Language of Motivation and Language of Actions [Reprint 2017 ed.] 9783110819298, 9789027923851

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JANUA LINGUARUM STUDIA M E M O R I A E N I C O L A I VAN WIJK D E D I C A T A edenda curai C. H. V A N S C H O O N E V E L D Indiana University

Series Maior,

67

LANGUAGE OF MOTIVATION AND LANGUAGE OF ACTIONS

by

MARIA NOWAKOWSKA

1973

MOUTON THE HAGUE • PARIS

© Copyright 1973 in The Netherlands. Mouton & Co. N.V., Publishers, The Hague. No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm or any other means, without written permission from the publishers.

LIBRARY OF CONGRESS CATALOG CARD NUMBER 72-94491

Printed in the Netherlands.

To Professor Anatol Rapoport

ACKNOWLEDGMENTS

I take this opportunity to thank those whose comments and criticism led to corrections and improvements: to Professor Anatol Rapoport, not only for his penetrating remarks, but also for his favourable reaction to the earlier draft of the manuscript, which provided a stimulus for further research; to Professor Tadeusz Kotarbinski, whose theory of actions provided a constant guideline, in particular in constructing the language of actions; to Mrs. Olga Katchan of the University of Sydney, for her careful linguistic comments; to Professor Zdzislaw Pawlak of the Mathematical Institute of the Polish Academy of Sciences and Mr. Anthony Elliott of the University of New South Wales for their comments concerning the formal aspects of the manuscript. M.N. Sydney, December 1971

PREFACE

It is probably no accident that a theory of grammatical structures can be so readily and naturally generalized as a schema for theories of other kinds of complicated human behaviour. An organism that is intricate and highly structured enough to perform the operations that we have seen to be involved in linguistic communication does not suddenly lose its intricacy and structure when it turns to non-linguistic activities. In particular, such an organism can form verbal plans to guide many of its non-verbal acts. The verbal machinery turns out sentences - and, for civilized men, sentences have a compelling power to control both thought and action. Miller and Chomsky 1

The principal aim of this book was to construct a formal system in which behaviour is treated as a certain language, and analysed by methods of mathematical linguistics. Very roughly, the latter deals with structures of sets of linearly ordered objects of some kind (letters, words, symbols, etc.). The methods used do not, however, depend on the nature of these objects and can be applied to non-linguistic behavioural phenomena also, provided only that they reveal a structure of the above type. This idea was explored in the book by constructing a general system called the language of actions, with an embedded subsystem called the language of motivation. In the first of these systems, actions (verbal or not) were identified with words of a certain vocabulary, while admissible strings of actions (i.e., for instance, such strings of actions which are physically possible to perform) played the role of sentences in the language. In this way, the set of strings of actions (relevant for the description of a given situation of interest) becomes formally identical with a certain language, and consequently, may be studied by the appropriate methods of mathematical linguistics. There is even more to that: outcomes of the strings of actions may be treated in much the same way as meanings of sentences, thus enriching the considered artificial language with a semantics. By defining various concepts within the above outlined framework, a fairly rich 1 "Finitary Models of Language Users", in Handbook of Mathematical Psychology, vol. II, R.D. Luce, R.R. Bush and E. Galanter, eds. (Wiley, New York, 1963), 419-491. For the quoted fragment, see p. 488.

10

PREFACE

deductive system of theorems was obtained. This system is developed in three principal directions. First, the study concerns the structural properties of sets of strings of actions which lead to a given result (or set of results). Here such concepts as various types of attainability, positive and negative decisive moments, complete possibility with respect to a set of outcomes, etc. are defined and characterized by theorems. Second, the analysis concerns structures of sets of results brought about by a given string of actions. Here the main concepts are those of various types of periodicity, and the theorems give a characterization of those strings of actions which produce results occurring periodically. Finally, the system was enriched by an additional primitive concept, namely that of the relation 'to be more ethical than', holding between outcomes. This relation makes it possible to introduce various types of ethical valuations of strings of actions which lead to sets of ethically diverse outcomes. Two types of such valuations, referred to as 'puristic' and 'liberal' were introduced and discussed. The starting point for defining the subsystem embedded in the language of actions, namely the language of motivation, was the assumption of the existence of a linguistic representation of motivation in natural language. Accordingly, the language of motivation was defined as the set of all sentences which contain special characteristic expressions, such as 'I want', 'I ought to', etc., which are used in evaluation, justification, explanation, or planning of actions. These expressions, called motivational functors, are assumed to represent some psychological continua (scales), forming a motivational space, in which every object or situation can be evaluated and classified. Moreover, it was postulated that in a choice situation, the decision is a function of orderings of alternatives along the dimensions of motivational space. This function, identified with the motivational structure of a given individual, has therefore the formal properties of Arrow's Social Welfare Function; such an interpretation permits one to draw some interesting conclusions from Arrow's Impossibility Theorem, as well as from certain results in the theory of voting. The analysis of the language of motivation concerned also the logical behaviour of motivational functors. This led to the construction of a motivational calculus, having the form of a set of rules of inference from sentences containing given motivational functors. This calculus is based on one primitive notion, namely that of semantic admissibility. The intuitive idea is that a (grammatically acceptable) sentence is semantically inadmissible if there exists no context in which it may be meaningfully applied, or if it can be applied only in rather unusual circumstances. Using the concept of semantic admissibility, semantic implication was defined (A implies semantically B, if the sentence 'A, but not B' is semantically inadmissible) to describe the usual inference from utterances containing motivational functors. The calculus obtained in this way bears some resemblance to modal, epistemic, and deontic logics, but differs from them in one respect: the point was not to construct an axiomatic system for one or more of the functors, but to devise a system of rules

PREFACE

11

of transformation, which - when applied to sets of motivational sentences - transform them into other sets, the latter representing a legitimate inference from the former. By constructing a set-theoretical model, it was shown that the suggested motivational calculus is logically consistent. Subsequently, the motivational calculus was used for obtaining formal rules of inference about motivational premises: roughly, if in an utterance of the form 'A, because of B\ B does not semantically imply A, but B complemented by C already jointly imply (semantically) A, then C may be attributed to the speaker, as a possible sentence (hypothesis) reflecting his intentions. This type of inference rule allows us to draw conclusions from utterances which have the form of an implication which is not valid, but whose premise can be complemented so as to make it valid. Next, concerning embedding the language of motivation into the language of actions, the latter was enriched by additional primitive concepts: of a set of sentences, and relations connecting these sentences either with strings of actions or with their results. These relations were intended to cover the intuitions of agreement between what was said in a sentence and what was done. The formal properties of these relations were chosen in such a way as to reflect certain semantic aspects of sentences involving the considered motivational functors (namely 'I want', 'I don't want', 'I ought to' and 'I shouldn't'). The relations between sentences and actions or their results allowed the definition of two concepts of consistency. One of them, called motivational, covers the intuition of overall consistency of 'mixed strings', containing both utterances and actions (in case of possibly more than one utterance). The second, called satisfiability, refers to sets of utterances and is defined by the requirement of the existence of a string of actions motivationally consistent with these utterances. The subsequent considerations proceed in three main directions. First, in the particular cases of utterances with functors 'I ought to' and 'I shouldn't' it was possible to give a characterization of sets of strings of actions motivationally consistent with these utterances: under some conditions, these sets constitute the 'context-free' languages. Second, the concept of motivational consistency was used to extend the set of inference rules presented in the motivational calculus, so as to cover the case of inference made on the basis of both actions and utterances. It seems that this approach opens up the possibility of the formalization of inference in psychology. Third, the analysis concerned the non-satisfiable sets of sentences, that is, such sets which cannot be embedded into a string of actions so as to obtain a motivationally consistent combined string. This describes a certain type of conflict, where it is not possible to satisfy all requirements specified in the utterances. An exhaustive taxonomy of conflicts between what a person wants, does not want, ought to do, and should not do was constructed, this taxonomy being built on the concept of maximally satisfiable sets of utterances.

12

PREFACE

Finally, we have outlined how the combined system of language of actions and language of motivation may be used to develop a theory of plans, as understood by Miller, Galanter, and Pribram in Plans and the Structure of Behavior; here the consecutive stages of the development of a plan of a certain string of actions play the same role as the consecutive stages of the generation of a given sentence, by consecutive applications of rewriting rules, as in mathematical linguistics. The concept of motivational consistency may be used to derive a hierarchy of such 'rewriting rules'. The order of presentation of the material in the book is the following. Chapters 1 and 2 are devoted to the analysis of language of motivation, from the linguistic and psychological (Chapter 1) and logical (Chapter 2) points of view. To make the book self-contained, Chapter 3 presents the basic concepts and results of mathematical linguistics; the first sections cover the theory of automata and generative grammars, as developed mostly in the U.S., Israel, and France. The remaining sections contain an outline of analytic models in linguistics, developed mainly in the U.S.S.R. and Roumania. Chapter 4 deals with the language of actions, while Chapter 5 is concerned with the combined languages of actions and motivation. Finally, Chapter 6 provides, on the one hand, an exhaustive summary and a discussion of the assumptions, main concepts, and results of the suggested models. On the other hand, it also tries to show the connections between the presented formal systems and the existing systems and theories. Particular points discussed are, among others, the relations with models of choice and motivation; with modal, deontic, and epistemic logic; with the theory of semantics; with von Wright's logic of actions and with Kotarbinski's praxiology. In view of the interdisciplinary character of the book, it may be of potential interest not only to psychologists, interested in the modelling approach to psychology, but also to linguists, logicians, and philosophers dealing with the theory of actions.

TABLE OF CONTENTS

Acknowledgments Preface

7 9

1. Language of motivation 1.1. Introduction 1.2. Distinguished classes of expressions of language of motivation . . . . 1.2.1. Class 1 : epistemic functors 1.2.2. Class 2: emotional functors 1.2.3. Class 3: motivational functors 1.2.3.1. Proper motivational functors 1.2.3.2. Normative (deontic) motivational functors 1.2.4. Class 4: intensional functors 1.2.5. Class 5: functors describing outcomes 1.3. Model of choice behaviour and structure of motivation 1.3.1. Assumptions 1.3.2. Motivational structure as a social decision function (Arrow's Social Welfare Function) 1.3.3. Consequences

17 17 19 19 24 27 27 31 37 38 39 39

2. Motivational calculus 2.1. Logical analysis of language of motivation. Motivational calculus. . . 2.1.1. The concept of admissibility of a sentence 2.1.2. The concept of semantic implication 2.1.3. General laws for semantic implications 2.1.4. Rules of semantic inference for some motivational functors . . 2.1.4.1. I know, I am certain, I think 2.1.4.2. I believe, I doubt 2.1.4.3. Some problems of ordering epistemic functors . . . . 2.1.4.4. l e a n , I cannot 2.1.4.5. I am glad that

48 48 50 52 52 53 53 57 58 60 63

40 44

14

CONTENTS

2.2.

2.3. 2.4. 2.5.

2.1.4.6. I prefer, I want, I don't want 2.1.4.7. I must, I ought to 2.1.4.8. Concluding remarks Logical consistency of motivational calculus 2.2.1. Propositional calculus 2.2.2. Modal logic 2.2.3. The logical status of the motivational calculus 2.2.4. Model interpretation of motivational calculus 2.2.5. Concluding remarks Reduction of functors Examples of analysis of composite sentences Motivational inference rules

65 67 70 71 71 74 77 79 89 90 91 93

3. Basic concepts of mathematical linguistics 3.1. Abstract definition of language 3.1.1. Examples 3.1.2. Basic models of mathematical linguistics 3.2. Synthetic models 3.2.1. Acceptors and languages 3.2.1.1. Finite state automata 3.2.1.2. Regular languages 3.2.1.3. Push-down acceptors 3.2.1.4. Properties of context free languages 3.2.1.5. Linear bounded automata 3.2.2. Generative grammars 3.2.2.1. Grammars and languages 3.2.2.2. Main types of grammars and their relations to types of languages 3.2.3. Problems of decidability 3.3. Categorial grammars 3.4. Analytical models 3.4.1. Division into distributional classes 3.4.2. Divisions and their derivatives 3.4.3. Typology of natural languages 3.4.4. Configurations 3.4.5. Subordination 3.5. An approach to semantics

95 95 96 97 98 98 99 101 107 112 114 117 117 119 121 123 124 125 127 128 129 129 130

4. Language of actions 4.1. Definition of the language of actions 4.1.1. Preliminaries 4.1.2. Language of actions and its semantics 4.1.3. Illustrative examples

132 132 132 133 135

CONTENTS

4.2. The basic subsets of the monoid D* 4.2.1. Inverse images. Attainability 4.2.2. Praxiological sets 4.3. Equivalence classes 4.3.1. General construction 4.3.2. Applications 4.4. Structural properties of sets of strings 4.4.1. The concept of complete possibility 4.4.2. Decisive moments 4.4.3. The concepts of the enforcing situation and the only-exit situation 4.5. Structures of sets of time-results 4.5.1. Introductory remarks 4.5.2. Sets of time-results R(u) 4.5.3. The concepts used for describing the sets R(u). Homonymity of a string of actions 4.5.4. The concept of time-traces 4.5.5. The concept of periodicity of time-traces 4.5.6. Periodicity of time-traces for concatenations of strings of actions 4.5.7. Synonymity of strings m and v 4.5.8. Suppressing actions 4.6. Uprightness of outcomes and uprightness of actions 4.6.1. Introductory remarks 4.6.2. Properties of the binary relation Eth 4.6.3. Properties of the quarternary relation ETH 4.6.4. Uprightness of strings of actions 4.6.4.1. Ethically puristic orderings of strings of actions . . . . 4.6.4.2. Ethically liberal orderings of strings of actions . . . . 4.6.5. Comments 4.6.6. Alternative interpretations of the relations Eth and ETH . . . 4.7. Concluding remarks 5. Language of actions with embedded verbal parts 5.1. The additional primitive concepts 5.2. Consistency between verbal and non-verbal actions 5.2.1. Preliminary concepts 5.2.2. Motivational consistency 5.3. Structure of motivationally consistent strings 5.3.1. Satisfiable strings of sentences 5.3.2. Properties of the sets O (a k) 5.3.3. Properties of the sets O' (a 5.3.4. Comments 5.3.5. Structural properties of motivationally consistent strings. . . . 5.4. Models of conflict

15

137 137 138 141 141 143 145 145 149 151 154 154 155 156 158 161 172 173 176 177 177 178 181 184 185 187 190 192 194 196 196 201 201 202 203 203 204 206 210 211 213

16

CONTENTS

5.4.1. Preliminaries 5.4.2. Maximal satisfiable sets 5.4.3. Indices of conflicts 5.4.4. Examples and comments 5.5. Motivational inference rules 5.6. Plans and grammars

213 214 216 222 224 227

6. Summary and discussion 6.1. Language of motivation 6.1.1. Motivational space 6.1.2. Motivational calculus 6.1.3. The natural language and the motivational calculus 6.1.4. Some comments on semantics of the language of motivation . . 6.1.5. Some further comments 6.2. Language of actions 6.2.1. The tacit assumptions underlying the construction of the suggested model 6.2.2. The main concepts introduced in the language of actions. . . . 6.2.3. Language of actions and von Wright's logic of actions . . . . 6.2.4. Final comments 6.3. The two languages combined 6.3.1. Additional primitive concepts and their intuitive justification. . 6.3.2. The main concepts and results 6.3.3. Discussion 6.4. Methodological remarks

230 231 231 235 237 239 242 243 244 247 249 251 251 251 253 254 256

Bibliography List of symbols Name index Topic index

257 264 265 268

1 LANGUAGE OF MOTIVATION

1.1. INTRODUCTION

The starting point for the considerations presented below is the assumption of the existence of a linguistic representation of motivation in natural language. This assumption does not depend on any particular definition of the concept of motivation, and is to be understood simply that some information about motivation (whichever way this concept may be defined) is contained in the sentences used for evaluation, justification, or explanation of decisions one makes or is planning to make. Even a cursory analysis of such sentences allows us to distinguish a set of characteristic expressions which appear in them. Such a set (far from complete) is presented in Table 1. Roughly, by language of motivation we shall mean here the set of all sentences which contain at least one of the expressions designated under number 3 in Table 1. The expressions of the remaining classes in this table play an auxiliary role, appearing in many, but not in all, sentences in the language of motivation. The subsequent analysis will be concerned mainly with some semantic aspects of expressions from Table 1; regarding their syntactic features, it will only be mentioned that, classified from the point of view of the theory of syntactic categories, all these expressions are functors. 1 It will be argued that these functors constitute linguistic representations of several scales, these scales forming a 'motivational space'.

1 The theory of syntactic categories was created (under the name semantic categories) by Husserl in the nineteenth Century, and subsequently developed by Lesniewski, Ajdukiewicz, Bar-Hillel, Lambek and others. For the most recent version see the section on categorial grammars in Chapter 3. Roughly, all syntactic categories other than those of names and sentences are generally called functors. The main aim of the theory is to design a symbolism for syntactic categories which would allow checking in a mechanical way whether or not a string of words constitutes a sentence.

18

LANGUAGE OF MOTIVATION TABLE 1 Functors of the language of motivation Class 1. Epistemic functors I I I I

can know how to may am able to

I know that I am certain that I am sure that I I I I

deduce am convinced that claim that think that

x X x X X

x x x x x I believe that I trust that

x x x x x It seems to me

x x x x x I suppose that

x x x x x

I doubt that

i

It is known that lt is certain that Certainly It is sure that Surely It is deduced that

x x x x x

It is claimed that It is thought that It is likely that It is probable that Probably It is believed that

x x x x x

J It appears that [ Apparently Seemingly i It is possible that 1 Possibly J It is supposed that | Supposedly It is conceivable It is doubtful that

Class 2. Emotional functors I am happy that I am glad that I find it pleasing that It does not matter whether It is a pity that It is bad that I am sorry that I am afraid that I am worried that I fear that It is difficult for me, because It is awkward for me, because

19

LANGUAGE OF MOTIVATION Class 3. Motivational functors 3a. Proper motivational functors I wish that I want I desire I prefer Hike I dream about 3b. Normative (deontic) motivational functors I must I ought to

It is necessary that One ought to One should It suffices that It is desirable that It is reasonable that It is good to It is proper that It is worthwhile

good

-bad

moral ethical esthetic useful true sensible

-

immoral unethical non-esthetic useless false senseless

necessary - unnecessary efficient - inefficient

Class 4. Intensional functors very, strongly, intensely, essentially, rarely, often, almost, quite, rather, ... Class 5. Functors describing outcome Unexpected, predicted, possible, impossible, doubtful, certain, uncertain, unbelievable, ... pleasant, unpleasant, good, hard, bad, difficult, ... desired, wanted, preferred, ... forced, necessary, rational, reasonable, ...

1.2. DISTINGUISHED CLASSES OF EXPRESSIONS OF LANGUAGE OF MOTIVATION 1.2.1. Class 1: epistemic

functors

This class comprises functors used for describing the state of knowledge, convictions, certainty, belief, etc., regarding the external world, as well as the subject's own abilities and possibilities.

20

LANGUAGE OF MOTIVATION

The latter are expressed by functors I I I I

can know how to may am able to.

The next group of functors, reflecting the speaker's state of knowledge about the external world or about himself, is divided into two subgroups: expressions in the first person singular, and the same expressions in the 'objectivized' stylization.2 For many functors of one group, there is a corresponding functor (or functors) in the other group. This group of functors, arranged roughly according to the decreasing subjective probability which they express, is I know that I am certain that I am sure that I I I I

deduce am convinced that claim that think that x x x x x x

I believe that I trust that x x x It seems to me

I

It is known that It is certain that Certainly It is sure that { Surely It is deduced that x x x x x It is claimed that It is thought that It is likely that It is probable that Probably It is believed that x x x x x lt appears that

{

i

Apparently Seemingly x x x It is possible that Possibly I suppose that It is supposed that x x x Supposedly I doubt that It is conceivable It is doubtful that. The above list, as all the following lists, does not pretend to be complete (for instance, it does not contain functors like 'I guess', 'perhaps', 'obviously', etc.). 2

Note, however, that the expressions from the right hand side may also be used for expressing a distance from the given judgment ('It is believed that certain numbers are unlucky' may well signify that the speaker does not share this view).

LANGUAGE OF MOTIVATION

21

It is assumed that the above list of functors constitutes a linguistic representation of the scale of subjective probability. This assumption is to be understood as follows: to each functor there corresponds an interval on the scale of subjective probability (intervals corresponding to different functors may overlap). The use of a given functor indicates, therefore, the value of the subjective probability 3 of the truth of the statement to which this functor is applied. Thus, for instance, in everyday usage, 'I suppose t h a t p ' indicates some more doubt than, say, 'I am certain that p\ In addition, the choice of the functor may also indicate the sources of information used in evaluation of the subjective probability. These sources may be knowledge, convictions, beliefs, superstitions, etc. For instance, President Nixon and a wife of a soldier fighting in Viet Nam may well have identical subjective probabilities regarding the truth of the statement 'The war in Viet Nam will terminate in 1971'; still, the first might use the functor 'I know that', while the second would perhaps use the functor 'I believe that', indicating her wishful thinking. The basic assumption that the above functors constitute a linguistic representation of the scale of subjective probability, is complemented by a series of assumptions referring to particular functors. At least some of them may be considered separately, as expressions which, together with their negations, represent two points on the scale characteristic for a given functor. To express modality, these functors are combined with intensional functors from Class 4. For instance, 'I believe that' and 'I do not believe that' may reflect two points on the scale of belief. Intermediate stages are then expressed by functors such as 'I rather believe that', 'I am inclined to believe that', etc., while the stages beyond the two points are expressed by functors such as 'I absolutely do not believe that', 'I believe firmly that', and so on. Clearly, to express the intensity ofjudgment about a given object, as expressed by a given functor, one can also use means outside linguistics, asking the subject to mark his opinion on a scale from 1 to 100, or from 0 to 1, etc. The content of the above cluster of epistemic functors has been analysed from different points of view. This is, of course, not surprising, since the underlying concept, namely probability (objective and subjective), has for a long time been in the centre of interest of research. Formally, probability is a numerical function defined on a class of events, 4 and satisfying certain axioms. For repetitive events, it expresses their frequencies of occurrence in long series of experiments; in this interpretation, probability has objective and empirical character. The concept of probability is used, however, also for expressing the 'degree of possibility' for events which do not have repetitive character, hence for such events which cannot have a frequency interpretation. 3

For an excellent account of the theories of probability see I. J. Good, The Estimation ofProbabilities (M.I.T. Press, Cambridge, 1965). 4 W. Feller, An Introduction to Probability Theory and its Applications, I (Wiley, New York, 1963).

