually seen as an integral p a r t of predicate logic. T h e r e is a n o t h e r extension t o increase t h e deductive power of predicate logic, one being neither so necesssary nor so undisputed, yet useful practically and interesting philosophically. It is t h e predicate calculus of second order or, shortly, second-order logic, from which t h e theory discussed so far is distinguished by t h e name of first-order logic. T h e vocabulary of t h a t first-order calculus, let it be recalled, contains, a p a r t from logical constants, individual variables (possibly, individual constants, too) and predicates. T h e s t a t u s of these predicates, symbolized in t h e foregoing exposition by capital letters, as ' P ' , ' Q ' , etc., was not discussed as yet. T h e present context gives us an o p p o r t u n i t y t o explain t h a t such letters should be int e r p r e t e d as predicate constants despite their being single letters, a n d not full-fledged expressions from a concrete vocabulary, say English. If we use letters in this role, it is for t h e sake of convenience; otherwise we would be b o u n d t o decide t o which domain t h e predicate calculus is t o be applied ( a n d so t o use predicates

3. Predicate logic compared with natural logic

131

concerning the chosen domain) instead of drawing attention to its universal applicability. Hence, such a letter functions as a predicate supposed to have a definite meaning which, however, is irrelevant to the validity of reasonings (being our sole concern in developing the calculus), and is therefore disregarded by us. In the second-order calculus, when we use single letters for t h e predicate category, we assign them a different role, namely t h a t of predicate variables; let Greek capital letters be such variables. Consequently, we need predicates of a higher order to be predicated of such variables, t h a t is to say, we need predicates to make sentential formulas out of such variables as their arguments. T h e appearance of t h a t second level explains why the new calculus is called second-order logic. The process of adding new levels can be continued to obtain next orders of logic. The addition of the second order of predicates has far-reaching consequences both in the technical and in t h e philosophical dimension. Technical advantages for mathematics are thoroughly discussed by Hilbert and Ackermann [1928] (a pioneering work in this field), also by Barwise [1977]; e.g., in arithmetic the induction principle can be conveniently stated in second-order predicate logic. As for philosophy, there is, for example, the well-known secondorder formalization of the Leibnizian principle of the identity of indiscernibles which runs as follows: x = y = \/ Φ (Φ(χ) = Φ(ν)). This formula opens a new prospect for the theory of identity, as it defines the predicate ' = ' in terms of equivalence and t h e general quantifier alone, without additional axioms. Its philosophical merits are obvious for those, say, who deal with the m a j o r problems involved in Leibnizian philosophy. Yet despite such a significance, it cannot be formalized in the first-order language. It should be noted, however, t h a t one takes advantage of t h e second-order language provided t h a t this language itself is not rejected by t h a t person for philosophical reasons. Objections raised by some critics are connected with the rule of introducing the existential quantifier (see above, Subsec. 2.2) which in second-order logic a m o u n t s to acknowledging the existence of abstract entities, the point decidedly objected to by nominalists. However, whether we succeed in

132

Six: The Predicate Calculus

either refuting or ignoring such objections or not, we have t o agree t h a t for some arguments, especially in philosophy, as in problems related to t h e Leibnizian principle, t h e second-order logic is an extremely useful device. T h e possibility of extending logic towards ever higher orders, i.e., categories of predicate variables, up to infinity, is an obvious a d v a n t a g e of predicate logic over natural logic. Only t h e means elaborated in symbolic logic are fit enough t o render t h a t infinite array of categories resulting in w h a t is called t h e predicate calculus of order omega. It is akin t o some versions of set theory which in its history proved to surpass t h e abilities of n a t u r a l logic. This extreme extension of predicate logic should be of special use in philosophical a r g u m e n t concerned with infinity, a realm so alien to n a t u r a l logic t h a t it may get lost in those new surroundings. On t h e other h a n d , even this far-reaching extension of logic towards a t r e a t m e n t of a b s t r a c t entities is not sufficient for some simple a r g u m e n t s dealing with a category of a b s t r a c t names. It is t h e category of names of properties. T h e a b s t r a c t entities of higher-order logics are not those which incessantly appear in philosophy, in humanities, and in every-day discourses, namely properties a t t r i b u t e d to individuals, and also t o other properties; t h e latter deserve to be called properties of higher orders but this analogy does not throw a bridge between predicate logic and natural logic. This discrepancy is worth a careful study, and for t h e present purposes let t h e following example illustrate t h e problem. A m o n g t h e most famous philosophical a r g u m e n t s are those stated by Descartes in his Discours de la methode. T h e sequence of a r g u m e n t s begins with t h e s t a t e m e n t CM (Cartesian Maxim) to t h e effect: (CM)

The good sense is a thing evenly distributed

among

humans

(in t h e original le bon sense, and bona ratio in a Latin version). Good sense is a property of individuals t h e possession of which by every individual can be rendered in first-order logic in t h e following form; (CM*)

VxS(x),

with t h e universe of h u m a n s and t h e predicate 'S' t o abbreviate t h e phrase '... possesses good sense'. In second-order logic we can

3. Predicate logic compared with natural logic

133

use a predicate having the predicate 'S' as its argument, but it is not what we need. What we need is the phrase 'good sense' alone to be used as the grammatical subject in a premise, and to denote that property of which another property is predicated, namely that of being distributed, and again of the latter property, namely distribution, it is predicated in the concise adverbial form that the distribution is even. Thus, something like natural logic of third-order is involved in such a short and simple statement, but it is not a logic likely to be rendered in a third-order predicate logic. Since (CM*) is expressible in natural logic, e.g., via English (as a language whose logic is part of natural logic), we can ask about logical relations holding between them. Obviously, the asterisked maxim follows from the other but not vice versa. If good sense is evenly distributed among humans, then each human is endowed with it; but from the latter there does not follow the fact of even distribution as stated in (CM). Neither inference nor the lack of inference can be ascertained by the third-order predicate calculus (as the only candidate, if any, to be authorized to settle these questions from the standpoint of theoretical logic). Since in predicate logic there are no syntactic and semantic categories for properties, properties of properties, and so on, there are no rules to guide and control inferences involving these categories. Nevertheless, in English, in French, in Latin, etc., there must inhere such rules, if not explicitly stated, then at least acting in an implicit way, so that we can be certain of the validity of such inferences, as well as capable of stating non sequitur, that is the lack of logical following, if it is the case. There is a means in predicate logic which makes it closer to natural logic as dealing with names of properties. It is the abstraction operator, i.e., one which transforms a sentential formula into a name of the class of those things which satisfy that formula; it does not bridge the gap in question but deserves to be mentioned in connexion with properties as abstract entities. 15 15

The abstraction operator plays a significant role in applications of logic, also to natural language, especially in its generalized form called lambda-operator. In spite of this role, which is of consequence from the rhetorical point of view, the discussion of this operator would exceed the limits of the present essay.

134

Six: The Predicate Calculus

T h e formal definition suited for one-argument predicates, where the cap above the variable plays the role of the operator and square brackets mark its scope, runs as follows:

Let ' 5 ' mean the same as above ('... possesses good sense'). This formula then means t h a t any human individual ζ belongs to the class of those possessing good sense if and only if he possesses good sense. The constituents of this equivalence describe the same state with t h e same words but in a different syntactical manner. On the left side there occurs the name 'class of ...' which does not appear on the other side. Obviously, whatever can be said in one of these ways can also be said in the other, and in this sense the extension of the language by adding the abstraction operator is inessential. Yet, when combined with the rule of introducing the existential quantifier, this definition results in the statement about t h e existence of a set, as do statements of second-order logic, e.g., the set of humans evenly endowed with good sense. Now we can predicate a property about t h a t set, e.g., t h a t it is nonempty, t h a t it contains more t h a n three members, etc. However, this does not help natural logic as concerned with properties of individuals (such as good sense), properties of such properties, etc. Classes as well as properties are abstract entities, and they are related to each other in an important way, yet they constitute different categories of abstract objects, irreducible to one another. This contrast emphasizes some features characteristic of natural logic which are not reflected in theoretical logic. These and other differences between the two logics require a diligent study in order to be explained and, hopefully, removed. Before such a study is undertaken, let it suffice to notice them to become aware t h a t for rhetorical purposes we must go beyond Some basic information on this subject is found in Logic [1981], esp. in the articles 'Combinatory logic' and 'Lambda-operator' by A. Grzegorczyk. A more advanced exposition of the lambda-calculus is given in Feys and Fitch [1969] and an instructive example of its linguistic applications is provided by Cresswell [1977],

135

3. Predicate logic compared with natural logic

theoretical logic, however sophisticated t h e a r r a y of ever-higherorder logics is. 16 3.4. T h e extensions which are discussed below are inessential, as mentioned above, in t h e sense t h a t when t h e e x t e n d e d logic is applied to a theory, t h e set of theorems derivable in it ( d u e t o t h a t logic) is t h e same as in t h e case of applying n o n - e x t e n d e d logic. This is not to m e a n , t h o u g h , t h a t t h e extension is of little use. Some extensions are necessary for reasons of practicality, t o develop m a t h e m a t i c s , as is, e.g., t h e introduction of function symbols, while other ones improve t h e technical side of a theory a n d , moreover, prove inspiring in a philosophical aspect. T h e first t o be discussed, b o t h for systematic a n d for historical reasons, are t h e theories of definite descriptions. Their beginnings go back to Frege [1893] a n d Russell [1905] b u t for present purposes it is enough to make use of t h e theory developed by Hilbert and Bernays [1934-39]. 17 T h e theory devised by David Hilbert ( t h e main a u t h o r ) is in accordance with t h e inferential (i.e., ruleoriented) approach a d o p t e d here in regard t o quantifiers, and constitutes a useful introduction to t h e later discussion of definition ( C h a p t e r Eight, Subsec. 2.3 and 2.4). This is why it has been selected for t h e present purposes. 3 TA(x) VxVy(A(x)AA(y)^x A(txA(x))

= y)

For purposes of t h e present discussion it is more convenient t o present t h e above rule in such a form as t o indicate possible occurrences of free variables z\,..., zn: (1)

3xA(zi,...,zn,x)

(2)

V x Vj,(A(zi,. ..,zn,x)AA(zu.

(3) 16

..,zn,y)-n

= y)m

A(icA(zi,...,zn,x)).

This reservation about the applicability of higher-order logics should be combined with attempts to take as much as possible from them for understanding and developing the logic of natural languages. Such an attempt is made by Gallin [1975], 17 A review of various approaches as a historical introduction to the problem is found in the article 'Definite description' by W. Marciszewski in Logic [1981].

136

Six: The Predicate Calculus

Formula (1) is called the existence condition, and formula (2), the uniqueness condition. This rule makes it possible to eliminate the existential quantifier in those cases in which it has been proved that the object satisfying the formula A exists and there is no other object satisfying A. Then we are allowed to introduce the following definite description: (4)

irA(zu..

.,zn,x),

and on the basis of (1) i (2) to obtain (3), and at the same time to define the function (5)

f(zlt...,zn)

=

iTA(zi,...,zn,x).

From (3) and (5) we can derive (6)

A(z1,...,z„,f(zi,...,zn)),

hence a formula in which the existential quantifier does not occur. Thus the elimination of the existential quantifier is an operation which consists in omitting the quantifier and replacing a variable (formerly bound) by the individual constant defined by the given description. However, we hardly have a definite description at hand when dropping the existential quantifier. Then we use a symbol, say 'a', to stand for any object which the predicate in question refers to, as prescribed by rule E E (cf. 2.2 above). It is not necessary for truth-preserving that a be unique, we content ourselves with its existence. Hence the rule which introduces 'a' to the language is like the formerly stated rule for definite descriptions, but with the difference that the uniqueness condition is omitted. An expression introduced to a language by so liberalized a rule is called indefinite description. In those natural languages which possess articles, the definite article is what forms a name being the counterpart of a definite description, and the indefinite article forms what corresponds to an indefinite description. In a language which lacks articles, as Latin, the counterpart of indefinite description can be characterized in terms of member supposition as discussed in Chapter Four (Subsec. 3-1). At the same time, a name which satisfies only the existence condition and is allowed to designate more than one object resembles

3. Predicate logic compared with natural logic

137

t h e familiar general n a m e s of traditional logic; its semantic interpretation is as t h a t of t h e non-empty predicate of which it has been formed. T h e t r e a t m e n t of general names as indefinite descriptions settles t h e question of t r a n s l a t i n g universal s t a t e m e n t s of traditional logic into class logic a n d predicate logic. T h e problem was raised in t h e discussion of t h e theory of classes ( C h a p t e r Four, Subsec. 2.1 and 2.2) a n d t h e r e two m e t h o d s of class-theoretical interpretation of t h e universal s t a t e m e n t s were presented, one of t h e m called strong interpretation, t h e other called weak interpretation. 1 8 Let us a d o p t t h e same distinction for universal s t a t e m e n t s rendered in predicate logic ( U A means universal s t a t e m e n t , t h e subscripts 'w' a n d 's' hint at t h e weak and t h e strong i n t e r p r e t a t i o n , respectively). [UA«,]

Mx{Ax-*Bx)

[UA,] Vx(Ax

Bx) Λ 3xAx.

T h e weak interpretation consists in t r e a t i n g U A as a s t a t e m e n t on non-existence. T h u s , 'Every Cretan is a liar', m e a n s t h e same as ' T h e r e are no C r e t a n s w h o are not liars'. This s t a t e m e n t would remain t r u e even if t h e r e were no C r e t a n s at all, i.e., if t h e t e r m ' C r e t a n ' were empty. T h a t interpretation is confirmed by an analysis of UA„,. Let t h e above sentence be rendered as [1] Vx(Cx -f

Lx),

where respective letters are abbreviations for t h e predicates 'is C r e t a n ' and 'is a Liar'). Now, using t h e rule EU (Elimination of Universal quantifier), we o b t a i n : [2] Ca —• La; T h e next step consists in expressing t h e above conditional in t h e following form: [3] ->(Ca Λ -.La); 18

See, e.g., Lesniewski [1992], vol. 1, p. 377. Illuminating historical data as to existential import of universal propositions are found in Kneale and Kneale [1962], and in Simons [1992], The latter also comments on this problem in Franz Brentano, recent free logics, and especially Lesniewski.

138

Six: The Predicate Calculus

after applying the IU rule (Introduction of Universal quantifier, see 2.2 above), we again have a quantified expression, viz.: [4] V*-.(Car Λ ->Lx) to the effect t h a t it is true of everybody t h a t he is not both a Cretan and non-Liar, t h a t is to say, t h a t : there does not exist anybody who is a Cretan and is not a liar. The last transformation is due to some relation between the existential and the universal quantifier, not discussed till now but being so intuitive t h a t it can be seen when comparing [4] and the following: [5]

->3x(Cx

A

-iLx).

T h e last formula demonstrates t h a t from any universal affirmative statement there follows a negative existential statement, i.e., a statement about the non-existence of any object having such-andsuch properties (here the properties of being a Cretan and of notbeing a liar). T h e relationship between predicate logic involving descriptions and the natural logic of articles (and similar constructions) requires a careful f u r t h e r study in two directions, the philosophical and the linguistic. T h e first can be exemplified by the extensive and thorough study by Ε. M. Barth [1974], the latter by a chapter in Hans Reichenbach's inspiring textbook [1948] trying to apply symbolic logic to natural language. Some authors look for still other ways to bring symbolic theoretical logic closer to natural languages. Peter Simons, for example, stated a program of modern theoretical logic in a way related to traditional logic and modernized according to S. Lesniewski's principles. 19 3.5. The theory of descriptions throws a bridge between predicate logic and the concept of function belonging to key concepts of 19

Simons [1992a] promises an introductory textbook to complete in some way Lesniewski's logic, which he sees as bridging the gap between traditional and modern logic. Cf. Simons [1992] (the chapters concerning Lesniewski's logic). An illuminating introduction to Lesniewski is due to G. Küng in Logic [1981] (see also Küng [1967]). As for Hilbert, whose approach is followed in this discussion, he did not develop the description theory towards applications to natural language, for he devised it as a step only in the proof-theoretical procedure of eliminating quantifiers.

139

3. Predicate logic compared with natural logic

mathematics. 2 0 Here is one of those points in which the great program of unifying mathematics and connecting it with logic, initiated by Frege and Russell, has been successfully accomplished. A function / is a rule which assigns to each member a; of a set X a unique element y from a set Y (not necessarily different from X). The set X is called the domain of/, the element y assigned to χ is called the value of / at x. In other words, a function is a rule for setting up a correspondence. To express a correspondence we need a two-place predicate; obviously, a description can be formed of a two-place predicate, provided it is, so to speak, a relational description. At the same time the requirement of uniqueness ensures, in each case of correspondence, the uniqueness of the value corresponding to each element. To adduce some natural-language counterpart to the transition from a predicate to a functional expression, let us consider the predicate 'is father of' used in the context (i)

of x, abbreviated as

y is the Father

F(y,x).

