Facing the Future
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FACING THE FUTURE Agents and Choices in Our Indeterminist Worl...

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Facing the Future

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FACING THE FUTURE Agents and Choices in Our Indeterminist World

NUEL BELNAP MICHAEL PERLOFF M I N G XU

With Contributions by Paul Bartha Mitchell Green John Horty

OXPORD

UNIVERSITY PRESS

2001

OXFORD U N I V E R S I T Y PRESS Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Pans Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan

Copyright © 2001 by Oxford University Press, Inc. Published by Oxford University Press, Inc 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press Library of Congress Cataloging-in-Publication Data Belnap, Nuel D , 1930Facing the future • agents and choices in our indeterminist world / Nuel Belnap, Michael Perloff, Ming Xu, with contributions by Paul Bartha, Mitchell Green, John Horty. P cm Includes bibliographical references and index. ISBN 0-19-513878-3 1 Agent (Philosophy) 2. Choice (Psychology) 3 Free will and determinism I Perloff, Michael II Xu, Ming III Title B105 A35 2001 128'4—dc21 00-064995

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

Preface This is a book about the causal structure of agency and action. It frames a rigorous theory by using techniques and ideas from philosophical logic, philosophy of language, and metaphysics with a small "m." This theory, which we sometimes call "the theory of agents and choices in branching time," describes agents as facing a future replete with real possibilities, some of which various agents realize by making choices. It is central to our theory that choices and the actions that they ground are radically indeterministic: Before an event of choosing, there are multiple alternatives open to the agent. Furthermore, since the choice is real, so must be the alternatives, and each alternative must be as real as any other. All we can say before the moment of choice is that the agent will make one of the open choices, leaving behind the unchosen alternatives. After the choice, it is correct to say that they were once possible, but are no longer possible. None of the possible choices is a mental or linguistic figment, nor is any a mere ghost image of "the actual choice." Given that the possibilities relevant for action are always possibilities for our future, the theory also refrains from appealing to "possible worlds" other than the one and only world that we all inhabit. These ideas are in some part rooted in common sense. Without help, however, common sense cannot seem to pull them together into a coherent whole. One of our principal aims is to carry out that job by articulating them in a completely intelligible exact theory. The resultant theory of agents and choices in branching time pictures the causal structure of our world as made up of alternative courses of events branching tree-like toward the future. Each branch point represents a choice event or chance event. On the one hand, each continuation from a branch point is individually possible; on the other hand, it is impossible that more than one of these continuations should be realized. If that sounds obscure, we agree: It is, we think, almost impossible to speak clearly and accurately about indeterminism except in the framework of a rigorously fashioned theory such as the one we propose. The theory of agents and choices in branching time is of real but limited interest without its application to understanding the language of action. We propose a certain linguistic form, a "modal connective," as being unusually helpful to anyone who wishes to think deeply about agents and their actions.1 1

Our usage here is common but not universal. A connective is any grammatical construe-

vi

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The form is "a sees to it that Q," where a names an agent and Q holds the place of a sentence; for example, we think of "Ahab sailed the seven seas" as "Ahab saw to it that he sailed the seven seas." The form is so important to our enterprise—all but three of our eighteen chapters are devoted to its study or use—that we give it an abbreviation, "[a stit: Q]" with "stit" as an acronym for "sees to it that." Stit theory explains the meaning of [a stit: Q] in the idiom of philosophical logic. In doing so, stit theory invokes a certain melding of the Prior-Thomason indeterministic semantics with Kaplan's indexical semantics. The combined semantics make [a stit: Q] roughly equivalent to "a prior choice by a guaranteed that Q." The stit idea is many-sided. We explore its grammar, semantics, and proof theory as logicians do. We delight in the fact that stit does not treat actions primarily as "things" to be counted or named. We explore some of the linguistics of stit (especially how and why its status as a modal connective lends itself to usefully complicated constructions), we consider some applications to difficult conceptual problems, and we argue the ability of stit to illuminate agency in a variety of ways. We look at some ways in which stit might be modified or generalized. In all of this, however, we try never to forget the central constraining thought: There is neither action nor agency nor doings without real choices, choices that find their place not merely in the agent's mind, but within the (indeterminist) causal order of our world. To see to it that Q, an agent must make a real choice among objectively incompatible future alternatives. When we say that an event may have many possible but incompatible outcomes, we thereby come down on the side of "hard" indeterminism as against determinism. There is no consensus on these ideas. Since the eighteenth century became understandably awed by the success of Newtonian science, the presumption of determinism has guided most of the philosophical and scientific explorations of both agency and nature. Taking determinism to be delivered by science as an unquestioned "fact," philosophers since Hume and Kant have worked at developing "compatibilist" theories that hold agency, in the guise of moral responsibility, to be compatible with what James called the block universe. Such theories have often taken possibilities as unreal: as arising from the mind, or from social practices, or from language—for example from consistency with the bits of language called "scientific laws." Our contrasting indeterministic presumption is eloquently expressed by the eminent paleontologist Stephen Jay Gould. I don't think that any deeper or more important principle pervades nature, and lies at the heart of all historical sequences, than this central but underappreciated notion of "contingency"—the great and tion that maps one sentence (or several sentences) into another sentence. For example, when you put sentences (i) "Ahab is captain" and (ii) "Ishmael is not captain" into the blanks of " and ," the result is another sentence, (iii) "Ahab is captain and Ishmael is not captain." That makes the "and" construction a connective. A truth functional connective is one like " and ": If for example you know whether each of (i) and (ii) is true, then you automatically know whether (iii) is true. A modal connective is defined negatively as being one that is not truth functional.