22

LANGUAGE OF MOTIVATION

Probabilities applied in such situations are generally called subjective.5 Thus, Ramsey, Keynes, and Jeffreys analysed various aspects of the degree of rational belief. Briefly, Ramsey6 claimed that subjective probability is connected with an event, and is based on knowledge, beliefs, convictions, etc. It can be measured by stakes one is willing to wager for the occurrence of the event in question. Both Keynes and Jeffreys present a formal approach to the degree of rational belief; they base their approaches on Bayes' formula and Bayes' rule as the principle of inductive inference.7 For Keynes,8 the concepts 'certain' and 'probable' describe various degrees of rational belief of the truth of a proposition, while probability is a logical relation between premises and conclusions drawn from these premises. While Keynes builds his theory in order to justify induction even in cases when numerical values of probabilities are not available, Jeffreys9 concentrates on an axiomatic approach to classes of situations when estimates of probabilities are available. According to him, probability is a primitive concept used for expressing the degree of belief which can be reasonably assigned to a proposition. Like Keynes, Jeffreys claims that one cannot speak of the probability of a proposition or a hypothesis as such; probabilities can only be determined on the basis of data collected up to a certain moment of time. Next, Russell analysed the concept of credibility (degree of credence). He writes: 10 I think, therefore, that everything we feel inclined to believe has a 'degree of doubtfulness', or inversely, a 'degree of credibility'. Sometimes it is connected with mathematical probability, sometimes not; it is a wider and vaguer conception. It is not, however, purely subjective. There is a cognate conception, namely the degree of conviction that a man feels about any of his beliefs, but credibility, as I mean it, is objective in the sense that it is the degree of credence that a rational man will give. Thus, Russell claims that credibility is not measurable, except in the case where objective probability exists, and in this case credibility ought to be equal to it. 5

For the terminology, see Good, Estimation of Probabilities. F.P. Ramsey, "Truth and Probability" (1926); in Ramsey: The Foundations of Mathematics (Kegan, London, 1931). 7 Bayes' formula expresses the probabilities a posteriori in terms of probabilities a priori and conditional probabilities:

8

P(„,A,

p{A\Hi)Pm P(,A\Hi)P(.H1) + ... + P(A | Hn)P(Hn) where Hi Hn are hypotheses which exclude one another and exhaust the set of all possibilities, P(Hi) are probabilities a priori of these hypotheses, A is an arbitrary event, and P(A\Ht) is the conditional probability of the event A given the hypothesis Hi. Bayes' formula gives probabilities a posteriori of the hypotheses H, given that A has occurred. The Bayes' rule asserts that P(Hi) = ... = P(Hn), i.e. that all hypotheses should be assigned the same probabilities a priori if these probabilities are unknown and there is no reason to think that these probabilities may be different. See Feller, Introduction. 8 J.M. Keynes, A Treatise on Probability (MacMillan, London, 1931). 9 H. Jeffreys, Theory of Probability (Oxford Univ. Press, Oxford, 1939). 10 B. Russell, Human Knowledge: Its Scope and Limits (Allen, London, 1948), 395. =

LANGUAGE OF MOTIVATION

23

Finally, Carnap 1 1 distinguishes the concepts of confirmability and testability. Proposition p is confirmable if it can be reduced to propositions directly testable by experiments or observations. Besides the points of view sketched above, one can find examples of combining logical and empirical interpretations of probability in philosophical literature. Thus, Ellis 12 claims that the whole family of terms concerning probability, such as relative frequency, degree of belief, subjective probability, etc., can be divided into those which concern probability as as empirical concept, and those which concern probability as a logical concept, i.e. a concept independent of the state of nature, and depending only on the language which we adopt for the description and rules of inductive inference. According to Ellis, logical and empirical probability constitute two languages of description of the same reality. He claims that there exists an intuitive feeling of the scale of probability, in much the same way as there exists a pre-thermometric scale of temperature. This point of view is supported by the fact that different people tend to draw the same conclusions from the same premises (or, at least, they can be taught to do so). In other words, there exists a certain consistency of human inference, conditioned by common experience. Thus, probability may be treated as a numerical magnitude, which exists objectively, and one can consider the problems of building scales for measurement of it; analysis of logical probability may be identified with analysis of the logic of the measurement of probability. In connection with this point of view, it is interesting to investigate how people assign probabilities to events in situations of uncertainty, in particular to events which occur only once and hence have no frequency interpretation. 13 It appears that people 'complement' their knowledge using the knowledge of probabilities or frequencies of events, which they consider similar or pertinent for the single event in question. This concept was, in a sense, utilised by Robbins, 1 4 who builds decision schemes which - absurd as it may seem - lead to better decisions in cases when one knows losses resulting from decisions in quite different, unrelated decision problems. Most of the empirical research concerning subjective probabilities was concentrated on utility theory, and will be discussed in connection with the third class of functors. Here one should only mention one line of research, namely that concerning the

11

R. Carnap, Meaning and Necessity (Chicago Univ. Press, Chicago, 1947). B. Ellis, Basic Concepts of Measurement (Cambridge Univ. Press, Cambridge, 1966). Incidentally, some mathematicians, for instance A. Renyi, hold the point of view that one can speak of the probability of a single event, as this event has some causes determining it. See A. Renyi, "Briefe über die Wahrscheinlichkeit", preprinted notes. 14 H. Robbins, "A new approach to the classical decision problem", in Induction; Some Current Issues, W. Kyburg and E. Nagel, eds. (Wesleyan Univ. Press, 1963). See also H. Robbins and E. Samuel, "Testing Statistical Hypotheses; the Compound Approach", in Recent Developments in Information and Decision Processes, R. Machol and P. Grey, eds. (MacMillan, New York, 1962). 12

13

24

LANGUAGE OF MOTIVATION

relations between subjective and objective probabilities. It was discovered, 15 that while subjective probability is an increasing function of objective probability, people generally display a marked conservatism, underestimating high probabilities. More precisely, people are conservative in the sense that their probability estimates are lower than those computed according to Bayes' formula. Finally, it is necessary to mention that the analysis of logical behaviour of some of the functors from the epistemic class served as a basis for creating various logical systems. First of all, one ought to mention Lewis, 16 who constructed a well known system of axioms (or, rather, systems of axioms) of modal logic, i.e. for the functor 'it is possible that'. It allowed him, among other things, to define the concept of necessity (p is necessary if it is not possible that not-/>), and the so-called strict implication ( p strictly implies q, if it is not possible that p and not-q). The system of modal logic will be presented in some detail in Chapter 2. Next, an exhaustive study of the concept of knowledge and belief was presented by Hintikka. 17 He constructed a system with a primitive modal operator K, where well formed formulas are of the form Kap, with a standing for the name of an individual, and p for a well formed formula (Kap is to be interpreted as an assertion that the individual a knows t h a t p is true). Hintikka's book had a great impact on epistemology, originating a series of studies, 18 mainly concerned with the functors 'I know' and 'I believe', and the acceptability of the so-called thesis 19 (roughly, a thesis which asserts that knowing implies knowing that one knows). 1.2.2. Class 2: emotional functors It is well known 2 0 that emotions can serve as drives, incentives, or can accompany the motivated behaviour (i.e. behaviour directed at a specific goal). For the present considerations (which serve mostly for illustrative purposes), out of a rich vocabulary of emotions, three dimensions were selected, connected with the 15 See, for instance, W. Edwards, "Conservatism in Human Information Processing", in Formal Representation of Human Judgments, B. Kleimutz, ed. (Wiley, New York, 1968). 16 See C.I. Lewis and C.H. Langford, Symbolic Logic (Century, New York, 1932). 17 J. Hintikka, Knowledge and Belief (Cornell Univ. Press, Ithaca, 1962). 18 See, for instance, H.N. Castañeda, "On Knowing (or Believing) that One Knows (or Believes)", Synthese 21 (1970), 187-203; R.M. Chisholm, "The Logic of Knowing", Journal of Philosophy 60, (1963), 773-795 and Theory of Knowledge (Prentice Hall, Englewood Cliffs, 1966); C. Ginet, "What Must Be Added to Knowing to Obtain Knowing That One Knows", Synthese 21 (1970), 163-186; R. Hilpinen, "Knowing that One Knows and the Classical Definition of Knowledge", Synthese 21 (1970), 109-132; E.J. Lemmon, "If I Know, D o I Know that I Know?", in Epistemology, A. Stroll, ed. (Harper and Row, New York, 1967), 54-83; Ch. Pailthorp, "Hintikka, and Knowing that One Knows", Journal of Philosophy 64, (1967), 487-500. 19 The term introduced by R. Hilpinen in "Knowing that One Knows". 20 See, for example, E.R. Hilgard, R.C. Atkinson, Introduction to Psychology (Harcourt, Brace and World, New York, 1967), 178-181.

LANGUAGE OF MOTIVATION

25

above relationships between motivation and emotions: pleasantness vs. unpleasantness, anxiety vs. its lack, and difficulty vs. easiness. The last two dimensions appear to be especially important in problem solving situations. It is not difficult to show linguistic representations of other dimensions of emotions, as distinguished, for instance, by Schlosberg,21 Block,22 or Pluchik, 23 by adding functors used for expressing anger, disgust, contempt, grief or surprise, sympathy, etc. Thus, a linguistic representation of the scale of pleasantness vs. unpleasantness can be, for instance, I am happy that I am glad that I find it pleasing that It does not matter whether It is a pity that It is bad that I am sorry that. Anxiety vs. its lack is reflected by the functors I am afraid that I am worried that I fear that. Finally, difficulty vs. easiness is described by the functors It is difficult for me, because It is awkward for me, because. In the first group of functors, the whole scale is represented, ranging from negative to positive values. In the remaining two groups of functors, they concentrate around the negative end-points of the scale, while there are no natural expressions indicating lack of anxiety, or lack of difficulty (the expression 'It is easy for me' carries much less emotional connotation than 'It is difficult for me'). As distinct from the class of epistemic functors, a remarkable fact is the scarcity of emotional functors, so to say, their 'separation' on the scales. Some effects of increasing the density of representation can be achieved by modification via intensional functors (like 'I am very glad that', etc.). Quite likely, this low 'density' of representation of the scale is connected with the difficulty of describing one's own emotions in the above three dimensions. On the other hand, as in the case of epistemic functors, one can consider (at least 21

H. Schlosberg, "The Description of Facial Expressions in Terms of Two Dimensions", J. Experimental Psychol. 44 (1952), 229-237 and "Three Dimensions of Emotion", Psychological Review 61 (1954), 81-88. 22 J. Block, "Studies in Phenomenology of Emotions", J. Abn. Soc. Psychol. 54 (1957), 358-363. 23 R. Pluchik, The Emotions: Facts, Theories, and a New Model (Random House, New York, 1962).

26

LANGUAGE OF MOTIVATION

some) functors separately, as expressions which, when combined with their negations, constitute specific scales. To express modality one would then use intensional functors, according to the accepted linguistic customs. These specific scales have some interesting properties: they can be more or less easily extended towards their end-points, while in some cases there appear 'gaps' in between. Thus, it seems that there are no ways of describing stages intermediate between 'I am glad that' and 'I am very glad that'. There are ways of weakening the first functor (such as 'I am rather glad that') or strengthening the second (such as 'I am awfully glad that'). One could also achieve the same effect by repeating the word 'very' ('I am very, very, very glad that'). Another way of 'stretching' the scale is to use metaphors; these consist of expressions with the functors 'such as', 'as', 'like', etc. ('I am happy as a lark'). The methods of stretching the scale mentioned above are not characteristic for motivational functors, and they apply to other functors as well. One can pose the question whether there exist end-points of these scales, or whether for each functor one can find one stronger than it. Another way of modifying functors used in natural language is to use the conditional. In a sense, it leads to negation of the reality of events described by expressions to which the functors are applied. To see that, it suffices to compare the expressions 'I am glad that' and 'I would be glad if'. The first describes the emotional state connected with a given event, while the second describes the anticipation of an emotional state which would take place if some event occurred. As a consequence of accepting three basic dimensions of emotions, each emotional state can be described by three 'coordinates': the first one assumes (simplifying the problem somewhat) three values, + , 0, and — (pleasant, neutral, unpleasant). The remaining two may assume two values each, namely - and 0 (causing anxiety, and not causing anxiety, and difficult - not difficult). Thus, each emotion can be classified by assigning to it a symbol like (0, —,—), etc. For instance, an examination would generally lead to an emotion of the type (—, — ,—), i.e. unpleasant, causing anxiety, and difficult. Theoretically, there are 3-2-2 = 12 possibilities of classifying emotions. Not all combinations will be represented, though, because of partial dependence of these dimensions. To obtain some simple laws concerning emotions, expressed in terms of the above three dimensions, it is worth while to make a complete list of all combinations and find out those combinations which correspond to the empty class of emotions. In the list below, next to each combination, an example is given of a situation which may (for some people at least) cause an emotional reaction of this combination. ( + , 0, 0) ( 0 , 0, 0) (—,0,0) ( + , —, 0) ( 0 , —, 0)

eating a cake brushing teeth stepping into a puddle watching a horror film on TV assumed to be empty

LANGUAGE OF MOTIVATION

27

0) appearing in court as a witness athletic training ) washing dishes (0 , 0 , - ) (-, 0 , - ) climbing up the stairs to the tenth floor participating in a sporting competition 5- ) (0 , t - ) assumed to be empty examination >- )

(+,

>

0, -

(+,

Denoting by a+, a0, b0> c0, and c_ the corresponding values of the coordinates, the analysis of empty classes of this list allows us to formulate the law a0 => ~ or equivalently ¿>- => ~

a0,

asserting that anxiety does not allow one to remain neutral; it 'polarizes' the dimension of pleasantness-unpleasantness so that anxiety causing situations exclude neutrality in this dimension. 1.2.3. Class 3: motivational functors The class of motivational functors considered in this section is divided into two groups: the first group, called proper motivational functors (with central functors 'I want' and 'I prefer'), reflects, roughly speaking, personal motives. The second group, called normative, or deontic motivational functors (with central functors 'I ought to' and 'I must'), reflects, first of all, motives of social character. 1.2.3.1. Proper motivational functors I wish that I want I desire I prefer Ilike I dream about. Except the functor 'I prefer', the above functors form a scale, giving a linguistic representation of the intensity of 'desires' and 'wants'. The functor 'I prefer' plays a special role, being intimately related to the models of choice, and hence also with motivation. Roughly speaking, in most of the models of choice it is assumed that the choice is always made so as to maximize some index, generally referred to as 'utility' in riskless

28

LANGUAGE OF MOTIVATION

choices, and 'expected utility' in choices involving risks. There arises the problem, therefore, of establishing the conditions for the existence of utility and finding the procedure to evaluate it experimentally, so as to be able to test the validity of the model by comparing the predicted and actual choices. One such set of conditions will be presented below. The intuitions concerning the functor 'I prefer' which serve as a basis for this construction, are the following: this functor allows us to build propositions whose general form is 'I prefer A to B\ The arguments which are to be put in place of A and B are either names of given objects, say au ..., aN, or situations in which these objects may appear with specific probabilities. The latter type of situation, to be called a lottery, is described by a system of probabilitiesp u ...,pN, where pt is the probability of obtaining the object at (these probabilities may be interpreted as objective probabilities, or as subjective probabilities; some of them may be 0, if a given object cannot appear in a given lottery). For instance, if objects au ..., aN are amounts of money, one can in general assume that preferences agree with values of these sums: 'I prefer at to a/ if, and only if, at represents a larger sum of money than aj. One can also assume that people can determine their preferences towards lotteries (such as state lottery, gambling, investment of money, etc.), basing these preferences on knowledge or intuitive feeling about probabilities of various outcomes (results ax, ..., aN). Formally, propositions 'I prefer A to B' concern arguments A and B having the form of probability distributions (pu ...,pN) on the set {av ..., %}. The construction of a utility scale consists of defining a numerical function u on the set {au ..., aN}, called utility, which satisfies the following requirement: proposition 'I prefer A to B\ where A = (pu ...,pN) and B = (qu ..., qN) is true if, and only + ... + if, u(A) > u(B), where u(A) = p{u(a^) + ... + pNu(aN) and u(B) — q^iaj + qNu(aN). In other words, the direction of preference is consistent with the expected values of the utility function u. Formally, the utility scale, in the sense defined above, can be interpreted as a model for a set of propositions concerning preferences. This set of propositions concerning objects or lotteries over objects cannot be arbitrary, otherwise the model would not exist (i.e. one could not define any utility function). It has to constitute a certain structure. The latter is defined by a set of axioms asserting, roughly speaking, some consistency among propositions of this set. To formulate these axioms, one introduces the concept of preferential equivalence: A is preferentially equivalent to B, or: indifferent, if the propositions 'I prefer A to 5 ' and 'I prefer B to A' are both true. Two axioms concern consistency in the direction of preference: they assert that preferences are transitive in the set of objects a 1 , ...,a N and in the set of lotteries over these objects. The next two axioms assert the possibility of a certain reduction: one of them asserts that in each lottery, in place of a prize (equal either to an object or to another lottery), one can substitute a lottery indifferent to this prize, and obtain as a result a

LANGUAGE OF MOTIVATION

29

lottery indifferent to the original one. The second axiom asserts that after such substitution one obtains a lottery (in which prizes may be other lotteries), which can be reduced, with the use of laws of probability theory, to a lottery in which the prizes are au

...,

aN.

The fifth axiom asserts a certain consistency in order of preferences: out of two lotteries, each containing only the most and the least preferred alternative (the best and the worst object among al, ..., aN), the one having higher probability of getting the most preferred alternative is preferred. Finally, the sixth axiom is of an existential character: it asserts that for any object from the set au ..., aN there exists a lottery, in which only the most and the least preferred objects appear as prizes, which is indifferent to this object (i.e. to a lottery which yields this object with probability one). The six axioms presented above (see Luce and Raiffa) 24 imply that the set of propositions expressing preferences is sufficiently regular, so that one can construct a model in the form of a utility function. This model has the following form: one assigns numbers to objects au ..., aN (utilities of objects) and interprets 'I prefer' as the relation > in the set of real numbers. To propositions of the system expressed by the functor 'I prefer' there correspond true propositions concerning inequalities between expected utilities (the expectations being taken with respect to given lotteries). The set of axioms quoted above is taken from Luce and Raiffa; 25 it is a modification of the first set of axioms for utility proposed by von Neumann and Morgenstern 26 . Generalizations for infinite sets may be found in Blackwell and Girschick. 27 Luce (see Luce and Raiffa) 28 suggested another set of axioms, covering the case of nondeterministic preferences, where the direction of preference may change from one occasion to another for the same person (this covers the case where the alternatives are 'close' to one another, so that the intuition for the direction of preference is vague). The above axiomatic systems provide a method of computing utilities, by observing values of probabilities (for different triplets of alternatives) for which a given person is indifferent between obtaining the middle alternative for certain, or participating in the lottery which will bring either of the two extreme alternatives. The determination of utilities may in turn be used for predicting choices in more complex situations, involving lotteries over alternatives (assuming that the choice is made so as to maximize the expected utility). This provides a test for the theory. In the empirical evaluations of utilities according to the scheme outlined above, one uses the objective probabilities, instead of subjective ones. This may lead to 24

R.D. Luce and H. Raiffa, Games and Decisions (Wiley, New York, 1957). See Luce and Raiffa, Games and Decisions. 26 J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton, N. J., 1944). 27 D. Blackwell and M.A. Girschick, Theory of Games and Statistical Decision Functions (Wiley, New York, 1957). 28 See Luce and Raiffa, Games and Decisions, Appendix 1. 25

30

LANGUAGE OF MOTIVATION

possible inadequacies, as clearly, in practical situations, the subjects would base their choices on expectations computed according to subjective probabilities. Here the difficulty arises, because the method of evaluating subjective probabilities, as suggested by Ramsey, involves observations of stakes in appropriate bets for the occurrence of the event in question. The stakes for these bets are sums of money, and the calculations are carried out under the assumption that utility increases linearly with money; this assumption has long been known to be false. The problem is, therefore, to design an empirical method which would yield the simultaneous measurement of both utilities and subjective probabilities. The theoretical foundation of such a method was given by Savage; 29 it is based on an event whose subjective probability is one-half (no assumption about utilities is required to test whether or not an event has subjective probability half). The knowledge of such an event allows one, in turn, to derive bounds for the utilities of different alternatives, and these may be used to obtain the subjective probabilities of other events. The experimental procedure for testing the theory of choice based on maximum subjective expected utility (SEU) has been carried out mostly by Edwards 30 and Davidson, Suppes, and Siegel.31 However, even this theory is open to criticism. It is based on the assumption that utilities of alternatives do not depend on the subjective probabilities of attaining them - the assumption utilized for computing the expected utility of risky choices (lotteries). Atkinson 32 suggested here another model, involving variables such as subjective probabilities of success and failure, incentive values of achieving success and avoiding failure, and motivation towards achieving success and towards avoiding failure. It is assumed that the incentive value of achieving success and avoiding failure is directly related to the subjective probabilities of success and failure. This allows one to write down the formula for what Atkinson calls motivation - an index to be maximized by the person who makes the choice (hence, in essence, the utility). In connection with the models of choice and utility theory, one should mention here also the results of Edwards 33 who discovered that in certain situations people display definite preferences (or aversions) towards certain probabilities, as well as preferences towards lotteries with a certain (subjectively optimal) amount of variance. These findings show that the model of choice according to which the expected utility is the only governing principle is inadequate, and needs to be revised and suitably amended. 29

L.J. Savage, The Foundations of Statistics (Wiley, New York, 1954). W. Edwards, "The Theory of Decision Making", Psychol. Bulletin 51 (1954), 380-417. 31 D. Davidson, P. Suppes, and S. Siegel, Decision Making: An Experimental Approach (Stanford Univ. Press, Stanford, 1957). 32 J.W. Atkinson, An Introduction to Motivation (Van Nostrand, Princeton, 1964). 33 See W. Edwards, "Probability Preferences in Gambling", Amer. Journal of Psychol. 66 (1953) 349-364 and "Variance Preferences in Gambling", Amer. Journal of Psychol. 67 (1954), 56-67. 30

LANGUAGE OF MOTIVATION

31

Returning to the problem of organization of the cluster of functors discussed in this section, one can formulate the hypothesis that the functors 'I wish', 'I want', 'I desire', etc. represent some fragments of utility scale (relatively to the considered set of objects). This hypothesis would imply the following regularities: if, say, someone uses the functor 'I wish' with respect to a given object, and the functor 'I wish very much', or 'I want' with respect to another object, then the second object has a higher utility than the first. The converse, of course, cannot be true: one can easily imagine that a man prefers to eat live frog than be hanged, but this does not mean that he 'wants to eat a frog' or 'wishes to be hanged'. Of course, the ordering of functors on the scale of utility, and the degree of preference expressed by a given functor may be different for different people; one can, however, expect a certain similarity of choices of functors in groups of people with the same cultural and linguistic training, i.e. for people with the same stylistic habits. 1.2.3.2. Normative (deontic) motivational functors Functors in this group are used in those propositions describing the consequences of accepted norms (general or specific) which have the form of injunctions, interdictions, advice, encouragement, or instructions. In other words, these functors appear in propositions serving the purpose of speeding up actions, delaying, or abandoning them, or in propositions which have the form of instructional implications. The group considered consists of three basic sub-groups, related to one another by certain interesting semantic connections. a I must I ought to

b It is necessary that One ought to One should It suffices that It is desirable that It is reasonable that It is good to It is proper that It is worthwhile

c good

-bad

moral - immoral ethical - unethical esthetic - non-esthetic useful - useless true - false sensible - senseless •

•

•

necessary - unnecessary efficient - inefficient. The central functor in these groups is 'I ought to' which serves, first of all, to express the conscious consequences of causal laws, and secondly, the norms, injunctions, etc., and their consequences. In the first sense this functor is used in the sentence 'If I want to read at night, I ought to turn on the lamp'. In the second sense, the functor is used in a sentence like 'If I want to be respected by others, I ought to live a righteous life'. The first two columns form a linguistic representation of a scale which could be

32

LANGUAGE OF MOTIVATION

called 'scale of restriction of freedom of action', or 'scale of admissibility of actions'. The first representation is more 'personal' (column a), and the second is more 'objective' (column b). The representation of this scale by functors of column a is rather sparse: the functor 'I ought to' can only be weakened into 'I rather ought to'. Further weakenings are obtained by replacing it by the negation of 'I must', i.e. by 'I don't have to', modified by intensional functors like 'I absolutely do not have to', etc. An increase of intensity of the functor 'I ought to' is also obtained by using 'I must': we have here expressions such as 'I necessarily must', etc. It is worth stressing that the linguistic representation of the scale of restriction of freedom of action by column a is not only scarce, but cannot be easily enriched by metaphors, either. This is due to the fact that the degree to which one 'must do something' is not available to non-introspective observations, hence it is difficult to point out an object or person who 'must do something' in a given degree. There are several expressions indicating lack of constraint (such as 'free as a bird'). To express the strength of compelling force, one uses the description of results which would follow if one would not do what one must. The more vivid and horrifying are these results, the stronger is the force expressed ('I must do that, otherwise they would crucify me'). The essential enrichment of the above scale is obtained by using functors from column b. These functors reflect the 'objectivized' force of constraint, from interdictions, to instructions and advice; they are used for explaining the reasons for applying a given functor from column a. Closely related to these two groups is the third (column c); generally, functors from column c are used for explaining the reasons for applying a given functor from column b. Functors from column c do not constitute any linear scale, as in the case of the preceding groups of functors; they constitute a hierarchy. The central functors 'good' and 'bad' form the most general linguistic representation of attitudes and evaluations. These functors occupy the top of the hierarchy, its highest level (the functors from column c are connected in pairs of functors opposite one another, so that the organization concerns pairs of functors, rather than separate functors). On the next level of hierarchy there are functors used in less general descriptions of the dimension 'good-bad', These pairs are connected with attitudes: egotistic (functors 'pleasant-unpleasant' discussed in class 2 of emotional functors), social (functors 'moral-immoral', 'ethical-unethical', 'esthetic-nonesthetic'), intellectual ('true-false'), and utilitarian ('useful-useless').34 Each of these pairs of the first level has, in turn, its more detailed description in terms of functors of the second level; part of such a description for the functors 'useful-useless' is given in column c. In connection with the discussed group of motivational functors, one can formulate 34

This classification is, of course, not disjoint.

