The Posing of Questions: Logical Foundations of Erotetic Inferences [1 ed.] 978-90-481-4595-9, 978-94-015-8406-7

Table of contents :
Front Matter....Pages i-xiv
Erotetic Inferences and How Questions Arise....Pages 1-33
Logical Theories of Questions....Pages 34-69
Questions....Pages 70-101
Semantics....Pages 102-126
Evocation of Questions....Pages 127-154
Generation of Questions....Pages 155-170
Erotetic Implication....Pages 171-208
Erotetic Arguments....Pages 209-230
Back Matter....Pages 231-250

Citation preview

TIlE POSING OF QUESTIONS

SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHll..OSOPHY OF SCIENCE

Managing Editor: JAAKKO HINfIKKA, Boston University

Editors: DIRK VAN DALEN, University of Utrecht, TheNetherlands DONALD DAVIDSON, University ofCalifornia, Berkeley TIiEO A.F. KUIPERS, University ofGroningen, TheNetherlands PATRICK SUPPES, Stanford University, California JAN WOLENSKI, Jagiellonian University, Krakow, Poland

VOLUME 252

ANDRZEJ WISNIEWSKI Adam Mickiewicz University, Poznan, Poland

THE POSING

OF QUESTIONS Logical Foundations of Erotetic Inferences

• Springer-Science+Business Media, B.V

Library of Congress Cataloging-in-Publication Data Wisniewski . Andrze j. The posing of questions : logical foundat ions of erotetic inferences I by Andrzej Wisniewski. p. cm . -- (Synt hes es library ; v. 252) Inc I udes bib l t ograph i ca I references and i ndex es . 1. Question (Logic ) I. Title . II. Series. BC199.Q4W57 1995 160--dc20 95-31035 CIP

ISBN 978-90-481-4595-9 ISBN 978-94-015-8406-7 (eBook) DOI 10.1007/978-94-015-8406-7

Printedon acid-free paper

All Rights Reserved

© 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995. Softcover reprint of the hardcover 1st edition 1995

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TO MY DAUGHTER

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. EROTETIC INFERENCES AND HOW QUESTIONS ARISE 1.1. Erotetic inferences . . . . . . . . . . . . . . . . . . . . . . . 1.2. The arising of questions from sets of declarative sentences 1.3. Evocation and generation. Requirements of adequacy 1.4. The arising of questions from questions and sets of declarative sentences 1.5. Erotetic implication. Requirements of adequacy . . . . . 1.5. Historical notes. . . . . . . . . . . . . . . . . . . . . . . . . 2. LOGICAL THEORIES OF QUESTIONS . . . . . . . . . . . 2.1. Basic terminology and notation 2.1 .1. Terms and declarative formulas 2.1.2. Metalanguage. Notation . . . . . . . . . . . . . 2.2. Logical theories of questions and answers . . . . . . . 2.2.1 . Questions: reductionism vs. non-reductionism 2.2.1.1. Radical reductionism . . . . . . . . . 2.2 .1.2 . Moderate reductionism. . . . . . . . 2.2.1.3. Non-reductionism 2.2 .2. Answers . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Harrah's analysis 2.2.3 .1. Questions 2.2.3 .2. Direct answers. . . . . . . . . . . . . 2.2.4 . Aqvist's analysis 2.2.4.1. Interrogatives 2.2.4.2. Direct answers . . . . . . . . . . . . . 2.2.5 . Hintikka's analysis 2.2 .5.1. Questions 2.2.5.2. Conclusive answers 2.2.6. Kubinski's analysis 2.2.6.1 . Simple numerical questions . . . . .

xi

. . . . 1 . . . . 1 3 12 14 . .. 25 . . . 26

. . . .. 34 . . . . . 34 34 . . . . . 35 . . . .. 37 . . . . 37 . . . .. 37 . . . .. 38 40 . . . . . 42 43 43 . . . .. 45 45 45 . . . . . 48 49 49 51 52 . . . . . 53

viii

CONTENTS

2.2 .6.2. Compound numerical questions 2.2.6.3. Propositional questions . . . . . . . . . . . . . 2.2.7. Belnap's analysis 2.2.7.1. Interrogatives 2.2.7.2. Direct answers "

56 59 62 62 67

3. QUESTIONS 70 3.1. Questions and direct answers 71 3.1.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . " 71 3.1.2 . Questions of the first kind . . . . . . . . . . . . . . . . . 72 3.1.3 . Questions of the second kind " 75 3.1.4 . Questions of the third kind 80 3.1. 5. Categoreally qualified questions 84 3.2. Questions of formalized languages and natural-language questions 86 3.2.1. Yes-no questions 87 3.2.2. Disjunctive questions and conjunctive questions 91 3.2.3 . Wh-questions 94 3.2.4 . The problem of completeness . . . . . . . . . . . . . .. 97 3.3 . General assumptions. The languages ;;e" and ;;e"" 99 100 3.3.1. The languages ;;e" and ;;e"" 4. SEMANTICS. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 . Basic concepts 4 .1.1. Interpretations, satisfaction and truth . . . . 4.1.2 . Normal interpretations and consistency . . . 4.2 . Entailment and multiple-conclusion entailment 4.2.1 . Entailment 4.2 .2. Multiple-conclusion entailment . . . . . . . . 4.3 . Erotetic concepts . . . . . . . . . . . . . . . . . . . . . 4 .3.1. Soundness, safety and riskiness 4.3.2. Just-complete answers and partial answers 4.3.3 . Presuppositions of questions 4.3.4 . Relative soundness. Normal questions and regular questions . . . . . . . . . . . . . . . . 4.3 .5. Self-rhetoricity and informativeness. Proper questions . . . . . . . . . . . . . . . . . . . . .

. . . .. . . . . . . . . . .

. . . .. . . . . . ,

. . . ..

102 102 102 104 106 106 107 113 113 114 115 118

. . . . . 120

ix

CONTENTS

122 4.4 . Semantics for the languages ;;eo and;;e" 4.4. 1. Semantics for;;eo 122 4.4 .2. Semantics for;;e" . . . . . . . . . . . . . . . . . . . . . 122 5. EVOCATION OF QUESTIONS 5.1. Definition of evocation . . . . . . . . . . . . . . . . . . . 5.2. Basic properties of evocation 5.2 .1. The evoking sets 5.2.2 . The evoked questions . . . . . . . . . . . . . . . . 5.2.3 . Evocation and relations between sets of d-wffs and relations between questions . . . . . . . . . 5.2.4 . Evoking and presupposing. Alternative definitions of evocation . . . . . . . . . . . . . . 5.2.5. The problem of compactness of evocation . . . 5.3. Metatheorems and examples . . . . . . . . . . . . . . . . 5.3.1. Evocation in the language ;;eo . . . . . . . . . . . 5.3 .2. Evocation in the Ianguage E"

. ..

127 127 129 129 130

. ..

132

. ..

. . . 136 . . . 139 . .. 141 . .. 141 145

6. GENERATION OF QUESTIONS 6.1. Definition of generation 6.2 . Basic properties of generation . . . . . . . . . . . . . . . . 6.2.1 . The generating sets . . . . . . . . . . . . . . . . . . 6.2.2. The generated questions . . . . . . . . . . . . . . . 6.2.3 . Generation and relations between sets of d-wffs and relations between questions . . . . . . . . . . 6.2.4 . Some alternative definitions of generation . . . . 6.2 .5. Compactness and other problems 6.3. Generation in the languages ;;eo and;;e" 6.3 .1. Generation in the language Z' . . . . . . . . . . . 6.3 .2. Generation in the language E"

155 156 .. 157 . . 157 . . 158 . . 159 . . 161 164 167 .. 167 168

7. EROTETIC IMPLICATION. . . . . . . . . . . . . . . . . . . . . .. 7.1. Definition of erotetic implication . . . . . . . . . . . . . . . . 7.2 . Basic properties of erotetic implication 7.3 . Implying and presupposing . . . . . . . . . . . . . . . . . . ..

171 171 173 177

x

CONTENTS

7 A . Alternative definitions of erotetic implication 7.5 . Some special cases: pure erotetic implication and strong erotetic implication . . . . . . . . . . . . . . . . 7.5 .1. Pure erotetic implication . . . . . . . . . . . . . 7.5 .2. Strong erotetic implication 7.6. Erotetic implication and problem-solving . . . . . . . 7.7. Erotetic implication and the reducibility of questions to sets of questions 7.7.1. The concept of reducibility 7.7.2 . Erotetic implication and reducibility 7.8. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 8. EROTETIC ARGUMENTS 8.1. Erotetic inferences, erotetic arguments and validity 8.2 . E-arguments and G-arguments 8.3 . Im -arguments and Ims-arguments 804. Validity and natural-language erotetic arguments 8.5 . Logics, schemata and rules 8.5.1. The idea of the logic of questions of a semantically interpreted formalized language 8.5.2. Schemata 8.5.3 . Rules 8.504. Some remarks about applicability 8.6 . The idea of the logic of questions of a semantically interpreted formalized language generalized BIBLIOGRAPHY

180 . . .. 183 . . . , 183 187 . . . . 189 194 194 197 . . . . 20a , 209 209 212 217 220 221 221 222 223 225 226

'.' . . . . . 231

INDEX OF SYMBOLS

239

INDEX OF SUBJECTS

242

INDEX OF NAMES

247

INTRODUCTION

This book is a study in the logic of questions (sometimes called erotetic logic). The central topics in erotetic logic have been the structure of questions and the question-answer relationship . This book doesn't neglect these problems , but much of it is focussed on other issues. The main subject is the logical analysis of certain relations between questions and the contexts of their appearance. And our aim is to elaborate the conceptual apparatus of the inferential approach to the logic of questions. Questions are asked for many reasons and for different purposes. Yet, before a question is asked or posed, a questioner must arrive at it. In many cases arriving at a question resembles coming to a conclusion: there are some premises involved and some inferential thought processes take place. If we agree that a conclusion need not be "conclusive", we may say that sometimes questions can play the role of conclusions. But questions can also perform the role of premises: we often pass from some "initial" question to another question. In other words , there are inferential thought processes - we shall call them erotetic inferences - in which questions play the roles of conclusions or conclusions and premises. The inferential approach to the logic of questions focusses its attention on the analysis of erotetic inferences. This book consists of eight chapters. In Chapter One we argue that special attention ought to be paid to those erotetic inferences, in which questions arise from the premises. We also argue that the concepts "a question arises from a set of declarative sentences" and "a question arises from a question and a set of declarative sentences" can be explicated in semantic terms and then we formulate some requirements of adequacy of the explications looked for. Finally, Chapter One includes some historical notes. In Chapter Two we introduce the basic terminology and notation used thorough the book. Chapter Two contains also a general overview of the existing logical theories of questions and answers as well as a more detailed exposition of the theories proposed by David Harrah, Lennart Aqvist, Jaalcko Hintikka , Tadeusz Kubinski and Nuel D. Belnap. There are two reasons for including this presentation . First , it seems that it can make the proposals presented in the next chapter more comprehensible and less arbitrary. Second, elementary introductions to the existing theories of questions and Xl

xii

INTRODUCTION

answers are hard to find in the literature. Yet, an impatient reader can easily skip the second part of Chapter Two. Chapter Three contains a new proposal about the analysis of questions. Questions of formalized languages are viewed as expressions of strictly defined forms; they are not reduced to expressions of other syntactical categories. Roughly, a question of a formalized language is an expression made up of two main constituents: the sign? and an object-language expression such that the equiform expression of the metalanguage designates the set of direct answers to the question. Questions of formalized languages represent natural-language questions; the relevant concept of representation is discussed and an analysis of some English interrogatives is provided . In Chapter Four we propose a certain semantics for formalized languages whose meaningful expressions are both declarative formulas and questions. This semantics is basically a version of model-theoretical semantics; some new concepts are introduced for the purposes of erotetic analysis. Among them are: the concept of soundness of a question in an interpretation of the language, the concepts of relative soundness and relative informativeness of a question, different concepts of presuppositions of questions, the erotetic concepts of safety, riskiness and self-rhetoricity . Normal questions, regular questions and proper questions are also defined and the concepts of justcomplete answers and partial answers are introduced. Many of the above erotetic concepts are defined by means of the concept of multiple-conclusion entailment; this concept also serves as one of the main tools of our further analysis. In Chapter Five the semantic concept of evocation of a question by a set of declarative formulas is introduced. The proposed definition of evocation is an explication of the concept "a question arises from a set of declarative sentences" analyzed in Chapter One. The properties of evocation are examined in detail and examples are presented. Chapter Six is devoted to generation of questions by sets of sentences. This concept is also defined in semantic terms. From the purely formal point of view generation is a special case of evocation: it is the evocation of risky questions. But the proposed definition can also be viewed as a new explication of the concept explicated previously in terms of evocation. In Chapter Seven the concept of erotetic implication is introduced. Erotetic implication is a ternary relation between a question, a (possibly empty) set of declarative formulas, and a question. As in the previous cases, it is defined in semantic terms. The proposed definition of erotetic implication is

INTRODUCTION

xiii

an explication of the concept "a question arises from a question and a set of declarative sentences" analyzed in Chapter One. The properties of erotetic implication are examined in detail and examples are presented. Some special cases of erotetic implication, namely, pure erotetic implication and strong erotetic implication are also analyzed. The concept of reducibility of a question to a non-empty set of questions is introduced and the connections between erotetic implication and reducibility are characterized. In Chapter Eight the results of the previous chapters are applied in the logical analysis of erotetic inferences. Two syntactical concepts of (abstract) erotetic arguments are introduced and the concepts of validity of these arguments are defined in terms of evocation and erotetic implication, respectively. The concept of the logic of questions of a semantically interpreted object-language whose meaningful expressions are both declarative formulas and questions is defined and discussed. Although this book is a study in the logic of questions, some of the problems discussed here may be of some interest to philosophers , cognitive scientists and computer scientists. This fact is reflected in the structure ofthe book: an attempt is made to apply the simplest possible notation and avoid excessive conciseness in presentation. Thus some parts of the book may be regarded as prolix by logicians. I do hope that they will forgive me that; after all, it is better to say too much than too little. Some results presented here were included in my book "Stawianie pytari: logika i racjonalnosc" published in 1990. The present book, however, is not a simple translation of its Polish predecessor: there are differences in the ways of presentation as well as in the arrangement and, more importantly, large parts of the material included here do not occur in the Polish version. Some of the new results were already published, into al. , in Journal of Philosophical Logic and Erkenntnis (cf. Bibliography), whereas the remaining ones are published here for the first time. lowe many thanks to many people. I am indebted to my teacher in logic, Tadeusz Kubinski, for his assistance at the early stage of my work on the logic of questions. I would like to thank Leon Koj for his continuous encouragement and help. Six chapters of this book were written during my stay as a Fulbright grantee in the Department of Philosophy, University of California, Riverside . I am indebted to the Fulbright Program for support. I am especially grateful to David Harrah for his comments, criticism and linguistic assistance; without his help this book would never have been completed. I am also indebted to Sylvain Bromberger and Theo A. F. Kuipers for discussions and comments on some parts of the material. I am

xiv

INTRODUCTION

grateful to Jon Ringen for his linguistic advices. Needless to say, this is the author who is responsible for all the mistakes.

March 1995

CHAPTER

1

EROTETIC INFERENCES AND HOW QUESTIONS ARISE

1.1. EROTETIC INFERENCES When we speak about inferences, we customarily assume that only declarative sentences can serve as their premises and conclusions. Questions are not declarative sentences; it is also a matter of discussion whether questions can be qualified as true and false. On the other hand, there are inferential thought processes in which questions play the role of "conclusions" whereas the "premises" are declarative sentences, declarative sentences and a question, or a question alone. It often happens that we arrive at a question on the basis of some previously accepted declarative sentences or sentence. When we accept that: (1.1) The theory of ideas is presented in the writings of Plato. (1.2) If the theory of ideas is presented in the writings of Plato, then it was invented by either Plato or Socrates. we arrive at the question: (1.3) Who invented the theory of ideas: Plato or Socrates? Similarly, when we accept that: (1.4) Each dialogue written by Plato contains some views of Socrates. (1.5) Timaios is a dialogue written by Plato. we arrive at the question: (1.6) What views of Socrates are contained in the dialogue Timaios? It also often happens that we arrive at a question when we are looking for the answer to another question on the basis of some data expressed by certain previously accepted declarative sentences or sentence. When the answer to the question: (1.7) Who discovered Grenada? is sought on the basis of the premise: (1.8) Grenada was discovered by the discoverer of Trinidad. we arrive at the question:

CHAPTER 1

2

(1.9)

Who discovered Trinidad?

Similarly, when we are looking for the answer to the question: (l .1O) Which town was the first capital of Poland? on the basis of the premise: (l .ll) The first capital of Poland is now a town which lies about fifty kilometers from Poznan. we arrive at the question: (1.12) Which towns lie about fifty kilometers from Poznan? Sometimes we pass directly from one question to another question . For example, when we are looking for the answer to the question: (l.13) Was Plato a pupil of Socrates and a teacher of Aristotle? we arrive at the question: (1.14) Was Plato a pupil of Socrates? The examples presented above show that the concept of inference may be generalized by introducing the concept of erotetic inference. At a first approximation an erotetic inference may be defined as a thought process in which we arrive at a question on the basis of some previously accepted declarative sentence(s) and/or a previously posed question. These examples also show that there are erotetic inferences of two kinds: the key difference between them lies in the type of premisses involved. In an erotetic inference of the first kind we pass from some previously accepted declarative sentence(s) to a question. In an erotetic inference ofthe second kind we pass from some previously posed question and possibly some previously accepted declarative sentence(s) to a question . (If an inference runs directly from a question to a question, we may say that the set of declarative premises is the empty set, so such inferences may be regarded as a special case of the erotetic inferences of the second kind.) The term "acceptance" is used here as a noncommittal cover term for a number of propositional attitudes, including rather "weak" ones; the declarative premises of an erotetic inference may belong to the questioner's knowledge or beliefs, but they also may be only hypothetically or even temporarily assumed. Similarly, a question playing the role of conclusion can be asked or posed (which implies some serious interest in looking for the answer), but it also can be - using Ajdukiewicz's terminology - a question which is only thought of. Although erotetic inferences occur in almost every process of reasoning, they have been systematically ignored by almost all logicians. The common attitude is to regard them as belonging to the "pragmatics" of reasoning, in

EROTETIC INFERENCES ANDHOWQUESTIONS ARISE

3

the very bad sense of "pragmatics" as referring to something that is not subjected to any objective rules. No doubt, there are erotetic inferences of this kind. But there are also erotetic inferences which have a well-established structure due to the existence of some logical relations between their premises and conclusions . Of course, premises and conclusions of various erotetic inferences can stand in various relations. But the most interesting are those erotetic inferences in which questions, which are the conclusions, arise from the premises: in the case of erotetic inferences arising seems to playa similar role as entailment in the case of standard inferences. To be more precise, as far as the erotetic inferences of the first kind are concerned, special attention should be paid to those in which questions, which are the conclusions, arise from the corresponding declarative premises. Similarly, it seems that the most interesting erotetic inferences of the second kind are those, in which questions performing the role of conclusions arise from the corresponding declarative premises and questions playing the role of the premises . In both cases the term "arising" is understood in a mainly descriptive, but also somewhat normative manner. The above statements sound mysterious unless the relevant concepts of arising are clarified . Let us then tum to the phenomenon of the arising of questions in a cognitive process. I

1.2. THE ARISING OF QUESTIONS FROM SETS OF DECLARATIVE SENTENCES Each inquiry may be viewed as a process of asking questions and looking for the answers to them. In most inquiries the asked questions are dependent upon the acquired or hypothetically assumed answers to the previously asked questions as well as upon some background knowledge: in the light of what has been established or assumed earlier some questions are admitted, whereas some other are not. Sometimes, however, certain further questions are not only admitted, but nearly suggested. These situations are usually referred to by saying that a given question arises from or is raised by what

1 Some speakersof Englishfind the expression "the arisingof a question" incorrect; some others disagree with them. We share the view of the second. Some speakers of English would prefer the expression "a questionarising" here; yet, this expressions in the context "the concept of a questionarising from ... " might misleadingly suggest that we are interested in a property of questions rather than in the underlying relationwhichholds betweena questionwhich arises from a given background and this background. Some languages (e.g. Polish) have a special expressionreferringto the lattercase ("powstawanie pytan") whichis distinctfrom that referring to the former case ("pytanie powstajace "),

4

CHAPTER 1

has already been established or assumed. The expressions cause to arise and gives rise to are also used in this context. Let us consider some examples. Let's imagine that at some stage of the investigation concerning the murder of JF Kennedy the following hypothesis has been assumed or even confirmed: (1.15) Some organization inspired the murder of JFK. We may say that the following question arises from (1.15): (1.16) Which organization inspired the murder of JFK ? Let's now imagine that during the same investigation also the following hypothesis has been assumed or confirmed: (1.17) If any organization inspired the murder of JFK, it was KGB, or the Cuban intelligence, or the New Orleans mafia. We may say that the following question arises from (1.17) together with (1.15) : (1.18) Which organization inspired the murder of JFK: KGB, the Cuban intelligence, or the New Orleans mafia? Let's now imagine, in tum, that at some stage of the inquiry the following hypotheses were assumed or accepted: (1.19) The murder of JFK was a political one . (1.20) In case of each political murder there is a plot which stands behind it. We may say that (1.19) together with (1.20) cause the following question to arise: (1.21) What plot stood behind the murder of JFK? Let us now tum to some investigation concerning investigation. Assume that the following hypotheses were assumed or confirmed in it: (1.22) If the investigation concerning the murder of JFK was honest and many-sided, the real culprit has been found. (1.23) The real culprit has not been found. It may be said that the following question arises from (1.22) along with (1.23) : (1.24)

Wasn't the investigation concerning the murder of JFK honest, or wasn't it many-sided? Let's now leave the area of criminal investigations and imagine a student of philosophy who finds out that: (1.25) Some Polish philosopher influenced Karl Marx.

EROTETIC INFERENCES AND HOW QUESTIONS ARISE

5

and then poses the following question: (1.26) Which Polish philosopher influenced Karl Marx? Again, we may say that the question (1.26) arises from the sentence (1.25). Our hypothetical student may in addition find out that: (1.27) If any Polish philosopher influenced Karl Marx, it was Cieszkowski, or Libelt, or Hoene-Wronski. In this situation he will presumably pose the question: (1.28) Which Polish philosopher influenced Karl Marx: Cieszkowski, Libelt, or Hoene-Wronski? Yet, it can be said that the question (1.28) arises from the sentences (1.27) together with (1.25) . At some stage of his investigations our hypothetical student may come to the conclusion that the Polish philosopher sought by him must have been a Hegelian who published his major works in some language different from Polish and known to Karl Marx; if in addition our student would find out that: (1.29) Cieszkowski was a prominent Polish Hegelian. (1.30) Each prominent Polish Hegelian published his major works in some foreign language. he presumably would pose the question: (1.31) In what foreign language did Cieszkowski publish his major works? This question , however, may be regarded as arising from the sentences (1.29) together with (1.30) . Our hypothetical student may also proceed by assuming the following : (1.32) If Hoene-Wronski was a prominent Polish Hegelian and elaborated on the philosophy of deed, he influenced Karl Marx. But sooner or later he will find out that: (1.33) Hoene-Wronski did not influence Karl Marx. In this situation he will presumably arrive at the question: (1.34) Wasn't Hoene-Wronski a prominent Polish Hegelian, or didn't he elaborate on the philosophy of deed? Again, we may say that the question (1.34) arises from the sentences (1.32) together with (1.33) . It is easy to find further examples of this kind. In what follows instead of speaking of the arising of a question from some sentence or sentences we will be speaking of the arising of a question from

6

CHAPTERl

a set made up of the corresponding sentences; this way of speaking, although sometimes clumsy (especially in the case when the relevant set is a singleton set), will allow us to avoid some misunderstandings. But is the arising of questions from sets of declarative sentences an empirical matter which varies from time to time and from question to question? Or are there some underlying - syntactical or semantical - relations whose existence justifies us in saying that a given question arises from some set of declarative sentences? It seems that the second possibility holds. Let us observe that in many cases we can qualify questions as arising from certain sets of sentences even if we do not know whether (or are not sure it) these sentences were in fact accepted by anyone. Coming back to the presented examples: it seems that even a person who has no idea how the real investigations proceeded would agree that the above-indicated questions arise from the corresponding sets of sentences. Moreover, in many cases it is the logical form of questions and declarative sentences that determines what questions arise from what sets of declarative sentences. In particular, this is true in the case of all our examples: it is easy to observe that if any expressions which occur in any of the above patterns were replaced (in a systematic manner) by expressions of the same syntactic categories, we could still legitimately say that the resultant question arises from the corresponding set of resultant declarative sentences. We can also go further in this direction. Let a, b, c stand for distinct names and P, R, S, T stand for distinct predicates. It seems that we can say that a question falling under the schema: (1.35) Which x is such that P(x)? arises int.al. from the singleton set which contains a sentence of the form: (1.36) For some x, P(x). On the other hand, the questions (1.16) and (1.26) may be regarded as falling under the schema (1.35), whereas the sentences (1.15) and (1.25) may be viewed as falling under the schema (1.36) (of course provided that the range of x is restricted in the appropriate manner and that we do not go very deep into the syntactic structures of the analyzed natural-language sentences; these assumptions will be adopted below too). We can also say that a question of the form:

(1.37)

Is it the case that P(a), or is it the case that P(b), or is it the case that P(c)? arisesint.at. from the set of sentences having the schemata: (1.38) For each x: if P(x), then x = a or x = b or x = c.

EROTETIC INFERENCES AND HOW QUESTIONS ARISE

7

(1.36) For some x, P(x). But the questions (1.18) and (1.28) may be viewed as having the form of (1.37) , whereas the sentences (1.17) and (1.27) may be regarded as falling under the schema (1.38). It can also be said that a question of the form: (1.39) Which x is such that Sea, x)? arises int.al . from a set of sentences whose elements have the form of: (1.40) For each y: if R(y), then for some x, S(Y, x) . (1.41) R(a). The questions (1.21) and (1.31) , however, can be regarded as falling under the schema (1.39). The sentences (1.19) and (1.29), in turn, can be viewed as falling under the schema (1.41), whereas the sentences (1.20) and (1.30) may be regarded as having the schema (1.40) . Finally, a question of the form: (1.42) Is it the case that rvRtc), or is it the case that ,T(c)? where"," is the negation connective, may be regarded as arising int.al. from a set which consists of sentences having the following forms: (1.43) If R(c) and T(c), then p. (1.44)

.. ., t, are terms of J, then an expression of the form F;"(tl> ..., to)' where F;" is a function symbol of J , is also a term of J . A closed term is a term with no individual variables . The closed terms will be also referred to as names.

LOGICAL THEORIES OF QUESTIONS

35

Atomic formulas of J are expressions of J of the form t, = t2 and of the form P;"(t l , . .. , tn) , where t" t2 , • • • , tn are terms. The set I', of declarative well-formed formulas (d-wffs for short) of the language J is the smallest set containing all the atomic formulas of J and having the following properties : (a) if A is in r J , then expressions of the form "'A , VXi A, :Ixi A are also in r J ; (b) if A , B are in r J , then expressions of the form (A --+ B) , (A v B), (A /I B) , (A == B) are also in r J • The d-wffs not containing free variables will be called sentences , whereas the d-wffs containing free variables will be called sentential functions . The freedom and bondage of variables are defined as usual. Each subset of the vocabulary of the language J which contains the connectives .., and --+, the universal quantifier V, both parentheses, all individual variables, at least one predicate symbol, the identity symbol = and possibly (and thus not necessarily) some other signs, such as other predicate symbol(s), the connectives v , /I, ==, the quantifier 3, individual constant(s), function symbol(s) or the comma will be called here efirst-order language with identity; in what follows by a first-order language we will mean a first-order language with identity. The concepts of term, closed term, atomic formula, declarative well-formed formula, sentence and sentential function , as well as the remaining syntactic concepts are defined for J in the same way as for any first-order language with identity. One general remark is in order here: since we are going to consider formalized languages whose meaningful expressions are both d-wffs and questions, it is convenient to identify a language with the set of its signs rather than with the set of its formulas. 2 .1 .2 . Metalanguage. Notation

We shall use the symbols t, t., .. . as syntactical variables which range over terms . Since we will be frequently speaking about closed terms, we shall introduce special metalinguistic variables for them. The letters u and v, possibly with indices, will be syntactical variables which represent closed terms. The letters A, B, C, D, with or without indices, will be metalinguistic variables for d-wffs. An expression of the metalanguage having the form Axi, ... Xi. refers to the sentential functions whose free variables are exactly the (explicitly listed) variables Xii' .. . , Xi. ' Whenever an expression of the form Axil . .. Xi. is used, it is assumed that the variables Xi" ... , Xi. are distinct. The letters P, R, possibly with indices, will be metalinguistic variables represent ing predicate symbols. The symbols X, XI' .. ., Y, Yh . .. ,

CHAPTER 2

36

Z. ZI• .. . will be metalinguistic variables for sets of d-wffs. The context of occurrence of the appropriate metalinguistic symbols will always determine which formalized language we have in mind. We adopt here the usual conventions for omitting parentheses. An expression of the form f l ~ f 2 is the abbreviation of an expression of the form ""(f( = (2) . An expression of the form t, ~ ... ~ fn. in tum. is the abbreviation of an expression of the form f) ~ f 2 II f) ~ f3 II . .. II f) ~

fn

II

f2

~

f3

II ... II

f n_1

~

fn'

We will also need the concept of universal closure of a d-wff. If A is a sentence . then the universal closure of A is equal to A itself. If A is a sentential function and X;I' • • •• X;. (where i, < ... < iJ are the all free variables of A. then the universal closure of A is of the form "lxi' ... "Ix;. A . The universal closure of a d-wff A will be designated by A. We assume that in the case of the analyzed languages the concepts of (proper) substitution of a term for a variable in 'a d-wff and of the substitutivity of a term for a variable are defined in the standard manner . The result of the substitution of a term f for a variable x, in a sentential function Ax; will be designated by A(X/f). By A(x;,1fl> ... , X;/fJ we shall designate the d-wff which results from a sentential function Ax;1 • • • xi. by the simultaneous (proper) substitution of the terms f ), .. . , f n for the variables X;I' .. . , xi.. respectively . In the metalanguage of any of the considered languages we assume the von Neumann-Bernays-Godel version of the set theory (we choose this version because we want to have the possibility of speaking about both sets and classes). We shall use the standard set-theoretical terminology and notation. In particular, by {Yl> ...• Yn} we shall designate the set made up of the elements YI> ... , Yn' The singleton (or unit) set having Y as its element will be referred to as {y} . The symbol" will "designate the empty set. The symbol ~ is the sign of inclusion, whereas the symbol C is the sign of proper inclusion. The symbol E is, depending on the context. the predicate of set membership or of class membership. The sign = will be also used as the metalingustic identity symbol. The symbols ~. g;. ~ mean: "is not an element", "is not included in" and "is not identical with". respectively . The symbols u, n, - are the signs of the union of sets. intersection of sets, and difference of sets, respectively. Sometimes we shall write X, Y instead of X u Y and X, A instead of X u {A}. A metalinguistic expression of the form {y: «IJ(y)} denotes the set of all objects satisfying the (metalinguistic) sentential function «IJ(y) . The expression "iff" is an abbreviation of "if and only if. " The symbol 0 indicates the end of a proof.

LOGICAL THEORIES OF QUESTIONS

37

The remaining symbols used in this book will be explained directly in the contexts of their first appearance. Let us finally add that the names of expressions (of an object language or of a metalanguage) will be formed here by preceding these expressions with words "term", "sentence", "question", etc. Phrases like "expression of the form" , "d-wff of the form", "question of the form", etc. , are used here in the role of Quine'an quasi-quotes. Thus, for example, the phrase "an expression of the form (A -. B)" means "an expression made up of the left parenthesis , a d-wff, the implication sign, a d-wff, and the right parenthesis." Of course, these conventions will not always be kept, but they will always be adopted in cases in which there may be a risk of a misunderstanding. In what follows we will be considering also some formalized languages which are not first-order languages. Yet, the terminology and notation introduced above will be also applied to these languages.

2.2. LOGICAL THEORIES OF QUESTIONS AND ANSWERS Any first-order language can be supplemented with a question-and-answer system. This, however, can be done in different ways. Before we shall introduce the concepts of question and direct answer accepted in the course of analysis pursued in this book, let us pay some attention to the existing logical theories of questions and answers. The following presentation is not an exhaustive one; we shall concentrate only on some main ideas and on the work of some logicians. Yet, it seems that such a presentation will make our proposals more comprehensible and less arbitrary. 2.2 .1. Questions: reductionism vs. non-reductionism As we read in one of the very few monographs on the logic of questions, "Different authors developing logical theories of questions accept different answers to the question 'What is a question?"'(Kubinski, 1971, p. 97). This statement, written down more than twenty five years ago, still gives us a realistic description of the situation within erotetic logic. To speak generally, the approaches to questions proposed by different logicians and formal linguists can be divided into reductionist and nonreductionist ones. Inside the reductionist approach, in turn, the radical and moderate standpoints can be distinguished. 2.2.1.1. Radical reductionism . According to the radical view, questions are not linguistic entities . The reduction of questions to sets of sentences or propositions is most often adopted here. Sometimes any set of sentences is

38

CHAPTER 2

allowed to be a question, but usually questions are identified with sets of answers of some distinguished category. Stahl (cf., e.g., Stahl, 1962) identifies questions with sets of their sufficient answers; these answers are declarative formulas of a strictly defined kind. Hamblin (cf. Hamblin, 1973) identifies a question with the set of its possible answers, whereas Karttunen (cf. Karttunen, 1978) identifies questions with sets of their true answers; in both cases the relevant answers are propositions in the sense of some intensional logic. Questions are also identified with functions defined on possible worlds; the set of values of a function of this kind consists of truthvalues, or of sets of individuals, or of sets of sets of individuals (cf., e.g., Tichy, 1978, and Materna, 1981). Also in this case some intensional logic serves as the basis of analysis. An analysis of questions and interrogatives in terms of (some versions of) Montague intensional logic is to be found e.g. in Groenendijk & Stokhof (1984). Some linguists developed the so-called categorial approach to questions: according to this view, questions are to be considered as functions from categorial answers to propositions (cf. , e.g., Hausser, 1983); a categorial answer may be a full sentence, but may also be a part of it, e.g. a noun phrase, an adverb, etc. There are philosophers of language (cf., e.g ., Vanderveken, 1991) who tend to identify questions with speech acts rather than with expressions. Let us stress, however, that in all of the above cases a distinction is made between an interrogative (or an interrogative sentence) and a question: whereas interrogatives are linguistic entities , questions are claimed not to be. Let us finally add that some linguists (cf. Keenan & Hull, 1973, and Hiz , 1978) proposed theories in which the semantically meaningful units are not questions , but question-answer pairs. Sometimes questions are also analyzed as ordered pairs consisting of interrogative terms and statements expressing the relevant presuppositions (cf. Finn, 1974).

2.2.1.2. Moderate reductionism. The moderate reductionist view considers questions as linguistic entities which, however, can be reduced to expressions of some other categories. To be more precise, it is claimed here that every question can be adequately characterized as an expression which is synonymous (or synonymous to some reasonable degree) to a certain expression of a different syntactical category. Or, to put it differently, each question can be adequately paraphrased as an expression belonging to some other syntactic category and then formalized within some logic which, although not primarily designed as the logic of questions, can thus be regarded as providing us with the foundations of erotetic logic. Some theorists propose the reduction of questions to declarative formulas

LOGICALTHEORIES OF QUESTIONS

39

of strictly defined kind(s) . Sometimes questions are identified with declarative formulas having free variables , that is, with sentential functions (cf. e.g., Cohen, 1929, or Lewis & Langford, 1932). But questions are also identified with sentences, that is, declarative formulas with no free variables . According to the early proposal of David Harrah (cf. Harrah, 1961, 1963) whether-questions are to be understood and then formalized as declarative sentences having the form of exclusive disjunctions, whereas which-questions should be identified with existential generalizations. We shall present Harrah's proposal in a more detailed way in Section 2.2 .3. Questions are also identified with imperatives of a special kind . The imperative-epistemic approach is the most popular here with Lennart Aqvist and Jaakko Hintikka as its most eminent representatives . According to Aqvist (cf. Aqvist, 1965, 1969, 1971, 1972, 1975), a question can be paraphrased as an imperative-epistemic expression of the form "Let it (tum out to) be the case that Ip", where Ip is a formula which describes the epistemic state of affairs which should be achieved . Pragmatically, a question is thus understood as an imperative which demands of the respondent to widen the questioner's knowledge . Questions are formalized within the framework of some imperative -epistemic logic; on the level of formal analysis we deal with interrogatives. Each interrogative consists of an interrogative operator and its arguments . Interrogatives are defined as abbreviations of certain formulas of the language of the considered imperative-epistemic logic. We shall present some elements of Aqvist's theory in Section 2.2.4. The imperative-epistemic approach to questions is also adopted by Jaakko Hintikka in his theory of questions and answers (cf. Hintikka, 1974, 1976, 1978, 1982a, 1982b, 1983). Hintikka interprets questions as requests for information or knowledge: according to his view, each question can be paraphrased as an expression which consists of the operator "Bring it about that" followed by the so-called desideratum of the question . The desideratum describes the epistemic state of affairs the questioner wants the respondent to bring about. Although the main ideas of Aqvist and Hintikka are similar , they are elaborated on in different ways. We shall present Hintikka's theory in a greater detail in Section 2.2.5. As far as the moderate reductionist view is concerned, the imperativeepistemic approach is the most widely developed one. Yet, there are also other proposals. In particular, there is an old idea (which goes back at least to Bolzano) that the paraphrase of a question should contain an optative operator. (It is worth noticing that Hintikka sometimes calls the operator

40

CHAPTER 2

"Bring it about that" an "imperative or optative operator. ") Apostel (cf. Apostel, 1969) claims that questions can be reduced to expressions which contain epistemic, deontic and alethic operators as well as the assertion operator; yet, Apostel also claims that the deontic operator "ought to" used by him should be replaced by some optative or imperative operator when a more adequate analysis of such operators will be available. In a short note Aqvist (cf. Aqvist, 1983) sketched an outline of the imperative-assertoric analysis of questions; according to this proposal, questions should be paraphrased (and then formalized) as imperatives which contain the assertoric operator "You tell me truly that" instead of the epistemic operator "I know that." Let us stress, however, that none of the above proposals has been elaborated on.

2.2 .1.3 . Non-reductionism. According to the non-reductionist approach, questions are specific expressions of a strictly defined form; they are not reducible to expressions of other syntactic categories.' The most widespread proposal here is to regard a question as an expression which consists of an interrogative operator and a sentential function. This view is accepted, among others, by Ajdukiewicz (cf., e.g., Ajdukiewicz, 1926, 1974), Hiz (cf. Hiz, 1962), Kubinski (cf. Kubinski, 1960, 1971, 1980, and other papers of him listed in the "Bibliography"), Koj (cf. Koj, 1972, 1987), Leszko (cf., e.g., Leszko, 1983, and to appear), and Harrah in his later papers (see, e.g., Harrah, 1973, 1975, 1976, 1984). Some prominent authors who only incidentally paid attention to questions also shared this view (cf., e.g., Carnap, 1937, Reichenbach, 1947, or Cresswell, 1965a, 1965b). Yet, the above idea is most widely elaborated on in the books and papers of Tadeusz Kubinski. Kubinski's analysis is mainly a syntactical one: questions of a formalized language are defined as expressions which consist of interrogative operators and sentential functions. Interrogative operators, in tum, consist of both constants and variables. The only free variables in the sentential functions which occur in questions are the variables of the corresponding interrogative operators; these variables are "bound" by the interrogative operators. The variables which occur in questions may belong to various syntactical categories. Roughly, the categories of variables indicate the (ontological) categories of objects which are asked about. For example, a question whose interrogative operator contains only individual variables asks about individuals. If the relevant variables run over sentential connectives, then the corresponding questions are about either the existence of some state(s) of affairs or some connection(s) between states of affairs. Questions with

LOGICALTHEORIES OF QUESTIONS

41

predicate variables, in turn, ask about properties or relations . When a question contains only sentential variables, it is a question about logical values (truth and falsity). Kubinski considers also "mixed" questions, that is, questions whose interrogative operators contain variables belonging to two or more different categories. We shall present some elements of Kubinski's theory in Section 2.2 .6. Although the "interrogative operator-sentential function" view is shared by most of the adherents of the non-reductionist approach to questions, there are also other proposals. Among them special attention should be paid to Nuel D. Belnap's theory of questions and answers (cf. Belnap, 1963, 1966, 1969a, 1969b, 1972, and Belnap & Steel, 1976). Belnap distinguishes between: (a) natural language questions, (b) interrogatives and (c) questions understood as abstract (set-theoretical) entities. Interrogatives are expressions of some formalized languages. They are not only formal counterparts of natural language questions, but they also express questions understood as abstract entities. A simple interrogative consists of the question mark?, the lexical subject and the.lexical request. The question mark is interpreted as a sign of the function which assigns to the lexical subject and the lexical request of a simple interrogative the corresponding (abstract) question; such a question consists, in turn, of the abstract subject and the abstract request. A compound interrogative can be obtained from simple interrogatives by performing some (logical of boolean) operations on them or their lexical subjects. The basic idea of Belnap's approach is that an interrogative "presents" a set of alternatives together with some suggestions or indications as to what kind of choice or selection among them should be made; the situation is analogous in the case of the corresponding questions. The function of the lexical subject of an interrogative is to offer the relevant (nominal) alternatives, whereas the role of the lexical request is to characterize the required kind of selection . The lexical request consists of three parts: the lexical selection-size specification, the lexical completeness-claim specification, and the lexical distinctness-claim specification. Roughly, the lexical selection-size specification informs how many (that is, how many exactly, how many at least and/or how many at most) of the alternatives offered by the lexical subject of the interrogative are called for. The lexical completeness-claim specification, in turn, informs about the amount of true nominal alternatives called for; there are interrogatives which call for all the true alternatives presented by them, but there are also interrogatives which do not demand so much. Finally , the lexical distinctness-claim specification tells whether the

42

CHAPTER 2

alternatives called for should be semantically different. We shall present some elements of Belnap's theory in Section 2.2.7 . Let us finish this section with one general remark. The controversy about the "nature" of questions is not only a conceptual one. If the radical reductionist view is correct, no logic (in the very sense of "logic", as opposed to "logical theory") of questions is possible. If the moderate reductionist view is correct, logic of questions should be developed only within the framework of some other philosophical logic. But if we accept the non-reductionist approach, the problem of building (or discovering, as a platonist might say) of the logic of questions remains open.

2.2.2. Answers Most logical theories of questions pay at least as much attention to answers to questions as to questions themselves. It is usually assumed that a question can have many answers; the phrase "answer: to a question" is not used synonymously with "the.true answer to a question." In other words, the analyzed answers are usually possible answers: their logical values are not prejudged . Yet, it is not the case that all possible answers are equally interesting to erotetic logicians. The standard way of proceeding is to define some basic category of possible answers. They are called direct answers (Aqvist, Belnap, Harrah, Kubinski) ,properanswers (Ajdukiewicz), sufficient answers (Stahl), conclusive answers (Hintikka and his associates), indicated replies (Harrah in his later papers), etc. Those "principal" possible answers (let us use this general terms here) are supposed to satisfy some general conditions, usually expressed in pragmatic (in the traditional sense of the word) terms. For example, direct answers in Kubinski's sense are "these sentences which everybody who understands the question ought to be able to recognize as the simplest, most natural, admissible answers to this question" (Kubinski, 1980, p. 12). Direct answers in Belnap's sense are the answers which "are directly and precisely responsive to the question, giving neither more nor less information than what is called for" (Belnap, 1969a, p. 124). Direct answers in Harrah's sense are replies which are complete and just-sufficient answers (cf. Harrah, 1963, p. 26 et al.). In the light of Hintikka's theory a reply is called conclusive just in case it completely satisfies the epistemic request of the questioner, that is, brings about the epistemic state of affairs the questioner wanted to be brought about. Let us stress that although the above conditions are formulated in pragmatic terms, some of the logical theories of questions define the principal possible answers to questions or interrogatives of formalized languages in terms of syntax and/or semantics. The questions (interrogatives)

LOGICALTHEORIES OF QUESTIONS

43

of formalized languages , however, are usually formalizations of natural language questions. Consequently , the principal possible answers are usually defined in such a way that the natural language sentences which correspond to them are those answers to the analyzed natural language questions which have the above-mentioned pragmatic properties . Or, to be more realistic, to define them in such a way is the aim of the enterprise. Yet, there are natural language questions which admit many readings and thus many formalizations. There are also natural language questions which seem to have no welldefined sets of answers ; why-questions are often recalled in this context. Moreover, most theories admit questions or interrogatives which have no counterparts in natural languages, but nevertheless have well-defined sets of the "principal" possible answers. It is not the case that only the principal possible answers are of interest to erotetic logicians. Most theories provide us also with definitions of other kinds of possible answers: partial answers, complete answers, just-complete answers, incomplete answers, corrective answers, etc. These answers are usually defined in terms of the "principal" answers; yet, it also happens that they (or some of them) are defined independently. Their definitions differ from theory to theory ; no well-established terminology has been elaborated yet. Let us finally add that sometimes replies which are not statements (e.g. noun phrases , nods, grunts) are regarded as answers; in most cases, however, answers are assumed to be statements and replies of other kinds are regarded as abbreviations of the corresponding statements. Let us now illustrate the above general remarks concerning questions and answers by some examples . We shall present here elements of five theories of questions and answers, namely, of the theories proposed by David Harrah , Lennart Aqvist , Jaakko Hintikka, Tadeusz Kubinski , and Nuel D. Belnap. These theories are among the most widely developed ones. Yet, our presentation will be far from completeness: we shall concentrate on the general methods of defining questions and the "principal " answers, leaving apart other topics discussed within the analyzed theories. 2.2.3 . Harrah 's analysis 2.2 .3.1. Questions. According to the early proposal of David Harrah (cf. Harrah, 1961, 1963) whether-questions are to be formalized as sentences having the form of exclusive disjunctions, whereas which-questions are to be formalized as existential generalizations . For example, the question :

CHAPTER 2

44

(2.1) Did you arrive by car, or by bus? has a formalization of the form: (2.2) (AI 1\ ...,A2) V (...,A I 1\ A~ The formal counterpart of the question: (2.3) Where did you buy your car? is of the form: (2.4)

3xi Axi

Sentences of the form (2.2) and (2.4) are special cases of whether and which-questions, respectively. The general definitions are as follows. Let A It ••. , An' where n ~ 1, be a finite sequence of sentences of some first-order language with identity whose vocabulary contains some closed terms and all the logical constants: ..., , -. , v , 1\ , == , V , 3. 1 Let us assume that for each i such that 1 ~ i ~ n, the expression A*i abbreviates the sentence: (2.5) «",««" ,(""A 1 1\ ...,A2) 1\ ... ) 1\ ...,Ai _l ) 1\ Ai) 1\ ...,Ai+I) 1\ .. . ) 1\ ...,An) The sentence:

«(.. .«A*I v A*2) v A*3) v .. .) V A*n_l) v A*n) is called the primedisjunction in the sequence A I, . . . , An. Whether-questions are then identified with sentences being prime disjunctions in at least twoelement sequences of sentences of the analyzed first-order language. Whichquestions, in tum, are identified with sentences (of the considered language) of the form: (2.6)

(2 .7)

3xi l

...