Its functional counterpart is the name 'the father of' in the context (ii) y = the father

of x, abbreviated as y =

f(x).

In this example (ii) is a rather deviant representative of natural language since the equality symbol hardly belongs to the vocabulary of ordinary English, which at this place would provide us with the verb 'is'. This deviation is deliberately committed for two reasons, namely, to distinguish (i) from (ii), which otherwise would take exactly the same form as (i), and to exemplify how a natural language can be freed from some ambiguities with the help of theoretical logic. The general method of making functions from descriptions is reported above (Subsec. 3.4) as the Hilbertian procedure of eliminating the quantifiers (see line (5)). Applying this method to the present example, we obtain: (iii) iyF(y,x)

=

f(x),

where on the left side one puts the description formed of the predicate formula (i), and on the right side, the name formed by the 20 This concept is also mentioned in Chapter Five, Subsec. 1.1 in connection with the idea of the truth-function.

140

Six: The Predicate Calculus

function symbol ' / ' . Since the left side denotes the unique y being the father of an x, it can be replaced by the single symbol y, and thus there holds: y = f(x). Despite the said ambiguity of the verb 'is', being either part of a predicate or a counterpart of the symbol ' = ' , natural logic avails itself of the concept of function in the sense explained above, connected with definite descriptions. This is due to the instinctive method of identifying an individual by descriptions which can be successively added as the need arises; e.g., to tell a John Smith from other so named individuals, we ask about his father's name, which amounts to using a relational definite description; if it does not suffice, we ask about a temporal (the birth time) and a spatial (the birth place) relation, and so on. This leads to the conclusion t h a t in this respect the predicate calculus with functions and natural logic remain in perfect agreement. The theory of descriptions yields encouraging evidence t h a t predicate logic and natural logic have much in common, and each of them can be better understood in t h e light of, or in contrast to, the other. Moreover, owing to predicate logic, natural logic becomes more conscious of its own laws and powers but also of its limitations, for instance those in dealing with infinity. On the other h a n d , natural logic challenges predicate logic by posing new questions, such as t h a t of dealing with orders of properties. To handle all these problems is a task for a further inquiry which should bring predicate logic closer to natural logic, in some points as close as t h e old syllogistic was, and, on the other hand, make natural logic still more sophisticated.

CHAPTER SEVEN

Reasoning, Logic and Intelligence 1. D o e s a logical t h e o r y i m p r o v e n a t u r a l intelligence? 1 . 1 . The outstanding philosopher Karl Popper once remarked that at the beginnings of human intellectual history it was the deeds of liars which incited a reflection on logic.1 We may assume that the first (and almost human) function of descriptive language as a tool was to serve exclusively for true description, true reports. But then came the point when language could be used for lies, for "storytelling". This seems to me the decisive step, the step that made language truly descriptive and truly human. It led, I suggest, to storytelling of an explanatory kind, to myth making; to the critical scrutiny of reports and descriptions, and thus to science. Each of the listed activities of the mind requires intelligence, be it a simple description, an inventive storytelling, or a critical scrutiny of either of them. Among the theories created in the early stage of our civilization, there emerged one which set the task of the critical scrutiny of descriptions, explanations and arguments. We now call it logic. Its core is formed by the theory of valid reasoning, while valid reasoning means obeying rules which ensure the truth of the conclusion from true premisses; thus they control truth-preserving ( s a l v a veritate) transformations. Logic equips us with the codification of such prescriptions, called inference rules, and with a conceptual apparatus to deal with them. Thus logic was always regarded as a device to support our natural intelligence in its critical activities. However, there was no point in speaking of natural intelligence before the rise of the research field concerned with artificial intelligence, abbreviated as AI. The AI theory is a branch of computer 1

See Popper [1976], p. 189 f.

142

Seven: Reasoning, Logic, and Intelligence

science which aims at understanding the nature of human and animal intelligence and specifically at creating machines capable of intelligent problem-solving. AI is closely connected with logic, which provides it with one of the main components of artificially intelligent systems (data-bases, heuristic rules, logical rules of inference). 2 Owing to the rise of AI and owing to the fact t h a t AI engineers are fully aware of the role of logic in their constructions, we may have a fresh look at the problem of how logic improves natural intelligence. Is it in the same manner which is applied in developing artificial intelligence (to endow it with more axioms, more rules, etc.), or some other way resulting, possibly, from insuperable differences between organisms and mechanisms? The question cannot be avoided, should we take the intentions of the founders of logic seriously. As reported in Chapter Three, in both the Cartesian and the Leibnizian trends in logic learned authors contended t h a t the study of logical precepts should improve natural human thinking; and t h a t complied with the purpose of Aristotle himself. Also nowadays people happen to believe t h a t one who is expert in logic should reason more efficiently than the rest of his neighbours; likewise, for instance, a person trained in mathematics surpasses the laymen in the ability of solving mathematical problems. The line of reasoning which leads to this view is roughly as follows. Logic is the theory concerning most general methods of solving problems, t h a t is the methods to be applied in any domain whatever, as are modes of reasoning, defining, etc. Provided t h a t such a theoretical knowledge concerning practice improves t h a t practice itself (as is the case in mathematics), it seems to follow t h a t among persons having at their disposal the same factual premisses, the logician has advantage over the others in the skill of reaching conclusions. T h a t skill, in t u r n , is characteristic of any 2

The notion of natural intelligence has already entered the conceptual repertoire of AI students as a new technical term. See, e.g., Callatay [1992], The phrase abbreviated as AI is construed either as the name of the field of research or the name of the subject matter of this research. To avoid confusion it is advisable to use upper case initials to refer to the former, and lower case letters to refer to the latter.

1. Does a logical theory improve natural intelligence?

143

keen mind. Namely, to reach a conclusion means to solve a previously stated problem (as can be seen in detective stories). A n d the ability of efficient problem-solving constitutes the core of what we call intelligence. Ergo, a flawless and efficient reasoning belongs to that core. When scrutinizing the validity of the above conclusion, I do not mean to challenge the prestige of logic or logicians. I see logic as an enormously significant factor in the development of our civilization; yet this is not to its advantage when it is expected to do things which it does not do, while its actual merits may happen to be overlooked. 'To take the bull by the horns', we should start from the fundamental distinction between verbalized reasoning and unverbalized reasoning. It is in the very nature of a logical theory that the forms of inference studied and codified by it are verbalized forms; hence the role of logic for the improvement of reasoning depends on how far verbalization is necessary for endowing reasoning with the required validity. To attack this problem, let us first take into account the skill of reasoning as found in mute animals. 1 . 2 . There is a widely known case, that has already became a classic, of unverbalized problem solution, viz., that of Köhler's chimpanzee Sultan which fitted a bamboo stick into another, after many attempts to solve the problem of grasping fruit that was out of his reach. In spite of the whole distance between a human and an ape, a human would react in a similar way, as it is the only correct solution, and similarly he would not need any verbalized inference. The whole process of reasoning can be done silently in one's imagination; it consists in processing a mental image of two things, viz., the stick and the fruit. Before the agent fits a bamboo stick into another, he tries this strategy in a wordless Gedankenexperiment which leads to the hypothesis that with an extended stick one would overcome the distance to the fruit. This encourages one to externalize this imagined action in the form of overt, actual, behaviour. Here we deal with a doubtless case of what I have termed o b j e c t u a l inference, also called material, in contradistinction to symbolic inference, also called formal (Chapter Two, Subsec. 3.1).

144

Seven: Reasoning, Logic, and Intelligence

T h e story shows that in some cases the use of words is not necessary for a reasoning to obtain an intelligent solution. However, one may argue that owing to a linguistic articulation of the problem and applying logical rules to it, the reasoning would be somehow enhanced. I intend to produce an example of such an articulation in order to examine the interaction of the objectual and the symbolic-logical component, and to estimate how much either contributes to the conclusion. This symbolic articulation will be of the kind called formalization, that is such that inference consists in processing symbols as geometrical forms, without any reference to their meanings, and each step in this process is explicitly legitimized by mentioning inference rules of the kind discussed in the preceding chapter (cf. Chapter Six, Subsec. 2.4). Such formalized proofs, when compared with those occurring in ordinary practice in mathematics and other fields, are long and cumbersome, hence they do not occur as arguments in a discourse, yet they prove indispensable for two other purposes, namely ( i ) for the research concerning certain logical properties of deductive systems, such as consistency, completeness of inference rules, independence of axioms from each other, etc., and ( i i ) for the application of computers to reasoning. T h e computer, or rather the program which the computer is fed with, is either a prover, i.e., a software to prove theorems, or a checker, i.e., a software to check the correctness of a proof produced by a human. In either case the proof in question is formalized since the computer is capable only of processing physical objects (which are configurations of electric pulses), and not of processing their meanings (if attached to physical entities). There is something remarkable in the case of checker as involved in the interaction between humans and computers. Namely, the proof to be checked should be fully formalized as required by the computer, and at the same time it should be comparable with proofs appearing in the human practice as far as its length and conspicuity is concerned (only then would the amount of effort put into producing the proof be surpassed by the advantages of automatic checking). T h e logical examination of Sultan's reasoning should consist in its formalization; the difference between objectual and symbolic

2. The internal logical code in human bodies

145

reasoning will then be most conspicuous, since formalization is the most perfect version of symbolization, i.e., verbalization in a symbolic language. To render Sultan's reasoning in a formalized way, I shall make use of a convenient method of formalization which has been devised as an interface between humans and computers in the process of automatically checking proofs. 2. T h e i n t e r n a l logical c o d e in h u m a n b o d i e s 2 . 1 . The system to be used for the present purpose consists in combining a checker with a many-sorted system of logic, t h a t is to say, a system which differs from t h e s t a n d a r d one presented in Chapter Six, by the fact t h a t it introduces local universes of discourse, i.e., varying from proof to proof, and admits of as many universes as one needs in the proof in question. 3 This trick alleviates the burden characteristic of formalization because it makes formulas considerably shorter. For, once having defined the sort of objects to be discussed, one is not bound to introduce predicates for denoting t h e classes introduced as sorts. The system to be used below is called Mizar MSE, the term 'Mizar' being its proper name (in a random way chosen after t h e name of a star), and the suffix 'MSE' being the abbreviation for Many-Sorted (predicate calculus with) Equality, t h a t is a version of Mizar belonging to the most elementary ones (the other versions provide the user with functions, metalinguistic devices, reference apparatus, etc.). It should be noted t h a t the formalized proof stated below is far from being typical of those usually produced in Mizar MSE, as the preparatory p a r t , i.e., t h a t which forms t h e section called 'environ' is unusually long when compared with t h e length of the proof itself. Usually, there is no point in producing such logically trivial demonstrations, but it is just t h a t logical triviality which should be shown in t h e present discussion. This demonstration was tested by Mizar's checker and assessed by it as valid, hence the formalization performed is faultless according to t h e standard of the adopted system. T h e proof is recorded literally in the way required by Mizar MSE so t h a t a reader interested in t h e 3

As for many-sorted logics, see Barwise [1977a].

146

Seven: Reasoning, Logic, and Intelligence

technical side could rewrite it and process the text in his computer, if previously equipped with the relevant software. 4 Every Mizar MSE proof starts from the section called 'environ', which includes assumptions (each marked with a number followed by the colon), to be used and referred to in the course of demonstration, as well as the definition of the universes over which the listed individual variables should range (the operation is marked by the phrase 'reserve [variables] for [sort of objects]'). As mentioned above, the sort names do the task of some predicates which would otherwise be inserted into formulas at the cost of a considerable increase of their length. In what follows there are the sorts, or universes, described as agent, length (attached to a stick), etc. (if we carried out the categorization in another way, it might prove more adequate, but not as short as desired for the conspicuity of our example). T h e thesis Τ to be proved is the instantiation of the consequent of the general law stated as (assumption) 1 in the form of a conditional (the prefixing phrase 'for ... holds' is the universal quantifier). T h u s the remaining assumptions should state two kinds of facts: (i) t h e existence of instances of t h e predicates occurring in the antecedent of 1, and this is done in the lines starting from t h e operator 'given' (the individualizing operator applied to a sort of individuals); (ii) t h a t the antecedent is satisfied by the given individuals, and this is being successively stated by assumptions 2, 3, and 4. Mizar MSE surpasses other systems of computer-aided reasoning since it allows variants of proof to simulate various habits or preferences of humans proving theorems. This feature proves crucial for our discussion, as we will be able to compare two variants and then to discuss the question as to which of them, if any, is closer to the inside process of reasoning as performed by Sultan or his human associates. 5 4

That software — created by Andrzej Trybulec — including the checker, the associated editor, etc., is freely distributed via e-mail (address romat@plearn, or filomat@plearn. 5

The variants are identical in the part preceding the section entitled 'proof'; these identical parts are repeated to facilitate their rewriting, or copying, by a potential Mizar M S E user wishing to check them by himself.

2. The internal logical code in human bodies

147

Here are the meanings of the abbreviated predicates. 'D[n,t,z]' for 'z is the Distance between η and t'; 'S[z,x,y]' for 'z is the Sum of χ and y'; l L[z,s]' for 'z is the Length of s'; 'R[n,t,s]' for 'n Reaches t using s'. Here is the proof in both variants (the adopted typeface is to imitate the shape of letters as seen on the computer screen, and to clearly distinguish the Mizar MSE text from the surrounding context). Variant A environ reserve reserve reserve reserve

η for t for x, y, s for

agent; agentsobject; ζ for length; stick;

1: for n,t,z,x,y,s holds (Dist[n,t,z] ft S[z,x,y] ft L[z,s] implies R[n,t,s]); given n' being agent; given t' being agentsobject; given x', y', z'

being length;

given s' being stick; 2:

Dist[n',t',z'];

3:

S [ζ',χ',y'];

4:

L [z',s'];

begin C: R [n', t', s' ] proof 5: (DistCn',t',z'] ft S[z',x',y'] ft L[z',s'] implies RCn'.t'.s']) by 1; 6: R[n',t',s'] by 5, 2, 3, 4; thus thesis by 6; end; Variant Β environ reserve η for agent; reserve t for agentsobject; reserve x, y, ζ for length;

148

Seven: Reasoning, Logic, and Intelligence

reserve s for stick; 1:

for n,t,z,x,y,s holds (D[n,t,z] & S[z,x,y] & L[z,s]

implies R[n,t,s]); given n' being agent; given t' being agentsobject; g i v e n x', y', z' being length; g i v e n s' being stick;

2:

D[n',t',ζ'];

3:

S [ζ', χ', y' ] ;

4:

L [ζ', s' ] ;

begin

T:

R[η',t',s']

proof thus thesis by 1, 2, 3, 4; end;

T h e difference between these variants consists in the number of derivation steps. Variant A, resembling more closely than Β the usual method of formalization, includes two such steps. T h e first results in formula 5 by (reference to) assumption 1 by virtue of the rule EU (Elimination of the Universal quantifier, called also instantiation), and the second results in formula 6 by 5, 2, 3, 4 by virtue of the detachment rule (ponendo ponens) (cf. Chapter Five, Subsec. 4.4). In variant Β there is only one derivation step in which instantiation and detachment merge into one operation. Which formalization is closer to the actual reasoning of an intelligent agent in the in question situation — this is a seminal question to be discussed below. 2 . 2 . T h e problem of the adequacy of formalized reconstruction of a reasoning does not arise at the level of ordinary courses in logic or ordinary textbooks, even in the more advanced ones. They take it for granted that a single inference step is controlled by exactly one rule, as exemplified in variant A. The question only then arises when we deal with a reasoning mechanism whose operations are discrete — as are single derivations, each marked by a separate line in a written formalized proof — but, unlike in that proof, they are not necessarily programmed according to the standard set of inference rules. This means that a single transition from one