LANGUAGE OF MOTIVATION

33

the following hypothesis: in decision situations, one first evaluates actions, their goals, and expected results in terms of functors from column c. Thus, to each evaluated object (action, goal, result, etc.) there corresponds an appropriate set of functors (adjectives) from column c. This set depends on the general attitude of the person making the evaluation. Next, to each set of such functors there corresponds a functor from column b, expressing a certain conclusion regarding the contemplated action (goal, result). Finally, the same conclusion is expressed in terms of functors from column a: the person making the decision also makes an evaluation as to what extent he ought to (must) do the contemplated action. The above hypothesis constitutes, in a sense, a model of the motivating role of attitudes and evaluations. It suggests another (linguistic) hypothesis, asserting the existence of a function, which assigns functors from column b to sets of functors from column c; more precisely, it asserts the existence of a certain class of propositions relating properties of objects (actions, goals, etc.) expressed by functors from column c with properties described in terms of functors from column b. The last hypothesis (asserting, so to say, the existence of a linguistic representation of the above psychological hypothesis), can be verified by analysing the structure of injunctions, interdictions, advice, etc., i.e. generally, propositions containing functors from columns b and c. The theory which deals with laws formulated in terms of functors from column b and some functors from column c is praxiology. Its basic aim, according to its founder (Kotarbinski) 35 is to construct and justify the norms of efficient actions; these norms, theorems of praxiology, are also called practical propositions, or practical instructions. As a rule, they have a form of implication, and contain functors from column b such as 'One ought to', 'It is reasonable that', etc., and functors from column c, such as 'effective', 'useful', and so on. The meaning of some functors from column b was analysed by Kotarbinski. He writes: 36 When we say "It is necessary", we point out a condition necessary in circumstances A, that is, such conditions without which in these circumstances C cannot occur. When we say "It suffices", we point out a condition sufficient in circumstances A, that is, such a condition which - when added to these circumstances - makes C occur. When we say "It is good", we point out an action, which - when added to circumstances A - makes the occurrence of C more probable than it would have been otherwise.

An exhaustive analysis of practical implications has been given by Leniewicz.37 35

T. Kotarbinski, Traktat o Dobrej Robocie (A Treatise on Good Work), in Polish (Ossolineum, Wroclaw, 1969, 4th ed.). 36 T. Kotarbinski, Rodzaje zdan prakseologicznych i sposoby ich uzasadniania (Types of praxiological propositions and ways of justifying them). In Polish. Kultura i SpoleczeAstwo 4 (1960). 37 L. Leniewicz, "Poj^cie dyrektywy praktycznej" (The concept of practical recommendation). In Polish. Prakseologia 31 (1968/69), 135-163. A more general approach to sentences involving the discussed functors, i.e. to orders, recommendations, interdictions, etc., may be found in H. Reichenbach, Elements of Symbolic Logic (The Free Press, New York, 1966).

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LANGUAGE OF MOTIVATION

He introduced the propositional functor § ('One ought to'), whose argument is a variable D from the set of descriptions (names) of actions. Besides that, he considers variables W for conditions, and C for goals, connecting them with the propositional functor !, to be read 'given'. The scheme of practical instruction is \{W, C) => §(£>);

verbally, 'Given the conditions W and goal C, one ought to perform the action D\ More precisely, Leniewicz enriches this scheme by introducing the concept of a criterion with respect to which one evaluates the achievement of a goal, and considers different variants of practical implications (choice of action for a given goal and conditions, choice of conditions for a given action and goal, and choice of goal for a given action and conditions). He interprets the functor § as 'One ought to', but he mentions that it may also be interpreted as 'It is reasonable that', etc., so as to obtain all stylistic variations of instructions. It appears that the above scheme, as suggested by Leniewicz, satisfies generally the requirements of a structural description of practical implications. This scheme may be somewhat enriched 38 by considering propositions containing functors from columns b and c. These propositions would have the following general form: 'Under such and such conditions, for such and such goal, it is necessary (one ought to, it is good to, ...) act in such and such a way, because under these conditions, and for this goal, such an action is proper (efficient, economical, ...)'. In another stylization, the same proposition has the form: 'If in such and such conditions and for such and such a goal, such and such an action is good (efficient, economical, ...), and in the actual situation one finds these conditions and wants to achieve precisely this goal, then one ought to (it is good to, it is reasonable to, ...) perform this action'. The functors from column c which appear in such propositions would play the role of criterion suggested by Leniewicz. To present formally the structure of such propositions one ought to introduce the quarternary relation R( W, C, D, Q) which holds between conditions W, goal C, action D and the set Q of functors of column c if, and only if, in conditions W, for goal C, the action D has the properties described by functors from the set Q. Denoting by § any of the functors from column b and using the functor ! in the 38 An extensive literature exists in Polish on this subject. See, for instance, T. Pszczofowski, "Prakseologiczna teoria ocen" (Praxiological theory of evaluations), Prakseologia 24 (1967); T. Pszczolowski, Prakseologiczne sposoby usprawniania pracy (Praxiological methods of improvement of work) (PWN. Warszawa, 1969); A. Podg6recki, "Metodyka formulowania wskazan i zalecen w naukach praktycznych w oparciu o wyniki badan nauk opisowo-eksplikatywnych (na przyktadzie socjotechniki)" [Methodology of formulating recommendations and advice in practical sciences based on results of studies in descriptive-explicative sciences (on an example of sociotechnique)], Prakseologia 28 (1967), 153-169; H. Stonert, "Charakterystyka twierdzen nauk praktycznych w aspekcie metodologicznym" (Methodological aspects of the characterization of theorems of practical sciences), Prakseologia 28 (1967), pp. 21-53.

LANGUAGE OF MOTIVATION

35

sense described by Leniewicz, the scheme of the considered proposition would be (*)

R(W, C, D, Q) & W , C)

§(D)

or, in equivalent form (**)

R(tV, C, D, Q) => [W,

C) => m i

In the above formulas, § is a variable from the set of functors of column b. Incidentally, in natural language some instructions are given in a shortened form: when Q contains functors of column c which express socially desirable properties, and when D concerns situations in which conditions W and goals C are easily guessed, one usually presents only the conclusion of the above implication, and the instruction is simply

One of the oldest examples of such a form of instruction is the Decalogue, where most of the instructions use the functor § = 'Thou shalt not'. If one succeeds in finding a sufficiently numerous set of instructions of the form (*), one could try the following conceptual construction, thus verifying the discussed hypothesis of the existence of a function which assigns functors of column b to sets of functors of column c. Let & be a hypothetical set of propositions of the form (*); it determines then the sets W, , and DeS) we have R(W, C, D, Q) => [l(W, C) => §(£)]. In other words, relation i¡/(Q, §) would hold between Q and § if whenever action D has properties described by functors from Q, then functor § applies to it, for all conditions W, goals C, and actions D. The linguistic hypothesis stated above asserts that this relation is a function with respect to the argument Q, i.e. to each Q there corresponds at most one element § of the set of functors of column b. If this hypothesis were true (at least with respect to a group of conditions, goals, and actions, and in a particular group of people), it would mean that for each action, the use of a functor such as 'One ought to' etc., is completely determined by properties of this action, designated by functors such as 'effective', 'moral', etc. In other words, one could obtain assertions which state that in a given group of individuals, all actions which are moral, ethical, and useful (say) are objects of instructions (expressed by the functor 'One ought to'), while all actions which are sensible, with uncertain effectiveness, are (say) objects of advice or recommendations. The truth of this hypothesis would also allow one to formulate the following

36

LANGUAGE OF MOTIVATION

psychological law of motivation: the stronger the functors of column b which are used in the assertion concerning an action, the stronger is the motive which this assertion inspires or breaks. There is no need to add that a similar construction may be carried out for instructions concerning the choice of conditions for given goal and actions, or choice of goals for given action and conditions, etc. In the above approach, the basic role would be played by the quarternary relation R(W, C, D, Q), which would have to be presented explicitly, i.e. one would have to be in possession of a list of laws asserting that in conditions W, for goal C, the action D has properties described by functors from the set Q. Whether the relation R{W,C,D,Q) holds between a particular combination of arguments W, C, D, and Q one has to decide on the basis of specific laws of a given domain of knowledge. Thus, if the conditions W were 'we want to eat the soup, but it is too hot to be eaten immediately', C was the goal 'to make the soup cooler faster than it would have cooled off by itself', and D is the action of stirring the soup with the spoon, then we would have R{W, C, D, {efficient, effective}), while we would not have, for instance R(W, C, D, {immoral, esthetic, true}). In this case, whether or not the relation R( W, C, D, Q) holds between particular W, C, D, and Q is inferred from the suitable law of physics. On the other hand, if W were 'being in possession of a license for a barber shop', C were the goal of 'making as much money from a barber shop as possible', and Q were the set consisting of the adjective 'sensible', then the relation R(W, C, D, Q) would hold for action D 'open the shop at such a place where you have the maximal number of prospective customers who are closer to your shop than to any other barber shop'. In this case, the conclusion is based on an economical law. The above examples concerned typical situations of the study of relation R(W, C, D, Q) for sets Q containing functors such as 'efficient', 'effective', 'sensible', etc., which are the basic object of interest of praxiology. In the case when Q contains functors such as 'moral', 'ethical', etc., to determine whether or not a given action D (in conditions W and for goal C) is 'moral', 'ethical', etc., one may sometimes refer to the analysis of norms accepted in the given social environment. Such an analysis would not yield, however, the information as to what extent these norms are internalized; nor would it allow one to differentiate the degrees to which adjectives such as 'moral' can be ascribed to a given action. The differentiation by means of an analysis of norms concerns only the end-points of the scale, i.e. it leads to distinguishing 'moral' and 'immoral' actions, etc.

LANGUAGE OF MOTIVATION

37

Edwards 39 performed experiments concerning attitudes, and in particular, social desirability. The procedure applied by him allows one to assign to every behaviour a numerical value on a scale of social desirability, on the basis of evaluations of this behaviour by the subjects. The measurement of social desirability served Edwards for testing the hypothesis asserting that the more extreme are the evaluations of a given behaviour, the more stable are they over a period of time (i.e. they do not change under repeated observations). Generally, the results of Edwards seem to indicate the truth of the following hypothesis: the more extreme (on the scale of social desirability) are the evaluations of a given goal or action, the smaller the freedom of decision, hence the smaller the contribution of motives expressed by proper motivational functors, and the larger the contribution of motives expressed by normative motivational functors. Finally, it is necessary to mention here that the logical behaviour of normative motivational functors is studied by deontic logic. 40 In analogy with modal logic, where the primitive functor L ('It is necessary') allows the definition of possibility ( p is possible, if ~ L ~ p), in deontic logic one introduces the deontic functor O ('ought'), and consequently, one defines the concept of permission (p is permitted if it is not the case that one ought notp). Von Wright combines deontic logic with a logic of actions, in an attempt to analyse concepts related to systems of norms, conflicts, etc. The presentation of his system, and comparison with the system suggested in this book, will be given in Chapter 4 41 . 1.2.4. Class 4: intensional functors

This class consists of functor-forming functors such as very, strongly, intensely, essentially, rarely, often, almost, quite, rather, ... which apply to some of the functors described previously, in order to modify their intensity. 39

See A. Edwards, The Social Desirability Variable in Personality Assessment (Dryden, New York, 1957). 40 See G.H. von Wright, An Essay in Deontic Logic and the General Theory of Actions (North Holland, Amsterdam, 1968) and also Norm and Action (Routledge and Kegan Paul, London, 1963). 41 One can easily imagine construction of similar languages as the language of motivation, for instance 'language of perception'. Clearly, one would have to introduce new functors, such as 'I see', 'It seems to me', etc. The basic sentences in such a 'language of perception' would connect the knowledge with the degree of uncertainty of this knowledge, and with emotions; such sentences would be 'a knows that S is P\ 'It seems to a that S is probably P\ 'a wants to believe that S is P', 'a is glad, because he sees P\ etc. In language of perception, the basic class of functors would be derived from 'I see' ('I perceive', and also 'I understand', 'I notice', etc.). The other classes would be the same as in language of motivation. Obviously, the studies of language of perception, as well as analogous languages of emotions, self-evaluation, etc., could lead to some progress in psychology, from the point of view of taxonomy of concepts and methods.

38

LANGUAGE OF MOTIVATION

With the help of intensional functors one can reduce the number of functors in particular classes. Thus, for instance, it is possible that 'to want very much' means the same as, say, 'to desire', and so on. Of course, the problem of such a reduction has to be settled on experimental grounds. It is known (Cliff),42 that adverbs such as 'rarely', 'often', and 'very' have properties of multiplicative intensification; there is a certain factor assigned to each of them, which expresses the degree of intensification. The knowledge of these factors, and the knowledge of the results of empirical studies concerning the above described identification, might lead to an assignment of numerical values to each functor on corresponding scales. Thus, if it turned out that 'to want very much' is the same as 'to desire', and if 'very much' would have the factor x, then the functor 'I desire' would represent x times as large an intensity of wanting as the functor 'I want'. 1.2.5. Class 5: functors describing

outcomes

This class is derived from classes 1, 2, and 3, and may be accordingly divided. It consists of name-forming functors applicable to name-arguments. For functors from class 1 we have unexpected, predicted, possible, impossible, doubtful, uncertain, unbelievable, ... For functors from class 2 we have pleasant, unpleasant, neutral, disturbing, good, hard, bad, difficult, ... Finally, for functors from class 3 we have (for subclass 3a) desired, wanted, preferred, ... and (for subclass 3b) forced, necessary, rational, irrational, reasonable, ... The organization of this class of functors is similar to the organization of classes 1, 2, and 3. One can expect that there exists a psychological law connecting the usage of functors from classes 1-3 with the use of functors from class 5. In loose formulation, this law may be expressed as follows: the description of an outcome is derived from the knowledge, desires, and emotions accompanying the actions which are to bring about this outcome.

43

See N. Cliff, "Adverbs as Multipliers", Psychol. Rev. 66 (1959), 27-44.

LANGUAGE OF MOTIVATION

39

1.3. MODEL OF CHOICE BEHAVIOUR A N D STRUCTURE OF MOTIVATION

1.3.1. Assumptions In the preceding sections it was assumed that the motivational functors presented in Table 1 form linguistic representations of a certain number of psychological continua, such as subjective probability, pleasantness vs. unpleasantness, degree to which one ought to do something, etc. These continua taken jointly may be treated as forming a motivational space. For the present considerations there is no need to list all of these continua; it will only be assumed that it is in principle possible to produce such a list, comprising a certain finite number of continua, some of them forming perhaps interval or higher type scales, 43 and some only ordinal scales. Moreover, it will be assumed that the set of these continua has the property that in every decision situation the choice between the alternatives is completely determined by the mutual relations of these alternatives with respect to the continua of the motivational space. In the assessment of such relations, some of the continua occasionally may not be applicable (e.g. subjective probability may not be used in the case of riskless choices); also, some continua may be used more than once, to assess various aspects of the contemplated alternatives on the same continuum (e.g. when contemplating visiting a couple, of which one is an enjoyable friend, and the other a loathsome creature; the perspective of seeing each of them may then be separately assessed on the'pleasuredispleasure' continuum). The aim of the considerations below will be to discuss the possible forms of the contributions to the decision arising from the ordinal type scales. From the formal point of view, the situation may be presented as follows. Given a finite set A of alternatives in a choice situation, the decision has the form of an ordering of elements of A, i.e. the form of a binary relation, say R, on A which is transitive and connected. Thus for all x, y, z e A (1) (2)

if xRy and yRz, then xRz, either xRy or yRx.

The ordering R is assumed to be a function of evaluations of elements of A on the continua of motivational space. In the case of continua of interval or higher type these evaluations have the form, say Tj, where j is the index of the scale in question, and Tj is a vector of scale values of elements of A. For ordinal type scales, the evaluations have the form of orderings of the set A, i.e. binary relations, say Rh where i is the index of the scale, and Rt satisfies conditions (1) and (2). 43

See, for instance, P. Suppes and J.L. Zinnes, "Basic Measurement Theory", in Handbook of Mathematical Psychology, vol. I, R.D. Luce, R.R. Bush, and E. Galanter, eds (Wiley, New York, 1963).

LANGUAGE OF MOTIVATION

40

Consequently, the model of choice may be written as R = (p{Tu T2, ..., Tm; Rlt R2, •••, R„) where

and = . Furthermore, for S c A, the symbol C(S) will denote the set { r x e S 1 and xRy for all y e 5}, i.e. C(S) is the set of all 'best' elements in S, according to the relation R. Similarly, Ci(S) will denote the set defined as C(S) with R replaced by Rt. 45

K.J. Arrow, Social Choice and Individual Values (Wiley, New York, 1963). Y. Murakami, Logic and Social Choice (Routledge and Kegan Paul, London, and Dover, New York, 1968). 48

42

LANGUAGE OF MOTIVATION

Clearly, if x, y e C(S), then xly, i.e. every two elements of C(S), are equivalent. Next, the finiteness of A, and hence of S, implies that C(S) is not empty: in every set there exists at least one 'best' element. Finally, the knowledge of C(S) for all S c A characterizes the relation R uniquely: indeed, taking for S the two-element subsets of A, one obtains the information whether xRy or yRx or both, for each pair (x, y). 2 (collective rationality). There exists at least one triple of alternatives in the set A, such that all possible orderings of this triple belong to the domain of definition off This condition asserts therefore that for some three alternatives, say a, b, c, and for every i, all orderings such as aPpPf, bPflPf, alpPf, aljblfi, etc., are permissible. In terms of the considered individual decisions, this condition asserts that in real life one can encounter choice situations in which some triple of alternatives may be ordered on the ordinal scales of the motivational space in any possible way. This assumption appears rather plausible. With a little imagination one can devise triplets of alternatives for which it is conceivable that they might be ordered in a pre-assigned manner on any psychological continuum. CONDITION

3 (positive association of social and individual values).47 Suppose that Ru ..., R„, R ' . . . , R'„ are two sets of orderings, and R, R' are the corresponding orderings f{Ru ..., R„) and f(R'u ..., R'n). Furthermore, let Ph P and P'h P' be the strict orderings corresponding to Ru R, R'i and R' respectively. Finally, let x0 be a fixed element of A. Assume that (1) for all x, y different from x0 andfor all i, we have xRty i f , and only i f , xR' ¡y; (2) for all y and i, if x0Riy, then x^R'tf, and if x0Pty, then xaP\y. Then xQPy implies x0P'y. We can interpret this condition as follows. Imagine that the orderings Rit ..., Rn induce the ordering R. Further, imagine that in each of the orderings Ru ...,/?„ the element x 0 was 'advanced' by a certain number of places in the ranking (possibly 0), while the order of other elements was unchanged. This leads to a new set of orderings R'i, ..., R'„ and a new resulting ordering R'. The condition asserts that all elements which were strictly preceded by x0 in the ordering R are also strictly preceded by XQ in the ordering R' (so that x0 does not move in the opposite direction from the orderings Rlt..., R„). In more concise form, neglecting the distinction between the strict and the weak relations and Rt, the condition states that if an element xQ is advanced in every individual ordering, it must also be advanced in the corresponding social ordering. CONDITION

47

Here 'individual* values correspond to evaluations on scales, and 'social' choice corresponds to the resulting overall ordering combining all these evaluations. For similar ideas, see Luce and Raiffa, Games and Decisions, chapter 14.