3xin Axil .. . Xi.

where n ~ 1 and Axil ... Xi. is not of the form 3xj B The underlying idea of Harrah's proposal is that questions are expressions of a questioner's incomplete knowledge about the subject matter; in an information-matching game played with statements the questioner expresses his limited knowledge by saying e.g. "This or that is the case" or "Some objects are so-and-so" and the respondent is supposed to widen the

I In the book Harrah (1963) questions are defined for the language of classical predicate calculus with identity and individual constants, but without function symbols. The proposed definitions,however, may be appliedin thecase of each first-orderlanguagewith identitywhose vocabularycontainssomeclosedtermsand the above-specified logicalconstants; the assumption concerning the existenceof closed termsallowsus to definedirect answers to which-questions.

45

LOGICAL THEORIES OF QUESTIONS

knowledge of the questioner by providing the correct direct answer. Harrah's proposal is most often interpreted as postulating the identification of questions with their presuppositions.

2.2.3.2. Direct answers. Harrah defines direct answers to questions in syntactic terms; on the other hand, when we take into consideration the role played by questions in the information-matching game described in the book Harrah (1963), we may say that these answers are undoubtedly the complete and just-sufficient replies . A direct answer to a whether-question of the form (2.6) is defined as a formula of the form A*i' where 1 ~ i ~ n, that is, as a conjunction having the form (2.5) . Direct answers to which-questions of the form (2.7) are defined as sentences falling under the schema: (2.8)

Vxil (XiI

.. .

Vxi•

= Ukl

A

(Axil .. .

A

Xi"

Xi"

==

«XiI

= UlI

A ... A

Xi"

= U'n)

v . .. V

= Uk..»)

where u lI ' .. . , u'n' , ukl' . .. , u lo, are closed terms. Thus, roughly, the direct answer A*i to a whether-question says that only Ai is the case, whereas the direct answer (2.8) to a which-question says that the objects designated by the following n-tuples of names: j and x" xj are the alphabetically earliest variables which are substitutable for x; in the sentential function Ax;; this sentential function contains no occurences of epistemic and imperative operators. A formula of the form (2.25), in tum, is the imperative-epistemic translation of an interrogative having the form: (2.26) ?EKBX; Ax; Let us stress that the interrogatives introduced above do not exhaust the list of interrogatives analyzed by Aqvist: we simply picked up some examples in order to illustrate the general idea of his approach. Let us also add that the above presentation is based mainly on the book Aqvist (1965); the paper Aqvist (1971) introduces some important refinements to the previous proposals.

2.2.4.2. Direct answers. One of the most interesting features of Aqvist's theory is that it deals primarily with questions and only secondarily with answers to them: it enables us to define and analyze various logical relations between questions or between questions and declarative sentences without assumimg what sentences count as (direct) answers to the analyzed questions. But Aqvist claims that a theory of answers can be developed within his framework as well. Following Harrah and Belnap, Aqvist uses the term "direct answers" for the "principal" possible answers to interrogatives. A direct answer is supposed to satisfy the request expressed by a question, that is, roughly, to guarantee, if true, the truth of the core of the interrogative; by the core of an interrogative we mean the epistemic formula which results from the imperative-epistemic translation of the interrogative by dropping the imperative operator!. Moreover, direct answers are supposed to do it in a straight and non-redundant way: each direct answer should afford a just sufficient condition for the fulfilment of the questioner's desire. These are the basic intuitions ; on the level of logical analysis Aqvist expresses them by introducing several technical concepts and formulating some necessary conditions which have to be fulfilled by each direct answer. Finally, Aqvist claims that such-and-such (syntactically characterized!) declarative formulas fulfill the specified requirements and thus can be regarded as direct answers to the analyzed questions. On the other hand, since these formulas are characterized in syntactic terms, the question-direct-answer relationship is

LOGICALTHEORIES OF QUESTIONS

49

effective; moreover, the concept of direct answer is non-contextual .

2.2.5. Hintikka's analysis 2.2.5 .1. Questions. Also Hintikka adopts the imperative-epistemic approach to questions (cf., e.g., Hintikka, 1974, 1976, 1978, 1982a, 1982b, 1983). To be more precise, Hintikka interprets questions as requests for information (knowledge): according to his view, each question can be paraphrased as an expression which consists of the operator "Bring it about that" followed by the so-called desideratum of the question. The desideratum describes the epistemic state of affairs the questioner wants the respondent to bring about. For example, the question: (2.1) Did you arrive by car, or by bus? should be paraphrased as: (2.27) Bring it about that I know whether you arrived by car or by bus. The question: (2.4) Where did you buy your car? should be paraphrased as: (2.28) Bring it about that I know where you bought your car. The question: (2.29) Who inspired the murder of Julius Caesar? is to be paraphrased as: (2.30) Bring it about that I know who inspired the murder of Julius Caesar. In general, the desideratum of a direct question consists of the expression "I know " within the scope of which the corresponding indirect question occurs . Thus desiderata of various questions contain such epistemic expressions as "know whether" , "know where", "know who", "know when", "know why", etc. These concepts of knowledge are explicated by Hintikka in terms of the concept of "knowing that". In doing this he makes use of his earlier results in epistemic logic, but also introduces some modifications and novelties to them (cf. Hintikka , 1983). One of the basic ideas of Hintikka's approach is that the question-forming words (i.e. "which" , "what", "where", "who", etc.) are analyzed as a kind of ambidextrous quantifiers, that is, quantifiers which can be construed either existentially or universally . For example, the desideratum of the question (2.29) can be analyzed either as:

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50

(2.31) 3x; (x; inspired the murder of J .e . the murder of J.e.») or as:

1\

3xj (xj

= x,

(2.32) \Ix; (x; inspired the murder of J.e.

~

3xj (xj

= x, 1\ K, (xj

1\

K) (xj inspired

inspired

the murder of J.e.))) where K, stands for the epistemic operator "1 know that" and x.; xj range over persons. By and large, the expression (2.31) describes the epistemic situation in which I know of some person(s) that he/she/they inspired the murder of J.e. (For (2.31) is equivalent to "3x; K, (x; inspired the murder of J.e.") The expression (2.32), in turn, describes the epistemic situation in which I know of each of the person(s) who inspired the murder of J.e. that he/she did it. The question (2.29) is a simple wh-question. The formal counterparts of desiderata of simple wh-questions fall under the scheme: (2.33) 3x; (Ax;

1\

3xj (xj

= x,

1\

K I A(x/x))

provided that the existential reading is assumed, or under the scheme: (2.34) \Ix; (Ax;

~

3xj (xj

= x,

1\

K, A(x/xj »)

provided that the universal reading is taken into consideration; in both cases xj is assumed to be substitutable for x; in Ax;.

The question (2.4) is to be analyzed in a similar way as the question (2.28) . The desideratum of the question: (2.1) Did you arrive by car, or by bus? under the existential (or disjunctive) reading of it is: (2.35) K, (you arrived by car) v K, (you arrived by bus) When the universal (or conjunctive) reading of (2.29) is assumed, the desideratum is: (2.36) (If you arrived by car ~ K I (you arrived by car» by bus ~ K, (you arrived by bus»

1\

(If you arrived

To speak generally, in the case of whether-questions the desiderata have the forms : (2.37) K, AI v .. . V K) An (2.38) (AI ~ K, AI) 1\ .. . 1\ (An ~ K, An) where (2.37) corresponds to the existential (disjunctive) reading and (2.38) corresponds to the universal (conjunctive) reading. Hintikka analyzes also questions different from simple wh-questions and whether-questions; yet, it seems that the above examples are sufficient to

LOGICALTHEORIES OF QUESTIONS

51

illustrate the basic ideas of his approach. Let us finally add that, unlike Aqvist, Hintikka does not pay much attention to the logical analysis of the imperative components of questions. It is assumed, however, that the imperative operator "Bring it about that" is conditional on the truth of the presupposition of a question; its force can be roughly expressed by the formula "assuming that the presupposition of the question is satisfied, bring it about that."

2.2.5.2 . Conclusive answers. The concept of conclusive answer plays the central role in Hintikka's theory of questions and answers. Roughly, a conclusive answer is a reply which brings about completely (that is, without the need of further explanations) the epistemic state of affairs the questioner wants to be brought about. Since this epistemic state of affairs is described by the desideratum of a question, we can also say that a conclusive answer brings about the truth of the desideratum. But since the desideratum describes the (desired) epistemic state of affairs of the questioner, the concept of conclusive answer is contextual: one and the same reply mayor may not be a conclusive answer depending on the knowledge of the .questioner. For example, the following sentence: (2.39) Sextus Fabius inspired the murder of Julius Caesar. is a conclusive answer to the (existentially interpreted) question: (2.29) Who inspired the murder of Julius Caesar? in case the questioner already knows who Sextus Fabius is, but (2.39) is not a conclusive answer if the questioner doesn't know the identity of Sextus Fabius . Yet, although the concept of conclusive answer is contextual. it doesn't mean that it is completely subjective and irregular. A reply B is a conclusive answer to a question Q if B together with the description of the questioner 's state of knowledge entails (in the sense of some underlying epistemic logic) the desideratum of the question Q.2 The main task of the theory of answers is to specify the conclusiveness conditions, that is, epistemic formulas which together with replies of strictly defined forms entail the desiderata of the analyzed questions. Having these conditions specified, we may say that a given reply is conclusive for'a given questioner if the appropriate conclusiveness condition(s) is satisfied by him/her (i.e, belongs to the description of his/her knowledge). For example, it can be shown that the conclusiveness condition for a reply of the form:

2 Let us recall that the reference to a questioner is already present in the logical form of a question.

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CHAPTER 2

(2.40) a inspired the murder of Julius Caesar.

(where a is a name of a person) to the existentially interpreted question (2.29) is (we leave the formalization apart): (2.41) I know who a is. So a given reply of the form (2.40) is a conclusive answer for me (or, to be more precise, is a conclusive answer to the existentially interpreted question (2.29) asked by me) if the corresponding sentence of the form (2.41) is true (is an element of the description of my actual state of knowledge). Note that conclusive answers need not be true. Hintikka specifies the conclusiveness conditions for many questions and replies to them.

2.2.6. Kubinski's analysis Questions considered by Kubinski are meaningful expressions of some formalized languages; these questions, however, are regarded as formal counterparts of natural-language questions or interrogative sentences. Kubinski's concept of question of a formalized language is purely syntactical: the leading idea of the analysis is that each question consists of an interrogative operator and a sentential function. Interrogative operators, in tum, consist of both constants and variables. The only free variables in the sentential functions which occur in questions are the variables of the corresponding interrogative operators ; these variables are "bound" by the interrogative operators. The principal possible answers to questions are called by Kubinski direct answers; they are defined in syntactic terms. Yet, in the case of relatively simple questions the appropriate definitions are formulated in such a way that each person familiar with the notation of first-order logic can really regard the just defined direct answers as (using Kubinski's words) "the simplest, most natural, admissible answers." When more complicated questions are considered, the assignment of direct answers is motivated by the possible readings of some question-forming expressions of natural language; the specification of sets of direct answers helps us to clarify their meaning. In order to illustrate the basic ideas of Kubinski let us concentrate on two categories of questions analyzed by him, namely, on the so-called numerical questions and propositional questions.

2.2.6.1. Simple numerical questions. Let us first take a look at the following table (k stands here for a positive integer):

LOGICAL TIlEORIES OF QUESTIONS

53

Table 1: (La) For some x;, Ax;. (2.a) For at least k X;,

Ax;.

(3.a) For more than k

x.; Ax;. (4.a) For exactly k

x;, Ax;.

(1.b) (l .c) (2.b) (2.c) (3.b) (3.c) (4 .b) (4.c)

For which [at least one] x.; Ax; ? For which [all] X;, Ax; ? For which [at least k] x;, Ax; ? Which are all [at least k] X; such that Ax; ? For which [more than k] x;, Ax; ? Which are all [more than k] X; such that Ax; ? For which [exactly k] x.; Ax; ? Which are all [exactly k] Xi such that Ax; ?

The expressions which occur in the right column of the above table represent the paraphrases of simple numerical questions; each such question consists of a sentential function Ax; and a simple numerical operator. The left column shows , however, that there is an analogy between the structure of simple numerical questions and the structure of some first-order (numerical) sentences. Yet , each numerical declarative sentence is paired with two simple numerical questions . Let us now be more formal. Assume that J* is a first-order language with identity whose vocabulary contains some closed terms and all the logical constants: -, , -. , v , A , == , V , 3 (cf . Section 2.1.1).3 Let us now extend the vocabulary of the language J* in order to obtain the vocabulary of a certain new language J t- The extension goes on by adding the following symbols: an infinite list of numerals 1,2,3, .. . , and the constants or is it the case that A2 , or is it the case that An? Is it the case that Al and is it the case that A 2 and is it the case that An?

'"

•••

Is it the case that A?; if so, is it also the case that B? It is the case that A; is it also the case that B? Each propositional question is accompanied with a set of direct answers to it. Again, direct answers are defined syntactically, but they are also assumed to be the simplest possible answers. The following table characterizes direct answers to propositional questions.

Table 5.

Question

[a]a A

Direct answers asA, ....,A

[!3"l!3n AI> ... , An

btAI> ... , An where i = 1,2, ..., n.

WWA .. ..., An

dnit ,... ,i>. AI> ... , An

[f]f AB

nAB,fAR,!AB MB, !AB

[1/]'1/ AB

Questions of the form [a]a A may be regarded as simpleyes-no questions (let us recall that asA is equivalent to A). Since btA .. ..., An is equivalent to asA; and hence to A;, questions of the form [!3n]!3n AI"'" An may be called disjunctive questions. A sentence of the form d nj , i>. A.. ..., An is equivalent to the conjunction BI A ... " Bn , where B, (l '~ r ~ n) is either of the form asAr or of the form ....,Ar (to be more precise, B, is equal to asAr if r is one of the indices iI' ..., ik' and equal to ....,A r otherwise; if the sequence ii ' ..., t, is empty, then B1 " • • • " B, is ....,A I " ••• " ....,AJ. Thus , roughly, questions having the form [o"lon AI ' ..., An ask about the logical value of each of the sentences AI, .. ., An' We shall call them conjunctive questions. Questions of the form [elf AB and [1/]7/ AB are called conditional questions. The direct answers to a question [e]e AB are the sentences !!AB, fAR, iAB,

62

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which, in tum, are equivalent to the sentences ""A, A /I B, A /I ""B, respectively; hence Kubinski calls them conditional questions withrevocable antecedents. The direct answers to questions of the form [1/]1/ AB are the sentences kAB, lAB, which are equivalent to A /I B and A /I ""B, respectively; thus these questions are called conditional questions withirrevocable

antecedents. Some examples may be helpful here. The formal counterpart of the question: (2.86) Is John handsome? is of the form [ala A. The question: (2.87) Is John handsome or intelligent? falls under the scheme [fj2]132 AB. But the question: (2.88) Is John handsome and is he intelligent? has a formal counterpart of the form [02W AB.s The question: (2.90) If Mary is John's mother, then is he the son of Peter? has two possible readings. Under the first reading the question (2.90) can be answered, int.al., by the sentence "Mary isn't John's mother"; under this reading the formal counterpart of (2.90) is of the form [e]e AB. The second reading does not allow for answering (2.90) with the negation of the "antecedent"; in this case the analyzed question falls under the scheme [1/] 1/ AB.

Let us stress that the questions described above do not exhaust the list of questions analyzed by Kubinski.

2.2.7. Belnap's analysis As in the case of Kubinski, also Belnap's approach to questions is a nonreductionistic one. Yet, Belnap's theory differs considerably from that of Kubinski and other adherents of the "interrogative operator-sentential function" approach.

2.2.7.1. Interrogatives. Belnap distinguishes between: (a) natural language questions, (b) interrogatives, which are expressions of some formalized languages, and (c) questions understood as abstract (set-theoretical) entities. Interrogatives are not only formal counterparts of natural language questions, but they also express questions understood as abstract entities.

5 The same can be said about the question "Is John handsome and intelligent?" provided that it is interpreted as a shortland of (2.88) .

LOGICAL THEORIES OF QUESTIONS

63

An interrogative may be simple or compound. The basic constituents of a simple interrogative are: the question mark ?, the lexical subject and the lexical request. The question mark is interpreted as a sign of the function which assigns to the lexical subject and lexical request of a simple interrogative the corresponding (abstract) question; such a question consists, in tum, of the abstract subject and the abstract request. Compound interrogatives result from simple interrogatives by performing some (logical or boolean) operations on them or their lexical subjects. Each interrogative presents a set of nominal alternatives together with some suggestions as to what kind of selection among them should be made. The relevant alternatives are determined by the lexical subject; the required kind of selection is characterized by the lexical request. In this short presentation we shall restrict ourselves to Belnap's analysis of interrogatives, leaving apart his analysis of questions understood as abstract entities . Moreover , we shall concentrate on the so-called elementary interrogatives, that is, whether-interrogatives and which-interrogatives. Elementary interrogatives constitute a subclass of simple interrogatives. Let J** be a first-order language with identity whose vocabulary contains int.al. all the logical constants ." s , v, -., ==, 3, V, some individual constant(s) and possibly some function symbol(s). Let us now extend the vocabulary of J** to the vocabulary of a certain new language J 3• The extension goes on by adding the following symbols: an infinite list of numerals I, 2, 3, ... , the question mark ?, and the following technical symbols: ;t , II (double stroke), - (dash), ; (semicolon). Terms and declarative formulas of J 3 are basically those of J** (basically, since generalized conjunctions of the form Al r; .. • /\ An and generalized disjunctions of the form Al v .. . v An are also permitted) . It is assumed that the grammar (syntax) of J 3 distinguishes in the class of sentential functions of J 3 with exactly one free variable a certain subclass: the elements of this subclass are called category conditions. Moreover , it is assumed that the grammar assigns to each category condition a non-empty class of names (i.e, closed terms) of J3: this class is called the nominal category determined by the corresponding category condition . (Some semi-formal example may be helpful here: if we agreed that "x is a country" is a category condition, then we might say that the nominal category determined by it consists of the names of countries). We leave apart the technical details of the above construction: for our purposes it suffices to say that the introduction of category conditions and nominal categories allows to reflect in J3 some properties of many-sorted languages without introducing many categories of

64

CHAPTER 2

variables. Category conditions will be designated below (on the metalanguage level) by Ctxi" ... , C"xin' By IC.h I we shall designate the nominal category determined by the category condition CjXir A metalinguistic expression of the form ~(Xil . . , Xi) refers to the sentential functions of J) whose free variables are at least the (explicitly listed) variables Xi" .. . , x in' To speak generally, an elementary interrogative is an expression of J) falling under the following (meta)schema: (2.91) ? (s c d) p where p is the lexical subject, s is the lexical selection-size specification, C is the lexical completeness-claim specification and d is the lexical distinctness-claim specification; s, C and d are constituents of the lexical request of the interrogative. The lexical subjectofa whether-interrogative is an expression of J) of the form: (2.92) (AI' ... , A,,) where AI' .. ., An are syntactically distinct sentences (d-wffs with no free variables), n ~ 1, and none of the sentences At, .. ., An is a conjunction of sentences from the set {Ah ... , An}. The sentences AI' .. ., An are called nominalalternatives presented by the corresponding whether-interrogative. The lexical subject of a which-interrogative is an expression of J) of the form : (2.93) (Ctxi " .. . , C,t-Xim; Xim+l' ... , Xin II ~(Xi' ... x in)) where n ~ 1, m ~ 0, n ~ m. The sentential function ~(Xil ... Xi.) is called the matrix ofthe interrogative; the variables are called queriables. A nominal alternative presented by a which-interrogative whose lexical subject has the form (2.93) is a d-wff of the form: (2.94) ~(X;/Uh .. . , xi/u,,)

where uj E /Chl for each 1 s j s m. In other words, a nominal alternative presented by a which-interrogative is a d-wff which results from the matrix of this interrogative by proper substitution of closed terms for the queriables; the terms substituted for the queriables which occur both in the matrix and category conditions of the interrogative must belong to the nominal categories determined by these category conditions. 6

6 Of course, if the lexical subject of a which-interrogative contains no category condition (i.e , if m = 0), this restriction does not apply .

LOGICAL THEORIES OF QUESTIONS

65

Lexical subjects determine the relevant (nominal) alternatives; the role of lexical requests is to characterize the required kind of selection among them. A lexical request consists of the lexical selection-size specification, the lexical completeness-claim specification and the lexical distinctness-claim specification. The lexical selection-size specification of an elementary interrogative is an expression of the form: (2.95) vW or of the form: (2.96) .where v, wEN and W ~ v. Intuitively speaking, v indicates the lower bound on the selection size of the alternatives called for, whereas W indicates the upper bound; the dash indicates that there is no upper bound. As far as the lexical completeness-claim specifications of elementary interrogatives are concerned, Belnap explicitly distinguishes only maximal (notation: V) and empty (notation: -) lexical completeness-claim specifications of these interrogatives . By and large, elementary interrogatives with maximal completeness-claim specifications call for all the true alternatives presented by them; if the completeness-claim is empty, the amount of true alternatives called for is unspecified. The lexical distinctness-claim specification of elementary interrogatives is either of the form ~ (in the case of the so-called non-empty distinctnessclaim specification) or of the form - (the empty distinctness-claim specification). By and large, if an interrogative contains the non-empty distinctnessclaim specification, it requires that the alternatives included in any of its (direct) answers should be different not only syntactically, but also semantically. Among elementary interrogatives only which-interrogatives can have non-empty distinctness-claim specifications. Let us now consider some examples. We will first analyze them in a semiformal way and then we will show what are the forms of the corresponding interrogatives . The interrogatives which formalize the following questions: (2.51) (2.52) (2.64) (2.56) (2.43) (2.59)

Which Which Which Which Which Which

Polish town is greater than Poznan? Polish towns are greater than Poznan? is the unique Polish town greater than Poznan? are all of the Polish towns greater than Poznan? Polish towns, at least four, are greater than Poznan? are all of the at least four Polish towns greater than

66

CHAPTER 2

Poznan? (2.97) Which Polish towns, at least four but at most five, are greater than Poznan? (2.98) Which are all of the at least four but at most five Polish towns greater than Poznan? have identical lexical subjects; they can be semi-formally represented by: (2.99) (x; is a town II X; is a Polish town greater than Poznan) and then formalized as an expression falling under the schema: (2.100) (C;x;IIAx;)

Yet, the above questions are different and so are their formal counterparts : the corresponding interrogatives have different lexical requests. Intuitively speaking, the questions (2.51) and (2.64) ask about exactly one town; in both cases the selection-size specification of the relevant interrogatives is But (2.64) calls for the only town greater than Poznan, whereas (2.51) calls for exactly one example of such towns. Hence the interrogative formalizing (2.64) contains the maximal completeness-claim specification V, whereas the completeness-claim specification of the interrogative which corresponds to (2.51) is -, i.e. the empty one. The situation is similar in the case of the questions (2.52) and (2.56): both of them ask about at least two towns greater than Poznan and thus the appropriate selection-size specifications are 2-' but (2.56) calls for all such towns and hence the completenessclaim specification of the corresponding interrogative is V, whereas (2.52) calls for at least two examples and hence - is the completeness-claim specification of the corresponding interrogative. The interrogatives which formalize the questions (2.43) and (2.60) have selection-size specifications 4-' The interrogatives which corresponds to the questions (2.97) and (2.98) have l as their selection-size specifications. Yet, in the case of formal counterparts of the questions (2.59) and (2.98) V is the completeness-claim specification, whereas the interrogatives formalizing the questions (2.43) and (2.97) have - as the appropriate completeness-claim specifications (on one construal of these interrogatives). We may say that the interrogatives which corresponds to the questions (2.52), (2.56), (2.43), (2.59), (2.97) and (2.98) have non-empty distinctnessclaim specifications. The lexical subject of the whether-interrogative which formalizes the question: (2. 101) Did you arrive by car ?

r

67

LOGICALTHEORIES OF QUESTIONS

is of the form: (2.102) (A, -,A) The lexical subject of the interrogative which is the formal counterpart to the question: (2.1) Did you arrive by car, or by bus? has the form : (2.103) (AI> A2 )

In both cases the selection-size specification is specifications are the empty ones.

II

lind the remaining

2.2 .7.2 . Direct answers . Belnap uses the term "direct answers" for the "principal" possible answers (cf. Section 2.2.2) to questions and interrogatives. Direct answers to elementary interrogatives are defined in purely syntactic terms. (To be more precise, this is true with respect to nominal direct answers; the corresponding real direct answers are semantic entities, similarly as questions are. Below we will be speaking only about nominal direct answers; the word "nominal" will be normally omitted .) The basic underlying idea, however, is that a direct answer is a d-wff which in some sense satisfies the conditions imposed by the lexical request of an interrogative. The lexical request consists of the lexical selection-size specification, the lexical completeness-claim specification and the lexical disctinctness-claim specification . Correspondingly, direct answers to interrogatives are build up of three consituents: lexical selections, completeness-claims and distinctnessclaims . For the sake of brevity we will be omitting the word "lexical" in our further presentation . Let Q be an elementary interrogative . By a selection connected with the interrogative Q (Q-selection for short) we will mean a d-wff of J 3 of the form:

(2.104)

BI

A

•••

A

B,

where k ~ 1 and BI> .. ., B, are nonequiform (i.e. syntactically different) nominal alternatives presented by Q (cf. Section 2.2.7 .1). Let Q be a whether-interrogative. Its (lexical) subject is thus of the form: (2.92) (AI' ..., An) and AI> .. ., An are the nominal alternatives presented by it. Let (2.105) Ail A •• • A Ai. be an arbitrary but fixed Q-selection. If at least one nominal alternative presented by Q does not occur in the selection (2.105) , the maximum

68

CHAPTER 2

completeness-claim with respect to this selection is defined as: (2.106) 'At, II ... II ,Atj where A ..., Atj are all the alternatives presented by Q which do not occur t" in the selection (2.105) and the sequence Atl' ... , Atj is a subsequence of the sequence AI, ... , An. If all the nominal alternatives presented by Q do occur in the selection (2.105), the maximum completeness-claim with respect to this selection is defined as the empty expression . Let Q be a which-interrogative. Its lexical subject has the form: (2.93) Let

(Ctxil, ... , CtXim; x im+ I'

... ,

xi.ll cfl(xi, ... x in))

(2.107) cfl(x;/Ull' .. . , x;/Ul n) II .. . II cfl(Xi/Uk l , ... , x;/ukn) be an arbitrary but fixed Q-selection. The maximum completeness-claim with respect to the selection (2.107) is defined as: (2.108) \lxil .. . \lxin (cfl(xi, . .. (Xii

=U

k l 1\ .. .

X in

x in) ... «Xii

= Ukn)) )

= Ul l

II ...

1\

Xin

= Ul n)

V ... V

Thus, roughly , a maximum completeness-claim with respect to a given selection is a formula which says that the selection includes all the true nominal alternatives presented by the interrogative . The distinctness-claim with respect to the selection (2.107) is defined as a conjunction of disjunctions of the form: (2.109) «uil ~ uj l ) v ... V t»; ~ ujn)) (1 ~ i < j ~ k) provided that k > 1; otherwise the appropriate distinctness-claim is identified with the empty expression. A (non-empty) distinctness-claim with respect to a given selection is thus a formula which says that the nominal alternatives included in the selection are really and not only syntactically different. Let us now adopt the the following notation : SkQ - a Q-selection which consists of exactly k conjuncts, max(SkQ) - the maximum completeness-claim with respect to SkQ, dist(SkQ) - the distinctness-claim with respect to SkQ. Direct answers to elementary interrogatives can now be characterized by means of the following table:

69

LOGICAL THEORIES OF QUESTIONS

Table 6

The form of an elementary interrogative Q ? (VWV "#-) P

The form of a direct answer to Q SkQ II max(SkQ) II dist(SkQ) where w ~ k ~ v. SkQ II max(SkQ) where k ~ v.

II

dist(SkQ)

SkQ II max(SkQ) where w ~ k ~ v. SkQ II max(SkQ) where k ~ v. SkQ II dist(SkQ) where w ~ k ~ v. SkQ II dist(SkQ) where k ~ v. SkQ where w

~

k

~

v.

Thus a direct answer to an elementary interrogative Q includes: (1) a Qselection which consists of as many? syntactically different nominal alternatives presented by Q as it is required by the lexical selection-size specification of Q, (2) the maximum completeness-claim with respect to this selection provided that Q has the maximal lexical completeness-claim specification, and (3) the distinctness-claim with respect to the considered selection provided that Q has the non-empty distinctness-claim specification . Let us finally note that the above presentation differs in some minor details from the original Belnap's presentation, but nevertheless leads to the same results.

7 To be more precise, if the selection-size specification is of the form :', the appropriate selection should consists of at least v, but at most w different nominal alternatives ; if the selection-size specification has the form ,-, the appropriate selection should consists of at least v distinct nominal alternatives, whereas the upper bound of them is undetermined .

CHAPTER

3

QUESTIONS

The considerations of the previous Chapter show that the question-andanswer systems can be built in different ways; these systems differ not only in technical details, but also in their philosophical and logical backgrounds. The controversy about the "nature" of questions is not a simple conceptual dispute; the accepted solutions to this problem determine the conceptual space for further research. If the radical reductionist view is correct, no logic of questions (in the very sense of "logic", as opposed to "logical theory") is possible . If the moderate reductionist view is correct, the logic of questions (whatever it might have been) should be developed within the framework of some other logic or logics. But the acceptance of the non-reductionist view leaves room for the construction of autonomous logic or logics of questions . We are not going to settle here the controversy about the nature of questions. The aim of this book is to present a prolegomenon to the inferential approach to erotetic logic; the construction of an autonomous logic of questions is one of the elements of the enterprise . For this reason we shall adopt here the non-reductionist perspective. What we really need is a method of constructing formalized languages whose meaningful expressions are both declarative formulas and questions , and within which some assignment of direct answers to questions is made. So we might have proceeded by adopting the methods of constructing questions or interrogatives developed by Belnap, or by Kubinski, or by some other adherents of the non-reductionist view (with some minor adjustments; see Section 3.3). But the already existing proposals do not exhaust the list of conceptual possibilities ; what is more important, they are rather complicated, at least at first glance. Thus we shall propose here some new approach, which is especially suited for the purposes of this analysis. Let us stress , however , that the acceptance of it is not a necessary condition of developing the inferential approach to erotetic logic; it only allows us to do it in a relatively simple way.

QUESTIONS

3.1.

71

QUESTIONS AND DIRECT ANSWERS

3.1. 1. Preliminaries Questions of formalized languages will be defined here in syntactic terms. The basic idea of our proposal is that each question is made up of two main constituents: the sign? and an object-language expression such that the equiform expression of the metalanguage designates the set of direct answers to the question. Let us stress that it is not tantamount to the reduction of questions to sets of direct answers: questions are still object-language expressions which have strictly defined forms. In order to enrich a formalized language with questions we have to supplement its vocabulary by some additional signs . These signs can be divided into erotetic constants and technical signs. The symbols?, { }, S, 0, U, W, T will be used here for the erotetic constants; the signs I (stroke) and, (comma) will be the technical signs. Having the erotetic constants and technical symbols at hand, we can formulate the erotetic formation rules, that is, rules which characterize what expressions of the analyzed language are questions of this language. The concept of question will be defined here for some class of formalized languages. This class consists of: (a) first-order languages with identity (cf. Section 2.2.1) enriched with the erotetic constants ? and { }, and (b) first-order languages with identity whose vocabularies contain infinitely many closed terms, enriched with: the erotetic constants ? and { }, at least one of the following erotetic constants : S, 0 , U, W, T, and, if necessary, the technical sign I (stroke). (c) first-order languages with identity whose vocabularies contain at least two closed terms and some unary predicate symbols which perform the role of category qualifiers, enriched with: the erotetic constants? and { }, at least one of the following erotetic constants: S, 0, U, and the technical signs [, ] (square brackets) , / (slash). When saying that the vocabulary of a given first -order language contains an infinite number of closed terms, we have in mind that among the signs of this language there are: (a) infinitely (denumerably) many individual constants, or (b) at least one function symbol and at least one individual constant. The assumption concerning the occurrence of infinitely many closed terms allows us to introduce some questions which have infinite sets of direct answers. The concept of category qualifier will be explained in the sequel.

CHAPTER 3

72

We assume that if the connectives 1\, V or ==, or the quantifier V are not among the primary signs of a language which fulfills the above conditions, then these symbols are introduced into the language by means of the appropriate (standard) definitions; it is also assumed that the technical sign , (comma) belongs to the vocabulary . 1 The concepts of expression, term, closed term, declarative well-formed formula (d-wft), sentence and sentential function , and the remaining syntactical concepts as well as the appropriate metalinguistic symbols are understood in the case oflanguages enriched with questions in an analogous way as in the case of the (basic) first-order languages. One additional remark is in order here . The signs: {, }, S, 0, U, T, W will be used below in two different roles : as erotetic constants and as metalinguistic expressions; we assume that the symbols equiform with the above erotetic constants do occur in the metalanguage and are understood there not as erotetic constants, but as expressions by means of which we can build names of certain sets of d-wffs (details will be explained below). This duality will not lead to any misunderstandings, however; the context of appearance will always determine their actual role at a given place.

3.1.2. Questions of the first kind Let ::e be an arbitrary but fixed language for which we want to define here the concept of question, that is, an arbitrary but fixed language which fulfills at least one of the conditions (a), (b) and (c) of the previous Section. A question of the first kind of the language ::e is an expression of this language of the form: (3.1)

?{AIt ... , A n },

where n > 1 and Alt • • • , An are nonequiform (i.e. syntactically distinct) sentences of ;e. . Thus in each question of the first kind there occur at least two nonequiform sentences (i .e . closed d-wffs) of the language as well as the erotetic constants ? and { }. If ? {Alt • •• , An} is a question of the first kind, then the sentences Alt ... , An are called direct answers to this question. Since each expression is a finite sequence of symbols, each question of the

1 According to the definition of a first-order language with identity given in Section 2.1 .1. the vocabulary of such a language need not contain the above connectives. the universal quantifier or the comma ; this is why we had to enter here the above pedantic remark .

73

QUESTIONS

first kind has a finite number of direct answers. Let us notice that the definition of questions of the first kind yields. first. that each such question has at least two direct answers. and second. that each finite and at least twoelement set of sentences of :£ is the set of direct answers to some question of the first kind of :£. The brackets { } belong to the vocabulary of the language :£. But they also occur in the metalanguage of :£. The finite set made up of the sentences AI' ... . An is usually designated by {A•• .. .. An}; this convention is also kept throughout this book. But the sentences A.. .. .. An are the direct answers to a question of the form ? {AI' .. .. An} . Thus we may say that each question of the first kind consists of the sign ? and some expression of the object language such that the equiform expression of the metalanguage designates the set of direct answers to the question. Again. let us stress that we do not identify here questions of the first kind with sets of direct answers to them: questions are linguistic entities of a strictly defined form. Yet. when a question of the first kind is given . it is extremely easy to say what sentences count as the direct answers to it. The concept of question of the first kind is very general. We may say that among questions of the first kind there occur. for example. expressions of the form: (3 .2)

? {A. ""A}

(3 .3)

? {A

(3.4)

? {...,A. A A B. A A ""B} ? {A f\ B. A f\ ""B}

(3.5)

A

B. A

A

...,B• ...,A

A

B• ...,A

A

""B}

where A . B are sentences. ? {A(x;!u). 3xi (Axi A Xi ;t u)} where Axi is a sentential funct ion with Xi as the only free variable and u is a closed term. If the considered language contains at least n distinct closed terms as well as unary and n-ary predicate symbols (where n > 1). then among questions of the first kind of this language there occur int.al. expressions having the form : (3 .6)

(3.8)

? {P(u). 3xi (P(xi) A Xi ;t u)} ? {P(u,)• .. .. P(un)}

(3.9)

? {P(u.)

(3.7)

Xi = U. Xi = UI

A V

VXi (P(Xi) -. Xi

= U.). P(U.)

Xi = UZ) • ...• P(U I ) Xi = UJ}

V ... V

A • •• A

A

P(U Z)

P(UJ

A

A

VXi (P(X;) -.

VXi (P(Xi) -.

74

CHAPTER 3

where u t , (3.10) (3.11)

•• • ,

u; (n

>

1) are nonequiform closed terms,

? {R(u 11,

• •• ,

? {R(u 11, = UII

1\ • • • 1\ X in

Xii

••• ,

u1n) , • • • • , R(ukl' .. .., ukn)} uln) 1\ VXil . •. v»: (R(xiI ,

R(ukl' .... , Ukn) 1\ X in

= Uln)

• •• ,

xin)

= U tn), •••, R(utl' ... , U 1n)

1\ VXil ... VXin

= Ukl

V • •• V (Xii

(R(xil , 1\ • ••

..... 1\ ••• 1\

•••, X;.) ..... (Xii 1\ X in

= U ll

1\

= ukn» }

where k > 1, n > 1 and R(ulI , ••• , u t.) , •• •• , R(uk" .... , ukn) are nonequiform sentences. The erotetic constants ? and { } are not regarded here as counterparts of any particular question-forming expressions of English or other natural languages . The basicfeature of questions of thefirst kind is that they have finite sets of direct answers; in natural languages questions having this property are formed by means of different expressions and grammatical transformations. Each quest ion of the form ? {AI> .. ., An} can be read "Is it the case that AI' or is it the 'case that A2, ••• , or is it the case that An?", but in some special cases a different reading is also possible or even legitimate. In particular, questions of the form: (3.2)

? {A, ,A}

can be read "Is it the case that A?". We shall call them simple yes-no questions. Questions falling under the schema: (3.6)

? {A(x/u), 3xi (Axi

1\ Xi

¢ U)}

may be read "Is it U that fulfills the condition Axi?" ; these question may be called focussed yes-no questions. IfAxi is an atomic sentential function Px., the corresponding question of the form (3.7) can also be read "Is it U that has the property P?". Questions of the form: (3.5)

? {A

1\

B, A

1\

'B}

can be read "It is the case that A; is it also the case that B?"; we shall call them conditional yes-no questions. Questions of the form: (3.4)

? {,A, A

1\

B, A

1\

'B}

can be read "Is it the case that A?; if so, is it also the case that B?". Questions having the form :

B, ,A 1\ 'B} can be read "Is it the case that A and is it the case that B?"; they may be (3.3)

? {A

1\

B, A

1\

'B, ,A

1\

called (binary) conjunctive questions. Questions of the form: (3.8)

? {P(u t ) , •• •, P(uJ}

75

QUESTIONS

can be read "Which of the u" ..., u, has the property P?"; questions falling under the scheme: (3.9)

? {P(u t ) Xi = Xi =

1\

U, V

VXi (P(xi) Xi

-+

Xi

= Uz),

U 1 V •.. V

=

... , P(u,) Xi = Un)}

U1),

P(u,)

1\ • • • 1\

1\

P(uz)

P(un)

1\

1\

VXi (P(Xi)

VXi (P(Xi)

-+

-+

can be read "Is it the case that UI> and only UI> has the property P, or is it the case that U 1 and Uz, and only they, have the property P, .... , or is it the case that U 1 and Uz and .. . and un' and only they, have the property P?". Similarly, questions falling under the schema (3.10) may be read "Which of the following n-tuples: 1) that satisfies a given sentential function . Each question of the form ? S(Ax;, ... x;.), where n > 1. can be read "Which n-tuple .... Xi/Un) and all the sentences of the form:

(3.15)

A(xi/u 1, ,

... ,

X;,/U 1,,)

/I

. .. /I

A(xi/ukl' .. ., x;'/ut..)

where k > 1 and A(x;,1u 1, , . .. , Xi/U 1,,) , ... , A(x;,1u k,• ... . xi/ u",) are nonequiform (i.e . syntactically different) sentences which belong to the set S(Axi , ••• Xi,,)' In other words. the set O(Ax;, ... x;,,) consists of all the sentences from the set S(Axi • • •• x;.,) and all the sentences which are conjunctions of nonequiform sentences from this set. By a direct answer to a question of the form ? O(Axi , • •• x;,,) we shall mean any sentence from the set O(Ax;, ... Xi.)' Open questions of the second kind fit the general pattern of the proposed approach: they consist of the sign? and object-language expressions such that the equiform metalanguage expressions designate the sets of direct answers. It is obvious that each open question of the second kind has an infinite number of direct answers. Again, the erotetic constants ? and 0 that occur in open questions of the

CHAPTER 3

78

second kind are not assumed to be formal counterparts of any specific question-forming expressions of natural languages. Each open question of the second kind of the form? O(Axi l • •• x,.,,) , where n > 1, can be read "What are some Xi.. .. ., X,." such that Axil .. . X,.,,?"; questions of the form? O(Axi) can be read "What are some x;'s such that Ax;?". We may say that open questions of the second kind are questions about objects or n-tuples of objects that satisfy a given sentential function; the difference between them and existential questions of the second kind is that the latter ask about one example of such objects or n-tuples of objects, whereas the open questions of the second kind may be answered by specifying either example or examples of such objects or n-tuples of objects. Yet, neither existential nor open questions of the second kind call for all the objects or n-tuples of objects that satisfy the relevant sentential functions; this role is performed by general questions of the second kind. A general question of the second kind of :£ is an expression of :£ of the form: (3.16) ? U(Axi l ... Xi.) where n ~ 1. Thus a general question of the second kind consists of the erotetic constants ? and U together with a sentential function enclosed in parentheses; its syntactical structure resembles the structure of existential and open questions of the second kind. . The symbol U occurs also in the metalanguage. On this level, however, its function is different. We assume that a metalinguistic expression of the form U(Ax i , . . . x,.,,) designates the set of sentences made up of all the sentences of the form: (3.17)

A(x;/u(> ... , xi/u,J Xi. = u,J

1\

"lXi' ... "Ix,." (Axil .. . Xi. -. XiI

= U1

1\ ... 1\

where A(xi,lu1, ... ,xju,J is an element of the set S(Axi, .. . x,.,,), and all the sentences of the form: (3.18)

A(Xi,lU I " ... ,xju l .) (Axi, .. . Xi. -. «Xii = f\ ... f\ Xi. = U/cn)))

1\ ... 1\

A(Xi,lUk.. .. .. ,x,.,,/u/cn) 1\ "Ix il .. . "Ix,." Xi. = U 1.) v ... V (Xii = Uk'

U 1, f\ ... f\

wherek> 1 andA(xi,lu lt , ... ,Xi.!U 1.), ... .,A(Xi,lUk...... ,xjU/cn) are nonequiform sentences from the set S(Axi , ... x,.,,) . By a direct answer to a question of the form? U(Axi l • • • Xi.) we mean any sentence from the set U(Axi l ... Xi.). Thus, again, a general question of the second kind consists of the sign? and an object-language expression such that the equiform metalinguistic expression designates the set of direct

QUESTIONS

79

answers to the question. By and large, a direct answer to a general question of the second kind is a sentence which says that the sentential function which occurs in the question is satisfied by some objects(s) or n-tuple(s) of objects and only by them. This is why a general question of the second kind can be regarded as a question about all the objects that satisfy a given sentential function. A general question of the second kind of the form? U(Axil ... Xi,,:' where n > 1, can be read "What are all of the Xii' .. ., Xi" such that Axil . . . Xi,,?"; questions of the form? U(Ax;) can be read "What are all of the x;'s such that Ax;?" . Again, the erotetic constants ? and U are not regarded here as counterparts of any specific question-forming expressions of natural languages; questions about all the objects satisfying a given condition are formulated in natural languages in different ways. Existential, open and general question of the second kind were defined above for a language whose vocabulary contains int.al. all the erotetic constants S, 0 , U. It may happen, however, that some language of the considered kind contains not all, but only some of the above erotetic constants. We assume that the questions of the second kind of such a language are the expressions which can be formed in the way described above by means of the erotetic constants which do occur in it. Generally speaking, all questions of the second kind are questions about objects or n-tuples of objects that satisfy a given sentential function; they differ as to the quantity of entities called for. This quantity, however, is specified in a quantifier-style manner: exactly one example, at least one example, and all the relevant examples. On the other hand, the existential quantifier means "there is at least one" which is tantamount to "there is one or more than one" ; the quantifier "there is exactly one" is definable in terms of the existential and universal quantifiers. In the case of questions, however, there is a substantial difference between asking about exactly one object (or n-tuple of objects) and about at least one object (a-tuple of objects): this is why we have distinguished here the existential and open questions as different basic categories of questions. In Kubinski's theory a similar distinction is expressed by the introduction of different interrogative operators; the same holds true in the case of Aqvist's approach. In Belnap's theory, however, questions about exactly one example and at least one example are distinguished by means of their selection-size specifications. The motivation for introducing the general questions of the second kind as opposed to existential and open questions of the second kind is similar to that which lies behind Hintikka's distinction between the existential and universal

80

CHAPTER 3

reading of desiderata of questions, and Belnap's distinction between interrogatives with maximum and empty completeness-claims; let us recall that also Kubinski distinguishes between questions about all the objects that satisfy a given sentential function and other questions .

