LOGIC, THOUGHT AND ACTION LOGIC, EPISTEMOLOGY, AND THE UNITY OF SCIENCE VOLUME Editors Shahid Rahman, University of Lille III, France John Symons, University of Texas at El Paso, U.S.A Editorial Board Jean Paul van Bendegem, Free University of Brussels, Belgium Johan van Benthem, University of Amsterdam, the Netherlands Jacques Dubucs, University of Paris I-Sorbonne, France Anne Fagot-Largeault, Collège de France, France Bas van Fraassen, Princeton University, U.S.A Dov Gabbay, King’s College London, U.K Jaakko Hintikka, Boston University, U.S.A Karel Lambert, University of California, Irvine, U.S.A Graham Priest, University of Melbourne, Australia Gabriel Sandu, University of Helsinki, Finland Heinrich Wansing, Technical University Dresden, Germany Timothy Williamson, Oxford University, U.K Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic At present, no single logical, semantical or methodological framework dominates the philosophy of science However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light on basic issues in the discussion of the unity of science This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity Logic, Thought and Action Edited by Daniel Vanderveken University of Quebec,Trois-Rivières, QC, Canada A C.I.P Catalogue record for this book is available from the Library of Congress ISBN 1-4020-2616-1 (HB) ISBN 1-4020-3167-X (e-book) Published by Springer, P.O Box 17, 3300 AA Dordrecht, The Netherlands Sold and distributed in North, Central and South America by Springer, 101 Philip Drive, Norwell, MA 02061, U.S.A In all other countries, sold and distributed by Springer, P.O Box 322, 3300 AH Dordrecht, The Netherlands Cover image: Adaptation of a Persian astrolabe (brass, 1712-13), from the collection of the Museum of the Historyy of Science, Oxford Reproduced by permission Printed on acid-free paper All Rights Reserved © 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Printed in the Netherlands In Memoriam J.-Nicolas Kaufmann 1941–2002 Contents Contributing Authors xi Introduction Daniel Vanderveken Part I Reason, Action and Communication The Balance of Reason Marcelo Dascal 27 Desire, Deliberation and Action John R Searle 49 Two Basic Kinds of Cooperation Raimo Tuomela 79 Speech Acts and Illocutionary Logic John R Searle and Daniel Vanderveken Communication, Linguistic g Understanding and Minimal Rationality in Universal Grammar Andr´ ´ Leclerc Part II 109 133 Experience, Truth and Reality in Science Truth and Reference Henri Lauenerr† 153 viii LOGIC, THOUGHT AND ACTION Empirical Versus Theoretical Existence and Truth Michel Ghins 163 Michel Ghins on the Empirical Versus the Theoretical Bas C van Fraassen 175 Part III Propositions, Thought and Meaning 10 Propositional Identity, Truth According to Predication and Strong Implication Daniel Vanderveken 185 11 Reasoning and Aspectual-Temporal Calculus Jean-Pierre Descl´es 217 12 Presupposition, Projection and Transparency in Attitude Contexts Rob van der Sandt 245 13 The Limits of a Logical Treatment of Assertion Denis Vernant 267 Part IV Agency, Dialogue and Game-Theory 14 Agents and Agency in Branching Space-Times Nuel Belnap 15 Attempt, Success and Action Generation: a Logical Study of Intentional Action Daniel Vanderveken 291 315 16 Pragmatic and Semiotic Prerequisites for Predication Kuno Lorenz 343 17 On how to be a Dialogician Shahid Rahman and Laurent Keiff 359 18 Some Games Logic Plays Ahti-Veikko Pietarinen 409 ix Contents 19 Backward Induction Without Tears? Jordan Howard Sobel Part V 433 Reasoning and Cognition in Logic and Artificial Intelligence 20 On the Usefulness of Paraconsistent Logic Newton C.A da Costa, Jean-Yves B´ziau, ´ and Ot´ ´vio Bueno 465 21 Algorithms for Relevant Logic Paul Gochet, Pascal Gribomont and Didier Rossetto 479 22 Logic, Randomness and Cognition Michel de Rougemont 497 23 From Computing p g with Numbers to Computing p g with Words — From Manipulation of Measurements to Manipulation of Perceptions Lofti Zadeh 507 Contributing Authors Nuel Belnap is Alan Ross Anderson Distinguished Professor of Philosophy and Professor of the History and Philosophy of Science at the University of Pittsburgh He has written chiefly in philosophical logic, having