LANGUAGE OF MOTIVATION

43

In the considered context, this condition appears to be unquestionable, provided the analysis is restricted to those ordinal scales in which one can designate one end point as 'better', in the sense that advancing an alternative toward this end point along each of the scales under consideration results, other conditions being equal, in an advancment of this alternative in the overall final ordering. It appears that such a 'common orientation' of scales is possible, at least for scales of emotion, as well as for the motivational (proper and deontic) scales. The fourth condition, crucial for Arrow's theorem, may be formulated as follows: 4 (independence of irrelevant alternatives). For any pair of alternatives, the social order regarding this pair is completely determined by the individual orders for this pair only. In Arrow's original formulation, this condition was required not only for pairs, but for any subset of the set of all alternatives, i.e. it was required that whenever Rt, ..., R„ and R'..., R'„ induce identical orders in the set S B. 7

MOTIVATIONAL CALCULUS

53

Next, the semantic implication satisfies the law of contraposition, i.e. if

then ~ B

~ A.

Indeed, A & ~ B and ~ B & ~ ~ A, which is the same as ~ B & A, are either both admissible, or both inadmissible. In the presentation of the specific semantic implications, we shall make use of the logical functors & and v , whenever practicable, in order to combine several formulae into one. Thus, we shall write A~

B&C

whenever A~l B and A~*s C. Similarly, we shall write (1)

A y B 1 C

if A 7 C and B ^ C. Note that the formula (1) can usually be interpreted only as a conventional way of expressing the fact that A ^ C and B ^ C, as the utterance A v B may be inadmissible. Thus, unless used for propositional arguments, the meaning of the symbols v and & in the formulae for semantic implications will be as follows: the symbol & appearing on the right hand side and the symbol v appearing on the left hand side should be interpreted as used in the conventional way of combining several implications into one, as described above. The symbol v on the right hand side will denote the usual alternative in the logical sense, while the symbol & on the left hand side, the linguistic conjunction of sentences. Finally, double implication A 7

5

will be used to express the fact that A ^ B and B"s A. 2.1.4. Rules of semantic inference for some motivational functors 2.1.4.1. I know, I am certain, I think In the formulae below, we shall denote by K, Cr, and T the functors 'I know that', 'I am certain that', and 'I think that'. Each of these functors applies to propositional arguments. We shall interpret 'I know that p\ to be denoted by Kp, as synonymous with 'I know that p is true', and we exclude from the consideration the use of 'I know' as in the sentence I know him quite well, where the argument is not a proposition. The functor K can be negated in two ways. First, as mentioned above, we denote by ~ Kp the sentence 'It is not true that I know that p\ Second, we shall introduce the

54

MOTIVATIONAL CALCULUS

functor K', to be read as 'I don't know', applicable to two or more propositional arguments. Accordingly, K'(p, q) is to be read as 'I don't know whether p or q\ Two types of negation will also be used for the functor T; the symbol ~Tp will stand for 'It is not true that I think that p\ and we shall introduce the functor T', to be read as 'I don't think that'. The functor T (and also 7") is interpreted here as an expression of judgment, as used in I think that he is ill, and not as referring to the process of thinking, as in I think of him quite often. We can now proceed with the presentation of the semantic inference rules. First, it seems quite obvious that we may formulate the following three laws (2) (3) (4)

K(p & q) 7 Cr(p & 7 T(p &q) 7

Kp&Kq Crp&Crq Tp & Tq.

The justification of the above formulae is quite simple; we shall illustrate it by analysing the first of them. Thus, one has to show that K(p Scq)^ Kp and K(p &q) 7 Kg. Indeed, consider the sentence of the form K(p & q) & ~Kp, i.e. 'I know that both p and q (are true), but it is not true that I know that p (is true)'. This sentence seems to be inadmissible, no matter what p and q are, which shows, by definition of the symbol 7> that K(p & q) 7 Kp. Similarly, K(p & q) 7 Kq, which yields formula (2). As already mentioned, formula (2) may be interpreted as a rule, which allows us to make the inference K3p & Kzq ('he knows that p and he knows that q') from the utterance K(p & q). In other words, if someone utters a sentence of the form 'I know that p and q\ one may infer he knows that p and he also knows that q. In the sequel, when discussing particular semantic implications, we shall sometimes refer to them as the inference rules which lead from sentences expressed in the first person singular to propositions expressed in the third person singular asserting something about the speaker's intentions. There is an interesting problem of converses to (2), (3), and (4). With the provisions discussed below, we have (5) (6) (7)

Kp& Kq 7 K(p & q) Crp&Crq 7 Cr(p & q) Tp & Tq 7 T(p & q).

The first of these laws, however, is true only if 'I know' is interpreted as intended, i.e. as an assertion of truth, and not as used sometimes instead of functors such as 'I heard that', 'I know that it is generally believed that', etc. The second of these laws, namely the inference concerning the functor Cr can be made only if the sentences Cr p and Cr q are combined into one utterance, i.e. if Cr p

MOTIVATIONAL CALCULUS

55

&Cr q stands for linguistic rather than logical conjunction. Generally, if the sentences Cr p and Cr q are uttered on two different occasions, one cannot infer that Cr3(p & q), i.e. that the speaker is certain that both p and q. In fact, by observing when we have Cr p & Cr q in the logical but not in the linguistic sense, and ~ Cr(p & q), one can obtain some information on how high the (subjective) probability of an event must be for a person to assert that he is certain that this event will occur. Indeed, imagine two events, say A and B, with probabilities a = P(A) and b = PCS), and assume that A and B are independent, so that the probability of the joint occurrence A n B is ab. Suppose now that the speaker knows the probabilities a and b, and also knows that the events A and B are independent. We may postulate the existence of a number r (possibly depending on the speaker), such that he will apply the functor 'I am certain that' to a description of an event only if his subjective probability of this event is at least r. If the subjective and objective probabilities did in fact coincide, and if it turned out that the subject applies the functor Cr to events A and B when confronted with them separately, but does not apply this functor when analysing the possibilities of the joint occurrence A n B, we would know that a > r, b > r, and ab < r, that is, (8)

ab < r < min {a, b).

A series of observations for different events A, B should allow us to obtain reasonable estimates of the number r. Of course, when performing this experiment one would have to take into account the fact that the subjective and objective probabilities do not coincide, and use instead of (8) the corresponding bounds obtained with the help of well-known relations between subjective and objective probabilities.8 Thus, if f(x) is the subjective probability of an event whose objective probability is x, the relation (8) would be replaced by (9)

f(ab)

< r < min

(f(a)J(b)).

Similar restrictions apply to rule (7); thus, (7) is true if the symbol & on the left hand side is interpreted as linguistic conjunction rather than logical conjunction. Let us now analyse some rules concerning the negation 'I don't know'. First, we have 9 (10)

K'(p,q):

K'(q,p),

since the sentence 'I don't know whether p or q, but it is not true that I don't know whether q or p' is inadmissible. 8 See for instance C.H. Coombs, R.M. Dawes, and A. Tversky, Mathematical Psychology (Prentice Hall, Englewood Cliffs, 1970). 9 A system for functors 'I know that' and 'I believe that' (considered in the next section) has been developed by J. Hintikka (see Knowledge and Belief). As already pointed out, his system differs primarily in the interpretation of the concept of implication used. Also, Hintikka considers the sentence 'I do not know whether p', which in our notation corresponds to K'(p, ~ p).

56

MOTIVATIONAL CALCULUS

Next, (11)

K'(p,q):~Kp,

and by (10) also (12)

K'(p, q) 7 ~Kq.

Combining (11) and (12) we may write (13)

K'(p, q) ; ~Kp & ~Kq.

The three rules above state that from the utterance 'I don't know whether p or q' one can infer that it is not true that the speaker knows that p, and it is not true that the speaker knows that q. By contraposition of (11) we have (14)

Kp 7 ~K'(p, q),

i.e. if the speaker knows that p, then it is not true that he doesn't know whether p or q. We may also state the rule (15)

~K'(p, q) ; K3p v K3q.

Here it is essential that the subscript 3 is used on the right hand side, as the alternative Kp v Kq is, in general, not admissible (this point has already been discussed in preceding sections). To verify (15) analyse the utterance ~K'3(p, q) & ~(K3p v K3q) that is, ~K'3{p, q) & ~K3p & ~K3q\ in words: 'It is not true that he doesn't know whether p or q, but it is also not true that he knows that p and it is not true that he knows that q\ Regarding the functors Cr and T we have (16)

Crp ;

~T~p,

or, by law of contraposition (changing/? and ~p): (17)

Tp 7

~Cr~p.

Thus, if someone claims 'I am certain that p\ then it is not true that he thinks that not p. Indeed, the sentence 'I am certain that p, but I think that not p' is not admissible. Next, we have (18)

Tp 7

T'~p.

or, by law of contraposition (changing/» and ~p):

MOTIVATIONAL CALCULUS

(19)

~Tp

7

57

~T~p,

is since the sentence 'It is not true that I don't think that p, but I think that not inadmissible. Consider for instance the sentence It is not true that I don't think he's dead, but I think that he is alive. Substituting p & q and p v q for p in (18) and using de Morgan's laws one obtains (20) (21)

T(p & q) 7 T'( ~p v ~q) T(p v q) 7 T\~p & ~q).

It ought to be noticed, however, that not all sentences of the form T(p v q) are admissible, as the functor 'I think' implies a degree of doubt; thus, it cannot be used for the case when p v q is a tautology, such as p v ~p. This may be illustrated by a sentence such as I think that their child is either a boy or a girl, which can be uttered only as a joke, or else in rather unusual circumstances. Furthermore, we have (22)

T'p & T'q 7 T'(p v q),

and a modified version of the converse to (18): (23)

T'p 7 ~Tp.

(It may be doubted, however, that the right hand side in (23) can be replaced by T~p, the latter being perhaps too strong. Thus, someone who says I don't think it will rain tomorrow will agree that it is false that he thinks that it will rain, but might object to phrasing it that he thinks it won't rain.) 2.1.4.2. I believe, I doubt We shall denote by B, B', and D the functors 'I believe that', 'I don't believe that', and 'I doubt that'. These functors, as the functors considered in 2.1.4.1, are applicable to propositional arguments. We list below several semantic implications for the functors B, B', and D, connecting these functors with conjunction and alternative. Regarding functor B we have 10 (24)

B(p & q) V Bp & Bq

and (25)

B(p v q) 7 7"(~/> & ~q).

10 It is worth mentioning that using the functors B and K, Hintikka (in Knowledge and Belief) introduces the concepts of doxastic and epistemic implications. According to him,p implies q doxastically (see p. 76) if the sentence B(p & ~ q) is indefensible, and p implies q epistemically (see p. 79) if the sentence K(p & ~ q) is indefensible. Hintikka introduces the first of these concepts for the purpose of analysing Moor's problem of saying and disbelieving, i.e. in an attempt to explain why a sentence of the form '/>, but I do not believe that p' is absurd.

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MOTIVATIONAL CALCULUS

For the functor B' we have, in general (26)

B'p :

T~p.

Substituting the conjunction p & q and the alternative p v q for p in (26), and using de Morgan's laws, we obtain (27)

B'(p &q) 7 T(~p

(28)

B'(p v q)~T(~p&

v

~q) ~q).

Moreover, (29)

B'p & B'q 7 B'(p v q) & 5 ( ~ p &

Finally, for functor D we have in general (30)

Dp ; T'p

and also (31) Dp & Dq 7 Z>0? v Of these rules, (25) and (29) seem to be most interesting. Concerning (25), it appears that if someone claims that he believes that p or q, one cannot infer that he believes p, or that he believes q, and one can only infer that he has some doubts regarding the conjunction ~p & ~q. Thus, when three candidates, say A, B, and C run for an election, C having a very small chance of winning, a person might well say I believe that either A or B will win; it implies that he doesn't think that C will win, but does not imply that he believes that A will win, or that he believes that B will win. Next, (29) is a shortened version of two implications, B'p & B'q 7 B'(p v q) and B'p & B'q 7 B(~p & ~q). To illustrate this, one can analyse the inference which can be drawn from an utterance such as I don't believe that rabies can be cured by taking aspirin, and I don't believe that they can be cured by uttering magic spells. It seems that the legitimate inference is that the speaker does not believe that rabies can be cured by either taking aspirin or uttering magic spells; also, one may infer that he believes that rabies cannot be cured by taking aspirin, and that they cannot be cured by uttering magic spells. 2.1.4.3. Some problems of ordering epistemic functors In Chapter 1 we discussed the problem of ordering the functors such as K, Cr, T, B, and D considered above, and also other functors of the same class, such as 'I am convinced that', 'I suppose that' (possibly modified by intensional functors, e.g. 'I am rather certain that', etc). It was suggested that to each of these functors there corresponds an interval on the scale of subjective probability. Clearly, if IF and IG are intervals on the subjective probability scale corresponding to functors F and G, then the relation IF c IG implies the semantic inference rule Fp 7 Gp. Assigning to each

MOTIVATIONAL CALCULUS

59

of the functors the midpoint of the interval corresponding to it, we arrive at an ordering of functors of the considered class. There is, however, a more interesting way of arriving at an ordering of the considered class of functors. Intuitively, we shall try to devise a method of ordering these functors according to the 'strength' of conviction which they represent. Note first that the most natural way of ordering functors, according to which F precedes G if Fp 7 Gp may be inapplicable, for there may exist pairs F, G such that neither Fp 7 Gp nor Gp 7 Fp. One could argue, for instance, that such a pair of functors is K and T, the sentence Kp being too definite to imply the rather vague Tp. Indeed, the sentence Kp & ~Tp is rather admissible (compare the somewhat heated denial: it is not true that I think that hepatitis is an unpleasant disease; I know it). A possible approach to the problem of ordering might be as follows. Consider the inference schemes from propositional calculus, or from traditional logic, such as I- p => q t- p

or

Vq

h p => q 1= ~q V ~p

or, say Barbara and Celarent modes 11 Every M is P (MaP) Every S is M (SaM)

No M is P (MeP) Every S is M (SaM)

Every S is P (SaP)

No S

is P (SeP)

Suppose now that with each of these inference schemata we associate the set of semantic inference rules obtained as follows: we apply functors from among K, Cr, T, B to each of the premises, and take as the left hand side of the semantic implication the conjunction of sentences thus obtained. Such a conjunction semantically implies the conclusion of the original inference scheme, to which a functor from the set {K, Cr, B, T} is applied. For instance, from the inference scheme ¥p=>q KP

we obtain semantic inference rules such as (32) (33) 11

K(p => q)&Crp^ Cr q T(p q) & Bp 7 Tq.

See for instance F.H. Parker and H.B. Veatch, Logic as a Human Instrument (Harper and Bros., New York, 1959), 302-303 (where, however, no traditional names of syllogisms are given). For the list of names, see K. Pasenkiewicz, Logika ogdlna (General logic), in Polish (Warszawa, PWN, 1968), 157-158.

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Similarly, the Celarent yields semantic inference rules such as (34) (35)

B(MeP) & Cr(SaM) 7 B(SeP) K(MeP) & T(SaM) 7 T(SeP)

and so on. Thus, rule (32) reads that if someone claims that he knows that p implies q, and that he is certain of the premise p, then he is also certain of the conclusion q. Similarly, the utterance 'I know that no M is P, and I think that every S is M' implies semantically that the speaker thinks that no S is P [rule (35)]. Imagine now that we collected a set, call it f , of such semantic inference rules. We may then say that a given ordering of the set {K, Cr, T, B} is compatible with T, if for every rule in f , the functor appearing in the conclusion of the rule does not precede in this ordering the latest among the functors used in the premises. For instance, if the set W consists of the single rule (32), then each of the orderings such as (K, Cr, B, T), (T, K, B, Cr), (T, B, K, Cr), etc., with K preceding Cr, is compatible with f . Now, if W consists of rules (32) and (35) (say), out of the three orders above only the first is compatible with W, as (35) requires that K precedes T. Thus, adding new rules may only decrease the set of compatible orderings of the set of functors. The following conjectures seem to be at least plausible: (1) as the set if increases by adding to it new semantic inference rules obtained in the manner described above, more and more orderings become actually eliminated as incompatible; (2) there will eventually remain exactly one ordering which will be compatible with all semantic inference rules obtained in the above manner; (3) the ordering described in (2) will be (36)

(K, Cr, B, T)

(4) every rule in W will be described by the following 'weakest link' principle: the functor applied to the conclusion coincides with the last [in the ordering (36)] among the functors used in the premises. In other words, the conclusion is as certain as the least certain of the premises. Clearly, the same method of ordering may be applied to other functors from the considered class of epistemic functors, such as 'I am sure that', 'I suppose that', etc. 2.1.4.4. I can, I cannot We shall denote by C and C' respectively the functors 'I can' and 'I cannot'. As distinct from the functors considered previously, C and C' do not apply to propositional arguments, but to verb phrases, or - more specifically - to descriptions of actions. By actions we shall also mean some psychological processes, as in I can imagine that, etc. In studying the semantic implications below, we shall represent sentences starting

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61

from 'I can' or 'I cannot' by specifying the object to which the action can (or cannot) be applied, and the prospective outcome of this action, i.e. a future state of the object. Accordingly, we shall write m(a, P) for '(to) make a to be P\ and we shall write C[m(a, i>)] and C'[m(a, P)] for 'I can make a to be P' and 'I cannot make a to be P\ respectively. Thus, in the sentence I can lend you this book the object a is the book in question, and the predicate P stands for 'is temporarily in your possession'. Similarly, in the sentence I cannot close this window the object a is the window, and the predicate P stands for 'is closed'. Sometimes the predicate P contains the specification of time, as in I can telephone him after six o'clock tonight, and sometimes it leaves the time unspecified, as in I cannot understand this passage. The object a sometimes coincides with the speaker, as in I can swim to the other side of the river. We shall use the conjunction and alternative according to the normal usage of connectives 'and' and 'or' in natural language, as applied to shorten two phrases whose objects or predicates coincide. Thus, m(a, P) & m(b, P) will be written as m{a & b, P), while m(a, P) & m{a, Q) will be written as m(a, P & Q). Similar notations 12 will be used for the alternative: m(a v b, P) will stand for m(a, P) v m(b, P) while m(a, P v 0 will stand for m{a, P) v m(a, Q). Finally, negation ~ may concern only the predicate P; we shall write m(a, ~ P) for '(to) make a to be ~P\ In the formulae below, symbols a, b will denote distinct objects, and symbols P, Q, distinct predicates. Moreover, we shall write P &.Q = 0 to denote the fact that predicates P and Q are disjoint in the sense that if a is P, then a is not Q and vice versa, Similarly, we shall write m(a, P) & m(b, 0 = 0 to denote the fact that making a to be P excludes making b to be Q, and vice versa. An analysis of sentences in which the speaker claims that he can or cannot do something, yields the following rules (37)

C[m(a, P) & m(b, 0 ] ; C[m(a, P)] & C[m(b, 0 ]

and, more specifically, (38)

C[m(a & b, />)] 7 C\m(a, P)] & C{m(b, />)]

(39)

C[m(a, P & 0 ] 7 C{m(a, J»)] & C[m(a, 0 ] ,

It seems that unless m(a, P) & m(b, Q) = 0, the converse to (37), and hence also to (38) and (39), is true, i.e. if a person claims that he can m(a, P) and that he can m(b, Q), 12

Here again one should point out that such notation does not adequately correspond to the way of combining phrases in natural language. Indeed, 'a and b are P' is not necessarily the conjunction 'a is P and b is P' (compare: John and Mary made friends). Similar examples may be found against the usage o f ' a is P and Q' for 'a is P and a is Q\ etc, (see Strawson, Introduction).

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and moreover, asserts it in one utterance, then he can make the conjunction m(a, P) & m(b, Q). It is important, however, that in the converse to (37), i.e. in (40)

C{m(a, P)] & C[m(b, 0 ] ; C[m(a, P) & m(b, 0 ]

the sign & on the left hand side is interpreted as a linguistic conjunction, rather than a logical one. Indeed, a person may be able to run from x to y in, say, less than 2 minutes, and he may be able to run from y to z in less than 2 minutes, but it does not imply that he is able to run from xto z via y in less than 4 minutes. Now, when m(a, P) & m(b, Q) = 0, the converse (40) cannot be true. In fact, it appears that the use of the connective 'and' in such cases may be inadmissible. Consider, for example, the sentence I can stay home all day Saturday, and I can go to a theatre on Saturday, a rather inadmissible utterance. However, if one is willing to accept this utterance as admissible, then it is usually taken to imply that the speaker can either stay at home all day, or that he can go to the theatre (on Saturday). In other words, in case of disjoint m(a, P) and m(b, Q) we have (41)

C[m(a, P)] & C[m(b, 0 ] ; C\m(a, P) v m(b, 0 ] ,

In such cases, it would seem more proper to use the connective 'or', i.e. (42)

C[m(a, P)] v C[m{b, 0 ] ; C[m{a, P) v m(b, 0 ] ,

It may be shown that in general the converse to (42) is false, i.e. that C\m{a, P) v v m(b, 0 ] does not imply the alternative C\m{a, P)] v C[m(b, 0 ] . Indeed, imagine a person shooting at a target divided into two sections, left and right, and assume that he has enough skill to be able to hit the target at will, but not enough skill to hit the desired part of the target. Then, if a stands for the bullet, and P and Q for predicates 'hit the left (right) section of the target', C[m(a, P) v m(a, 0 ] is simply C\m(a, P v 0 ] , that is, the sentence / can hit the target. In this case C[m(a, P)] stands for / can hit the left section of the target, and similarly for C\_m(a, 0 ] ; we have then C\m(a, P v 0 ] but not C[m(a, P)], and not C[m(a, 0 ] . There is an interesting usage of the sentence of the form C[m(a, P) v m(a, 0 ] when P c Q; the sentence then is partially 'redundant', as logically m(a, P) implies m(a, 0 . If such a sentence is uttered, in spite of its redundancy, the speaker's intentions are that C\m(a, 0 ] , and that making a to be P is possible, though he cannot guarantee it. This usage is illustrated by examples such as I can cure him, or at least make him suffer less or I can kill the enemy, or at least disable him. Next, regarding the functor C' we have (43)

C'[m(a, P)] 7 ~C[m(a, P)].

It is important, however, to note that 'I cannot' may also be used as synonymous with 'I shouldn't', 'I oughn't', etc., and 'I can' is also used sometimes as expression of permission, or lack of constraint (hence synonymous with 'I may'). The law (43) holds

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63

if on both sides the functors are interpreted in the same way, i.e. either both referring to physical possibility, or both to the presence or absence of constraints. Next, we have the following semantic inference rules: (44)

C'\m(a, P) & m{b, 0 ] 7 C[m(a, P)] & C\m{b, 0 ] ,

and (45)

C'\m(a, P) v m(b, 0 ] ; C'[_m(a, P)] & C'\m(b, 0 ] ,

Thus, the utterance I cannot afford to buy a house and a car implies that the speaker can in fact afford to buy a house, and that he can afford to buy a car (though not both of these items). Similarly, I cannot afford to buy a house or a car implies that the speaker cannot afford to buy either of them, i.e. that he cannot afford to buy a house, and that he cannot afford to buy a car. We shall, however, not include (44) in the system, as it would lead to the following troubles: putting m(b, Q) = m(a, P) in (44) we obtain C'\m{a, P)] 7 C[m(a, />)], while (43) gives C'\m(a, P)] 7 ~C[m{a, P)]. This means that if both (43) and (44) are included in the system, it becomes inconsistent, unless we make an obviously unacceptable assumption that every utterance of the form C"[m(a, P)] is inadmissible. Thus, if we want to have the system consistent, and allow the admissibility of utterances C'[m(a, P)], one of the laws (43) or (44) has to be removed. Finally, if P c Q, i.e. if every object which is P is also Q, we have (46)

C[w(a, P)] 7 C[m(a, 0 ]

and (47)

C'[m(a, 0 ] 7 C'[m(a, P)].