2. The internal logical code in human bodies

149

s t a t e of a mechanism or a u t o m a t o n t o its next s t a t e is not b o u n d t o correspond t o a single logical rule. T h e fact t h a t variant Β is also accepted by t h e Mizar MSE checker as a valid inference, hence t h a t it is implementable by a computer equipped with t h a t checker, d e m o n s t r a t e s t h a t t h e r e is a reasoning device (namely t h a t devised by t h e Mizar MSE designer) which may correctly arrive at t h e same conclusion with t h e use of different algorithms of inference. This awareness could have arisen only after t h e meeting of logic with c o m p u t e r s . T h e main hero of t h e first such meeting was J o h n von Neum a n n , t h e American m a t h e m a t i c i a n of H u n g a r i a n origin and Germ a n training (in t h e Hilbert school), regarded as t h e f a t h e r of t h e digital c o m p u t e r . His concise book The Computer and the Brain (1st edition [1957]) s u m m e d u p t h e first phase of experiences concerning relations between logic, language, m a t h e m a t i c s , t h e comp u t e r and t h e brain. He concluded with t h e following s t a t e m e n t s (p. 81 f. of t h e edition of 1979). It is only proper to realize that language is largely a historical accident. The basic human languages are traditionally transmitted to us in various forms, but their very multiplicity proves that there is nothing absolute and necessary about them. Just as languages like Greek or Sanscrit are historical facts and not absolute logical necessities, it is only reasonable to assume that logic and mathematics are similarly historical, accidental forms of expression. They may have essential variants, i.e., they may exist in other forms than the ones to which we are accustomed. Indeed, the nature of the central nervous system and of the message systems that it transmits indicate positively that this is so. [...] Thus logic and mathematics in the central nervous system, when viewed as languages, must structurally be essentially different from those languages to which our common experience refers. Von N e u m a n n ' s point expressed in t h e above s t a t e m e n t s is of utmost i m p o r t a n c e for logic, and specially for logic viewed from the rhetorical point of view proposed in this essay. To emphasize this point I resorted t o t h e trick of examining t h e supposed reasoning of an animal in t e r m s of predicate logic in its Mizar MSE version. T h e fact t h a t t h e reasoning is carried out by an a p e does not prevent a generalization since a h u m a n placed in such a situation is supposed t o behave in a similarly intelligent way (it is why we admire Sultan for his human-like performance), while t h e

150

Seven: Reasoning, Logic, and Intelligence

fact that a dumb creature is capable of such inferences settles the vital question: whether language is necessary for reasoning. The answer in the negative paves the way for the problem stated by von Neumann: if reasoning belongs to those processes which may occur either (i) without a linguistic counterpart or (ii) accompanied, or even supported, by verbalized inferences, then we should ask about the latter if the same logic controls both reasoning processes. Von Neumann conjectures the answer in the negative. If he is right, then in our arguments we should take into account both the logic involved in a language and that extralinguistic logic which is more the work of Nature than of Culture (though the latter may have some feedback influence, as in any interaction between biological and cultural domains). The point claimed as fundamental for the present essay is identical with that suggested by von Neumann. In the preceding chapters as many logical theories have been presented as is necessary to state and to develop this key point. To express it conveniently, let me resort to the term code, endowed with so broad an extension as to cover both linguistic and non-linguistic systems. Two features are common to both of them: each is organized as a syntactic system governed by formation and transformation (processing) rules, and each of them imparts a structure to the processes of sending, transmitting and receiving signals between units of a system which it controls, be it a society, a nervous system, a computer set, or genetic machinery. From the point of view of a human observer, a language used for communication can be said to be outside as functioning among the members of a society, while some processes which occur in human bodies, e.g., as impulses on the nerve axons, are said to take place inside them. Correspondingly, I shall use the terms external code and internal code. The notion of internal code can be regarded as a 'technological' alternative to what Fodor [1975] and other philosophers of artificial intelligence suggested to name the language of thought. Fodor argued that mental processes involve a medium of mental representation, and that this medium is like a language, e.g., thoughts are like sentences. The point of the present essay agrees with Fodor's representationism in acknowledging mental or neural counterparts of linguistic units, but there

2. The internal logical code in human bodies

151

is the important difference which has been already stated in t h e comments pertaining to von Neumann's hypothesis. 6 When applying the idea of internal code to processes of reasoning we again shall take advantage of the Mizar MSE reconstruction of Sultan's inference. In t h e described computer simulation there is a linguistic layer, namely t h e text put into memory and appearing on the screen, to which t h e physical processing of configurations of electric pulses corresponds. T h e former occurs in an outside code, viz., a language for communication between t h e computer and its user, while the latter occurs in a machine language, hence in an internal code. Now we are able to articulate the Mizar MSE lesson: it clearly appears t h a t inferences can be perfectly made in the code of a machine, be it a computer, be it a brain. H u m a n inferences may have linguistic expression, and this is a fact of historic consequence; for a verbalized reasoning can be subjected to criticism from t h e outside, by the party to a dialogue, and this circumstance (as rightly emphasized by Karl Popper) is the one to which our civilization owes its enormous drive. However, with an inside code there can occur entirely silent reasonings, i.e., having no linguistic counterparts. This is why humans as reasoning animals share this ability with other animals and with some machines. These facts throw light on the role of theoretical logic for t h e improvement of natural intelligence. Logical theories are hardly necessary to enable us to reason, even no language is necessary for t h a t purpose, but both language and logic are unavoidable t o examine reasonings with respect to the t r u t h of their conclusions, and this includes examination of logical validity. This is what is meant by Popper in his text mentioned at the beginning of this Chapter. A logician, in spite of his education, may prove a less skilled reasoner, and thereby a less intelligent being, t h a n , e.g., a great detective; yet, if we need to say why we praise such a detective we have to resort t o the conceptual a p p a r a t u s of a logical theory (which Sherlock Holmes also did when he wished to explain reasons of his successes); logicians are those who should be appreciated for providing us with such theories. It is by no means t h e 6 An extensive discussion of the 'language of thought' hypothesis is found in Sterelny [1991].

152

Seven: Reasoning, Logic, and Intelligence

whole merit of logical theories. Their equally significant contribution consists in providing us with an outside logical code. It may be much different from the internal logical code of a machine (in particular of an animal or human body) but having been created by humans it makes it possible to raise questions, make comparisons and try analogies which give researchers a chance of understanding the logic of internal codes. The point developed in this Section to the effect that wordless reasonings play an essential role in the lives of humans and animals, has a peculiar feature which is worth noticing. On the one hand, this point appears evident in the light of the facts discussed here, on the other hand, it is vehemently attacked by the large camp of learned authors who have called themselves behaviourists. These deny the existence of any internal states of a biological machine, hence thinking must be regarded by them as silent speaking, the latter being an external behaviour. Reasoning is a kind of thinking, and therefore, according to the behaviourist doctrine, there does not exist any reasoning without a verbal expression. The above conclusion is obviously wrong in the light of experiments like that of Köhler and also in the light of everyday observations of animals. Hence it would be a waste of time to engage in polemics in that matter, but the influence of the behaviourist doctrine upoi. some circles makes it reasonable to mention it in the context of this discussion. Let this point be addressed to the followers of behaviourism as the challenging question of how they interepret their view in the era of computers which must have internal states and operate according to an internal code because they have been so constructed by humans. As the constructors of reasoning machines, we know that such an internal code is necessary for a machine to perform inference operations. This is not the least important reason to believe in a logical code in human bodies. 3. T h e problem of generalization in t h e internal c o d e 3.1. Both reconstructions of Sultan's inference offered in the preceding section may be criticised as too sophisticated as far as nonhuman reasoners are concerned. At the same time one may propose

3. The problem of generalization in the internal code

153

the more general argument t h a t so simple a situation should also be handled with simpler logical means in the case of h u m a n reasoners. The point I am to vindicate is to t h e effect t h a t to start from a general assumption, as in both variant A and in variant Β (in the preceding Section), though not necessary for the validity of inference (see variant C below), gives us an insight into nonverbalized reasoning, and hence is a process of great significance for the task of influencing other minds. To better state t h e problem, let me recall both t h e predicates employed in the formalized reconstruction and t h e general conditional stated with their help as the first assumption. 'D[n,t,z]' is to mean 'z is the Distance between η and t ' ; l S[z,x,y]' is to mean 'z is the Sum of χ and y'; l L[z,s]' is to mean 'z is the Length of s'; 'R[n,t,s]' is to mean 'n Reaches t using s'. The assumption reads as follows: 1: for n,t,z,x,y,s holds (D[n,t,z] L· S[z,x,y] &c L[z,s] implies R[n,t,s]). The objection which may be raised concerns t h e feature of universality characteristic of this statement. It may be argued t h a t anybody who is to solve a problem like t h a t of Sultan has to cope with a concrete situation in which there appears an individual fruit, and individual stick, and so on. Why should he generalize his observation in the form of such a universal conditional, t h a t is a general law, and in a moment later descend from this universal statement to its concrete instance, describing just t h e situation he is dealing with from the very beginning? To discuss this question, again let us take advantage of Mizar's flexibility in formalizing inferences. Using it, we can, as in a laboratory experiment, change a factor to watch its connection with other factors. Now let us change the set of assumptions (i.e., the content of the environ section). The use of the operator 'given', which is to create the names of concrete objects, is not delayed to the proof section, as was the case in variants A and B, but is made at the start, i.e., in the environ section. This corresponds to the fact of perceiving the said objects in the first moment of facing t h e problem by the reasoner. Then the assumed conditional referred to as 1, concerning the connection between t h e facts listed in the antecedent and t h a t mentioned in the consequent, takes the form

154

Seven: Reasoning, Logic, and Intelligence

in which variables ranging over sorts (as in the former variants) are replaced by their substitutions, being like proper names. The proof, so reformulated, runs as follows. Variant C environ given given given given 1:

n' being agent; t ' being agentsobject; x ' , y ' , z' being length; s' being stick;

Dist[n',t',z']

& S[z',x\y']

& L[z',s']

implies R [ n ' , t ' , s ' ] ; 2: D i s t C n ' , t ' , z ' ] ; 3: 4:

S[ζ',χ',y']; L [z',s'];

begin C: R [ n \ t \ s ' ] proof t h u s t h e s i s by 1, 2, 3 , end;

4;

Also this variant is accepted by Mizar MSE.7 A comparative discussion of variants A, Β and C is like a thought experiment in which we imagine how reasoning would run if we were placed in Sultan's position. Mizar MSE makes us sure that the successive variants of verbalization comply with the requirements of logical validity as checked by the algorithm of formalization. Now it is up to us to decide which variant is closest to our thought process as imagined in this Gedankenexperiment. (An advantage 7 For the sake of experiment let all the lines starting with 'given' be left behind at their former places (i.e. in the proof section, as in A and B). Then the system gives the error message which reads sorry, 11 errors detected, and when asked to list the errors it lets the user know: unknown variable identifier. T h e announced number of errors equals the number of the occurrences of the letters marked with the apostrophe in formula 1. The error consists in the fact that they were not previously declared as variables (including also indefinite constants), hence they are not recognized by the system. The operator 'given' introduces alphanumerical sequences (here single apostrophized letters) in the role of (indefinite) constants, and so authorizes their use in the proof, while the same role is played by 'reserve' for genuine variables.

3. The problem of generalization in the internal code

155

of Mizar MSE over other systems of computer-aided reasoning consists in its ability to simulate various ways of deriving the same conclusion from the same assumptions, these ways being close to the mathematical practice, while in other checking programs each of them sticks to only one set of derivation rules as fixed in t h e system in question.) When comparing variant Β with A (as stated in the preceding Section), we encounter t h e question whether t h e transition from the assumption to the conclusion takes two steps (as in A), namely instantiation and detachment, or only one, in which these two operations are, so to say, merged (as reconstructed in variant B). This question sheds light on a possible difference between the logic of our brain and the historically developed system known as firstorder predicate logic, the difference conjectured by von N e u m a n n (cf. this Chapter, Subsec. 2.2). In this case it may cross our mind t h a t the difference consists simply in t h e simultaneity of mental operations as opposed to the sequential character of a formalized proof, in which each operation is recorded as a separate line. If this supposition is confirmed, it should have a considerable impact upon the ways of applying predicate logic to rhetorical practice. It may prove t h a t an 'orderly' stated reasoning, one arranged in t h e manner resembling a formalized demonstration, is hardly understandable for someone endowed with a natural logical skill, because one's own mechanism is faster and more efficient. A more involved question derives from the comparison of t h e present variant C, in which the assumption is a concrete conditional, with t h a t feature shared by A and Β which consists in assuming a general conditional. Which reconstruction is closer to the reality of the human brain or mind? This rather sophisticated issue is the subject m a t t e r of the next Section. 3.2. The answer to t h e question ending t h e preceding passage may seem obvious, even trivial. It would be to t h e effect t h a t an adequate reconstruction involves the concrete conditional, for all the elements to be dealt with in problem-solving, such as a stick, etc., are individual entities, and those are the only ones mentioned in the conclusion. Hence, the argument would run, the act of generalization resulting in t h e universal sentence (about any fruit,

156

Seven: Reasoning, Logic, and Intelligence

any stick whatever etc.) is wholly superfluous; to climb to the top of generality and then immediately to climb down the plain of concreteness is no reasonable strategy if what one needs is to handle the concrete alone. Yet let us look at the question at quite a different angle. True, when facing a problem like that of Sultan, I am not bound to employ the universal statement; but am I able to abstain from doing that? Let us also note a rhetorical side to the question. Among the most frequent errors in argumentation is that of hasty generalization, i.e., a generalization not supported by a suitable body of facts. But if the drive to generalize is so irrestible, we should rather encourage a critical generalization than blame people for having that drive. Returning to the main point of the discussion, let us note that the question has an age-old tradition, and take advantage of some thoughts of our esteemed ancestors. That tradition revived in the work of the Dutch mathematician, logician and philosopher Evert Willem Beth (1908-1964). Its most extensive treatment is found in the book by Beth and Piaget [1966].8 The book starts from the issue which Beth calls the LockeBerkeley problem. I shall call it the Locke-Kant problem after the names of those philosophers who contributed most to the stating and discussing of the question (some authors used, in this context, to mention Berkeley as a strong opponent of Locke's position, but Berkeley's own answer was so vague that it can be disregarded in the present discussion). The history of the problem mainly involves Descartes, Locke and Kant. Each of them tried to solve a puzzle which, according to Beth, has been successfully solved by modern predicate logic as dealing with generalization and instantiation. Descartes contributed to the issue with fitting comments on the nature of mathematical reasoning but without noticing the difficulty discovered later by Locke. Descartes' point is that our mind is so constituted by nature that general propositions are formed of 8

That the book was written together with the famous Swiss psychologist concerned with logical thinking, Jean Piaget, proves Beth's interest in the borderline between logic and psychology and his anticipation of what is presently pursued under the name of cognitive science.

3. The problem of generalization in the internal code

157

the knowledge of particulars. However, it is not enough to acknowledge that we cannot do without the knowledge of particulars. In the reasoning about a concrete object (e.g., a triangle) we must be able to reason about any object whatsoever in order to justify the generalization we attempt in our proof. It would appear — Beth comments — that, according to Descartes, it is the essence of the triangle, and not any triangle whatsoever, which is the object of the intuition. It was Locke who asked what it is that justifies the transition from the particular to the general. In Beth's reformulation, more precise than the original statement, we have to do with two connected but different questions, namely: (1) Why do we introduce into the demonstration of a universal mathematical proposition an intermediate phase which relates to a particular object? (2) How can an argument which introduces an intermediate phase nevertheless give rise to a universal conclusion? Locke's solution consists in introducing the idea of the general object, e.g., the general triangle. For instance, when demonstrating that for any triangle the sum of the angles is equal to two right angles we refer the conclusion to, as Locke puts it (cf. Beth [1970], p. 43): the general idea of triangle, ... for it must be neither oblique nor rectangle, neither equilateral, nor scalenon; but all and none of these at once. In Beth's comment (quoted before the above excerpt from Locke) we find the phrase 'the idea of general triangle' while in Locke's original we read 'the general idea of triangle'. Obviously, these phrases are not equivalent, the latter is acceptable for empiricists and nominalists while the former (that in Beth's paraphrase) is not. However, it may serve as a convenient abbreviation. The situation will then be like that described by Twardowski in the following comment: 9 Vorerst sei noch bemerkt, dass wir behufs Vereinfachung des Ausdruckes statt von Gegenständen der allgemeinen Vorstellungen [...] von allgemeinen Gegenständen sprechen werden. 9

Twardowski [1894], p. 106, italicized by W.M.