Preface

vii

liberating truth that tiny inputs, virtually invisible and risibly impotent in appearance at the outset, can cause history to cascade down any route in a vast array of entirely different pathways. (Gould 1999, p. 30) This book neither argues for indeterminism nor tries to pick holes in arguments for compatibilism. Our project assumes the indeterminism of the causal order in which agency is embedded, it assumes that actions are based on real choices, and it assumes that choices are therefore not predetermined. Our goal is not to persuade, but to make these ideas intelligible. Although numerous philosophers share our general point of view, not many exact theories share these assumptions and aims. Our strategy is to concentrate almost exclusively on the objectively causal side of indeterminism and agency, which already presents enough difficulties without bringing in noncausal concepts. We therefore lay aside many deeply important aspects of agency and choice that involve intentions, propositional attitudes, or other mental phenomena. We look for ways in which applications of stit theory can engender a better understanding of agency. Seven examples: (i) an analysis of refraining that clarifies how it can be both a doing and a not-doing; (ii) an analysis of imperatives that emphasizes their agentive content; (iii) an extended treatment of deontic logic that insists that obligations and permissions (a) are directed to agents capable of making choices, and (b) are embedded in the indeterministic causal order of our world; (iv) fresh analyses of promising and of assertion, analyses that argue the unwisdom of the doctrine that among all the objective possibilities, a unique course of events constitutes the one and only "actual future"; (v) an exploration of the causal side of the requirement on action that the agent "could have done otherwise"; (vi) the causal structure of joint agency; and (vii) a generalization of stit theory to strategies considered from a causal point of view. In sum: Holding the extra-mental and extra-linguistic status of incompatible possibilities as given, and supposing that the future sometimes depends upon an agent's choices among incompatible options, we offer a tense-modal theory intended to describe some causal aspects of agency in our indeterministic world. Guidance on reading this book. In the spirit of Carnap, each chapter begins with some introductory remarks in order to permit easy skipping of topics not of present interest to the reader. We append the following largescale structural notes as additional guidance. The book divides into six parts of varying degrees of technicality, followed by an appendix. Each of these may be characterized as follows. Part I introduces stit theory. We intend the chapters in this part to be accessible to all those with an interest in our topic. Portions of chapter 2 and chapter 5 do, however, involve willingness to put up with some elementary logical constructions, and chapter 2 offers a brief explanation of the theory of agents and choices in branching time that underlies stit theory. In this part we introduce key grammatical and semantic features of stit, including the distinction

viii

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of the deliberative stit from the achievement stit. Largely concentrating on the achievement stit, the various chapters of part I suggest applications with the help of many pictures, make comparisons with some other work on agency, and, beginning with Anselm's work in 1100, give a little history of the modal logic of agency. This part also contains applications of stit theory to imperatives and to promising. Part II supplies precisely and in detail the nuts and bolts—or, more aptly, roots and branches—of the theoretical structure that supports our account of agents, actions, and our indeterminist world. The three chapters of this part are foundational in character, and involve substantially more rigor, though not much more mathematics. They stress conceptual analysis rather than theoremproving. This part more than any other focuses on the problems faced by any indeterministic theory. Here we argue against the beguiling but harmful doctrine of "the actual future," which says that among the many courses of events that might come to pass, there now exists a privileged such course that will actually do so. This foundational part examines, postulate by postulate, the theory of agents and choices in branching time, and explains in detail the semantic subtleties required of a language spoken in an indeterminist world. Part III offers two applications of the achievement stit: One chapter aims to illuminate the dark idea of "could have done otherwise," and another considers the causal aspects of joint agency. These chapters are a little more technical. Part IV is of the same level of technicality as part III: It offers applications of the deliberative stit, chiefly to help in elaborating such deontic concepts as obligation and permission, which, we believe, are in much need of a theory of agency. Part V uses the already-established theoretical structure of agents and choices in branching time in order to develop an austere (causal, not normative) account of strategies as a kind of generalization of stit. One chapter in this part connects our theory of strategies to Thomason's deontic kinematics. Part V proves a theorem or two, though much of it is, again, conceptual analysis. Part VI provides the technical backbone of stit theory, including proofs of decidability, soundness, and completeness. The chapters of this part are required reading for those who wish to investigate or develop the logical and mathematical properties of theories of agency similar to ours. The appendix gathers, for easy reference, most of the various theses, structures, postulates, definitions, semantic ideas, and systems that are introduced elsewhere and are employed throughout the book. We use boldface to refer to certain of these items: Look in the appendix for its sections §l-§9, for stit theses Thesis 1-Thesis 6, for postulates Post. 1-Post. 10, for definitions Def. 1-Def. 20, and for axiomatic concepts Ax. Conc. 1-Ax. Conc. 3. Although in each section of this book we feel free to refer to any other section, it may be useful to indicate the following dependency-structure among the various parts. Parts I (introduction to stit), II (foundations of indeterminism), and VI (proofs and models) are almost entirely independent. The reader's primary interests may therefore be allowed to determine with which of parts I, II, or VI it is best to begin. Part III (applications of the achievement stit) and part IV

Preface

ix

(applications of the deliberative stit) are mutually independent, whereas each presupposes familiarity with part I. Part V (strategies) requires chapter 2 of part I as minimum background.