2.1.4.5. I am glad that We shall now consider semantic implications for one of the functors from class 2, i.e. from the class of emotional functors, namely for the functor 'I am glad that'. This functor applies to propositional arguments. It will be more convenient, however, to specify the form of the argument in a manner similar to that used in paragraph 2.1.4.4. for the functor 'I can'. Accordingly, in the proposition which appears as an argument for 'I am glad that', we shall distinguish its object, a, and predicate P. We shall use the symbol G(a, P) to denote the sentence 'I am glad that a is P ' (will be P, was P, etc). In the sequel, (a, P) will denote the proposition 'a is P', and we shall apply functors such as K and T to these propositions. Finally, we shall denote by {a, P)G the passive form of the sentence G(a, P), i.e. 'a being P makes me glad'. We can now formulate the semantic implications concerning the functor G. First of all, we have (48)

G(a, P) 7 K3(a, P) v T3(a, P),

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i.e. the sentence 'I am glad that a is P' implies that the speaker either knows that a is P, or at least thinks so. This may be illustrated by the sentence such as I am glad that London is the capital of France. If this sentence is accepted, it must also be accepted that the speaker has some gaps in his knowledge of geography, and is convinced that London is in fact the capital of France. Next, we have (49)

G[(a, P) v (b, 0 ] :7 (a, P)G3 & (b, Q)G3

i.e. if someone claims that he is glad that an alternative holds, then each of the parts of the alternative must have the property of making him glad. Specifically, if either objects or predicates in (49) coincide, we obtain (50)

G(a v b, P) 7 {a, P)G3 & (b, P)G3

and (51)

G(a, P v Q) ; {a, P)G3 & (a, Q)G3.

The intuitive justification of rule (49) is the following. The utterance of an alternative signifies usually that the speaker does not know which of the components of the alternative holds; thus, is someone asserts that he is glad that some alternative holds, the message which he conveys is that he will be glad in either case (though, perhaps, not in the same degree) - and this is precisely the content of laws (49)-(51). As an example, consider the sentence I am glad that he'll come during the weekend, the inference being that his coming on Saturday will make the speaker glad, and so will his coming on Sunday. It is interesting to observe that from the sentence in which the speaker asserts that he is glad because of a conjunction, one cannot make any inference about the components of the conjunction. In other words, the sentence G\(a, P) & {b, 0 ] does not imply (a, P)G3, nor does it imply (b, Q)G3. Indeed, a person who misplaced his gloves may well assert that he is glad because he found them (he found both the left and the right glove); however, this does not imply that finding one glove makes him glad, unless he already has the other one. Using (51) and de Morgan's laws, one can obtain the following laws concerning the negation: consider the sentence G(a, ~ (P & 0 ) which is equivalent to G(a, ~ P v v ~ 0 , hence, by (51), to (a, ~P)G3 & {a, ~Q)G3. In other words, we have (52)

G{a, ~{P & 0 ) : (a, ~P)G3 & (a, ~Q)G3.

Thus, I am glad that it is not cold and raining implies that the speaker is glad that it is not cold, and also that he is glad that it is not raining.

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2.1.4.6. I prefer, I want, I don't want We shall now consider proper motivational functors 'I prefer', 'I want', and 'I don't want', to be denoted by Pf, W, and W' respectively. Regarding the functor W, we shall denote by fV(a, P) the sentence 'I want a to be P\ Similarly, W'(a, P) will stand for 'I don't want a to be P\ The functor Pf applies to pairs of arguments of the form considered; thus, we shall write Pf(a, P; b, Q) for the sentence 'I prefer a to be P than b to be Q\ In most cases of the usage of functor Pf in the natural language, either the objects or the predicates coincide, that is, in most cases the considered sentence has the form Pf(a, P; a, Q) or Pf(a, P; b, P). As an example of the first usage we may take the sentence I prefer coffee with sugar to coffee without it. The usage with P = Q, i.e. the second of the above, is Iprefer pears to apples, the predicate P standing for 'to be eaten by me'. However, the usage with a + b and P Q is not excluded, as in I prefer coffee with sugar to tea without it. It seems that the utterance Pf(a, P; b, Q) in the case when a b and P # Q indicates that in some sense the preferences for objects a and b, and for predicates P and Q are opposite one to another. The justification of this is that if it were not so, there would be no point in expressing the direction of preference with distinct predicates. Thus, for instance, the sentence I prefer good apples to bad pears implies that if the choice was made between good pears and good apples, the pears would be preferred (here it is implicitly assumed that the predicate 'good' is better than the predicate 'bad'). If one could define preferences towards objects separately, and towards predicates separately, then - denoting these preferences by the same symbol Pf - we would have (53)

Pf(a, P; b, Q) 7 [Pf3(a, b) & Pf3(Q, i>)] v [Pf3(b, a) & Pf3(P, Q)].

Preferences for predicates, as appear above, may be determined empirically for certain classes of predicates, for instance, colours. We have also (54)

Pf(a, P; b, P) & Pf(a, Q; b, Q) 7 Pf(a, PwQ;b,Pv

Q).

Thus, if a person claims that he prefers to see film A to film B on the 5 P.M. program, and that he prefers to see film A to film B on the 8 P.M. program, then he prefers to see film A to film B on either of the programs. Next, regarding the alternative, we have the following law (55)

Pf(a, P;b v c,P) 7 Pf(a, P; b, P) & Pf(a, P; c, P).

This is in accordance with the use of the connective 'or' in the natural language, where

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it reflects a degree of uncertainty. Thus, if someone claims that he prefers a to be P to either b or c being P, then he must prefer (a, P) to (b, P) and also (a, P) to (c, P). To give an example, I prefer beef to either lamb or pork means that the speaker prefers beef to lamb, and that he prefers beef to pork. The preferences in which there appear alternatives either of objects or of predicates constitute the basis for the construction of the utility scale. The alternatives then have the form of 'lotteries' assigning probabilities to various events, and preferences concern pairs of lotteries. The events described in these lotteries have the form of pairs {a, P), and the lottery assigns probabilities to events of this type either for the same predicate P and various objects a, or for the same object a and various predicates P. One of the possible sets of axioms implying the existence of a utility scale was presented in Chapter 1. The problem of utility and its connection with preference relation has been studied extensively in the literature, and we shall not discuss it in this section. We shall now investigate the properties of the functors W and W'. For the first of these functors and the conjunction we have (56)

W(a, P&Q)

7 W{a, P) & W{a, Q)

W(a &b,P)

7 W(a, P) & W(b, P).

and (57)

The converses to (56) and (57) are not true. First, the sentence W(a, P) & W(a, Q) may be admissible even if P & Q = 0, when the properties expressed by P and Q are sufficiently desirable. Then the sentence W(a, P&Q) is not admissible, and the conjunction W(a, P) & W(a, Q) & ~ W{a, P & Q) is admissible, which shows that the converse to (56) does not hold. It may happen, however, that P and Q do not exclude one another, and yet, the person may want a to be P, want a to be Q, but may not want a to be both P and Q. Such a situation occurs if the joint combination of P and Q has some undesirable 'side effects'. Thus, some men may want their wives to be intelligent, want them to be attractive, but not both. Similar reasoning shows that the converse to (57) is also false. Thus, a woman may want to have a as her husband, and she may also want to have b as her husband, and yet be unwilling to commit bigamy. We have, however, (58)

W(a, P) & W(a, Q) 7 W(a, P v Q)

and (59)

W{a, P) & W(b, P) 7 W(a v b, P).

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67

The converses to (58) and (59) are, of course, false. Indeed, someone may say that he wants to have an international competition won by a competitor of a specific nationality, and yet be indifferent regarding various competitors of this nationality, so that he would object to making the inference that he wants a specific person of this nationality to win the competition. For the functor W', i.e. 'I don't want', the implications bear some resemblance to de Morgan's laws. Generally, we have (60)

W(a, P) 7 W'(a, ~P).

It seems, however, that the converse is false: imagine two candidates, a and b, running for an election. Then if someone wants a to win (predicate P), he doesn't want b to win (i.e. a to lose). However, if someone doesn't want a to win, he may well agree that for this to happen b must win, but he might object against phrasing it that he wants b to win, so to say, on his own merit; it may be that out of two bad candidates, b presents a 'lesser evil'. As consequences of (60) we obtain (61)

W(a, P & Q) ; W'{a, ~i> v

~Q)

and (62)

W(a, P v 0 ;

W'(a, ~P & ~Q).

We have also (63)

W'(a, P) & W'(b, Q) 7 W'l(a, P) v (b, 0 ] ,

hence from (57), replacing its right hand side by the conjunction W'(a, ~ P ) & W'(b, ~P) [by (60)] and using (63): (64)

W(a &b,P)

7 W\a v b, ~P).

Finally, connecting functors W, W', and Pf, we have an obvious law (65)

fV(a, P) & W'(b, Q) 7 Pf(a, P; b, Q)

Clearly, the converse does not hold: a person may prefer to die of pneumonia than of cancer, but it does not imply that he wants to die of pneumonia. 2.1.4.7. I must, I ought to To complete this section, we shall briefly analyse some functors from normative motivational class, namely the functors 'I must' and 'I ought to', to be denoted by M and O. Both of them apply to the same type of argument as the functor 'I can', i.e. the argument has the form m(a, P), standing for '(to) make a to be P\ We shall there-

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fore denote by M[m(a, P)] the sentence 'I must make a to be P \ and similarly, 0\m{a, P)] will stand for 'I ought to make a to be P'. It appears that the laws for functors M and O are almost identical, and we shall therefore discuss them jointly. Thus we have (66)

M\m{a, P) & m(b, 0 ] 7 M\m(a, P)] & M[m(b,

0]

and (67)

0\m{a, P) & m(b, 0 ] 7 0\m(a, P)] & 0[m(b, 0 ] .

As an example, consider the sentence I must give the children medicine, and put them to sleep, which implies that the speaker feels that she must do both. The same sentence with 'I must' replaced by 'I ought to' serves as an illustration of (67). The converses to (66) and (67) require us to distinguish two cases. When the actions m(a, P) and m(b, Q) do not exclude one another, we have simply (68)

M[m(a, P)] & M[m(b, 0 ] 7 M[m(a, P) & m(b, 6 ) ]

and (69)

0\m{a, i>)] & 0[m(b, 0 ] 7 0\m(a, P) & m(b, 0 ] ,

The above examples of putting children to sleep and giving them medicine may serve as illustrations of these laws. When m(a, P) and m(b, Q) exclude one another, we have (70)

M[m(a, />)] & M\m(b, 0 ] 7 M{m{a, P) v m(b, 0 ]

and (71)

0\m(a, />)] & 0\m{b, 0 ] 7 0[tn(a, P) v m(b, 0 ] .

To illustrate these laws consider the sentence I ought to tell George openly to go to hell, but I ought to be careful not to appear unfriendly, a dilemma requiring making a choice of action of getting rid of George for good, or putting up with him. The same sentence with 'I ought to' replaced by 'I must' illustrates (70). It would seem that when m{a, P) and m(b, Q) exclude one another, the utterances M[m(a, P)] & M[m(b,

0]

0\m{a, P)] & 0\m(b,

0]

and

signify the existence of a conflict. It is interesting to observe that in the same situation of m{a, P) and m(b, Q) excluding one another, the utterance 0[m(a, P)] & M[m{b, 0 ]

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indicates how the conflict was solved, i.e. which decision was taken: that one, which is attached to the functor 'I must'. Consider, for example, I ought to go home, but I must stay a little longer in the office and I ought to stay a little longer in the office, but I must go home. The first sentence implies that the speaker decided to remain in the office, while the second, that he decided to go home immediately. Finally, it is worth considering the negations of functors M and O. There are two types of negation: we shall denote by M' and O' the functors 'I mustn't' and 'I oughtn't' ('I shouldn't'). These two functors, however, express a restraining force as strong as the compelling force expressed by the corresponding functors M and O. Another type of negation is obtained by expressing lack of constraint; thus, we shall denote by H' the functor 'I don't have to'. The functors M' and O' behave differently under conjunction and under alternative. Thus, we have (72)

M'\m(a, P) v m(b, 0 ] ; M'[m(a, />)] & M'[m(b,

0]

and (73)

0'\m(a, P) v m(b, 0 ] 7 0'[m(a, />)] & 0'[m(b,

0],

To illustrate the fact that 'restraining' functors like M' and O' when applied to the alternative imply the conjunction of constraints, consider the sentence13 Neither shalt thou desire thy neighbour's wife, neither shalt thou covet thy neighbour's house, his field, or his manservant, or his maidservant, his ox, or his ass, or any thing that is thy neighbour's. For the conjunction, the sentence M'[m(a, P) & m(b, 0 ] does not imply M'[m(a, /•)], nor does it imply M'[m(b, g ) ] (and the same is true for O'). To illustrate it, consider the sentence I mustn't marry Paul and Peter, which does not mean that the speaker feels that she mustn't marry Paul (or Peter). The sentence, as it stands, reflects only the speaker's attitude towards bigamy (at least as regards Paul and Peter). The use of M and M' (and also O and O') reveals then a symmetry, in the sense that neither M[m(a, P) v m{b, 0 ] nor M'\m(a, P) & m(b, 0 ] 13

Deuterotomy, Chapter 5: 21.

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implies anything about actions m(a, P) or m(b, Q) separately. The implications are conditional: the first sentence implies "If I don't make a to be P, then I must make b to be Q", while the second implies "If I make a to be P, then I mustn't make b to be Q". Finally, for the functor H' ('I don't have to') we have, first of all (74)

H'[m(a, P)] 7 ~M[m(a, P)] & ~M'[m(a, />)],

i.e. the functor H' expresses lack of compelling force or constraint with respect to m(a, P). Thus, (74) is equivalent to the conjunction of two implications (75)

H'[m(a,P)-] 7

~M[m(a,P)-]

and (76)

H'[m(a, P)] 7 ~M'[_m(a, P)].

Substituting the conjunction in (75) and using the third person singular, we obtain (77)

H'[m{a, P) & m(b, 0 ] 7 ~M3[_m(a, P) & m{b, 0 ] ,

that is, using (66) and de Morgan's laws, (78)

H'[m(a, P) & m(b, 0 ] 7 ~M3[m(a, P)] v ~M3[m(b,

0],

Similarly, for the alternative we obtain from (76) and (72) (79)

H'[m(a, P) v m(b, 0 ] 7 ~M'3{m{a,

P)] v ~M\[m{b,

0]

which shows the symmetry between the functors M and M'. 2.1.4.8. Concluding remarks The semantic implications of this section were expressed in a certain symbolism used for representing sentences containing functors of motivational language. In designing this symbolism, it was necessary to introduce some simplifications, that is, to neglect certain aspects of the sentences being represented, and concentrate only on a few selected aspects. Thus, the only features represented in the symbolism were (a) the particular motivational functor used, and (b) the simple structural information about the argument, specifying whether it is a conjunction, alternative, or negation. Basically there is no trouble in introducing a more complicated symbolism,14 in 14

No symbolism, however, can ever be adequate: even the 'symbolism' consisting of simply writing down the sentence is not entirely satisfactory for analysing inferences made from utterances, as it fails to capture the intonational features.

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71

which one can code, for example, tenses of verbs used, invent some special symbols for questions, or represent some specific information about grammatical structure of the considered sentence. The simplifications in symbolism determine the restrictions of the complexity of semantic implications which can be analysed within the given symbolism. On the other hand, it is precisely because of the restrictions that one obtains a set of laws sufficiently general, and consequently sufficiently regular, to constitute a consistent system. The system presented above is clearly incomplete, as the main intention was to illustrate the methods of approach to the problem rather than attempt to give a detailed and thorough solution. The considerations can be continued in at least three directions. First, one could try to find further semantic implications for the same functors; second, one could analyse new functors, from motivational language or from outside it. Third, one could introduce a more complex symbolism, trying to capture other features of sentences, and consequently analysing more subtle semantic implications. Admittedly, the system presented (as any other, more complex) is only an approximation to the reality which it attempts to describe. One ought to remember, though, that an approximation is the condition sine qua non for the very existence of a system. 2.2. LOGICAL CONSISTENCY OF MOTIVATIONAL CALCULUS

In this section we shall analyse the problem of logical consistency of the system of implications introduced in the preceding section. In order to do that, it will be necessary to outline briefly the problems of consistency in logic, as studied in connection with various logical systems, such as the classical propositional calculus, or different systems of modal logic. This will enable us to formulate the particular problem of consistency which we are investigating, and determine the logical status of the presented set of rules. 2.2.1. Propositional calculus We begin with the simplest case, namely that of propositional calculus. In formalizing it, one starts from the concept of a well formed formula (wff). Roughly speaking, to determine the set of wff's, one has to supply a method of checking whether a given string of symbols is a wff or not. This may be accomplished in various ways, the most commonly used being the following. First, one specifies the basic set of primitive symbols, together with some definitions which enable us to replace some strings of symbols by others. For instance, one may take as primitive symbols the letters p, q, r, ... with or without subscripts, parentheses ) and (, and symbols v and ~ . The class of well formed formulas can now be defined by the following three conditions:

72 (1) (2) (3)

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a letter standing alone is a wff; if a is a w f f , so is ~ P at a P £

v 0; (a => P) & 08 =*• «)•

These definitions introduce new symbols &, =>, and o , and determine their usage. In the terminology of mathematical linguistics (discussed in Chapter 3) letters p, q, r, ... and symbols v , (,),&,=>, o constitute a certain alphabet, and wff's form 'grammatically correct' expressions of a certain language. Given the set of all wff's, one can consider various subsets of it. Among many possible methods of distinguishing sets of wff's, one plays a special role. This method consists of distinguishing a certain subset of wff's, its elements called axioms, and specifying the so-called transformation rules, that is, rules which allow us to transform sets of wff's into a wff. For a given set si of axioms and set 3k of transformation rules, we can define the set C(sf, 0£) as the set such that (a) (b)

every wff in si is in every wff which results from application of transformation rules from @t to wff's in C(si, 3k) is in C( si, 3k,).

The set C(si, 01) may be called the set of consequences of the set si of axioms (for the given set of transformation rules). Generally, elements of C(si, 3$) are called theses of the system; those theses which are not axioms (i.e. are in C( si, 3k) but not in si) are called theorems. Usually, the set 3k of transformation rules is fixed, and consists of two rules: (a) the formula which results from substituting any wff in place of a given symbol in a thesis is itself a thesis, provided that at each occurrence of this symbol, it is replaced by the same wff (this is the so-called Rule of Uniform Substitution)', (b) if a and a => jS are theses, so is P (this is the so-called Rule of Detachment, or Rule of Modus Ponens). Having thus fixed the set 01 of transformation rules, one can study the properties of sets C ( s i ) = C(si, 01) of consequences of various sets of axioms. One can define consistency of the set si of axioms by requiring that for no wff a, both a and ~ a are in C(si). Moreover, if for each wff a, either a or ~ a is in C(si), it

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is called complete. In this definition, completeness is a syntactic concept, as it does not refer to any interpretation. The main object of the axiomatic approach to propositional calculus is to provide a set si of axioms such that C(si) would coincide with a special class of wff's, namely with the class of valid wff's, or tautologies. This set of wff's is defined by specifying an algorithm which allows us to determine whether or not a given wff is a tautology (is valid). The algorithm consists of computing the so-called truth value of a wff for given truth values of propositional variables (single letters) appearing in the formula. This method, since it is widely known, will not be described here. A wff is a tautology, if its truth value equals 1 for any of the possible combinations of truth values of propositional variables in it. In other words, a wff is a tautology, if it is true in every model, the latter being identified with a substitution of truth values of propositional variables. If V is the class of all valid wff's, then any set si of axioms such that C(si, 0t) = ~f is said to be complete. In other words, the set si is complete if every valid formula is a thesis of the system C ( s i ) , and every thesis is valid (note that in this definition, completeness is a semantic concept, as it is defined through an interpretation of the system). Various such systems of axioms have been constructed. For instance, Lukasiewicz15 gave the system consisting of the following three axioms: (p*>q) => [(r) => (p=>r)~] (~p=>p) =>/> P => (~P=>p q=>p v q pyq => qvp (,q=>r) => [(p=>q)

=> 0>=>r)]

and [p v (qv/•)]

=> i(qwp)

v r)

(the last axiom, however, was subsequently shown to be unnecessary).17 15

See, for instance, K. Pasenkiewicz, Logika. A.N. Whitehead and B. Russell, Principia Mathematica, 3 vols. (Cambridge Univ. Press, Cambridge, 1910-1913). 17 Historical information about propositional calculus may be found in A. Church, Introduction to Mathematical Logic, vol. I (Princeton Univ. Press, Princeton, 1956), 155-156. 18

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74

2.2.2. Modal logic The situation in modal logic is very similar to that in propositional calculus, though somewhat more complex. First, in addition to the set of primitive symbols of propositional calculus, one adds the symbol L and the following rule for wff's:

if a is a wff, so is La. Next, one introduces the definitions

Ma

a

and a -3 p = ~ M ( a & ~jS).

La and Ma are read as 'a is necessary' and 'a is possible', respectively. The symbol a -3 J? is read as 'a strictly implies /?'. Next, to the set of rules of transformation, consisting of the Rule of Uniform Substitution and Rule of Modus Ponens, one adds the Rule of Necessitation: if a is a thesis, so is La. Various systems of modal logic are now obtained by adding to the set of axioms of propositional calculus (either of the above two sets of axioms serves the purpose equally well) some axioms involving the functor L (or M). One of these systems is the so-called system T based on the additional axioms

Lp=>p Up=>q)

=> (Lp=>Lq).