co-authored The Logic of Questions and Answers (Yale University Press, 1976) with Thomas Steel, Entailment: the Logic of Relevance and Necessity (Princeton University Press) vol I (1975) with Alan Ross Anderson and vol II (1992) with Anderson and J Michael Dunn, The Revision Theory of Truth (MIT Press, 1993) with Anil Gupta, and Facing the Future: Agents and Choices in our Indeterminist World (Oxford University Press, 2001) with Michael Perloff and Ming Xu Jean-Yves B´ ´ eziau is now Professor of the Swiss National Science Foundation at the Institute of Logic of the University of Neuchˆ aˆtel He got a PhD in mathematical logic in Paris and a PhD in philosophy in S˜ ao Paulo He worked as a research fellow in Brazil, Poland and California (UCLA, Stanford) His main interests are paraconsistent logic, universal logic, philosophy of logic and philosophy of mathematics He has written with Newton da Costa and Otavio Bueno the book Elementos de teoria paraconsistente de conjuntos, Cle-Unicamp, Campinas, 1998, 188pp and more than 50 papers in Journals and collective books Ot´ ´ avio Bueno is Associate Professor of Philosophy at the University of South Carolina His main research area is in philosophy of science, philosophy of mathematics, philosophy of logic, and paraconsistent logic He has published papers in many journals and collections including Philosophy of Science, Synthese, Journal of Philosophical Logic, Studies in History and Philosophy of Science, British Journal for the Philosophy of Science, Analysis, Erkenntnis, History and Philosophy of Logic, and Logique et Analyse He is the author of two books: Constructive Empiricism: A Restatement and Defense (CLE, 1999), and Elements xii LOGIC, THOUGHT AND ACTION of Paraconsistent Set Theory (CLE, 1998) with Newton da Costa and Jean-Yves B´´eziau Newton da Costa is retired Professor of Philosophy at the University of Sao ˜ Paulo, Brazil, and currently Visiting Professor of Philosophy at the Federal University of Santa Catarina, Brazil His main research interests are non classical logic, model theory, foundations of inductive inference and the philosophy of science His publications include Logiques Classiques et non Classiques (Paris, Masson, l997), El Conocimiento Cientifico (Mexico, UNAM, 2000) and Science and Partial Truth (Oxford University Press, 2003), as well as some other books and numerous papers in specialized journals Marcelo Dascal is a Professor of Philosophy at Tel Aviv University, Israel He has taught in major universities in Europe, the Americas, and Australia He has been a fellow of the Netherlands Institute of Advanced Studies (Wassenaar), of the Institute of Advanced Studies (Jerusalem), Leibniz Professor at the Center for Advanced Studies (Leipzig), and is currently Gulbenkian Professor at the University of Lisbon (PT) He was awarded the Alexander von Humboldt Prize for 2002-2003 His main research areas are the philosophy of language and communication, the philosophy of mind, pragmatics, the history of modern philosophy, and the study of controversies As a Leibniz specialist, he has published La s´ ´emiologie de Leibniz (Paris, 1978), Leibniz: Language, Signs and Thought (Amsterdam, 1987), and has co-edited Leibniz and Adam (Tel Aviv, 1991) and Leibniz the Polemicist (Amsterdam, forthcoming) In the area of pragmatics and the philosophy of language he has published Pragmatics and the Philosophy of Mind, vol 1: Language and Thought (Amsterdam, 1983) and Interpretation and Understanding (Amsterdam, 2003); he has edited Dialogue — An Interdisciplinary Approach (Amsterdam, 1985) and co-edited Philosophy of Language — A Handbook of Contemporary Research (Berlin, 1991, 1995) He is the founder and editor of the journal Pragmatics & Cognition and of several book series Jean-Pierre Descl´ es is professor of Computer Sciences and Linguistics at Sorbonne University He is vice president of the Acad´ ´emie Internationale de Philosophie des Sciences His research interests are in the domain of the logic and natural languages; cognition and language; time, tense and space; logic of object determination His publications include Langages applicatifs, langues naturelles et cognition (Paris, Herm`es 1990) and articles in journals