3.1.4 . Questions of the third kind As we have shown in Chapter 2, in many logical theories of questions much attention is paid to questions which, to speak generally, specify in a detailed manner the number of entities called for; Belnap's interrogatives with selection-size specifications different from II and 1-' Kubinski's numerical questions, as well as many interrogatives in Aqvist's sense are cases in point here. We shall now show how such questions can be represented within our approach. Assume that 5£ is an arbitrary but fixed language of the considered kind which contains the signs W, T among its erotetic constants and the sign I (stroke) among its technical symbols. In other words , let 5£ be a first-order language with identity whose vocabulary contains infinitely many closed terms, the erotetic constants: ? ,{ }, W, T, and the technical symbol I. It is neither assumed nor denied here that the vocabulary of 5£ contains the erotetic constants S, 0, or U; it is assumed here, however that all the logical constants: ..." -., s ; v , ==. 3. V and the comma do occur (as primary or defined symbols) in ;e. Like questions of the first and second kind, also the questions of the third kind which we are going to define will consist of the sign ? and objectlanguage expressions such that the equiform metalanguage expressions will designate the sets of direct answers. The syntactical structure of these questions may seem complicated at first sight; the definitions proposed below will become more comprehensible if we begin with the characteristic of the relevant sets of sentences, i.e . sets of direct answers . The sign W occurs in the object language as well as in the metalanguage. On the object-language level W is an erotetic constant; its metalinguistic role is different. Let Ax; be an arbitrary but fixed sentential function of the language 5£ which contains X; as the only free variable. We assume that a metalinguistic expression of the form : (3.19) (x, ... X,+k_1 IAx) where k ~ 1 and r is the least index such that for each s ~ r, the variable x. is substitutable for x; in Ax;, designates the infinite sequence of sentential

81

QUESTIONS

functions Bt , B2 ,

•••

of the language ~ defined as follows:"

B; = A(x;!x,) " ... "A(x;lx'+HII-J "X, ~ x,+t "X, ~ X,+2 " " x, ~ X,+Hn-2 " X,+I ~ X,+2 " ... " X,H+n-3 ~ x,+Hn.2

where B, reduces to A(x;lx,) if k = 1. On the metalanguage level an expression of the form: (3.20) W(x, ... x'H_dAxj) designates the set of all the sentences which result from the sentential functions that belong to the sequence (x, ... X,H_IIAxj) by proper substitution of nonequiform closed terms for nonequiform free variables. In other words, a set W(x, ... x,H_tIAxj), where k > 1, is made up of the sentences falling under the schema: (3.21) A(x;!u,) " " A(x;!urn) 1\ U t ~ ... ~ urn where m ~ k and u , urn are nonequiform (i.e . syntactically different) closed terms. A set W(x,IAxj) consists of all the sentences of the form A(x;!u) together with all the sentences falling under the schema (3.21), where m > 1 and u t , ... , urn are nonequiform closed terms. The following remarks can make the above symbolism more comprehensible. An expression of the form W(x, ... x,H_dAxj) contains exactly k distinct variables left of the stroke . On the other hand, the set W(x, ... X,H_IIAxj) is made up of the sentences which are conjunctions of, first, conjunctions of at least k distinct substitution instances of the sentential function Axj , and second, the appropriate distinctness-claims; if k = 1, all the sentences from the set S(Ax;) also belong to it. Thus the number of variables which occur left of the stroke indicates what is the lower bound of nonequiform closed terms which should be substituted in the sentential function which occurs right of the stroke . The sentences in the set W(x, ... x,H_dAxj) say that the objects designated by the substituted closed terms satisfy the sentential function Axj and are distinct (if more than one term was substituted); in each case at least k such objects are taken into consideration. The sign W occurs in the language ~ as an erotetic constant. An existential question of the third kind of the language ~ is an expression of :£ of the form : (3.22) where k

4 X," _I

? W(x, .. . x,H_tIAxj) ~

1 and r is the least index such that for each s

~

r, the variable

The condition imposed on the index r guarantees the substitutivity of the variables x,• ...• for Xi in Ax i•

CHAPTER 3

82

Xs is substitutable for x, in Ax;.5 By a direct answer to a question of the form? W(x, ... x,+k_IIAx;) we mean any sentence from the set W(x, .. . x,+k_IIAx;). When we take into consideration the explanations presented above , we may say that an existential question of the third kind is a question about at least k distinct objects that satisfy a given sentential function. A question of the form? W(x, ... x'+k_dAx;) can be read "What are at least k different x;'s such that Axft", As in the case of questions of the first and second kind, however, we do not regard the erotetic constants ? and W as formal counterparts of any specific question-forming expressions of natural languages; questions about at least k distinct objects satisfying a sentential function can be formed in natural languages in different ways. Although the definition of existential question of the third kind is somewhat complicated, concrete questions of this kind are relatively simple. Let us assume that in the language :£ there occur the predicate symbols PI I and P 12 • The following expressions are examples of existential questions of the third kind of :£:

(3.23) (3.24)

? W(xd PII(X I» ? W(xjx2IPll(Xl»

? W(X2 X3 I'v'x i p 12(X" X2» (3.26) ? W(X5X6X7 13x4 PI2(X4 , XI» Let us now introduce the concept of general question of the third kind. As above , we shall start from the characteristics of sets of direct answers. Let Ax; be an arbitrary but fixed sentential function of :£ with x, as the only free variable. Assume that a metalinguistic expression of the form:

(3.25)

(3.27)

(x, ... X,+k_1 1/ Ax;)

where k ~ I and r is the least index such that for each s ~ r, the variable Xs is substitutable for x; in Ax;, designates the infinite sequence of sentential functions CI , C2 , ... of the language :£ defined as follows:

C.

= A(x/x,)

... 1\

'v'X,+k+._1

A(x/X,+k+._2)

1\ ... 1\

x, ¢ X,+k+.-2

1\

X,+I ¢ X,+2

(A(x/x,+k+._I)

-+ X,+k+._1

1\

x, ¢ X,+I 1\ x, ¢ X,+2 1\ X,+k+.-3 ¢ X,+k+._2 1\

1\ ... 1\

= x,

V .. . V

where C\ reduces to: A(x/x,)

5

Cf. footnote 4.

1\

'v'X,+1 (A(x/x,+,)

-+

X,+I

= x,)

X,+k+._1

= X,+k+._2)'

QUESTIONS

in case k

= 1. A metalinguistic

83

expression of the form:

(3.28) T(x, .. . x,+k_lIIAx;) designates the set of all the sentences of the language ;;£ which result from the sentential functions of the sequence (x, .. . X,+k_l II Ax;) by proper substitution of nonequiform closed terms for nonequiform free variables. Generally speaking, an element of the set T(r, '" X,+k_1 II Ax;) is thus a conjunction of three basic constituents: (1) a sentence being the conjunction of at least k nonequiform substitution-instances of the sentential function Ax;, (2) the sentence which says that the object designated by the substituted closed terms are different, and (3) the sentence which says that the objects designated by the substituted closed terms are the only objects that satisfy the sentential function Ax;. To be more precise, this is the case when k > 1; if k = 1, the analyzed set contains also the sentences which are substitution instances of the sentential function A(x;!x,) /\ VX,+1 (A(x;!x,+,) -+ X,+I = x,). The role of the variables that occur left of the strokes is thus similar as in the case of expressions of the form (3.20) (cf. above). The symbol T was used above in its metalinguistic role. But T occurs in the object language as an erotetic constant by means of which we may construct general questions of the third kind. A general question of the thirdkind of the language ;;£ is an expression of ;;£ falling under the schema:

(3.29) ? T(x, ... x,+k_IIIAxi ) where k ~ 1 and r is the least index such that for each s ~ r, the variable X s is substitutable for Xi in Axi • A direct answer to a question of the form ? T(x, .. . X,+k_1 II Ax;) is a sentence which belongs to the set T(x, ... X,+k_111 Ax;). A question of the form? T(x, ... X,+k_l II Ax;) can be read "What are all of the at least k different x;'s such that Ax;", Since the language ;;£contains infinitely many closed terms, each question of the third kind has an infinite number of direct answers. So far we have assumed that the language for which we defined existential and general questions of the third kind contains the erotetic constants W and T . It may happen, however, that some language of the considered kind contains only one of these erotetic constants; in this case we assume that questions of the third kind of it are the expressions which can be formed in the way described above by means of the erotetic constant which does occur in the language.

CHAPTER 3

84

3.1.5. Categoreally qualified questions. Let ~o be a first-order language with identity whose vocabulary contains at least two closed terms and some unary predicate symbols; it is assumed that all the logical constants: "" ..., 1\, v, ==, 3, V and the comma do occur (as primary defined symbols) in ~o~ .. Let us now extend the vocabulary of ~o to the vocabulary of some new language ~. The extension goes on by adding the erotetic constants ?, { }, at least one of the following erotetic constants: S, 0 , U, and the technical signs: [, ] (square brackets) , I (slash). Terms and declarative well-formed formulas (d-wffs) of ~ are those of ~o. We assume, however, that the grammar of ~ distinguishes in the set of unary predicate symbolsof ~ some non-empty subsetof category qualifiers. An atomic sentential function of the form P(xj) , where P is a category qualifier of ;e, is called category condition of ;e. We also assume that the grammar of ;e assigns to each category condition of ;e some at least twoelement set of closed terms of ;e; this set is called the nominal category determined by thiscategory condition. Theassignment of nominal categories is made in such a way that if P(xj ) and P(xj ) are category conditions which differ only as to their variables (i.e. if i ;t J), the nominal categories determined by them are identical. Some analogy may be helpful here. We may think of the category qualifiers as of formal counterparts of such expressions as "is a place", "is a person", "isa time-period", etc.; the appropriate nominal categories would then consistof the formal counterparts of names of places, names of persons, names of time-periods, etc., respectively. The basic ideaof that construction is similar to that of Belnap (see Chapter 2, Section 2.7.7); for the sake of simplicity, however, we do not allow compound category conditions. Assume that the vocabulary of ~ contains the erotetic constant S. By a categoreally qualified existential question ofthe second kind of ~ we.shall mean an expression of ~ having the form: (3.30) ? [p.(xi l ) ]

•••

[P,,(x..)]/S(Axi l

•••

x..)

where n ~ 1 and Ph ... , P" are category qualifiers of~. On the metalanguage level by [P(xi ) ] we shall designate the nominal category determined by the category condition p(x i ) . An expression of the metalanguage of the form: (3.31)

[p.(xi l ) ]

...

[P,,(x..)]/S(Axi,

...

x..)

will designate the set made up of all the sentences of ~ which result from the sentential function Axil ••• x.. by proper substitution of closed terms for the free variables XiI' • • • , x.. in such a way that a closed term substituted for

QUESTIONS

85

the variable x;. - where 1 :s k :S n - belongs to the nominal category determined by the category condition Pix;.). In other words, elements of the set (3.31) fall under the schema: (3 .32)

A(xi/u t , . .. , x;/u.)

where u, - for 1 :S k :S n - is a closed term that belongs to the set [Pix;.)]. By a direct answer to a question of the form? [Pt(xil)] ... [P.(xin)]/S(Axil ... Xi.) we shall mean any sentence that belongs to the set [Pt(Xil)] .. . [Po (Xin)]/S(Axil .. . xin)· Notice that since each nominal category contains at least two closed terms, each categoreally qualified question of the considered kind has at .least two direct answers. A question of the form? [p\(x i.)] [P.(xin)]/S(Axil ... xin), where n > 1, can be read: "Which n-tuple Y ~ Yt and X IF Y, then x, IF

COROLLARY 4.2.

r,

SEMANTICS

109

COROLLARY 4.4 . If X u 2 1 IF fu ~ for any 2\, ~ such that 2\ n ~ = o and 2 1 u ~ = 2, then X IF f. Corollaries 4.2 and 4.3 are immediate consequences of Definition 4.6. For the proof of Corollary 4.4 assume that X non IF f . So there exists a normal interpretation ~ of:£ such that ~ FX and for each B E f, ~ non FB. Let 2\ be the set of all the d-wffs of:£ which are true in ~ and let ~ be the set of all the d-wffs of:£ which are not true in ~. We now have X u 2 1 non IF f u ~ as required. Thus me-entailment in any of the considered languages is a multipleconclusion consequence in the sense of the monograph Shoesmith & Smiley (1978) (see also below). A relation IF of me-entailment is said to be compact if whenever X IF f there exist finite subsets XI of X and f l of f such that X\ IF ft . We have: COROLLARY 4.5. Me-entailment in :£ is compact iff entailment in :£ is

compact. Proof: (~) By Corollary 4.1. (~) Let us first observe that if entailment in :£ is compact, then the class of normal interpretations of :£ fulfills the following condition: (:) for each set of d-wffs X of :£ : if each finite subset of X has a normal model, then the set X has a normal model. For , if there is a set of d-wffs, say, XI' such that each finite subset of Xl has a normal model, but XI has no normal model, then XI is an infinite set; moreover, XI entails in 5£ some contradictory sentence, say, A, which is not entailed in :£ by any finite subset of XI' It follows that entailment in :£ is not compact; so if entailment in :£ is compact, the condition (:) holds. Since the compactness of entailment yields the condition (:), it suffices to prove that if X IF f and the condition (:) holds, then there are finite sets Xl' f t such that XI ~ X, f l ~ fand XI IF fl ' Moreover , it suffices to consider the cases in which X or f are infinite sets. Assume that f is an infinite set. Let uc(Y) be the set of universal closures of the d-wffs of f (cf. Chapter 2, Section 2.1.1). Let us designate by ,uc(Y) the set of negations of the sentences of the set uc(Y) . Suppose that X IF f . Hence X u ,uc(y) E Inc. By condition (:) we get that there exists a finite and non-empty subset 2 of the set X u ,uc(y) such that 2 E Inc. There are three possibilities : (a) 2 ~ X, (b) 2 ~ ,uc(Y), (c) 2 ~ X u ,uc(Y), where 2 g;. X and 2 g;.'uc(Y). If the possibility (a) holds, then for each finite subset f l of the set f we have 2 IF f\; at the same time 2 is a

110

CHAPTER 4

finite subset of the set X. If the possibility (b) takes place, then some finite subset of the set Y is me-entailed by the empty set and thus also by each finite subset of the set X. It is obvious that if the possibility (c) holds, then some finite subset of the set Y is me-entailed by some finite subset of the set

X. Assume that X is an infinite set. If Y = 0, then X E Inc . By condition (:) we get that there exists a finite subset XI of the set X such that XI E Inc. Thus XI IF 0; on the other hand, 0 is a finite subset of Y. If Y ~ 0 , we proceed analogously as in the case in which Y is an infinite set. 0 Thus me-entailment in a language is compact just in case entailment in this language is compact. Let us stress that Corollary 4.5 speaks of any language of the considered kind. Since we neither assume nor deny here that entailment in a language is compact, the same holds true in the case of meentailment: we leave room for different possibilities . Let us also note: COROLLARY 4.6 . If AI> ..., An are sentences, then {AI v .. . v An} IF {AI> .. ., An} · COROLLARY 4.7. If AI' ... , An are sentences, then: X IF {AI' .. ., An} iff X FA I V ... V An. According to Corollary 4.6, a finite and non-empty set of sentences of a given language is me-entailed in this language by a disjunction of all its elements (to be more precise, by a singleton set which contains this disjunction). Corollary 4.7 says that a finite and non-empty set of sentences is me-entailed by a set of d-wffs X just in case the set X entails some disjunction of all the elements of this set. Let us stress that the corollaries 4.6 and 4.7 describe some general properties ofmc-entailment: they are true with respect to any language of the considered kind. Let us also stress that the assumption that AI> ... , An are sentences is essential: in the case of sentential functions the situation is different. For, let's consider a sentential function P(x;) of some language and let's assume that there is a normal interpretation ~ of the language such P(x;) is satisfied in ~ only by some ~­ valuations, but not by all of them. The set {P(x;) v -'P(x;)} does not meentail in the analyzed language the set {P(x;), -, P(x;)}, but does entail the sentence P(x;) v -'P(x). To give a more concrete example: let's imagine that P stands for "is a prime" and the domain of the interpretation consist of all the natural numbers. Recall that the universal closure of a d-wff A is referred to as A. The following is a consequence of the corollaries 4.3, 4.5 and 4.7 :

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COROLLARY 4.8. If entailment in ;;e is compact, X is a set ofd-wffs of;;e and Y is a non-empty set of d-wffs of;;e, then X 11= Y iff either there is A E Y such that X FA or there are AI' ... , An E Y such that X F AI v . .. V An. Thus if entailment is compact, me-entailment of a non-empty set of d-wffs Y reduces either to entailment of the universal closure of a single d-wff of Yor to entailment of some disjunction of universal closures of d-wffs of Y. It does not mean, however, that the concept of me-entailment is superfluous: we will be considering also such languages, in which entailment (and thus me-entailment as well) is not compact.

Historical note. The idea of multiple-conclusion consequence goes back to Gentzen (1934); one of the possible ways oflooking at a valid sequent ofthe form Ai, ... , Am f- B\, ..., B; is to construe it, to speak generally, as stating multiple-conclusion entailment of the set made up of the formulas referred to by BI> ... , B, from the set made up of the formulas referred to by AI> ... , Am. Under this interpretation the turnstile f- is a relation symbol and a calculus of sequents is a (single-conclusion) metacalculus for a multiple-conclusion object-calculus. Yet, there is also another possibility: a sequent AI , .. ., Am f- BI, ''' , Bn is a notation for a formula Al A ... A Am --+ B I V ••• V B, and Gentzen's calculi of sequents are variants of the corresponding conventional calculi. Shoesmith and Smiley claim that Gentzen interpreted his calculi of sequents in this latter way. If this is so, it is Camap who for the first time introduced the concept of multiple-conclusion entailment (cf. Carnap, 1943; Carnap uses the term "involution"). The concept of multiple-conclusion consequence was incorporated into the general theory of logical calculi by Dana Scott (1974). Multiple-conclusion consequence and related concepts (multiple-conclusion calculus, multipleconclusion rules , etc.) are analyzed in detail in the monograph Shoesmith & Smiley (1978); for developments see also the book Zygmunt (1984). Let us add that the approach presented by Shoesmith and Smiley is much more general than ours : it is not restricted to first-order languages supplemented with model-theoretical semantics. Assume that L is a formalized language and let us designate by Form(L) the set of formulas of L; the only condition imposed on Form(L) is the non-emptiness clause. A relation f- between sets of formulas of L is a called a multiple-conclusion consequence if and only if f- fulfills the following conditions for any X, Y, Z ~ Form(L):

(C1)(Overlap) : (C2)(Dilution):

If X n Y

If

X

~

"¢

0, then X

XI> Y

~

f-

Yl and X

Y. f-

Y, then x,

f-

Yl'

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If

Xu ZI f- Y u ~ for any ZI' ~ such that ZI = 0 and ZI u ~ = Z, then X f- Y This is the syntactical concept of multiple conclusion-consequence. But there is also a semantical one. Assume that L is supplemented with some semantics rich enough to define some (relativized) concept of truth for formulas. A partition of L is an ordered pair < T, U> such that T n U = o and T u U = Form(L); we may think of the elements of the set T of a partition < T, U> as of consisting of truths in the sense of the underlying semantics and of the set U as of consisting of untrue formulas. Let .:i be a class of partitions of L. A relation f- between sets of d-wffs of L is the multiple-conclusion consequence relation characterized by .:i iff for each partition < T, U> E .:i and for each E f-, X n U ~ 0 or Y n T ~ 0 : this is the semantical concept of multiple-conclusion consequence. It may be proved that each relation being a multiple-conclusion consequence in the syntactical sense of the word is a multiple-conclusion consequence in the semantical sense, that is, a multiple-conclusion consequence characterized by some class of partitions of the language. It can also be proved that for each class of partitions of the language, the multiple-conclusion consequence characterized by this class is a multiple-conclusion consequence in the syntactical sense of the word, that is, fulfills the conditions (C 1) , (C2) and

(C3)(Cut for sets):

n

~

(C3 ) . For proofs, see Shoesmith & Smiley (1978) , p . 30 .

.What we have called above multiple-conclusion entailment in ;;e would presumably be called by Shoesmith and Smiley multiple-conclusion consequence characterized by the class of normal interpretations of ;;e, or, to be more precise, by the class of partitions of the set of d-wffs of ;;e determined by the class of normal interpretations of ;;e. By means of the concept of me-entailment we can define certain useful erotetic concepts in a simple, general and natural way; as a matter of fact, this concept will be the most useful tool of our further analysis. To our best knowledge, the idea of applying the concept of me-entailment in erotetic logic appeared for the first time in the papers Buszkowski (1987) and Wisniewski (1987)3 and in a marginal form in the dissertation Wisniewski (1986); see also the papers Buszkowski (1989) and Wisniewski (1989) for more extended expositions . For the application of this concept in erotetic

3 The paper Wisniewski (1987) is an abstract of the paper presented at the VIII International Congress of Logic. Methodology and Philosophy of Science. Moscow 1987. For some mysterious reasons the organizers retyped the manuscript before printing without making any proof reading. As a result this paper presumably wins the world record for misprints .

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logic see also the book Wisniewski (l990a) and the papers Wisniewski (l990b), (l991a), (l991b), (1993), (l994a) and (l994b). 4.3 .

EROTETIC CONCEPTS

4.3.1. Soundness, safety and riskiness The semantic concepts introduced so far do not pertain to questions. Let us now define some semantical erotetic concepts which will be useful in our further analysis. We assume that the language ;£ for which we define these concepts is an arbitrary but fixed language for which we defined in Chapter 3 the concept of question ; moreover, we assume that the semantics for the "declarative" part of ;£ is of the kind characterized in the previous sections of this Chapter. The first important step is a negative one: we do not assign here truth and falsity to questions. The reason is that it is doubtful whether all questions can express thoughts and describe states of affairs. Yet, we shall introduce here a more neutral semantic concept of soundness of a question in a given interpretation of a language. DEFINITION 4.7. A question Q of ;£ is sound in an interpretation ~ of the language ;£ iff at least one direct answer to Q is true in ~. The basic idea of this definition was suggested by Sylvain Bromberger (cf. Bromberger, 1992, p. 146). Soundness understood in the above sense is called by Belnap (nominal) truth ; yet, we prefer to use here the more neutral term. There are questions which are sound in each normal interpretation of the language, questions which are sound only in some such interpretation(s) and questions which are not sound in any normal interpretation of the language. Following Belnap, we shall introduce here the concepts of safety and riskiness of a question. DEFINITION 4.8. A question Q of ;£ is said to be safe iff Q is sound in each normal interpretation of ;£; otherwise Q is said to be risky. Thus a safe question is a question which has at least one true direct answer in each normal interpretation of the language. Let us observe that safe questions might also have been defined as questions whose sets of direct answers are me-entailed by the empty set; we can easily prove: COROLLARY 4.9. A question Q of;£ is safe iff dQ is me-entailed in ;£ by

the empty set. Questions having at least two direct answers which contradict each other

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and questions which have tautologies among their direct answers are paradigmatic examples of safe questions. We shall use the symbol Sa/for the set of safe questions of a considered language, adding a suffix where necessary.

4.3 .2. Just-complete answers and partial answers Direct answers were defined above in syntactic terms. One of the consequences of this fact is that a sentence which is equivalent to a direct answer need not be a direct answer; on the other hand, such a sentence may perform the same pragmatic functions as a direct answer. Let us then introduce the semantic concept of just-complete answer; the definition given below is a slight modification of that proposed by Belnap." DEFINITION 4.9. A sentence A of :£ is a just-complete answer to a question Q of :£ iff there is a direct answer B to Q such that B entails in :£ the sentence A and A entails in :£ the answer B. In other words, a just-complete answer is a sentence which is equivalent to some direct answer. We shall use the symbol cQ for the set of justcomplete answers to a question Q. It seems natural to call a partial answer to a question any sentence which is not equivalent to any direct answer to the question, but which is true if and only if a true direct answer belongs to some specified proper subset of the set of all the direct answers to the question. In other words, a partial answer is a sentence which is neither direct nor just-complete answer, but whose truth guarantees that a true direct answer can be found in some "restricted area" and whose truth is guaranteed by this fact. By means of the concept of me-entailment we can express this intuition as follows : DEFINITION 4.10. A sentence A of:£ is a partial answer to a question Q of:£ iff A is not a just-complete answer to Q and there exists a non-empty proper subset Y of the set of direct answers to Q such that: (i) Y is meentailed in :£ by A and (ii) A is entailed in :£ by each element of Y. Let us stress that our concept of partial answerhood differs from those analyzed by Harrah, Belnap, and Kubinski. The set of all the partial answers to a question Q will be referred to as pQ. Let us observe that if a question has more than two direct answers, then each disjunction of at least two but not all direct answers which is not

4

Cf. Belnap & Steel (1976), p. 126.

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equivalent to any single direct answer is a partial answer to the question. If entailment is compact, then each partial answer is either a disjunction of at least two but not all direct answers or a sentence which is equivalent to such a disjunction. The above definition yields, however, that questions with exactly two direct answers have no partial answers (let us recall that each question was assumed to have at least two direct answers). But, looking from the pragmatic point of view, a question which has exactly two direct answers requires a selection of one of them and does not leave room for a partial selection. Moreover, such a question can have answers of other kinds: incomplete, corrective, etc. We will not define here, however, these concepts of answers: this is not necessary from the point of view of the needs of our analysis.

4.3.3. Presuppositions of questions The concept of a presupposition of a question is defined in various logical theories of questions in different ways. We will accept here the definition of this concept given by Nuel D. Belnap. DEFINITION 4 .11. A d-wff A of 5£ is a presupposition of a question Q of 5£ iff A is entailed in 5£ by each direct answer to Q. The main advantage of the above definition is that is expresses a clear logical intuition : a presupposition of a question is a d-wff whose truth is necessary for the soundness (having a true direct answer) of the question. The set of presuppositions of a question Q will be referred to as PresQ. It can happen that the truth of some presupposition is not only necessary, but also sufficient condition of soundness of the question. Let us then introduce the concept of prospective presupposition of a question: DEFINITION 4 .12. A presupposition A of a question Q of 5£ is a prospectivepresupposition of Q iff A me-entails in 5£ the set of direct answers to Q. To speak generally . a prospective presupposition is thus a presupposition which, if true, guarantees the existence of a true direct answer to a question .' The set of prospective presuppositions of a question Q will be denoted by PPresQ. The set of presuppositions of a question is always nonempty (at least the tautologies of the language belong to it), but it need not be the case with the set of prospective presuppositions . There are languages of the considered

5

In Belnap's tenninology a prospective presupposition is a d-wff which expresses the

presupposition of the question.

CHAPrER4

116

kind which contain questions that have no prospective presuppositions; an example will be given below. But each question which has only finitely many direct answers has a non-empty set of prospective presuppositions: by Corollary 4.6. a disjunction of all the direct answers is a prospective presupposition of it. Let us also add that if a question has prospective presuppositions, all of them are equivalent. Prospective presuppositions should be distinguished from maximal

presuppositions. DEFINITION 4 .13 . A presupposition A of a question Q of ;:e is a maximal presupposition of Q iff A entails in ;:e each presupposition of Q. A maximal presupposition is thus a presupposition which entails any presupposition. The set of maximal presuppositions of a question Q will be referred to as mPresQ. There is no general reason why each question of any language of the considered kind should have a maximal presupposition. But each question whose set of direct answers is finite does have maximal presuppositions: any disjunction of all the direct answers to it perform this function. Let us also observe that if a question has prospective presuppositions, all of them are maximal. This is due to : COROLLARY 4 .10 . PPresQ ~ mPresQ. Proof: Assume that A E PPresQ and that A r£ mPresQ. So there is a presupposition B of Q such that A non FB; it follows that there is a normal interpretation 9 of the considered language such that 9 FA and 9 non FB. But B is a presupposition of Q; so for each C E dQ we have 9 non F C. Thus A non 11= dQ and hence A r£ PPresQ. We arrive at a contradiction. 0 Corollary 4.10 cannot be strengthened to identity : there are languages of the considered kind which contain questions that have maximal presuppositions which are not prospective presuppositions. Some example may be helpful here. Let us extend the language ;e' (see Chapter 3, Section 3.3.1) by introducing into it existential questions of the second kind: this new language will be designated below by ;e". Assume that the class of normal interpretations of ;:e" consists of all the interpretations of the language. Let us now consider the following existential question of the second kind: (4 .1)

? S(P.·(x l »

Clearly the sentence: (4 .2)

3x. p.·(x.)

is a presupposition of (4.1), but is not a prospective presupposition of it :

SEMANTICS

117

there are (normal) interpretations of the language in which the sentence (4.2) is true , but no sentence of the form P8u) (i.e. no direct answer to (4.1» is true. At the same time (4.2) is a maximal presupposition of (4.1). For, let us assume that there is a presupposition , say, A, of (4.1) which is not entailed in ;;e"# by the sentence (4.2). So the set: (4.3)

{3x1 Pll(X\), ,J}

where J is the universal closure of A, has a model. But it can be proved that if X is a set of d-wffs of ;;e"# such that there are infinitely many individual constants of ;;e"# which do not occur in the d-wffs of X, then X has a model if and only if there is an interpretation ~ = of ;;e"# which is a model of X and fulfills the following condition : (i) for each y E M there exists a closed term u of ;;e"# such that for each ~-valuation s, u(l[s] = y . The proof goes along the lines of the Henkin-style proof of Godel's theorem of the existence of a model: the only difference is that we use the individual constants which do not occur in the d-wffs of X as the "witnesses". Clearly there are infinitely many individual constants of ;;e"# that do not occur in the d-wffs of the set (4.3); so there is an interpretation, say, ~', of ;;e"# which fulfills the condition (i) and which is a model of the set (4.3). It follows that for some sentence of the form p\\(u), ~' is also a model of the set: (4.4)

{

3x, PI \ (Xl), ,A, PI I (u) }

But this is impossible, since PI1(U) is a direct answer to (4.1) and thus entails the sentence J (let us recall that A was assumed to be a presupposition of the question (4.1» . We arrive at a contradiction: so (4.2) is a maximal presupposition of (4.1). On the other hand, (4.2) is not a prospective presupposition of the analyzed question. The example analyzed above is instructive for some other reason as well: it presents a language which contains questions that have no prospective presuppositions. If the question (4.1) of ;;e"# had a prospective presupposition, this presupposition would be entailed by the maximal presupposition (4.2). So (4.2) would be a prospective presupposition of (4.1); since it is not, the question (4.1) has no prospective presuppositions . Let us stress, however , that we do not claim here that existential questions of the second kind cannot have prospective presuppositions in any language. The concepts of presupposition introduced above are defined by means of the concepts of entailment and me-entailment in a language and there are languages of the considered kind in which existential questions of the second kind do have prospective presuppositions . The language ;;e"" is a case in point here (see

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below) . Let us finally introduce the concepts offactual presupposition and maximal factual presupposition, which will be useful in our analysis of generation . DEFINITION 4.14. A presupposition A of a question Q of :£ is a factual presupposition of Q iff A is a synthetic d-wff of :£. DEFINITION 4.15. A factual presupposition A of a question Q of :£ is a maximal factual presupposition of Q iff A entails in :£ each factual presupposition of Q. It is not the case that each question has factual presuppositions. Moreover, there is no general reason why each question which does have factual presuppositions should have maximal factual presuppositions. The sets of factual presuppositions and maximal factual presuppositions of a question Q will be referred to as Pres,Q and mPres,Q, respectively. 4.3.4 . Relative soundness. Normal questions and regular questions

Let us now introduce the concept of relative soundness of a question, which seems to be of basic importance to erotetic logic. The underlying intuition is that a question Q is sound relative to a set of d-wffs X just in case the question Q must have a true direct answer if all the d-wffs in X are true . This intuition can be expressed in terms of me-entailment via DEFINITION 4.16. A question Q of ;e is sound relative to a set of d-wffs X of :£ iff the set X me-entails in :£ the set of direct answers to Q. In other words, Q is sound relative to X just in case there is no normal interpretation of the language in which all the d-wffs in X are true, but no direct answer to Q is true. If Q is sound relative to a singleton set {A}, we say that Q is sound relative to the d-wff A. Let us stress that the concept of relative soundness introduced above must be carefully distinguished from the concept of soundness of a question in an interpretation ofthe language introduced in Section 4.2.1: these are different concepts. A safe question is sound relative to any set of d-wffs. This is the trivial case; in order to distinguish the non-trivial cases let us introduce the following concept: DEFINITION 4.17. A question Q of ;e is madesound by a set of d-wffs X of :£ iff X me-entails in :£ the set of direct answers to Q although the set of direct answers to Q is not me-entailed in :£ by the empty set. Thus Q is made sound by X just in case Q is not sound relative to the

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empty set, but Q is sound relative to X. It is clear that only risky questions can be made sound by sets of d-wffs. Each question has a non-empty set of presuppositions; the truth of these presuppositions is a necessary condition for having a true direct answer. Yet, when we look in the opposite direction, the truth of all the presuppositions need not be a sufficient condition of having a true direct answer : there are languages of the considered kind in which some questions do not have this property . The language 5£'# mentioned in Section 4.3.3 is a case in point here : since the sentence (4.2) is a maximal presupposition of the question (4.1) but not a prospective presupposition of it, it happens that all the presuppositions of (4.1) are true in some normal interpretation of the language, but no direct answer to (4.1) is true in it. In order to distinguish questions whose presuppositions guarantee the existence of a true direct answer from the remaining ones let us introduce the concept of normal

question:" DEFINITION 4.18 . A question Q of 5£ is said to be normal iff the set of direct answer- to Q is me-entailed in 5£ by the set of presuppositions of Q. A normal question is thus a question which is sound relative to the set of its presuppositions. Note that since being a normal question depends on meentailment in a language (and thus basically on the conditions imposed on the class of normal interpretations), the same question occurring in one language may be normal in it without being normal in some other language. Again , the question (4.1) gives us a simple example here : as far as the language 5£'# is concerned, (4 .1) is not normal in it. As we shall see, however, (4.1) is normal in the case of the language 5£" . But any question of the first kind is normal in any language ; moreover, each safe question is normal. The remaining questions , however, mayor may not be normal : it depends on the semantics of the language. Let us now introduce a more specific concept of regular question. To speak generally, a question Q will be called regular if there is a single presupposition of Q whose truth implies the existence of a true direct answer to Q. To be more precise, we adopt DEFINITION 4.19. A question Q of 5£ is said to be regular iff there is a presupposition A of Q such that A me-entails in 5£ the set of direct answers to Q.

6 Buszkowski , who in fact introduced this concept (cf. Buszkowski, 1989), uses here the term correct question .

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In other words, a question is regular if it is sound relative to some of its presuppositions . Note that each question of the first kind is regular. We also have: COROLLARY 4.11. A question Q is regular iff Q hasprospective presuppo-

sitions. COROLLARY 4 .12. Each regular question is normal. COROLLARY 4 .13. If entailment in ;;£ is compact, then each normal

question of;;£ is regular. Corollary 4.11 shows that the concept of regular question can be defined in terms of prospective presuppositions. Corollaries 4.12 and 4 .13 imply that "being a normal question" and "being a regular question" coincide if entailment in a language and thus also me-entailment in it are compact. Yet, there are languages of the considered kind in which entailment is not compact. 4.3.5. Self-rhetoricity and informativeness. Proper questions Let us now define the concept of a self-rhetorical question. DEFINITION 4.20. A question Q of ;;£ is said to be self-rhetorical iff there is a direct answer A to Q such that A is entailed in ;;£ by the set of presuppositions of Q. By proposing the above definition we are not going to explicate the general notion of rhetoricity of a question: this notion is clearly a pragmatic one (in the traditional sense of the word). Our aims are limited: we attempt to explicate the concept of "rhetoricity for logical reasons". The following are examples of self-rhetorical questions: questions having tautologies as direct answers, questions all of whose direct answers are contradictory sentences, questions which have only one direct answer being a synthetic sentence. We shall use the symbol Ret for the set of self-rhetorical questions of a language, adding a suffix where necessary. Each self-rhetorical question is normal in the sense of Definition 4 .18. In order to distinguish normal questions which are not self-rhetorical from the remaining ones we introduce: DEFINITION 4 .21. A question Q is proper iff Q is normal but not selfrhetorical . Let us finally introduce a certain concept of informativeness of a question. Looking from the intuitive point of view, a question is informative relative to a given set of d-wffs just in case each direct answer to the question conveys some information which cannot be legitimately drawn from the

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analyzed set of d-wffs . In order to explicate this notion let us first introduce the concept of content of a set of d-wffs. Let X be a set of d-wffs of :£. The content of the set X (in symbols : Ct(X» is defined as follows: DEFINITION 4.22. Ct(X) = {A: A is entailed in :£ by X and A is not a tautology of 5£} In other words, the content of X consists of all the d-wffs which are nontrivially entailed by X (i.e. are not tautologies)." The relativized concept of informativeness of a question can now be defined as follows: DEFINITION 4 .23 . A question Q is informative relative to a set of d-wffs X iff for each A E dQ, Ct(X) C Ct(X U {A}). Thus a question Q is informative relative to a set of d-wffs X just in case the content of X is a proper subset of the content of any set which results from X by adding a direct answer to Q: any such set entails more non-trivial consequences than the initial set X. It is easily seen, however, that Q is informative relative to X if and only if no direct answer to Q is entailed by

X. By proposing the above definition we identify the possibility of extracting information from a set of d-wffs with the entailment of the sentence conveying this information. No doubt, there is plenty of idealization in such an approach; consequently, Definition 4.23 presents an idealized concept of informativeness of a question. The same holds true in the case of our definition of self-rhetoricity. We can easily prove : COROLLARY 4 .14 . A question Q is not self-rhetorical relative to the set of presuppositions of Q.

iff Q is informative

Let us finally note : COROLLARY 4.15. A question Q is proper iff Q is normal and Q is informative relative to the set of presuppositions of Q. Thus a proper question is a question which is both normal and informative relative to the set of its presuppositions.

7 The definition proposed above is similar to that given by Popper (cf. Popper, 1959, p. 120), who, however, uses the concept of derivability.

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4.4.

SEMANTICS FOR THE LANGUAGES

;e' AND ;e"

All the definitions of semantic concepts introduced above remain schematic until both the considered language and the class of normal interpretations of this language will be characterized in detail. Let us now propose some semantics for the languages ;e' and ;e" described in Chapter 3, Section 3.3.1: these are the languages from which we will be taking examples of evoked, generated and implied questions.

4.4.1. Semantics for ;e' To speak generally, the language ;e' results from the language J of classical predicate calculus with identity, individual constants and function symbols (cf. Chapter 2, Section 2.2.1) by adding questions of the first kind; they are the only questions of ;e'. We assume that each interpretation of ;e' is a normal interpretation of ;e'. Thus the class of normal interpretations of ;e' is equal to the class of all the interpretations of the initial language J of classical predicate calculus with identity (where the concept of interpretation of J and the remaining "nonerotetic" semantic concepts are defined for J in the same way as in the case of languages enriched with questions). Thus we have: COROLLARY 4.16. Entailment in ;e' reduces to logical entailment. COROLLARY 4.17. Each tautology of ;e' is a logically valid formula of J. COROLLARY 4.18 . Entailment in ;e' is compact. COROLLARY 4.19. Me-entailment in ;e' is compact. Since questions of the first kind are the only questions of ;e', we also have: COROLLARY 4.20. Each question of ;e' has prospective presuppositions and maximal presuppositions. COROLLARY 4.21. Each question of ;e' is normal and regular. For proofs, it suffices to observe that a disjunction of all the direct answers to a question of the first kind Q is a prospective presupposition of Q; hence by Corollary 4.10 Q has maximal presuppositions, by Corollary 4.11 Q is regular and thus by Corollary 4.12 also normal .

4.4.2. Semantics for

s:

The language ;e" differs from the language ;e' only as to its "erotetic" part: among the questions of ;e" there occur not only questions of the first kind, but also questions of the second kind and questions of the third kind . To speak generally, questions of the second kind are questions about

SEMANTICS

123

objects or n-tuples of objects that satisfy a given condition: exactly one example, at least one example or all the examples are called for. Questions of the third kind , in tum , may be regarded as questions about at least k different objects that satisfy a given condition or about all of the at least k different objects that satisfy a certain condition. Looking from the purely syntactical point of view, however, direct answers to questions of the second and third kind are substitution-instances of some sentential functions. On the other hand, one of the peculiarities of model-theoretical semantics is that it admits interpretations of a language in which some sentential functions are satisfied by some valuations, but nevertheless no sentence being a substitution-instance of these sentential functions is true : the reason is that some elements of the domains of the analyzed interpretations have no names (are not assigned to any closed terms). As a result certain questions of the second and third kind have no true direct answers in some interpretations of ;;e"" in which, to speak generally, there exist the objects or n-tuples of objects called for. In the case of natural languages, however, the situation is different: when saying that a question about objects or n-tuples of objects that satisfy a given condition has no true (direct) answer, we usually have in mind that there are no such objects or n-tuples of objects. Moreover , in the case of natural language questions we usually have in our disposal names of the entities called for or some method of forming their names. In order to avoid the consequence that a question of the second or third kind has no true direct answer only for the reason that some elements of the domain of an interpretation have no names, we assume here that normal interpretations of the language ;;e"" are those interpretations in which each element of the domain has a name, that is, is assigned to some closed term.8 By means of the semantic concepts introduced above normal interpretations of ;;e"" can be precisely defined as follows: DEFINITION 4.24 . An interpretation ~ = of ;;e"" is a normal interpretation of ;;e"" if and only if the following condition holds : (.) for each y E M there exists a closed term u of ;;e"" such that for each ~-valuation s, u~[s] = y. If u is a closed term, than for any interpretation ~ and for any ~ ­ valuations s, s' we have u~[s] = u~[s']; the value of a closed term in an

8 A similar idea was suggested by Belnap (cf. Belnap, 1963, Chapter 7.7, or Belnap & Steel, 1976, p. 73). Belnap, however, does not apply it in his general considerat ions.

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124

interpretation is not dependent upon valuations and depends only on the interpretation function. Thus we may say that an interpretation ~ of ;eoo is normal just in case each element of the domain of ~ is a value of some closed term of the language. Or, to put it differently, each element of the domain has a name. As an (almost) immediate consequence of Definition 4.24 we get that for each normal interpretation ~ of ;;eoo, a sentence of the form 3xj l ••• 3xjn Axj l ... xjn is true in ~ if and only if at least one sentence of the form A(x;/u lo ... , xi/un) is true in ~. Thus we can easily prove ( F!£OO and IF!£oo stand below for entailment and me-entailment in ;;eoo, respectively): THEOREM 4.1. 3xil

.. .

3xjn Axj l

xin IF!£OO S(Axj l

...

.xjn) iff X F!£OO 3xj ,

xjn) .

3xjn Axj l .. . xin• By Corollary 4.3, Theorem 4.2 and the appropriate definitions we also get:

THEOREM 4.2. X IF!£oo S(Axj l

THEOREM 4.3. XIF!£oo O(Axil .. . x jn)

iff X F!£oo 3x

...

j , . .. .

3xjn Axil ... x jn '

Let (#)

3 '::?kxj Axj

be an abbreviation of a d-wff of the form: (4.5)

3x,

3x,H_I(A(x;!x,) /\ ... /\ (Ax;!x,H_I)

r;

x, ~ X,+I /\ x, ~ X'+2

r; /\ x, ~ X,H_I /\ X,+\ ~ X,+2 /\ ... /\ X,H_2 ~ X,H_I), where r is the least index such that for each s '::? r, the variable Xs is substitutable for x. in Axj ; if k = 1, then (4.5) reduces to 3x, A(x;!x,). An expression of the form 3 '::?kxj Axj can thus be read "there exist at least k x, such that Ax;" . The following theorems are true:

THEOREM 4.4. 3 '::?kxj Axj IF!£OO W(x, ... x,H_,IAxj ) . THEOREM 4.5. X IF!£OO W(x, ... x,H_t1Axj )

iff X F!£OO

3 '::?kxj Axj •

By Corollary 4.6, Theorem 4.1, Theorem 4.3, Theorem 4.4, Corollary 4.10, Corollary 4.11, Corollary 4.12 and some well-known metalogical results we get: COROLLARY 4.21: Questions of thefirst kind of ;;eoo, existential and open

questions of the second kind of ;;eoo andexistential questions of thethirdkind of ;;eoo have prospective presuppositions and maximal presuppositions; moreover, each such question is normal and regular. The situation is different, however, in the case of general questions of the second and third kind of ;;eoo.