(This system was introduced by Feys.) 1 8 Verbally, if p is necessarily true, then p is true, and if p strictly implies q, then if p is necessarily true, 1 9 so is q. By adding to the axioms of system 7" the axiom

Lp => LLp (necessity of necessary truth; if p is necessarily true, then it must necessarily be so), one obtains the so-called system S 4 . By adding to the axioms of system T the axiom

Mp => LMp R . Feys, " L e s logiques nouvelles des modalités", Revue Néoscholastique de Philosophie 40 (1937), 517-553, in particular 533-535. 19 It is easy to see that p - 3 q is equivalent to L(p => q). Indeed, p - 3 q iff ~ M(p & ~ q), hence (by definition of the symbol M), iff ~ ( ~ L{~ (/> & ~ ?))). Using both de Morgan's and the double negation laws of propositional calculus, the last formula is easily seen to be equivalent to L{~ p v q), i.e. to Hp => q) in view of the definition of the symbol =>. 18

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75

one obtains the so-called system S5 (the last axiom asserts that if p is possible, then p is necessarily possible). The definition of consistency remains, of course, the same: the set Sk) of consequences of axioms s i (by applying the rules in must be such that for no wff a, it contains both a and ~a. The problem of validity in modal logic is, in a sense, analogous to the problem of validity in propositional calculus. First, one has to define the truth tables for the functors M and L, analogous to the truth tables for the functors &, v , etc. in propositional calculus. Roughly, Ma has the value 1 if there exists a substitution of truth values of propositional variables in a which gives it the value 1, and La has the truth value 1 if a has the truth value 1 under any substitution. These substitutions are referred to as 'worlds', representing all conceivable states of affairs relevant to the formula in question. Thus, a is possible if it is satisfied in at least one world, and it is necessary if it is satisfied in every conceivable world. The validity of a formula is defined by the requirement that it is true in every model, i.e. that it is true in every conceivable world (is necessarily true). Equivalently, a formula a is valid, if there is no model (world) in which its negation ~ a would be satisfied. This enables us to construct an algorithm for testing the validity of a given formula. This algorithm consists of a series of well defined steps aimed at constructing an example of a world in which the formula is not satisfied. If all such steps fail, it means that there is no world in which the negation of the formula in question holds, hence it is satisfied in every conceivable world, and consequently valid. The algorithms differ somewhat for systems T, S4, and S5, the difference lying in the interpretation of the term 'conceivable world'. We shall illustrate the procedure by an example. 20 Consider, for instance, the formula (1)

L(p=>M(q=>r))

=>

M(q=>(Lp=>Mr)).

Now, if there is a world in which (1) is false, then in this world (call it Wj), we must have (2)

L(p=>M(q=>r))

and (3)

~ M(q=>(Lp=>

Mr)).

Next, (2) implies that in the world vvx (4) 20

p => M(q=>r),

See G.E. Hughes and M.J. Cresswell, An Introduction to Modal Logic (Methuen. London, 1968), 84-85.

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while from (3) it follows that (5)

~(q=>(Lp=>Mr)).

Thus, from (5) we deduce that in the world Wj (6)

r).

Thus, the world w1 has the property that the implication q => r is false (as q = 1, r = 0), but M{q=>r) is true.

For this to hold there must exist another world, call it w2, in which q => r is true. Let us now investigate the properties of this hypothetical world w2. Since [by (7)] Mr is false and Lp is true in the world w2 must have the property that r = 0 and p = 1 in it. This, however, contradicts (3). Indeed, if ~ M(q=>(Lp => Mr)), then q => (Lp=>Mr) must be false in every world, in particular in w2. Thus, q = 1 in vv2, which together with r = 0 shows that q => r is false in w2, contrary to the supposition. This shows that formula (1) is valid, as there can be no world in which its negation is satisfied. An alternative approach to the problems of validity in modal logic has been suggested by Hintikka. 21 He introduced the concept of model sets. The intuitive idea is that instead of worlds, one may consider classes of sentences describing these worlds. Formally, these classes of sentences, called model sets, are simply sets of formulas satisfying certain conditions, such as: if a & /? belongs to a model set, so do a and /?; if a is a formula in a model set, then ~ a is not in the set. In analogy with the interpretation of the functor M in the terminology of 'possible worlds', one of the conditions for model sets is that if Met belongs to some model set, then there exists an alternative model set which contains a. Now, a given formula of modal calculus is satisfiable, if it could be a member of some model set in a model system, and it is valid if its negation is not satisfiable, i.e. if it cannot be an element of any model set in any model system. 21

J. Hintikka, "The Modes of Modality", Acta Philosophica Fennica (1963). Modal and Manyvalued Logics, 65-81.

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2.2.3. Logical status of motivational calculus We shall start from the discussion of the problem: to what extent can the set of semantic implications introduced in section 2.1.4 be regarded as a logical system, in the same sense as propositional calculus, or systems of modal logic. Repeating briefly the requirements, to define a system one has to specify the set of primitive symbols together with the set of rules for recognizing well formed formulas (wff's), i.e. 'grammatically correct' strings of symbols. Next, a (finite) set s i of well formed formulas, called axioms, is chosen, and a set 0t of transformation rules, leading from wff's to wff's is specified. The system22 is now identified with the set of all consequences of axioms, the consequences obtained by applying the rules from either to axioms, or to formulas already proved to be in the system. Next, the system is called consistent if it does not contain any pairs of wff's such that one of them is the negation of the other. The above concepts refer to the so-called uninterpreted systems. However, systems are usually built with a specific interpretation in mind, this interpretation determining the class of wff's, called valid wff's. The question then arises to find a set si of axioms and the set 0t of rules, such that the system determined by sf and 0t coincides with the class of valid wff's. Such systems are called complete. In our case, it ought to be clear that the motivational calculus does not satisfy the above requirements for a system, in the sense that the set of primitive symbols and formation rules is not given explicitly. Conceivably, one could take as primitive symbols all variables such as p, q, ..., a,b, ...,P,Q, ..., and also all constants in formulas for semantic implications, namely &, v , (, ), K, K', T, J ' , ..., and also symbols m (appearing in formulas such as m(a, PJ) and the subscript 3. Simple formation rules (or, to be more precise, rules for distinguishing formulas which are well formed) governing the use of parentheses, subscript 3, etc. can be easily formulated. The main difficulty, however, lies in eliminating some strings of consecutive functors: it is even not clear which strings, if any, ought to be eliminated (for instance, is the combination B'Cr = 'I don't believe that I am certain that' to be treated as admissible or not?). Of course, one could accept all combinations of such type as constituting parts of wff's; the sentences with inadmissible combinations would then automatically imply semantically any sentence whatever. Indeed, if A is an inadmissible sentence, so is A & ~B no matter what B, hence A 7 B according to the definition of semantic implication given in section 2.1.2. This would amount to accepting that gibberish implies everything, as in logic, where an implication with a false premise is always true. 22

These are the requirements for the so-called (uninterpreted) formalized systems. There are very few systems which are fully formalized; for instance, most branches of mathematics are not yet fully formalized. Thus, in a somewhat less restrictive usage, the term 'system' is applied to any set of objects (formulas, assertions, laws, theorems, etc) which reveals some type of internal organization.

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Regarding the transformation rules, one can proceed as in the propositional calculus, and accept the Uniform Substitution Rule (utilized in fact for deriving some of the semantic implications in section 2.1.4), and the Rule of Modus Ponens in the form: if a and a."s /? are asserted, then ¡5 may also be asserted. Finally, concerning the axioms, the situation is more delicate. Roughly, two ways of approaching this problem are possible. First, one may ask for a set si of semantic implications which would have the property that all other semantic implications may be derived from those in si. In other words, one may ask for a set of axioms which would generate all semantic implications from section 2.1.4 (as well as some other assertions). Naturally, the meaningful question would be to require the construction to have some 'minimal' property in an appropriate sense, that is, the set of axioms ought to be parsimonius in some sense or other. This approach, then, would follow the standard paths of various branches of mathematics or logic, where at a certain stage of accumulation of knowledge (in the form of specific theorems) one asks for a systematization of the domain, that is, for a set of basic facts and principles which imply all the remaining ones. In this formulation, therefore, we would treat the particular semantic implications from section 2.1.4 as known facts (valid formulas), and ask for a system of axioms which would imply (among others) all of these formulas. There is, however, an alternative approach to the problem of axiomatization, more in agreement with the original idea of treating our implications as tools for making inferences from utterances in natural language. To present this approach formally, we denote by si x the set of semantic implications of section 2.1.4. Consider now a set Sf of sentences uttered by a given person on a given occasion {.¥ may consist of a single sentence, or of a string of sentences constituting a discourse). Denote by C(£f) the set of all consequences of the set S? obtained by applying rule of Modus Ponens with implications in si l. More precisely, the set C(£f) is defined by the conditions (a) (b)

if a is in y, then a is in C(£f); if a. is in C(£f) and a ~ [} is a semantic implication obtained by a substitution from an implication in si u then ft is in C(£/').

Thus, in this approach, one may say that sets form sets of axioms, and semantic implications from si t appear not as axioms, but transformation rules. Thus, the sets C(£f) would play the role of various systems obtained by varying the sets of axioms. In other words, in this approach, by changing the sets Sf one obtains (for a fixed set si t ) a family of systems C(£?). In the sequel we shall be interested mainly in the latter approach, in particular in the consistency problems related to it. The concept of consistency, as explained in the preceding section, was defined for

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systems, i.e. for sets of consequences C(Sof axioms i f . We shall now extend this concept to the considered case of sets of transformation rules. Intuitively, we want to require that the rules in s i 1 do not introduce any inconsistency into the sets C(y), that is, Q{if) may be inconsistent only if if is inconsistent. In other words, we want to define the consistency of transformation rules by requiring that they map consistent sets if into consistent sets C{if). Clearly, to make this definition meaningful, one has to have an independent definition of consistency of sets i f , other than that applicable to sets C(if) (the latter being that C(^) is not to contain any sentence together with its negation). Thus, we shall say that a set if of sentences is consistent, if there exists a model for it, in the sense that it describes a possible state of affairs ('world'). We can now state the requirement of consistency for our set x of semantic implications: we require namely that the mapping if -> C(if) preserve consistency, in the sense that if if is a consistent set of sentences, so is C(if). 2.2.4. Model interpretation of motivational calculus a. Introductory remarks. The model will have the form of a world of hypothetical speakers who use the language to express their thoughts, convictions, beliefs, desires, etc. so that all implications of section 2.1.4 are satisfied. Thus, if A ^ B is any of the semantic implications of section 2.1.4, the model must be such that whenever A is 'true', or 'satisfied', so is B. This necessitates the introduction of a concept of 'truth' of a given sentence for a given person. We shall construct a set-theoretical model of 'subjective truth' of a given sentence for a given person in such a way that for any implication A 7 B of section 2.1.4 and any speaker, if A reflects his subjective truth, so does B. Naturally, as sentences A and B may involve various functors, we shall have to construct specific interpretations of 'subjective truth' for each functor separately. Let m denote the set of speakers, its members to be denoted by the terminal letters of the Latin alphabet, x, y, z,..., and let if denote the class of sentences describing all possible states of affairs of a fragment of reality, and the speaker's relations with this fragment of reality. We imagine a particular moment of time, fixed throughout the considerations. Given a speaker x e and a functor F, let ifr then ~p $ S^r.xi for every person x e yd,x

c

yT\X)

and sentence p, if p e £fDiX, then p e ¿fT\x (that is,

•

A similar interpretation may be given to the semantic implications of section 2.1.4 which involve functors applicable to arguments other than propositional, that is, which are applicable to appropriate verb phrases. The corresponding laws will assert 24

In other words, if sentences Kp and Kg reflect the 'subjective truth' of person x, the same is true for the sentence K(p & q) and conversely.

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81

that the classes of verb phrases considered have some closure properties, or that they are related one to another in some specific way. The problem of constructing a model for motivational calculus may now be stated as the problem of constructing classes of sentences (or verb phrases) satisfying the properties described by semantic implications from section 2.1.4. As all the above rephrasings of semantic implications are preceded by the universal quantifier 'for every person x' it will be sufficient to construct such classes separately for every person x. Accordingly, we shall drop the subscript x from notations in order to simplify them, and restrict consideration to a fixed person x. Also for simplicity, the model will be presented in successive steps, covering groups of functors one at a time. b. Model: epistemic functors. Formally, the model will be described as an ordered set of six objects (6)

(X, P, vT, vB, vCr,

VD>

where X is a fixed nonempty set, to be called the space, P is a probability distribution on X, and vT, vB, vCr, vD are subsets of X satisfying the relation vT c vB n vCr n vD

Fig. 1

(see fig. 1). We interpret letters p, q, ... in formulas of sections 2.1.4.1 and 2.1.4.2 as sets in X. The symbol when applied to symbolsp,q, ... will be interpreted as the set-theoretical complementation v ; when applied to formulas preceded by functors (such as ~Tp, etc), it will be the logical negation. Similarly, the symbol & is interpreted as intersection n of sets, if applied to propositional arguments in formulas for semantic implications; otherwise, it will be interpreted as logical conjunction. Finally, the symbol v when applied to proposition-

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al arguments will be interpreted as set-theoretical union u , and otherwise as logical alternative. The symbol 7 will be interpreted as logical implication. Finally, the subscript 3 is to be neglected, i.e. formulas with this subscript are to be interpreted as analogous formulas without it. The interpretation of symbols K, Cr, T, T, B, B', and D is the following: Ka Cm 7a r'oc Bx B'a Da.

5 = = =

P(a) = 1 a => vCr a => v r a c Vj a vB 5 « c v ^ d f =a c

Finally, K' is to be applicable only to a, ft such that an/? = 0, a vCp>„ n vDiU

and V

T,U n vGiU = 0.

The considerations of the preceding section were devoted to the interpretation of the symbols appearing in the formulas of motivational calculus in terms of the above model (for a fixed person), and to showing that the particular semantic implications of section 2.1.4 are satisfied in the model. The basic idea was to identify the arguments appearing in formulas involving the considered functors as sets in X, and to interpret the symbols &, v , and ~ either as corresponding logical functors or as set-theoretical operations, depending on their place in the formula. This identification has been accomplished in two steps. For functors K, Cr, T, T, B, B', D and also G, the arguments were propositions, and were simply interpreted as subsets of X. For the remaining functors, C, C", W, W', Pf, M, M', O, O', and H', the arguments were interpreted as intersections of sets v(a), v(b),... assigned to objects, and setsP, Q,... assigned to predicates. In all cases, except for the functor G, there was no need for distinguishing between functors in the first and third person singular. Apart from the functors K and K' which were interpreted in terms of a probability distribution on the underlying space X, the remaining functors (except for G, Pf and H') were interpreted by choosing a subset vF for a given functor Fand using one of the following two schemes: (1)

Fa. = a => vF

90 (2)

MOTIVATIONAL CALCULUS Fa d-f

a r).

Also (b)

r ( ~ r =>

p v g ))

and (c)

T ' ( ~ /•=>(/>=> ?))

and so on. For example, consider the utterance / doubt that this scientist is sufficiently creative, and I doubt that he is intelligent enough; however, I don't think that he is unable to write a good paper. In this case p = the scientist in question is sufficiently creative, q = the scientist in question is intelligent enough, r = the scientist in question is unable to write a good paper. Then the sentence in italics may be paraphrased, using (a), (b), and (c), as follows: (a)

I don't think that the fact that the given scientist is not sufficiently creative and not sufficiently intelligent implies that he is unable to write a good paper.

(b)

I don't think that if the given scientist is able to write a good paper, it means necessarily that he is sufficiently creative, or sufficiently intelligent.

(c)

I don't think that the fact that a given scientist is able to write a good paper implies that if he is not sufficiently creative, then he must be sufficiently intelligent.

As another example, consider the implication G[(a, P) & (b, 0 ] 7 { - K3l(a, P) & (b, 0 ] => r 3 [ ( a , P) => ~ (b, 0 ] } . Indeed, by (48) we may write G [(a, P) & (b, 0 ] ; {K3[(a, P) & (b, 0 ] v T3[_(a, P) & (b, 0 ] } . The right hand side may be written as an implication ~ K3[(a, P) & (b, 0 ] => T3 [(a, P) & (b, 0 ] , and, using (23), the conclusion of it may be written as r3{~[(a,JP)&(ft,0]} or

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93

r 3 [>,/>)=> ~ (b, Q)1 As an illustration, consider a sentence uttered by John I am glad that Peter is not seriously ill, and that he will come tomorrow for dinner. Using the law derived above, this sentence implies that if it is not true that John knows that Peter is not seriously ill and will come for dinner tomorrow, then John doesn't think that if Peter is not seriously ill, then he won't come for dinner. 2.5. MOTIVATIONAL INFERENCE RULES

Consider now a sentence in which the speaker justifies or explains why he had made, or intends to make, a certain decision. In most cases such sentences may be paraphrased to a general form iq, because of p\ 'p, therefore I will q\ etc. Let us agree to treat p as a premise and q as a conclusion, and consider the implication pi q. There are then three possibilities. First, the implication p 7 q may be true, i.e. the sentence '/> and not q' may be inadmissible. Second, it may happen that the implication p ^ q is false, but that the implication p & r 7 q is true for some sentence r from a fixed class R of sentences. Finally, it may happen that every implication of the form p & r ^ q is false, regardless of r, provided re lilt seems that the second case is both most common and also most interesting from the theoretical point of view. In what follows, R denotes a fixed class of sentences, possibly depending on the sentences p and q, each of them representing, roughly speaking, the description of a conceivable state of the speaker regarding his knowledge, motives, convictions, beliefs, etc. For simplicity, let us agree to call persons whose utterances of the form lq, because of />' satisfy either the first or the second condition (i.e. either p 7 q holds, or p & r ^ q holds for some r e R) motivationally consistent. Further, let us distinguish the first and the second case by referring to 'explicit' or 'implicit' motivational consistency. The problem may now be formulated as follows: suppose that we observe an utterance which is implicitly motivationally consistent. What can be inferred about the implicit premises (as expressed by the sentence r 'complementing' the premise p) ? A more detailed theoretical analysis will be presented in Chapter 5, where the considerations will include not only the utterances, but also actions; here we shall merely outline the main ideas. The point is that in general, the additional premise r need not be unique: in consequence, if someone utters the sentence 'q, because of p" with the false implication p 1 q but true p&r^q, then in general r cannot be taken as expressing his intentions, unless r is in some sense or other unique. The inference may therefore concern only such a sentence r' which appears as a component in every sentence reR which converts a false implication p 7 q into a true implication p8cr"s q. Formally, let # ( p , q, R) denote the class of all sentences reR such that p &r 7 q

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is true, i.e. such that the utterance p &r & ~q is not admissible. Further, let W'ip, q, R) be the class of all sentences s such that every re, ), (, etc. (and also symbols L and M in case of modal logic). The set V* will consist of all finite strings of symbols of V, and we can distinguish at least two harps in V* which are of interest: one consisting of those strings which are well formed formulae, and the other consisting of all those strings which are not only well formed, but also valid. Actually, in this case we may study also other harps over the same alphabet, by studying various axiomatic systems, such as for instance systems T, SA and of the modal logic. Finally, a special interpretation, to be studied in detail in the next chapters, is obtained by identifying elements of V with actions; then V* consists of all strings of actions (where a string of actions is interpreted as performing one action after another). Various harps in V* which are of interest would be: those consisting of strings which are physically possible to perform, or those consisting of strings which are causing the occurrence of a given outcome at a given moment. 3.1.2. Basic models of mathematical

linguistics

All harps described above have a certain structure; in other words, the strings which belong to a given language reveal regularities which are not, in general, present in strings which do not belong to the language. The 'amount of regularity' of structure in languages may vary considerably: this regularity is very conspicuous and relatively simple in artificial languages, such as languages of logical calculi, somewhat less pronounced in natural languages treated as strings of words, and still less pronounced in natural languages treated as strings of phonemes. The main object of mathematical linguistics is to analyse properties of various harps. Roughly, the models analysed here may be divided into two main classes: the analytic and the generative (synthetic). The synthetic approach is represented most strongly in the U.S., Israel, France, and Czechoslovakia (Chomsky, Ginsburg, Bar-Hillel, Schiitzenberger, Culik). The analytic approach is mostly represented in the U.S.S.R. and Roumania (Gladki, Kulagina, Marcus). The starting point in the study of analytic models is a given harp; the object is to characterize the structural properties of its strings. In the analytic approach one

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BASIC CONCEPTS OF MATHEMATICAL LINGUISTICS

introduces various concepts which may serve as tools for analysing a given harp, such as distributional classes, partitions, etc, and studies relations between various properties of harps, strings, etc., expressed in terms of these concepts. In the second class of models, the generative, one starts from certain methods of defining harps, through their grammars or through their acceptors. A grammar, having the form of a set of rules for constructing strings, defines the harp (or, language) implicitly, as the class of all strings which may be generated by means of the rules of a given grammar. The acceptor is a certain abstract automaton which can 'read' strings of symbols and 'accept' or 'reject' them. The language of a given acceptor is then defined implicitly as the harp consisting of all strings which are accepted by the automaton. As will be explained in the next section, these two concepts, of grammar and of acceptor, are very closely related. The main point is that the more complex the rules of grammar - or the more complex the automaton - the more complex are the regularities of the language defined by it. In effect, by considering grammars (or acceptors) with increasing degrees of complexity, one obtains harps which more and more resemble natural languages. In this way, natural languages may be studied through the analysis of a series of harps, each being a somewhat better approximation of the language under consideration than the preceding one. The analysis of relations between analytic and synthetic models consists of, first, establishing analytic properties of harps defined through a given grammar, and conversely, finding which (if any) grammar, or grammars generate a harp with specific analytic properties. The division into analytic and generative models is not a very sharp one: in fact, there exist models of a mixed, generative-analytic type (like, for example, the categorial grammars). 3.2. SYNTHETIC MODELS 3.2.1.

Acceptors and languages

As already pointed out, acceptors are abstract automata, whose object is to determine whether or not a given string of symbols belongs to a given language. Alternatively, they may be interpreted as generators of languages. Thus, an automaton as considered here is not a technical device, but a mathematical notion, such as, say, function or number. It was originally introduced in connection with the problem of algorithms and decidability. In this section we shall present three main classes of acceptors, starting from the simplest: finite state automata (also called Rabin-Scott automata), push-down automata, and linear bounded automata. As will be seen in the next paragraph, they correspond to three types of grammars, namely the right linear grammars, contextfree grammars, and phrase structure grammars.

BASIC CONCEPTS OF MATHEMATICAL LINGUISTICS

99

3.2.1.1. Finite state automata Finite state automata, or Rabin-Scott automata 5 can be described intuitively as devices consisting of a 'head', or 'control unit', capable of assuming one of a finite number of internal states ('memory configurations'), and capable of reading (or 'scanning') symbols of a given alphabet written on a tape, one symbol at a time being read by the automaton. After reading a symbol, the automaton switches to the next state (depending on the present state and the symbol just read) and begins reading the next symbol to the right of the one just read. In addition, one state is singled out as the initial (or 'start') state, and a subset of the set of all states, its members being called final states, is distinguished. One can imagine that a given string of symbols is written on a tape, the automaton is set in the initial state and begins reading the leftmost symbol. If after completing the reading of the string the automaton is in one of the final states, the string is accepted, otherwise it is rejected.6 The set of all strings accepted is called the language of the automaton. Formally we have DEFINITION

1. By a finite state acceptor we mean a quintuple

S/

= • S, called next state, or one-step transition function.

We interpret M(s, v) as the state to which the control unit switches after it read the symbol ve Vbeing in the state se S. The function M may be extended inductively to a function M*: S x V* S as follows: (a) (b)

M*{s, e) = s for all s e S; M*(s, yv) = M(M*(s, y), v) for s e S, y e V*, veV.