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Acknowledgments Our contributors deserve special thanks for giving us permission to use their materials: P. Bartha for contributing chapter 11 and for co-authoring chapter 12; M. Green for co-authoring chapter 6; and J. Horty for co-authoring §2A of chapter 2, which has its source in Horty and Belnap 1995. This essay is also used in a substantially different way in Horty 2001, a book elaborating the normative aspect of choice. Horty 2001 should be consulted by every reader of this book. We add that each contributor has also helped in other ways too numerous to count. Many other people have helped in the research reported in this book. We apologize for the simple alphabetical listings that we offer as sincere but inadequate thanks. The following assisted with the development of ideas and with the preparation of the manuscript during their respective terms as Alan Ross Anderson fellows: M. Allen, D. Bruckner, C. Campbell (who caught many dozens of errors and infelicities during a final proof-reading), K. Davey, U. Ergun (who created many of the figures), C. Hitchcock, C. Jones, J. MacFarlane (who produced the initial draft of the index), J. Roberts, L. Shapiro, A. Staub, V. Venkatachalam (who was responsible for the initial draft of the bibliography), and M. Weiner. In addition, a number of others have helped over the years, especially by commenting on or offering us fresh information in connection with the separate articles on which this book is based: G. Antonelli, L. Aqvist, C. Bicchieri, R. Brandom, M. Brown, D. Davidson, D. Elgesem, R. Gale, A. Gupta, D. Henry, I. Humberstone, D. Kaplan, M. Lange, W. Lycan, D. Makinson, J. Moore, R. Neta, I. Porn, K. Schlechta, K. Segerberg, S. Sterrett, J. Thomson, D. Turner, D. Vanderveken, F. von Kutschera, D. Walton, H. Wansing, and anonymous referees for several journals. Finally, we thank two anonymous OUP referees for encouraging us to turn a heap of articles into a book.

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Contents I

Introduction to stit 1 Stit: A canonical form for agentives 1A Agentives 1B Stit: Simple cases 1C Grammar of the modal logic of agency 1D Mini-history of the modal logic of agency 1E Conclusion and summary

. . . . .

3 5 9 14 18 26

2 Stit: Introductory theory, semantics, and applications 2A Theory and semantics: The two stits 2B Applications of stit, with many pictures

. .

28 29 39

3 Small yet important differences from earlier proposals 3A Von Wright 3B Chisholm 3C Kenny 3D Castaneda 3E Davidson 3F Conclusion

. . . . . .

59 60 65 68 74 78 81

4 Stit and the imperative 4A The theory of fiats 4B Ross's paradox and stit 4C Chellas's theory 4D Agentive constructions 4E Negations of imperatives 4F The many varieties of imperatives 4G Embedding imperatives 4H Conclusion

. . . . . . . .

5 Promising: Stits, claims, and strategies 5A From stit to promising 5B From RR to promising 5C Strategic content of promises and word-givings

97 . 98 . 107 . 116

82

82 84 85 87 89 92 94 96

xiv

II

Contents

Foundations of indeterminism

6 Indeterminism and the Thin Red Line 6A Preliminary considerations 6B Parameters of truth 6C The assertion problem 6D The Thin Red Line 6E Time's winged chariot hurries near

. . . . . . , .

133 134 141 156 160 170

7 Agents and choices in branching time with instants 7A Theory of branching time 7B Theoretical reflections on indeterminism 7C Theory of agents and choices 7D Domain

. . . .

. . . .

177 177 203 210 219

8 Indexical semantics under indeterminism 8A Sources 8B Structure parameters: The "world" of the speakers 8C Interpretation and model: The "language" of the speakers . 8D Points of evaluation, and policies 8E Generic semantic ideas 8F Semantics for stit-free locutions 8G Clauses for stit functors

, . . . . . .

220 . 221 . 226 . 227 . 228 . 234 . 239 . 247

9 Could have done otherwise 9A Could have been and might have been 9B Could have done and might have done 9C Might have been otherwise 9D Might not have done it 9E Could not have avoided doing 9F Could have prevented 9G Could have refrained 9H Might have refrained 9I Had available a strategy for not doing 9J Summary

. . . . . . . . . .

255 . 257 . 257 . 259 . 260 . 261 . 262 . 263 . 265 . 268 . 269

10 Multiple and joint agency 10A Preliminary facts 10B Other-agent nested stits 10C Joint agency: Plain and strict 10D Other-agent nested joint stits

. . . .

271 . 272 . 273 . 281 . 290

III

Applications of the achievement stit

Contents

IV

xv

Applications of the deliberative stit

11 Conditional obligation, deontic paradoxes, and stit 11A Technical preliminaries 11B Semantics of obligation 11C Completeness 11D Conditional obligation 11E Oc-statements versus cO-statements 11F The Good Samaritan 11G Contrary-to-duty obligations 11H Problems with the proposed semantics of obligation . . . .

295

12 Marcus and the problem of nested deontic modalities 12A The parking problem 12B The form of obligations 12C The Anderson/dstit simplification 12D The form of prohibitions 12E Generalized prohibitions 12F Generalization on agents 12G Temporal generalization 12H The outer ought

318 318 319 321 322 325 333 335 335

V

296 298 303 304 306 309 312 314

Strategies

13 An austere theory of strategies 13A Nature of austere strategics 13B Review of choices in branching histories 13C Elementary theory of strategies 13D Favoring 13E Application to finding a strategy for inaction

. . . . .

14 Deontic kinematics and austere strategics 14A Basic concepts 14B From Thomason's deontic kinematics to austere strategics 14C From austere strategics to Thomason's deontic kinematics 14D Remarks

. . . .

VI

341 342 344 345 356 359 364

365 368 370 376

Proofs and models

15 Decidability of one-agent achievement-stit theory with refref 15A Preliminaries 15B Companions 15C Soundness: Validity of refref equivalence 15D Companion sets

381

. . . .

382 385 390 392

xvi

Contents 15E Alternatives and counters 15F Semi-ref-counters 15G Completeness and finite model property

. . 397

. . 401 . . 408

16 On the basic one-agent achievement-stit theory 16A Preliminaries 16B Soundness 16C Companion sets and their alternatives 16D Construction of preliminary structures 16E Completeness

. . . . .

415 . 415 . 417 . 419 . 424 . 428

17 Decidability of many-agent deliberative-stit theories 17A Preliminaries 17B Soundness 17C Completeness and compactness 17D Finite model property

. . . .

. . . .