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125

The following lemma is a consequence of some well-known metalogical results: LEMMA 4.1. Let X be a set of d-wffs of:£" such that there are infinitely many individual constants of :£"" that do not occur in the d-wffs of X. Then X has a model iff X has a normal model. The proof of the implication (=*) goes along the lines of the Henkin-style proof of Godel 's theorem of the existence of a model: we use entailment in :£"" as the consequence relation in proving the analogue of Lindenbaum's Lemma and the individual constants that do not occur in the d-wffs of X as the "witnesses". Lemma 4.1. implies : COROLLARY 4.22. A finite set of d-wffs of :£"" has a model iff it has a

normal model. CQROLLARY 4.22. Each tautology of:£" isa logically validformula ofJ. COROLLARY 4.23. Entailment in :£"" reduces to logical entailment in case of finite sets of d-wffs. Corollary 4.23 can be generalized to some infinite sets of d-wffs (at least to those which fulfil the antecedent of Lemma 4.1), but it cannot be generalized to any infinite set of d-wffs. Let us consider the set: (4.6)

{3x\P t 1(x \) } U S(-,p\l(X\»

There is no normal interpretation of :£"" which is a model of the set (4.6) . For, let us assume that the set (4.6) has a normal model. So the sentence :lx\P11(X\) is true in some normal interpretation of :;e"". Thus by Theorem 4.1 some sentence of the form p\t(u), where u is a closed term , is true in this interpretation. But this is impossible, since the negation of the considered so the set (4.6) has no normal sentence belongs to the set S(-,P8x j model. But since (4.6) has no normal models, it entails in :£.., among others , the sentence P\'(a1). This sentence, however, is not logically entailed by the set (4.6) . Let us finally note the following "negative" result :

»;

COROLLARY 4.24. Both entailment in :£.. and me-entailment in :£"" are

not compact. Proof: Let us again consider the set (4.6). Since (4.6) has no normal models, it entails in :£"" a contradictory sentence, say, pt1(a\) fI -,P\\(a t ) . But it is easily seen that each finite subset of (4.6) has a normal model; so the sentence P8a\) fI -,p\1(a1) is not entailed in :£"" by any finite subset of the set (4.6) . So entailment in :£"" is not compact ; hence by Corollary 4.5

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also me-entailment in ;;e** is not compact. 0

CHAPTER 5

EVOCATION OF QUESTIONS

In this Chapter we shall define the semantic relation of evocation of a question by a set of declarative formulas. We shall also examine the basic properties of this relation. The proposed definition provides us with one of the explications of the intuitive concept of the arising of a question from a set of declarative sentences (cf. Chapter 1). In the general part of our considerations we assume that the language for which we define the concept of evocation is an arbitrary but fixed formalized language :£ of the kind considered in Chapter 3, that is, a certain first-order language with identity enriched with questions in the way presented in Chapter 3. We also assume that the semantic concepts pertaining to the language :£ are defined in the way presented in Chapter 4. Unless otherwise indicated, the metalinguistic symbols used below pertain to the language :£ (thus, for example, I~ stands for me-entailment in :£, Taut for the set of tautologies of :£, and so forth). For the sake of brevity, the qualifications "in 5£", "of :£", etc., will be normally omitted; thus, for example, instead of "me-entails in::i " we write "me-entails" and similarly in other cases. Let us stress, however, that what we are going to define below is the relation of evocation in a language :£ of a question of:£ by a set of d-wffs of 5£; this concept will be defined by means of the concept of me-entailment in :£. 5.1. DEFINITION OF EVOCATION The relation E of evocation ofa question by a set of declarative formulas is defined by: DEFINITION 5.1. E(X, Q) iff (i) X I~ dQ, and (ii) for each A E dQ, X non I~ A. According to the above definition, a question Q is evoked by a set of dwffs X if and only if the set X me-entails the set of direct answers to the question Q, but no singleton set which contains a direct answer to the question Q is me-entailed by the set X. The clause (i) of the above definition

128

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is fulfilled just in case Q is sound in each normal interpretation of the language in which all the d-wffs in X are true. Thus if all the d-wffs in X are true (in some normal interpretation of the language; in what follows we will normally be omitting this provision, also with respect to soundness), then Q is sound, i.e. has a true direct answer. It is neither assumed nor denied here, however, that X consists of truths. The clause (ii) is fulfilled if and only if no direct answer to Q is entailed (in the standard sense of the word) by X. If E(X, Q), then the set X is said to be the evoking set, whereas the question Q is said to be the evoked question. We shall write E(A" ... , An' Q) instead ofE({A" ... , An}, Q). It is easily seen that the relation of evocation has the properties required by the conditions (Cd. 1), (Cd.2), (Cd.3) and (Cd.4) of Chapter 1. The condition (Cd .3) is fulfilled due to the clause (i) of Definition 4.1 , whereas the condition (Cd .2) is fulfilled due to the clause (ii). At the same time we can easily prove:

If E(X, Q), then no direct answer to Q belongs to X. COROLLARY 5.2. If E(X, Q), then X entails each presupposition of Q.

COROLLARY 5.1.

For the proof of Corollary 5.1 it suffices to observe that if some direct answer to Q belonged to X, the clause (ii) would not be fulfilled. Thus we may conclude that the condition (Cd. I) of Chapter 1 is satisfied . For the proof of Corollary 5.2 let us assume that X evokes Q and that there is a presupposition of Q, say, A, such that X does not entail A. So there is a normal interpretation ~ of the considered language such that ~ is a model of X and A is not true in ~. But A is a presupposition of Q and thus is entailed by each direct answer to Q; hence no direct answer to Q is true in ~ . It follows that X does not me-entail the set dQ; thus X does not evoke Q. We arrive at a contradiction. Therefore X entails each presupposition of Q; it follows that the condition (Cd.4) is fulfilled as well. Thus we may say that Definition 5.1 provides us with an adequate explication of the analyzed concept of the arising of a question from a set of declarative formulas (with respect to the requirements (Cd. 1), (Cd.2) , (Cd.3) and (Cd.4)) . Presumably the intuitive sense of the introduced concept is best characterized by: THEOREM 5.1. E(X, Q) iff Q is sound relative to X and Q is informative

relative to X. Proof: Recall that Q is sound relative to X iff X IF dQ. Assume that E(X, Q) . Then for each A E dQ we have X non 1= A.

EVOCATION OF QUESTIONS

129

Therefore dQ n Taut = 0. At the same time for each A E dQ we have X, A t= A . Since dQ n Taut = 0, then for each A E dQ we get Ct(X) C Ct(X U {A}). Thus Q is informative relative to X. Assume that Q is informative relative to X. Let us suppose that for some A E dQ we have X t= A. Hence X t= A (where A is the universal closure of A). Since Q is informative relative to X, there exists B $. Taut such that X non t= B and X, A t= B; therefore X, A t= B. Thus X t= A -00 B. But if X t= A, then X t= B. We arrive at a contradiction. 0 Let us notice that evocation takes place int.al. in the situation in which some partial answer to a question is me-entailed by a set of d-wffs, but at the same time no just-complete answer to the question is entailed by this set. This is due to THEOREM 5.2. If X entails some partial answer to Q and X entails no justcomplete answer to Q, then E(X, Q) . Proof: Assume that there is aCE pQ such that X t= C. Since C is a partial answer to Q, then by Definition 4.10 there exists an at least twoelement proper subset Y of the set dQ such that C IF Y. Thus X IF Y. Since Y C dQ, then X IF dQ by Corollary 4.3. Let's now assume that for each B E cQ we have X non t= B. Since by Definition 4.9 dQ S;; cQ, it follows that for each A E dQ, X non t= A. Thus by Corollary 4.1 for each A E dQ, X non IFA. Hence E(X, Q). 0 Theorem 5.2 cannot be strengthened to equivalence for any language of the considered kind ; there are languages, however, for which the corresponding equivalence holds . Let us now examine the basic properties of evocation . 5.2. BASIC PROPERTIES OF EVOCATION 5.2 .1. The evoking sets

We shall first prove some theorems which describe the sets of d-wffs which can evoke questions. THEOREM 5.3 . If E(X, Q) , then X is incomplete. Proof: If E(X, Q), then X entails no direct answer to Q. Direct answers are sentences. Thus it suffices to prove that the set X does not entail the negation of at least one direct answer to Q. Assume that E(X, Q) and that for each A E dQ, X t= ...,A. By the definition of me-entailment we get X non IF dQ . We arrive at a contradiction . 0

130

CHAPfERS

Theorem 5.3 implies that complete sets of d-wffs do not evoke any questions. It seems that this is correct: complete sets are sometimes described as sets which answer all questions (expressed in the appropriate language). On the other hand, the domain of the relation of evocation consists of all the incomplete sets of d-wffs of the analyzed language. This is due to Theorem 5.3 and THEOREM 5.4. Each incomplete set ofd-wffs evokes at least one question. Proof: If X '1. Cp, then there is a sentence A such that X non F A and X non F ...,A. It is easily seen that the question whose direct answers are the sentence A and the sentence ""A, exclusively, is evoked by the set X. 0 Theorem 5.3 and Theorem 5.4 yield THEOREM 5.5. A set of d-wffs X is incomplete if and only if X evokes at least one question. Thus the concept of completeness can be described and even defined in terms of evocation. Since each inconsistent set ofd-wffs is complete, then Theorem 5.3 yields THEOREM 5.6. Each evoking set is consistent. According to Theorem 5.6, each evoking set has a normal model. As an immediate consequence of Theorem 5.6 we get: COROLLARY 5.3. IfE(X, Q), then X contains no contradictory d-wff, We will discuss the significance of Theorem 5.6 and Corollary 5.3 for the theory of erotetic arguments in Chapter 8. 5.2 .2. The evoked questions Let us now examine what questions can be evoked by sets of d-wffs. THEOREM 5.7. If E(X, Q), then Q is not self-rhetorical. Proof: If E(X, Q), then for each A E dQ, X non FA and by Corollary 5.2 each normal model of X is a model of PresQ. Suppose that Q E Ret. Thus for some A E dQ we have PresQ F A; hence X F A and thus non E(X, Q). 0 Thus no evoked question is self-rhetorical (in the sense of Definition 4.20); moreover, self-rhetorical questions are not evoked by any sets of dwffs. Theorem 5.7 together with the definition of the concept of being a selfrhetorical question yield COROLLARY 5.4. Each "evoked question has at least two direct answers which are not contradictory sentences. COROLLARY 5.5. No direct answer to an evoked question is a tautology.

EVOCATION OF QUESTIONS

131

It follows that COROLLARY 5.6. Each evoked question has at least two direct answers

which are synthetic sentences. A question each of whose direct answers is a contradictory d-wff is called a completely contradictory question. I Thus as a consequence of Corollary 5.4 we get THEOREM 5.8. Completely contradictory questions are not evoked by any

sets of d-wffs . Questions which have only tautologies as direct answers are sometimes called completely tautological questions.2 Thus as a consequence of Corollary 5.5 we get: THEOREM 5.9. Completely tautological questions are not evoked by any

sets of d-wffs. Let us stress that Corollary 5.5 does not imply that safe questions cannot be evoked. The reason is that there are safe questions whose direct answers are not tautologies. Recall that a question Q is called normal iff PresQ Il= dQ. In other words, a question is normal if and only if it is sound relative to the set of its presuppositions . We shall prove THEOREM 5.10. Let Q be a normal question. Then Q is evoked by some set of d-wffs iff Q is not self-rhetorical. Proof: Since we have already proved Theorem 5.7 , it suffices to prove that each normal but not self-rhetorical question is evoked by some set of dwffs. If Q is not self-rhetorical , then no direct answer to Q is entailed by the set PresQ of presuppositions of Q. But if Q is normal, we also have PresQ Il= dQ. Thus E(PresQ, Q). 0 Theorem 5.10 shows that in those languages in which each question is normal the range of the evocation relation consists of all the questions which are not self-rhetorical. There are languages, however, in which some questions are not normal. Recall that a proper question is a question which is normal but not selfrhetorical .

1

Cf. Kubinski (1980) , p. 66

2

Cf. Kubinski (1980), p. 66 .

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132

Theorem 5.10 yields

THEOREM 5.11. Each proper question is evoked lJy at least one set of dwffs. Let us finally pay some attention to the evocation of safe questions. The following theorem is true:

THEOREM 5.12. If X

~

Taut and Q is a safe but not self-rhetorical

question, then E(X, Q). Proof: If X ~ Taut and Q E Saf, then X IF dQ. Since Q ~ Ret, then dQ n Taut = 0 . But by assumption we also have X ~ Taut. Therefore for each A E dQ, X non IF A. Hence E(X, Q). 0 According to Theorem 5.12 , each subset of the set of tautologies (the empty set included!) evokes any safe but not self-rhetorical question. Let us also note

THEOREM 5.13. If E(X, Q) and X ~ Taut, then Q has no factual presuppositions. Proof: If E(X, Q), then by Corollary 5.2 X entails each presupposition of the question Q. Assume that Q has factual presuppositions. Since each factual presupposition of Q is entailed by each direct answer to Q, it follows that there is a normal interpretation of the analyzed language in which all the direct answers to Q are not true. But if E(X, Q), then X IF dQ; thus X g; Taut. 0 Theorem 5.13 implies that the empty set and sets of tautologies evoke only such questions, which have no factual presuppositions. 5.2.3. Evocation and relations between sets of d-wffs and relations between questions We can easily prove

THEOREM 5.14 . If X and Yare sets of d-wffs such that for each normal interpretation ~: ~ 1= X iff ~ 1= Y, then E(X, Q) iff E(Y, Q). According to Theorem 5.14 sets of d-wffs which have exactly the same normal models evoke exactly the same questions.

THEOREM 5.15. Let E(X, Q) and Y

~

X. Then E(Y, Q)provided that

YIF

dQ .

According to Theorem 5.15 a question which is evoked by a set of d-wffs is also evoked by each subset of this set which me-entails the set of direct answers to the question.

THEOREM 5.16. Let E(X, Q) and X

~

Y. Then E(Y, Q) provided that for

EVOCATION OF QUESTIONS

each A

E

133

dQ, Y non 1= A.

According to Theorem 5.16, if X evokes Q and X is a subset of Y, then Yevokes Q if no direct answer to Q is entailed by Y. On the other hand, it need not be the case that a question Q evoked by a set of d-wffs X is also evoked by a set of d-wffs which includes X: although the set of direct answers to Q is me-entailed by each set which includesX, it nevertheless can happen some of such "larger" sets entail direct answers to Q. In other words, questions evoked by a given set of d-wffs are evoked only by those larger sets which even potentially do not answer them. Thus we may say that evocation is not "monotonic". On the other hand, questionswhich arise from a given set of sentences do not arise from those larger sets which answer these questions; so the above conclusion seems to be an intuitive one. We shall now examine the relations which connect evocation and some basic relations between questions.3 Let us first introduce the concept of equipollence of questions. We say that a question Q2 is equipollent to a question QI if and only if there exists a bijection i: dQ2 .... dQ, such that for each A E dQ2' A is equivalent to i(A) (i.e. to the corresponding element of dQ,), In other words, Q2 is equipollent to Q! just in case there exists a one-to-one mapping of the set of direct answers to Q2 into the set of direct answers to Q. such that the corresponding direct answers to Q2 and QI are equivalent. Let us first prove LEMMA 5.1. If (i) for each B E dQ2 there is A E dQ, such that B 1= A, and (ii) for each A E dQI there is B E dQ2 such that A F B, then E(X, Q2) iff E(X, QI)' Proof: Assume that E(X, Q2) ' Thus X IF dQ2 and for each B E dQ2, X non IF B. Hence by (i) we get X IF dQ!. At the same time by (ii) we have X non IF A for each A E dQI' Therefore E(X, Q,). The reverse implication can be proved in an analogous way. 0 As an immediate consequence of Lemma 5.1 and the definition of equipollence we get THEOREM 5.17. If a question Q2 is equipollent to a question QI' then E(X, Q2) iff E(X, QI)' Thus we may say that equipollent questions are evoked by the same sets of d-wffs.

3

These relations were defined by Tadeusz Kubinski (cf. Kubinski, 1980, pp. 58 and 80) .

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CHAPTER 5

In order to define the erotetic relations of being stronger and being weaker we have to introduce (following Tadeusz Kubinski) the concept of regular entailment. 4 The relation h of regular entailment is defined by DEFINITION 5.2. (i) Let X (/. Inc. Then X 1=, A iff X 1= A. (ii) Let X E Inc. Then X 1=, A iff A E Contr. Thus a set of d-wffs X regularly entails a d-wff A if and only if (i) X is consistent and X entails A , or (ii) X is inconsistent and A is a contradictory d-wff. The relation of regular entailment is a consequence operation (in the sense of Tarski) in each so-called homogeneous family of sets of d-wffs; a family e of sets of d-wffs is said to be homogeneous if and only if (a) e is empty, or (b) e is nonempty and each element of e is consistent, or (c) e is nonempty and each element of e is inconsistent. We say that a question Q2 is weaker than a question Ql if and only if Q2 is not equipollent to QIo but there is a surjectionj: dQI ... dQ2 such that for each A E dQI' A hj(A). In other words, Q2 is weaker than Q 1just in case Q2 is not equipollent to QIo but there is a mapping of the set of direct answers to QI into the set of direct answers to Q2 such that each direct answer to QI regularly entails the corresponding direct answer to Q2' Note that if the relation of being weaker were defined by means of the concept of (standard) entailment, we would obtain the following counterintuitive consequence: each question is weaker than any completely contradictory question (i.e . a question each of whose direct answers is a contradictory sentence). THEOREM 5.18 . Let E(X, QI) and let Q2 be a question weaker than QI' Then E(X, Q2) if X entails no direct answer to Q2' Proof: If Q2 is weaker than QI' the following condition is fulfilled: (i) for each A E dQI there is B E dQ2 such that A 1=, B. If B is regularly entailed by A, then B is entailed by A (of course, the reverse implication need not be true). Thus the following condition is also fulfilled: (ii) for each A E dQI there is B E dQ2 such that A 1= B. Assume that E(X, Ql)' Therefore X IF dQI' But since the condition (ii) is

4 For a detailed description of the properties of regular entailment see the paper Kubinski (1964); for this concept see also the papers Kubinski (1966), (1987), or the books Kubinski (1971), pp. 48-49, or Kubinski (1980), pp. 55-56.

EVOCATION OF QUESTIONS

135

fulfilled, then by the definition of me-entailment we get X IF dQ2' By assumption for each B E dQ2 we have X non l= B. Hence X non IF B for each B E dQ2' Thus E(X. Q2)' 0 Theorem ' 5.18 yields that a set of d-wffs which evokes a question Ql evokes also all the questions weaker than QI whose direct answers are not entailed by this set. A question Q2 is stronger than a question Q. if and only if the question Q. is weaker than the question Q2 and the question Q2 is not weaker than the question Q1 ' THEOREM 5.19 . Let E(X, QI) and let Q2 be a question stronger than Q1' Then E(X, Q2) provided that X IF dQ2' Proof: If Q2 is stronger than Qh the following condition is fulfilled: (i) for each B E dQ2 there exists A E dQI such that B l= A. So if E(X, Q1) and Q2 is stronger than Qh then by (i) and Corollary 4.1 we have X non IF B for each B E dQ2' But by assumption we also have X IF dQ2' Thus E(X, Q2)' 0 According to Theorem 5.19 a set of d-wffs which evokes a question Q1 evokes also those questions stronger than QI. whose sets of direct answers are me-entailed by this set. A question Q2 is said to be equivalent to a question QI if and only if dQ2 = dQ1 ' Since equivalent questions are also equipollent, as a consequence of Theorem 5.17 we get THEOREM 5.20. If a question Q2 is equivalent to a question Q" then E(X, Q2) iff E(X, Q1)' Equivalent questions are thus evoked by the same sets of d-wffs. A question Q2 is included in a question QI if and only if the set dQ2 is a proper subset of the set dQI' One can easily prove THEOREM 5.21. Let E(X, QI) and let Q2 be a question which is included in QI' Then E(X, Q2) if X IF dQ2' Theorem 5.21 says that a set of d-wffs which evokes a question Ql evokes also those questions included in Q1 whose sets of direct answers are meentailed by this set. A question Q2 includes a question Ql if and only if the question Q1 is included in the question Q2'

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CHAPTERS

THEOREM 5.22. Let E(X, QI) and let Q2 be a question which includes QI' Then E(X, Q2) if for each B E dQ2 - dQI' X non 1= B. Theorem 5 .22 says that a set of d-wffs which evokes a question QI evokes also those questions that include QI whose direct answers which are not direct answers to QI (the remaining direct answers, so to say) are not entailed by this set. Let us finally note COROLLARY 5.7. If the set of direct answers to a question Q2 is a subset of the set of direct answers to a question QI and each direct answer to QI entails some direct answer to Q2' then E(X, Q2) iff E(X, Ql)' Corollary 5.7 is an immediate consequence of Lemma 5.1.

5.2.4. Evoking andpresupposing. Alternative definitions of evocation One may ask whether to say that a question is evoked by a given set of dwffs is always tantamount to saying that all the presuppositions of the question are entailed by this set whereas its direct answers are not. Well , sometimes it is the case, but the general answer is negative . Let us now analyze this in detail. Recall that a question Q is normal iff PresQ IF dQ, i.e. the set of direct answers to Q is me-entailed by the set of presuppositions of Q. A question Q is regular iff for some A E PresQ, A IF dQ. We can prove: THEOREM 5.23. Let Q be a normal question. Then E(X, Q) iff X entails each presupposition of Q and X entails no direct answer to Q. Proof: Since X entails no direct answer to Q just in case X does not meentail any singleton set which contains a direct answer to Q, and by Corollary 5.2 an evoking set entails all the presuppositions of the evoked question, it suffices to prove that if Q is normal, then dQ is me-entailed by X if X entails each presupposition of Q. Assume that Q is normal and that X entails each presupposition of Q. Thus each normal interpretation which is a model of X is also a model of the set of presuppositions of Q. But since Q is normal, then each normal interpretation which makes true all the presuppositions of Q also makes true at least one direct answer to Q. Thus the set X me-entails the set of direct answers to Q. 0 Thus as long as normal questions are considered, evocation takes place just in case each presupposition is entailed and no direct answer is entailed . But we can also prove the following (meta)theorem

EVOCATION OF QUESTIONS

137

THEOREM 5.24 . The following conditions are equivalent: (a)

each question is normal,

(b) for each question Q: E(X, Q) iff X entails each presupposition of Q and X entails no direct answer to Q. Proof: (a) =t (b). By Theorem 5.23. (b) =t (a). It suffices to assume that there is a non-normal question, say, Q(, and to put X = PresQ (. 0

Theorem 5.24 shows that the first clause of the definition of evocation can not always be replaced by the clause "X entails each presupposition of Q" ; since there are languages in which some questions are not normal and some of such questions can be evoked, there are cases in which entailing all the presuppositions is insufficient for evocation even if no direct answer is entailed . But sometimes entailing a single presupposition is sufficient. This is due to: THEOREM 5.25 . Let Q be a regular question. Then E(X, Q) iff X entails someprospective presupposition of Q and X entails no direct answer to Q. Proof: Let us recall that to say that X entails no direct answer to Q is tantamount to saying that no singleton set which contains a direct answer to

Q is me-entailed by X. (=t) If Q is a regular question, then Q is normal and Q has prospective

presuppositions. So if E(X, Q), then by Theorem 5.23 there is a prospective presupposition of Q which is entailed by X. (~) Let A be a prospective presupposition of Q entailed by X. Since A meentails the set of direct answers to Q, it follows that X me-entails the set of direct answers to Q. 0 Thus as long as regular questions are concerned, evocation takes place just in case some prospective presuppos ition is entailed and no direct answer is entailed . But we can also easily prove THEOREM 5.26. If the following condition holds: (e) for each question Q: E(X, Q) iff X entails some prospective presupposition of Q and X entails no direct answer to Q,

then each evoked question is regular. Since there are languages which contain non-regular questions and some of them can be evoked, the condition (e) need not be true with respect to each language of the considered kind . Let us now prove

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THEOREM 5.27 . If entailment is compact, then the following equivalence holds: (ee) E(X, Q) iff there are AI' ... • An E dQ (n > 1) such that X pAl v .. . V An andfor each A E dQ. X non pA. Proof: If there are AI' ... , An E dQ (n > 1) such that X pAl v ... v An. then - since direct answers are sentences - X I~ dQ by Corollary 4.7 and Corollary 4.3. At the same time for each A E dQ we have X non pA iff X non I~ A. Thus it suffices to prove that if entailment is compact and E(X. Q). then there exists an at least two-element finite subset Y of the set dQ such that X I~ Y; if this is so. then by Corollary 4.7 there are A I. . .. , An E dQ (n > I) such that X p AI V ••• v An. If E(X, Q). then X I~ dQ. If entailment is compact, then by Corollary 4.5 also me-entailment is cempact . Thus there is a finite subset Y of dQ such that X I~ Y. Suppose that Y = 0. Therefore X E Inc . Thus by Theorem 5.6 it is not the case that E(X, Q). Hence Y "¢ 0. Suppose"that Y is a singleton set. Therefore for some A E "dQ we have X p A; thus non E(X, Q). Hence Y has at least two elements. 0 Theorem 5.27 shows that if entailment is compact, evocation takes place just in case some disjunction of at least two direct answers is entailed but no single direct answer is entailed. But, again, there are interesting languages of the considered kind in which entailment is not compact; for example, the language 5£** has this property. Let us observe that Theorem 5.27 yields that for these languages in which entailment (and thus also me-entailment) is compact we may define evocation according to DEFINITION 5.3. El(X, Q) iff (i) there are AI> ...• An E dQ (n > 1) such that X p AI V ••• v An' and (ii) for each A E dQ. X non p A. Our previous results imply that if evocation was defined according to Definition 5.3, it would still have the properties required by the conditions (Cd. I). (Cd.2), (Cd.3) and (Cd.4); thus Definition 5.3 may be regarded as providing us with an adequate (with respect to the above conditions) explication of the analyzed notion of the arising of a question from a set of declarative sentences for those languages of the considered kind in which entailment is compact. Similarly. Theorem 5.25 shows that for those languages which include only normal questions evocation may be defined according to

EVOCATION OF QUESTIONS

139

DEFINITION 5.4 . E 2(X, Q) iff (i) for each B E PresQ, X l= B, and (ii) for each A E dQ, X non l= A . Theorem 5.27 yields, in tum , that if a language contains only regular questions, evocation may be defined by DEFINITION 5.5 . E 3(X, Q) iff (i) for some B E PPresQ, X l= B, and (ii) for each A E dQ, X non l= A . The definitions 5.3 , 5.4 and 5.5 have one important property in common: they do not make use of the concept of me-entailment, Let us recall, however, that there are languages of the considered kind in which entailment is not compact, or which contain non-normal or non-regular questions. 5.2 .5. The problem of compactness of evocation The evoking sets can be either finite or infinite. Similarly, the evoked questions can have both finite and infinite sets of direct answers. But one may ask whether evocation has the "compactness" property , i.e . whether a set of d-wffs evokes a question if and only if some finite subset of it evokes this question, and whether a question is evoked if and only if some question whose set of direct answers is a finite subset of the set of direct answers to the principal question is evoked. The general solution to the above problem is negative; as we have pointed out, a question evoked by some subset of a given set of d-wffs need not be evoked by the entire set, and a question included in a given question evoked

by some set of d-wffs need not be evoked by this set. But in some languages evocation manifests a kind of "restricted compactness". A question Q is said to be finite if and only if the set of direct answers to Q is finite; by an infinite question we mean a question whose set of direct answers is infinite but denumerable. We may prove THEOREM 5.28 . If entailment is compact and E(X, Q), then there are: a finite subset Y of X and a finite question QI whose set of direct answers is a subset of the set of direct answers to Q such that E(Y, Qt). Proof: It suffices to consider the situations in which X is an infinite set or Q is an infinite question. Assume that E(X, Q) , where X is an infinite set. If entailment is compact, so is me-entailment: thus there must exist a finite subset Y of X such that Y 11= dQ. Since Y is a subset of X, no direct answer to Q is entailed by Y. So Y evokes Q. Assume that E(X, Q), where Q is an infinite question. Therefore X 11= dQ.

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If entailment is compact, then also me-entailment is compact. Thus there exists a finite subset Z of dQ such that X IF z. The set Z has at least two elements; otherwise we would have had X E Inc or X F A for some A E dQ, and thus non E(X, Q). Since direct answers are sentences, Z is a set of sentences. In each of the considered languages questions of the first kind do occur; moreover, each finite and at least two-element set of sentences is a set of direct answers to some question of the first kind. Thus there is a finite question QI such that dQI s dQ and X IF dQI' Since E(X, Q) and dQI ~ dQ, then for each B E dQI we have X non IF B. Therefore E(X, QI)' 0 Thus we may say that compactness of entailment yields not only the compactness of me-entailment, but also a kind of "restricted compactness" of evocation . It cannot be said, however, that evocation has this property in each language of the considered kind; in particular, evocation does not have the analyzed property in the language ;;e". 5 One additional remark is in order here. Theorem 5.28 shows that if entailment in some language is compact, then infinite questions of this language are superfluous, at least with respect to evocation. Yet, entailment in the language ;;e' is compact. Thus Theorem 5.28 seems to justify our decision as to introducing into the language ;;e' only finite questions. Let us finally pay some attention to the property of evocation described by THEOREM 5.29. Let E(X, Q).lfany ofthe following conditions is fulfilled: (*) entailment is compact and Q is a normal question, (**) Q is a regular question then there exists a d-wff C such that (i) X F C, (ii) for each A E dQ, A F C, and (iii) E(C, Q). Proof: Since by Corollary 4.13 each normal question is regular if entailment is compact, it suffices to consider the case in which Q is a regular question. Assume that E(X, Q), where Q is regular question. Therefore by Theorem 5.25 there exists a prospective presupposition, say, C, of Q, such that X F C. But PPresQ ~ PresQ; so for each A E dQ we have A F C. Since C is a prospective presupposition, we also have elF dQ. Since E(X, Q), then by Theorem 5.7 Q f£. Ret; therefore C non F A for each A E dQ. Thus E(C,

»

, An example may helpful here: the question (#) ? S(P11(XI is evoked in the language :£•• by the set {3.x t PII(X t)}. but no finite question included in (#) is evoked by this set.

EVOCATION OF QUESTIONS

141

Q). 0

Theorem 5 .29 says, first , that if entailment is compact, then for each normal evoked question there exists a kind of "intermediate" formula: a formula which evokes the question and is entailed both by the evoking set and by each direct answer to the evoked question. Theorem 5.29 also says that the situation is analogous in the case of each evoked regular question. Let us stress that the property of evocation described by Theorem 5.29 is important from the point of view of the logical analysis of erotetic inferences. Theorem 5 .29 implies that, roughly, in some cases erotetic arguments whose conclusions are evoked by the premises can be split into arguments of two kinds: standard deductive arguments which lead to some "intermediate" formulas (in practice those formulas are usually the relevant prospective presuppositions) and erotetic arguments which lead from the intermediate formulas to questions. We shall discuss this in detail in Chapter 8.

5.3. METATHEOREMS AND EXAMPLES

5.3.1. Evocation in the language ;e* So far we were assuming that the language for which we defined evocation is an arbitrary but fixed language of the considered kind. Let us now be more concrete and let us pay some attention to evocation in the language ;e*, that is, to evocation of questions of the language ;e* by sets of d-wffs of this language. Of course, evocation in ;e* is defined by means of the concept of me-entailment in ;eo. We will be using the symbol E::e* instead of the symbol E in order to stress that we are considering evocation in the language ;eo. The symbols Ip::e* and F::e* refer to me-entailment and entailment in ;eo, respectively . Let us recall that since each interpretation of ;e* is assumed to be a normal interpretation of this language, entailment in ;e* reduces to logical entailment. The language ;e* contains only questions of the first kind . Let us first prove the following (meta)theorem: THEOREM 5.30. E::eo(X, ? {AI> ... , An}) iff (i) X F::e* AI V ... V An' and (ii) for each A E {AI> .. ., An}, X non F::e* A. Proof: If Q is of the form ? {AI' ... , An}, then dQ = {AI> , An}· Since AI, ... , An are sentences, then by Corollary 4.7 X Ip::e* {Ail , An} iff X F::e* AI V '" V An. At the same time for each A E {AI' ... , An} we have X non Ip::e* A iff X non F::e* A. 0 According to Theorem 5.30, a question of the first kind is evoked in the

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142

language :£" by a set of d-wffs X if and only if X entails in :£" a disjunction of all the direct answers to this question. but no single direct answer to it is entailed in :£" by X. We shall now present some theses which say that questions of ;;e" of a strictly defined form are evoked by some finite sets of d-wffs of :£0which contain d-wffs of strictly defined forms. Let us assume that the metalinguistic symbols F. Fl' F2 , • •• represent nonequiform (i.e. syntactically different) atomic sentences of the language z " of the form Pt (u\, .. .• un), where u t • • • • , u; are closed terms. Let P, R be metalinguistic variables which represent predicate symbols of :£0. The theses given below are immediate consequences of Theorem 5.30 and some wellknown metalogical facts. We shall not comment on them; let us only recall that although each question of the form ? {At, .. ., An} can be read "Is it the case that A I' or is it the case that A 2• • • • , or is it the case that An?", in some special cases a different reading is also possible. In particular, questions of the form ? {A, ,A} can also be read "Is it the case that A?", whereas questions of the form? {A 1\ B. A 1\ 'B} can be read "It is the case that A ; is it also the case that Bl" , Questions of the form? {A 1\ B, A 1\ , B. ,A 1\ B, ,A 1\ 'B} can be read "Is it the case that A and is it the case that m il; questions having the form? {,A . A 1\ B. A 1\ 'B} can be read "Is it the case that A?; if so. is it also the case that B?". (5.1)

E~o(F

(5.2)

E~o(Ft

v F2 , ? {FI

1\

F2 , '(F t

1\

F2)}) .

(5.3)

E~"(FI v F2• ? {F J

-.

F2• '(Ft

-.

F2)}) ·

(5.4)

E~o(FI

1\

F2• ' (F t

1\

F2)}) .

(5.5)

E~o(Fl -. F2• ? {F I , 'Ftl ).

(5.6)

E !£o(Ft -. F2• ? {F 2• 'F2}) ·

(5.7)

E~o(Fl

v ,F. ? {F. ' F }).

-. F2• ? {F t

== F2• ? {F t

(5.8) E!£o(F I == F2• ?

1\

F2• ' (F t

1\

F2)}) .

{Ft v F2• ,(FI v F2) } ) .

EVOCATION OF QUESTIONS E~.(FI

A

(5 .10) E~.(FJ

--+

(5.9)

•••

>

(5.19) where m

(5.20) where m

>

Fm• ? {Fl • ..., Fm}).

F1 v ... v Fm• ? {F --+ FI ,

••••

F --+ Fm}).

A

•••

A

Fm--+F.? {F1--+F• ..., Fm--+F}).

F J v ... v Fm• F. ? {FI ,

... ,

Fm}),

1.

(5 .21) E~.(FI where m > 1. (5.22) E~.(FI where m > 1.

(5.23)

V

1.

E~.(F --+

>

F•• "'(F1 A

1.

E~.(FJ

>

••• A

1.

(5.18) E~.(F --+

where m

F. --+ F, F. ? {F1 A

F. F2 --+ ..,F. ? {F. "'F}) .

(5.17) E~.(FI v ... where m

A

143

A

.. .

A

A ... A

E~.(P(ul)' ... .

Fm--+ F

A

..,F. ? {..,F!> ... , ..,Fm}).

Fm--+ F, ..,F. ? {"'F I ,

. .. .

..,Fm}).

P(U.) , ? {Vx; P(x;)• ..,Vx; P(x;)}).

••• A

F.)}).

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144

(5 .24) E;e.(P(u .) /\ R(u.), .. ., P(uJ /\ R(un) , ? {Vx j (P(xj ) ..... R(xj ) ) , ",Vxj (P(x;) ..... R(xj ) )}) , where P ;c R. (5 .25) E;e'(P(u.), .. ., P(un) , "'P(u n+ I), ? {Vx j (xj ;c un+ i :" P(xj ) ) , 3xj (xj ;c un+ 1 where Un+ l is not equal to any of u 1, ••• , Un'

/\

"'P(x j )) }) ,

(5 .26) E;e.(P(u.) /\ R(u.), ... , P(U n) /\ R(un) , P(Un+ l ) /\ ..,R(un+ 1) , ? {VXj (Xj ;C Un + • ..... (P(Xj ) ... R(x j ))) , 3xj (Xj ;C Un + 1 /\ P(Xj ) ..,R(xj )) }), where P ;c R, and un+ 1 is not equal to any of U\o .. . , un'

(5.27) E;e.(P(u ,) , P(U2) , ? {u . = U2, U1 where U\o u2 are distinct closed terms ,

;C

/\

u2}),

(5.28) E;e.(Vx; (P(x;) :; (xj = u. v V x, = uJ) , ? {u 1 ;c ;C un' "'(U . ;C ;C uJ}), where n > 1 and u , u; are nonequifonn closed terms. (5.29) E;e,("'(u 1 ;c . . . ;c uJ , ? {u 1 = U2' U 1 = U3' ... , u 1 U3, .. ., Un_I = Un}), where n > 1 and u 1, .. . , u; are nonequifonn closed terms ,

(5.30) E;e.(3xj P(x j ) , ? {P(U), 3xj (P(x j )

/\

= u., U2 =

x, ;c u)}).

Let us recall that a question of the form ? {P(U), 3xj (P(xj ) can be read "Is it U that has the property P? " . (5.31) E;e.{F, /\ ... /\ Fm

.....

/\

x, ;c

u)}

3xj P(x;), F., .. ., Fm ,

? {P(u), 3xj (P(x;) /\ x, ;c u)}), where P does not occur in the sentences F., ... , Fm •

(5.32) E;e.(3xj (P(x j ) /\ (xj = u 1 V ... V x, = un)), ? {P(U1) , where n > 1 and U\o .. . , u; are nonequifonn closed terms.

. .. ,

P(uJ}

145

EVOCATION OF QUESTIONS

Let us recall that a question of the form? {P(u,) • ...• P(un}} can be read "Which of the u\> ...• u; has the property P?". (5.33) Elfo(FI A .. . A Fm -+ :Ix; (P(X;) A (X; = UI V .. . V X; = uJ), Fl' ... , Fm , ? {P(U.), .. .. P(un )}) . where n > 1, U\> .... u; are nonequiform closed terms and F; ~ P(uk ) for 1 ~ i ~ m, 1 ~ k s n.

(5.34) Elfo(-'VX; (Xi

= U.

v .. , v X;

= u; -+ P(X;», ?

{-,P(U.), ....

-,P(u n )}) ,

where n

>

1 and u•• ...• u, are nonequiform closed terms.

(5.35) Elfo(VX; (x; ? {-,P(U I ) • where n > 1. P(uk) closed terms .

= u. .. . .

~

v ... v x,

= u, -+ P(X;»

-+

F. -,F.

-,P(u n)}) .

F for 1

~

k

~

nand U\> .... un are nonequiform

One can easily find further theses of the above kind. We leave it to the Reader . 5.3.2. Evocation in the language ;eoo

Let us now switch to the language ;e" and analyze what questions of this language are evoked by what sets of d-wffs of it. As above, we shall use the symbol Elf" instead of the symbol E in order to stress that we are speaking about evocation in the language ;;eoo. that is. evocation of questions of ;;e" by sets of d-wffs of this language. Of course. evocation is now defined by means of me-entailment in the language ;e". The following (meta)theorem can be proved in the same way as its analogue pertaining to the language ;;eo; the symbol Flf" refers to entailment in the language ;e": THEOREM 5.31. Elf"(X,? {A., .... An}) iff (i) X Flf" AI v .. . V An' and (ii) for each A E {A., ...• An}, X non Flf"" A. Each d-wff of ;e"" is a d-wff of ;e" and each question of the first kind of ;e" is a question of the first kind of ;e"". As far as finite sets of d-wffs of ;e"" are concerned , entailment in ;e"" reduces to logical entailment; the situation is analogous in the case of some infinite sets of d-wffs (cf. Chapter 4. Section 4.4.2 for details). On the other hand. entailment in ;e" also reduces to logical entailment. Thus if we replace in the theses (5.1) - (5.35) the

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146

symbol E~. by the symbol E~ ••, we will obtain theses which characterize evocation in the language ;;e••. Let us stress, however, that such a step is not

»

always legitimate. For example. we may say that the infinite set S(P11(X 1 of sentences evokes in the language ;e. the question:

(I)

? {Vx1 PII(X 1). -,Vx 1 Pll(X t ) }

but we cannot say that the set S(P 11(X I» evokes the question (I) in the language ;e". The reason is that the set S(Pll(X I» entails in the language ;e•• the direct answer VX1 PII(X I) to the question (I) (let us recall that each element of the domain of a normal interpretation of ;e•• has a name) . As far

as finite sets of d-wffs are concerned, evocation in ;e. yields evocation in ;e••; when we take into consideration infinite sets of d-wffs, the situation depends on the form of the analyzed set. The language ;e•• contains not only questions of the first kind . but also questions of the second and third kind. Let us now prove THEOREM 5 .32 .

E~ ••(X.

? S(Axil ... Xi,.»

iff

(i) X F~ .. :Ixit • • • :Ixin Axil ' " Xin' and (ii) for each B E S(Axil .. . x in). X non F~ .. B. Proof: If Q is of the form? S(Axit •• • x in) . then dQ

= S(Axil ... xin).

By Theorem 4 .2 we get X Il=~ .. S(Axil ... Xi,.) iff X F~·· :Ixi l ... :Ixin Axit . .. x in• At the same time for each B E S(Axit .. • Xi) we have X non l1=x•• B iff X non Fx•. B. 0 According to Theorem 5.32, an existential question of the second kind of the form ? S(Axit • • • Xi,.) is evoked in ;e•• by a set of d-wffs X if and only if the set X entails in ;e•• the existential generalization :Ixi l ... :Ix;n Axil . .. x;n of the sentential function Axil .. . x in that occurs in the question, and the set X does not entail in ;e•• any sentence from the set S(Axil Xi,.), that is , any sentence that results from the sentential function Axil Xi" by proper substitution of closed terms for the free variables. As above . let us assume that the metalinguistic symbols F, F l , F2• .. . represent nonequiform sentences of ;e•• having the form Pt(u l • .. . , u,J, where u l , . .. . u, are closed terms . The letters P, R are metalinguistic variables which range over predicate symbols of ;;e... The letter f will be used below as a metalinguistic variable which represents function symbols of ;e... Let us also recall that each existential question of the first kind ? S(Axit ... Xi,,)' where n > I. can be read "Which n-tuple 1.

x.)

.. .

--+ 3x,

... ,

x.))),

.. . 3x. Ptx, ... , x.)),

= n.

(5.45) E!£••(Vx.+ 1 VX.+m3x1 ? S(R(u" , um) 1\ P(x t , where P .,c. R if m = n.

3x. (R(x.+ 1, ,

x.))),

... ,

x.+ m)

1\

P(x" ... , x.)),

148

CHAPTERS

(5.46) E'f ••(Vx, .. . VXn (3xn + , P(xl ,

Xn , xn + t ) /\ vs.; VXn+2 (P(x!> ... , Xn , xn+ l ) /\ P(x!> ... , Xn , xn +2) xn + 2 = xn + I», ? S(P(ut , •• • , u., xn + I») · ... ,

(5.47) E'f••(Vx t .. . VXn (3xn+ t P(x l , .. . , Xn , Xn + l ) /\ VXn+ t VXn +2 (P(x!> ... , Xn , Xn + l ) /\ P(x!> ... , Xn , Xn +2) Xn + t = Xn+ 2», ? S(P(ut , .. . , u.; Xn + l ) /\ VXn +2 (P(u t , -. Xn + 2 = xn + t») ·

-.

-.

... ,

u., Xn +2)

VXn (P(x!> ... , xJ -. 3xn + t (R(xn + t ) /\ f(x t , •• •, xJ Xn + t», Ptu., ... , uJ , ? S(R(xn + t ) /\ f(u t , ... , uJ = Xn + I»·

(5.48) E'f••(Vxt

(5.49) E'f••(3x , ? S(P(X h

.. .

3xn (P(x!> ... , xJ /\ f(x l , ... , xJ , xJ /\ f(x h .. . , xJ = u» .

=

= u),

(5.50) E'f••(VXn + 1 (R(xn + ,) -.3x, ... 3xn (P(xl , , xJ /\ f(x!> .. ., xJ Xn + I», R(u), ? S(P(x!> ... , xJ /\ f(x!> , Xn) = u».