dealing with combinatory Logic, theoreti- 530 LOGIC, THOUGHT AND ACTION Figure 15 Depth of explicitation In a summarized form, the rules governing fuzzy constraint propagation are the following (Zadeh, 1996a) (A and B are fuzzy relations Disjunction and conjunction are defined, respectively, as max and min, with the understanding that, more generally, they could be defined via t-norms and s-norms (Klir and Yuan, 1995; Pedrycz and Gomide, 1998) The antecedent and consequent constraints are separated by a horizontal line.) Conjunctive Rule Conjunctive Rule X is A X is B X is A ∩ B (X ∈ U, Y ∈ B, A ⊂ U, B ⊂ V ) X is A Y is B (X, Y ) is A × B Disjunctive Rule Disjunctive Rule X is A (A ⊂ U, B ⊂ V ) A is A Y is B (X, Y ) is A × V ∪ U × B or X is B X is A ∪ B where A × V and U × B are cylindrical extensions of A and B, respectively Conjunctive Rule for isv X isv A X isv B X isv A ∪ B Projective Rule Surjective Rule (X, Y ) is A Y is projV A where projV A = supu A X is A (X, Y ) is A × V Derived Rules From Computing with Numbers to Computing with Words Compositional Rule 531 Extension Principle X is A (X, Y ) is B Y is A ◦ B (mapping rule)(Zadeh, 1965; 1975) X is A f (X) is f (A) where A ◦ B denotes the composition of A and B where f : U → V , and f (A) is defined by µf (A) (ν) = sup µA (u) u|ν=f (u) Inverse Mapping Rule Generalized modus ponens f (X) is A X is A if X is B then Y is C X is f −1 (A) Y is A ◦ (¬B) ⊕ C where µf −1 (A) (u) = µA (f (u)) where the bounded sum ¬B ⊕ C represents Lukasiewicz’s definition of implication Generalized Extension Principle f (X) is A q(X) is q f −1 (A) where µq (ν) = supu|ν=q(u) µA (f (u)) The generalized extension principle plays a pivotal role in fuzzy constraint propagation However, what is used most frequently in practical applications of fuzzy logic is the basic interpolative rule, which is a special case of the compositional rule of inference applied to a function which is defined by a fuzzy graph (Zadeh, 1974; 1996) More specifically, if f is defined by a fuzzy rule set f : if Xis Ai then X is Bi , i = 1, , n or equivalently, by a fuzzy graph Ai × Bi f is i and its argument, X, is defined by the antecedent constraint X is A, then the consequent constraint on Y may be expressed as mi ∧ Bi , Y is i where mi is a matching coefficient, mi = sup(Ai ∩ A), which serves as a measure of the degree to which A matches Ai 532 LOGIC, THOUGHT AND ACTION Syllogistic Rule: (Zadeh, 1984) Q1 A’s are B’s Q2 (A and B)’s are C’s (Q1 ⊗ Q2 )A’s are (B and C)’s, where Q1 and Q2 are fuzzy quantifiers; A, B and C are fuzzy relations; and Q1 ⊗ Q2 is the product of Q1 and Q2 in fuzzy arithmetic Constraint Modification Rules: (Zadeh, 1972; 1978) X is mA → X is f (A), where m is a modifier such as not, very, more or less, and f (A) defines the way in which m modifies A Specifically, if m = not then f (A) = A (complement) if m = very then f (A) = 2A (left square), where µ 2A (u) = (µA (u))2 This rule is a convention and should not be constructed as a realistic approximation to the way in which the modifier very functions in a natural language Probability Qualification Rule: (Zadeh, 1979b) (X is A) is Λ → P is Λ, where X is a random variable taking values in U with probability density p(u); Λ is a linguistic probability expressed in words like likely, not very likely, etc.; and P is the probability of the fuzzy event X, expressed as P = U µA (u)p(u) du The primary purpose of this summary is to underscore the coincidence of the principal rules governing fuzzy constraint propagation with the principal rules of inference in fuzzy logic Of necessity, the summary is not complete and there are many specialized rules which are not included Furthermore, most of the rules in the summary apply to constraints which are of the basic, possibilistic type Further development of the rules governing fuzzy constraint propagation will require an extension of the rules of inference to generalized constraints As was alluded to in the summary, the principal rule governing constraint propagation is the generalized extension principle which in a schematic form may be represented as f (X1 , , Xn ) is A q(X1 , , Xn ) is q(f −1 (A)) 533 From Computing with Numbers to Computing with Words