158

Seven: Reasoning, Logic, and Intelligence

W i t h the proviso t h a t in what follows I shall use the phrase 'general object'; e.g., instead of saying t h a t the perception ( Vorstellung) of a stick employed to reach a fruit is general, the stick itself will be said to be general. However, the observation t h a t we have to do with the phenomenon of generality in h u m a n thinking does not solve the problem of the validity of generalizations. It was Immanuel Kant who suggested a thought-provoking solution which resulted from his theory of mathematical cognition as specifically differing from empirical and metaphysical cognition. It was Beth who duly appreciated K a n t ' s contribution by quoting his interpretation of the Euclidean proof that the sum of the angles of a triangle is equal to two right angles (Beth praises Kant for describing the mathematical procedure in question 'in an arresting manner'). The use of symbolic letters in Euclid corresponds to Jaskowski's notion of the indefinite constant and to some procedures of Gentzen and Beth. It expresses this remarkable combination of concreteness and generality which so attracted Descartes' and K a n t ' s attention. 1 0 The statement to be proved is to the effect: The sum. of the angles of a triangle is equal to two right angles. T h e proof starts from an expression like t h a t Let ABC be any triangle or Let ABC be an arbitrary triangle, to express the generality of the object under consideration which is represented by a figure like this: A

Ε

K a n t ' s reasoning runs as follows. 10

See Beth [1970], Chapter IV entitled 'The ff. The original form of the proof is found in I. Kant, Kritik der reinen Vernunft [1781] a after Beth and Piaget [1966]) is Kant [1933],

Problem of Locke-Berkeley', p. 42 Elements, Book I, Proposition 32; 716. The English version (quoted p. 579.

3. The problem of generalization in the internal code

159

Now let the geometrician take up these questions. He at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of his triangle and obtains two adjacent angles, which together are equal to two right angles. He then divides the external angle by drawing a line parallel to the opposite side of the triangle, and observes that he has thus obtained an external adjacent angle which is equal to an internal angle - and so on. In this fashion, through a chain of inferences guided throughout by intuition, he arrives at a fully evident and universally valid solution of the problem. It is remarkable t h a t in K a n t ' s exposition t h e proof does not contain any references t o axioms or previously d e m o n s t r a t e d theorems, otherwise t h a n it was practised by Euclid, w h o arranged his proofs in a sequence starting from those deduced directly from axioms and postulates. K a n t does not appeal t o deduction b u t to a m a t h e m a t i c a l intuition concerned with spatial relations. 3 . 3 . T h e above example is fit t o do at least two jobs. It is to illust r a t e t h e main problem of this and t h e following sections, i.e., t h e issue of intelligent generalization. A p a r t from t h a t main purpose, t h e above specimen of reasoning is perfectly suited t o exemplify a n o t h e r point concerning h u m a n inferences, crucial for t h e theory of argument; let us mention it before resuming t h e main t h e m e . It is t h e fact t h a t t h e skill of reasoning much d e p e n d s on its subject, t h a t is t o say, it may successfully handle a certain s u b j e c t , a n d to fail when dealing with a n o t h e r . For instance, one may have a highly developed spatial intuition which makes him a perfect reasoner in geometrical deductions, and at t h e same time he may totally fail in reasonings concerning social life. It is so because our reasonings are usually objectual, and only exceptionally symbolic. 1 1 This means t h a t t h e success of one's reasoning mainly d e p e n d s on one's familiarity with t h e object in question a n d one's skill in transforming t h a t object mentally in t h e direction determined by t h e problem t o be solved. Such skills are d u e t o inborn abilities as 11

These concepts were introduced in Chapter Two, Section 3, and mentioned in Three, Subsections 1.1, 3.4., the latter providing them with an important historical context.

160

Seven: Reasoning, Logic, and Intelligence

well as training in the domain in question. This fact seems to be of little advantage to symbolic (i.e., formal) logic, as it is far from proving its necessity for reasoning and arguing. Yet, on the other hand, only a professional logician can detect such basic features of human intelligence, since the notion of objectual inference appears against the contrastive background of symbolic (formal) inference, and that can only be precisely defined in theoretical logic. The rhetorical moral to be drawn is to the effect that, in trying to win an audience over to our own position, we should first recognize not only the audience's beliefs but also their ability of objectual inference in the domain of objects to be handled in arguments; if we lack such a skill, we should either give up or, if possible, give the audience the necessary training. Having so taken advantage of the example suggested by Kant, let us return to the issue of correct generalization, which is to be examined in this example, in accordance also with Kant's and Beth's intentions. 4. W h a t intelligent generalization d e p e n d s o n 4.1. After having thus reviewed the historical output regarding the problem of general objects and generalization, we shall consider the solution starting from that of Kant, and at the same time taking advantage of modern logical tools (as suggested by Beth) and of the notion of internal logical code (as suggested by von Neumann). The text, as quoted below, in which Kant proposes his solution, should be read in the German original, otherwise we would miss the suggestiveness of the Kantian terminology. 12 Then I shall comment on it in the form of an English paraphrase. Die einzelne hingezeichnete Figur ist empirisch und dient gleichwohl, den Begriff unbeschadet seiner Allgemeinheit, auszudrücken, weil bei dieser empirischen Anschauung immer nur auf die Handlung der Konstruktion des Begriffs, welchem viele Bestimmungen, z.E. der Grösse, der Seiten und der Winkel, ganz gleichgültig sind, gesehen und also von diesen Verschiedenheiten, die den Begriff des Triangels nicht verändern, abstrahiert wird [...]. Die Mathematik [...] eilt sogleich zur Anschauung, in welcher sie den Begriff in concreto betrachtet, aber doch nicht empirisch, sondern 12

I. Kant, Kritik

der reinen Vernunft [1781], A 713ff.

4. What intelligent generalization depends on

161

bloss in einer solchen, die sie a priori darstellt, d.i. konstruiert hat, und in welcher dasjenige, was aus den allgemeinen Bedingungen der Konstruktion folgt, auch von dem Objekte des konstruierten Begriffs allgemein gelten muss. T h e single figure drawn on a sheet of p a p e r is s o m e t h i n g experiential, and yet it does t h e d u t y of expressing t h e concept of triangle without any loss of generality. It is so because in this experiential perception one takes into account only t h e o p e r a t i o n of constructing t h e concept of triangle while disregarding those properties which are irrelevant t o t h e concept u n d e r c o n s t r u c t i o n , i.e., those which do not alter this concept, as are, e.g., size, sides a n d angles. M a t h e m a t i c s strives for intuitive perception ( A n s c h a u u n g ) in which it deals with a concept in concreto. However, this concreteness does not a m o u n t t o a sensory perception. It is j u s t t h a t perception which has been a priori introduced, t h a t is c o n s t r u c t e d , by m a t h e m a t i c s . W h a t e v e r results from t h e general construction postulates must universally hold also for t h e o b j e c t of t h e concept so constructed. T h e general postulates of construction (allgemeine Bedingungen der Konstruktion referred t o by K a n t ) can best be exemplified by what Euclid calls 'requirements' a n d is usually rendered by 'postulates'. In Book One these are as follows. ( E l ) A straight line may be drawn from any one point t o any o t h e r point. (E2) A t e r m i n a t e d straight line may be p r o d u c e d t o any length in a straight line. (E3) A circle may be described from any centre, at any distance from t h a t centre. This interpretation of t h e K a n t i a n term Bedingungen provides us with a fitting illustration of w h a t may be called t h e intuitionistic approach, as is t h a t of Descartes and K a n t , in contradistinction t o t h e logical approach t o m a t h e m a t i c a l d e m o n s t r a t i o n s . W h a t in t h e former is seen as a certain ability given a priori t o intuitively construct m a t h e m a t i c a l entities, should, according t o t h e l a t t e r , be verbalized as a deductive system. These a p p r o a c h e s are not b o u n d to contradict each o t h e r , they m a y prove c o m p l e m e n t a r y in t h e sense t h a t t h e intuitionistic approach pertains t o t h e internal

162

Seven: Reasoning, Logic, and Intelligence

code while the logical one to the external code such as a historically shaped language of mathematics. There is a strong convergence between these codes; necessarily so, as there must be a feedback between them such t h a t they influence each other. However, they are not identical and a significant difference is shown in the discussed problem of generality. In the mental process as described by K a n t , i.e., as a process somehow conditioned by an internal code, the concrete and the general constitute a single whole: the general is seen as if through the concrete. However, in a written text of mathematical demonstration, even when as close to intuitive perception as in Elements, these aspects are separated. In the above (footnote 10) mentioned proof of theorem 32 (unlike in the K a n t ' s discussion of the same fact) the demonstration starts with a concrete triangle which is given the proper name ABC, and ends with a generalization in the form of the universal statement: if a side of any triangle be produced, etc. However, it should be asked by what right we pass from a singular to a universal statement. The same step, when made in a reasoning concerning empirical objects, is blamed by deductive logic as the error of non gequitur, t h a t is a lack of entailment; and even in more tolerant inductive logic a generalization procedure based on a single fact is regarded as wrong. Thus we return to Kant: only t h e presence of the general in the concrete justifies generalization. There is no mystery in such a merger if we only agree t h a t what is successively formulated in the external code as a statement about a singular object and as an entailed statement about a general object, is, in the internal code, given simultaneously. Machines — and brains are machines too — are radically different from sheets of paper on which we write down our demonstrations; there are many simultaneous processes in a machine, while a sheet of paper is only capable of receiving sequences of symbols successively line by line. 4.2. In the above discussion of the Locke-Kant problem, the issue of generalization and general objects was restricted to mathematical concepts, in accordance with the intentions of those philosophers who raised and tried to solve it. However, the reasoning

4. What intelligent generalization depends on

163

under study, t h a t which enabled Sultan to solve his problem, is concerned with the empirical world. May we derive, then, any advantage from the discussion concerning t h e domain of mathematics? Fortunately, it is mathematics which has most to do with the empirical world, and Sultan's problem is partly mathematical. The other discipline which provides us with ideas involved in the examined reasoning is praxeology which, as duly regarded by Ludwig von Mises, is like mathematics in its dealing with concepts given a priori. 13 Both mathematics and praxeology appear at the level of animal thinking, therefore Sultan can act as the main character of our story. His reasoning lies within the limits of the capability of animal thinking carried out in an internal code; at the same time it can be expressed in a human language, hence in an external code, and can logically be formalized in a way capable of being checked by a computer. Having provided several variants of formalization, we are now able to ask which of them is closer to Sultan's supposed inference and which is more probably like t h a t of humans, especially with regard to the issue of generalization. As a result of such a comparison, we should be better equipped to grasp the role of language for the validity of generalization. Before we proceed to discuss the above question we should do more justice to the sophistication of Sultan's inference which was so far dealt in a much simplified manner. It was deliberately simplified to reveal the basic logical structure at the cost of some finer details, but we should not stop at so elementary a level. Once more, let me recall the main assumption (as occurring in variants 1 and 2) and the meanings of the predicates concerned. 'D[n,t,z]' is to mean 'z is t h e Distance between η and t'; 'S[z,x,y]' is to mean 'z is the Sum of χ and y'; 'L[z,s]' is to mean 'z is the Length of s'; 'R[n,t,s]' is to mean 'n Reaches t using s'. The assumption reads as follows: 13

This is Ludwig von Mises' [1949] idea endorsed by the present author. See Chapter Five, Subsec. 2.2, where the idea of the apriori character of action theory is briefly discussed.

164

Seven: Reasoning, Logic, and Intelligence

1: for n,t,z,x,y,s holds (D[n,t,z] L· S[z,x,y] & L[z,s] implies R[n,t,s]). Assumption 1 involves two mathematical predicates D, and S, belonging to geometry (the latter interpreted as the sum of two sections), one predicate to coordinate the mathematical object length with the physical object stick, and one praxeological predicate R. This is a simplification, since no mention is made of the method of the physical carrying out of addition. The perception t h a t such a physical operation is possible is what constitutes the creative element in the reasoning and substantiates assumption 1. T h e following text, which adds necessary praxeological elements, should be a better approximation to the actual course of reasoning carried out in an internal code. If the distance between the point being the end of stick one and the goal point (i.e., the point to be reached, e.g., in order to grasp the desired fruit) equals the length of stick two, then the stick three whose length is the sum of the lengths one and two can be used to reach the goal and it can be made of sticks one and two through putting one of them into another. T h e phrase (predicated of the extended stick) 'can be used to reach' expresses the idea of a means or a tool, hence another praxeological notion. The one formerly used was t h a t of the goal, and this pair clearly exemplifies t h a t kind of obviousness and unavoidability which is characteristic of basic mathematical notions. The statement t h a t to attain a goal one should devise means which do not exclude each other is as obvious and necessary as t h a t a straight line may be drawn from any point to any other point. Both yield, each in its own domain, general postulates of construction (allgemeine Bedingungen der Konstruktion, as Kant called them). For example, if one says 'You cannot have your cake and eat it, too', one applies t h e above praxeological postulate when interpreting the goal as a kind of happiness while the eating of a cake and the preserving of the same as means which exclude each other; wherefore — one concludes with geometry-like necessity — the success of such an action requires a choice between alternative means. Owing to t h a t likeness, mathematical and praxeological notions can be treated on an equal footing with respect to criteria of correct generalization. Thus the solution of the Locke-Kant problem, the issue primarily concerned with mathematical general objects, can

5. The role of a theory for intelligent generalization

165

also be adopted in solving praxeological ones; and those two kinds alone occur in Sultan's problem. 1 4 5. T h e role o f a t h e o r y for i n t e l l i g e n t g e n e r a l i z a t i o n 5 . 1 The tendency toward generalization is so f u n d a m e n t a l and so irrestible even at the animal level t h a t it should be seen as an instinctive drive. Therefore I shall term it instinct for generalization. It serves the instinct for survival since experience is necessary for an individual to survive, and generalization is the core of experience, even at t h e primitive level controlled by the laws of conditioned responses. E.g., a dog once hit with a stick tends to avoid other sticks, even those having different length, shape, etc., hence he must have acquired the general notion of a stick. From t h a t biological level we rise to the logical level when we note that t h e instinct for generalization is by no means infallible. There arises the question of the criteria of correctness, and once more it can be seen how errors contribute to t h e rise of logic; likewise, according to Popper, do lies (see the beginning of this Chapter). One may object t h a t because of this unreliability of generalization, the drive to generalize does not deserve the name of an instinct (as suggested above). However, I do not think t h a t the feature of infallibility should be involved in the concept of instinct; those who think otherwise can in the present context treat t h e label 'instinct' as a convenient abbreviation which is to hint at an irrestibile force, actually involved in t h e animal tendency to generalizations. It is this force which must be seriously treated by logic; so far its textbooks warn of generalizations (unless supported by statistical methods). Yet in their everyday lives people as well as animals incessantly generalize (without any recourse to statistics), 14

When claiming such similarity between mathematics and praxeology one should face the following question: why has mathematics developed into an omnipresent science whose results, owing to enormous chains of deduction, are farthest from being trivial, while praxeology lacks any age-old tradition, and its statements remain at the level of platitudes? The answer may be sought in the fact that human deeds (being the subject matter of praxeology) are hardly measurable objects, and that prevents non-trivial inferences.

166

Seven: Reasoning, Logic, and Intelligence

and nevertheless they succeed to survive in spite of the warnings of logicians. The strategy for logic which derives from the above stated assumptions is as follows. The logician should acknowledge the instinct for generalization as basically right, and then provide people with means to minimalize the risk of error (instead of discouraging them from generalizing at all). These means, though, cannot be restricted to statistical methods, as those are applicable only in special cases where one is able to make a measurement. The answer to be accepted as at least as old as Karl Popper's critique of the neopositivistic program for science. There is only one method, which is as far from being satisfactory as is democracy in the matters of a political system, but like democracy is the only one which is feasible. This method consists in creating theories to be confronted with facts of experience; the better a theory stands up to such confrontation the more it proves reliable. The discipline which most successfully rises to the occasion is mathematics (including formal logic). Contrary to the neopositivistic doctrine, there is no convincing reason to count mathematics as being radically different from empirical sciences; there are rather differences of degree, and mathematics is found at the t o p of the scale of reliability (the case of theoretical physics being so close to mathematics exemplifies t h a t law of continuity). So we come again to the problem of generalization. Mathematical general objects are so very general t h a t they find a gigantic, one may say astronomical, number of applications in every domain of the universe, hence their existence proves to have been tested with the uttermost success. Let us take, for example, the success story of the number zero; we cannot live without zero, hence its existence is beyond any doubt (once upon a time, in the era of Roman numerals, people lived without it, but what a poor life it was!). It may be that mathematics has a certain advantage over t h e other branches of our knowledge which, possibly, derives from t h e fact t h a t all animals, including humans, have enormous inborn capabilities of computing. Mathematics used by a small bee, or even a still smaller ant, is comparable with t h a t functioning in huge computers. With animals other t h a n humans, practical mathematics cannot be transformed into theoretical mathematics

5. The role of a theory for intelligent generalization

167

while with h u m a n s it can, a n d so they have a considerable initial capital to s t a r t with. A similar benefit must be enjoyed by some concepts being at t h e b o t t o m of other disciplines. For instance, linguistics as a theory of communication a n d praxeology as a theory of action may stem from i n n a t e practical abilities of communication a n d of action, respectively; out of such a rough material, fine a n d successful theoretical concepts may be made, even if their success yields t o t h a t of m a t h e m a t i c s d u e t o t h e lesser degree of measurability of t h e subject m a t t e r . T h u s t h e correctnes of a generalization should be j u d g e d by t h e degree of its success. Let t h e following example, again concerning t h e life of monkeys, explain t h e point. A chimpanzee was taught a specially devised sign language so that he could request an apple, a plum, or a banana. At the same time he learned to name some qualities of objects, as hard, soft, warm, cold, short, and long. Once the animal was given a nut. It WEIS appreciated as very tasteful, but the monkey had no means of expression to communicate his request for more nuts. Then, one day, he found how to handle the problem. He asked for a hard plum. Was it a right generalization? If suggested by a h u m a n b o t a n i s t , it would certainly be wrong because of its inadequacy for b o t a n i c classification. B u t with regard to the monkey's purposes it should be accepted as flawless a n d , moreover, deserving t o be admired for its linguistic ingenuity. Even if this story were hardly subs t a n t i a t e d , it could be used as a fable t o exemplify t h e p o s t u l a t e d relativism in t h e assessment of generalizations, as for t h e present purposes t h e ascertainment of its credibility is not necessary. 1 5 To conclude this p a r t of discussion it should be said t h a t t h e reliability of a generalization depends on t h e context of t h e theory 15

Unfortunately, I cannot quote the original source of this story, as I found it reported in a daily newspaper without any references. Some psychologists with whom I discussed the case were sceptical about it, but I do not see reasons for scepticism. The demarcation line between a human and an animal mind will not be blurred until a monkey by himself suggests a new sign for nuts and defines it in terms of 'hard' and 'plum'. Only that would mean having the innate idea of a language, as humans are supposed to have, while the remarkable performance reported in the story may be explained by a combination of conditioned reflexes with the functioning of a logical gate as that for conjunction.