=

Let us now consider open questions of the second kind . THEOREM 5.33. E'f••(X, ? S(Axi, ... x;,,» iff E'f••(X, ? O(Axi, ... Xi.»· Proof: The set O(Axil .. . Xi.) consists of all the sentences from the set S(Axil .. . x;,,) and all the sentences having the form of a conjunction of nonequiform elements of this set. So S(Axil .. . Xi.) £: O(Axil ... Xb,) and for each C E O(Axi, ... Xi.) there exists B E S(Axi, ••• Xi.) such that C F'f.' B. Therefore by Corollary 5.7 E'f" (X, ? S(Axi, ... x;,,» iff E'f"(X, ? O(Axil ... x;,,» . 0 According to Theorem 5.33, an existential question of the second kind having the form ? S(Axi l ... Xi.) and the corresponding open question of the second kind? O(Axil ... x;,,) are evoked by exactly the same sets of d-wffs. It follows that the necessary and sufficient conditions of evocation of existential and open questions of the second kind are identical . Thus if we replace iri the formulas (5.36) - (5.50) the symbol S by the symbol 0, we will obtain theses which describe evocation of some open questions of the second kind. We leave it to the Reader. Let us now consider existential questions of the third kind. One can easily prove

EVOCATION OF QUESTIONS

149

THEOREM 5 .34 . E9!**(X, ? W(x,IAx;») iff (i) X F9!** 3xi Axi , and (ii) for each B E S(Axi ) , X non F9!** B. Proof: By Corollary 5.7 and Theorem 5 .32 . 0 Recall that questions of the form? W(x, .., X,H_IIAxi) , where k > 1, can be read "What are at least k different x/s such that Axi?" , whereas an expression of the form 3 ~kxi Axi can be read "there exists at least k Xi such that Axi . "

>

1, then E9!"(X, ? W(x, ... X,H_IIAxi)) iff (i) X F9!** 3 ~kxi Axi , and (ii) for each B E S(A(x;!x,) /\ .. . /\ A(X;!X,H_I) /\ x, ~ X,+I /\ X, ~ X'+2/\ .. . 1\ X, ~ X,H_I 1\ X,+ I ~ X,+2 /\ .. . /\ X,H_2 ~ X,H_l) () W(x, .. . X,H_IIAxi ) , X non F9!** B. Proof: Let us designate by cI> the set S(A(x;!x,) /\ ... /\ A(X;!X,H_l) /\ x, ~ X,+I /\ x,~ X,+2 1\ . " /\ x, ~ ,H-l /\ X,+I ~ X,+2 /\ ... /\ X,H_2 ~ X,H_l) () W(x, ... X,H_IIAxi) . If E9!**(X, ? W(x, .. . X,H-I!Axi ) ) , then X 1l=9!** W(x, .. . X,H_IIAxi) ; therefore X F9!" 3 ~kxi Ax; by Theorem 4.5. At the same time for each C E W(x, .. . X,H_IIAx;) we have X non IF9!** C. Since cI> £; W(x, ... X,+k_lIAxi)' then for each B E cI>, X non IF9!** B and thus X non F9!** THEOREM 5.35. Ifk

B. If X F9!** 3 ~kxi Axi, then X IF9!** W(x, .. , X,H_IIAxi) by Theorem 4.5 . For each C E W (x, .. . X,+k_1 IAxi ) there is B E cI> such that C F9!** B. Thus if for each B E cI>, X non F9!** B, then also for each C E W(x, .. . x,+k_IIAx;), X non IF9!** C. Therefore E9!**(X, ? W(x, ... X,H_IIAxi)). 0 By applying Theorem 5.35 and some well-known metalogical results one can easily show that, into al., the following theses are true.

.. . /\ Fm -. 3 ~kxl P(XI), F I , ... , Fm, xkIP(x1) ) ) , where P does not occur in the sentences Flo ... , E; if k = 1.

(5.52) E9!**(F1

? W(x 1

/\

...

CHAPTER 5

150

(5.55)

E~,,(Vxl

? W(x 1

(5.56)

VXn 3~kxn+1 P(xl , • • • , xn' xn+ I ) , xtIP(u 1, .. . , un' Xl))) '

E~,,(VxI

VXn (R(xl , .. . , Xn) .... ? W(x\ xtIP(x l »), where P ¢ R if n = 1 and k = 1. (5.57) E~ ••(3~kxn+' VXI ? W(x I ... xtIR(u\, where P ¢ R if n = 1 and k

(5.59)

E~,,(,VXI (R(xl )

....

», R(u

3~kxn+1

VXn (R(xh .. . , xJ , uJ " P(X I ))) , = 1.

P(xl )) , ? W(xIIR(x l )

P(xn+ 1

n),

l , ... ,U

»,

"

P(xn+ I

"

,P(XI ))) ,

where P ¢ R. (5 .60) E~,,(VxI (R(x l ) .... P(x l )) " F I " ? W(x.lR(x t ) " ,P(xt ))) , where P does not occur in F and P ¢ R.

.. .

"

F m "" F, ,F, F lo .. . , F m,

We shall not formulate here the necessary and sufficient conditions of evocation of general questions of the second and third kind . By and large, these questions have true direct answers provided that there exists finitely many objects satisfying the sentential functions which occur in them. It is impossible, however, to express in the language ::e" the fact that a given set is finite. But we will formulate sufficient conditions of evocation of some general questions of the second kind ; we will also formulate here sufficient conditions of evocation of general questions of the third kind. Let Ax; be an arbitrary but fixed sentential function of the language ::e" which contains x, as the only free variable. Let us assume that an expression of the form : (*)

3 ~ kx, Ax;

is an abbreviation of the expression having the form:

EVOCATION OF QUESTIONS

(5.61) Vx, ... VX,+k (A(x;!x,) /\ ... /\ A(x;!x,+k) V ••• V

x, = x,+k

V

X,+l = X'+2

151

x, = X,+l V x, X,+k_l = X,+k),

-+

V •• • V

= X,+2

where r is the least index such that for each s ~ r, the variable Xs is substitutable for Xi in the sentential function Ax;. An expression of the form 3 ~kx; Ax; can be read "there exists at most k x, such that Ax;." Let us now prove the following LEMMA 5.2. For each kEN: {3x; Ax;, 3 ~kx; Ax;} 1l=!£OO U(Ax;) . Proof: Let ~ be a normal interpretation of the language ;;eoo such that ~ F 3x; Ax; and ~ F 3 ~kx; Ax;. If ~ F 3x; Ax;, then by Theorem 4.1 at least one sentence of the set S(Axi ) is true in the interpretation ~. The number of such sentences may be finite; it is also possible that infinitely many sentences of the set S(Ax;) are true in ~.

Suppose that finitely many sentences of the set S(Axi ) are true in ~. Thus there are exactly j such sentences, for some fixed j E N. Hence for some sentence(s) of the form A(x;!u\), ... , A(x;!uj ) , where u 1, • • • , uj are nonequiform closed terms, we have ~ F A(x;!u1) /\ • •• /\ A(x;!uj ) (of course if j = 1, then A(x;!ut ) /\ ••• /\ A(x;!uj ) reduces to A(x;!ut Suppose that ~ non F Vx; (Axi -+ Xi = U 1 V • • • V X; = u) . It follows that ~ F 3x; (Ax; /\ Xi ;iC U 1 /\ ••• /\ X; ;iC uj ) . So by Theorem 4.1 there exists a closed term, say, u, such that ~ F A(x;!u) r; U ;iC U\ /\ • • • r; U ;iC uj ' The term U is distinct from each of the terms Ul' ••. , uj - otherwise we would have had ~ non F A(x;!u) /\ U ;iC u 1 r; •• • /\ U ;iC uj ' Thus the sentence A(x;!u) is thej+l-th sentence of the set S(Ax;) which is true in the interpretation ~. On the other hand, by assumption there are exactly j such sentences. We arrive at a contradiction. Therefore

».

~

FA(x;!u

uj ) .

1)

/\

• ••

/\

A(x;!u) /\ Vx; (Ax;

-+

x,

= U1 V

• •• V

x,

=

So there is B E U(Ax;) such that ~ FB. Suppose that infinitely many sentences of the set S(Axi ) are true in the interpretation ~ . There are two possibilities: (a) there exists an infinite subset M' of the domain of the interpretation ~ such that for each y E M'

CHAPTERS

152

and for each ~-valuation s such that y = s, we have ~ FAx; [S] , or (b) the poss ibility (a) does not hold , but there exists a finite and non-empty subset M" of the domain of the interpretation ~ such that for each y E M" and for each ~ -valuation S such that y = s, we have ~ FAx; [s]. Suppose that the situation (a) takes place . Therefore the sentence 3 ~k+ Ix; ~ ; so ~ non F 3 ~kx; Ax;. We arrive at a contradiction. Hence the poss ibility (b) holds. Since the subset M " is finite and non-empty , it has exactly m elements, for some fixed mEN. At the same time m ~ k. It follows that for some mEN, where m ~ k, we have

Ax; is true in

~

F 3~ mx; Ax; 1\

3~mx;

F 3x, .. . 3x,+m_1

(A(x;!x,)

Ax;,

and hence ~

1\ ••• 1\

A(x;!X,+m_l)

1\

x, ~ ... ~

X,+m_l) 1\ Vx, .. . VX,+m (A(x;!x,) 1\ • • • 1\ A(x;!x,+m_l) 1\ A(x;!x,+m) 1\ x, ~ ... ~ x,+m_l -+ x,+m = x, V ... V x,+m = X,+m_l)'

where -r is the least index such that for each s ~ r, the variable x, is substitutable for x. in Ax;; if m = 1, then the first conjunct reduces to 3x, A(x;!x,)) and the second conjunct reduces to VX,VX'+l (A(x;!x,) -+ X,+I = x,). By applying Theorem 4.1 we get

1\

A(x;!x,+t)

F A(x;!u.) 1\ • •• 1\ A(x;!u",) 1\ VX,+m (A(x;!x,+m) -- X,+m x,+m = uJ for some closed terms u 1' ... , um' Therefore ~

= u.

V

~

FA(x;!u t)

1\ ••• 1\

A(x;!uJ

V x, = um)' So there is B E U(Ax;) such that 5.2. 0

~

1\

Vx; (Ax; -+ x,

= u t v ...

F B. This completes the proof of Lemma

We can now prove THEOREM 5.36.

If

X he" 3x; Ax;, for some kEN, X F;e" 3 ~kx; Ax;, and for each C E S(Ax;), X non F;e" C, then E;e"(X, ? U(Ax;)) . Proof: If X F;e" 3x; Ax; and for some kEN, X F;e•• 3 -skx, Ax;, then X IP;e.. U(Ax;) by Lemma 5 .2 . For each B E U(Ax;) there is C E S(Ax;) such that B F;e•• C. Thus if for each C E S(Ax;) we have X non F;e" C, then we also have X non IP;e•• B for each B E U(Ax;). Therefore E;e"(X, ? U(Ax;)). (i) (ii) (iii)

o

EVOCATION OF QUESTIONS

153

Thus a question of the form? U(Ax;) ("What are all of the x;'s such that Ax;? ") is evoked in the language ;;e•• by a set of d-wffs X if the set X entails the sentence 3x; Ax; which says that that the condition Ax; is satisfied by at least one object, the set X entails some sentence of the form 3 ~kx; Ax; which says that at most k (where kEN) objects satisfy the condition Ax;, and the set X entails no sentence from the set S(Ax;), that is, entails no sentence which says that a given object satisfies the condition Ax;. Let us now consider general questions of the third kind. Let an expression of the form: (**)

3(k)x; Ax;

be an abbreviation of an expression having the form: (5.62)

3~kx;

Ax; " 3~kx; Ax;,

where kEN. An expression of the form 3(k)x; Ax; can be read "there are exactly k x, such that Ax;." We shall prove LEMMA 5.3 . For each m ~ k: 3(m)x; Ax; I~!£ •• T(x, ' " x,+k_lIlAx;). Proof: Let ~ be a normal interpretation of the language ;;e•• such that for some m ~ k, ~ ~ 3(m)x; Ax;. If k = 1 and m = 1, then ~ ~ 3x, A(x/x,)

r; VX,VX'+I

(A(x/x,) " A(x/x,+I) - X,+I

= x,).

Thus by Theorem 4 .1 there is a closed term, say, u, such that ~ ~ A(x/u) " V X,+I (A(x/x,+I) - X,+I

= u).

The above sentence is an element of the analyzed set T(x, II Ax;). Assume now that k > 1. Therefore m > 1. By assumption we have ~ ~ 3x, .. . 3x,+m_1 (A(x/x,)

r; .. . "

A(x/X,+m_l) " x, ~ ... ~

X,+m_l) " Vx , ... VX,+m (A(x/x,) " ... " A(x/X,+m_l) " A(x/x,+m) x, = X,+I V x, = X,+2 V •• , V x, = x,+m V X,+ l = X,+2 V . • • V x,+m_1 = x,+m) '

It follows that ~ ~ 3x, ... 3x,+m_1 (A(x/x,) "

X,+m_l) " Vx, . .. VX,+m (A(x/x,) " x, ~ ... ~ x,+m_1 - x,+m = x, V

r;

• ••

A(x/X,+m_l) " x, ~ ' " ~ "A(x/X,+m_l) "A(x/x,+J " V x,+m = X,+m_I) '

Thus by Theorem 4.1 there are nonequiform closed terms u1, that

••• ,

um such

154

CHAPTER 5

VX,+m (A(x/x,+m) -. X,+m

= Ut

V ••• V

X,+m

= uJ .

The above sentence is an element of the analyzed set T(x, ... x,+k_IIIAxi) . 0 Thus for the general questions of the third kind we get

THEOREM 5.37. Iffor some m ~ k, J( he"" 3(m)xi Axi, andfor each B E S(Axi) , X non I=~"" B, then E~""(X, ? T(x, '" x,+k_tIlAx;» . Proof: Theorem 5.37 is an immediate consequence of Lemma 5.3 and the fact that for each A E T(x, ... X,+k_1 II Axi) there exists B E S(Axi) such that A F~" B. 0 According to Theorem 5.37, a general question of the third kind ? T(x, .. , x,+k_IIIAx;) ("What are all of the at least k different Xi'S such that Ax;?") is evoked in the language 5£"" by a set of d-wffs X, int.al., if the set X entails some sentence of the form 3(m)xi Axi, where m ~ k, that is, a sentence which says that there is exactly m objects satisfying the condition Axi , and X entails no sentence falling under the schema A(x/u) , that .is, entails no sentence which says that the object designated by u satisfies the condition Axi • When speaking here about entailment we have in mind entailment in the language ;;e"". By means of Theorem 5.36 or Theorem 5.37 and some well-known metalogical results one can easily find examples of general questions of the second and third kind evoked by sets of d-wffs of the analyzed language . We leave it to the Reader .

CHAPTER 6

GENERATION OF QUESTIONS

The aim of this Chapter is to introduce the concept of generation of a question by a set of declarative formulas. We shall also characterize the basic properties of generation. There are two possible ways of looking at generation. From the purely formal point of view it is a special case of evocation: it is the evocation of risky questions. But the proposed definition of generation can also be viewed as a new explication of the intuitive concept of the arising of a question from a set of declarative sentences. This explication is also an adequate one; yet, now adequacy is relativized to a new set of criteria . In the case of evocation the conditions (Cd. 1), (Cd.2), (Cd.3) and (Cd.4) formulated in Chapter 1 played the role of requirements of adequacy; as far as generation is concerned, the relevant set of requirements of adequacy consists of the conditions (Cd. 1), (Cd.2), (Cd.4) and (Cd.5) . Let us recall that the condition (Cd.5) is stronger than the condition (Cd.3). Consequently, the range of generation is narrower than the range of evocation. On the other hand, it seems that generation is more interesting from the point of view of possible applications; we shall discuss this in Chapter 8. As in the case of evocation, we assume that the language for which we define the concept of generation is an arbitrary but fixed formalized language 5£ of the kind considered in Chapter 3, that is, a certain first-order language with identity enriched with questions in the way presented in Chapter 3. We also assume that the semantic concepts pertaining to the language 5£ are defined in the way presented in Chapter 4; unless otherwise indicated, the metalinguistic symbols used below pertain to the language 5£ and the qualifications "in 5£" , "of 5£", etc. will be normally omitted. As in the case of evocation, however, we define here the concept of generation in a language 5£ of a question of 5£ by a set of d-wffs of 5£; this concept is defined by means of the concept of me-entailment in 5£.

CHAPTER 6

156

6.1. DEFINITION OF GENERATION The relation G of generation of a question by a set of declarative formulas is defined by: DEFINITION 6.1 G(X, Q) iff (i) X IF dQ, (ii) 0 non IF dQ, and (iii) for each A E dQ, X non IF A. According to Definition 6.1, a question Q is generated by a set of d-wffs X if and only if the set of direct answers to Q is me-entailed by the set X, although the set of direct answers to Q is not me-entailed by the empty set and no singleton set that contains a direct answer to Q is me-entailed by the set X. If G(X, Q), then the set X is said to be the generating set, whereas the question Q is said to be the generated question. Similarly as in the case of evocation, we shall write G(Ah ... , A., Q) instead of (;({A h . .. , A.}, Q). It is easily seen that the relation of generation has the properties required by the conditions (Cd.1), (Cd.2), (CdA) and (Cd.5) formulated in Chapter 1. The clause (iii) of Definition 6.1 is fulfilled just in case no direct answer to Q is entailed by X; so the requirements (Cd. 1) and (Cd.2) are fulfilled . Since each question generated by a given set of d-wffs is also evoked by this set, then by Corollary 5.2 each presupposition of a generated question is entailed by the generating set; so the requirement (Cd A) is fulfilled as well. The clause (ii) of Definition 6.1 yields that there is a normal interpretation of the language in which all the direct answers to a generated question are not true; so a generated question does not have a true direct answer in every case. But the clause (i) of Definition 6.1 yields that a generated question does have a true direct answer in each normal interpretation of the language in which all the d-wffs in the generating set are true; so we may say that a generated question must have a true direct answer if all the d-wffs in the generating set are true. Thus the conjunction of clauses (i) and (ii) of Definition 6.1 warrants that the requirement (Cd.5) is also satisfied. Therefore we may say that Definition 6.1 provides us with a new explication of the analyzed concept of the arising of a question from a set of declarative sentences: this new explication is adequate with respect to the conditions (Cd. 1), (Cd.2), (CdA) and (Cd.5). But we may also look at generation as on a special case of evocation . It is obvious that the following is true: COROLLARY 6.1. G(X, Q)

iff

E(X, Q) and Q

~

Saf.

GENERATION OF QUESTIONS

157

Thus generation may be viewed as evocation of risky questions. Presumably the intuitive meaning of the concept of generation is best characterized by THEOREM 6.1 . G(X, Q) iff Q is made sound by X and Q is informative relative to X. Theorem 6.1 is an immediate consequence of Theorem 5.1 together with Corollary 6.1 and the definitions 4.16 and 4.17. . Let us now analyze the basic properties of generation. 6.2 . BASIC PROPERTIES OF GENERATION

6.2.1. The generating sets We can easily prove THEOREM 6.2 . IfG(X, Q) , then X

¢

0.

THEOREM 6.3 . IfG(X, Q), then X ¢ Taut. According to theorems 6.2 and 6.3 neither the empty set nor sets of tautologies can generate questions. Let us recall that the situation is different in the case of evocation (cf. Theorem 5.12) . On the other hand, like in the case of evocation, we can easily prove: THEOREM 6.4. Each generating set is consistent. As a consequence of Theorem 6.4 we get COROLLARY 6.2. If G(X, Q) , then X contains no contradictory d-wff. Let us now prove THEOREM 6.5. IfG(X, Q), then X is incomplete and contains at least one

synthetic d-wff. Proof: If G(X, Q), then by Corollary 6.1 and Theorem 5.3 X f/; Cp, At the same time by Theorem 6.3 X ¢ Taut and by Corollary 6.2 dQ n Contr = 0 . Therefore X n Synt ¢ 0. 0 Theorem 6.5 says that each generating set is incomplete and contains at least one synthetic d-wff. It follows that both complete sets of d-wffs and sets of tautologies do not generate any questions. But also the following theorem is true: THEOREM 6.6. Each incomplete set of d-wffs that contains at least one

synthetic d-wff generates at least one question . Proof: If X f/; Cp , then there is a sentence, say, A, such that X non FA and X non F'A. If X n Synt ¢ 0, then there is a synthetic sentence, say, B, such that X

FB

(if each of the synthetic d-wffs of X is a sentential

CHAPTER6

158

function, B may be identified with the universal closure of some synthetic sentential function of X). Each language of the considered kind contains conditional yes-no questions; moreover, direct answers to such questions can be built up of any sentences of a language. On the other hand, it is obvious that the conditional yes-no question? {B /\ A, B /\ ""A} is generated by the set X. 0 As a consequence of Theorems 6.5 and 6.6 we get THEOREM 6.7. A set of d-wffs X generates at least one question incomplete and contains at least one synthetic d-wff.

iff X is

Theorem 6.7 shows that the domain of the generation relation consists of all the incomplete sets of d-wffs of the considered language which contain at least one synthetic d-wff. On the other hand, incompleteness and containing some synthetic components seem to be the basic features of sets of d-wffs which can be carriers of nonanalytic knowledge. 6.2.2. The generated questions

Let us now analyze the questions which can be generated by some sets of d-wffs. As a consequence of Corollary 6.1 and Theorem 5.7 we get THEOREM 6.8. If G(X, Q), then Q is risky but not self-rhetorical. A completely contradictory question is a question each of whose direct answers is a contradictory d-wff; a completely tautological question is a question each of whose direct answers is a tautology. Theorem 6.8 yields: COROLLARY 6.3. If G(X, Q), then Q is neither a completely contradictory question nor a completely tautological question . Thus both completely contradictory questions and completely tautological questions are not generated by any sets of d-wffs. Theorem 6.8 also implies COROLLARY 6.4 . No direct answer to a generated question is a tautology. COROLLARY 6.5. Each generated question has at least two direct answers which are synthetic sentences. Let us add that the synthetic direct answers to a generated question cannot contradict each other. This is due to THEOREM 6.9. IfG(X, Q), then there are no A , B E dQ such that for each ~ 1= A iff ~ non 1= B . normal interpretation Proof: It suffices to observe that if there are A, B E dQ such that for each normal interpretation ~, ~ 1= A iff ~ non 1= B, then Q E Saf and thus

o.

GENERATION OF QUESTIONS

159

non G(X, Q). 0 One remark is in order here. Although Theorem 6.9 implies that simple yes-no questions cannot be generated, it does not imply that conditional yesno questions and focussed yes-no questions cannot be generated. On the other hand, as we pointed out in Chapter 3, Section 3.2.1, many natural language yes-no questions could be formalized as conditional or focussed yes-no questions . Recall that a question Q is normal iff PresQ IF dQ; Q is proper iff Q is normal but not self-rhetorical . THEOREM 6.10. Let Q be a normal question. Then Q is generated by some

set of d-wffs iff Q is risky but not self-rhetorical. Proof: Since we have already proved Theorem 6.8, it suffices to prove that each normal, risky and not self-rhetorical question is generated by some set of d-wffs. If Q is normal, then PresQ IF dQ. If Q is risky, then 0 non IF dQ; if moreover Q is not self-rhetorical, than the set PresQ entails no direct answer to Q. Therefore G(PresQ, Q). 0 Theorem 6.10 implies that in those languages in which each question is normal the range of generation consists of all the risky but not self-rhetorical questions. Let us recall, however, that there are languages of the analyzed kind which contain non-normal questions. Let us finally note THEOREM 6.11. Each proper but risky question is generated by at least

one set of d-wffs. Proof: It suffices to observe that each proper but risky question is generated by any singleton set which contains a prospective presupposition of the question . 0

6.2.3. Generation andrelations between setsofd-wffsandrelations between questions It is obvious that, like in the case of evocation, sets of d-wffs which have exactly the same normal models generate exactly the same questions. We can easily prove THEOREM 6.12. If X and Yare sets of d-wffs such that for each normal interpretation ~ : ~ 1= X iff ~ 1= Y, then G(X, Q) iff G(Y, Q). We can also easily prove the following analogues of Theorems 5.17 and

CHAPTER 6

160

5.20: 1 THEOREM 6.14. If a question Q2 is equipollent to a question QI' then G(X, Q2) iff G(X, QI)' THEOREM 6.15. If a question Q2 is equivalent to a question QI> then G(X, Q2) iff G(X, QI)' According to the above theorems, both equipollent and equivalent questions are generated by exactly the same sets of d-wffs. The following theorems are analogues of Theorems 5.15 and 5.16: THEOREM 6.16. Let G(X, Q) and Y ~ X. Then G(Y, Q) provided that YH= dQ. THEOREM 6.17 . Let G(X, Q) and X ~ Y. Then G(Y, Q) provided that for each A E dQ, Ynon FA. Theorem 6.16 says that a question generated by a given set of d-wffs is also generated by each such set included in it which me-entails the set of direct answers to the question. Theorem 6.17 says, in tum, that a question generated by a set of d-wffs is also generated by those larger sets which do not entail any direct answer to the question. This condition is a necessary one: so generation is not "monotonic" . We can also prove THEOREM 6.18 . Let G(X, QI) and let Q2 be a question weaker than Qt. Then G(X, Q2) if Q2 is risky and for each B E dQ2' X non F B. THEOREM 6.19. Let G(X, QI) and let Q2 be a question which includes QI' Then G(X, Q2) if Q2 is risky and for each B E dQ2 - dQI> X non F B. According to Theorem 6.18, a set of d-wffs which generates a question Ql generates also those questions weaker than QI> which are risky and whose

direct answers are not entailed by this set. Theorem 6.19 says, roughly, that a set of d-wffs which generates a given question generates also those questions "larger" than it, which are still risky and whose "new" direct answers are not entailed by this set. Let us stress that the "riskiness" assumption is indispensable; the situation is different in the case of evocation (see Theorems 5.18 and 5.22). The following theorems are analogues of theorems 5.19 and 5.21:

t

5.2.3.

For the definitions of the analyzed relations between questions see Chapter 5. Section

GENERATION OF QUESTIONS

161

THEOREM 6.20 . Let G(X, QI) and let Q2 be a question stronger than QI' Then G(X, Q2) if X 11= dQ2' THEOREM 6.21. Let G(X, QI) and let Q2 be a question included in QI' Then G(X, Q2) if X 11= dQ2" Thus a set of d-wffs which generates a question QI generates also those questions stronger than QI and included in QI whose sets of direct answers are me-entailed by this set.

6.2.4. Some alternative definitions of generation There are languages for which we can define the concept of generation without applying the concept of me-entailment. Let us recall that a factual presupposition is a presupposition being a synthetic d-wff . We shall prove THEOREM 6.22 . Let Q be a normal question. Then G(X, Q) iff Q has

factual presuppositions, X entails each factual presupposition of Q and X entails no direct answer to Q. Proof: (~) If G(X, Q), then E(X, Q). On the other hand, if Q is a normal question, then by Theorem 5.23 X evokes Q just in case X entails each presupposition of Q and X entails no direct answer to Q. Thus it suffices to prove that if G(X, Q) , then Q has factual presuppositions . Assume that G(X, Q). Suppose that PresQ ~ Taut. Since Q is normal, then PresQ 11= dQ; therefore 011= dQ and hence non G(X, Q). SO PresQ rt Taut. If follows that there is a presupposition, say, A , of Q, such that A fl Taut . Suppose that A E Contr. Since direct answers are sentences, it follows that dQ £ Contr. Therefore non G(X, Q) by Corollary 6.3. So A (/; Contr. Thus A E Synt; hence Q has factual presuppositions. (4=:) If Q has factual presuppos itions, then 0 non 11= dQ. If for each A E dQ, X non 1= A, then also for each A E dQ, X non 11= A. Since Q is normal, then PresQ 11= dQ. Thus it suffices to prove that X entails each presupposition of

Q. Suppose that X does not entail some presupposition of Q. Since X entails each presupposition of Q being a tautology and by assumption X entails each factual presupposition of Q, it follows that some presupposition of Q is a contradictory d-wff. But direct answers are sentences; so each direct answer to Q is a contradictory sentence. Therefore each synthetic d-wff is a factual presupposition of Q. Since X entails each factual presupposition of Q, it follows that X is inconsistent. Thus X entails each direct answer to Q. We

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arrive at a contradiction . Therefore X entails each presupposition of Q. 0 Theorem 6.22 says that generation of a normal question takes place just in case the question has factual presuppositions, each factual presupposition of it is entailed and no direct answer to it is entailed. Thus for those languages in which each question is normal we may define generation according to DEFINITION 6.2 . G1(X, Q) iff (i) Prest!? ~ 0, (ii) for each B E Prest!?, X 1= B, and (iii) for each A E dQ, X non 1= A . There are languages, however, which contain non-normal questions; moreover, some of these questions can be generated. Let us also note THEOREM 6.23: IfG(X, Q) iffG 1(X, Q) , then each question having factual

presuppositions is normal. Proof: Assume that there is a question, say, Q, such that Q has factual presuppositions, but is not normal. Thus PresQ non IF dQ ; hence 0 non IF dQ and the set Prest!? does not entail any direct answer to Q. On the other hand, Prest!? entails each factual presupposition of Q. SO G·(Prest!?, Q) . But since Q is not normal, then Prest!? non IF dQ; therefore non G(Prest!?, Q) .

o

We can also prove THEOREM 6.24. Let Q be a normal question. Then G(X, Q) iff Q is risky, X entails each presupposition of Q and X entails no direct answer to Q. Proof: By Theorem 5.23, Corollary 6.1 and Corollary 4.9. 0 THEOREM 6.25. The following conditions are equivalent: (a) each question is normal, (b) for each question Q: G(X, Q) iff Q is risky, X entails each presupposition of Q and X entails no direct answer to Q. Proof: (a) ~ (b). By Theorem 6.24. (b) ~ (a). Assume that there is a non-normal question, say, Qt- Since Q. is non-normal, then Q. is risky and not self-rhetorical. It is obvious that the set PresQ. of presuppositions of QI entails each presupposition of Q. and entails no direct answer to Qr- On the other hand, since QI is not normal, then the set PresQ. does not me-entail the set of direct answers to Q. and thus does not generate Qi - 0 We shall now prove

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163

THEOREM 6.26. Let Q be a regular question. Then G(X, Q) iff X entails some maximal factual presupposition of Q and X entails no direct answerto

Q. Proof: Since each regular question is normal, the implication (~) is a consequence of Theorem 6.22 and the definition of the concept of maximal factual presupposition. On the other hand, since Theorem 6.22 is true and each regular question is normal, then in order to prove the implication (~) it suffices to show that each regular generated question has maximal factual presuppositions. Assume that G(X, Q) . Since Q is regular, it has prospective presuppositions . Let B be a fixed prospective presupposition of Q; thus B IF dQ. Since G(X, Q), then B fl Taut. Suppose that B E Contr. Therefore Q E Ret; hence by Theorem 6.8 non G(X, Q). Therefore B E Synt; it follows that B E PresrQ. Suppose that B fl mPresrQ. So there exists a factual presupposition, say, A, of Q such that B non FA. Therefore B non IF dQ. We arrive at a contradiction. Thus B E mPresrQ; so Q has maximal factual presuppositions.O According to Theorem 6.26 a regular question is generated just in case some of its maximal factual presuppositions is entailed, but no direct answer to it is entailed . Let us stress, however, that there are regular questions which have no maximal factual presuppositions and thus cannot be generated (regular safe questions are case in point here). Moreover, there are languages which contain non-regular questions. But if a language contains only regular questions (as, for example, the language ;e* does), then generation in this language can be also defined by DEFINITION 6.3 . G2(X, Q) iff (i) for some B E mPresrQ, X F B, and (ii) for each A E dQ, X non FA. Let us observe that both Definition 6.2 and Definition 6.3 do not apply the concept of me-entailment. Let us finally note that if some language contains only regular questions, generation in this language can also be defined differently . This is due to: THEOREM 6.27 . Let Q be a regular question. Then G(X, Q) iff there is a prospective presupposition of Q which is not a tautology and is entailed by X, and X entails no direct answer to Q. Proof: (~) If Q is regular, then Q is normal and Q has prospective presuppositions . So if G(X, Q), then by Theorem 6.24 there is a prospective presupposition,

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say, A, of Q such that A is entailed by X. Suppose that A E Taut. Since A 11= dQ, then 011= dQ and thus non G(X, Q). We arrive at a contradiction . So A fl Taut. (¢) It suffices to observe that if there is a prospective presupposition of Q which is entailed by X and is not a tautology, then ~ non 11= dQ and X 11= dQ.

o

6.2.5. Compactness and otherproblems Like in the case of entailment, we can prove the following: THEOREM 6.28. If entailment is compact and G(X, Q), then there exist: a finite subset Y of X and a finite question Q. whose set of direct answers is a

subset of the set of direct answers to Q such that G(Y, QI)' Proof: By Corollary 6.1, Theorem 5.28 and the fact that if Q is risky, and dQ. is a subset of dQ, then also Ql is risky. 0 Theorem 6.28 says that the compactness assumption yields for generation the same kind of "restricted compactness" as in the case of evocation. Let us stress, however, that even if entailment is compact it cannot be said that, first, a set of d-wffs generates a question if and only if some finite subset of it generates this question: it may happen that no direct answer to a question is entailed by a given finite subset of the considered set, but the whole set entails some direct answer(s) to the question. Second, it cannot be said that a question is generated if and only if some finite question whose set of direct answers is a subset of the set of direct answers to the primary question is generated : it can happen that no direct answer to a finite question is entailed by a given set of d-wffs, but nevertheless this set entails some direct answer(s) to the corresponding "larger" question. Let us now prove THEOREM 6.29. Let Q be an infinite question. Then if entailment is compact and X generates each finite question included in Q, then G(X, Q). Proof: If X generates each finite question included in Q, then X 11= dQ. Suppose that X entails some direct answer(s) to Q. Thus X entails a direct answer to some finite question included in Q; hence this question is not generated by X. But by assumption each finite question included in Q is generated by X. So X entails no direct answer to Q. Suppose that 011= dQ. Since entailment is compact, so is me-entailment; it follows that for some finite and non-empty subset Y of dQ we have 0 11= Y. The set Y must have at least two elements; otherwise some finite question(s) included in Q would not have been generated. But each finite and at least two-element set of sentences is a set of direct answers to some question. So if 0 11= Y, then at

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165

least one finite question included in Q is not generated by X. We arrive at a contradiction. So 0 non I~ dQ. Therefore G(X, Q). 0

If entailment is compact, then the consistency of all the finite subsets of a given infinite set guarantees the consistency of the whole set. Theorem 6.29 shows that if entailment is compact , then the generation of all the finite questions included in a given infinite question guarantees the generation of this question. The compactness assumption has also other interesting consequences. We may prove THEOREM 6.30 . If entailment is compact and Q is a normalquestion, then G(X, Q) iff X entails some maximal factual presupposition of Q and X entails no direct answer to Q. Proof: By Theorem 6.26 and Corollary 4 .13 (if entailment is compact, then each normal question is a regular one) . 0 We can also prove: THEOREM 6.31. If entailment is compact, Q is a normal question and G(X, Q) , then there exists a d-wff C E Synt such that: (i) X F C, (ii) for each A E dQ, A F C, and (iii) G(C, Q) . Proof: If entailment is compact and Q is a normal question, then by Corollary 4.13 and Corollary 4.11 Q has prospective presuppositions. Let C be a prospective presupposition of Q. By Theorem 6.24 we have X F C; since C is a prospective presupposition of Q, we also have C Ip dQ and for each A E dQ, A F C. Since G(X, Q), then by Theorem 6.8 Q r£ Ret; so for each A E dQ, C non FA. Moreover, if G(X, Q), then 0 non I~ dQ. Therefore G(C, Q). On the other hand , by Theorem 6.5 we get C E Synt.

o

THEOREM 6.32 . If Q is a regular question and G(X, Q), then there exists

a d-wff C E Synt such that: (i) X F C, (ii) for each A E dQ, A

F C, and

(iii) G(C, Q) . Proof: Similar to that of Theorem 6.31. 0

Theorem 6.31 says that if entailment is compact, then for each normal generated question there exists an "intermediate" synthetic formula which is entailed both by the generating set and by each direct answer to the generated question, and which itself generates the question. Theorem 6.32

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says that the situation is analogous in the case of each generated regular question. Let us observe that the "intermediate" formula has the following property: if it is not true, then both no direct answer to the generated question is true and at least one d-wff in the generating set is not true. Thus we may say that the analyzed formula can play the role of a "potential falsifier" of the generated question and the generating set. When viewed in the perspective of logical analysis of erotetic inferences, theorems 6.31 and 6.32 show that in some cases an erotetic inference whose conclusion is generated by the premises can be split into a standard inference which leads from the premises to the "intermediate" formula and an erotetic inference which leads from the "intermediate" formula to the question; we shall come back to it in Chapter 8. Let us designate by .dQ the set of negations of all the direct answers to a question Q, that is, the set {.A: A E dQ}. We shall prove

THEOREM 6.33. IfG(A I, ... , An' Q), then: (i) there exists A E {AI> ... , An} such that .dQ 1= .A, or (ii) G(.dQ, ? {.AI> ... , .An}). Proof: Assume that G(A I, .... An' Q). Therefore {AI' .. ., An} 11= dQ. There are two possibilities : (a) for some A E {AI> ... , An}, A 11= dQ, or (b) for each A E {AI' ... . An}, A non 11= dQ. If the possibility (a) takes place, then by the definition of me-entailment for some A E {AI> An} we have .dQ 1= .A. Assume that the possibility (b) holds. Since {AI> An} 11= dQ, then the set {AI. . . . , An} has at least two elements; so also the set {.A I' .. .• • An} has at least two elements . Hence ? {.A I> .. . , .An} is a question; moreover, by the condition (b) no direct answer to this question is entailed and thus also me-entailed by the set .dQ. Since {AI' .... An} 11= dQ, then {A I ' ... , An} 11= dQ. Therefore .dQ 11= {.A I ' . .. , .An} by the definition of me-entailment. Suppose that 0 11= {.A I' ... , .An}. Hence 0 1= .A I v .. . v .An. It follows that .A I v .. . v .An E Taut; therefore Al /I .. . /I An E Contr and hence {AI> .. . . An} E Inc. Thus by Theorem 6.4 non G(AI> . .. , An' Q). We arrive at a contradiction. So 0 non 11= {.AI> .. ., .An} . Therefore G('dQ.? {.A I> ... , .An}). 0 Theorem 6.33 yields

THEOREM 6.34. If G(AI • .. . , An' Q) andfor each A E {AI' .. .. An}, A non 11= dQ, then G('dQ. ? {.AI> .. .. • An})· Theorem 6.33 says that if a finite set of d-wffs {AI • .. ., An} generates a question Q, then the set of negations of all the direct answers to Q either entails the negation of universal closure of some d-wff in {AI> ... , An} or

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generates the question whose set of direct answers is made up of the negations of the universal closures of the d-wffs in the set {AI' .. .. An}; according to Theorem 6.34 the second possibility takes place if the set of direct answers to Q is not me-entailed by any single d-wff in {AI ' . ..• An}. One remark is in order here . A generated question is always risky and thus it can happen that no direct answer to it is true. In this situation, however. the negations of direct answers are true (let us recall that direct answers are sentences, not sentential functions) . According to Theorem 6.33. the set of negations of direct answers to a generated question Q either determine which d-wffin the generating set {AI' ...• An} is not true or generates a consecutive question about it. 6.3. GENERATION IN THE LANGUAGES

;;e'

AND

;;e"

6.3.1. Generation in the language ;;e' Let us now consider generation in the exemplary languages ;;e' and ;;e" described in Chapter 3, Section 3.3 .1. We begin with the language ;;e'. Of course, generation in ;;e' is defined according to the pattern presented by Definition 6.1 and by means of the concept of me-entailment in;;e·. We shall use the symbol G£e. for generation in ;;e' and the symbols Taut£e' for me-entailment in ;;e', entailment in ;;e' and the set of tautologies of ;;e', respectively. Let us recall that entailment in ;;e' reduces to logical entailment and the tautologies of ;;e' are simply the logically valid formulas of the classical predicate calculus with identity. individual constants and function symbols. The only questions of ;;e' are questions of the first kind. Let us prove:

IF£e" F£e"

THEOREM 6.35. G£e'(X, ? {AI> ..., An}} iff (i) X Al V • • • V An> (ii) Al V • •• V An f/. Taut£e" and (iii) for each A E {AI> .... An}, X non he. A. Proof: If G£e'(X, ?{AI• ..., An}}, then by Corollary 6.1 E£e'(X, ? {AI> .. ., An}}; so by Theorem 5.30 the conditions (i) and (iii) are fulfilled . Assume that Al V V An E Taut£e" Therefore Al V . .. V An. Hence {AI' An} by Corollary 4.7 and thus non G£e'(X, ? {AI' .. ., An}} . We arrive at a contradiction. So AI v .. . v An f/. Taut£e" If the conditions (i) and (iii) are fulfilled, then E£e'(X, ? {AI> ... , An}} by Theorem 5.30 . If moreover AI v ... v An f/. Taut£e" then non Al v .. . v An; hence by Corollary 4 .7 non {AI' .. ., An} . Thus by Corollary 4 .9 ? {AI' ... , An} is not a safe question. Therefore G(X. ? {AI' ... . An}) by

F£e'

0 F£e'

01F£e'

0

IF£e'

0

F£e'

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Corollary 6.1. 0 According to Theorem 6.35, a question of the first kind is generated in;;e· by a set of d-wffs X of ;;e. if and only if X entails some disjunction of all the direct answers to the question, this disjunction is not a tautology and the set X does not entail any single direct answer to the question. When speaking about entailment, we have in mind entailment in ;;e. and thus essentially logical entailment; similarly, by a tautology we mean here a tautology of ;;e. and thus essentially a logically valid formula. By applying Theorem 6.35 and some well-known metalogical results we can easily find what questions of ;;e. are generated in ;;e. by what sets of dwffs of this language. And it can be shown that if we replace in the theses (5.11) - (5.22) and (5.29) - (5.35) formulated in Chapter 5 the symbol E;e. by the symbol G;e., wi will obtain exemplary theses which characterize generation in ;;e".

6.3.2. Generation in the language ;;e•• Generation in the language 5£•• is defined by means of the concept of meentailment in ;;e". Below we use the symbol G;e.. for generation in ;;e•• and the symbols I~;e.., F;e", Taut;e.. for me-entailment in ;;e••, entailment in ;;e•• and the set of tautologies of ;;e••, respectively. For the characteristics of these concepts see Chapter 4, Section 4.4.2. The language ;;e•• contains questions of the first kind and questions of the second and third kind . The following theorem can be proved in an analogous way as Theorem 6.35: THEOREM 6.36 . G;e••(X, ? {AI> ... , AnD iff (i) X F;e.. Al V ... V An, (ii) AI · V .. , V An f£ Taut;e", and (iii) for each A E {AI> ... , An}, X non F;e•• A. In the case of finite sets of d-wffs entailment in ;;e•• reduces to logical entailment and thus to entailment in ;;e•. Moreover, Taut;e.. is equal to Taut;e•. So by replacing in the theses (5.11) - (5.22) and (5.29) - (5.35) the symbol E;e.. by the symbol G;e•• we obtain some exemplary theses which characterize generation in ;;e•• of certain questions of the first kind of this language. For existential questions of the second kind we have

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169

»

? S(Axil ... xin iff (i) X I=~ •• 3xil ... 3xj• Axil'" Xj. , (ii) ~il .. . 3xj • Axil ... x; f£ Taut~ ••, and (iii) for each B E S(Axil ... xin) , X non h .. B. Proof: Similarly as in the case of Theorem 6.35 (we use Theorem 4 .2 instead of Corollary 4 .7 and Theorem 5.32 instead of Theorem 5.30). 0

THEOREM 6.37 .

G~ ••(X,

For open questions of the second kind we can easily prove

»

THEOREM 6.38 . G~ ••(X, ? S(AxiJ '" xin iff G~ ••(X, ? O(Axil .. . xin». Thus the necessary and sufficient conditions of generation in 5£" of existential and open questions of the second kind are exactly the same. Let's now consider existential questions of the third kind . THEOREM 6.39. G~ ••(X, ? W(x,IAxj » iff (i) X I=~ •• 3xj Axj , (ii) 3xj Axj f£ Taut~ ••, and (iii) for each B E S(Axj ) , X non I=~ •• B.

»

THEOREM 6.40 . If k > I , then G~ ••(X,? W(x, ... x,+k_I!Axj iff (i) X I=~ •• 3 ~kxj Axj, (ii) 3 ~kxj Ax; f£ Taut~ ••, and (iii) for each B E S(A(x;!x,) r; • . • "A(x;!x,+k_I) "X, ¢ X'+1 " x, ¢ X,+2 " ... " x, ¢ X,+k.1 "X,+I ¢ X'+2 " ... " X,+k_2 ¢ X,+k_I) n W(x, .. . X,+k_IIAxj ) , X non I=~ •• B The proofs of the above theorems go along the lines of the proof of Theorem 6.35 : the difference is that we use theorems 4.4 and 4.5 instead of Corollary 4.7 and theorems 5.34 and 5.35 instead of Theorem 5.30 . Let us now consider general questions of the second and third kind. Since 3xj Ax; is a presupposition of both a question of the form ? U(Ax;) and a question of the form ? T(x, ... x,+k_IIIAx;), then by Corollary 6.1 and theorems 5.36 and 5.37 we get: THEOREM 6.41. If (i) X I=~ •• 3xj Ax;, (ii) 3x; Axj f£ Taut~ ••, (iii) for some kEN, X I=~ •• 3 5:kxj Axj , and (iv) for each C E S(Axj ) , X non I=~ •• C,

then G~ ••(X, ? U(Ax;».

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THEOREM 6.42. If (i) for some m ~ k, X F!£•• 3(m)x; Ax;, (ii) 3x; Ax; f£ Taut!£••, and (iii) for each B E S(Ax;), X non F!£•• B, then G!£••(X, ? T(xr .. . xr+k_IIIAxj By applying the above theorems and some well-known metalogical results one can easily find examples of questions of the second and third kind generated in ;e.' by sets of d-wffs of this language. In particular, it can be shown that if we replace in the theses (5.36) - (5.60) the symbol E!£•• by the symbol G!£••, we will obtain exemplary theses which characterize generation in ;;e".

».