18 Doing and refraining from refraining 18A Preliminaries 18B Main results

451 . . 452 . . 454

Appendix: Lists for reference 1 Stit theses: Thesis 1-Thesis 6 2 Structures 3 BT + I + AC postulates: Post. 1-Post. 10 4 Branching-time-with-instants definitions: Def. 1-Def. 9 . . 5 Agent-choice definitions: Def. 10-Def. 14 6 Basic semantic definitions: Def. 15-Def. 16 7 Derivative semantic definitions: Def. 17-Def. 20 8 Grammar 9 Axiomatics concepts: Ax. Conc. 1, Ax. Conc. 2, and Ax. Conc. 3

435 435 439 441 445

459

. . . . . . . .

. 459 . 460 . 461 . 463 . 466 . 468 . 469 . 471

. . 472

Bibliography

475

Index

483

Part I

Introduction to stit

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1 Stit: A canonical form for agent ives Among the topics of discussion in this world, none are more common than those concerning the achievements and refrainings, obligations and prohibitions, successes and failures of the agents with whom we share a common space.* "What happened when so-and-so did that?" we ask, or "What should have been done?" Biographies and narrative histories, which form a sizable segment of our reading materials, have as their central concern the doings of agents, their obligations and prohibitions, the outcomes of their choices, and the range of things from which they refrain. Philosophers have long sought to find a distinction between those sentences which attribute agency and those that do not in the verbal configurations we commonly use to talk about such matters. If such a distinction existed it should be a relatively simple matter to uncover some general principle in everyday speech to differentiate Joshua fit the battle of Jericho from Joshua survived the battle of Jericho. We find, to our dismay, that we are no closer now to a linguistic litmus test for agency than was Aristotle. After all, it would have been of some importance for Aristotle's peers to decide between an agentive interpretation of Alexander succeeded to the throne of Macedonia and a non-agentive interpretation. For if Alexander was agentive in the matter of his becoming king, then there was a prior choice of his which led directly to * With the kind permission of Theoma, Belnap and Perloff 1988 is the basis of each section of this chapter except §1D. That section is drawn from Belnap 1991, for the use of which we thank the International Phenomenological Society.

3

4

Introduction to stit

that outcome, and he was likely guilty of regicide; while if he was not agentive in the matter of becoming king, then there was no choice of his that guaranteed his succession to the throne. J. L. Austin said that "The beginning of sense, not to say wisdom, is to realize that 'doing an action', as used in philosophy, is a highly abstract expression—it is a stand-in used in the place of any (or almost any) verb with a personal subject ..." (Austin 1961, p. 126). In his essay he tried to throw light on the question of "doing an action" by looking at the range of cases in which excuses are offered both in everyday usage and in the law, and to arrive at a proper vocabulary for action by "induction" on the proper uses of words. Many years have passed, the lesson has been learned, and it is time for philosophy to go beyond the mere beginnings of sense and progress toward a deeper understanding of an agent doing an action. How then should we proceed? The Austin legacy is of course a particularly rich one that is being carried on in a variety of ways. One way attempts to explain certain facts about language in terms of propositional content and illocutionary forces: "speech act theory" as represented for example by Searle and Vanderveken. That is not our program, as we make clear in many places hereafter, suggesting that a theory of "speech acts" should begin with a prior theory of acts. Accordingly we follow in the wake of Austin's suggestion that when faced with the question of the meaning of a word or a term such as "doing an action," we should "reply by explaining its syntactics and demonstrating its semantics" (Austin 1961, p. 28). A suggestion of this book is that the next step in the progression toward greater sense and wisdom is to have available the sort of clean and well-honed linguistic resource that Austin, and other philosophers, have realized to be necessary. We think that the most promising path to a deeper understanding of an agent making a choice among alternatives that lead to action is to augment our philosophical language with a class of sentences whose fundamental syntactic and semantic structures are so well designed and easily understood that they illuminate not only their own operations but the nature and structure of the linguistic settings in which they function. An example of a doing-an-action sentence that Austin might have had in mind is Ahab sailed in search of Moby Dick.

(1)

It has a personal subject, "Ahab," and an action-like verb, "sailed," and seems to be describing an action in Austin's deliberately wide sense. We take (1) not only to be true, but to be agentive for Ahab, for Ahab's sailing in search of Moby Dick was a direct result of a choice he made among alternatives available to him. On the other hand, although the perfectly ordinary sentence, The Pequod sailed in search of Moby Dick

(2)

is surely true, and though we may be hard pressed to say exactly why, we are not hard pressed to say that it is not agentive: It does not even have a personal subject. Consider now

1. Stit: A canonical form for agentives Ishmael sailed in search of Moby Dick.

5 (3)

Ishmael signed on as a member of the ship's company in total ignorance of Ahab's vengeful purpose; are we then to say that (3) is agentive for Ishmael, but false, on the consideration that both "Ahab" and "Ishmael" are appropriately "personal" subjects? Or is (3) true on analogy with the example about the Pequod? English is not to be trusted in these waters. Since English fails to serve us as an adequate pilot, what we want is a resource sensitive to the difference between those cases using an action-like verb in which agency is ascribed, and those cases in which there is merely the appearance of agency. With decent Austinian respect and regard for the structure of the language, we propose, insofar as is possible, to locate such a resource within English itself, a resource that will also allow us to become clearer about the relation between agentive sentences in their declarative uses and agentives in their imperative uses; for surely Mr. Starbuck, hand me yon top-maul, and

(4)

Mr. Starbuck handed Ahab the top-maul

(5)

are, in context, more than accidentally related. In fact, except for contextdetermined indexicals, they are the same agentive sentence in two different appearances: (4) is an imperative issued to Mr. Starbuck by Ahab, while (5) is an agentive declarative whose truth or falsity is intimately connected to the satisfaction or failure of satisfaction of the imperative.