In this expression, X1 , , Xn are database variables; the term above the line represents the constraint induced by the IDS; and the term below the line is the TDS expressed as a constraint on the query q(X1 , , Xn ) In the latter constraint, f −1 (A) denotes the pre-image of the fuzzy relation A under the mapping f : U → V , where A is a fuzzy subset of V and U is the domain of f (X1 , , Xn ) Expressed in terms of the membership functions of A and q(f −1 (A)), the generalized extension principle reduces the derivation of the TDS to the solution of the constrained maximization problem µq (X1 , , Xn )(ν) = sup (u1 , ,un ) (µA (f (u1 , , un ))) in which u1 , , un are constrained by ν = q(u1 , , un ) The generalized extension principle is simpler than it appears An illustration of its use is provided by the following example The IDS is: most Swedes are tall The query is: What is the average height of Swedes? The explanatory database consists of a population of N Swedes, Name , , Name N The database variables are h1 , , hN , where hi is the height of Name i , and the grade of membership of Name i in tall is µtall (hi ), i = 1, , n The proportion of Swedes who are tall is given by the sigma-count ( Zadeh, 1978b) Count (tall – Swedes / Swedes) = N µtall (hi ) i from which it follows that the constraint on the database variables induced by the IDS is N µtall (hi ) is most i In terms of the database variables h1 , , hN , the average height of Swedes is given by hi have = N i 534 LOGIC, THOUGHT AND ACTION Since the IDS is a fuzzy proposition, have is a fuzzy set whose determination reduces to the constrained maximization problem µhave (ν) = sup µmost h1 , ,hN N µtall (hi ) i subject to the constraint N ν= hi i It is possible that approximate solutions to problems of this type might be obtainable through the use of neurocomputing or evolutionarycomputing-based methods As a further example, we will return to a problem stated in an earlier section, namely, maximization of a function, f , which is described in words by its fuzzy graph, f ∗ (Fig 10) More specifically, consider the standard problem of maximization of an objective function in decision analysis Let us assume – as is frequently the case in real-world problems – that the objective function, f , is not well-defined and that what we know about can be expressed as a fuzzy rule-set f: if X is A1 then Y is B1 if X is A2 then Y is B2 if X is An then Y is Bn or, equivalently, as a fuzzy graph Ai × Bi f is i The question is: What is the point or, more generally, the maximizing set (Zadeh, 1998) at which f is maximized, and what is the maximum value of f ? The problem can be solved by employing the technique of α-cuts ( Zadeh, 1965; 1975) With refererence to Fig 16, if Aiα and Biα are α-cuts of Ai and Bi , respectively, then the corresponding α-cut of f ∗ is given by Aiα × Biα fα∗ = i From this expression, the maximizing fuzzy set, the maximum fuzzy set and maximum value fuzzy set can readily be derived, as shown in Figs 16 and 17 From Computing with Numbers to Computing with Words Figure 16 Figure 17 535 α-cuts of a function described by a fuzzy graph Computation of maximizing set, maximum set and maximum value set A key point which is brought out by these examples and the preceding discussion is that explicitation and constraint propagation play pivotal roles in CW This role can be concretized by viewing explicitation and constraint propagation as translation of propositions expressed in a natural language into what might be called the generalized constraint language (GCL) and applying rules of constraint propagation to expressions in this language – expressions which are typically canonical forms of propositions expressed in a natural language This process is schematized in Fig 18 536 LOGIC, THOUGHT AND ACTION Figure 18 Conceptual structure of computing with words The conceptual framework of GCL is substantively differently from that of conventional logical systems, e.g., predicate logic But what matters most is that the expressive power of GCL – which is based on fuzzy logic – is much greater than that of standard logical calculi As an illustration of this point, consider the following problem A box contains ten balls of