Next, we have

5

See M.O. Rabin and D. Scott, "Finite Automata and their Decision Problems", IBM J. Res. Develop. 3 (1959), 114-125. 6 This definition of acceptance is somewhat different from the one used by Chomsky [see N. Chomsky, "Formal Properties of Grammars", in Handbook of Mathematical Psychology, R.D. Luce, R.R. Bush and E. Galanter, eds. (Wiley, New York, 1963)]. Instead of introducing the concept of a final state, Chomsky considers automata which accept strings by first returning to the initial state.

100

BASIC CONCEPTS OF MATHEMATICAL LINGUISTICS

DEFINITION

2. The string x e V* is said to be accepted by the automaton si if

M*(s0, x) e F and 3. By language of the automaton si (or language generated by si) we mean the set of all strings in V* accepted by si, i.e. DEFINITION

L{si) = {x e V*: M*(s0, x) e F}. Finally, we have 4. A subset L }.

In this case, each letter a in the string may be either erased, or replaced by the string AB, each letter b may be replaced either by AA or by BCD, while each letter c must be replaced by D. Then, for instance, r(baca) will consist of eight strings: AAD, AAABD, AADAB, AAABDAB, BCDABDAB.

BCDD, BCDABD,

BCDDAB,

The main theorem about substitution mapping states that it leads from regular languages to regular languages. More precisely, we have If A is a regular language, andfor every veV the set T(V) is also a regular language, then t(A) is a regular language. THEOREM 4 . 1 3

It may be worthwhile to illustrate the application of the last theorem. Let A and B be two regular languages, and let L^ be the language consisting of two one-letter strings, a and b: L! = {a, b}. Let r(a) = A, T(b) = B. Clearly, the language L1, as a finite language, is regular, and we conclude that the language t(L x ) = A u B is regular. In a similar way, taking L2 = {ab} we obtain the result that the product AB is regular. b. Algebraic characterization of regular languages. We shall now present theorems which give an intrinsic characterization of regular languages (these theorems belong properly to analytic linguistics, and will be discussed again in the next paragraph). Consider the monoid V* and an equivalence relation,14 say R, in V*. The relation R determines a partition of V* into classes of equivalent elements. The number of equivalence classes is called the index of relation R. The equivalence relation R is called right invariant if for all x,y,ze the condition xRy implies xzRyz.

DEFINITION 6.

V*

Thus, right invariance of equivalence of strings means that whenever two strings 13 The theorem was first proved by Y. Bar-Hillel, M. Perles, and E. Shamir, "On Formal Properties of Simple Phrase Structure Grammars", Z. Phonetik. Sprach. Kommunikationsforsch. 14 (1961), 143-172. Reprinted in Chapter 9 of Y. Bar-Hillel, Language and Information (Addison Wesley, Reading, Mass., 1964). 14 A binary relation is called an equivalence relation if it is reflexive (xRx), symmetric (if xRy, then yRx), and transitive (if xRy and yRz, then xRz).

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105

x and y are equivalent, so are the strings obtained by concatenating them (from the right) with any string z. A characterization of the class of regular languages is contained in the following The set A e D* x D*: wtdw2 e A}. Clearly, C{d, A) = C(d\ A) if and only if d ~A d'. The set C(d, A) plays the role of the class of all contexts wi-w1 which allow the action d, in the sense that the resulting string w^dw2 is in A.

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LANGUAGE OF ACTIONS

We may extend the definition of ~ A -equi valence to elements of D*: for M, V e D* we write u ~A v if for any w1,w2eD* we have wluw2 e A, if and only if w1vw2 e A. Similarly, C(u, A) will be defined as the set of all pairs with w1uw2 e A. For simplicity, we use the same symbol ~ A for the equivalence in D and in D*. In the sequel, we shall be concerned mainly with the case of the set D, the considerations for D * being analogous. 2. We say that the action d e D is parasitic with respect to A, or simply: A-parasitic, if there exist no strings wu w2 e D* such that wldw2 e A. DEFINITION

Clearly, we have PROPOSITION 1.

All A-parasitic actionsform a single A-cell.

Let 91 = {A0, Av, ..., Anj be the class of all /4-cells in the partition of D into ~ A equivalent classes. We shall assume that A0 is the cell of all ^4-parasitic actions. As in Chapter 3, we shall denote by 91* the monoid of all finite strings of elements of 91, i.e. all finite sequences of /1-cells. Given a string u e D* we denote by SA(u) the string in 91* defined as the string of ^4-cells corresponding to the consecutive actions of u, the lengths of pauses being neglected. The mapping SA. D* -* 91* is a homomorphism, i.e. for all u, v e D * we have6

The string SA(u) will be called the A-structure of the string u. Given a set B ©*. Let AQ and B*0 be the monoids over the cells of /1-parasitic and 5-parasitic strings (thus, A* consists of strings {e, A0, A0A0, A0A0A0, ... and similarly for B*0). We shall prove THEOREM 1.

For any sets A, B c D*, if SA(B) {z) > np then in view of the properties (a)-(c) of the relation p (see section 4.1), we have z e R ~1 (sj, «,). It follows that if for K = maTii^j^ptj we have K and h1(z)-...-hp(z) = 1, then z belongs to all setsi?~ 1 (s;, n}), and £ _ 1 ( 2 ) # 0 contrary to the assumption. | We adopt the following 1. The string z e L leaves the complete possibility with respect to Q, if h ^ z ) = ... =/i p (z) = 1. DEFINITION

Imagine a housewife who in the morning starts the preparation for a dinner party, to be given in the afternoon. She has to clean the house, polish the silverware, glasses, etc., and also prepare food for dinner. Let us imagine that she has only two alternative dishes for dinner, say A or B, and that these alternatives contain different components which have to be bought. Also, let us imagine that for certain reasons (e.g. financial) the preparation of both alternatives is impossible. Then one of the possible strings z which leaves the complete possibility with respect to A and B is cleaning and other preparations which do not require specific components for the dishes A or B. | Let H ( 2 ) be the set of all strings which leave the complete possibility with respect to Q. EXAMPLE.

Since 0 e H ( 0 ) , the set H ( Q ) is nonempty, and we may define (1) Lib Q = max {q>(z)\z e H(g)}. By Theorem 1 above, we have Lib Q < K = m a x ^ , - ^ whenever the set Q is not strongly attainable. Intuitively, strings z in H ( g ) have the property that whatever time-result in Q one chooses, z may be continued so as to lead to this time-result. The value Lib Q equals the longest possible duration of time during which one can postpone the decision as to the choice of time-results in Q to be attained, without excluding the possibility of attaining any of the time-results. The situation used as an example here will be completely artificial, and its main object will be to illustrate the formal aspects of the concepts introduced. Let us suppose that the set of actions contains three actions, to be denoted by letters a, b, and c. Moreover, let us neglect for simplicity the pauses between actions, and let EXAMPLE.

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147

us assume that performing actions a, b, and c take 3,2, and 1 unit of time, respectively. In other words, cp(a) = 3, (p(b) = 2 and Lib Q, since for the value on the left hand side the maximum is taken over a larger set than for the value on the right. The practical implications of the last formula are quite obvious: to increase the length of time during which one can postpone the decision committing one to, or

LANGUAGE OF ACTIONS

149

excluding any of the time-results, yet retaining the possibility of eventual commitment to any of the time-results, one should make this set of time-results as small as possible. 4.4.2. Decisive moments Generally, by a decisive moment (with respect to a given time-result) we shall mean any moment at which it is possible to originate a string of actions which uniquely determines the occurrence or non-occurrence of this time-result, regardless of the subsequent strings. Depending on the availability of various types of strings, four types of decisive moments will be distinguished. We start from introducing the auxiliary concepts of the sets Elim z (j, n) and Det z (i,ri). 2. We say that ueD* eliminates, given z, the time-result (s,ri), in symbols, u e Elim z (i,ri), if zu e L and

DEFINITION

(a)

R-zHs, n) # 0;

(b)

R-J(s, ri) = 0.

3.8 We say that u e D* determines, given z, the time-result (s, n), in symbols, u e Det 2 (s, n), if zu e L, and

DEFINITION

(a)

there exists v e D* with zv e L, cp(z) + (p(v) > n and zv $ R~i(s, ri);

(b)

given any we D* such that zuw e L and cp(z) + (p(u) + cp(w) > n, we have zuw e R~1(s, ri).

The intuitive interpretation of definitions 2 and 3 seems to cover closely what one should expect from terms 'eliminate' and 'determine'. Indeed, u eliminates (s, ri) given z, if after z the occurrence of (s,ri)is possible, while after zu it is not possible. Similarly, u determines (s, n) given z, if after z it is still possible that (s, n) does not occur, while after zu it must occur, regardless of any subsequent actions. We can now present the definitions of various types of decisive moments 4. The termination of string zeL leads to - a negative decisive moment with respect to (s, ri), if Elim z (i, ri) # 0; - a positive decisive moment with respect to (s, ri), if Det z (s, ri) # 0; - a quasi-decisive moment with respect to (s, ri), if Elimz(s, n) 0 and Det z (i, ri) ^

DEFINITION

# 0; - an ultimate decisive moment with respect to (s, ri), if it leads to a quasi-decisive moment, and moreover, for any ue D* with zu e L, we have either u e Elim z (i, ri) or u e Det z (s, ri). 8 The last condition is simply the spelled out condition zuR(s, ri), or zu e ri). Indeed, note that in this case zuw e ri) may be replaced by zuw p (.?, ri) (as the moment n falls before the end of the string zuw), and then (b) coincides with the definition of the relation R for zu and (s, ri).

150

LANGUAGE OF ACTIONS

Using notations similar to those in the preceding example, let us assume that a fragment of a graph of actions is of the form:

Then for the time-result E, action a leads to a positive decisive moment: if one chooses the continuation a (i.e. one performs ad), then independently of the subsequent actions, the time-result E will occur (possibly combined with some other time-results). Next, the moment of termination of the string ab is a negative decisive moment: if one chooses the continuation abb, then the time-result E is completely eliminated. Termination of the string aba is a quasi-decisive moment: one can choose continuation abaa which is certain to lead to E, continuation abac eliminating E, and continuation abab which can be extended either to E, or to non-is. Finally, termination of the string abab is the ultimate decisive moment: each continuation leads either to E or to non-£ (though, perhaps, not immediately). | The connection between the concept of complete possibility and the concept of the sets Elim 2 (i, n) is contained in 2. The termination of the string z e L leaves the complete possibility with respect to the set Q = {(s^ n^, ..., (sp, np)} of time-results, if and only if for every decomposition z = z{z2...zm we have, for all j = 1, ..., p andk = 1 m THEOREM

(2)

zk £ Elimziz2...zfc_

nj).

If condition (2) does not hold for some j and k, then there is no possibility of attaining the time-result (Sj, ns) after completing z1.. ,zk, hence also after completing z. Conversely, if (2) holds for all j and k, we have in particular, for /c = m PROOF.

z

m $ ElimZl...Zm_, (sj, nj),

hence there exists Vj e D* such that zt...zmVj — zvj e R'1 (Sj, nj). This shows that after performing z one can still attain the time-result (sj, nj). | We shall now give some examples of decisive moments taken from everyday life.

151

LANGUAGE OF ACTIONS

A negative decisive moment for the time-result 'to win a million in next week's lottery' is the last moment at which one can still buy a ticket for the lottery; any action other than buying the ticket eliminates the time-result. Next, a moment preceding pushing the appropriate button in an atomic weapon headquarters is a positive decisive moment for the time-result 'extinction of mankind in the near future'. This example shows that a positive decisive moment does not mean that the time-result is in any sense positive. Quasi-decisive moments are probably most common in everyday life. For the time-result such as 'fried egg at time f, the quasi-decisive moments are all those which precede t by a sufficient number of minutes, provided the egg is still raw in the frying pan; one can fry it, attaining the time-result, break it, eliminating the time-result, or (say), make coffee, leaving the possibility of attaining the time-result or eliminating it. Finally, examples of ultimate decisive moments can be found in histories of wars, where it does sometimes happen that the application of a certain continuation of action (a special manoeuvre) at the proper moment leads to the time-result 'victory', while all other continuations, including application of the same manoeuvre at a later moment, lead to the time-result 'battle lost' (i.e. eliminate the time-result 'victory'). 4.4.3. The concepts of the enforcing situation and the only-exit situation In this section we shall present two closely related concepts, namely that of the enforcing situation and that of the only-exit situation. These concepts are discussed in Chapter V of Kotarbiiiski. 9 The intuitive idea connected with these concepts is the following. We speak of an enforcing situation, when the string of actions performed up to a given moment must be continued in a uniquely determined manner, irrespective of the results which may occur. A man who climbed to the top of a ski-jumping tower and started to slide down is in an enforcing situation: he must jump and land, regardless of the subsequent results. To give an example, in figure 3 the enforcing situation occurs at each of the nodes of the graph, except the last; after each of the strings z, za, zab, and zabc the next action is determined.

d

Fig. 3

9

Traktat of Dobrej Robocie (Treatise on Good Work), in Polish, (Ossolineum, Wroclaw, 1969).

152

LANGUAGE OF ACTIONS

Next, the only-exit situation occurs when the string performed up to a given moment is such that in order to achieve a given time-result, the string must be continued in exactly one way. Thus, it is not an enforcing situation, as, in general, there may be more than one manner of continuing, but only one continuation leads to the desired time-result. To give an example, consider a criminal faced with, say, 10 years of prison, unless he 'squeals on' his accomplices, in which case he will receive only a 3 year sentence. Then, for the time-result 'to be free in 3 years' the only continuation is to betray the accomplices. Other continuations are possible, but do not lead to this time-result. Graphically, the only-exit situation (with respect to the time-result denoted by E) is presented on figure 4: it occurs at each of the interior nodes of the thick line. To define the concept of enforcing situation, we shall introduce the auxiliary concept of unicursal string.

Fig. 4

Intuitively, a string u will be called unicursal, if after originating it there are no other possibilities but continuing it until the end. Formal definition of a unicursal string is somewhat complex: the point is to express the fact that every admissible string, whose beginning coincides with the concatenation of z and a beginning of a unicursal string, must have a continuation, and either the initial part of this continuation coincides with the terminal part of the unicursal string, or else, the whole of the continuation constitutes an initial part of a continuation of the unicursal string. Referring to fig. 3, abed is a unicursal string. It means that after terminating, say, za, all continuations must have the form of either zab, zabc, zabed, or else, zabed, followed by a certain (not necessarily unicursal) string. | Formally, we express it as follows: 5. String u E D* is unicursal given z, if zu e L, and whenever u = xy with 0, then every string in L whose beginning is zx must be of the form either

DEFINITION

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153

zxywx (completing u by performing^, and then continuing Wj), or zxv wherey = vw2 (performing a part of the continuation y of the string u). It follows that we have If u is a unicursal string given z, and u = xy, then if 0, then x is a unicursal string given z; if cp(y) > 0, then y is a unicursal string given zx.

PROPOSITION 1.

Indeed, if in the conditions of the proposition we have (p(x) > 0 and x = x'x" with (p(x') > 0, then the requirements of Definition 5 are satisfied for the decompositions of u into u = xy and u = x'(x"y). Thus, x satisfies the requirements of Definition 5 for an arbitrary decomposition x — x'x" such that cp(x') > 0. This shows that x is a unicursal string given z. If we have u = xy with (p(y) > 0, and y = y'y", with 0, such that x e R ~zl (s, n) implies x = zuv for some v e D*.

DEFINITION

In other words, in an only-exit situation given z with respect to (s, n) there is exactly one way of continuing z (namely u) which may possibly lead to the time-result (s, n). It follows from Definition 7 that we have 2. If the termination of the string z leads to an only-exit situation with respect to the time-result (s, ri), then it leads also to a negative decisive moment, unless it leads to an enforcing situation. PROPOSITION

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LANGUAGE OF ACTIONS

Indeed, if the termination of z leads to an only-exit situation (with respect to (s,«)), and does not lead to an enforcing situation, then (a) there exists u such that 0 and x e R'^is, n) implies x = zuv for some veD*;

(b) there exists u' eL such that zu' e L and strings u and u' have distinct beginnings (as z does not lead to an enforcing situation). Thus, zu'v^R-1 (s, n) for any v e D*, which means that u' e Elim z (j, «); therefore, | the termination of z leads to a negative decisive moment. Clearly, the converse is not true: at a negative decisive moment there are continuations which eliminate completely the time-result (s, «), but it is not necessary that the remaining admissible strings have exactly one common beginning. This is illustrated on the diagram shown in figure 5.

On both figures, the termination of z leads to a negative decisive moment with respect to (s, n). In the left hand diagram it leads also to an only-exit situation, while it is not so on the right hand diagram: there are two continuations, namely a and b, which do not eliminate the time-result (s, n).

4.5. STRUCTURES OF SETS OF TIME-RESULTS 4.5.1. Introductory

remarks

The main objects of interest in section 4.4 were sets R~1(s, n) of strings leading to a given time-result (s, n). Various properties of these sets have been analysed; nonemptiness of sets R -1 (s, n), or their unions or intersections, led to the definitions of various types of attainability; structural properties of strings from these sets, as expressed by the relations ec and Ec served for the definitions of praxiological sets of strings; finally, the property of containing subsets of strings with the same beginning led to the definition of complete possibility and the related concept of decisive moments. In this section, the main interest will be centered on the sets R(u) of time-results occurring as consequences of performing a string u.

LANGUAGE OF ACTIONS

155

Generally, we shall try to connect the properties of strings u with properties of the sets R(u). More precisely, we shall study conditions under which one can make an inference about the set R(u1u2---un) of time-results of performing u1u2...un given the information about the properties of the sets Riu^, R(u2), ..., R(u„). 4.5.2. Sets of time-results R(u) The set R (u) was defined as R(u)

= {(s, n): s e S, n e N, uR(s, «)}•

Consequently, the following relation describes the connection between sets R(u) and R'^s, «): for all u e L, s e S, n e N, we have u e R ~1 (s, n) i f , and only i f , (s, n) 6 R(u). As an illustration of the application of the above relation, we prove the following simple PROPOSITION

1. If u e Prax Q, then Q n zv e L

and

R(zv)

156 (4)

LANGUAGE OF ACTIONS

(S, n)e

n

1)6 D* (u), then for every s e Horn w, the result s is transient with respect to u.

PROPOSITION

Similarly, we have 6. If min Tr u > cp (u), then for every s e Horn u the result s is delayed with respect to u.

PROPOSITION

Thus, if max Tr u < cp (u), then all outcomes of u are transient, while if min Tr u > >

min N2 and the number c is a multiple of the common period T. In situation (a), the carrier of the sum N t u N 2 is N l , the period equals the value of c, and the number of repetitions for the set u N2 is 2. This situation is depicted on the figure below. N,

N! X XX

X XX

X XX

X XX

N'[

X XX

X XX

N Nx u N2 of type (c, 2) Fig. 10

In situation (b), the carrier for the sum Nt u N2 is the period remains equal to T, and the number of repetitions depends on the numbers qu q2, and c. This situation is depicted in fig. 11.

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LANGUAGE OF ACTIONS

T N, XX X

XX X

N2 XX X

XX X

N'l

XX X

N'i ^

c = 2T

Nt of type (T, 4) N2 of type (T, 3) u N2 of type (T, 5)

Fig. 11

The proof of Theorem 5 is the following. Consider first case (a). By assumption, Nt and N2 are periodic of the same type (T, q), hence Nl

= X)

N2 =

i =0

9

{j

TjtN*2.

j=o

Since N2 — rCN'{, we can write N2 = \ j j=

0

=

l / VjtNI ^ =

^

*JT W )

=

tj ^T^i') = j=0

Therefore Ni u N2 = Nj, u tcAT2, hence A^ u JV2 is periodic with type (c, 2), the carrier being Nt. The first assumption (a) guarantees that max N, < min N2 — min xcNv Let us now consider case (b). Suppose that type (T, q'). Then Ni = \ j TjtN';, N2="[J j=0 J=0

is of the type (T, q) and N2 is of the

tjTN2.

Since N2 = tcN'{, we can write N2 =

; =0

VjtWi)

=

U1 j=0

Since in case (b) we have c = mT, we can write N

2 = '(J Z(.j + m)T^ij= 0

+

170

LANGUAGE OF ACTIONS

If max JVj > min N2 (assumption (b)), then one of the sets •zu+m)TN'l sum is a subset of A^. Thus, Nt u N2 =

in the last

U TjTN'i, j= o

for some h < q' + q, which was to be shown. | Theorem 5 gives conditions for periodicity of the sum of two periodic sets of the same period, whose carriers are identical, in the sense that one of them is a translation of the other (we may say that they are congruent). The last condition is not necessary, as may be seen from the following example: let Nt = {1,3; 11, 13; 21, 23} (N'l = {1, 3}, T! = 10, qt = 3) and N2 = {2, 5; 12, 15; 22, 25} (N% = {2, 5}, T2 = 10, q2 = 3). Then the set Nt u N2 = {1, 2, 3, 5; 11, 12, 13, 15; 21, 22, 23, 25} is a periodic set with carrier N'[ u = h = q i = 3). In general, we have

= {1, 2, 3, 5}, T = Tl = T2 = 10, and q =

If the periodic sets Nl and N2 with carriers N'{ and N2 are of the same type (T, q), and THEOREM 6 .

max (NL u N%) < min ( T t N I u

ItN'2\

then the sum Nt u N2 is periodic, with carrier N'[ u N2 and of the type (T, q) (i.e. both the period and the number of repetitions are the same for each component). Theorem 6 is illustrated by the example above. To prove Theorem 6 we may proceed as follows. Since N^ and N2 are of the same type (7", q), we may write NtuN2= =

U1 j= 0

"(j TjtN'1 u j= 0

'J j=0

=

V ^jtNI j=0

U TjtN^)

=

u N'O,

where the last equation follows from the fact that the sum of translations of two sets equals the translation of the sum of these sets. Thus, the set N1 u N2 is periodic, of type (T, q), and its carrier is u N2, since the condition imposed on the carrier and its translation in the definition of periodicity is guaranteed by the assumption of the theorem. I

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171

In Theorems 5 and 6 the period T was common for both sets and N2. This condition, however, is not necessary, as can be seen from the following example. Let Nt = {5, 6; 10, 11; 15, 16; 20, 21; 25, 26; 30, 31} (carrier N'{ = {5, 6}, period Ti = 5 , q i = 6), and N2 = {4, 6, 7; 14, 16, 17; 24, 26, 27} (carrier N2 = {4,6,7}, T2 = 10, q2 = 3). Then the sum N

1

U N

2

=

{4, 5 , 6 , 7 , 10, 1 1 ; 14, 15, 16, 17, 2 0 , 2 1 ; 2 4 , 2 5 , 2 6 , 2 7 , 3 0 , 3 1 }

is a periodic set with carrier N* = {4, 5, 6, 7, 10, 11} = Nl u

u x5N'[

period T = T2 = 10 and q = q2 = 3. This situation is covered by THEOREM 7. If N1 and N2 are periodic of types ( 7 \ , ) and (T2, q2) with carriers N[ and N2 such that (1) the period T2 is an integer multiple of the period Tu i.e. T2 = kTx for some k; (2) the number qx equals kq2; (3) the set

(10)

N* = N2 u N'{ u xTlN'[ u x2TiN'l

u ... u r ( t _

satisfies the condition max N* < min xT2N* then the sum Nt u N2 is periodic. The carrier of N^ u N2 is the set N* defined by (10), and both the period and the number of repetitions are the same as for the set with the greater period, i.e. they are equal to T2 and q2 respectively. Thus, under the conditions of Theorem 7, the carrier of the union of two sets is equal to the carrier N2 of the set with greater period, plus the translations of the carrier N'l of the set with smaller period by 0, 1, ..., k— 1 multiples of To prove Theorem 7 one has to show that 'U1

j= 0

*JT2N*

= N I V

N2.