1A

Agentives

Let us accordingly begin with the following convention: The agentive form that we are about to introduce shall be set off with square brackets [ ... ]. It shall have two open places as indicated, the first to take an agent term, the second to take a declarative sentence (the declarative complement of the new form). The point about the second open place is nontrivial: Having noted that declarative sentences can either ascribe agency or not, we specifically include as possible declarative complements for the second open place both those sentences that do ascribe agency and those that do not ascribe agency. The resulting squarebracket sentence is to say that the proposition expressed by the declarative complement is guaranteed true by a prior choice of the agent. So [the carpenter ... Ahab has a new snow white ivory leg] is to be agentive for the carpenter, and is to say that he is the agent in the matter of Ahab's having a new snow white ivory leg. With what verb or verb phrase shall we replace the ellipsis in that sentential form (it was after all only elliptical)?1 Among the candidate English verb phrases that history suggests are the following: 1

Numerous other philosophers have considered this or similar questions, including at least Anderson, Aqvist, Bennett, Chellas, Chisholm, Danto, Davidson, Fitch, Hamblin, Hilpinen,

6

Introduction to stit i. brings it about that ii. makes it the case that Hi. causes it to be the case that iv. is responsible for the fact that v. lets it be the case that vi. allows it to be the case that vii. takes steps in order that

viii. behaves so that in consequence ix. sees to it that As you can see, these are all grammatically acceptable; but items (i)-(iii) suggest that causal processes are either at work or hovering in the background, whereas we wish to de-emphasize this suggestion. Items (iv)-(vi) give the impression of moral judgment and ethical responsibility, and whereas those are important ideas (none more so), it is inappropriate to build them into the foundation of this enterprise. Items (vii) and (viii) are closer to the mark in their straightforward association of an agent and something made true by the agent, but even these candidates might suggest to some ears that we want you to concentrate on a second, prior action performed by an agent. The English verb form (ix), sees to it that, has, to our ears at least, fewer of the obvious defects of the others, and sees to it that has the definite advantage of suggesting alternatives and choices. So our preferred sentence form is [a sees to it that Q], which for logical emphasis we abbreviate as [a stit: Q], referring to such a sentence as a stit sentence. We intend that "Q" here take up a sentential position, as is proper for a complement of "sees to it that." (Note: In early chapters we use "Q," whereas in later chapters we use "A" or "5," and so on, with exactly the same intention. The slight difference in notation derives from the diversity of the origins of the various chapters, and in order to avoid fresh mistakes, we have thought it best not to enforce uniformity in this particular regard.) It is a consequence of what has just been said that a sentence [a stit: Q] with a as its subject is always agentive for a. Let us enter this important idea as the "agentiveness of stit thesis." Hohfeld, Humberstone, the Kangers, Kenny, Lindahl, Makinson, Melden, Needham, Porn, Talja, Thalberg, and von Wright. Bibliographic access to the literature with which we are familiar can be gained through Thalberg 1972, Aqvist 1984, Bennett 1988, Makinson 1986, and the bibliography of this book. Although we do not undertake to discuss all of these contributions, we do discuss some of them, especially in §1D and chapter 3. Above all, we should be remiss if we failed to note that NB and MP were each introduced to this family of ideas by our teacher, Alan Ross Anderson—see Anderson 1962 and Anderson 1970.

7

1. Stit: A canonical form for agentives AGENTIVENESS OF STIT THESIS. (Stit thesis. Reference: Thesis 1) Q] is always agentive for a.

[a stit:

This is the first of six "stit theses"; you can find a list in §1 of the appendix. By this thesis we intend the descriptive claim that the English sentence "a sees to it that Q," which we abbreviate as [a stit: Q], always describes a as an agent carrying out an action. The complement, Q, may be agentive for a or not. This almost equally important claim is our "stit complement thesis." STIT COMPLEMENT THESIS. (Stit thesis. Reference: Thesis 2) grammatical and meaningful for any arbitrary sentence Q.

[a stit: Q] is

The two theses together emphasize the double aspect of stit: Q in [a stit: Q] may be any sentence at all, but [a stit: Q] is always agentive for a—perhaps difficult to interpret, perhaps contradictory, perhaps merely false, but always agentive for a. So [Ishmael stit: Ishmael sailed on board the Pequod] is agentive for Ishmael and a true sentence, while [Ishmael stit: Ishmael refused to share a room at the Spouter Inn] is equally agentive for Ishmael, but false. The stit sentence [Queequeg stit: the Pequod is fitted out for its voyage], which has a non-agentive as its declarative complement, is agentive for Queequeg but false, for it is not due to Queequeg, but to Peleg, that the Pequod is fitted out for its voyage. At this point you may fairly ask how stit treats other agentive sentences, that is, sentences other than stit sentences that ascribe an action to an agent. Our answer is that if such a sentence is truly agentive, then it can appropriately be paraphrased as a stit sentence. Furthermore, if it is not truly agentive, then to attempt such a paraphrase will tend to reveal this fact. We are therefore led to advance the "stit paraphrase thesis." STIT PARAPHRASE THESIS. (Stit thesis. Reference: Thesis 3) Q is agentive for a just in case Q may usefully be paraphrased as [a stit: Q]. Therefore, up to an approximation, Q is agentive for a whenever Q [a stit: Q}. Given this thesis, we may use the equivalence Q [a stit: Q] as a helpful test. Contrariwise, the helpfulness of the test tends to support the thesis, rough as it is. In any case, we intend this strategy, clarification by paraphrase, as neither definitional nor reductive. It is rather an attempt to isolate, by way of a canonical form, a particular set of English sentences in order to study them more closely as they interact with each other and with other parts of language

8

Introduction to stit

in different linguistic environments, just as a biologist might tag a particular organism to follow its activities as it interacts with members of its own species and with other species in various physical environments. Analogically, most of us are comfortable in saying that an English sentence is conditional if it can be paraphrased into the canonical form "if Q1 then Q2," and then revealing that the sentence The Prophet will tell Queequeg and Ishmael about Ahab provided they stop a minute is in fact conditional by paraphrasing it as If Queequeg and Ishmael stop a minute then the Prophet will tell them about Ahab. EXAMPLE. The sentence (1) is agentive since it can be paraphrased by prefixing an "Ahab saw to it that": [Ahab stit: (1)].