CHAPTER

7

EROTETIC IMPLICATION

The aim of this Chapter is to introduce the concept of erotetic implication. Erotetic implication is a semantic relation between a question, a (possibly empty) set of declarative formulas, and a question . Its definition may be regarded as an explication of the intuitive concept of the arising of a question from a question and a set of declarative sentences (cf. Chapter 1). As in the case of previous chapters, in our general considerations we assume that the language for which we define the concept of erotetic implication is an arbitrary but fixed formalized language :f of the kind considered in Chapter 3; the semantic concepts pertaining to the language :f are defined in the way presented in Chapter 4. Unless otherwise indicated, the metalinguistic symbols used below pertain to the language :f ; the qualifications "in :f" , "of:f" will be normally omitted . Let us stress, however, that what we are going to define here is the concept of erotetic implication in the (arbitrary but fixed) language :£; this concept will be defined by means of the concept of me-entailment in :f. 7.1. DEFINITION OF EROTETIC IMPLICATION Recall that the sign C stands for "is a proper subset of." DEFINITION 7. 1. A question Q implies a question Q, on the basis of a set of d-wffs X (in symbols: Im(Q, X, Q,)) iff (i) for each A E dQ: X, A IF dQ" and (ii) for each B E dQI there exists Y C dQ such that Y ~ 0 and X, BIF Y. The relation 1m defined above will be called erotetic implication . According to Definition 7.1 , a question Q implies a question Q, on the basis of a set of d-wffs X if and only if, first, for each direct answer A to the question Q, the set of direct answers to the question Q, is me-entailed by the set X u {A}, and second, for each direct answer B to the question Q, there

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exists a non-empty proper subset Y of the set of direct answers to the questionQ such that Y is me-entailed by the set X u {B}. If Im(Q, X, QI)' then the question Q is said to be the implying question, the question QI is called the implied question, and the set X is said to be the set of auxiliary d-wffs. We shall write Im(Q, At, ... , An> Q\) instead of Im(Q, {AI> ... , An}, QI)' It is easily seen that the relation of erotetic implication defined above has the properties required by the conditions (Cd.6) and (Cd.7) formulated in Chapter 1. The following are immediate consequences of Definition 7 .1 and the definition of me-entailment: COROLLARY 7.1. Let Im(Q, X, QI)' Then for each normal interpretation ~ : if at least one direct answer to the implying question Q is true in ~ and all the d-wffs in X are true in ~, then at least one direct answer to the implied question QI is true in ~. COROLLARY 7.2 . Let Im(Q, X, QI)' Then for each direct answer B to the implied question QI there exists a non-empty propersubset Yof the set of direct answers to the implying question Q such that for each normal interpretation ~: if B is true in ~ and all the d-wffs of X are true in~, then at least one answer in Y is true in ~.

Thus we may say that Definition 7. 1 provides us with an adequate explication (with respect to the conditions (Cd.6) and (Cd.7)) of the concept of the arising of a question from a question and a set of declarative sentences. The concept of erotetic implication is purely semantical. Let us, however, add some general comments here. First, it may be said that by the clause (ii) of Definition 7.1 each direct answer to the implied question narrows down (together with the auxiliary d-wffs) the class of possibilities offered by the implying question . Or, to put it differently: each of the direct answers to the implied question if it is true and if all the auxiliary d-wffs are true gives us a guarantee that some true direct answer to the implying ("initial") question can be found in some "limited area", that is, in a certain proper subset of the set of all the direct answers to the implying question (in some cases this proper subset may be a singleton set; it depends on the logical form of the questions involved). In other words, the implied question is in each case (i.e. regardless of the fact which of its direct answers will appear to be acceptable) cognitively useful with respect to the implying question . Thus, to speak generally, finding a true direct answer to the implied question is always a profit. On the other hand, the clause (i) of Definition 7.1 guarantees that the implied question must have a true direct answer if the implying

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173

question has a true direct answer and the set of auxiliary d-wffs consists of truths. In other words, the implied question is secured by the implying question and auxiliary d-wffs, Let us stress , however, that we neither assume nor deny here that the implying questions have true direct answers and that the auxiliary d-wffs are true . But it may be proved that if all the auxiliary d-wffs are true , then either both the implied question and the implying question have true direct answers or none of them have such answers . By Corollary 7.1 and Corollary 7.2 we get: COROLLARY 7.3. Let Im(Q, X, Q\). Then/or each normal interpretation ~ being a model 0/ X: at least one direct answer to the implied question QI

is true in ~ if and only if at least onedirect answer to the implying question Q is true in ~ . Thus the posing of implied questions is also a method of "checking" our initial questions and declarative premises: if it will occur that an implied question has no true direct answer, then the initial question or/and some declarative premise(s) should be revised . 7.2 . BASIC PROPERTIES OF EROTETIC IMPLICATION Let us now characterize the basic properties of erotetic implication. As an immediate consequence of Definition 7.1 we get: THEOREM 7 .1. Im(Q, X, Q). According to Theorem 7.1, each question implies itself on the basis of any set of d-wffs . It follows that each question can be both an implied question and an implying question. It also follows that any set of d-wffs (the empty set included!) may be a set of auxiliary d-wffs, One can easily prove THEOREM 7.2. If Im(Q, X, Q,) and X s Y, then Im(Q , Y, Q,). Theorem 7.2 says that a question implied on the basis of a given set of dwffs X is also implied on the basis of any set which includes X. Thus we may say that erotetic implication is "monotonic" with respect to sets of auxiliary d-wffs. As we have pointed out, the situation is different in the case of evocation and generation (cf. theorems 5 .16 and 6.17) . Recall that a question Q is said to be sound relative to a set of d-wffs X iff X \1= dQ. Both the implying question and the implied question mayor may not be sound relative to the set of auxiliary d-wffs . But we may prove that an implied question is sound relative to the set of auxiliary d-wffs just in case the implying question is sound relative to this set.

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THEOREM 7.3. Let Im(Q, X, QI)' Then QI is sound relative to X iffQ is sound relative to X. Proof: (~) Assume that XIp dQI and that X non Ip dQ. It follows that there exists a normal interpretation 9t in which all the d-wffs of X are true, but no direct answer to Q is true. Since X Ip dQI' then at least one direct answer to QI is true in 9t . If Im(Q, X, QI), then for each B E dQl there exists a non-empty proper subset Y of dQ such that X, B Ip Y. It follows that at least one direct answer to Q is true in 9t . We arrive at a contradiction. (~) Assume that XIp dQ and that X non Ip dQl' It follows that there exists a normal interpretation 9t of the language in which all the d-wffs in X are true , no direct answer to Q1 is true and at least one direct answer to Q is true. On the other hand, if Im(Q, X, QI), then for each A E dQ we have X, A Ip dQI ' Thus if 9t is a model of X and at least one direct answer to Q is true in 9t, then at least one direct answer to QI is true in 9t as well. We arrive at a contradiction. 0 The following theorem is also true: THEOREM 7.4. /fIm(Q, X, Ql)' then: (i) for each A E dQ, the question QI is sound relative to the set X, A, (ii) for each B E dQI' the question Q is sound relative to the set X, B. Proof: By Definition 7.1, Definition 4.16 and Corollary 4.3. 0 According to Theorem 7.4, the implied question is sound relative to any set which consists of the auxiliary d-wffs and a direct answer to the implying question, and the implying question is sound relative to any set made up of the auxiliary d-wffs and a direct answer to the implied question. Let us now prove: THEOREM 7.5 . /fIm(Q, X, QI), Q E Saf and X 5; Taut, then Q1 E Sal. Proof: If Im(Q, X, Ql) and Q E Sal, then XIp dQ. Therefore XIp dQI by Theorem 7.3 . But since X 5; Taut, then QI E Saf. 0 Theorem 7.5 says that only safe questions are implied by safe questions on the basis of any subset of the set of tautologies (the empty set included) . THEOREM 7.6. /fIm(Q, X, Ql) and Q E Sal, then Xlp dQI ' Proof: If Q E Sal, then X Ip dQ. So X Ip dQI by Theorem 7.3. 0 According to Theorem 7.6, safe questions imply only such questions, which are sound relative to the sets of auxiliary d-wffs, THEOREM 7.7. /fIm(Q, X, QI) , QI E Sal and X £ Taut, then Q E Sal. Proof: Similarly as in the case of Theorem 7.6. 0

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Thus only safe questions can imply safe questions on the basis of sets of tautologies or the empty set. THEOREM 7.8. Let Qt E Sa!. Then Im(Q, X, QI) iff for each B E dQl there exists a non-empty proper subset Y of dQ such that and X, B IF Y. Proof: It suffices to observe that if Qt E Saf, then for each A E dQ we have X, A IF dQI'O According to Theorem 7.8 a safe question Ql is implied by a question Q on the basis of a set of d-wffs X if and only if for each direct answer B to Qt there is a non-empty proper subset of the set of direct answers to the question Q which is me-entailed by the set X u {B} (that is, the clause (ii) of Definition 7.1 is fulfilled). Thus it is not the c¥e that a safe question is implied by each question on the basis of any set of d-wffs. One can easily prove THEOREM 7.9. If X E Inc, then Im(Q, X, Qt) . Thus a given question on the basis of an inconsistent set of d-wffs implies each question. Recall that a completely contradictory question is a question each of whose direct answers is a contradictory d-wff. THEOREM 7.10. If Im(Q, X, Qt), X ~ Taut and Q is a completely contradictory question, then Q, is a completely contradictory question. Proof: If Im(Q, X, Qt), then by Theorem 7.4 for each B E dQt we have X, B IF dQ . Thus if dQ ~ Contr and X ~ Taut, then for each B E dQI and for each normal interpretation ~ we have ~ non 1= B. Direct answers are sentences. Therefore dQt ~ Contr, that is, QI is a completely contradictory question. 0 According to Theorem 7.10, a completely contradictory question on the basis of a set of tautologies or the empty set implies only completely contradictory questions. THEOREM 7.11. Let Q, be a completely contradictory question. Then Im(Q, X, Qt) iff for each A E dQ : X, A E Inc. Proof: Assume that Im(Q, X, Q,). Then for each A E dQ we have X, A IF dQ,. Since dQt ~ Contr, then for each A E dQ it is the case that X, A E Inc. Assume that for each A E dQ it is the case that X, A E Inc. Then for each A E dQ we have X, A IF dQl ; so the clause (i) of Definition 7.1 is fulfilled. Since dQl ~ Contr, then for each B E dQ, we have X, B E Inc; hence for each non-empty proper subset Yof the set dQ we have X, B IF Y.

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Since each question has at least two direct answers, the set dQ has nonempty proper subsets. Thus also the clause (ii) of Definition 7.1 is fulfilled . Therefore Im(Q, X, QI)' D Theorem 7.11 says that a completely contradictory question is implied if and only if each set which consists of the auxiliary d-wffs and a direct answer to the implying question is inconsistent. One can easily prove: THEOREM 7.12 . Im(Q, X, Q\).

If a question Q is equipollent to a question Q) , then

THEOREM 7.13. If a question Q is equipollent to a question Q2' then Im(Q , X, Q\) iff Im(Q2' X, QI)' THEOREM 7.14. If a question QI is equipollent to a question Q2' then Im(Q , X, QI) iff Im(Q, X, Q2)' Theorem 7.12 says that a question Q implies each question equipollent to it on the basis of any set of d-wffs. According to Theorem 7.13, equipollent questions imply the same questions on the basis of a given set of d-wffs. Theorem 7.14 says that equipollent questions are implied by the same questions on the basis of a given set of d-wffs . Since equivalent questions are also equipollent, the situation is analogous in the case of equivalence of questions . Let us now prove THEOREM 7.15. If a question QI is stronger than a question Q and for E dQ, X, A IF dQ" then Im(Q, X, QI)' Proof: If a question Q\ is stronger than a question Q, then for each B E dQI there is A E dQ such that B t= A. Thus we may say that for each B E dQI there exists Y C dQ such that Y ~ 0 and X, B IF Y. Since by assumption for each A E dQ we have X, A IF dQI' then Im(Q, X, QI)' D

each A

According to Theorem 7.15, a question Q implies on the basis of a set of d-wffs X those stronger questions whose sets of direct answers are meentailed by each set made up of the elements of X and a direct answer to Q (i.e, those stronger questions which are sound relative to any set which consists of the elements of X and a direct answer to Q). THEOREM 7.16. If a question QI is weaker than a question Q andfor each B E dQI there exists a non-empty propersubset Y of dQ such that X, B IF Y, then Im(Q, X, QI)' Proof: It suffices to observe that if a question QI is weaker than a question Q, then each direct answer to Q entails some direct answer to QI' D

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Theorem 7.16 says that a question implies on the basis of a given set of d-wffs those weaker questions, for which the clause (ii) of Definition 7.1 is fulfilled with respect to the analyzed set of d-wffs. Let us also note THEOREM 7.17 . If a question Q is included in a question QI andfor each B E dQI - dQ there exists a non-empty proper subset Y of dQ such that X, B \1= Y, then Im(Q, X, QI)' THEOREM 7.18 . If a question Q includes a question QI andfor each A E dQ - dQI> X, A 11= dQI> then Im(Q, X, QI)' We omit the simple proofs of the above theorems. (

7.3. IMPLYING AND PRESUPPOSING Let us now examine the connections between the concepts of presupposition of a question and erotetic implication. One may ask whether arriving at an implied question is always tantamount to arriving at a cognitively useful question which is sound relative to the set made up of the presuppositions of the implying question and the auxiliary dwffs. Sometimes it is the case. We can prove: THEOREM 7.19. Let Q be a normal question. Then Im(Q, X, QI) iff (i) X u PresQ 11= dQI, and (ii) for each B E dQI there exists a non-empty proper subset Y of dQ such that X, B 11= Y. Proof: Recall that a question Q is normal iff PresQ 11= dQ. Observe also that the condition (ii) is equal to the clause (ii) of Definition 7.1. (~) Assume that X u PresQ non 11= dQI' Thus some normal interpretation ~ is a model of the set X u PresQ, but makes false each element of dQI' But Q is normal; so at least one direct answer to Q is true in ~. Thus by Corollary 7.1 at least one direct answer to QI is true in ~ as well. We arrive at a contradiction. (~) Assume that for some A E dQ we have X, A non 11= dQI' So there is a normal interpretation ~ which is a model of the set X u {A} and in which each direct answer to QI is not true. But since A is true in ~ , ~ is also a model of the set PresQ. Therefore X u PresQ non 11= dQt. 0 But the general answer to the initial question of this paragraph is negative; we can also prove :

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THEOREM 7.20. The following conditions are equivalent: (a) each question is normal, (b) for any questions Q, Q\: Im(Q, X, QI) iff (i) X u PresQ IF dQI' and (ii) for each B E dQI there exists a non-empty proper subset Y of dQ such that X, B IF Y. Proof: (a) ~ (b) . By Theorem 7.19. (b) ~ (a). Assume that there is a question, say, Q, which is not normal. Clearly Im(Q, PresQ, Q). But since Q is not normal, we also have PresQ u PresQ non IF dQ. 0 It is worth noticing, however, that the following theorems are true: THEOREM 7.21. Let Q and Q\ be normal questions. Then Im(Q, X, QI) iff (i) for each C E PresQI: X u PresQ 1= C, and (ii) for each B E dQI there exists a non-empty proper subset Y of dQ such that X, B IF Y. Proof: It suffices to observe that for each normal question QI and each set of d-wffs Z we have Z IF dQI iff Z 1= C for each C E PresQI; on the other hand, if Q is normal, then by Theorem 7.19 X u PresQ IF dQI iff for each A E dQ, X u {A} IF dQ\ . 0 Theorem 7.21 shows that in the case of normal questions arriving at an implied question is tantamount to arriving at a cognitively useful question whose presuppositions are entailed by the set consisting of the presuppositions of the implying question and the auxiliary d-wffs. THEOREM 7.22. Let Q and Q\ be regularquestions. Then Im(Q, X, Q\) iff (i) there exist a prospectivepresupposition A of Q and a prospective presupposition C of QI such that X, A 1= C, and (ii) for each B E dQI there exists a non-empty proper subset Y of dQ such that X, B IF Y. Proof: (~) Assume that Im(Q, X, QI)' Each regular question is normal, so by Theorem 7.19 X u PresQ IF dQI' Since QI is regular, it follows that for some C E PPresQ, we have X u PresQ F C. But also Q is regular; so PPresQ ~ 0 . Let A be a fixed element of PPresQ. For each normal interpretation ~ we have : A is true in ~ iff ~ is a model of the set PresQ. So X, A 1= c. (~) Assume that there exist A E PPresQ and C E PPresQI such that X, A C. Since QI is regular, it follows that X, A IF dQI' But for each normal interpretation ~ we have : A is true in ~ iff ~ is a model of the set PresQ. So X u PresQ IF dQ\ . Since each regular question is normal and the condition (ii) holds, then Im(Q , X, Q\) by Theorem 7.19. 0

F

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According to Theorem 7.22, in the case of regular questions arriving at an implied question is tantamount to arriving at a cognitively useful question whose certain prospective presupposition is entailed by a certain prospective presupposition of the implying question together with the auxiliary d-wffs. This result cannot be generalized, however. This is due to THEOREM 7.23. Thefollowing conditions are equivalent: (a) each question is regular, (b) for anyquestions Q, QI: Im(Q, X, QI) iff (i) there exist A E PPresQ and C E PPresQI such that X, A 1= C, and (ii) for each B E dQ. there exists a non-empty proper subset Y of dQ such that X, B IF Y. Proof: Straightforward. 0 In order to go on we need: LEMMA 7.1. If entailment is compact, then a set of d-wffs Z me-entails some non-empty proper subset Y of dQ iff Z entails some direct or partial answer to Q. Proof: Assume that entailment is compact. Thus by Corollary 4.5 also me-entailment is compact. If Z IF Y and entailment is compact, then - since direct answers are sentences and hence are equal to their universal closures - by Corollary 4.8 the set Z entails some single direct answer to Q or some disjunction of direct answers to Q. Suppose that no single direct answer to Q is entailed by Z. So there is a disjunction, say, AI v ... V Ak , of at least two (distinct) direct answers to Q which is entailed by the set Z; moreover, since Y is a proper subset of dQ, the set {At> ... , Ak } is also a proper subset of dQ. Direct answers are sentences; so by Corollary 4.6 AI v ... V Ak IF {AI> ..., Ak } . At the same time for each A E {At> ..., Ak } we have A 1= AI v ... V Ak • So by Definition 4.10 the disjunction AI v ... V Ak is a partial answer to Q; this partial answer, however, is entailed by the set Z. If Z entails some direct answer to Q, then - since each question has at least two direct answers - Z entails some non-empty proper subset of dQ. If Z entails some partial answer to Q, then by Definition 4 .10 Z me-entails some non-empty proper subset of dQ. 0 We can now prove: THEOREM 7.24 . If entailment is compact and Q, QI are normal questions, then Im(Q, X, QI) iff (i) there is a presupposition A of Q such thatfor some maximal presupposition D of QI ' X, A 1= D, and (ii) for each B E dQI there exists C E dQ u pQ such that X, B 1= c.

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Proof: (=*) By Corollary 4.13, Theorem 7.22, Corollary 4.10 and Lemma 7.1. (~) Since we have already proved Lemma 7.1, it suffices to prove that the condition (i) yields the clause (i) of Definition 7.1 in case Q, QI are normal questions and entailment is compact. Assume that there is a presupposition A of Q such that for some maximal presupposition D of Q., X, A 1= D. Since QI is normal and each normal interpretation of the language which makes true any maximal presupposition of QI is also a model of the whole set of presuppositions of QI> then X, A IF dQI' Assume that there exists a direct answer B to Q such that X, B non IF dQ!. It follows that there is a normal interpretation ~ which is a model of the set X, B, but which makes false all the direct answers to QI' Since B is true in ~, then ~ is a model of the set of presuppositions of Q. SO the presupposition A of Q is true in ~. It follows that X, A non IF dQI' We arrive at a contradiction. Thus for each direct answer B to Q we have X, B IF dQi- 0 According to Theorem 7.24 if entailment in the language is compact, then erotetic implication takes place between a normal question QI and a normal question Q on the basis of a set of d-wffs X just in case a certain maximal presupposition of Ql is entailed by some set made up of the d-wffs of X and a presupposition of Q. and each set made up of the d-wffs of X and a direct answer to QI entails some direct or partial answer to Q. 7.4 .

ALTERNATIVE DEFINITIONS OF EROTETIC IMPLICATION

Theorems 7.19, 7.21 and 7.22 yield that for those languages which contain only normal questions or only regular question erotetic implication can be defined in a way different from that proposed by Definition 7.1 . Some other possibilities exist for those languages of the considered kind in which entailment (and thus also me-entailment) is compact. But let us first prove THEOREM 7.25. If (i) for each A E dQ: X, A 1= B1 V ••• V B, for some BI> ... , B, in dQ. (k ~ I), and (ii) for each B E dQI there exists a finite proper subset {AI> .. ., Am} (m ~ 1) of dQ such that X, B 1= A I V .. . V Am' then Im(Q, X, QI)' Proof: Direct answers are sentences. Thus if the condition (i) holds, then by Corollary 4.7 and Corollary 4.3 for each A E dQ we have X, A IF dQiOn the other hand, the clause (ii) of Definition 7. I is an immediate consequence of the condition (ii). 0

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According to Theorem 7.25 if each set which consists of the elements of a set of d-wffs X and a direct answer to a question Q entails some disjunction of direct answers to a question QI> and each set which consists of the elements of the set X and a direct answer to the question Q. entails either a single direct answer to Q or a disjunction of at least two but not all direct answers to Q, then Q implies Q\ on the basis of X. Theorem 7.25 can be strengthened to equivalence for those languages in which entailment is compact. Let us now prove THEOREM 7.26. If entailment is compact, then Im(Q, X, Q.) iff (i) for each A E dQ: X, A P B. v '" V Bk for some B., ... , B, in dQI (k ~ 1), and (ii) for each B E dQ. there exists a finite proper subset {AI> ... , Am} (m ~ 1) ofdQ such that X, B pA. v ... v Am. Proof: Assume that entailment is compact. (~) If Im(Q, X, Q.), then for each A E dQ we have X, A I~ dQ\. Since entailment is compact and direct answers are sentences and thus are identical with their universal closures, then by Corollary 4.8 we get the condition (i). Similarly, if entailment is compact, then the clause (ii) of Definition 7 .1 together with Corollary 4.8 yield the condition (ii) . ( ... , B, in dQ. (k ~ 1), and (ii) for each B E dQ. there exists C E dQ U pQ such that X, B P C, then Im(Q, X, QI)' Proof: Direct answers are sentences . Thus if the condition (i) holds, then by Corollary 4.7 and Corollary 4.3 for each A E dQ we have X, A I~ dQ\ . Let B be a direct answer to Qt. Assume that for some A E dQ, X, B pA. It follows that X, B I~ {A}. On the other hand , each question has at least two direct answers; thus the set {A} is a (non-empty) proper subset of dQ. Assume that for some C E pQ we have X, B P C. Since C is a partial

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answer to Q, then by Definition 4 .10 there exists a non-empty proper subset Z of dQ such that C If Z; so X, B If z. 0 According to Theorem 7.27, if each set which consists of the elements of a set of d-wffs X and a direct answer to a question Q entails some disjunction of direct answers to a question QI> and each set which consists of the dwffs of X and a direct answer to QI entails some direct or partial answer to Q, then Q implies QI on the basis of X. Again, this can be strengthened to equivalence for those languages of the analyzed kind in which entailment is compact. The following theorem is true : THEOREM 7.28 . If entailment is compact, then Im(Q, X, Ql) iff (i) for each A E dQ: X, A F B( v ... V B, for some BI> ... , B, in dQI (k ~ 1), and (ii) for each B E dQI there exists C E dQ U pQsuch that X, B F C. Proof: (~) By Theorem 7.26, Definition 7.1 and Lemma 7 .1. (~) By Theorem 7.27. 0 Thus for those languages in which entailment is compact we can also define erotetic implication as follows: DEFINITION 7 .3 . 1m 2(Q, X, QI) iff (i) (ii)

for each A E dQ: X, A FBI v: «, V B, for some BI> .. ., B, in dQl (k ~ 1), and for each B E dQI there exists C E dQ u pQsuch that X, B F C.

Let us recall , however, that there are interesting languages of the analyzed kind in which entailment is not compact; the language ;;e"" gives us a simple example here. For some languages erotetic implication can be defined in terms of propositional implication defined by Belnap (cf. Belnap & Steel, 1976, p . 122; see also Chapter 1, Section 1.6) . Like the concept of multipleconclusion entailment, the concept of propositional implication is also a generalization of the standard concept of entailment. Following Belnap , let us temporarily assume that a question Q of a language ;;e of the considered kind is said to be true in an interpretation ~ of this language if and only if at least one direct answer to Q is true in ~ (as we see, Belnap's concept of truth for questions coincides with our concept of soundness of a question) . Let us call a quasiformula of ;;e any d-wff or question of ;;e . Then, using H, H' for sets of quasiformulae of;;e and h for quasiformulae of ;;e, we define : H H'-propositionally implies h if and only if h is true in each normal interpretation of ;;e in which all the quasiformulae in the set HuH' are true. We may prove that if a question Q X-propositionally implies a question

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QI and for each direct answer B to QI there exists either a direct answer to Q which is X-propositionally implied by B or a question Q2 which is Xpropositionally implied by B and whose set of direct answers is a proper subset of the set of direct answers to the question Q, then Q implies QI on the basis of X. This result can be strengthened to.equivalence for these languages, in which for each question Q the following condition is fulfilled : (#) for each at least two-element proper subset Y of dQ there exists a question Q2 such that dQ2 = Y. This condition, however, seems to be rather unnatural when we take into consideration questions with an infinite number of direct answers. As a matter of fact, the condition (#) is not satisfied by most of the languages of the analyzed kind.

7.5. SOME SPECIAL CASES: PURE EROTETIC IMPLICATION AND STRONG EROTETIC IMPLICATION

7.5.1. Pure erotetic implication Erotetic implication is a relation between a question, a set of declarative formulas (d-wffs), and a question. It is neither assumed nor denied that the appropriate set of declarative formulas is non-empty. Thus by means of the concept of erotetic implication we can define the relation "a question implies a question." DEFINITION 7.4 . A question Q implies a question QI (in symbols: Im(Q, QI» if and only if Im(Q, 0 , QI)' In other words, Q implies QI just in case Q implies QI on the basis of the empty set of d-wffs. The relation defined above may be called pure erotetic implication. The intuitive content of the concept of pure erotetic implication can be best expressed as follows: a question Q implies a question Ql just in case QI must have a true direct answer if Q has such an answer, but moreover each direct answer to QI narrows down the class of possibilities offered by Q. Let us now pay some attention to selected properties of pure erotetic implication. By Corollary 7.3 and Definition 7.4 we get COROLLARY 7.4. Let Im(Q, QI)' Then for each normal interpretation is:

at leastone direct answer to the implying question Q is truein ~ if and only if at least one direct answer to the implied question Ql is true in ~ . Thus in the case of pure erotetic implication the implying question and the implied question are sound in the same normal interpretations of the

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language (provided that any of them is sound in some normal interpretation at all); moreover, these questions are not sound in the same normal interpretations of the language . Corollary 7.4 yields: THEOREM 7.29 .lfIm(Q, Q.), then i . (i) Q is safe iff Q. is safe, (ii) Q is risky iff Q. is risky. (iii) Q is completely contradictory iff Q. is completely contradictory. Thus safe questions can be implied (in the sense of Definition 7.4) only by safe questions and imply only safe questions; the situation is analogous in the case of riskiness. It is not the case, however , that a safe question implies each safe question . By Theorem 7.8 and Definition 7.4 we get: THEOREM 7.30. Let QI E Saf. Then Im(Q, Q.) ifffor each BE dQ. there exists a non-empty proper subset Y of dQ such that B IF Y. Let us also note THEOREM 7.31. lfIm(Q, Q.) and Q is a normal but risky question, then Q. is made sound by the set of presuppositions of Q. Proof: By Theorem 7.19, Definition 7.4, Definition 4.17 and Theorem 7.29.0 According to Theorem 7.31 a question implied by a normal but risky question is made sound by the set of presuppositions of the implying question, that is, to speak generally, does not have a true direct answer in every case, but must have a true direct answer in case all the presuppositions of the implying question are true. We can also prove a more general result: THEOREM 7.32. Let Q be a normal but risky question. Then Im(Q , Q\) iff (i) Q. is madesoundby the set of presuppositions of Q, and (ii) for each B E dQ. there exists a non-empty proper subset Y of dQ such that B IF Y. Proof: By Theorem 7.19, Definition 7.4, Defmition 4.17 and Theorem 7.29.0 We shall now prove: THEOREM 7.33 . lfIm(Q, QI), then PresQ = PresQ•. Proof: Let A be a presupposition of Q and let B be a direct answer to Q\. If Im(Q , Q\), then B me-entails some non-empty proper subset Y of dQ. Suppose that B non FA. SO there is a normal interpretation ~ in which B is true and A is not true. But A is a presupposition of Q and thus is entailed

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by each direct answer to Q. Thus no d-wff in the set Y is true in ~ ; it follows that B doesn't me-entail Y. We arrive at a contradiction. Therefore BFA. But B was an arbitrary direct answer to Qt: so A is a presupposition of QI as well. Hence PresQ ~ PresQI' Let C be a presupposition of QI and let D be a direct answer to Q. If Im(Q, QI), then D IF dQt. But C is entailed by each direct answer to QI; so each normal interpretation which makes true some direct answer to QI makes true the presupposition C as well. It follows that D F C; so C is also a presupposition of Q. Therefore PresQ I ~ PresQ. But since we also have PresQ ~ PresQ" then finally PresQ = PresQt. 0 Theorem 7.33 shows that in the case of pure erotetic implication the implied question and the implying question are presuppositionally indistinguishable, i.e . have exactly the same presuppositions. Let us stress, however, that presuppositionally indistinguishable questions need not be identical: for example, safe questions have the same presuppositions, but may have different sets of direct answers. Theorem 7.33 has interesting consequences. Let us first prove: THEOREM 7.34. If Im(Q, QI) and Q is a normal question, then QI is a

normal question. Proof: Let Q be a normal question and let Im(Q, QI)' Thus by Theorem 7.33 PresQ = PresQI; at the same time by Theorem 7.19 we have PresQ IF dQI' Therefore PresQI IF dQI; so QI is a normal question. 0 Thus as far as pure erotetic implication is concerned, normal questions can imply only normal questions. We may also prove THEOREM 7.35 . If Im(Q, QI) and QI is a normal question, then Q is a

normal question. Proof: Assume that Im(Q, QI) and that QI is a normal question. Since QI is normal, then PresQI IF dQI' But by Theorem 7.33 PresQ = PresQI; so PresQ IF dQI' Assume that Q is not normal. It follows that there is a normal interpretation ~ in which all the direct answers to Q are not true, but which is a model of the set PresQ. But if no direct answer to Q is true in ~, then by Corollary 7.4 no direct answer to QI is true in ~ as well. So PresQ non IF dQI' We arrive at a contradiction. So Q is normal. 0 Thus we may also say that in the case of pure erotetic implication normal questions can be implied only by normal questions. When we take into consideration resular questions, the situation is analogous. The following theorems are true:

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THEOREM 7 .36.

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If Im(Q. Ql) and Q is a regular question, then Qt is a

regular question. Proof: Assume that Im(Q. Ql) and that Q is a regular question. So there is a presupposition. say, A. of Q such that A 11= dQ. By Theorem 7.33 A is also a presupposition of Qt. Assume that A non 11= dQI. So there is a normal interpretation ~ such that A is true in ~ and no direct answer to Ql is true in ~. Thus by Corollary 7.4 no direct answer to Q is true in ~; so A non 11= dQ. We arrive at a contradiction. Hence A 11= dQI and therefore A is a prospective presupposition of QI. SO by Corollary 4.11 QI is a regular question. 0 THEOREM 7.37.

If Im(Q. QI) and QI is a regular question, then Q is a

regular question. Proof: Similar to that of Theorem 7.36. 0 By means of the theorems presented in the previous sections of this Chapter we may characterize further properties of pure erotetic implication. We leave it to the Reader. Let us only note: THEOREM 7.38. If entailment is compact and Q is a normal question, then Im(Q, QI) iff (i) there exist a presupposition A of Q and a prospective presupposition C of QI such that A F C, and (ii) for each direct answer B to QI there exists D E dQ u pQ such that B 1= D. Proof: (=*) Assume that Im(Q. Ql). Since Q is normal and entailment is compact.

then by Corollary 4 .13 Q is regular; so by Theorem 7.36 Qt is also regular. Thus by Theorem 7.22 and Definition 7.4 there exist a presupposition A of Q and a prospective presupposition C of Qt such that A F c. Each regular question is normal; so by Theorem 7 .24 and Definition 7.4 for each direct answer B to QI there exists D E dQ u pQ such that B F D. (~) Assume that there exists a presupposition A of Q and a prospective presupposition C of QI such that A F C. Thus A 11= dQl and therefore PresQ 11= dQt . Assume that for each direct answer B to Ql there exists D E dQ u pQ such that B FD. Thus by Lemma 7.1 for each direct answer B to Qt there exists a non-empty proper subset Yof dQ such that B 11= Y. But by assumption Q is normal. So Im(Q. Ql) by Theorem 7.19 and Definition 7.4.0 According to Theorem 7 .38 if entailment is compact . then the relation of pure erotetic implication holds between a normal question Q and a question Qt just in case some presupposition of Q entails some prospective presupposition of QI and each direct answer to QI entails some direct or partial

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answer to Q.

7.5.2 . Strong erotetic implication Let us now pay some attention to a certain special case of erotetic implication understood as a ternary relation, that is, to strong erotetic implication. It can happen that the second clause of the definition of erotetic implication is fulfilled only for the reason that the set of auxiliary d-wffs entails some non-empty proper subset(s) of the set of direct answers to the implying question . In such a case the implied question is secured by the implying question and the auxiliary d-wffs, but its cognitive usefulness may be doubtful since these are essentially the auxiliary d-wffs which enable us to narrow down the class of possibilities offered by the implying ("initial ") question. In order to distinguish these cases from the cases in which the second clause is fulfilled for "erotetic" reasons we shall define the following concept: DEFINITION 7.5. A question Q strongly implies a question QI on the basis of a set of d-wffs X (in symbols: Ims(Q, X, iff (i) for each A E dQ: X, A IF dQ.. and (ii) for each B E dQI there exists a non-empty proper subset Y of dQ such that X non IF Y and X, B IF Y. It is obvious that each question strongly implied by another on the basis of a given set of d-wffs is also implied by them. The converse implication , however, need not be true. It is also easy to observe that there are questions which cannot be either strongly implying questions or strongly implied questions. Completely tautological questions (i.e . questions which have only tautologies as direct answers) are cases in point here. One can easily prove

Q.»

THEOREM 7.39. ifIms(Q, X, Q.), then dQ g; Taut and dQ. n Taut

= 0.

Similarly, there are sets of d-wffs which cannot bear the relation of strong erotetic implication. For example, inconsistent sets of d-wffs (that is, sets which have no normal interpretations as models) cannot constitute sets of auxiliary d-wffs in the present case. This is due to THEOREM7.40. IfIms(Q, X, Q.), then X f/. Inc. Thus it can hardly be said that the concept of strong erotetic implication is an explicatum for the analyzed concept of the arising of questions from questions and sets of declarative sentences; looking from the intuitive point of view, it seems that each question can arise from some question on the basis of some set of d-wffs, and each question can give rise to some question(s). Yet, the concept of strong erotetic implication seems to be the

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most interesting from the point of view of possible applications. The reason is that in the case of strong erotetic implication these are precisely the direct answers to the implied question which enable us to narrow down (on the basis of the auxiliary d-wffs) the class of possibilities offered by the initial (i.e, implying) question. The basic properties of strong erotetic implication are similar to that of erotetic implication. In particular, strong erotetic implication has the properties characterized by the corollaries 7.1, 7.2, 7.3, and the theorems 7.3, 7.4, 7.5, 7.6, 7.7, 7.10, 7.13, 7.14 . As far as the problem of "reflexivity" of strong erotetic implication is concerned, we have: THEOREM 7.41. Ims(Q, X, Q) iff Q is informative relative to X. Proof: (~) Assume that Ims(Q, X, Q) and that Q is not informative relative to X. Thus there exists a direct answer, say, A, to Q such that X FA. Yet, meentailment has the following property: (*) if X F A, then for each Z ¢ 0: if X, A IF Z, then X IF Z. For assume that X F A and that there is a non-empty set of d-wffs, say, ZI' such that X, A IF Z, and X non IF Zt. Thus there is a normal interpretation ~ which is a model of X, but in which each d-wff of Zt is not true. Since X F A, then A is true in ~. It follows that X, A non IF Zt. If Ims(Q, x, Q), then for some non-empty proper subset Yof dQ we have X, A IF Yand X non IF Y. By (*) it follows that X non FA. We arrive at a contradiction . ( then Ims(Q, X, QI). THEOREM 7.45 . If a question Q includes a question Qt, Q is informative relative to X andfor each A E dQ - dQt, X, A IF dQt, then Ims(Q, X, QI).

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THEOREM 7.46. If a question Q is included in a question Qt, Q is informative relative to X and for each B E dQI - dQ there exists a non-empty proper subset Y of dQ such that X, B 11= Yand X non 11= Y, then Ims(Q, X,

e.»

If we replace in the theorems 7.8, 7.16, 7.19, 7.20, 7.21, 7.22, 7.23 the clause "for each B E dQI there exists a non-empty proper subset Y of dQ such that X, B 11= Y" with the clause "for each B E dQt there exists a nonempty proper subset Y of dQ such that X non 11= Y and X, B 11= Y" and consequently the sign 1m with the sign Ims, we will obtain theorems which characterize properties of strong erotetic implication. Further theorems of this kind can be obtained from theorems 7.24 and 7.28 by enriching the clause "for each B E dQI there exists C E dQ U pQ such that X, B 1= C" with the provision "X non 1= C" and then by replacing the symbol 1m with the symbol Ims . One can also easily prove:

THEOREM 7.47 . If (i) for each A E dQ : X, A 1= B, V • • • V B, for some .. . , B, in dQI (k ~ 1), and (ii)for each B E dQt there exists a finite proper subset {AI' , Am} (m ~ 1) of dQ such that X, B 1= Al v . .. V Am and X non 1= Al v v Am. then Ims(Q, X, QI)' THEOREM 7.48 . If entailment is compact, then Ims(Q, X, QI) iff (i) for each A E dQ : X, A 1= B I V .. . V B, for some B I , .. . , B, in dQI (k ~ 1), and (ii)for each B E dQI there exists a finite proper subset {At> , Am} (m ~ 1) of dQ such that X, B 1= At v .. . v Am and X non 1= At v v Am. BI ,

7.6. EROTETIC IMPLICATION AND PROBLEM-SOLVING The concept of erotetic implication is an explicatum for the analyzed notion of the arising of a question from a question and a set of declarative sentences . There is an old idea, however, that each well-posed initial "big " question can be answered by posing auxiliary ("small") questions whicharise from it and answering these questions step-by-step. Is it possible to reflect this idea by means of the concept of erotetic implication? Surprisingly enough, the answer is affirmative. Let us recall that a simple yes-no question is a question of the form ? {A, ""A}; the sentences A and ...,A are called the affirmative direct answer and the negative direct answer, respectively . We may prove:

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THEOREM 7.49 . If entailment is compact and Q issoundrelative to X, then there exists afinite sequence eI> ofsimple yes-no questions such that: (a) each question of eI> is implied by Q on the basis of X, (b) each set made up of direct answers to thequestions ofel> which contains exactly onedirect answer to each question ofel> entails along with X some direct answer to Q, and (c) each non-logical constant that occurs in a direct answer to a question of eI> occurs in some direct answer to Q. Proof: If entailment in a language is compact, then me-entailment in this language is also compact. So if Q is sound relative to X (i.e , X IF dQ holds), then X me-entails some finite subset(s) of dQ. Let Z be a fixed finite subset of the set of direct answers to Q such that X IF z. There are three possibilities: (1) Z is the empty set, or (2) Z is a singleton set, or (3) Z contains at least two sentences. Assume that the third possibility holds. Let us establish some enumeration A I' ... , An among the elements of Z. We define the following finite sequence eI> = QI' .. ., Qn-I of simple yes-no questions:

Qi = ? {Ai' ...,Ai} where 1 :::; i :::; n-l. Let Qj (1 :::; j :::; n-l) be a question of eI>. It is obvious that for each A E dQ we have X, A IF {Aj , ...,Aj} and hence X, A IF dQj' On the other hand, Aj is entailed by the set X, Aj and is also a direct answer to Q. Since each question has at least two direct answers, it follows that the set Xu {Aj } me-entails the proper subset {AJ of dQ. Furthermore, since X IF Z, i.e . X IF {AI' .. ., An} and n > 1, then the set X, ...,Aj me-entails the non-empty proper subset {At, ... , An} - {Aj} of dQ. Hence Q implies Qj on the basis of X; so each question of eI> is implied by Q on the basis of X. Let Y be a set made up of direct answers to the questions of eI> which contains exactly one direct answer to each question of eI>. We have two possibilities: (a) Y contains at least one affirmative direct answer to some question of eI>, i.e, for some 1 :::; j :::; n-l, Aj E Y, or (b) Y consists of the negative direct answers to the questions of eI>, i.e . Y = {""At> ... , ...,An_ I } . In the case (a) the set X u Yentails some direct answer(s) to Q, since the affirmative direct answers to the questions of eI> are at the same time direct answers to Q. If the possibility (b) holds, then - since X IF {At> ... , An} - the set X u Y (i.e , X u {...,At> .... ...,An_ l }) entails the direct answer An to Q. Thus each set made up of direct answers to the questions of eI> which contains exactly one direct answer to each question of eI> entails along with

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X some direct answer to Q. On the other hand, it is obvious that each nonlogical constant that occurs in a direct answer to a question of cfl occurs in some direct answer to Q. Assume now that the possibility (2) holds. i.e . that Z is a singleton set. Let B be the element of Z and let Ql = ? {B, -'B} . It is obvious that for each A E dQ we have X, A IF dQ i - On the other hand, since X IF Z and thus X IF {B}, then both X, B IF {B} and X, -,B IF {B}; so Q implies Q. on the basis of X. Furthermore, the one-element sequence made up of the question QI satisfies the conditions (a), (b) and (c). Assume finally that the possibility (1) holds, i.e . that Z is the empty set. Since X IF Z, it follows that there is no normal interpretation which is a model of X. Let C be a fixed direct answer to Q and let Q2 = ? {C, -, C} . By Theorem 7.9 we get Im(Q, X, Q2); on the other hand, the one-element sequence made up of the question Q2 meets the conditions (a), (b) and (c). 0 Viewed pragmatically , Theorem 7.49 yields that if entailment is compact, then each question Q which is sound relative to a given set of d-wffs, say, X, can be answered by answering a finite number of simple yes-no questions which are implied by Q on the basis of X (and thus arise from Q on the basis of X) and then inferring some direct answer to Q from the accepted answers to the yes-no questions and the initial set X; what is more, this can be done regardless of which answers to the yes-no questions appear to be acceptable. On the other hand, simple yes-no questions are customarily regarded as the simplest questions, at least epistemologically . Some general remarks are in order here. The proof of Theorem 7.49 shows how the appropriate yes-no questions can be found; yet, it also shows that there are always many finite sequences of simple yes-no questions which meet the conditions (a), (b) and (c). Both the decision as to which sequence of implied yes-no questions should be taken into consideration and the order in which the implied yes-no questions should be considered is a matter of heuristics . Moreover, we do not want to say that answering an initial question by answering simple yes-no questions implied by it and the appropriate declarative premises is always the most efficient method of solving the problem posed by the initial question. What is important , however, is that we can always proceed this way if entailment is compact and our initial question is sound relative to some previously accepted premises . Note that it is the compactness assumption that guarantees that the sequence cfl which meets the conditions (a), (b) and (c) of Theorem 7.49 is finite. Yet, the compactness assumption is dispensable in case the initial

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question has a finite number of direct answers; if, however, entailment in a language is not compact, it may happen that only infinite sequences of simple yes-no questions correspond to some question which is sound relative to a given set of d-wffs , As a general result we have

If Q is sound relative to X, then there exists a sequence of simple yes-no questions such that: (a) each question of cI> is implied by Q on the basis of X, (b) each set made up of direct answers to the questions of cI> which contains exactly one direct answer to each question of cI> entails along with X some direct answer to Q, and (c) each non-logical constant that occurs in a direct answer to a question of cI> occurs in some direct answer to Q; if moreover Q is afinite question or entailment is compact, then there exists a finite sequence cI> of simple yes no-questions which meets the conditions (a), (b) and (c) . THEOREM 7.50. cI>

Proof: Let us recall that any question of the considered kind has at most denumerably many direct answers. So for each question there are sequences without repetitions whose terms are all the direct answers to the question. We choose a fixed sequence AI> A 2 , ... of this kind and then define the sequence cI> = QI' Q2' ... of simple yes-no questions in the following way :

Qi = ? {Ai+I' ...,Ai+l} The rest of the proof runs along the lines of the first part of the proof of Theorem 7.49. The resultant sequence of simple yes-no questions is finite if the initial question is finite; on the other hand , we have already proved Theorem 7.49. 0 So far we have considered the case in which an initial question is sound relative to some set of d-wffs. But since a question evoked by a set of d-wffs is also sound relative to this set, all that have been said above can also be said about the situation in which the initial question is evoked by a given set of d-wffs. In the case of evocation, however, we can also prove some more specific result. But let us first prove LEMMA 7.2 . If entailment is compact and Q is evoked by X, then there exists a finite and at least two-element subset Y of dQ such that Y is me-entailed by X and no propersubset of Y is me-entailed by X. Proof. If entailment is compact, then also me-entailment is compact. It is obvious that if Q is evoked by X and me-entailment is compact, then the following conditions are fulfilled: (a) X me-entails some finite subset(s) of dQ, and (b) each finite subset of dQ which is me-entailed by X has at least two elements.