168

Seven: Reasoning, Logic, and Intelligence

in which it is involved. If it fits well into t h e theory (it makes an effective use of the means of the theory, it increases the explanatory and predictive power, etc.), and the theory itself is well-confirmed (as is, specially, mathematics in its age-long history), then the proposed generalization deserves to be accepted. For instance, t h e monkey's theory t h a t the new object (called ' n u t ' by humans) is a hard plum makes perhaps the best use of the means of the language at its disposal, and fittingly predicts the behaviour of h u m a n mentors. Therefore the general concepts of a plum and of a hard object prove fittingly formed, and the new generalization introducing the concept of a hard plum should be also assessed as correct — in spite of its deficiencies in the context of another theory, such as that developed by human botanists to cover an enormously wide range of facts. Thus t h e logical merits of generalizations, either deliberate or instinctive, should be judged by the cognitive usefulness of t h a t theory to which they contribute. 5 . 2 . In order to explain the notion of the cognitive use of a t h e o r y for the present purposes, I shall resort to a rhetorically relevant case study. 1 6 From a primitive example of generalization discussed above we should pass to a more sophisticated one. Let it be the concept of European as occurring in so many contexts which are specially relevant to exemplify the rhetorical approach t o logic. This example is to show how much the assessement of defensibility of generalization is theory-dependent. In some circles it is claimed t h a t the concept of European involves a reference to Christian values, in other circles no such claim is made, and in still others t h e same claim is contested. 16

A more general explanation would need too much time; the problem has a long history related to the ideas of pragmatism in philosophy and methodology of sciences, hence it would require a historical discussion exceeding the intended limits of this essay. The strategy of resorting to case studies as the equivalent of a systematic exposition agrees with the point of this essay that the insight into essential features of an object should be provided by suitably chosen individual cases.

5. The role of a theory for intelligent generalization

169

Let us examine some a r g u m e n t s for t h e position listed first. These will be taken from t h e article by Georges H o u r d i n , t h e founder of t h e Catholic weekly La Vie.17 The argument starts from the realization that to be a European involves endorsing the tendency to limit national sovereignty in a certain way. Then the argument runs as follows: this tendency can be substantiated only by the Christian postulate of making peace combined with the Christian readiness for sacrifices, especially those made by nations for the sake of a more universal and peaceful society. The conclusion is t o the effect t h a t to be a European involves endorsing postulate of making peace combined with the Christian

the Christian readiness for

sacrifices. The author lists three great personalities, the founders of European unity, all of them having been inspired by these Christian ideas, namely the German Konrad Adenauer, the Italian (formerly the Austrian) Alcide de Gaspari, and the Lotharingian Frenchman Robert Schuman. T h e mentioning of these three personalities a d d s a b o d y of historical facts t o H o u r d i n ' s theory of being a E u r o p e a n . T h e s e facts should have been predicted by t h e theory, hence their being t h e case confirms it to some e x t e n t . T h a t a prediction comes t r u e does not a m o u n t t o t h e definite verification of a theory, yet this increases t h e defensibility of t h e generalization involved since t h e theory proves cognitively useful. However, in order t o j u d g e t h e cognitive use of a theory we need more t h a n t h e existence of confirming instances. T h e theory should s t a n d u p t h e test of denying instances, conveniently called counterexamples. If such instances are listed, t h e theory may be defended by their r e f u t a t i o n , or by their r e i n t e r p r e t a t i o n , or by proving their irrelevance, or by a modification of t h e theory itself, nevertheless an action should be taken t o f u r t h e r t h e discussion. As for t h e case under study, t h e s t a t e m e n t t o be tested (italicized in t h e above indented passage) should be interepreted as a universal s t a t e m e n t t o t h e effect: Every European is one endorsing the Christian postulates [etc.]. In other words, Christianity is a necessary condition of E u r o p e a n i s m . To check this proposition by search for counterexamples, we need a more precise definition w h a t it means t o be a E u r o p e a n . It is t h e defender of t h e thesis 17

The article was published in La Vie in September 1992; I read it in a translation which failed to give the exact date.

170

Seven: Reasoning, Logic, and Intelligence

who should cope with this task. Suppose t h a t he offers only a partial definition (as a complete one is indeed difficult) to the effect t h a t those eminent politicians who contributed to t h e foundations of a united Europe were certainly Europeans. This answer makes it possible to look for counterexamples. As far as the opponent knows, Winston Churchill and Charles de Gaulle were eminent politicians who considerably contributed to establishing a united Europe. Did they endorse the said Christian postulates? It is now up to the defender either to prove the Christian affiliation of the said persons or to withdraw, or else modify, his claim — unless he calls in question the relevance of the given counterexamples (which may on occasion be reasonable, too, but should be hardly expected in this case). Should the defender succeed in counting Churchill and de Gaulle as Christian-inspired politicians, then the defended theory would score more points as far as its cognitive use is concerned. Since t h e main thesis of this theory is identical with the tested generalization, the latter proves more defensible t h a n it had been before applying t h e testing procedures. Any case study, though enjoying the merit of inspiring concreteness, has a darker side which consists in its incompleteness, and t h a t may lead to a misleading one-sided picture. It is up to the author of such a study to minimize this drawback through a wellconsidered choice of the instances to be examined. I tried to do my best when hinting both at the phase of confirmation and the phase of a t t e m p t e d refutation, and also at the role of definitions and the role of a body of facts. If I still failed, then a critique of t h a t failure should advance the understanding of the logic of generalization.

6. Logic and geography of mind: mental kinds of reasoning 6.1. Logic is able to greatly contribute to the geography of the mind and t h a t , in t u r n , helps cognitive rhetoric, i.e., rhetoric addressed to an intelligent and benevolent audience (cf. One, Subsec. 1.3.). Note that the contention of this essay is t h a t rhetorical activity needs logic most as (i) the foundation for a descriptive theory of mind, also as (ii) a language to express a critique of

6. Logic and geography of mind: mental kinds of reasoning

171

arguments, and least as (iii) a means to improve abilities to reason. Though logic is perfectly fit for the third task as well, there is no urgent demand for such services on the side of rhetoric, for natural intelligence proves sufficient to correctly make everyday inferences. On the other hand, these difficulties occur in the mutual understanding of arguments which seem to be insuperable, which encourages us to learn more a b o u t human minds, more that can be expected either from folk psychology or from academic psychology. Then one requires stronger cognitive methods, and these may come from logic. 1 8 One of the reasons to s u b s t a n t i a t e this view is that logic provides us with ideal types or reasoning, defining, etc., which enable the use of ordering relation in the set of mental acts. For instance, one uses several examples to explain the meaning of a term in the course of an argument. Such a behaviour is disapproved of by folk logic which in every case d e m a n d s complete and precise definitions. 1 9 Modern theoretical logic, though, is not so rigorous; it helps the other side to understand, e.g., the partner's instinctive tendency to use examples as approximating the ideal of definition in the degree necessary for the argument in question; at the s a m e time logic provides the parties to the discussion with means of assessing whether such an approximation is actually relevant to the point being defended. (Other advantages which the philosophy of the mind may take over from theoretical logic will be mentioned below in this Section.) 6 . 2 . A philosophical m a p of the mind owes much to the concept of a f o r m a l i z e d proof. At least four zones of the mind concerned 18 To emphasize this point, let me mention an argument between Dr. Karol Wojtyla (at that remote time the f u t u r e Pope was a lecturer of ethics at Lublin Catholic University) and myself concerning the relation between logic and ethics. He defended the view that ethics gives more insight into the human mind than psychology does. I shared his remarkable, even paradoxical, point that a normative approach may grant us more descriptive knowledge than a descriptive one does, yet I decidedly preferred logic to ethics in that role. 19 What I call folk logic (imitating the already existing concept of folk psychology) is to great extent s h a p e d by entries found in general dictionaries and popular encyclopedias which in a simplified manner reflect the s t a t e of logic in the first half of the 19th century.

172

Seven: Reasoning, Logic, and Intelligence

with reasoning can be distinguished through using this concept as a frame of reference. The exposition of predicate logic in this essay aimed at this concept as one of its main objectives, while the three-variant formalization of Sultan's reasoning carried out in this Chapter provides examples of how formalization can help in the hypothetical reconstruction of mental processes. The notion of formalized proof enables us to use an ingenious trick which may be called physicalization of logic. For this purpose we should reserve the term 'proof' for something as visible and tangible as physical objects are, while the word 'reasoning' would be reserved for a mental process capable of being recorded and expressed by a proof. Thus a proof is a physical object, e.g., a sequence of three-dimensional (though perceived rather as twodimensional) signs made out of dried ink, or a record on a magnetic tape, etc. It is just this trick which accounts for the fact that proofs can be handled by computers, and that people can think of developing this capability even towards artificial intelligence. Logical correctness of a proof is defined by a set of inference rules (such as those discussed in Chapter Six, Section 2); what is crucial about such rules is the fact that the transformations which they define are physical transformations of shapes of symbols without any reference to their meanings. Owing to such physical concreteness, we have a solid basis to define a set of notions in terms of formalization procedure. First, as suggested above, let reasoning recorded as a formalized proof be called formalized reasoning. Now we are able to define the concept of intuitive reasoning as one which is (i) non-formalized but is (ii) acceptable according to certain standards maintained by experts, in particular mathematicians. This reference to standards of mathematical intuition agrees with von Neumann's previously mentioned view on the historical relativity of logic and mathematics. It may still be better understood in the context of such theories as that expressed by Wilder [1981], e.g., in the following statement. "[The concept of] 'proof' in mathematics is a culturally determined, relative matter. What constitutes proof for one generation, fails to meet the standards of the next or some later generation. Yet the mathematical culture of each generation possesses generally accepted standards for proof. At any given time, there exist cultural norms for what constitutes an acceptable proof in mathematics." (p. 40, Sec. 10 'The relativity of mathematical rigor').

6. Logic and geography of mind: mental kinds of reasoning

173

The adjective 'intuitive' was unnecessary before t h e emergence of the notion of formalized proof or formalized reasoning; previously it meant simply a proof, or reasoning, without any additional feature. Once the concept of intuition in reasoning emerged together with t h a t adjective, a new mental space was discovered and opened to inquiry. Now being aware t h a t a formalized proof deals with symbols (as physical objects) alone, we become able t o pose the question of what intuitive reasoning deals with. We begin to realize t h a t we must deal with objects themselves, namely those objects for which respective symbols stand for. But if we deal with objects themselves, somehow being presented to our minds, may we not sometimes (i.e., in some points of a proof) do without symbols at all? Is it not so t h a t we need symbols and sentences only as steps of a ladder which even if placed differently, or in a lesser number, would equally enable to reach the top? T h e answer in the affirmative is obvious to those who practise proving theorems. It is the answer like t h a t : we cannot do without any ladder at all, but the same result can be achieved with different ladders, i.e., with different wording, greater or lesser gaps in wording left to be filled up by a reader, etc. Owing to such a reflection, t h e present a u t h o r felt authorized to introduce the notion of objectual reasoning ( C h a p t e r Two, Section 3) in order to hint at this aspect of intuitive reasoning, which consists in its dealing with objects instead of with symbols. Thus, 'objectual resoning' stands for the same class as does the term 'intuitive reasoning', and analogous equivalence holds for the terms 'symbolic reasoning' and 'formalized reasoning'; the former member of each opposites hints at the matter of transformations (objects vs. symbols), while the latter at the way of processing (intuitive, or mental, vs. physical). The awareness t h a t there exist mental objects of reasoning which are representations of some extramental objects (physical or abstract ones), which we owe to the logical theory of formalization, proves crucial from the rhetorical point of view. Now one can realize t h a t the addressee of his argument may mentally live in a world of objects very different from t h a t of his own — in spite of speaking the same ordinary language, common to both sides. Hence a failure of one's argument may be a hint t h a t first of all

174

Seven: Reasoning, Logic, and Intelligence

the partner's mental world should be recognized, and only then is the time ripe to look for convincing arguments, relevant to the results of such recognition. The next mental domain which we can explore owing to the concept of formalized proof is t h a t of instinctive reasoning as described above in the example of Sultan's logical performances. Again, a significant rhetorical moral should be drawn from the theory built around this concept. Note t h a t in the case of a disagreement occurring in an intuitive reasoning, its sources can be investigated through an exchange of messages concerning the conceptual world of each party to the dialogue. Yet a disagreement whose sources go to the deep s t r a t u m of instinctive reasoning cannot be detected in a similar way, because the reasoner himself is not aware of the course and premises of his reasoning; this is why he cannot contribute to the mutual understanding between him and his partner. Hence a partner intent upon understanding the other one must possess more logico-psychological skill and devote more time to investigations in order to decipher the subconscious reasoning of the other side t h a n is necessary in the case of conscious intuitive inferences. An example of such deciphering was given above in studying Sultan's case. The adopted method consisted in several tentative reconstructions of the supposed process of reasoning, and t h a t , due to a precise formalization, revealed several possibilities of valid inference (presented as variants A, Β and C). W h e n facing the choice between the variant including generalization and t h a t lacking generalization, we resorted to the more general conjecture t h a t there is an instinct for generalization which is part of the instinct for survival (such as the generalization involved in learning). Hence generalization is expected to appear in any perception, also t h a t providing the assumption of scrutinized inference. Such theoretical constructions are unavoidable where no introspective or other experiential d a t a can be used. Obviously, their result is only hypothetical but such hypotheses, when subjected to suitable criticism, lead to the next steps of research, and so advance our understanding of the domain under investigation. T h u s we have listed three ways of reasoning, all of them being relevant to the rhetorical point of view, namely instinctive,

7. Formal ('blind') reasoning and artificial intelligence

175

intuitive, and formalized reasoning. T h e first is developed without any words or symbols; in the second a wording is necessary and essential but does not match the whole content of reasoning; in the third, the wording is adequate for the content, so that one can check its validity without any regard to the meanings of the symbols involved. It should be noted that in the last case the symbols are not devoid of meaning; but even if the meanings are disregarded, the validity of reasoning or its lack can be stated by a purely mechanical check, i.e., one taking into account only physical transformations of symbols controlled by specially devised inference rules. Thus formalized reasoning has, so to speak, two faces, one turned towards the human ability of understanding meanings, the other towards a mechanical device to check validity by tracing physical transformations. The picture of two faces leads to the question of what will happen if the human-oriented face disappears and there remains only the machine-oriented one. T h e kind of inference which one deals with in such a situation is called formal reasoning. There is a difference and a similarity between formal and formalized reasoning. A formalized reasoning is one which has, so to speak, been given a feature of formality without losing the feature of having a meaning, while a formal reasoning is deprived of the latter. Thus the next point to be considered is formal reasoning, an issue related to the problem of artificial intelligence.