(6)

By Thesis 3, then, since (1) "Ahab sailed in search of Moby Dick" is correctly paraphrased as (6) [Ahab stit: Ahab sailed in search of Moby Dick], both the original and its longer transform are not only true but agentive for Ahab. EXAMPLE. It seems a travesty to paraphrase the sentence about the Pequod, (2), by [Pequod stit: (2)]. Indeed, on their ordinary readings, (2) seems true, while its attempted stit paraphrase seems false. Furthermore, it appears to us that this difference helps us see the non-agentiveness of (2)—helps us to see, that is, that (2) does not ascribe an action to an agent and is therefore non-agentive. EXAMPLE. It strikes us that (3) has two quite ordinary readings, an agentive reading in analogy with the Ahab example, (1), and a non-agentive reading in analogy with the Pequod example, (2). For many purposes the difference is irrelevant or at least of negligible importance; but if one is discussing serious matters of, for example, ethical responsibility, then the difference counts. It matters in such cases whether or not (3), as employed in a particular discussion, can be paraphrased by [Ishmael stit: (3)]. If the non-agentive reading of (3) seems forced, take others. Ishmael sailed over the seas

(7)

is presumably agentive, while Ishmael sailed over the side of the Pequod,

(8)

in its most natural use, would not be agentive. Our agreement on these verdicts tracks our agreement on the effort to paraphrase with stit: It seems right to claim that (7) [Ishmael stit: (7)], and it seems wrong to claim that (8) [Ishmael stit: (8)]. Observe also that there are certainly circumstances under which one would take

1. Stit: A canonical form for agentives [Ishmael stit: (8)]

9 (9)

as true (perhaps he deliberately saw to it that he went flying over the side). Nevertheless, this possibility should not be confused with passing the stit paraphrase test for agentiveness. The truth of the stit sentence (9) does not imply that its complement, (8), is agentive, nor that the stit sentence is equivalent to its complement. Only when (8) is taken as meaning the same as (9) is it plausible to take (8) as ascribing agency to Ishmael.

1B

Stit: Simple cases

We now use the strategy of clarification by paraphrase to demonstrate how stit sentences can help us better to know our way around in some areas in which agency counts. Consider for example Ahab found the White Whale.

(10)

Certainly it is true, but is it agentive? In order to answer we have to address ourselves specifically to the question: Did Ahab see to it that Ahab found the White Whale? As we have seen already, Ahab was agentive in his search for Moby Dick, but was he agentive in finding it? We think not. Although Ahab was a participant in guiding the Pequod to the ultimate outcome, and the major participant, his actually finding the White Whale was due in large part to chance, to natural forces beyond his control. "Time and tide flow wide," remarks Mr. Starbuck. "The hated fish has the round watery world to swim in." Thus, though (10) is true, it is false when paraphrased as [Ahab stit: Ahab found the White Whale]. Therefore, insofar as (10) is true, it is non-agentive; and insofar as it is agentive, it is not true. Consider the following pair: Queequeg struck home with his harpoon.

(11)

Queequeg's harpoon struck home.

(12)

Some careful speakers of English might always use (11) as an agentive and (12) as a non-agentive, whereas most of us are liable to use them interchangeably, sometimes in one way and sometimes the other. But [Queequeg stit: Queequeg struck home with his harpoon]

(13)

[Queequeg stit: Queequeg's harpoon struck home]

(14)

and

come to much the same thing. Such differences between (11) and (12) as there are disappear when they are embedded into stit contexts; the stit sentences (13) and (14), regardless of the uncertainties of their complements, are transparent with respect to agency.

10

Introduction to stit

1B.1

Imperatives

"Clear away the boats! Luff!" cried Ahab. Treatment of such imperatives is by no means tangential to our concerns. We endorse the view that C. H. Hamblin expresses in his masterful study: Imperatives are not only among the most frequent of utterances; they are also, surely, among the most important. If the human race had to choose between being barred from uttering imperatives and being barred from uttering anything else, there is no doubt which it should prefer. (Hamblin 1987, p. 2) Hamblin additionally reports that a full twenty percent of Shakespeare is in the imperative mood. (We happily acknowledge that a study of Hamblin's book set us under way and helped us avoid some threatening reefs.) Imperatives, in our usage, constitute a grammatical category. Following an established tradition, however, we think of each use of an imperative as having a force and a content. With regard to force, Ahab's imperatives may have been orders or commands, which many think the only possibilities; but Ahab might instead have been inviting, requesting, suggesting, advising, ... the helmsman to luff. Putting force to one side, however, we are after the content, about which stit theory has a definite and (as it happens) helpful opinion that we sum up in the "imperative content thesis." IMPERATIVE CONTENT THESIS. (Stit thesis. Reference: Thesis 4) Regardless of its force on an occasion of use, the content of every imperative is agentive. For example, Luff! can have its content represented as The helmsman luffs, which in turn, since it is agentive, can be paraphrased as The helmsman sees to it that he luffs. Thus, Luff! can be paraphrased as Helmsman, see to it that you luff! In this case the application of our thesis is easy because The helmsman luffs, which looks to be the most plausible content for the imperative, is already agentive. Still, there is more to be learned. To luff is to see to it that the bow of the boat is heading directly into the wind. Accordingly, the content of the imperative Luff! can be put into the canonical form [Helmsman stit: the helmsman luffs]

(15)

or equivalently, [Helmsman stit: the boat is headed into the wind].