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Let Ybe a fixed subset of dQ which meets the conditions (a) and (b). It may happen that no proper subset of Y is me-entailed by X. Yet, let us assume that X me-entails some proper subset(s) of Y. Let ¢ be the family of all the proper subsets of Y which are me-entailed by X. Since Y is finite, the family ¢ is also finite; moreover, each member of ¢ is a fmite set. Thus there is a natural number n such that some set in ¢ contains exactly n elements and no set in the family ¢ contains less than n elements. By the condition (b) we have n ~ 2. Since ¢ is the family of all the proper subsets of Y which are me-entailed by X, it follows that there is a subset of Y - and hence of dQ - which contains at least two elements, is me-entailed by X and whose proper subsets are not me-entailed by X. 0 We can now prove THEOREM 7.51. If entailment is compact and Q is evoked by X, then there exists a finite sequence II> of simple yes-no questions such that: (a) each question of II> is both evoked by X and strongly implied by Q on the basis of X, (b) each set made up of direct answers to the questions of II> which contains exactly one direct answer to each question of II> entails along with X somedirect answerto Q, and (c) each non-logical constant that occurs in a direct answer to a question of II> occurs in some direct answer to Q. Proof. Let Y be a fixed finite subset of dQ which meets the conditions of the consequent of Lemma 7.2. Let us establish some enumeration AI, A2 , ... , An of the elements of Y. Thus Y = {A lo .. ., An} . We define the following finite sequence II> = Qlo ... , Qn-I of simple yes-no questions: Q; =? {A; , ...,Ai } where 1 ~ i ~ n-l. Let Qj (1 ~ j ~ n-l) be a question of 11> . Since Qj is a simple yes-no question, then X IF dQj' The direct answer Aj to Qj is also a direct answer to Q; thus Aj is not entailed by X. Suppose that X F ""Aj • Since X IF {A lo .. ., An}, it follows that X me-entails the proper subset {A lo ..., An} - {AJ of Y. But no proper subset of Y is me-entailed by X. So X non F ...,Aj • Thus X evokes Qj; so each question of II> is evoked by X. It is obvious that for each A E dQ we have X, A IF {Aj , ""Aj} and hence X, A IF dQj' On the other hand, Aj is entailed by the set X, Aj and is also a direct answer to Q. Moreover, Aj is not entailed by X alone. Since each question has at least two direct answers, it follows that the set X, Aj me-entails the proper subset {Aj } of dQ, which is not me-entailed by the set X. Furthermore, since X IF {AI' ... , An} , then the set X, ...,Aj me-entails the proper subset {AI' .. ., An} - {Aj} of dQ. On the other hand, no proper subset of {A lo ... , An} is me-entailed by X alone. Hence Q strongly implies Qj on the

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basis of X; so each question of ~ is strongly implied by Q on the basis of X. We may show in an analogous way as in the proof of Theorem 7.49 that each set made up of direct answers to the questions of ~ which contains exactly one direct answer to each question of ~ entails along with X some direct answer to Q. It is also obvious that, due to the construction of the sequence ~, each non-logical constant that occurs in a direct answer to a question of ~ occurs in some direct answer to Q. 0 The definition of evocation provides us with an explication of the concept of the arising of a question from a set of declarative sentences. Thus the intuitive content of Theorem 7.51 may be expressed as follows: if entailment (or equivalently : me-entailment) is compact, then each question Q which is evoked by a given set of d-wffs X and thus arises from X can be answered by answering a finite number of simple yes-no questions which arise from X and are strongly implied by Q on the basis of X and thus also arise from Q on the basis of X; what is more, this can be done regardless of the fact which answers to these yes-no questions will appear to be acceptable. But again, there are always many sequences of simple yes-no questions which meet the conditions (a), (b) and (c) of Theorem 7.51. Moreover, answering an initial evoked question in a way suggested by the proof of Theorem 7.51 need not be the simplest available method of solving the problem posed by the initial question. 7.7. EROTETIC IMPLICATION AND THE REDUCffiILITY OF QUESTIONS TO SETS OF QUESTIONS

7.7 .1. The concept of reducibility The problem of reducibility of questions has different aspects. First, one can speak of reducibility of questions to expressions of some other kind: declaratives, imperatives, epistemic imperatives, alethic modalities, etc. As we pointed out in Chapter 2, there are many attempts of developing a logic of questions within the frameworks of other philosophical logics; the imperative-epistemic approach is currently the most popular one. The second aspect of the problem of reducibility of questions is the reducibility of questions of one kind to questions of another kind. The relevant concept of reducibility, however, may be understood here in two different ways; we may label them the "one-to-one case" and the "one-to-many case", respectively. To be more precise, one can speak of: (a) reducibility of a (single) question of some kind to a (single) question of another kind, or (b) reducibility of a (single) question of some kind to a set of questions of some

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kind or kinds. These concepts do not coincide, although the first may be regarded as a special case of the second. It is a little surprising that erotetic logicians paid more attention to the "one-to-one case" than to the "one-tomore case"; this is surprising since in order to find the correct answer to an initial question we usually pass to a number of auxiliary questions and try to answer them . Generally speaking, the "one-to-one case" concept of reducibility has been clarified in two ways : as an equivalence within a given calculus (see e.g . Aqvist, 1965) or as some equivalence relation between questions which is defined in terms of (set-theoretic or semantic) relations between sets of their direct answers or in terms of relations between sets of presuppositions (cf. mainly Kubinski, 1980, but also Belnap & Steel, 1976). There are, however, other approaches as well. In Wisniewski (1994b) the concept "a question is reducible to a non-empty set of questions" is defined and its basic properties are examined . This concept is essentially a normative one, but not without some descriptive component. The intuitions which underlie the proposed definition can be briefly described as follows . First, an initial question and the questions to which the initial question is reducible must be mutually sound. To be more precise, it is required that if a question Q is reducible to a set of questions ir, then Q has a true direct answer if and only if each question in ir has a true direct answer. It is not assumed , however, that all initial questions must have true direct answers; it is only required that if an initial question has such an answer, then all the questions to which it is reducible also have true direct answers and vice versa. This equivalence must be fulfilled, but in some cases possibly for trivial reasons (such as that all the questions taken into consideration always have true direct answers, i.e. are safe, or do not have true direct answers at all). The second requirement is the efficacy condition: it is required that an initial question can always be answered by answering the questions to which it is reducible. To be more precise, it is required that if a question Q is reducible to a set of questions ir, then each set made up of direct answers to the questions of ir which contains exactly one direct answer to each question of ir must entail some direct answer(s) to Q. Hence any "combination" of direct answers to the questions of ir can provide us with some direct answer to Q and thus by answering the questions of ir we can always answer the initial question Q, regardless of the fact which direct answers to the questions of ir will appear to be acceptable. The last requirement is the relative simplicity condition: all questions to which a given question is reducible are supposed to be no more complex than the initial question. The notion of complexity of a question is understood here

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as follows: a question Q is said to be more complex than a question QI if and only if Q has more direct answers than QI (i.e. the cardinality of the set of direct answers to Q is greater than the cardinality of the set of direct answers to QI)' Thus it is required that if a question Q is reducible to a set of questions '1', then no question in '1' has more direct answers than the question Q. The precise definition of the analyzed concept of reducibility is given by DEFINITION 7.6. A question Q is reducible to a non-empty set of questions '1' iff (i) for each direct answer A to the question Q, for each question Q; of '1': A me-entails the set of direct answers to Q;, and (ii) each set made up of direct answers to the questions of '1' which contains exactly one direct answer to each question of '1' entails some direct answer to Q, and (iii) no question in '1' has more direct answers than Q. For brevity, the non-emptiness clause will be omitted in the sequel. It is easily seen that the concept of reducibility introduced by Definition 7.6 meets all the requirements listed above. The efficacy condition and the relative simplicity condition are fulfilled due to the clauses (ii) and (iii) of Definition 7.6, respectively. On the other hand, the following is true: COROLLARY 7.5. If a question Q is reducible to a set of questions '1', then

for each normal interpretation ~: Q is soundin ~ iff each question of '1' is sound in ~ . Proof: Assume that Q is reducible to '1'. (=*) If Q is sound in a normal interpretation ~, then there is a direct answer to Q which is true in ~. This direct answer, however, me-entails sets of direct answers to all the questions of '1'. Thus each question of '1' is sound in ~. (~) If each question in '1' is sound in a normal interpretation ~, then there is a set which contains exactly one direct answer to each question of '1' such that all the elements of this set are true in ~. On the other hand, this set entails some direct answer to Q. Thus Q is sound in ~. 0

Thus if Q is reducible to a set '1', then Q and the questions in '1' are mutually sound. It is obvious that each question Q is reducible to the singleton set which has Q as its member. Thus each question is reducible to some set of questions. Observe also that the relation of reducibility is "transitive" in the sense of the following:

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COROLLARY 7.6. Let ir be a non-empty set of questions. Let p. be a sequence of non-empty sets of questions such that each question of ir is reducible to exactly oneelement of p. andfor each element r of p. there is a question of ir which is reducible to the set r . Then if a question Q is reducible to the set of questions ir, then Q is also reducible to the set of questions which is the union of all the sets which are elements of the sequence p. . Proof. It is obvious that the clauses (i) and (iii) of Definition 7.6 are fulfilled if the above conditions hold. Let M be the set of questions which is the union of all the sets which are elements of the sequence p. . Let Z be a set made up of direct answers to the questions of M which contains exactly one direct answer to each question of M . Since each question of ir is reducible to some set of questions of p., it follows that .the set Z entails some direct answer to each question of ir. But the question Q is reducible to the set of questions ir. Thus Z entails some direct answer to Q. 0 For further properties of the relation of reducibility see Wisniewski (l994b) . Note, however, the following consequence of Corollary 7.6 : COROLLARY 7.7: If a question Q is reducible to a set of questions ir, then: (a) Q is safe iff each question in ir is safe, and (b) Q is risky iff at least one question in ir is risky. Thus, safe questions can be reduced only to such sets of questions, which contain safe questions and no risky questions; a risky question, in tum, can be reduced only to a set of questions which contains at least one risky question.

7.7.2. Erotetic implication and reducibility Let us now examine the connections between reducibility and erotetic implication; when speaking below about erotetic implication, we will have in mind pure erotetic implication. First, it is worth emphasizing that Theorem 7.50 together with the definitions 7.6 and 7.4 yield 1:

1 Recall here that each question of the considered kind has at least two direct answers; so there are no questions which have less direct answers than simple yes-no questions.

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THEOREM 7.52: Each safequestion Q is reducible to some set of questions madeup of simple yes-no questions which are implied by Q; each finite safe

question Q is reducible to some finite set of questions made up of simple yes-no questions which are implied by Q. THEOREM 7.53: If entailment is compact, then each safe question Q is reducible to some finite set of questions madeup of simple yes-no questions which are implied by Q. Theorems 7.52 and 7.53 seems to explicate the old idea that each safe question is reducible to a set of simple yes-no questions which arise from it. On the other hand, the following are immediate consequences of Theorem 7.52 and Corollary 7.7: THEOREM 7.54: A question Q is safe iff Q is reducible to some set of simple yes-no questions which are implied by Q. THEOREM 7.55 : A finite question Q is safe iff Q is reducible to some finite set of simple yes-no questions which are implied by Q. Theorem 7.53 together with Corollary 7.7 yield THEOREM 7.56: If entailment is compact, then a question Q is safe iff Q is reducible to some finite set of simple yes-no questions which are implied by Q. Thus the concept of safety can be characterized or even defined in terms of reducibility and pure erotetic implication . Let us now switch to risky questions. Corollary 7.7 implies that risky questions cannot be reduced to sets made up of simple yes-no questions . However, we can prove that in some cases they can be reduced to sets made up of implied conditional yes-no questions . Let us recall that a conditional yes-no question is a question of the form ? {A " B, A " -,B}, where A, B are sentences. We can prove THEOREM 7.57 : Each regular risky question Q is reducible to some set of questions madeup of conditional yes-no questions which are implied by Q. Proof. Let Q be a regular risky question. The set dQ has at least two elements and is either finite or infinite but denumerable. Let Alt A 2 , ... be a fixed sequence without repetitions whose terms are all the direct answers to Q. Since Q is regular, then the set of direct answers to Q is me-entailed by some presupposition of Q and thus also by the universal closure of this presupposition. Yet, the universal closure of a presupposition is also a presupposition (let us recall that free variables are understood here in the "generalizing" manner). So we may say that the set of direct answers to Q

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is me-entailed by some presupposition of Q being a sentence. Let B be a fixed presupposition of Q such that B is a sentence and B IF dQ. We define the following set of conditional yes-no questions:

• = {Q: dQ = {B " Ai' B " ,Ail. where i > l}. Since each question has at least two direct answers, the set 'I' is non-empty . Since B is a presupposition of Q. then for each direct answer A to Q and for each question Qi of 'I' we have A IF dQi' Let Y be an arbitrary but fixed set made up of direct answers to the questions of 'I' which contains exactly one direct answer to each question of '1'. If Y contains some direct answer of the form B " Ai' where i > I , then Yentails some direct answer to Q. If Y consists of all the direct answers of the form B " 'Ai' where i > 1, then - since B me-entails dQ - the set Yentails the direct answer Al to Q. There is no question which has less than two direct answers. Thus Q is reducible to '1'. Let Qj = ? {B " Aj , B " 'Aj} be an arbitrary but fixed question of'l'. It is obvious that the direct answer B " Aj to Qj entails the direct answer Aj to Q and thus me-entails the non-empty proper subset {Aj} of dQ. On the other hand, since B IF dQ, then the direct answer B " 'Aj to Qj me-entails the non-empty proper subset dQ - {Aj} of the set of direct answers to Q. Since Qj is an element of '1', then - according to what has been said above each direct answer to Q me-entails the set of direct answers to Qj' Therefore Q implies Qj; so each question of 'I' is implied by Q. 0 Observe that if Q is a finite question, then the set 'I' defined in the proof of Theorem 7.57 is finite. On the other hand. each finite question is regular: any disjunction of all the direct answers to a finite question is a prospective presuppositions of this question . The following theorem can be proved in a similar way as Theorem 7.57: THEOREM 7.58: If Q is a finite risky question, then Q is reducible to a finite set of questions made up of conditional yes-no questions which are implied by Q.

Recall that a question Q is said to be normal iff PresQ IF dQ. The following theorem is also true : THEOREM 7.59: If entailment is compact and Q is a risky but normal question, then Q is reducible to a finite set of questions made up of conditional yes-no questions which are implied by Q.

Proof: If Q is normal, then PresQ IF dQ. If entailment is compact, then also me-entailment is compact; it follows that there are finite subsets Yof PresQ and X of dQ such that Y IF X. Since Q is risky. Y is not empty. Let

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A be a fixed conjunction of all the elements of Y (if Y is a singleton set, A is equal to the element of Y). Let A be the universal closure of A. It is clear that both A and A are prospective presuppositions of Q; thus A IF x. If the set X is the empty set, then A is a contradictory sentence and hence each direct answer to Q is a contradictory sentence. Let B be an arbitrary but fixed sentence and let QI = ? {A 1\ B, A 1\ ' B}. It is obvious that Q is reducible to the set {QI}; moreover, Q implies QI' Assume that X is a singleton set and that C is the element of X. Since each direct answer is a sentence, then C is a sentence. Let Q2 = ? {A 1\ C, A 1\ ,C} . It is readily seen that the question Q is now reducible to the set {Q2} (since A IF C, then A 1\ ,C is a contradictory sentence and thus entails any direct answer to Q). What is more, Q implies Q2' Let X = {A h ..., At}, where k > 1. We shall define the followsig set of conditional yes-no questions:

'I'

= {Q:

dQ

= {A

1\

Ai'

A

1\

,A;}, where i

> 1}.

It may be shown in a similar way as in the proof of Theorem 7.57 that the question Q is now reducible' to the set 'I' and that each question of 'I' is implied by Q. 0 Let us call a binary question any question which has exactly two direct answers . The considerations presented above allow us to say that the following is true: THEOREM 7.60: (i) Each regular question Q is reducible to some set of binary questions which are implied by Q; (ii) Each finite question Q is reducible to some finite set of binary questions which are implied by Q; (iii) If entailment is compact, then each normal question Q is reducible to some finite set of binary questions which are implied by Q. Proof. This follows from theorems 7.52 , 7.53 , 7.57, 7.58, 7.59 on the basis of the fact that each safe question is normal and regular. 0 7.8 . EXAMPLES So far we have assumed that the language for which we defined erotetic implication was an arbitrary but fixed formalized language of the kind characterized in Chapter 3. Let us now switch to some concrete languages and formulate some exemplary theses which describe what quest ions of these languages are implied by what questions of them on the basis of certain sets of d-wffs. We shall start with the language ~o; let us recall that the only questions of ;eoare questions of the first kind (cf. Chapter 3, Sections 3.1.2

201

EROTETIC IMPLICATION

and 3.3.1) and that each interpretation of ;;e' is a normal interpretation of this language. Of course, erotetic implication in ;;e' is defined by means of the concept of me-entailment in ;;e'. We shall use the symbol Im~. for erotetic implication in ;;e'. A metalinguistic expression of the form Bx; ... x;", like an metalinguistic expression of the form Axil .. • xin, refers to the sentential functions whose free variables are exactly the (explicitly listed) variables Xi" .. . , xin' In order to simplify formulas instead of A(xi/ul> ... , xi/u,J we shall write Au! ... u., We assume that nonequiform metalinguistic expressions referring to direct answers represent nonequiform sentences. The members of sets of auxiliary d-wffs will be simply listed and not enclosed in parentheses. By applying some well-known metalogical results one can easily prove that, among others, the following theses are true: (7.1)

Im~.(?

{A

f\

B, A

f\

..,B, ..,A

f\

B, ..,A

f\

"'B}, ? {A, "'A}).

(7 .2)

Im~.(?

{A

f\

B, A

f\

"'B, ..,A

f\

B, ..,A

f\

"'B}, ? {B, "'B}).

(7.3)

Im~.(?

{"'A, A

(7.4)

Im~.(?

{A, ..,A}, A == B, ? {B, "'B}).

(7 .5)

Im:£.(? {A, ..,A}, B -. A, C -. ..,A, B v C, ? {B, C}).

(7.6)

Im~.(?

{A, "'A}, B, ? {B -. A, "'(B -. Am.

(7.7)

Im~.(?

{A v B, "'(A v B)}, ..,A, ? {B, "'B}).

(7.8)

Im~.(?

{A

(7 .9)

Im~.(?

{A -. B, "'(A -. B)}, A, ? {B, ..,B}).

f\

f\

B, A

B, "'(A

f\

f\

"'B}, ? {A, "'A}).

B)}, A,? {B, "'B}).

(7.10) Im~.(? {A == B, "'(A == B)}, A -. B, ? {B -. A, "'(B -. A)}). (7.11) Im~.(? {A

f\

B, A

f\

"'B}, A, ? {B, "'B}).

202

(7. 12) Im~.(? {Alt where n > 1 and 1

CHAPTER 7 • •• ,

~

An}, AI v ...

V

An' ? {Ai' ,A;}),

V

An' ,(AI

i s;n .

(7 . 13) Im~.(? {AI' ..., An}, AI v ... ? {,A It . •• , ,An}), where n > 1.

(7 . 14) Im~.(? {AI' .. ., An}, B .... AI v .. . where n > 1 and 1 s i ~ n. (7.15) Im~.(? {Alt . . • , An} , B .... A I v .. . ? {,A I' ... , -.An}), where n > 1. (7. 16) Im~.(? {AI' ..., An} ,' BI == A it where n > 1.

••. ,

AJ ,

1\ •• • 1\

V

An' B, ? {Ai ' ,Ai})'

V

An' B, , (AI

Bn == An' ? {BI ,

(7 . 17) Im~.(? {AI' .. ., An}, BI .... AI' ..., B; .... An' BI v ... ? {BI , . .. , BnD , where n > 1. (7 .18) Im~.(? {AI' .. ., An}, B .... A I v .. .

? {B, ,B}), where n > 1 and 1 < i

~

V

AJ ,

1\ • • • 1\

...,

V

Bn}),

Bn,

Ai_I' ,B .... Ai v ...

V

An'

Ai_I' C .... Ai v ...

An' B

-

n.

(7. 19) Im~.(? {AI' ..., An} , B .... A I v .. . V C, ? {B , C}), where n > 1 and 1 < i ~ n.

V

V

(7 .20) Im~ .(? {AuI I .•• UI., ... , AUkl ... Uk,,}, 3xil ... 3xi• (Axil ... Xi. 1\ «XiI = U lI 1\ ••• 1\ Xi. = U 1.) V • • • V (Xii = Ukl 1\ • •• 1\ Xi. = Uk,,»), ? {AU fl ... Uf", ,Aufl ... Uf,,}), where k > 1 and 1 s I ~ k.

EROTETlCIMPLICATION

203

(7.2 1) Im~*(?

{Au ll • •• u 1n, ... , Auk, • •• u kn} , B ..... 3xil . . . 3xin (Axil .. . Xin /I « Xii = U II /I /I X in = U t .) V . .. V (Xi' = Ukl /I .. . /I X in Ukn))) , B, ? {Au tl U/n' ,Au/I .. . u/n} ) ,

=

where k

>

1 and 1

S;

l s; k.

(7 .22) Im~*(? {Aut , «Xii = u ll /I

/I

»), , Vx i l

Ukn

u kl /I .. . /I

3xil

U 1n, ... , AUkl . .. Ukn},

xin =

= Uti

V Xin«X il

xin =

V (Xii

U t.) V /I

u kn) ..... Axil .. . X in) ,

3xin (Axil · .. . xin /I Uk' /I /I xin

=

=

xin U t.) V {,AUt , ... U tn,

/I

?

.. .

=

V (Xii

=

, ,Aukl

... ukn}) , where k

>

1.

(7.23) Im;e*(?

{Vx i l .. . V x i" (Axil'" X in =: « Xii = U 11 /I .. . /I X in

V .. . V (Xii

=: «Xii

=

= Ukl

/I

... /I X in

U t , /I ... /I X in

=

= Ut,'» ) ,

..... «Xii Ukn»

),

=

U t i /I .. . /I X in

(7. 24) Im;e*(?

= Ukl

/I

.. . /I X in

=

?

{ VXil ... VXi" (Axil

.. . , V Xi l ... V Xi" (Axil '" V (Xii

where k

>

= Ukl

/I

.. . /I Xi"

=

U kl /I

... /I X in

Xi" "" « Xii

=

=

= U II = U II

U 1, /I

= Ukn» )}, Au t , ' " X in ..... (Xi i = U 11 /I X in ..... «Xii = U 11 /I .. . /I X in

X in

=

{VXil ... V X in (Axi l '"

Ukl /I ... /I Xin /I .. . /I Xi"

= Ukn))) }) .

{VXi l .. . VXi" (Axil ... Xi" =: (Xii

.. . , V XiI ... V Xi" (Axi , V ... V «Xii Ukl /I

?

U t.) V .. . V (Xii

, VXi l .. . V X in (Axi, .. . X in ..... « Xii

V .. . V (Xii

Ukn ,

=

,VXil .. . V X in (Axil'"

U, .) V .. . V (Xii

Ukn))) }, AU II . .. U 1n /I . .. /I Auk, ... Ukn,

= U 1n)

/I

... /I X in /I Xi"

=

X in

=

= Utn) = U tn»,

U 1n)

U 1n, ... , AU kl ... /I X in /I X in

= U 1n»,

= U t ,,)

V ...

= Ukn))) }) ,

1.

Let us now switch to the language !£**. Erotetic implication in !£** is defined in terms of me-entailment in !£** (cf. Chapter 4, Section 4.4 .2). We shall use the symbol Im;e.. for erotetic implication in !£**. The language !£** contains, among others, quest ions of the first kind. The theses presented above were chosen in such a way that if we replace in them the symbol Im;e* with the symbol Im;e**, we will obtain theses which characterize erotetic implication in !£**. The language !£** contains also questions of the second and third kind. One can easily pro ve that the following theses are true :

CHAPTER 7

204

(7.25)

Im~ ••(? S(Axi, ... x in) , 3xil .. . 3xin Axil ... Xin, ? {Au) ... Un'

....,Au! ... Un}).

(7.26)

Im~.. (?

S(Axii ... Xin), B .... 3xi l ? {Au! ... u., ""'Au) ... Un}) .

...

3xin Axil '" Xin, B;

(7 .27) Im~ .. (? W(x, ... x,H_IIAxi ) , 3 ~hi Axi , ? {Au! 1\ ... 1\ AUm U) ~ ... ~ Um, ""'(Au) 1\ ... 1\ AUm 1\ U) ~ ... ~ uJ}), where k > 1, m ~ k and ul> ... , u.; are nonequifonn closed terms,

1\

(7.28) Im~ ••(? W(x, x,H_dAx j ) , B .... 3~hi Ax;. B, ? {Au! 1\ ... 1\ AUm 1\ u! ~ ~ Um, ""'(Aul 1\ ... 1\ AUm 1\ u l ~ ... ~ uJ}), where k > 1, m ~ k and uI , ... , u.; are nonequifonn closed terms.

== Axil .. . X;.),

(7.29)

Im~ .. (?

(7.30)

Im~ ••(? S(Axi, . .. X;.), B .... 3xi, .. . 3xi• (Axil'" Xi. 1\ «Xi'

S(Bxil .. . x in) , VX il ... VX in (Bx; ... x;. ? {Au! ... u.; ....,Au 1 ... un}).

Xi. = U)J V ... V (Xii = Uk' 1\ ... 1\ Xin = Ut.))), C"" 3xi, ... 3xi• (Axil '" x.; 1\ «Xii = V!I 1\ ... 1\ Xi. = V (Xii = V" 1\ ... 1\ Xi. = VSn))) , B V C, ? {B, C}).

=U

1, 1\

.. . 1\

(7.31)

Im~ ••(? S(Axii

Xin = U1.) V ? {Au ll ... Uln, 1\

where k

(7.32)

> 1.

X;.), VXil ... VX in (Axil'" Xin .... «Xii V (XiI = Uk' 1\ ... 1\ Xin = Ut.))), , AUkl ... Uk-})'

VIJ V ...

= UII

1\ ...

VXil ... "Ix;. (Axil ... x in == «Xii = Xin = Uln) V V (Xii = Ukl 1\ ... 1\ Xin = Ut.))), C"" VX il ... VX in (Axil'" Xin == «XiI = VII 1\ .. . 1\ Xi. = Vln) V ... V (Xii = VSI 1\ ... 1\ Xin = Vsn))) , B V C, ? {B, C}).

Im~ .. (? U(Axil .. . x in), B

UII

1\ ... 1\

(7 .33) Im~••(? U(Axil ... Xin), B .... VXil

U 11 1

1\. .. 1\

Xi. =

U)

) I.

V

VX in (Axil .. . x.; V

(Xii =

U)

== «XiI = 1\ ... 1\

t)

EROTETIC IMPLICATION

»)

Xin=

U

• • • 1\

Xin =

1

'.

v

U

u,

V . .. V «XiI

=

U,

I.

) V ... V (XiI

Xin =

1\

VI

1\ ... 1\ Xin

X;.

V .. . V

=

IW I

(7. 34) Im!£••(? T(r, . .. x,H_I IIAx;), B C, ? {3(m)xi Ax i, 3(n)x i Axi}), where m ~ k, n ~ k and m "¢ n.

-+

=

»)

1\ .. . 1\ Xin

l"

V

1\ .. . 1\

"

=

V

1\

'" »),

U, '.

= V

=

1\ . •• 'I

C -+ 'Ixil ... 'Ix in (Axil'" Xin == « Xii

V ... V (Xii

205

z•.•

» ), B

V

Xin = ·V

V

=

C, ? {B, C}).

3(m)xi Ax i• C -+ 3(n)xi Ax i, B V

(7.35) Im!£••(? S(Axil ... x in), 'Ix il .. . 'IXu. (Axil ... X;. == ? S(Bx il .. . Xi.» '

Bx., ... Xi.)'

(7.36) Im!£••(? U(Axil .. . Xi.)' 'Ixil ... 'Ixin (Axil .. . Xin == Bx.. ... Xin) , ? U(Bxil .. . x in»· (7 .37) Im!£••(? T(r , ... X,H_I IIAx;), 'Ixi (Axi

? T(r , .. . x,+k_1 1I Bx j».

== Bx

j) ,

(7 .38) Im!£••(? S(Axil .. . x in) , 3xil ... 3xin Bx; ... x in, 'Ix il .. . 'Ix;. (Bx il ... x;. -+ Axil' " Xi.) , ? S(Bxil . .. Xi.» '

(7.39) Im!£••(? W(x, ... x,H_dAx;), ? W (x, .. . x, +m_dBx i», where m ~ k. (7.40) Im!£.. (? S(Axil X;., ? S(,Axil

3~mxi

Bx; 'Ixi (Bx, -+ Ax;),

xin), 3xil .. . 3xin Axil .. . X;., 3xil ... 3x;. 'Axil .. . Xin

».

CHAPTER 7

206

(7.4 1)

Im~ ••(?

S(Axil ... xin) , 3xil ... 3xin Axil'" Xin, 3xi, ... 3xin Bx; ... .Axil ... Xin) , ? S(Bx i, ... Xin» ·

Xin, YXi, ... YXin (Bx it • • • Xin -+

».

(7.42)

Im~ .. (?

S(Axi) , 3 ~kxi Axi , ? U(Axi

(7.43)

Im~ .. (?

S(Axi) , 3;;::kxi Axi , ? W(x, .. . x,H_t1Axi» .

S(Axi) , 3xi Ax i, Y Xi (Ax; -+ Bxi) , 3 ~mxi Bx.; ? T(x, ... x,H_111 Bxi where m ;;:: k. (7.45)

Im~ .. (?

»,

xj = Xi» , 3(l )xi Axi , Y Xi (Ax; -+ Bx), 3 ~kxi Bx.; ? U(Bx i where xj is substitutable for Xi in Axi . (7 .46) Im~ ••(? S(Ax i

1\

(7 .47) Im:e••(? S(Axi

1\

-+

Y Xj

(A(x/xj )

»,

= Xi»' :.. X,H_l II Bx»,

VXj (A(x;lx)

Bx), 3(m)xi Bx.; ? T(x,

where m ;;:: k and

-+

-+

xj is substitutable for

xj

Xi in

3(l)xi Axi, VXi (Axi

Axi .

(7 .48) Im~ ••(? U(Axi), 3(m)xi Ax i, ? T(x, ... X,H_l II Ax i» , where m ;;:: k.

(7.50) Im~ ••(? T(x, ... xrH_1IIAxi), 3(m)xi Ax i, 3xi .Axi , ? S(.Axi» , m ;;:: k.

where

Among the implied questions characterized by the above theses there are no open questions of the second kind and (with the exception of theses (7.39) and (7.43» existential questions of the third kind . Yet , one can easily prove

EROTETIC IMPLICATION

207

LEMMA 7.3 : (i) lm'foo(Q, X, ? O(Axi, •• • Xi.)) iff lm,£oo(Q, X, ? S(Axi, • •• Xi.» . (ii) lm'foo(Q, X, ? W(xrIAx i» iff lm'foo(Q, X, ? S(Axi)). (iii) lm'foo(Q, X, ? W(x r ." xr+k_IIAxi» iff Im'foo(Q, X, ? S(A(x;!xr) " ... " A(x;!Xr+k_l ) s x, ~ xr+ I " x, ;e xr+2 " .. . " x, ;e Xr+k_1 " xr+ 1 ;e xr+ 2 " ... " Xr+k- 2 ;e Xr+k_I»' where k > 1. By applying Lemma 7.3 and some of the above theses one can easily find exemplary theses which present open questions of the second kind and existential questions of the third kind as implied questions. Open questions of the second kind and eXistentfal questions of the third kind of the form? W(x r IAxi ) have not occured so far as implying questions . Yet, exemplary theses pertaining to these questions can be obtained from some already presented theses by means of the following lemma : LEMMA 7.4: (i) If Im'foo(? S(Axi, QI)'

Xi..)' X, QI), then Im'foo(? O(Axii ... xin) , X, . (ii) Iflm,£oo(? S(Axi), X, QI), then Im'foo(? W(xrIAx i) , X, QI)' Let us finally note some theses which characterize pure erotetic implication; these theses are consequences of Lemma 7 .3 and Theorem 7.1 : (7 .51) Im 'foo(? S(Axii (7.52) Im'foo(? O(Axii

...

...

...

Xi.), ? O(Axi,

.. .

Xi..) '

Xi.) ' ? S(Axi,

...

Xi.» '

(7.53) Im,£oo(? S(Ax;), ? W(xrIAx i». (7.54) Im'foo(? W(xrIAx i) , ? S(Ax;» . (7.55) Im'foo(? S(A(.x;!xr) " ... " A(X;!Xr+k_I) s x, ;e x., I " .r, ;e x r+2 ... " x, ;e Xr+k-I " xr+ 1 ;e xr+ 2 " ... " Xr+k-2 ;e Xr+k_I) , ? W(x r ... xr+k_IIAxi», where k > 1.

"

208

CHAPTER 7

(7.56) Im!£••(? W(xr ... xr+t_1IAx;), ? S(A(x;!xr) 1\ ••• 1\ A(X;!XrH_1) ;c x r+1 1\ x, ;c x r+2 1\ .•. 1\ x, ;c XrH .1 1\ x r+1 ;c x r+2 1\ ••• XrH_2 ;c XrH _1 where k > 1.

»,

1\

xr

1\

(7.57) Im!£••(? O(Ax;), ? W(xrIAx;».

(7.59) Im!£.. (? O(A(x;!x r)

1\ ••• 1\ A(X;!XrH_1) 1\ XrH_ 1 1\ Xr+1 ;C Xr+2 1\ • •• ? W(Xr ... x rH_t1Ax;». where k > 1. 1\ • • • 1\

x,

;C

x, ;c 1\

Xr+1 1\ x, ;C Xr+2 XrH_2 ;C XrH_1) ,

(7.60) Im!£••(? W(xr ... XrH_1 IAx;), ? O(A(x;!x r) 1\ ... 1\ A(X;!XrH_1) ;c x r+1 1\ .r, ;c x r+2 1\ ... 1\ x, ;c xrH . ! 1\ x r+1 ;c x r+2 1\ ... XrH_2 ;c XrH_1» ' where k > 1.

1\ 1\

xr

CHAPTER

8

EROTETIC ARGUMENTS

8.1. EROTETIC INFERENCES , EROTETIC ARGUMENTS AND VALIDITY As was pointed out in Chapter 1, we often arrive at a question on the basis of some previously accepted declarative senteflce or sentences. We also often arrive at a question when we are looking fur the answer to another question on the basis of some data expressed by certain previously accepted declarative sentences or sentence. And sometimes we arrive at a question just when we are looking for the answer to another question. Some examples were presented in Chapter 1, Section 1.1; here are others : (I .a) John writes three books during one year. But if John writes three books during one year, then he is a monk or a bachelor or he has a very patient wife. So is John a monk, or a bachelor, or has he a very patient wife?

(l.b)

Is John a monk, or a bachelor, or has he a very patient wife? Well, if John is a monk or a bachelor or has a very patient wife, then he is single or married. But if John is single, then he is a monk or a bachelor. On the other hand, if John is married, then he has a very patient wife. So is John married or single?

(I .c) Does John write a lot and does he read a lot? So does John write a lot? By an inference we usually mean a process of passing from some declarative sentence(s) to a declarative sentence. Examples of the kind presented above show, however, that the concept of inference can or even should be generalized by introducing the concept of erotetic inference. At a first approximation an erotetic inference may be defined as a thought process in which we arrive at a question on the basis of some previously accepted declarative sentences(s) and/or a previously posed question. These examples also show that there are erotetic inferences of two kinds: the key difference lies in the type of premises involved. In the case of erotetic inferences of the first kind the premises are declarative sentences. The premises of an erotetic inference of the second kind consist of an (initial) question and possibly of some declarative sentence(s); erotetic inferences which do not involve any declarative premises can be regarded as a special case of the erotetic inferences of the second kind. Erotetic inferences are thought processes. Yet, logic does not deal with thought processes, at least not directly. But we may introduce here more

CHAPTER8

210

"safe" concepts of (abstract) erotetic arguments; these arguments are counterparts to erotetic inferences in the same way as arguments understood in the usual manner are counterparts to standard inferences. By an erotetic argument of thefirst kind we shall mean an ordered pair which consists of a non-empty and finite set of declarative formulas and a question. By an erotetic argument of the second kind we shall mean, in tum, an ordered triple which consists of a question, a finite (possibly empty) set of declarative formulas and a question. If is an erotetic argument of the first kind, then the formulas of X are called premises of this argument and the question Q is called its conclusion. Similarly, if < Q, X, QI> is an erotetic argument of the second kind, then the question Q is said to be the erotetic premise or the initial question of this argument, the formulas of X and the question Q, are called declarative premises and the conclusion of this argument, respectively. An erotetic argument of the second kind which does not involve declarative premises is said to be a pure erotetic argument. In what follows we will be using the expressions "e-argument", "e.-argument" and "e--argument" instead of the long expressions "erotetic argument" , "erotetic argument of the first kind" and "erotetic argument of the second kind" , respectively. We will sometimes use the expression "question/conclusion" as an abbreviation of the expression "question which is the conclusion. " The attitude towards erotetic inferences shared by almost all logicians is that these inferences belong to the "pragmatics" of reasoning, in the very bad sense of "pragmatics" as referring to something that is not subjected to any objective rules. An average logician would probably say that asking questions is a method of expressing our curiosity and curiosity is subjected to almost everything but not logical rules. But asking or posing a given question is one thing and arriving at it is another. No doubt, there are erotetic inferences which may be regarded as completely irregular , at least at first sight. But there are also erotetic inferences which fall under a common formal schema. For example, it is easily seen that the following e.-argument has the same formal structure as (l.a): (2.a)

Andrew works twelve hours a day. But if Andrew works twelve hours a day, then he is a businessman or a young logician or a slave in a gold mine. So is Andrew a businessman, or a young logician, or a slave in a gold mine?

Similarly, the following

~-argument

has the same formal structure as

(l.b):

(2.b)

Is Andrew a businessman, or a young logician, or a slave in a gold mine? Well, if Andrew is a businessman or a young logician or a slave in a gold mine, then he is a workaholic or he is forced to work. But if Andrew is a workaholic, then he is a

EROTETIC ARGUMENTS

211

businessman or a young logician . On the other hand, if Andrew is forced to work, then he is a slave in a gold mine. So is Andrew forced to work, or is he a workaholic?

It is also easily seen that the following e-argumenr has the same formal

structure as (l .c): (2 .c)

Is Andrew rich and is he handsome ? So is Andrew rich?

But the existence of formal schemata of e-arguments is only one part of the story: what is more important from the logical point of view is that some e-arguments seem to be intuitively valid, whereas some others seem not to be. In particular, the e-arguments presented above may be regarded as intuitively valid; the same holds true in the case of the examples presented in Chapter 1, Section 1.1. But the impression of intuitive validity is not the only feature shared by those e-arguments: in each case we can also legitimately say that the question which is the conclusion arises from the premises. Moreover, the relevant concepts of the arising are exactly those analyzed in the previous chapters of this book: it is easily seen that the conditions (*) , (**) and (****) formulated in Chapter 1, Section 1.2 are fulfilled by the premises and conclusions of the analyzed et-arguments . Similarly, a moment's reflection will show that the conditions (#) and (##) formulated in Chapter 1, Section 1.4 are fulfilled by the premises and conclusions of the ~-arguments analyzed above. We can press further in this direction. In Chapter 1 we presented examples of questions which arise (in the analyzed sense of the word) from some given declarative sentences. But if we will treat those sentences as premises, we will see that erotetic inferences which lead to the arising questions are intuitively valid; the situation is analogous in the case of examples which pertain to the arising of questions from questions and declarative sentences. There is an old idea that logic is the science of argument. According to this idea, the main task of logic is to provide - in some way or another - the schemata of valid arguments, and handling this task presupposes some idea of validity. So if we agree that erotetic arguments can or even should be analyzed within the logic of questions, one may ask: how should we define validity in the case of erotetic arguments? In the case of e-arguments the appropriate notion of validity is given neither by God nor by Tradition; so some more or less arbitrary decision has to be made. Since the concept of validity is a nonnative one, its definition cannot be derived entirely from empirical phenomena; so some regulative principle is needed. The careful analysis of intuitively valid e-arguments and

212

CHAPTERS

the coincidences between intuitive validity and the arising of questions do suggest a conception of validity . The conception we are going to propose here is that: an erotetic argument of thefirst kind is validjust in case the

question which is the conclusion of this argument arises from the premises in the sense of being evoked by the set. of premises. Similarly, an erotetic argument of the second kind is valid just in case the question which is the conclusion of this argument arises from the initial question and the declarative premises in the sense of being implied by the initial question on the basis of the set of declarative premises. Let us stress again: this is a decision. Yet, everything said in Chapter 1 in defense of explicating the intuitive notions of the arising of questions by means of the concepts of evocation and erotetic implication can now be said in support of this decision. By and large, an erotetic argument in which the the question/conclusion is evoked by the premises leads to a question which is sound relative to thepremises and is not logically redundant with respect to the premises. An erotetic argument in which the question/conclusion is implied by the premises , in tum, leads to a question which is secured by the premises and which is in each case cognitively useful with respect to the

premises. We now express the ideas sketched above in a more detailed and formal way.

8.2.

E-ARGUMENTS AND G -ARGUMENTS

Assume again that :£ is an arbitrary but fixed formalized language of the kind considered in Chapter 3, that is, to speak generally, a first-order language with identity enriched with questions in the way described in Chapter 3. Assume that the semantics for:£ is of the kind characterized in Chapter 4. In what follows, questions and d-wffs as well as the premises and conclusions of the e.-arguments analyzed are all assumed to be expressions of :£ . Similarly, by evocation and generation we will mean evocation and generation in :£ . The concept of evocation can be used to distinguish the following class of erotetic arguments of the first kind: DEFINITION 8.1. An erotetic argument of the first kind

is an

E-argument iff the question Q is evoked by the set X. The E-arguments may be called question evoking arguments . We shall consider the E-arguments as valid erotetic arguments of the first kind .

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The properties of evocation were characterized in detail in Chapter 5. We can now use the theorems and corollaries proved in Chapter 5 in order to characterize the possible premises and conclusions of E-arguments as well as the semantic relations between the premises and conclusion of an Eargument. The details can be constructed feasibly from the discussion in Chapter 5, so we shall concentrate here only on some substantial points and leave the rest of the enterprise to the Reader. Note first that the premises of an E-argument need not be true: there are E-arguments with only true premises, but there are also E-arguments which involve false premises. As we pointed out in Chapter 1, the declarative premises of erotetic inferences may be items of knowledge or belief of the questioner , but they also may be only hypothetically or even temporarily assumed. In the case of an E-argument we may say, however, that if all the premises are true (in a given normal interpretation of the language; in what follows this provision will be assumed), then - by the definition of evocation - at least one direct answer to the question/conclusion is true . The question

which is the conclusion of an E-argument is sound relative to the (set of) premises of this argument. Yet, the information provided by the premises is always insufficient to determine which of the direct answers to the question/conclus ion is true: no direct answer to the question is entailed by the premises and thus even if the premises are true, the question/conclusion cannot be answered in a logically legitimate way by deriving a direct answer from them. The question which is the conclusion of an E-argument is not logically redundant withrespect to thepremises of thisargument. Moreover, Theorem 5.1 shows that in case of E-arguments the question/conclusion is informative relative to the premises. It follows that when we enrich the set of accepted premises with that direct answer to the question/conclusion which will finally appear to be acceptable, we will have the possibility of deriving new, nontautological consequences. This possibility is warranted in each case, that is, regardless of which of the direct answers finally appears to be acceptable. Nothing said above implies that the question/conclusion of an E-argument must have a true direct answer in every case. We only claim that such a question must have a true direct answer if all the premises are true. There are cases, however , in which this condition is fulfilled for trivial reasons: the question which is the conclusion is safe. In order to distinguish the cases which are non-trivial in this respect from the remaining ones we introduce the following concepts:

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DEFINITION 8.2. An erotetic argument of the first kind is a

G-argument iff the question Q is generated by the set X. DEFINITION 8.3 . An erotetic argument of the first kind is an E'-argument iff is an E-argument, but not a G-argument. According to Corollary 6.1, generation is the evocation of non-safe (i.e . risky) questions; thus each G-argument is an E-argument and thus a valid e l argument. Moreover, the conclusion of a G-argument is always a risky question which, however, is made sound by the (set of) premises of this argument. On the other hand, each E'-argument has a safe question as the conclusion. G-arguments may be called question generating arguments. It may occur that the question which is the conclusion of a G-argument has no true direct answer in that particular interpretation of the language which is accepted by an inquirer . If that occurs, we may say that at least one of the premises accepted by the inquirer who performs the corresponding erotetic inference is not true (again, in the interpretation of the language accepted by himlher). Thus by asking or posing questions which are conclusions of G-arguments a questioner may also check his/her initial assumptions; moreover, this can lead to the revision of (some of) them. Let us recall in this context that Theorem 6.33 shows that the set of negations of all the direct answers to the question/conclusion of a G-argument either entails the negation of the universal closure of some particular premise of this G-argument or generates a question whose set of direct answers consists of the negations of universal closures of all the premises; according to Theorem 6.34 only the second possibility holds if no single premise meentails the set of direct answers to the question which is the conclusion. Thus if it appears that no direct answer to the question/conclusion of a Gargument is true, the inquirer may use the set of negations of all the direct answers to this question either to determine what particular premise is not true or as the set of premises of a new G-argument whose conclusion is a question that asks, roughly, which of the premises is not true. Of course, since the set of premises of an e.-argument is always finite, the second move is possible if the initial question is finite or entailment in the language is compact (cf. Theorem 6.28). There are E-arguments and G-arguments which have only one premise, but there are also e-arguments of the analyzed kind which have more than one premise. It is interesting, however, that if any of the following conditions holds: (i) Q is a normal question and entailment is compact, or (ii) Q is a regular question, then each E-argument which has Q as the

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conclusion and which involves many premises can be split into two arguments: a standard (deductive) argument whose premises consist of the premises of the initial E-argument and whose conclusion is a prospective presupposition of Q, and a single-premise E-argument which leads from the conclusion of the former argument to Q. This result follows from Theorem 5.29. Theorems 6.31 and 6.32 show, in turn, that the situation is analogous in the case of G-arguments. However, the prospective presupposition of Q which serves as the premise of the "final" E-argument or G-argument need not occur among the premises of the "initial" E-argument or G-argument: it may happen that it is only entailed by them. Furthermore, the conditions (i) and (ii) mentioned above are essential and it seems that there is no general reason for which each many-premises E-argument can be split into a standard argument and a single-premise Esargumeat. Since it is not the case that each question can be evoked and it is not the case that each set of d-wffs can be an evoking set, there are questions which cannot be conclusions of E-arguments and there are finite sets of d-wffs which do not constitute sets of premises of any E-arguments. Theorem 5.3 yields that complete sets of d-wffs cannot serve as sets of premises of Earguments; Theorem 5.4 implies that each incomplete finite set of d-wffs constitutes the set of premises of some E-argument. Although at first sight these results look plausible, this impression disappears when we take into consideration the fact that each inconsistent set of d-wffs is complete. According to the commonly accepted idea, many problems arise from inconsistencies; on the other hand, there is no E-argument whose set of premises is inconsistent. This doesn't mean, however, that we cannot describe the phenomenon of the arising of problems from inconsistencies in terms of E-arguments . Recall, first, that evocation is not "monotonic" . Observe also that a question can be evoked by a consistent subset of an inconsistent set. The typical situation in which we say that a question arises from an inconsistency can be briefly described as follows: there is a set of accepted premises, say, {AI' ..., An}, which entails some conclusion, say, B. (For simplicity let us assume that AI> ... , An' Bare sentences I and that B is not entailed by any single sentence from the set {AI' ... , An}2). On the other hand, there are strong reasons (empirical, for example) for the accepta nee of the negation of B. Of course, since the set X = {A I ' .... , An' A I /I . • •

I If this condition is not fulfilled , in what follows we should speak about universal closures of the above d-wffs,

2

Otherwise no erotetic argument is needed.