various sizes of which several are large and a few are small What is the probability that a ball drawn at random is neither large nor small? To be able to answer this question it is necessary to be able to define the meanings of large, small, several large balls, few small balls and neither large nor small This is a problem in semantics which falls outside of probability theory, neurocomputing and other methodologies An important application area for computing with words and manipulation of perceptions is decision analysis since in most realistic settings the underlying probabilities and utilities are not known with sufficient precision to justify the use of numerical valuations There exists an extensive literature on the use of fuzzy probabilities and fuzzy utilities in decision analysis In what follows, we shall restrict our discussion to two very simple examples which illustrate the use of perceptions First, consider a box which contains black balls and white balls (Fig 19) If we could count the number of black balls and white balls, the probability of picking a black ball at random would be equal to the proportion, r, of black balls in the box From Computing with Numbers to Computing with Words Figure 19 537 A box with black and white balls Now suppose that we cannot count the number of black balls in the box but our perception is that most of the balls are black What, then, is the probability, p, that a ball drawn at random is black? Assume that most is characterized by its possibility distribution (Fig 20) In this case, p is a fuzzy number whose possibility distribution is most, that is, p is most Figure 20 Membership function of most Next, assume that there is a reward of a dollars if the ball drawn at random is black and a penalty of b dollars if the ball is white In this case, if p were known as a number, the expected value of the gain would be: e = ap − b(1 − p) Since we know not p but its possibility distribution, the problem is to compute the value of e when p is most For this purpose, we can employ the extension principle (Zadeh, 1965; 1975), which implies that the possibility distribution, E, of e is a fuzzy number which may be expressed as E = a most − b(1 − most) For simplicity, assume that most has a trapezoidal possibility distribution (Fig 20) In this case, the trapezoidal possibility distribution of E can be computed as shown in Fig 21 538 LOGIC, THOUGHT AND ACTION Figure 21 Computation of expectation through use of the extension principle It is of interest to observe that if the support of E is an interval [α, β] which straddles O (Fig 22), then there is no non-controversial decision principle which can be employed to answer the question: Would it be advantageous to play a game in which a ball is picked at random from a box in which most balls are black, and a and b are such that the support of E contains O Figure 22 Figure 23 Expectation of gain A box with balls of various sizes and a definition of large ball Next, consider a box in which the balls b1 , , bn have the same color but vary in size, with bi , i = 1, , n having the grade of membership 539 From Computing with Numbers to Computing with Words µi in the fuzzy set of large balls (Fig 23) The question is: What is the probability that a ball drawn at random is large, given the perception that most balls are large? The difference between this example and the preceding one is that the event the ball drawn at random is large is a fuzzy event, in contrast to the crisp event the ball drawn at random is black The probability of drawing bi is 1/n Since the grade of membership of bi in the fuzzy set of large balls is µi , the probability of the fuzzy event the ball drawn at random is large is given by (Zadeh, 1968) P = n µi On the other hand, the proportion of large balls in the box is given by the relative sigma-count (Zadeh, 1975b; 1978) Count (large.balls / balls.in.box ) = n µi Consequently, the canonical form of the perception most balls are large may be expressed as µi is most n which leads to the conclusion that P is most It is of interest to observe that the possibility distribution of P is the same as in the preceding example If the question were: What is the probability that a ball drawn at