(16)

The two stit sentences are equivalent, but while the complement of the former is agentive, the complement of the latter doesn't mention the agent at all.2 Unlike Luff! the imperative 2

"Complement" versus "content" sometimes sounds confusing. The (grammatical) complement of [a stit: Q] is Q. Its (semantic) content is an agentive proposition.

1. Stit: A canonical form for agentives Be on deck at dawn!

11 (17)

does not show its content so obviously; the stit apparatus, however, increases in value as problems become more complex. Consider (17) as addressed to the helmsman. The obviously non-agentive The helmsman is on deck by dawn

(18)

cannot, by our thesis, represent its content; we need an agentive form, and fortunately there is already something at hand: the stit sentence, [The helmsman stit: the helmsman is on deck at dawn]. This, in turn, can be transformed back into the imperative Helmsman, see to it that you are on deck at dawn, which we take to be an accurate paraphrase of the original imperative (17). This is important because you might be tempted to think that (18), inasmuch as it displays an agent as its subject, is the content of the imperative (17); but there are many ways in which it might be true that the helmsman is on deck at dawn, while false that he sees to it that he is. Because the content of every imperative must be agentive, a helpful paraphrase will not merely have the name of the addressee as its subject: That name will appear as subject of a sentence of the form [a stit: Q}. We therefore have the following: Since every imperative has agentive content, and since every agentive can be paraphrased as a stit sentence, it must be that every imperative will have a stit paraphrase of the form [a stit: Q]—sometimes with Q an agentive (hence a possible content of an imperative) and sometimes not. These brief remarks do not suffice. We return twice to the topic of imperatives, once in §1C, and again in chapter 4.

1B.2

Deontic contexts

The traditional account (see, e.g., Aqvist 1984) has it that deontic statements have one of the forms

Oblg:Q,i.e. it is obligatory that Q, Frbn:Q, i.e. it is forbidden that Q, or

Perm:Q, i.e. it is Permitted that Q where Q is a declarative. 1-1 REMARK. (On notation) The standard deontic notation is Op, Fp, and Pp. We depart from the standard for good reason: We shall in the course of this book deal with many other modalities, several of which would have an equal right to single letters such as 0, F, and P. We therefore use abbreviations for deontic and other modalities that we hope are short enough for the eye to take

12

Introduction to stit

in at a glance and long enough for the mind to remember. (It helps that we do not often use these longer expressions in the course of complicated calculations in which extreme brevity is a decided advantage.) Among its many virtues, this approach allows such great latitude that it can be connected to every possible action by any agent. On the other hand, agents have in this grammar no distinguished place; if invoked at all, the agent's name is only an accidental feature of the declarative complement. It is an easy mistake to think that Oblg:Q says more than it actually does say. It does not say, for example, who, if anyone, is obliged to see to it that Q. Consider It is forbidden that cooks be on the bridge. It is tempting to read this as though it imposes a prohibition on the cook, but it doesn't, as we can see from the analogous It is forbidden that dogs be on the bridge; in either case the sentence form is not fit to tell us who is to see to it that no cooks (or dogs) are on the bridge. While a deontic language of declarativesentence complements may be satisfactory for impersonal oughts, many within the tradition have seen that agents need to be treated with more care. Perhaps foremost among those who have argued that these standard alternatives are inadequate is Castaneda, for example, in Castaneda 1974 (see additional discussion in §3D). Some deontic logicians have suggested the step of changing the grammar so that Oblg:, Perm:, and Frbn: are taken to be adjectives modifying actionnominals. The fundamental forms are then taken as Oblg:a, i.e., a is obligatory, Frbn:a, i.e., a is forbidden, or Perm:a, i.e., a is permitted, where a stands in place of a term designating an action. So Oblg:a might be instanced by Sailing is obligatory, or Frbn:a by Bringing the boat into the wind is forbidden. Since actions are always the actions of agents, this step is in the right direction, but it still fails appropriately to recognize the agent. Furthermore, there is a considerable loss in expressive power, since clearly understood declarative sentences, Q, are so much easier to come by than are clearly understood action-nominals, a. In order to regain the headway lost in the move from sentences to action-nominals, some have tried adding negative doings such as not-sailings, or disjunctive doings such as luffing-or-flensing; but even with these additions, there remain two features missing from the overhauled model: first, the flexibility that follows from permitting an arbitrary declarative within the scope of a deontic statement, and second, the grammatical means to identify and keep track of the agent. We propose a combination having the strengths of both the declarativesentence complement plan, Oblg:Q, and the action-nominal complement plan,