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A An ..... B, ....,B} is inconsistent, it evokes no question and an e-argument which has the elements of X as the premises and the question ? {....,A I> • • • , ....,An } as the conclusion is not an E-argument. But the proper subset XI = {At A .. . A An ..... B, ""'B} of X evokes the question? {....,AI> , ""'AnP and the e.-argument which leads from the premises Al A .. . A An B, ....,B to the question? {""'A I , ... , ""'A n} is an E-argument! Let us now ask: is the simultaneous acceptance of all the elements of the occurently inconsistent set X possible? Although this can happen to some subjects, it seems that in a typical situation of the above kind the sentences AI' .. ., An become kept in suspense.'What really happens is that an inquirer is aware of the fact that B is entailed by the set {A" ... , An} and thus accepts the implication AI A .. . A An ..... B: this is his/her first premise. The second premise employed is ....,B; the question? {"",..1\, ... , ....,An } is the conclusion. The sentences AI> .. ., An do not play the role of premises in the erotetic inference which is consecutively performed by an inquirer : this inference involves only the sentences AI A .. . A An ..... B 'and ....,B as the premises. And, let us recall, the corresponding e-argument is an E-argument. This is one of the ways in which we can describe the arising of problems from inconsistencies in terms of E-arguments . Since E-arguments are defined in terms of evocation, then by Theorem 5.7 there is no E-argument that culminates in a self-rhetorical question . It follows that neither completely contradictory questions nor completely tautological questions are conclusions of E-arguments; it also follows, however, that no direct answer to the question/conclusion of an E-argument is a tautology. One may argue that this last consequence excludes among the possible conclusions of E-arguments both questions of the form "Is A a tautology?" and questions about entailment from a given set of premises. This argument rests on a mistake, however. Questions of the above kind are metalogical: even if A is a tautology of some language, the direct answer "A is a tautology" is not a tautology of the same language; similarly , a sentence of the form "B is entailed by AI> .. ., An", if true, is not a tautology of the object language. In other words: metalogical questions need not be analyzed as self-rhetorical and thus may appear as conclusions of some metalogical Earguments.

3 Since B is entailed by the set {A,• .... An} without being entailed by any single element of it. no direct answer to the above question is entailed by X" On the other hand, it is obvious that the set of direct answers to the analyzed quest ion is me-entailed by XI'

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Let us finally stress that the riskiness of the questions which are the conclusions of G-arguments is not the only important feature that distinguishes them from E'-arguments. According to Theorem 6.5, among the premises of a G-argument there always occurs at least one synthetic d-wff; the situation may be different in the case ofE'-arguments. Thus the premises of a G-argument are always at least partially nonanalytic, whereas the question/conclusion is risky, not self-rhetorical and both sound and informative relative to the premises.

8.3.

1m-ARGUMENTS AND ImS-ARGUMENTS

Let us assume once more that 5£ is an arbitrary but fixed formalized language of the kind considered in Chapter 3 and that the semantics for 5£ is of the kind characterized in Chapter 4. When speaking below about questions and d-wffs, we mean questions and d-wffs of 5£; ez-arguments analyzed below are assumed to have premises and conclusions which are expressions of 5£. By erotetic implication, pure erotetic implication and strong erotetic implication we mean the above relations in 5£. By applying the concept of erotetic implication we can distinguish the following class of ez-arguments: DEFINITION 8.4. An erotetic argument of the second kind is an 1m-argument iff the question QI is implied by the question Q on the basis of the set of d-wffs X. We shall consider the 1m-arguments as valid erotetic arguments of the second kind. The Im-arguments will be also called question implying arguments. Again, almost everything which has been said above about the properties of erotetic implication can now be used in the characterization of Imarguments. We, therefore, highlight only those properties of Im -arguments which seem most important from the point of view of possible applications . First, note that by Corollary 7.1 an 1m-argument which has a sound question (i.e. one having a true direct answer) as the erotetic premise and only true d-wffs as the declarative premises has a sound question as the conclusion : this holds for each normal interpretation of the language. Of course, this does not exclude there being Im-arguments which involve unsound questions as erotetic premises and/or untrue d-wffs as declarative premises: what is excluded is that a question which is the conclusion of an Im-argument has no true direct answer in case the question which is the erotetic premise is sound and all the declarative premises are true. In other

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words, the question which is the conclusion of an Im-argument is secured by the premises. This is not all, however. Corollary 7.3 yields that if all the declarative premises of an Im-argument are true, then the question/conclusion is sound if and only if the question which is the premise is sound (again, this result is restricted to normal interpretations of the language). Thus Im-arguments which involve true declarative premises but unsound erotetic premises cannot lead to sound conclusions. So if it will appear that the question/conclusion has no true direct answer, but all the declarative premises are true, an inquirer should suppress his/her initial question; if, however, the logical value of the declarative premises is not beyond doubt, the negative result concerning the question/conclusion opens the debate about soundness of the initial question and the logical values of the declarative premises. Let us now assume that the erotetic premise of an 1m-argument is sound and that all the declarative premises of this argument are true. It follows that the question which is the conclusion of the analyzed Im-argument is sound, i.e. has at least one true direct answer. Suppose now that the inquirer who performs the corresponding erotetic inference establishes that some selected direct answer to the question/conclusion is true (i.e . finds a certain true direct answer to the question which is the conclusion). The cognitive situation of the inquirer is now better than before: the second clause of the definition of erotetic implication yields that the true answer just found is correlated with some proper subset of the set of direct answers to the initial question which contains some true answer(s) to this question (given the current assumption that the declarative premises are true). So the inquirer need not consider all the "alternatives" offered by the initial question, but may concentrate his/her attention on some narrower set of them with a guarantee of success: sometimes this narrower set is even a singleton set. Observe that it is completely irrelevant here which direct answer to the question/conclusion appears to be true. By the second clause of the definition of erotetic implication each direct answer to the question/conclusion can be used that way: each set made up of the declarative premises and a direct answer to the question/conclusion me-entails some non-empty proper subset of the set of direct answers to the initial question. So we may say that each direct answer to the question which is the conclusion of an 1m-argument is potentially cognitively useful and thus the whole question is in each case cognitively useful with respect to the premises. One point must be clarified here. It is not the case that an inquirer who passes from his/her initial question on the basis of some declarative premises

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to an implied question simultaneously makes an attempt to answer this question and then uses the answer just found (if any) in order to answer the initial question. An inquirer who performs an erotetic inference which is essentially an Im-argument arrives at a question which is both secured by the premises and is cognitively useful with respect to them. This question has the above properties because it is implied (in the sense of erotetic implication) by the premises. Looking for the answer to the question/conclusion and applying the obtained results are further matters. In most cases finding a true or justified direct answer to a question which is the conclusion of an Imargument is tantamount to finding an auxiliary premise which can then be used either to solve the initial problem (this can happen, e.g., if the answer together with the declarative premises entail some single direct answer to the initial question) or to make progress towards a solution. Observe that if the set Z made up of the declarative premises and the justified direct answer to the question/conclusion entails no single direct answer to the initial question, but some finite set Yof direct answers to the initial question is me-entailed by Z, a question whose set of direct answers is equal to Y is evoked by the set Z. So an inquirer may proceed by performing the corresponding erotetic inference of the first kind and then looking for the answer to the question which is the conclusion of this inference; in some cases this can be done by performing a consecutive erotetic inference of the second kind in which the elements of the set Z will play the role of the new declarative premises and the evoked question would be the new initial question (i.e. the new erotetic premise) . Let us finally distinguish two special categories of 1m-arguments. DEFINITION 8.5. An erotetic argument of the second kind is a pure erotetic 1m-argument iff X is the empty set and the question Q implies the question QI (i.e. Im(Q, QI) holds). DEFINITION 8.6. An erotetic argument of the second kind is an Ims-argument iff the question Q strongly implies the question QI on the basis of the set X (i.e . Ims(Q, X, Q\) holds). Thus a pure erotetic argument is an 1m-argument which involves no declarative premises and an Ims -argument is an 1m-argument in which the conclusion is not only implied, but also strongly implied by the premises. By and large, we may say that the conclusion of an Ims-argument is both secured by the premises and is in each case cognitively useful with respect to the premises in a non-redundant and thus substantial way. Since both pure erotetic 1m-arguments and Ims-arguments are Im arguments, they may be considered as valid erotetic arguments of the second

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kind.

8.4. VALIDITY AND NATURAL-LANGUAGE EROTETIC ARGUMENTS The concepts of validity of erotetic arguments were defined above for arguments worded in a formalized language. Yet, the concepts of validity introduced above can also be applied - but in an indirect way - to erotetic arguments expressed in natural languages. Questions and d-wffs of a formalized language of the considered kind represent some declarative sentences and questions of natural languages. The relevant concepts of representation were characterized in Chapter 3. One might say that a question Q of a natural language is evoked by a set of sentences X of this language just in case the formal counterpart of Q is evoked by the set of formal counterparts of the elements of X, and similarly in other cases. Yet, saying this would be misleading: some caution is needed here. The reason is that there is no evocation and erotetic implication in general: there is always evocation in some fixed formalized language and erotetic implication in some fixed formalized language. So the proper formulation is: a question Q of a natural language is evoked on the ground ofaformalized language :£ by a set X of declarative sentences of the analyzed natural language just in case the question Ql of :£ which is a formal counterpart of the question Q is evoked in :£ by a set X, of d-wffs of :£ made up of the formal counterparts of the sentences of X (and similarly for erotetic implication, generation, etc. in the case of natural-language questions and sentences). The evaluation of erotetic arguments worded in a natural language amounts to the evaluation of validity of their formal counterparts. Yet, questions and declarative sentences can be formalized in various languages. Moreover, a naturallanguage question may be evoked on the ground of one formalized language by some set of natural-language sentences without being evoked by it on the ground of some other formalized language. The situation is similar in the cases of generation, erotetic implication, strong erotetic implication and pure erotetic implication . It follows that a natural-language erotetic argument may be viewed as valid if formalized in one language and as invalid if formalized in some other language. But one may say that the situation here does not differ in a substantial way from that known from other branches of logic .

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8.5. Lootcs, SCHEMATA AND RULES

8.5.1. The idea of the logic of questions of a semantically interpreted formalized language If we agree that the E-arguments and the Im-arguments are valid erotetic arguments , then providing the schemata of these arguments becomes the natural task of erotetic logic. And it would be desirable if the logic of questions would also provide us with rules which govern valid erotetic inferences. Yet, to explain how erotetic logic can accomplish these tasks we must first clarify the concept of the logic of questions of a given semantically interpreted formalized language of the analyzed kind. Assume again that :£ is an arbitrary but fixed language of the kind considered above, that is, a first-order language (with identity) enriched with questions in the way described in Chapter 3. Assume also that the semantics for:£ is of the kind described in Chapter 4. In what follows, references to evocation, erotetic implication, generation, etc. will be references to these relations in :£. Let us now construct some new language L(:£) in the following way: (a) the vocabulary of L(:£) results from the vocabulary of:£ by adding two new constants, say, ,. and >-; and (b) the well-formed formulas (wffs for short) of the language L(:£) are expressions having one of the following forms: (i) AI'" An ,. Q where n ~ 1, A I> ... , An are d-wffs of :£ and Q is a question of :£,

Q AI ... An >- QI where n ~ 0, AI> .. ., An are d-wffs of:£ and Q, QI are questions of :£ .4 (ii)

The truth-conditions for the wffs of L(:£) are as follows: a wff of the form AI'" An ,. Q is true iff E(Z, Q), where Z is the set made up of the d-wffs AI' " '' An; (00) a wff of the form Q AI .. . An >- QI is true iff Im(Q, Z, QI)' where Z is the set made up of the d-wffs AI' .. ., An • s The logic of questions Log(:£) ofa language :£ can now be defined as the set of all the true well-formed formulas of the language L(:£). Since :£ was assumed to be an arbitrary language of the analyzed kind, (0)

= 0, then the corresponding formula is of the form Q > QI'

4

If n

S

If n = 0, then Z is of course the empty set.

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there are many logics of questions in the sense of the proposed definition (or, to put it in different words, there are logics of questions of various languages). But the theses of the logic of questions of each concrete language tell us what questions of this language are evoked by what finite sets of d-wffs of it, and what questions of this language are implied by what questions of it on the basis of what finite sets of d-wffs of the language. Let us stress that the general idea of constructing object-language logics of questions in the way proposed above is due to Kubinski (cf., e.g., Kubinski, 1980). Yet, the relations described by the theses of Kubinski's systems are different from those analyzed above; these are such relations between questions as, for example, equipollence, being stronger than, being weaker than, being independent from, etc." What is more important, the systems of Kubinski cannot be viewed as theories of erotetic arguments. Logics of questions constructed according to the above pattern are deductive systems in the sense of Tarski; we can always define a consequence operation such that the analyzed logic is closed under it. For details see Kubinski (1980), p. 59-60.

8.5 .2. Schemata Assume now that Log(;£) is the logic of questions of some arbitrary but fixed language ;£ of the considered kind. When we have a concrete thesis of Log(;£) having the form Al ... An ,. Q, the following:

(#)

Q may be regarded as a schema of E-argument (and thus of a valid erotetic argument of the first kind) provided by the logic Log(;£) . Similarly, if Q AI'" An >- QI (where n > 0) is a concrete thesis of Log(;£), the following:

6

Cf. Chapter 5, Section.5.2.3 .

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Q AI An (##)

QI is a schema of Im-argument (and thus of a valid erotetic argument of the second kind) provided by the logic Log(;E). (Let us stress that (#) and (##) only show what is the structure of the analyzed schemata. In each concrete schema there occur concrete, i.e. object-language, questions and d-wffs of ;E. But ;E is a formalized language and each of its d-wffs represents a class of declarative sentences of a natural language whose elements have a strictly defined form. Similarly, each question of ;E represents a class of questions of a natural language. This is why the term "schema" is appropriate here although the constituents are always concrete expressions.) The situation is analogous in the case of theses of the form Q >- Ql' It can be said that each logic of questions understood in the way described above provides the schemata of valid erotetic arguments in the same way as sentential logics or predicate logics provide the schemata of valid arguments having only declaratives as premises and conclusions. To press the analogy further , it can be said that any logic of questions of the analyzed kind may be regarded as a theory of erotetic arguments.

8.5.3. Rules Providing the schemata of valid e-arguments is one thing, formulating the rules which govern valid inferences is another. Rules are introduced on the metalanguage level. If a formalized language is given, then, from a purely formal point of view, a rule is a relation (in the set-theoretical sense of the word) between sets of expressions and single expressions. The general concept of an erotetic rule may be defined in a similar way. Again, let ;E be an arbitrary but fixed language of the considered kind (i.e ., roughly, a semantically interpreted first-order language enriched with questions); as before questions and d-wffs are the questions and d-wffs of;E. An erotetic rule of the first kind (in short : e-rule) is a binary relation between non-empty finite sets of d-wffs and questions, i.e. a set of ordered pairs of the form < {Bt> ... , Bn} , Q>. If our previous analysis of validity of e-arguments is (at least partially) correct, among all the e-rules special attention should be paid to question evoking rules, in short: E-rules. They can be precisely defined as follows:

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DEFINITION 8.7. An e-rule R is an E-rule iff for each set of d-wffs X and for each question Q such that R(X, Q), the question Q is evoked by the set X. Similarly, an erotetic rule of the second kind (or ~-rule) is a ternary relation between questions, finite sets of d-wffs, and questions, i.e . a set of ordered triples of the form < Q, X, QI >, where X is a finite set of d-wffs of :£. Again, if our previous considerations concerning the validity of e-arguments are correct, special attention should be paid to the question implying rules, in short : Im-rules, They are defined by: DEFINITION 8.8. An ~-rule R is an 1m-rule iff for each set of d-wffs X and for any questions Q, QI such that R(Q, X, QI), the question QI is implied by the question Q on the basis of the set X. Observe that in order to introduce a concrete E-rule it suffices to prove on the metalanguage level a thesis which says that questions of :£ of a strictly defined form are evoked by some finite sets of d-wffs which contain d-wffs of :£ of strictly defined forms; the case of 1m-rules is similar. In particular, the theses (5.1) - (5.60) presented in Chapter 5 and the theses (7.1) - (7.60) presented in Chapter 7 enable us to introduce some E-rules and 1m-rules; we leave it to the Reader. But the theses which enable us to introduce rules are metatheorems about the underlying logic Log(:£) (the logics Log(!£") and Log(!£*") in the case of our examples). Thus, in some sense, the E-rules and Im-rules are given by the logic Log(:£) . Of course, since we have logics of questions of different languages, we also have different collections of E and 1m-rules; each such collection is given by a certain logic of questions of the considered kind. The E-rules and 1m-rules should not be confused with the rules of proof of the appropriate (object-language) logic of questions. We have defined the logic of questions of a language :£ as the set of all true well-formed formulas of some kind and these formulas are neither d-wffs nor questions of :£. But of course the problem ofaxiomatization of the logics of questions of the analyzed kind suggests itself. So far it is an open problem. But if the E-rules and Im-rules are not rules of proof, what is their cognitive status? One may argue that some of them govern some actually performed erotetic inferences. But they also may be regarded as the rules which govern bringing new questions into a discourse. To be more precise, the E-rules may be viewed as the rules for introducing questions which are sound relative to the premises and which at the same time are not logically redundant in relation to them. The Im-rules, in tum, may be regarded as the rules which govern bringing into a discourse further questions which, on the

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one hand, must be sound if the initial questions are sound and the relevant declarative premises are true . and. on the other hand. which are in each case cognitively useful with respect to the initial questions. The E-rules and Im-rules, like most (if not all) logical rules, do not tell us which of them should be applied in a given cognitive situation. Yet, these rules provide us with tools of reconstruction of erotetic steps that occur in almost every process of reasoning; aspects which so far have been most often regarded as subjective and irregular. This is not to say. however, that the analyzed rules can be applied only that way. Another possibility lies in using them in reconstructions of some schematic trains of thoughts that aim at solving problems of a given kind or even in designing such trains. For an example of such an analysis see Kuipers & Wisniewski (1994). As far as possible applications are concerned. two sub-categories of the rules characterized above seem to be the most interesting: the G-rules and the Ims -rules, Roughly. an E-rule is a G-rule just in case its consequent introduces a generated question ; similarly. an 1m-rule is an Ims-rule if its consequent introduces a strongly implied question. These concepts can be precisely defined as follows: DEFINITION 8.9. An e-rule R is a G-rule iff for each set of d-wffs X and for each question Q such that R(X. Q) . the question Q is generated by the set

X. DEFINITION 8.10 . An ~-rule R is an Ims-rule iff for each set of d-wffs X and for any questions Q. Q, such that R(Q, X, QI). the question QI is strongly implied by the question Q on the basis of the set X. By and large, the G-rules (we may call them question generating rules) are rules for bringing into a discourse risky questions which nevertheless are sound relative to the premises (i.e. are made sound by the premises) and are not logically redundant in relation to them. The Ims-rules, in tum. are rules for bringing into a discourse questions which have all the pragmatic advantages of strongly implied questions. Finally. one can distinguish pure erotetic 1m-rules. We leave their definition to the Reader. 8.5.4. Some remarks about applicability As we have pointed out. it is the logic of questions of a language that provides us with the schemata of valid erotetic arguments as well as with the appropriate erotetic rules. But the key concepts of evocation and erotetic implication in a language (as well as of generation in a language, strong erotetic implication in a language, pure erotetic implication in a language)

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were defined above in terms of me-entailment in the language and thus basically by means of the concept of normal interpretation of the language. It follows that the range of applicability of the schemata of valid erotetic arguments provided by a given logic of questions and of the appropriate erotetic rules can be restricted: the semantic warranties (relative soundness, cognitive usefulness, etc.) offered by the schemata and rules are always restricted to the normal interpretations of the underlying language. The situation here may differ from that known from classical logic. As long as standard arguments are concerned, the schemata of valid arguments (where validity is defined in terms of logical entailment) provided by classical logic guarantee the preservation of truth for each interpretation of the language. If each interpretation of the underlying "erotetic" language is regarded as normal (as it happens, for example, in the case of the language :;eO), the appropriate semantic warranties pertain to all interpretations; if, however, only interpretations satisfying some special conditions are considered normal, the warranties pertain only to interpretations which fulfill these conditions. Such a situation may have practical consequences. For instance, if it is not the case that an inquirer has in his/her disposal the names of all the objects called for or a method of forming their names, the application of some Erule(s) provided by the logic of questions of the language :;e" may lead him/her to a question which is unsound (has no true direct answer) although all his/her declarative premises are nevertheless true.

8.6.

THE IDEA OF THE LOGIC OF QUESTIONS OF A SEMANTICALLY INTERPRETED FORMALIZED LANGUAGE GENERALIZED

SO far we have restricted ourselves to first-order languages enriched with questions in the way presented in Chapter 3 and supplemented with some model-theoretical semantics of the kind characterized in Chapter 4. Let us now conclude by sketching a more general perspective. Assume that we have at our disposal some formalized language L which consists of two parts: assertoric and erotetic. The assertoric part of L is a standard formalized language: a propositional language, a first-order language or some higher-order language. The concept of a declarative well-formed formula (d-wft) of L is defined in the standard way (with respect to the concrete form of the assertoric part of L) ; L includes d-wffs which are sentences (d-wffs with no free individual or higher order variables) and possibly d-wffs which are not sentences, that is, sentential functions. Let us also assume that the assertoric part of the language L is supplemented with some semantics (see below). The vocabulary of the

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erotetic part of L contains the vocabulary of the assertoric part and some expressions which enable us to form questions of L. These questions are not d-wffs, but we here leave open the way of constructing questions of L. This can be done in many ways: according to the general idea presented in Chapter 3, but also, e.g., in the way proposed by Belnap or Kubinski (cf. Chapter 2). Yet, we do assume that the grammar (syntax) of the erotetic part of L is built in such a way that the following general condition is met: to each question of L the grammar of this language assigns an at least two-element set of sentences of L which are direct answers to the question. Direct answers are thus defined syntactically; however, we will also think of them as of the possible and just-sufficient answers to the question. The assertoric part of L is either a propositional language or a first or higher-order language; all of these languages can be supplemented with various semantics. Yet, whatever the semantics of the assertoric part of L would be, it contains some definition of the concept of truth for d-wffs. Of course since we are dealing with logical semantics, truth is defined with regard to some theoretical constructs: models of some kind, or algebraic structures, or matrices, or games, etc. Nevertheless, if some concept of truth for d-wffs is given, the d-wffs can be divided into true and untrue (where "untrue" is not always equal to "false"), and since truth is defined with respect to a class of appropriate constructs, this can be done in many ways. So the accepted semantics determines various partitions of the assertoric part of the language. A partition of the assertoric part of the language L (or a partition of L for short) is an ordered pair < T, U> of sets of d-wffs of L such that: (a) T and U are disjoint, and (b) T u U is the set of all the d-wffs of L (cf. Shoesmith & Smiley (1978); cf. also Chapter 4 of this book). In what follows we only assume that the assertoric part of the language L is supplemented with some semantics which is rich enough to define some (relativized) concept of truth, we do not, however, decide what is the particular form of this semantics. We also assume that the class of admissible partitions of L is defined in some way or another, and that, regardless of the fact how this class is defined, the following necessary conditions hold: (a) each admissible partition is a partition determined by the accepted semantics, (b) the intersection of the first elements of all the admissible partitions is a non-empty set (elements of this intersection may be thought of as valid

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d-wffs).? The class of admissible partitions may be identical with the class of partitions determined by the underlying semantics, but may also be a proper subclass of it. We may think of the set T of an admissible partition < T, U> of L as consisting of truths in the sense of the underlying semantics, but for our present purposes the purely technical concept of truth of a d-wff in a given partition of the language is sufficient. We say that a d-wff A of L is true in a partitionM = < T, U> of L if and only if A E T; otherwise A is said to be untrue in M. If the class of admissible partitions of L is fixed, it is possible to define the semantical concept of multiple-conclusion consequence characterized by the class of admissible partitions of L (cf. Shoesmith & Smiley (1978); cf. also Chapter 4, Section 4.2.2); we shall use here the term "multipleconclusion entailment (me-entailment) in L" for this concept. We say that a set of d-wffs X of L multiple-conclusion entails in L a set of d-wffs Y of L if and only if whenever all the d-wffs in X are true in some admissible partition of L then at least one d-wff in Y is true in this partition of L. The concept of single-conclusion entailment in L as well as other basic semaIitical concepts can be defined accordingly; we use the concept of truth in an admissible partition instead of the concept of truth in a normal interpretation. The concepts of evocation and erotetic implication can now be defined as follows :

7 Some additional comments and examples may help to clarify matters here. There exist rather "strange" partitions of a given language. For example, , where At is the set of all the atomic d-wffs of L and Rem is the set of non-atomic d-wffs of L. is clearly a partition of L, but it can hardly be said that it is an admissible partition. The reason is that cannot be regarded as a partitiondeterminedby the underlyingsemantics. But it need not be the case that each partition determined by the accepted semantics is an admissible one; the admissible partitions can be supposed to satisfy some additional conditions. For example, one can require that each admissible partition of a language whose assertoric part contains the language of Peano's arithmetics (PA for short) should include all the theses of PA among the "truths" of this partition. As a result all the theses of PA will be valid d-wffs of the language; the whole construction, however, has aimedat it. We do not exclude- but also not assume - that admissible partitions of L are defined in this or some similar manner. As far as the languages considered in the previous chapters of this book are concerned, one may think of their admissible partitions as of those determined by normal interpretations. To be more precise, a partition < T, U> of such a languageis admissible iff there is a normal interpretationof it such that the set Tconsists of all the d-wffs which are true in this interpretationand the set U consists of all the remaining d-wffs.

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DEFINITION 8.11. A question Q of L is evoked in L by a set of d-wffs X of L iff (i) the set of direct answers to Q is me-entailed in L by the set X, and (ii) for each direct answer A to Q, the set {A} is not me-entailed in L by the set X. DEFINITION 8.12. A question Q of L implies in L a question QI of L on the basis of a set of d-wffs X of L iff (i) for each direct answer A to Q, the set X u {A} me-entails in L the set of direct answers to QI> and (ii) for each direct answer B to QI there exists a non-empty proper subset Y of the set of direct answers to Q such that Y is me-entailed in L by the set Xu {B}. The concepts of generation in L , strong erotetic implication in L and pure erotetic implication in L can be defined accordingly. By means of the concepts of evocation and erotetic implication defined in the above manner we can define the (general) concepts of question evoking argument , questions implying argument , question evoking rule and question implying rule; their definitions are analogous to that proposed in the previous sections of this chapter . It is obvious that the basic properties of evocation and erotetic implication which enabled us to call question evoking arguments and question implying arguments valid erotetic arguments are preserved in the case of any language of the considered kind. Yet, it is not the case that the analogues of all the theorems proved in the previous chapters of this book hold for evocation and erotetic implication in any language; there are properties of evocation and erotetic implication which are strongly dependent on the underlying semantics. Moreover, the set of admissible partitions can be determined by some nonclassical logic. The concept of the logic of questions of a language L can now be defined analogously as in Section 8.5.1. Each logic of questions of this kind provides us with some schemata of valid erotetic arguments and with some erotetic rules ; each of them can be regarded as a theory of erotetic arguments. Yet, there are worse theories and better theories . It is a matter of further studies which logics of this kind are especially interesting both from the intuitive point of view and from the point of view of possible applications. It is possible that logics of questions based, generally speaking, on some nonclassical underlying "assertoric " logics will appear to be the most interesting. Yet, it seems to be a reasonable strategy to start the exploration of a new territory with the help of well-tried tools: this is why the considerations of this book were basically kept within the limits of first-order

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languages enriched with questions and supplemented with extensional semantics.

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Synthese 74: 65-90.

INDEX OF SYMBOLS

N,34 J , 34 "", 34 - , 34 v,34 /1 ,34 "' ,34 V, 34 3,34

iff, 36 0 ,36 !,46

K,46

=,34, 36 XI ' X 2, .. •,

aI' a2'

34

, 34

PIn, Pt , 34 FIn, F2n , 34 t, t., ,35 u, u" , 35 v, v" , 35 A, A" , 35 B, B I , •• • , 35 C, C" , 35 D ,D" , 35 Axil ••• X.. , 35 P, PI' ,35 R, R" ,35 X, X" , 35 Y, Y" , 35 Z, ZI' ,36 t l ;>! t2 , 36 t l ;>! .. . ;>! tno 36 .4,36 A(x/t) , 36 A(xi/t" .. ., x..ltn ) , 36 0, 36 {YI ' ..., Yn}, 36

{y}, 36 £;, 36 C,36 E,36

E,36 !t,36 ;>!,36 u ,36

n ,36 -, 36

x, 50 J* , 53 J ,,53 kSxi,54 k< xi ,54 la" 54 Cx,,54 (k S )xi, 54 (k.221 Log (9'-). 221

241

INDEX OF SUBJECTS

admissible partitions, 227-228 answers, 42-43 - conclusive, 51 - direct - - to a question of the first kind , 72 - - to an existential question of the second kind,76 - - to an open question of the second kind,77 - - to a general question of the second kind, 78 - - to a categoreally qualified existential quest ion of the second kind, 84-85 - - to an existential question of the third kind, 81-82 - - to a general question of the third kind, 83 - - to a natural language question, 8, 86 - - in Aqvist's theory , 42 , 48-49 - - in Belnap 's theory, 42, 68-69 - - in Harrah's theory , 42, 45 - - in Kubinski's theory, 42, 55-56, 61 - just-complete, 114 - partial, 114-115 - principal possible , 42-43 arising - of a question from a set of declarative sentences, 4-12 - of a question from a question and a set of declarative sentences, 14-24 assertoric operator, 40 atomic formulas, 35 auxiliary d-wffs , 172 auxiliary premises, 18 binary questions, 200 categoreally qualified questions, see questions , categoreally qualified category conditions, 63, 84 category qualifiers, 84 closed terms , 34 compactness - of entailment, 107 - of me-entailment, 109

complete sets, 107 completely contradictory question, 131 completely tautological question, 131 completeness-claims, 68 completeness/incompleteness,erotetic, 9798 compound numerical quest ions, 57-59 conclusive answer, see answer, conclusive conclusiveness conditions, 51 conditional questions , 61-62 - with irrevocable antecedents, 62 - with revocable antecedents, 62 conditional yes-no questions, 74 , 90 conjunction connective, 34 conjunctive questions, 61 , 74, 91-95 consistent set, 106 content of a set of d-wffs , 121 contradiction, 105 contradictory d-wff , 105 core of an interrogative, 48 cut for sets, 112 declarative premises , 210 declarative well-formed formula, 35 desideratum of a question, 49-50 dilution , III direct answers, see answers, direct disjunction connective, 34 disjunctive questions , 61, 91-94 distinctness-claim, 68 domain of an interpretation, 103 d-wff , seedeclarative well-formed formula entailment - in a language, 106 - - in the language ;e', 122 - - in the language ;e", 125 - logical , 106 - multiple-conclusion in a language, 108, 111-113 ,228 - - multiple-conclusion in the language 122 - - multiple-conclusion in the language ;e", 124-126 - regular, 134

z:

INDEX OF SUBJECTS epistem ic operators, 46, 49-50 equipollence of questions , 133 equivalence - connective, 34 - of d-wffs, 107 - of questions, 135 erotetic arguments - of the first kind, 210 - of the second kind, 210 - pure , 210 erotetic constants , 71 erotetic implication - in a language , 171,229 - - in the language ;e', 200 - - in the language ;e" , 203 - pure, 183 - strong , 187 erotetic inferences - of the first kind, 2, 209 - of the second kind, 2, 209 erotetic premise , 210 erotetic rules - of the first kind, 223 - of the second kind, 224 evocation of questions - in a language , 127,229 - - in the language ;e', 141 - - in the language ;e" , 145 - on the ground of a formalized language, 220 evoked question , 128 evoking set, 128 existential quantifier, 34 existential questions, see questions of the second kind, existential, questions of the third kind, existential, and questions, categoreally qualified, existential of the second kind e-arguments, see erotetic arguments e-arguments. see erotetic arguments of the first kind e2-arguments , see erotetic arguments of the second kind e-rules , see erotetic rules of the first kind e2-rules , see erotetic rules of the second kind E-argument, see question evoking argument

243

E'-argument, 214 E-rule, see question evoking rule finite question, 139 first-order language, 35 - with identity, 35 focussed yes-no questions, 74, 89-90 function symbols, 34 general questions, see questions of the second kind, general, and questions of the third kind, general generation of questions - in a language, 156 - - in the language ;e', 167 - - in the language ;e" , 168 - weak, 32 generated question, 156 generating set, 156 G-argument , see question generating argument G-rule, see question generating rule hypothetical yes-no questions , 91 identity symbol, 34 imperative-assertoric analysis ofquestions, 40 imperative-episternic approach to questions, 39 imperative-epistemic translations of interrogatives, 46 imperative operators, 46, 50 implication - connective, 34 - erotetic, see erotetic implication - propositional, see propositional implication implied question , 172 implying question, 172 incomplete set, 107 inconsistent set, 106 infinite question, 139 informativeness , 121 initial question, 18, 210 interpretation function, 102 interpretation of a language , 102 - normal, see normal interpretations interrogative model of inquiry, 32

244

INDEX OF SUBJECTS

interrogative operators, 40, 45-48, 52-61 interrogative quantifiers, 58 interrogatives, 38-42 , 45-48 - compound, 63 - elementary, 63-67 - simple, 63 - whether, see whether-interrogatives - which, see which-interrogatives See also questions 1m-argument, see question-implying argument Ims-argument, 219 just-complete answers , see answers , justcomplete language - J, 34 - J* , 53 - J** , 63 - J" 53-54 - J 2 , 59-60 - J), 63-64 - ::t:, WI - ;;e" , WI - ;;e.', 116 lexical - completeness-claim specifications , 41 , 65 - distinctne ss-claim specifications , 41, 65 - request , 41 , 63 - selection-size specifications , 64-65 - selection, 67 - subject , 41 , 63 - - of a whether-interrogative, 64 - - of a which-interrogative, 64 logical entailment, see entailment , logical maximal presupposition, see presupposition of a question; maximal maximal factual presupposition, see presupposition of a question, maximal factual me-entailment, see multiple-conclusion entailment metalinguistic variables , 35-36 model of a set of d-wffs, 104

multiple-conclusion consequence, 111-112 multiple-conclusion entailmen t, see entailment, multiple-conclusion names , see closed terms negation connective , 34 nominal - alternatives, 64 - categories, 63, 84 non-logical constants, 102 non-reductionism, 40-42 normal interpretations, 104-105 - of the language 5£., 122 - of the language 5£••, 123 normal model of a set of d-wffs , 106 normal questions , 119 numerical questions , see simple numerical questions, and compound numerical questions open questions of the second kind, see questions of the second kind, open overlap , III partial answers, see answers, partial partition of a language, 112, 227 predicate symbols , 34 presupposition of a question , 115 - factual , 118 - maximal , 116 - maximal factual , 118 - prospective, 115 proper questions, 120 propositional implication, 30, 182 propositional questions , 60 pure erotetic argument, see erotetic argument, pure pure erotetic 1m-argument, 219 pure erotetic implication, see erotetic implication, pure quasiformulas , 30, 182 queriables, 64 question evoking argument, 214 question evoking rules , 223-224 question generating argument, 214 question generating rules, 225 question implying argument, 217

INDEX OF SUBJECTS quest ion implying rules, 224 question/conclusion, 210 questions, 37-42 - binary, see binary questions - categoreally qualified, 84-85 - - existential of the second kind , 84 - completely contradictory, see completely contradictory questions - completely tautological , see completely tautological questions - conditional, see conditional questions - conjunctive, see conjunctive questions - finite , see finite question - infinite, see infinite question - normal , see normal questions - numerical, see compound numerical questions and simple numerical questions - of the first kind, 72 - of the second kind - - existential, 76 - - - categoreally qualified, see questions, categoreally qualified, existential of the second kind - - general , 78 - - open , 77 - of the third kind - - existential , 80-81 - - general, 83 - proper, see proper questions - propositional, see propositional questions - regular, see regular questions - risky, see risky questions - safe , see safe questions - self-rhetorical, see self-rhetorical questions - wh, 94-97 - - simple, see simple wh-questions - whether, 44,50,64-67,91 -95 - which, 44, 56-59, 64-67 , 95-97 - yes-no , see yes -no questions, conditional yes-no questions, focussed yes-no questions , hypothetical yes-no questions, simple yes-no questions See also interrogatives Q-selection, see lexical selection

245

reducibility of a question to a set of questions , 196 reductionism - moderate, 38-40 - radical, 37-38 regular entailment , see entailment, regular regular questions, 119 relations between questions, 133-136 relative soundness, 118 requests - abstract, 63 - lexical, see lexical requests risky questions , 113 safe questions, 113 satisfaction, 103-104 self-rhetorical questions , 120 sentence , 35 sentential function, 35 simple interrogatives, 63 simple numerical operators, 53-54 simple numerical questions, 52-54 simple wh-questions, 50 simple yes-no questions, 74, 61 soundness - of a question in an interpretation, 113 - relative , see relative soundness subjects - abstract , 63 - lexical, see lexical subjects synthetic d-wffs, 105 tautologies - of a language, 105 - - of the language 5£., 122 - - of the language 5£••, 125 terms, 34 - closed , see closed terms truth - of a d-wff in an interpretation, 104 - of a d-wff in a partition, 228 - of a question, 30, 113, 182 universal quantifier, 34 universal closure of a d-wff, 36

246

INDEX OF SUBJECfS

validity - of erotetic arguments of the first kind, 212 - of erotetic arguments of the second kind, 217 - of natural-language erotetic arguments, 220 value of a term, 103 valuation (9-valuation), 103 whether- interrogatives, 64-67 which-interrogatives, 64-67 wh-questions, see questions, wh whether-question, see questions, whether which-questions , see questions, which yes-no questions , 87-91, see also conditional yes-no questions, focussed yes-no questions, hypothet ical yes-no questions , simple yes-no questions

INDEX OF NAMES

Ajdukiewicz, K. 2, 40, 42 Apostel, L. 40 Aqvist, L. 31,39,40,43,45-49,79,80, 195 Batog, T. 106 Belnap, N. D. 27,29-31 ,41-42,43,6269, 70, 75, 79, 80, 84, 89, 90, 91, 97, 99, 100, 105, 113, 114 Beth, E. 32 Bolzano, B. 39 Bromberger, S. xiii, 25, 27-28, 31, 113 Buszkowski, W. 112,119

Karttunen, L. 38 Keenan, E. 38 Kiefer, F. 89 Kleiner, S. A. 31 Koj, L. xiii, 40, 89 Kubinski, T. xiii, 28-29, 37, 40-41, 43, 53-62, 70, 75, 79, 80, 91, 97 . 99, 100, 114, 131, 133, 134, 195,222, 227 Kuipers, T. A. F. xiii, 32, 225 Langford, C. H. 39 Leszko, R. 40, 57 Lewis, C. I. 39 Lindenbaum, A. 125

Carlson, L. 31 Cantor, G. 98 Carnap, R. 12, 40, III

Materna, P. 38 Montague, R. 38

Cohen, F. S. 38 Cresswell, M. J. 40 Collingwood, R. G. 27

Peano, G. 104, 228 Prior, A. 26 Prior, M. 26

Finn, V. 38

Reichenbach, H. 40 Ringen, J. xiv

Garrison, J. W. 32 Gentzen, G. III Groenendijk, J. 31,32,38 GOdel, K. 117 Hajicova, E. 89 Halonen, I. 32 Hamblin, C. 38 Harrah, D. xiii, 13,28 ,29,31,39,40, 42, 43-45, 98, 114 Harris, S. 32 Hausser, R. R. 38 Henkin, L. 117 Hintikka, J. 27,31 ,32,39,42,43,4952,79 Hintikka, M. B. 32 Hii, H. 38, 40 Hull, R. 38

Scott, D. 107, III Shoesmith, D. J. 107,109,111 -112,227 Sintonen, M. 32 Smiley, T. J. 107,109,111-112,227 Stahl, G. 38, 42 Steel, Th. B. 27,30,41,90, 114, 123, 182,195 Stokhof, M. 31,32,38 Tarski, A. 134, 222 Tichy, P. 38

Vanderveken, D. 38 W6jcicki, R. 107 Zygmunt, J. 107, III

SYNTIIESE LIBRARY 198. 1 Wolenski, Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X 199. R. W6jcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations. 1988 ISBN 90-277-2785-6 200. 1 Hintikka and M.B. Hintikka, The Logic ofEpistemology and the Epistemology of Logic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6 ISBN 90-277-2808-9 201. E. Agazzi (ed.), Probability in the Sciences. 1988 ISBN 90-277-2814-3 202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical Knowledge. 1989 ISBN 0-7923-0131-5 ISBN 0-7923-0135-8 204. A. Melnick, Space, Time, and Thought in Kant. 1989 205. D.W. Smith, The Circle ofAcquaintance. Perception, Consciousness, and Empathy. 1989 '\ ISBN 0-7923-0252-4 206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of Arthur W. Burks. With his Responses, and with a Bibliography of Burk's Work. 1990 ISBN 0-7923-0325-3 207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study in Husserl, Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4 208. T.C. Meyering, Historical Roots of Cognitive Science. The Rise of a Cognitive Theory of Perception from Antiquity to the Nineteenth Century. 1989 ISBN 0-7923-0349-0 209. P. Kosso, Observability and Observation in Physical Science. 1989 ISBN 0-7923-0389-X 210. 1 Kmita, Essays on the Theory ofScientific Cognition. 1990 ISBN 0-7923-0441-1 211. W. Sieg (ed.), Acting and Reflecting. The InterdisciplinaryTurn in Philosophy. 1990 ISBN 0-7923-0512-4 ISBN 0-7923-0546-9 212. J. Karpinski, Causality in Sociological Research. 1990 213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 1991 ISBN 0-7923-0823-9 214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein's Philosophy of Psychology. 1990 ISBN 0-7923-0850-6 215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and Epistemological Implications of the Work of W.V.O. Quine and of N. Goodman. 1990 ISBN 0-7923-0904-9 216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability. Philosophical Perspectives. 1991 ISBN 0-7923-1046-2 217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the ISBN 0-7923-1322-4 Universe. 1991 218. M. Kusch, Foucault's Strata and Fields. An Investigation into Archaeological and Genealogical Science Studies. 1991 ISBN 0-7923-1462-X 219. C.J. Posy, Kant's Philosophy ofMathematics. Modem Essays. 1992 ISBN 0-7923-1495-6 220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomy and Connectionism. 1992 ISBN 0-7923-1519-7 ISBN 0-7923-1566-9 221. lC. Nyiri, Tradition and Individuality . Essays. 1992 222. R. Howell, Kant's Transcendental Deduction. An Analysis of Main Themes in His Critical Philosophy. 1992 ISBN 0-7923-1571-5

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