7. Formal ('blind') reasoning and artificial intelligence 7 . 1 . Nowadays logic proves able to contribute to what people used to call artificial intelligence or A I for short (in a way predicted by Leibniz — see Chapter Three, Subsec. 3.3). The AI theory is a branch of computer science which aims at understanding the nature of human and animal intelligence and specifically at creating machines capable of intelligent problem-solving. A I should work in a way similar to the following. The computer may be given a d a t a base, i.e., a systematically organized store of relevant facts (fed by a human designer) equipped with retrieval facilities, or it may have to learn some of

176

Seven: Reasoning, Logic, and Intelligence

these facts. T h e machine is t o generalize a n d compare, discover relations, and predict possible outcomes of actions. To solve such tasks t h e machine needs heuristic procedures — like those used by h u m a n s in searching for unknown goals according t o some known criterion (as discussed, e.g., by Polya [1971]) — as well as inference rules supplied by logical calculi. It is claimed by AI theorists t h a t in b o t h points there is a considerable analogy between t h e reasoning of a h u m a n being and t h e reasoning of a machine. Is t h a t claim right? This is t h e crucial question t o be settled in dealing with t h e problem of import of logical calculi for rational cognition a n d rational communication, t h e latter presupposing t h e former, a n d b o t h including the factor of critical arguing, so much stressed by Popper. 2 0 Two alternative strategies of simulating h u m a n reasoning should be considered in AI research. Success or failure in this enterprise should provide us with evidence to help t h e u n d e r s t a n d i n g of hum a n reasoning processes. T h e strategies in question are related t o w h a t formerly, viz., in C h a p t e r Two, Section 3, was discussed as t h e opposition between objectual, or material, and symbolic, or formal, inference (i.e., reasoning); by virtue of this distinction material inference may also be called informal. In w h a t follows, first t h e notion of o b j e c t u a l inference will be examined in some examples and comments, then it will be compared with w h a t we know a b o u t formal inference d u e to predicate logic. Both kinds of reasoning will be discussed with regard both t o n a t u r a l and to artificial intelligence. 7.2. T h e idea of artificial reasoning had been anticipated long before t h e c o m p u t e r became able to materialize it in a physical shape. T h e most f a m o u s forerunners were Hobbes and Leibniz. T h e latt e r invented an ingenious m e t a p h o r of blind thinking to render 20

There is a vast multitude of consequential problems of communication which cannot be even mentioned in this book, deliberately confined to the issues of the nature of formal and informal, conscious and unconscious reasoning as applied in an argumentative discourse. Fortunately, the reader may consult the penetrative essay on rational discourse by Posner [1982]; e.g., its analysis of the use of sentence connectives in natural language fittingly completes what has been said on this subject in the preceding chapter of this essay.

7. Formal ('blind') reasoning and artificial intelligence

177

the nature of what nowadays we call formal reasoning. 21 A recent counterpart of that metaphor can be found in John R. Searle's [1980] much discussed story of a Chinese room which describes a Gedankenexperiment. It makes sense to recall both stories so that the idea of formal inference could appeal to our imagination. Leibniz's concept of blind thinking may better be explained in the context of his point regarding the material side of mental processes. If the brain — he claimed — were blown up to the size of factory, so that we could stroll through it, we should then see the content of thoughts. When commenting on this point in terms of modern science, one might say something like that. Should we know enough (as we do not yet) about the neural mechanism, we would be actually capable of recognizing thoughts produced by that mechanism — as if someone placed inside a huge mechanical calculator would be able to read results of arithmetical operations recorded in positions of rotating cogwheels, configurations of cogs, etc. Numbers can be read from such mechanical states with no worse result than from inscriptions on a sheet of paper. Since linguistic units can be arithmetized, that is, given a numerical representation (as is evident in the functioning of computers), those inferences which are not about numbers can also be expressed as sequences of digits; a look at those sequences shaped as physical states should then reveal their numerical meaning (as five fingers may mean the number five) and thus reveal the non-numerical meaning coded in them. There is no materialistic extremism in Leibniz's approach, there is just the awareness that one can coordinate physical objects with abstract objects, and so identify the latter on the basis of the former. Similar coordination — it is assumed — must have been done by Nature in human and animal bodies, and it is how the story of the visit to the brain should be construed. Let us imagine now that the process of transformation of numerical data is so involved that the observer placed inside a brain or a computer is not able to follow the corresponding transformations of abstract objects, for instance, meanings of expressions. 21

The discussion of this notion in the context of the 17th century ideas is found in Chapter Three, Subsec. 3.2.

178

Seven: Reasoning, Logic, and Intelligence

However, he notices the input and the output, and being sure that the processing mechanism complies with relevant criteria, such as rules of arithmetical operations, logical rules of inference, etc., he can rely on the final read-out even if he is not capable of checking the correctness of any of the intermediate steps. Now imagine that the person in question is not an observer of a brain or a computer but its user. Then he uses his device to find the final solution without being engaged in approaching it himself and step by step. Thinking based on the belief in the reliability of the final result produced by the symbol-processing device is, so to say, b l i n d t h i n k i n g , hence Leibniz used to call it caeca cogitatio. To put it another way, the contact of the mind with objects discussed is not direct, but takes place through such signs as those instruments of thinking which represent objects assigned to them. In this mode of thinking the thing itself is not present in the person's mind; operations on large numbers are a simple example of this. As long as we remain in the sphere of small numbers, for instance when multiplying three by two, we still can be guided by some image of the object itself; e.g., we imagine an arbitrary but fixed triple of things and join to it one triple more. Such an operation can be performed physically or mentally even if we do not have at our disposal symbols of the numerical system. It is otherwise when we have to multiply numbers of a dozen or so digits each. In such a case we are deprived of that visual contact with them and have to rely on sequences of figures which represent them. Such sequences are physical objects assigned to numbers as abstract objects, and operations on figures are unambiguously assigned to the corresponding operations on numbers. For instance, juxtaposition, that is writing the symbols '2', '·', '3' one after another, is an operation on signs, and the corresponding operation on numbers consists in multiplying two by three. Since objects themselves are not 'seen' by the mind in this mode of computing, and the mind's attention is focussed on their symbolic representations, the phrase 'blind computing' fittingly describes the situation. Leibniz believed that other mental operations, especially reasoning, can also be performed on symbolic representations of their objects, and thus one would deal with 'blind

7. Formal ('blind') reasoning and artificial intelligence

179

t h i n k i n g ' , in p a r t i c u l a r , 'blind reasoning'. T h i s is why he so intensely tried t o create a logical calculus which could be h a n d l e d by a logical m a c h i n e , analogously t o t h e a r i t h m e t i c a l calculus successfully h a n d l e d by his a r i t h m e t i c a l machine. 7.3. T h o u g h n o m o d e r n a u t h o r applied Leibniz's m e t a p h o r t o arificial intelligence, it nicely fits into t h e present s t a t e of A I research ( t h e adjective ' p r e s e n t ' is a concession t o t h o s e writers w h o claim t h a t , in t h e f u t u r e , c o m p u t e r t h i n k i n g should b e indistinguishable f r o m t h a t of h u m a n s ) . T h i s s t a t e is a d e q u a t e l y reflected in Searle's [1980] t h o u g h t e x p e r i m e n t m e n t i o n e d above, which was m e a n t t o challenge t h e following claims of strong A I ( ' s t r o n g ' m e a n s able t o m a t c h h u m a n abilities): 1. t h a t t h e m a c h i n e can literally b e said t o understand a story told by a h u m a n which is d e m o n s t r a t e d by correct a n d non-trivial conclusions d r a w n by t h e machine f r o m t h a t story, a n d 2. t h a t w h a t t h e machine a n d its p r o g r a m d o explains t h e h u m a n ability of u n d e r s t a n d i n g t h e story, as displayed by d r a w i n g conclusions. In t h e following story Searle a t t e m p t s t o show t h a t t h e s e claims are far from being s u b s t a n t i a t e d . Suppose that I am locked in a room and given a large batch of Chinese writing. Suppose furthermore (as is indeed the case) that I know no Chinese, either written or spoken, and that I'm not even confident that I could recognize Chinese writing as Chinese writing distinct from, say, Japanese writing or meaningless squiggles. Now suppose further that after this first batch of Chinese writing I am given a second batch of Chinese script together with a set of rules correlating the second batch with the first batch. The rules are in English, and I understand these rules as well as any other native speaker in English. They enable me to correlate one set of formal symbols with another set of formal symbols, and all that 'formal' means here is that I can identify symbols entirely by their shapes. (Italics by W.M., cf. Subsec. 7.2 above.) Now suppose also that I am given a third batch of Chinese symbols together with some instructions, again in English, that enable me to correlate elements of this third batch with the first two batches, and these rules instruct me how to give back certain Chinese symbols with certain sorts of shapes in response to certain sorts of shapes given to me in the third batch. Unknown to me, the people who are giving me all these symbols call the first batch 'script', they call the second batch a 'story', and they call the third batch 'questions'. Furthermore,

180

Seven: Reasoning, Logic, and Intelligence

they call the symbols I give them back in response to the third batch 'answers to the questions', and the set of rules in English that they gave me, they call 'the program'. Now just to complicate the story a little, imagine that these people also give me stories in English, which I understand, and then they ask me questions in English about these stories, and I give them back answers in English. Suppose also that after a while I get so good at following the instructions for manipulating the Chinese symbols and the programmers get so good at writing the programs that from the external point of view — that is, from the point of view of somebody outside the room in which I am locked — my answers to the questions are absolutely indistinguishable from those of native Chinese speakers. Nobody just looking at my answers can tell that I don't speak a word of Chinese. Let us also suppose that my answers to the English questions are, as they no doubt would be, indistinguishable from those of other native English speakers, fo. the simple reason that I am a native English speaker. From the external point of view — from the point of view of somebody reading my 'answers' — the answers to the Chinese questions and the English questions are equally good. But in the Chinese case, unlike the English case, I produce the answers by manipulating uninterpreted formal symbols. As far as the Chinese is concerned, I simply behave like a computer; I perform computational operations of formally specified elements. For the purpose of the Chinese, I am simply an instantiation of the computer program. W h a t Searle refers to as 'answers to the questions' can be identified with conclusions drawn from the text according to appropriate processing rules. For example, in another story told to exemplify such procedures it is said of a man t h a t he ordered a hamburger, was very pleased with it, and as he left the restaurant he gave the waitress a large tip. Then the listener of the story (it may be a computer as well) is asked the question 'Did the man eat the hamburger?', and if he answers 'yes' it means t h a t he draws a right conclusion from the set of d a t a which contains those reported in the story and some other ones stored in the listener's memory as the knowledge about connections between some kinds of situations — such as t h e fact t h a t a man who is pleased with his food is likely to consume it; logical inference rules to be applied to such d a t a are the same as those appearing in Sultan's reasoning, variant A, viz., instantiation and detachment. Thus the Chinese room story illustrates the n a t u r e of formal reasoning as one being carried out without any resort to the mean-

7. Formal ('blind') reasoning and artificial intelligence

181

ing of the text in question. This story not only contributes to our realizing that there are four kinds of mental behaviour in the process of reasoning (instinctive, intuitive, formalized, and formal), and so provides us with a mind-logical map for rhetorical purposes, but has also a direct rhetorical moral. Is it not the case that some people in particular situations behave like the inhabitant of the Chinese room, even when their native language comes into play? I mean repeating some slogans and applying correct inference rules to them, yet without understanding either them or their consequences. Suppose that the slogans in question read: 'Laputians are wicked', 'every red-haired is a Laputian', 'the wicked should be punished'. Then one easily jumps to the conclusion 'all red-haired should be punished' even without understanding all the terms involved in the syllogism. In fact, at least some terms must be understood, for instance, in order to be able to identify red-haired and to inflict the punishment on red-haired people. However, there may be gaps in understanding, and these do not invalidate the formal correctness of reasoning. Such a thoughtless reasoning is not purely formal, but the more it approximates a formal one (as an extreme), the more it tolerates gaps in understanding meanings, and in this sense the concept of formal reasoning contributes to explaining some forms of mental conduct encountered in society. Obviously, it is not cognitive rhetoric which may take advantage of such phenomena but rather that which deserves to be called demagogic rhetoric. Thus, in a sense, where natural intelligence decreases, it becomes more similar to artificial intelligence. The last remark, when taken in the context of the Chinese room story, hints at a challenge to the partisans of strong Artificial Intelligence. In order to vindicate their claims, they should create entities able to simulate not only formal mode of reasoning but also all the other ones, that is instinctive, intuitive, and formalized. The last — let it be reminded — complies with the formal inference rules but has a semantic interpretation as well. But even if such a success is not very likely, the actual achievements of Artificial Intelligence prove very instructive from the rhetorical point of view.

182

Seven: Reasoning, Logic, and Intelligence

W h a t Leibniz imagined as blind thinking, what the champions of modern logic developed as the theory of formal proof and formal system, becomes physically materialized in t h e machines and programs devised in our times. And this is also a step towards developing a logic relevant to rhetorical purposes.

CHAPTER EIGHT

Defining, Logic, and Intelligence 1. T h e o s t e n s i o n p r o c e d u r e a s a p a r a d i g m o f d e f i n i t i o n 1 . 1 . According to t h e official doctrine found in textbooks of logic, there are genuine definitions called normal (see Section 2 below) in which t h e term defined and t h e defining phrase have identical extensions, and there are some deviant sorts which do not meet this identity requirement (as, e.g., a terminological explanation by examples in which the sum of extensions does not match the extension of the term so explained). Among t h e latter there is the procedure of introducing new terms through ostending exemplary objects which these terms stand for; this procedure is called ostensive definition. This disregarding of ostension by the official doctrine may be explained by taking into account mainly the practice of mathematicians who do not deal with entities capable of being perceived by the senses; at the same time they enjoy the privilege of being able to formulate most precise normal definitions of abstract objects which are constructed by them and therefore are present, so to speak, inside their minds and do not need to be observed from the outside. 1 On the other hand, every normal definition, 1

This view is certainly right as far the professional practice of mathematicians is concerned. However, some basic mathematical notions which precede a professional treatment of them, such as is the notion of cardinal numbers, and that of natural numbers, have something to do with ostension. A nice example can be found in the book by Freudenthal [1960], where the design of lessons of a cosmic language to be addressed to another civilization starts from ostensive definitions of names of some numbers (objects to be displayed are radio signals grouped in one-element sets, two-element sets, etc.).