The left hand side, after substituting the expression for N* equals I)1

*JT2N*

j= 0 =

2-1 j4 -U 0

=

?

U

j = 0 U jqi-l = 0ft—rU=1 0

^

u

*U

r=

VRRM) 0

i j t A ^ N ' O .

=

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LANGUAGE OF ACTIONS

The first of these sums equals, by definition, N2. In the second sum, after substituting T2 = kTx one obtains Î2-1 k- 1 u u w t r r ^ r ) j =0 r =0

=

«2-1 t - i u u vuk+mM. j =0 r= 0

Using the identity from the proof of Theorem 5, the last sum equals Î2*-1 U s= 0 since q2k =

=

Nu

Thus, the proof is complete.

J

4.5.6. Periodicity of time-traces for concatenations of strings of actions In section 4.5.5 we discussed the concept of periodicity of time-traces, and we characterized two classes of situations: one class comprised situations when periodicity was desirable, and the other when it was undesirable. The methods leading to lack of periodicity consist, as already mentioned, of randomization. It is intuitively obvious that the methods of attaining periodicity should consist of concatenating a given string with itself, i.e. repeating this string. Under some conditions, to be stated below, the time trace of a given result s of the string u...u (repeated k times) should equal the union of k repetitions of the time trace of s of a single string u, translated by the duration q> (u). The conditions mentioned above consist of requiring, first, that the time-results of the string u performed after if = u...u (r times) should be the same as those of the string u alone, except the delay of the length (p(t/). The second condition, connected with the definition of periodicity, should ensure that the successive time-traces of the repetitions of u do not overlap. We shall separate the first condition as a definition of the concept of ¿-independence of two strings. 11. Let u, v, and uv be in L, and let s e S. We say that v is s-independent of u if the condition vR(s, n) implies uvR(s, n + (p(u)).

DEFINITION

We may now formulate Let u,u2,u3, ..., w* e L and s e S (here u' = u...u ( j times)). Assume that dim Tr H < (u), and number of repetitions k. THEOREM 8. (s)

PROOF.

We have Tr ( s ) « 2 = Tr (s) w u r„ (u) Tr (s) M

LANGUAGE OF ACTIONS

173

in view of the assumed «-independence. Similarly, Tr ( s ) w 3 = Tr ( s ) w 2 u T29WTTwU

= Trw u T„ (a) Tr (s >H u T 2 „ (u) Tr (s >m

and so on. Let k = maxTr (s) M and r — min Tr(s)w. Then k—r = dim Tr(s)w < cp(u), that is, k < r + cp(u). But the right hand side of the last inequality equals min TV(U)TT(S)U, which shows that max Tr (s) H < min T 9(a) Tr (s) «. Thus, the successive copies of the set Tr(s)w do not overlap, which completes the proof. It is worth remarking that the condition dim Tr(s)w < q>(u) is essential, as can be seen from the following example. Let 5 . Then, under the assumptions of the theorem Tr ( s ) « 3 = {2, 8} u {2 + 5, 8 + 5} u {2 + 10,8 + 10} = {2,7, 8, 12, 13, 18} and the last set is not periodic. Indeed, as a 6-element set, it can be divided either into 3 two-element sets, or into 2 three-element sets. However, {12, 13, 18} is not a translation of {2,7, 8}, nor is {8,12} a translation of {2, 7}. 4.5.7. Synonymity of strings u and v In natural languages, by synonymous expressions one means expressions whose meanings overlap, at least partially. According to the linguistic interpretation accepted here, strings of actions can be treated as expressions in a certain artificial language, and their results (sets of timeresults) as the meanings of these expressions. Thus, the meanings of expressions u and v are the sets R(u) andi?(v) of time-results. We can therefore say about two strings of actions that they are synonymous if their meanings overlap, at least partially, in a certain manner (specific for a given goal of actions). In other words, two strings of actions are synonymous, if for a given goal it is irrelevant which of these strings is performed (thus, they can be replaced by one another). Since in our interpretation the scopes of meanings of u and v are the sets R(u) and R(v), hence sets of a particular type (subsets of the Cartesian product S x N), we can interpret the concept of synonymity in different ways, by studying mutual relations between the sets R(u) andi?(v). To describe the positions of sets R(u) and R(v), we shall use the sets Horn and Tr introduced in the preceding sections. First of all, when the temporal aspects constitute only a secondary object of interest, and attention is concentrated on sets of consequences of u and v, one can study synonymity by analysing the sets Horn u and Horn v of outcomes caused by strings u and v. Thus, the intersection

174

LANGUAGE OP ACTIONS

Horn u n Horn v equals the set of those outcomes which occur when both u and v are performed. This intersection equals therefore that set of results, with respect to which u and v can be replaced by one another (provided, as already mentioned, that temporal aspects are neglected). For instance, suppose that it is known that two types of cure may be applied to a certain illness - say, a series of antibiotic injections (string u) or a turpentine rub (string v). Both these strings lead to the outcome'return to health', though not equally quickly. This result belongs therefore to Horn u and to Horn v, hence to the intersection Horn u n Horn v. From the point of the outcome 'return to health' both strings u and v are synonymous (one can be replaced by the other). Clearly, sets Horn u and Horn v are different (say, u may cause also outcomes such as avitaminosis, allergy, etc., which may be absent if string v is applied). The last example shows that to evaluate the synonymity of two strings u and v, it is important to consider not only the intersection Horn u n Horn v, but also the differences Horn«—Horn v and Horn v—Horn u. The meaning of these differences is illustrated in the figure below (for simplicity it is assumed that the sets S of outcomes and N of time-moments are continuous).

Horn M—Horn v Horn wnHom v Horn v—Horn u Fig. 12

The first difference, namely Horn u—Horn v, comprises those results which occur if u is performed, but do not occur if v is performed, and similarly for the second difference. Thus, for instance, the result 'avitaminosis' belongs to Horn u—Horn v, while results such as 'itching and burning on the back' belong to Horn v—Horn u. Similarly, if one is interested in results occurring at a particular time n, then for the description of synonymity one can use the sets Hom(B)w n Hom (n) v, Hom(n)M— Hom (B) v, and Hom (n) v—Hom (B) w. This is illustrated on the figure below. In this figure, the sets Hom (n) w, Hom (n) v, their intersections and differences, are marked on the axis S. It ought to be mentioned that for the time-moment n' we have Horn'" '« n Hom ( n )v = 0, hence each of the differences equals the first of its terms.

175

LANGUAGE OF ACTIONS S

Hom(n)M—Hom(n)v (n)

(n)

(n)

(n)

Hom MnHom v

Horn ( n K •Horn ^

v

Horn v—Horn M n'

n

n"

•N

Fig. 13

On the other hand, for the time-moment n" we have Horn'" "'« c Hom (n ")v, hence Horn'""1!/ n Horn'""'» = Horn'""'«, and H o m ^ ' ^ - H o m * " " ^ = 0 (there are no results in the first of these two sets which are not in the second). If our main object of interest is not only the outcome s, but also the moments of its occurrence, then for the description of synonymity of strings u and v, one may use the sets Tr(s)w and Tr (s) v. For instance, in the example considered above, if s denotes the outcome 'to become healthy', then for string u (antibiotics) this outcome occurs starting from a certain moment w0, i.e. the time-results (s, n0), (s,n0 + 1), ... occur. For string v we have a similar sequence of time-results, but originating at a later moment n$ > n 0 . Thus, Tr(s)w = { « : « > n 0 }, while Tr (s) v = { « : « > n^}. In this case Tr (s) v v-Tr (s) m (CD) Fig. 14

176

LANGUAGE OF ACTIONS

Finally, in view of the fact that the sets Tr (s) w and Tr ( s ) v are sets of integers, one can consider one more type of synonymity of strings u and v (with respect to the outcome s). Intuitively, we would like to describe the situation when u and v have identical time-traces of the outcome s except for a translation. As an example, imagine that seeing a certain film causes some emotions (e.g. fear) at well defined moments from the beginning of the film. Then, from the point of view of the outcome s = 'to experience fear during the film' the string u = 'go to the 3 P.M. program' and v = 'have dinner and go to the 8 P.M. program' give time-traces of the outcome s translated one with respect to the other by 5 hours. These time-traces are disjoint, but they may be treated as identical in some respect, and - as a consequence - they may be treated as resulting from synonymous strings u and v. Generally, in this case the sets Tr (s) w and Tr (s) v satisfy the condition Tr(s)t< = x c Tr (s) v for some c. In this type of synonymity, the main object of interest is the value of the constant c. 4.5.8. Suppressing actions Most of the considerations of the preceding sections were expressed in terms of the relation R. In introducing this relation, we tried to express (in terms of the underlying relation p) the fact that the occurrence of s at the moment n falling after string u is terminated, i.e. such that n > cp (u), may be attributed to string u. One of the possible ways of doing that, utilized until now, consists of introducing the relation R : the timeresult (s, n) is due to u (that is, uR(s, «)) if (s, n) occurs regardless of the string performed after u. This concept at times appears to be somewhat too strong, ruling out many cases in which one would intuitively agree to attribute the time-result (s, n) to performing the string u, even though u and (s, n) are not in the relationR. For instance, imagine that a given bridge is to be blown up; the dynamite charges are attached to the pillars, and connected with the fuse, whose burning time is, say, 2 minutes. Then the action 'lighting the fuse' performed at time t may be regarded as causing the time-result 'bridge destroyed at time t + 2 minutes'. Imagine, however, that after lighting the fuse it is possible to extinguish it, or cut it at a place yet unburnt. Then the above action and time-result are, technically speaking, not in the relation R. These considerations justify the introduction of the following definition: 12. Given a string ue L and time-result (s, n), we write up(s, n) if (1) n < (u). Denote by M(u, n) the set of all strings v such that q> (v) > >n — (p(u) and uv e L. For a fixed seS, we may then divide the set M(u, n) into two sets, M+s (u, n) and M~ (u, n), where v e M+s (u, n) if uvp(s, n) and v e M~ (u, n) otherwise. If up(s, n) and v e M~ (u, n), we say that v suppresses the time-result (s, n). We have then PROPOSITION 10. Let ueL

and (s, n) be such that n > (u) is in M+s (u, n), while uR (s, n) i f , and only i f , M(u, n) = = M+ (h, n), i.e. ifM~ (u, n) = 0.

In the example of blowing up the bridge, if u consists of lighting the fuse, and (s, n) is 'bridge destroyed 2 minutes after the termination of u\ the set M+s (u, n) will consist of strings such as 'waiting for 2 minutes', 'run for cover and then wait', etc., while the suppressing strings (in M~(u, n)) would be such as, for instance, 'cut the fuse so as to separate the burning part from the rest', etc. Generally, the relative sizes of the sets M+(u, n) and M~(u, n) may serve as a measure of how strongly performing u determines the occurrence of (s, n). Indeed, if the set M~(u, n) is small compared with M*(u, n), then only a few strings v are such that if performed after u, they prevent the occurrence of (s, n). The relation R may be regarded as the extreme case, when the set M~ (u, n) of suppressing actions is empty. 4.6. UPRIGHTNESS OF OUTCOMES AND UPRIGHTNESS OF ACTIONS

4.6.1. Introductory remarks In the considerations of the preceding sections, we dealt either with structures of strings leading to a given time-result or set of time-results, or with the structure of the set of time-results of a given string. In terms of the introduced system of concepts, all outcomes of the set S were treated identically: there was no way of distinguishing 'bad' and 'good' results, or 'ethical' and 'unethical' ones. Thus, the distinguishing of strings was also relative with respect to a given outcome s, or set of outcomes Q, and we could only distinguish among efficient strings those which are more or less economical with respect to the given result, and longer or shorter in duration. At present we shall enrich the system of primitive notions (D, L, (p, S, p} by one more concept; namely we shall introduce a binary relation Eth in the set S x S, i.e. a relation which holds between pairs of outcomes from S.

178

LANGUAGE OF ACTIONS

The relation Eth will be called the ethical dominance. The symbol SjEth s2 will be interpreted as denoting the fact that the outcome is ethically no better than the outcome s2 (or: outcome s2 is ethically no worse than the outcome i j ) . Relation Eth is an additional primitive notion in our system. The justification for introducing such an additional notion lies in the fact of the existence of such a relation in (at least) some sets of results: this can be observed in the human ability to discriminate pairs of outcomes with respect to their ethical value. This ability is due to social learning of the valuation of outcomes in ethical categories, dictated either by legal or religious systems, or by certain sub-cultural patterns. It ought to be clear that some individual differences may exist between the degree to which the norms are internalized, and in the sources of these norms. In other words, a person may internalize more or fewer norms, more or less deeply, and the norms arising from distinct sources may be internalized in various proportions. Due to the complex structure of systems of ethical norms, the inter-individual consistency of ethical valuations of outcomes (or actions) may differ for different outcomes; in some cases, however, one can speak of practically complete consistency, at least for some sub-cultures (e.g. 'manslaughter' is always ethically lower than 'theft', except perhaps in some gangs). Regarding other outcomes, consistency may not be so complete; this will occur either if the outcomes differ little with respect to their ethical value, or when they concern distinct domains of behaviour. As an example of a pair difficult to compare because of a small difference, consider 'opening someone's love letter' and 'opening someone's business letter'. As an example of difficulties in comparison because of distinct domains of behaviour, consider 'writing poison-pen letters' and 'torturing animals'. It should be stressed, that still more pronounced difficulties are encountered when the outcomes (or actions) are ethically positive. This is clear, because all ethical systems teach, first of all, how to differentiate among socially bad actions. For a given set of outcomes S, the empirical determination of the relation Eth consists of indicating all pairs j 1 ( s2 for which jjEth s2, i.e. for which is ethically no better than s2• A given pair, s1,s2 can be treated as being in the relation ^ E t h s2 if a given person, the object of interest, or a person accepted as a referee, claims that s t Eth s 2 . One can also interpret SjEth s2 as an opinion of the majority of a given group. 4.6.2. Properties of the binary relation Eth In the sequel, it will be assumed that Eth is a binary relation in S, satisfying the following conditions: ASSUMPTION 1. Relation Eth is transitive-, for any su s2, s3 in S, if s1Eths2 j 2 E t h i 3 , then j x Eths 3 .

2. Relation Eth is connected: for every s1,s2inS both conditions s j Eth s2 ands2Eth s1 hold.

ASSUMPTION

and

at least one and perhaps

179

LANGUAGE OF ACTIONS

Thus, assumptions 1 and 2 guarantee that every two outcomes from set S can be compared in the sense of the relation Eth (Assumption 2), and that these comparisons are consistent (Assumption 1). Relation Eth determines a certain ordering in the set of outcomes S. To describe this ordering, it is necessary to introduce two concepts related to Eth, namely the concepts of ethical equivalence and ethical dominance. 1. Outcomes su s2 e S are ethically equivalent, to be written ^ i f i i E t h j j and j j E t h s j . DEFINITION

S Eth

s2,

We have Under assumptions 1 and 2, the relation of ethical equivalence = Eth is an equivalence in S, i.e. it is reflexive, symmetric, and transitive.

THEOREM 1.

Thus, the set S splits into disjoint classes of ethically equivalent elements. Next, we define the relation of ethical dominance as follows: 2. The outcome sl e S is ethically worse than the outcome s2 e S (i.e. s2 ethically dominates s to be written as DEFINITION

•^l

Eth S2

if i t E t h s2 but not s 2 Eth st. From Definition 2 it follows that we have 2. The relation -3Eth °f ethical dominance is antireflexive, antisymmetric, and transitive, i.e. for alls1,s2, s3e Swe have

THEOREM

not

-3Eth ¿i;

ifsi ~~3Eth s2, then not s2 -3Eth s^, if* i Eth s2 and s2 -3Eth i3, then st -3Eth

i3.

Thus, properties of the relation Eth described in Assumptions 1 and 2 lead to an ordering of outcomes from set S, or more precisely, to a linear ordering of sets of equivalent outcomes. Thus, one can define a numerical function E(s) on the set S, such that -EXsjl ) < E(s2) if, and only if SjEth s2. Such a function constitutes, therefore, a numerical representation of the ethical values of outcomes, i.e. it constitutes a scale of ethical values, as defined by Stevens.12 Unfortunately, without additional assumptions (besides 1 and 2), such a scale is only of an ordinal type. This means that the knowledge of the values ¿s(si) and E(s2) for outcomes and s2 allows us to infer 12

S.S. Stevens, "Measurement, Psychophysics and Utility", In Measurement: Theories", C.W. Churchman and F. Ratoosh, eds. (Wiley, New York, 1959).

Definitions

and

180

LANGUAGE OF ACTIONS

only which of these outcomes is ethically worse. One cannot, however, draw any conclusions about the ethical values of outcomes from the results of arithmetical operations performed on the numerical values E(s). For instance, from the fact that E(s2)-E(si) < E(sA) — E(s3) one cannot infer that the difference (whatever this term may mean as applied to pairs of outcomes) between s2 and s1 is less than that between s 4 and s3. This is related to the fact that assumptions 1 and 2 are very weak, and allow a large degree of arbitrariness in the choice of the function E(s): it is only required that the condition £ ( i i ) < E(s2) should hold if, and only if, SjEth s2, otherwise the values of E(s) may be arbitrary. In order for a scale of a higher type to exist, it is necessary to have a relation different from the binary relation Eth. To gain an insight into what type of relation is necessary, let us assume that ethical values can be measured on, say, an interval scale. In this case it would be admissible to infer about ethical values on the basis of comparison of the differences E(s2)—E(s1) and E(s4)—E(s3) for two pairs of outcomes, i s 2 and s3, sA. In other words, for each four outcomes sit s2, s3, s 4 in S it would be meaningful to ask: 'is the difference in ethical sense between s2 and less or equal than the difference between sA and s3 ?'. The meaningfulness of questions of this type, and a sufficiently high degree of consistency in answering, implies the existence of a quarternary relation holding between pairs of pairs of outcomes (sj, s2) and (s3, s4). This relation is simply the set of all quadruplets s l 5 s2, s3, s4 of outcomes from S for which the answer to the above question is 'yes'. Similarly as in the case of the relation Eth, one could interpret it either as the answer of a given person, or as an opinion of the majority in a given group. It seems that for some quadruplets of outcomes from the set S (as for pairs of outcomes discussed above), human intuitions are so strong, and also consistent, that one is justified in speaking of such a relation. For instance, for the outcomes = steal $ 1000 s2 = eat lunch (ethically neutral) j 3 = steal $ 10 ¿ 4 = murder a man for robbery the majority of people would probably agree that the difference between eating lunch and stealing $ 1000 is less than the difference between stealing $ 10 and murder. Intuitively, murder is highly negative ethically, theft is negative, too, but not so much, and stealing $ 1000 is somewhat more unethical 13 than stealing 'only' $ 10. It should be stressed that the empirical determination of the above quarternary

13

This is an obvious simplification, as the 'evils' are considered out of context. In fact, one should consider all deeds together with the situations in which they occur. Thus, for instance, stealing $ 10 may be 'better' or 'worse' than stealing $ 100, depending on why it was stolen, and how rich was the person from whom it was stolen.

LANGUAGE OF ACTIONS

181

relation is much more complicated than the determination of the binary relation Eth. For instance, for the quadruple ix s2 s3 s4

= kill a man subjecting him first to sadistic torture — achieve the same as in but without torture = embezzle some money — tear the wings off flies

one could probably claim that SjEth s2, i 2 Eth j 3 , and i 3 Eth s 4 (hence also j t Eth s3, SiEth sA and s 2 Eth i 4 ) . Thus, st is ethically no better (or even strictly worse) than s2, which is worse than s3, which is worse than s 4 . But comparing the differences between, say, s3 and with the difference between s4 and s2 is much more difficult, and one can expect inconsistencies in opinions, or even difficulties in expressing them. Is the difference (quite clear) between embezzlement and murder with sadistic tortures less or more than the (equally clear) difference between tearing off the wings of flies and 'ordinary' murder? As before, the difficulties increase when we consider quadruplets of outcomes consisting both of ethically positive and ethically negative outcomes. For instance, if sy and s2 are as before, murder with and without torture, and s3 = helping an old woman in crossing the street; s'4 = risking one's own life to save someone from a fire, then obviously .^Eth s2, i^Eth s3> and i 3 Eth and the differences between s 3 and slt and between and s2 become even clearer, On the other hand, comparison of these differences becomes more difficult. It appears, however, that for a fairly homogeneous set of outcomes, one can assume the existence of such a quarternary relation. This argument is supported by the findings of Ekman, 14 who discovered empirically that people evaluate the seriousness of crimes as if a ratio scale existed for it. 4.6.3. Properties of the quarternary relation ETH Independently of the difficulties in the empirical determination of the above discussed quarternary relation, one can formulate conditions for such a relation implying the existence of an interval scale, allowing comparisons between two sets of outcomes by means of the averages of their ethical values. One can find in the literature several sets of axioms for such quaternary relations implying the existence of an interval scale. For the purpose of the present considerations, the most suitable seems to be the system presented by Suppes and Zinnes 14

G. Ekman, "Moral Judgments", Perceptual and Motor Skill 15 (1962), 3-9. See also C.H. Coombs, R.M. Dawes, and A. Tversky, Mathematical Psychology (Prentice Hall, Englewood Cliffs, N.J., 1970), 45-59 for the results of the repetition of Thurstone's experiment on scaling the seriousness of offenses.

182

LANGUAGE OF ACTIONS

concerning finite sets.15 We shall present this system, using notations adapted for the present purpose. Thus, assume that S' l + ( - 5 > 2 + 1 0 - 1 + 50-1 n £ (u, E) = = 9 1 + 2 + 1 + 1 as the numbers of elements in the sets Tr (Sl) w, Tr(S2'w, Tr (Se) u, and Tr(Ss)M are 1, 2,1, and 1, respectively. We can now define the relation of ethical dominance of actions: 10. The string ue Lis ethically worse, in the average sense, than the string v e L, in symbols u -3eYH v> if

DEFINITION

$(u, E*)