random is small, the answer would be P is n νi where νi , i = 1, , n, is the grade of membership of bi in the fuzzy set of small balls, given that n µi is most What is involved in this case is constraint propagation from the antecedent constraint on the µi to a consequent constraint on the νi This problem reduces to the solution of a nonlinear program What this example points to is that in using fuzzy constraint propagation rules, application of the extension principle reduces, in general, to 540 LOGIC, THOUGHT AND ACTION the solution of a nonlinear program What we need – and not have at present – are approximate methods of solving such programs which are capable of exploiting the tolerance for imprecision Without such methods, the cost of solutions may be excessive in relation to the imprecision which is intrinsic in the use of words In this connection, an intriguing possibility is to use neurocomputing and evolutionary computing techniques to arrive at approximate solutions to constrained maximization problems The use of such techniques may provide a closer approximation to the ways in which human manipulate perceptions Concluding Remarks In our quest for machines which have a high degree of machine intelligence (high MIQ), we are developing a better understanding of the fundamental importance of the remarkable human capacity to perform a wide variety of physical and mental tasks without any measurements and any computations Underlying this remarkable capability is the brain’s crucial ability to manipulate perceptions – perceptions of distance, size, weight, force, color, numbers, likelihood, truth and other characteristics of physical and mental objects A basic difference between perceptions and measurements is that, in general, measurements are crisp whereas perceptions are fuzzy In a fundamental way, this is the reason why to deal with perceptions it is necessary to employ a logical system that is fuzzy rather than crisp Humans employ words to describe perceptions It is this obvious observation that is the point of departure for the theory outlined in the preceding sections When perceptions are described in words, manipulation of perceptions is reduced to computing with words (CW) In CW, the objects of computation are words or, more generally, propositions drawn from a natural language A basic premise in CW is that the meaning of a proposition, p, may be expressed as a generalized constraint in which the constrained variable and the constraining relation are, in general, implicit in p In coming years, computing with words and perceptions is likely to emerge as an important direction in science and technology In a reversal of long-standing attitudes, manipulation of perceptions and words which describe them is destined to gain in respectability This is certain to happen because it is becoming increasingly clear that in dealing with real-world problems there is much to be gained by exploiting the tolerance for imprecision, uncertainty and partial truth This is the primary motivation for the methodology of computing with words (CW) and the From Computing with Numbers to Computing with Words 541 computational theory of perceptions (CTP) which are outlined in this paper Acknowledgement The author acknowledges Prof Michio Sugeno, who has contributed so much and in so many ways to the development of fuzzy logic and its applications References Berenji H.R (1994) “Fuzzy Reinforcement Learning and Dynamic Programming”, Fuzzy Logic in Artificial Intelligence (A.L Ralescu, Ed.), Proc IJCAI’93 Workshop, Berlin: Springer-Verlag, pp 1–9 Black M (1963) “Reasoning with Loose Concepts”, Dialog 2, pp 1–12 Bosch P (1978) Vagueness, Ambiguity and all the Rest, Sprachstruktur, Individuum und Gesselschaft (M Van de Velde and W Vandeweghe, Eds.) 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Dialogue and Games, to the logic of action, dialogues and language games and the last part, Reasoning and Cognition in Logic and Artificial Intelligence, to the role and formal methods of logic and. .. History and Philosophy of Logic, and Logique et Analyse He is the author of two books: Constructive Empiricism: A Restatement and Defense (CLE, 1999), and Elements xii LOGIC, THOUGHT AND ACTION