184

Eight: Defining, Logic, and Intelligence

also in mathematics, and every ostensive definition, as well as every other definition consists in the coordinating of an expression with a thing, be it a physical object, an abstract object, or still something else. The presence of a thing in the process of defining is most conspicuous in the case of ostensive definitions, and in this sense the 'deviant' case of ostensive definition, paradoxically, can be regarded as a paradigm of definition in general (cf. Subsec. 3.1 below). Therefore it is advisable to discuss ostensive definition at the outset, even if some orthodox logicians should perceive this as turning things upside down. Ostension is closely related to the instinct for generalization which was discussed earlier (Seven, Subsec. 5.1). To see this connection (to be discussed below in Subsec. 1.2), let us first consider an example of ostension. Suppose that one is entrusted with the task of giving a name to a thing which is shown to him. This is not a typical case, ä typical ostension takes place when a person who knows an expression teaches another person its meaning (in the language in question) by hinting at a suitable specimen of the relevant kind and by uttering the statement 'the name of this object is such-and-such'. However, this non-typical case in which only one person is involved is more convenient to start with. Such a procedure is nicely reported in the following passage of the Bible. The Lord God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them; and whatever Adam called every living creature, that was the name thereof. (Gen. 2, 19) This definitional situation consists of the following elements. There is a thing which appears to us, a name we introduce to call that thing, and a performative sentence to express the creative FIAT: let the thing be called 'such-and-such'. The phrase encircled here with commas is a variable for a metalinguistic name, i.e., the name of that name which we give the thing. Thus, there was presumably a moment in which Adam said: let this be called 'tiger'. The 'let' form is to express the act of terminological creation. It has to be replaced by another form when the creation is accomplished, and one wishes to communicate its results to someone

1. The ostension procedure as a paradigm of definition

185

else. This may have been the situation of A d a m and Eve (created after A d a m ' s definitional activities). Having once called an animal a tiger, Adam may have communicated his definition to Eve with words like those: (OQ) The name of this object is 'tiger'. Or, simply (OU) This is a tiger. T h e existence of two such forms, referred t o as OQ (for Ostensive definition which Quotes a term) and OU (for Ostensive definition which Uses a term) should not induce us to think t h a t there are two kinds of definition corresponding to these forms, one dealing with the expression 'tiger', the other with the tiger itself. Obviously, t h e form OU has two functions, and in the other function it expresses only the recognition an object as a tiger without any definitional purposes. It is a context, or a special intonation (which A d a m might have used when teaching Eve the language he invented) or a typographical device (as the abbreviation ' d f ' for 'definition' preceding the statement in question) which indicates the definitional role in which a statement of t h e form OU is equivalent to a statement of the form OQ. It is a reasonable advice, both logical and rhetorical, t h a t one should preferably use the form OQ rather t h a n OU to avoid misunderstandings, unless the above-listed means (to hint at the definitional function of the uttered statement) are sufficiently readable for the audience. On the other h a n d , t h e form OU should be admitted as a more economical stylistic variant for the convenience of people suitably skilled in linguistic expression and interpretation. 1.2. In the ostensive defining there manifests itself the instinct for generalization (cf. Seven, Subsec. 5.1), i.e., a tendency which, like other instincts, is rather a biological drive t h a n a cultural norm; it often works independently of the will of t h e subject in question, even if the subject is not aware of its functioning. One cannot help generalizing, this propensity is not only part of the animal instinct for survival but also a creative cognitive force in h u m a n s . True, there occur frequent and serious errors in generalizing. However, this fact should not be dramatized, for — as a rule — each generalization is a hypothesis to be tested and, possibly, to

186

Eight: Defining, Logic, and Intelligence

yield t o another hypothetical generalization which in the given moment would better stand up to its test. Contrary to the opinion of Aristotle and even of the 19th century philosophers of science, h u m a n cognition is a risky enterprise in which certainty is the ideal extreme to be sought but not to be enjoyed in t h e actual world (in this perspective logic and mathematics are closest to certainty and thus constitute the actual extreme). On the other hand, this situation is not so uncomfortable as it may seem because generalizations should be relativized to questions posed at a given time, and may match these questions, even if they are not able to stand u p to new ones. Thus Newton's mechanics was perfectly sufficient with regard to the questions and needs of science and technology for two centuries, and in this limited range it continues to be valid; it is the range of, so to speak, the medium-sized world, i.e., neither t h a t of the macrocosmic (cosmological) nor t h a t of the microcosmic (subatomic) scale. As we shall see, the merits and flaws of our everyday prescientific generalizations (to be dealt with by cognitive rhetoric) should be subjected t o analogous relativization: a generalization which is wrong for some purposes may prove adequate for some other testing questions or purposes, and in this sense it may enjoy a limited certainty (it is why our reason, despite being so fallible, may succeed in handling matters of everyday life). T h e nexus between generalization and definition can conveniently be shown in the case of ostensive definitions. When Adam in Paradise gave animals their names, in each case he must have decided why, i.e., in view of which properties, should the given name belong to the animal in question. Should a creature be called 'tiger' for its having four legs alone, or for having also such and such size, colour, shape, and habits? The more properties are included into the content of the term being defined, the greater t h e possibility of falsifying t h e generalization, since grouping them under one term is equivalent to the statement t h a t each of them implies each other. Suppose t h a t the creature in question brought to Adam by God was a male, and physical features of a male were included by Adam in the content of the term 'tiger'; a biological theory involving the statement t h a t all tigers (as characterized by the earlier listed properties) are males would be totally unable to account for the procreation of tigers, and so it would fail to stand

1. The ostension procedure as a paradigm of definition

187

u p to a crucial test, b u t for some limited purposes of practical life in Paradise it might work. T h e procedure of ostension shows u p still o t h e r features of any definition, namely t h e fact t h a t definitions must be introduced on t h e ground of a theory. This s t a t e m e n t may seem unsettling t o those who share t h e empiristic doctrine, especially in its neopositivistic version, t h a t there is a basis of empirical concepts which are absolutely primitive, and this basis consists of expressions having been introduced t o t h e language in t h e ostensive way. Should one succeed in d e m o n s t r a t i n g t h a t even ostensive procedures m u s t presuppose some theoretical assumptions, t h e n a fortiori this result would hold for more advanced definitions. 2 T h e above-mentioned language for cosmic intercourse, devised by Freudenthal [1960], provides us with an instructive example of conceptual presuppositions necessary for efficient ostensive procedures. If a cosmic receiver of t h e set of three radio signals accompanied by a conventional signal t o d e n o t e t h e n u m b e r t h r e e (as exemplified by t h e three-signal set) is t o correctly interpret t h e message, he must have previously had t h e idea of c o m m u n i c a t i o n , especially t h e idea of defining, t h e n t h e idea of ostensive defining, and t h e idea of n u m b e r ; t h e last is specific t o t h e reported case, t h e other ones belong t o every ostensive procedure. T h e ideas necessary in A d a m ' s case were those of colour, s h a p e , size, etc., beacuse naming something tiger for having yellowish f u r with black b a n d s presupposes t h e awareness t h a t yellow and black are colours. In this context I preferred t h e t e r m 'idea' over ' c o n c e p t ' , following t h e choice of Leibniz w h o reserved conceptus for b e t t e r defined and more precise t h o u g h t s and idea for those remaining vague, even not fully verbalized and realized, b u t nevertheless efficiently controlling cognitive processes. If an empiricist helplessly asks where such preconceptions come from, he should blame himself for denying any kind of i n n a t e knowledge or inborn cognitive skills. T h a t t h e r e is a knowledge or skill recorded in animal cells is no s u p e r n a t u r a l mystery to be renounced by modern enlightened minds. A n d these internal records 2

This problem is treated more extensively by Kotarbmska [1960] and by Marciszewski [1966].

188

Eight: Defining, Logic, and Intelligence

are pre-theoretical seeds of later consciously and verbally developed theories (following Stoics and Augustine, we might call them rationes seminales, i.e., a kind of cosmic software). Since there are enormously many degrees of conscious realization and verbalization, for brevity I subsume those pre-theoretical ideas under the concept of theory, though the deeper we go into the biological (as opposed to cultural) layers, the less reason there is to speak of concepts or theories. The next comment, necessary to explain the set of notions proposed for handling definitions, is concerned with the relation between concepts and theories. Again, this relation is considered against the background of the above mentioned Leibnizian lex continui enabling one to see connections which would otherwise have remained unnoticed. Let it be noted that if there is a set of primitive concepts, i.e., such that none of them is defined in terms of earlier introduced ones, then these concepts, in order to be understandable, have to be interconnected with one another in a way which contributes to explaining the content of each of them. Such interconnections must be expressed in the form of some propositions (even if unverbalized), as can best be seen in sets of axioms of deductive theories. Obviously, such a set of propositions must have consequences, and those together with axioms form a theory. Owing to the said lex continui we can consider various degrees of awareness associated with such concepts, degrees of verbalization and precision, and the deeper we go 'back' (from our conscious cultural level) the less there is actuality and the more mere potentiality, but we should not fail to investigate that 'underground world' if we intend to reach the best possible understanding of the phenomena of cognition and communication. 2. N o r m a l definitions of predicates and n a m e s 2.1. The term normal definition stands for a statement which makes it possible to eliminate a newly introduced expression by replacing it with older ones. A normal definition is a genuine definition in the traditional sense, hence when speaking about a definition without any adjective we mean a normal definition. Normal definitions have the form of equivalences or identities (as discussed in Chapter Six); the symbol of equivalence or of

2. Normal definitions of predicates and names

189

identity, sometimes accompanied by the abbreviation 'df', is said to be a definitional functor. The expression being defined, say A, which occurs on the left side of the definitional functor is called the definiendum, and the expression used for defining what occurs on the right side is called the definiens. The form of a definition depends on the syntactic category of the expression to be defined. 3 Names and name-forming functors can be defined by identities, while definitions of sentence-forming functors (such as connectives, predicates, etc.) must take on the form of an equivalence. Let the following formulas exemplify the typical forms of definitional equalities and equivalencies. Of Of Of Of Of

a a a a a

name-forming functor: 'the successor of χ: S(x) — χ + 1. name: 1 = 5(0). name-forming functor: χ — y=z = z + y = x. sentential connective: (p —• q) = (-ιρ V q). predicate: χ is even iff χ is divisible by 2.

In these examples definitions are formulated without any formal index distinguishing them from theorems. Such an index is not necessary when its function is indicated by the context. Otherwise we write 'df' either as the subscript (or the superscript) of the definitional functor or as the prefix of the whole definitional formula (the latter is advisable when combined with a numbering of successive definitions), for instance:

S(x) -df x + 1, Df.l S(x) = dfx + 1.

In a natural language we have at our disposal some convenient stylistic variants of definitional forms, sometimes assisted by such typographical devices as italicizing the definiendum, for example: (a) a number is said to be even if it is divisible by two; (b) 'even' denotes a number divisible by two; (c) 'even' means the same as 'divisible by two'. In the first of these examples the context of the construction '... is said to be ... if ...' is to indicate that we deal with an equivalence which otherwise would be stated in a more clumsy form using the phrase 'if and only if' or, for brevity, 'iff'. One may also use the form: 3

T h e notion of syntactic category is sketched in Chapter Six, Subsec. 1.1.

190

Eight: Defining, Logic, and Intelligence

(d) a number is even iff it is divisible by two. The last form, that is one which does not use either quotations marks or devices like the phrase 'it is said', provides an opportunity for pointing to the distinction between object language, i.e., a language dealing with extralinguistic entities, and metalanguage, i.e., a language describing another language. A typical device to form a metalanguage is quotations marks in the form of, e.g., inverted commas. With the help of inverted commas we form names of expressions, for instance the first word in example (b) is a metalinguistic name of an expression which refers to even numbers (as mathematical entities). Also the phrase 'is said to be' as used in example (a) concerning an act of speech belongs to the metalanguage of English. It is held by some logicians (e.g., Kotarbinski [1929]) that a genuine normal definition is a nominal definition, i.e., a terminological statement about the meaning of an expression; in other words, that it is a metalinguistic statement. This claim evidently disagrees with some practices of mathematicians. A typical form to be found in mathematical papers, textbooks, etc., can be exemplified by the following definition: A right angle is any one of the four angles between two lines that intersect so that adjacent angles between them are equal. (Mathematics [1975], p. 150). There occurs, however, a corresponding metalinguistic form; e.g., in Euclid himself the same definition of a right angle is worded with the phrase 'is called a right angle' which follows the description of properties of the object in question (Book One, definition 10). Why are mathematicians, who should be most sensitive to the preciseness of the language they use, so careless t h a t they ignore the distinction between an object language and a metalanguage? Should their habit be imitated in rhetorical practice, or rather the latter should prove more cautious? It is the latter option which is advisable, and the reason is that whenever a mathematician defines the name of an object he can be sure t h a t the object for which the name stands for does exist; then the metalinguistic form and the object-linguistic form are equivalent; this issue will be discussed later, in the context of correctness conditions of the definitions of names. First, however, we should consider the rules of defining predicates.

2. Normal definitions of predicates and names

191

2 . 2 . To be correct, a definition of an η-place predicate Ρ should take t h e form P(x i, ...,xn) = A and satisfy t h e following conditions: (1) x i , . . . , x n are distinct variables, t h a t is, every variable may occur only once in t h e definiendum. (2) No free variables other t h a n x\,..., xn occur in t h e definiens A, t h a t is, every variable which occurs free in t h e definiens should also occur free in t h e definiendum. (3) In t h e definiens, t h e non-logical c o n s t a n t s should be either primitive or previously defined in t h e theory. T h e following comments may explain t h e meaning of these conditions. Condition 1 would be violated if, for instance, t h e binary predicate of t h e set-theoretical inclusion were defined as follows: (X C X) =XDX

= X.

T h e obvious failure arises in t h a t we are actually defining t h e oneplace predicate 'included in itself'; hence t h e symbol ' c ' could not be eliminated from contexts like 'TV c W (read Ν - n a t u r a l numbers, R - real n u m b e r s ) . T h e need of applying this condition to definitions in a n a t u r a l language is hardly felt since n a t u r a l languages do not have devices analogous t o longer sequences of variables (even if there are some c o u n t e r p a r t s of single variables). Condition 2 would be violated by t h e following a t t e m p t t o define t h e synonym of sentences: Df(syn): A is said t o be synonymous with Β if t h e r e are such inference rules in a language L, t h a t A is derivable f r o m Β a n d Β is derivable from A. In this a t t e m p t e d definition Df(syn) t h e variable L occurs free in t h e definiens a n d does not occur in t h e definiendum at all, c o n t r a r y to condition 2. Let Ao a n d Bo be t h e sentences s u b s t i t u t e d for A and Β, respectively, and let L\ and L2 be t h e names of languages s u b s t i t u t a b l e for L occurring in t h e definiens (alone). S u p p o s e t h a t AQ and BQ are derivable from each other in L\, b u t not in L2· However, this lack of derivability in one of these cases c o n t r a d i c t s t h e consequence of Df(syn) t o t h e effect t h a t derivability has t o hold for any case whatever, according t o t h e law of binding a free

192

Eight: Defining, Logic, and Intelligence

variable in the consequent of a conditional, provided it is not free (i.e., it either does not occur at all or occurs as bound) in the antecedent. This law runs as follows: (C Ξ ß(ar)) —• ( C —• VxD(x)), where χ does not occur in C as a free variable. Condition 2 can be applied to some structures of a natural language. The error illustrated by Df(syn) can be reproduced in English as follows: " T w o expressions are said to be synonymous if they are derivable from each other in a language". This obviously leads to such inconsistencies as the one following from the formula Df(syn); for instance, there is a language T E , viz., a Theological version of English, in which the expressions 'something is a religious truth known to people but unattainable by human reason' and 'something is a truth made known by God' are derivable from each other; by virtue of the above counterpart of def(syn) it follows that these sentences are synonymous in an absolute sense (i.e., not relativized to a language, say T E ) which is a wrong conclusion. This error arises from using the phrase 'a language', being a counterpart of L as a free variable, in the definiens. It will be rectified if one introduces the same variable into the definiendum and prefixes the whole definition by the universal quantifier to bind that variable in such a way as this: for any language, two expressions are said to be synonymous in it if they are derivable from each other in it. Here the pronoun 'it' plays the role of a variable ranging over languages, while the universal quantifier 'any' ensures that in both occurrences 'it' refers to the same language. As for Condition 3, a typical infringement of what it requires is called the vicious circle in defining. Let this fallacy be compared with the vicious circle in reasoning which consists in using the statement to be proved in the role of a premise; e.g., the view that Leonardo's paintings are masterpieces is deduced by someone from the view that Leonardo was a man of genius, and when he is asked to substantiate this view, then he mentions the fact that Leonardo created artistic masterpieces. The same example, suitably reformulated, can illustrate the definitional vicious circle: let the expression 'a man of artistic genius' be defined by the expression 'a man who is able to create masterpieces', and then 'masterpiece' be defined as 'a product of artistic genius'. In spite of being so obviously erroneous, such slips are frequent in everyday arguments, certainly because of deficiencies of human attention and memory.

2. Normal definitions of predicates and names

193

However, we should not be over hasty in blaming such a manner of defining; sometimes it may have a rational, even if unconscious, justification overlooked by logical pedants. Namely, concepts of natural language are subjected to frequent modifications caused by contexts of their use, therefore what resembles the fallacy of vicious circle may reasonably contribute to old meanings by completing it with new contexts. Such concepts as 'man of genius' and 'masterpiece' form two contexts (even if created on the pattern of vicious circle) which shed light upon each other as far as their meaning and extension are concerned. The meaning of the term, e.g., 'man of genius' is far from being precise; it is vaguely felt rather than duly defined, hence a hint at the relation of its meaning to the meaning of the term 'masterpiece' may contribute to making each of them less vague. Such a defence of circularity should not be adopted in the case of circular reasoning, but the case of defining is different because of the mentioned interaction between meanings. This problem will be discussed further later on, with reference to the theory of axiomatic definitions. Now, continuing the survey of normal definitions we should deal with definitions of names and functors. 2 . 3 . In the language of modern logic unlike in natural languages there is a sharp demarcation line between predicates and names, which is also reflected in the theory of definition.4 Since the term 'name' is needed both to cover individual names in predicate logic and general names functioning in natural languages, it is advisable in the present context to resort to the term 'individual constant' as a more precise counterpart of the term 'name'. An expression is said to an individual constant if it denotes an individual as does, e.g., a proper name or (in a suitable context) a personal pronoun. To be correct, a definition of an individual constant c in the form c — y = A must satisfy conditions 2 and 3 (as above); condition 1 is disregarded since in this form there cannot occur more In natural language this difference is blurred by the fact (discussed in Chapter Six) that the same lexical item may function in different syntactic roles, especially as a grammatical subject, hence as a name, and as a predicate. On account of this fact, the logical theory of definitions should be adjusted to natural languages. 4

194

Eight: Defining, Logic, and Intelligence

than one variable. Furthermore, such a definition should satisfy the condition of definiteness combining the condition of existence with the condition of uniqueness. The term 'condition of definiteness' is not usual in logical texts since existence and uniqueness are never separated in them. When applying the logical theory of definitions to natural languages, we are sometimes bound to consider them separately, hence for the cases in which they are inseparable it is convenient to have a special term; for that role I suggest 'condition of definiteness' since the definite descriptions are those terms which always satisfy both conditions. A theory affected by a definition infringing upon the definiteness condition has to incur contradiction. If the existence condition is violated, that is, no entity is referred to by an individual constant, then contradiction results from the law of existential generalization: ψ{α)

%φ{χ)\

the corresponding inference rule is that of Introducing the Existential Quantifier. Obviously, if 'Pegasus' is used as an individual constant in the sentence, e.g., "Pegasus is a wise horse', then it follows that there is an individual which is a wise horse. If the uniqueness condition is infringed, then there are at least two different individuals named with the same individual constant, say α and b, both named 'c'. This means that c = a and c = 6, hence a = b which denies the assumption of their difference (that is, a / b) and thus makes the theory inconsistent. 2.4. Why is the definiteness condition obligatory for individual names and is not for predicates? What about general names which are ignored by modern predicate logic, but not ignored by traditional logic nor by natural languages? The answer to the first question should help to answer the latter. Predicate logic is closely tied with set theory, which provides logic with the entities which its symbols stand for. There is an astonishing simplicity in the set-theoretical ontology. Two categories of objects correspond to an infinite variety of syntactic constructions, namely individuals and sets (which in some contexts I prefer

2. N o r m a l definitions of predicates and names

195

to call classes for some useful associations). This miracle is possible thanks to the fact that sets themselves can form an infinite hierarchy and be ordered in various ways. Against this background certain predicates are seen as expressions denoting sets of some sets. For instance, some arithmetical predicates, such as those concerning addition, multiplication, etc., are construed as denoting sets of ordered triples (i.e., three-element sets) of numbers; thus the predicate iSum(z,x,y)\ to be read iz is the sum of χ and y\ refers to the infinite set of triples of, say, natural numbers, as (0,0,0), (1,0,1), (1,1,0), (2,1,1), (2,0,2) and so on, in infinity. This description is concerned with predicates referring to relations since relations are defined in set theory as ordered sets. A s for non-relational, that is, one-argument (one-place, unary) predicates, they refer to sets of individuals, e.g., the predicate 'is an odd number' is predicated of arithmetical individuals being odd numbers. In this conceptual setting, predicates are never deprived of reference; each of them enters the language of an applied predicate calculus as coordinated with a set, which may, in particular, be the empty set. There is no such privilege granted to individual constants, and therefore it is necessary for each such constant to prove its non-emptiness, i.e., that it has a reference. A s for the condition of uniqueness, an analogous condition should be stated for predicates (though logic textbooks fail to state it, presumably because of its obviousness); the infringent of this condition could be called double-denotation fallacy, that is the error of attaching two different sets (as denotations) to one predicate expression (the same designation could be applied to the case of individual constants which fail to satisfy the uniqueness condition). A l s o this error leads to a contradiction. Suppose that ( i ) there are two different sets A and B, that is to say Α φ Β, and ( i i ) they are assigned to the same predicate ' P ; then it follows ( f r o m ii) that: yx(P(x) = (xe >1)), and V r ( P ( x ) ξ ( ι £ Β)); hence V x ( ( x e Α Ξ (χ € Β ) ) , i.e., A = Β, contrary to the assumption ( i ) . Natural-language arguments can hardly be expressed in predicate logic. Instead of predicates we use general names, and also

196

Eight: Defining, Logic, and Intelligence

individual constants can be replaced by suitably transformed (e.g., by articles) general names. This discrepancy between the modern logical orthodoxy and the logic practised since time immemorial in natural languages (roughly approximated by Aristotelian syllogistic) is a thought-provoking phenomenon. It may be due to deep differences between a spoken and a written language; it is only in t h e latter t h a t one may take full advantage of the technique of variables whose application range is being suitably restricted by predicates, while in natural languages the range in question is being indicated by a general name (this is just a conjecture to be developed and tested in an opportune context). Though the semantics of general names seems to exceed t h a t substantiated by set-theoretical ontology, the latter can be expected to provide us with some d a t a useful for the semantics of general names, and for corresponding rules of definition to deal with existence and uniqueness. We shall restrict our attention to what may be called genuine or proper general names in contradistinction to apparent ones, analogously to the distinction between genuine individual constants and apparent ones as, e.g., 'Pegasus'; they are alike with respect to some linguistic (syntactic) rules of use, but not with respect to certain logical (semantic) rules. I shall employ the traditional term universal (as a noun, not an adjective) t o distinguish the references of genuine general names. Let the (supposed, as yet) reference of a general name be termed a 'universal'. Let a set be said to be a universal if it meets the following requirements: (i) it is non-empty; (ii) it is potentially infinite, t h a t is, it is not defined by the enumeration of its elements b u t by a property common to all elements, so t h a t any new object possessing t h a t property is counted as a member of the set; (iii) it is embedded in a relational structure of sets being the domain of a theory endowed with a desired explanatory power. The above list of requirements is adjusted to some common practices and intuitions. Universale are traditionally meant as being related j u s t to essential properties, such as t h a t of being a m a n , and not accidental ones, such as being an inhabitant of Berlin (which does not satisfy either (ii) or (iii)). Obviously, condition (iii) is burdened with a vagueness which in many cases makes it impossible to decide whether we have to do with a universal or

3. The holistic doctrine of definition

197

not, but for the present purposes it suffices that at least in some cases the question can be settled. Thus defined, the concept of universals hints at a possibility of stating the condition of definiteness for general names. This requirement is met if there is a universal referred to by the name and, at the same time, it is unique; uniqueness, as in the case of predicates, consists in attaching exactly one universal to the name in question, i.e., in avoiding the double-denotation fallacy. Owing to the introduction of the concept of universal to provide general names with a reference, the new item should be added to the list of forms of normal definitions as given at the outset of this Section. As yet, we have no formal means to present the structure of the definition of a general name because of the lack of appropriate means in predicate logic, in which the syntactic category of general names, as construed above, does not occur. The closest neighbour is the category of names made from predicates with the help of Hilbert e-operator, in whose definition there explicitly occurs item (i) from among the three above-mentioned conditions. 5 Unfortunately, the language of predicate logic does not suffice to express the two other conditions. In particular item (iii) as resorting to some concepts of methodology of sciences cannot be handled solely in terms of predicate logic. Its significance from the rhetorical point of view has to be shown in connection with a philosophical doctrine about mind, language and logic which is outlined in next sections. 3. T h e holistic doctrine of definition 3.1. The traditional theory of definition, going back to Aristotle, claims that each definition characterizes a species of things by mentioning its genus (kind) and what distinguishes the species in question from the other species of the same genus. This was stated in the famous maxim definitio fit per genus et differentiam speci5

This operator forms a name to be used in the member supposition, i.e. referring to an arbitrary element of the class denoted by a predicate, provided that the predicate in question is not empty. See Hilbert and Bernays [1934-39], Beth [1959], Bell [1993].

198

Eight: Defining, Logic, and Intelligence

ficam. Genus and species referred to in this context belong to the category of universale discussed in the previous section. At t h e same time, it was hardly possible to refuse the label of definition to some other kinds of statements, e.g., terminological explanations as found in dictionaries, hence the characterization of a species of things through a kind and a specific difference has been called classical definition to distinguish it from other types of definition. It held a distinguished position in the structure of scientific or philosophical theories, while other kinds played only an auxilary role. In this sense the traditional theory can be termed a monistic d o c t r i n e of definition. On the other hand, with the decline of Aristotelian metaphysics, upon which this monistic doctrine was based, there appeared another kind of monism. This new doctrine grew so dominant t h a t it took over even the label of 'traditional conception' from the old Aristotelian doctrine. According to this new tradition, the only function of a definition consists in its being a convention with regard to the use of a language. Such a convention results in introducing a new expression or combination of symbols, called definiendum, to replace another combination of expressions, called definiens, whose meaning is already known in terms of certain already existing expressions. Thus definitions are considered as a kind of shorthand which can be, theoretically at least, dispensed with altogether; it has no relation to the content of the theory considered, but only to the language in which it is expressed. However, this second monism proved as untenable as was the first. Not only the fact t h a t the definiteness condition ties a definition to an extra-linguistic reality, but even more the existence of recursive definition in mathematics (see Section 4 below) compelled theorists of definition to acknowledge its kinship with some statements about things. 6 Such being the case, it was tempting to adopt a pluralistic position which admitted of more t h a n one kind of definition with respect to the question of what definitions refer to, whether to things or to expressions of a language. T h a t position was typically represented by Ajdukiewicz [1958] and [1974], in the latter expressed 6

See, e.g., Curry [1958]; the above characterization of the second monistic doctrine is taken from the beginning of that paper.

3. The holistic doctrine of definition

199

even in the title ('Three concepts of definition'). The term nominal definition was reserved for definitions of expressions, and the term real definition for definitions of things. This compromise, though, creates a new problem, also handled by Ajdukiewicz, namely the question of what should be regarded as common to real and nominal definition: finally, the only feature shared by them is that each of them is a univocal characterization of something, the former of things, the latter of expressions; this is too little, indeed, to deserve a common denomination. To disentagle us from these problems, let us try to see the role and the corresponding structure of definition in a new way. This new approach starts from asking about the structure of the predicate 'is definition'. It is usually taken for granted that it should be a binary, i.e., two-argument, predicate of the structure 'D(a;, j/)', where '£>' means 'is the definition of', 'x' represents the statement being the definition, and 'y' represents something being defined, either an expression or a thing. Should we stick to that point as an irrevocable dogma? Once having agreed that the procedure of ostension should be a paradigm for any definition altogether, one is prepared to describe acts of defining by the following six-argument predicate D(x,e,L,c,t,r), that is to say: χ is the definition of the expression e^ (definiendum in language L) by characterization c intended to introduce the thing t into the domain of L, and to produce the internal record r of t in the mind-body of an addressee. An internal record (cf. above, Subsection 1.2) is conceived as belonging both to one's body, where data must be somehow physically stored (as in computer memory), and to one's mind by which the data can be read out in appropriate conditions (e.g., in an act of remembering). The internal record is produced by what is above called characterization, and comprises such various stimuli as verbal descriptions, drawings, models, etc.; in an ostensive procedure it can be said that we deal with a model, namely a physical instantiation of a general object (i.e., a universal). This approach takes into account all the elements involved in the process of defining in their mutual relations forming a whole,

200

Eight: Defining, Logic, and Intelligence

it can thus be termed the holistic doctrine of definition, opposed to both monism and pluralism. Its holistic nature consists also in taking into account all the three kinds of semiotic relations, viz., syntactic, semantic and pragmatic connections. A definition defines am expression (syntactic aspect), introduces a thing (semantic aspect), and produces an internal record (pragmatic aspect). According to the holistic approach, a nominal definition (cf. 2.1 above) should rather be called an asemantic definition to point at the lack or irrelevance of the semantic component. The term 'real definition' proves then unnecessary as any complete definition is concerned with a thing belonging to extra-linguistic reality. In certain situations some members of this whole prove irrelevant to the purpose in question and are then disregarded. For instance, when we propose a new term to abbreviate a longer phrase (e.g., 'radius of a circle' for 'straight line drawn from the centre to the circumference'), when the existence of the thing denoted by the latter is granted (as is granted the existence of a circle's radius owing to postulate 3 in Book One of Elements), then the lack of a mention about the thing itself does not do any harm to our theorizing. In other cases, the explicit mention of the thing in question must be made in the form of the proof of its existence. All the six factors occur in the ostensive definition. What is remarkable about it is that the characterization c, which is usually provided by a verbal description, in the case of ostension is provided by a physical object bearing a special relation to the introduced thing t\ this relation consists in instantiating the thing by the physical object shown in the role of characterization. The occurrence of this relation helps us to realize how the thing t is referred to by the expression e. For instance, when ostensively defining the thing 'yellow' (being, presumably, a universal) one produces, e.g., a piece of lemon and utters the phrase 'this is yellow' so that the lemon provides the intended stimulus on the retina and somewhere deeper in the brain so producing an internal record to represent the thing. Whether such stimulating characterization is provided by a physical object or by a verbal description depends on the stage of construction of the language in question. To begin with, we absolutely need ostension, but later on a verbal description can do

3. The holistic doctrine of definition

201

as well. After the notion of lemon and t h a t of colour are introduced to t h e language, t h e procedure of ostension can be replaced by a purely verbal explanation, as 'yellow is a colour like t h a t of a lemon'. T h u s the ostensive definition is not deprived of anything t h a t essentially belongs to the act of defining, while the fact t h a t t h e thing introduced in somehow instantiated in this act helps us to realize t h a t basic t r u t h about definitions t h a t not only a piece of language b u t also a piece of reality is what we deal with in defining. 3 . 2 . The lesson t h a t definitions deal also with things, and not with language alone, is necessary to understand the role of definition in theorizing, especially in developing scientific theories. Were every definition just a verbal move which makes our texts shorter but does not advance our knowledge about the world, t h e strategies of cognition and of communication would be something very different from what they actually are. 7 This claim may seem to oppose t h e commonly acknowledged postulate t h a t every normal definition should meet the criterion of eliminability, to the effect t h a t it be possible to replace any statement containing a defined expression by an equivalent statement not containing t h a t expression. In orthodox expositions of the logical theory of definition, this criterion is accompanied by t h a t of non-creativity which states t h a t a definition should not function as an axiom, t h a t is, whatever is provable in a theory on t h e basis of the axioms with the definition added to them has also to be provable without t h a t definition. 8 To avoid misunderstandings it should be stressed, first, t h a t these criteria have been stated to hold for deductive formalized theories, and no one has dared to claim their validity for empirical 7

Cf. requirement (iii), of Subsection 2.4 above, that the definition of a thing named with a general name should increase the explanatory power of a theory. For mere shorthand we need not such a serious term as 'definition'; the term 'abbreviation' would do, as the situation is not much different from that in which the abbreviation Ά Ι ' replaces the term 'Artificial Intelligence'. 8

See, e.g., Suppes [1957], Slupecki and Borkowski [1967], Grzegorczyk [1974], Logic [1981], These criteria are emphasized by Polish authors, in accordance with the teachings of S. Lesniewski; in the Anglo-Saxon world they became wider known due to P. Suppes who explicitly quotes Lesniewski.

202

Eight: Defining, Logic, and Intelligence

theories; second, even in regard to formalized theories it should be remembered t h a t Lesniewski had his own rigorous doctrine of formalization which r a t h e r postulated a reform of mathematical practice t h a n tried to do justice to it. This practice, however, does not obey t h e criterion of non-creativity, as duly observed by Curry [1958] (in spite of his formalistic tendencies), in one case at least, namely in t h e case of a recursive definition, called also inductive definition or definition by induction. T h e creative function of definitional recursion is conspicuous in its introducing a new object which otherwise would not a p p e a r altogether in t h e domain in question. Such creation is performed in two steps. In t h e initial step it is unconditionally specified which objects belong to a given set, e.g., t h a t 0 belongs to t h e set of n a t u r a l numbers; in t h e induction step it is specified which objects belong t o t h e set in question provided t h a t t h e objects listed in t h e initial condition belong to it. As an example take t h e definition of addition ( S is t h e successor f u n c t i o n ) . y+ 0= y the initial step y + S(x) = S(y + ζ) the induction step T h e object introduced by t h e above definition belongs to t h e category of functions, or operations, which together with n a t u r a l n u m b e r s , as individuals, inhabit t h e domain of P e a n o arithmetic (so called a f t e r Giuseppe Peano, 1858-1932, who first axiomatized t h e arithmetic of n a t u r a l numbers). Functions are genuine m a t h ematical entities which are ordered sets, e.g., t h e function of addition is identical with t h e set of ordered pairs resulting (successively, so t o speak) f r o m 0 and 0, 0 and 1, 1 a n d 0, 1 and 1, and so on into infinity. W i t h o u t t h e above inductive definition, no such object would enter t h e domain of arithmetic. One may ask whether t h e listed pair of equations constitutes a normal definition; t h e answer is as follows: t h e literal form of normal definition can be trivially obtained when t h e equations are preceded by a clause like t h a t : " T h e function