1. Stit: A canonical form for agentives

13

Oblg:a. We propose to focus on deontic statements that have only agentives for their complements and thus can always be paraphrased into one of the following forms. Oblg:[a stit: Q], i.e., a is obligated to see to it that Q Frbn:[a stit: Q}, i.e., a is forbidden to see to it that Q Perm:[a stit: Q}, i.e., a is permitted to see to it that Q Though Perm:[a stit: Q], for example, is not agentive (in our technical sense), we intend it as quasi-agentive in the loose sense that it involves both an agent and an agentive, and in the stricter sense that like an agentive, it has the agent itself as a recoverable part of its intension. Our proposal calls, then, for a deontic language enriched by the restriction that the complements of obligation, permission, and prohibition be limited to stit sentences, a proposal that forms part of the "restricted complement thesis." RESTRICTED COMPLEMENT THESIS. (Stit thesis. Reference: Thesis 5) A variety of constructions concerned with agents and agency—including deontic statements, imperatives, and statements of intention, among others—must take agentives as their complements. Recall that (i) stit sentences always express an action, (ii) stit sentences never lose or misplace the agent, and (TO) there are no grammatical or metaphysical or semantic restrictions on the declaratives that may be put in place of Q. With these features of stit in mind, applying Thesis 5 to deontic constructions retains, we think, the most valuable features of both the declarative-sentence complement account, Oblg:Q, and the action-nominal complement account, Oblg:a. For example, in context the burden of the prohibition of cooks (or dogs) from the bridge might be Oblg:[the Third Mate stit: no cooks (or dogs) are on the bridge]. Or we might have Frbn:[the cook stit: the boats are lowered]. Question: Does this imply Oblg:[the cook stit: ~(the boats are lowered)]? The application of stit theory to deontic concerns is a topic deserving treatment at length. We return to it explicitly in §2B.9, chapter 11, chapter 12, and chapter 14, as well as more indirectly in chapter 4 (via imperatives) and in chapter 5 (via the idea of promising).

14

1C

Introduction to stit

Grammar of the modal logic of agency

The formal grammar of stit is easy: A singular term goes in one blank, and a sentence in the other. The grammar of English agentives, especially that of imperatives, is more delicate. The first topic is embedding. We all know that declarative sentences can either occur stand-alone or be embedded in a larger context. One may encounter the following standing alone: The Prophet will tell Queequeg and Ishmael about Ahab. One may also find the selfsame declarative as the consequent of a conditional: If they will stop a minute, then the Prophet will tell Queequeg and Ishmael about Ahab. One should bear in mind that exactly the same possibilities exist for the two other "moods" of English, the interrogative and the imperative. With respect to interrogatives, one has for example the two stand-alone interrogatives Did Queequeg and Ishmael give their informed consent to sailing after the White Whale? and

Whom did the Prophet tell about Ahab? One also has these selfsame interrogatives embedded in a (declarative) "dependence" construction: Whether they gave their informed consent to sailing after the White Whale depends on whom the Prophet told about Ahab. When interrogatives are embedded, we tend to call them "indirect questions," a terminology that distracts from the chief point: The purely grammatical phenomenon of the very same interrogative sentence occurring now as stand-alone and now embedded is precisely analogous to declaratives occurring in these ways. Finally, and crucially for our purposes, the English grammar of imperative sentences exhibits precisely the same behavior. For example, start with the stand-alone imperative Mr. Starbuck, hand me yon top-maul. (Of course the "Mr. Starbuck" may be dropped when context makes clear the addressee, and "me" must be supplanted by "Ahab" when the imperative is used by some third party.) Then each of the following is a way to embed this selfsame imperative in a larger context. Mr. Flask, request Mr. Starbuck to hand me yon top-maul.

1. Stit: A canonical form for agentives

15

Ahab ordered Mr. Starbuck to hand him yon top-maul. Mr. Starbuck carried out Ahab's order to hand him yon top-maul; or at least Mr. Starbuck handed Ahab yon top-maul. Did Ahab advise Mr. Starbuck to hand him yon top-maul? Ahab demanded that Mr. Starbuck hand him yon top-maul. Ahab demanded that yon top-maul be handed to him by Mr. Starbuck. Mr. Starbuck refused to hand Ahab yon top-maul. Mr. Starbuck refused Ahab's request (order, advice) to hand him yon top-maul. Mr. Starbuck is obligated (permitted, forbidden) to hand Ahab yon topmaul. There is clearly no doubt, then, that imperatives can be either stand-alone or embedded. In English, embedding any of declaratives, interrogatives, or imperatives is a complicated matter requiring at least inversion and context-driven switching of pronouns. Our deepest comment is this: Embedded imperatives are in truth embedded imperatives. That is, they are constituent or embeddable grammatical transforms of the very same imperative sentences. In a logically perspicuous language, they would be the very same sign designs. Furthermore, agentive declarative sentences are, we think, also syntactic variants of imperatives, whether stand-alone or embedded. Both Mr. Starbuck handed Ahab yon top-maul and the clause embedded in That the top-maul was handed to Ahab by Mr. Starbuck was of no concern to Mr. Flask represent variants on exactly the same theme. The intimidating variety of syntactic structures is a good reason for thinking about stit. In all these cases the paraphrase into canonical form allows us to see that whatever the complications of natural language, we have one and the same agentive sentence, its surface grammatical form varying from context to context. We may put this as a thesis, our last: the "stit normal form thesis." STIT NORMAL FORM THESIS. (Stit thesis. Reference: Thesis 6) In investigations of those constructions that take agentives as complements, nothing but confusion is lost if the complements are taken to be all and only stit sentences. Our recommendation is that both imperatives and agentive declaratives be normal-formed as stits. What, then, of the grammar of stit itself? We think that the best thing to say is that a stit sentence itself is both a declarative and

16

Introduction to stit

an imperative.3 A stit sentence can be embedded wherever a declarative or an imperative can be embedded. We have seen the variety of English sentences that are helped by paraphrase as stits. Let us consider the other direction. Given a stit sentence, [a stit: Q], we should expect to read this piece of notation differently in English, depending on how it is used. Here are some paradigmatic examples of readings of [a stit:

Q}.

• As a stand-alone imperative: a, see to it that Q! • As a stand-alone declarative: a sees to it that Q, a is seeing to it that Q, a saw to it that Q. • As an embedded imperative: a to see to it that Q, for a to see to it that Q, that a see to it that Q, a's seeing to it that Q. • As an embedded declarative: a sees to it that Q, that a see to it that Q.4 These subtle alterations required by English grammar are really required; but they tend to obscure rather than reveal the fact that we can locate the selfsame agentive in a variety of contexts: sometimes as a stand-alone declarative, sometimes as an embedded declarative, sometimes as a stand-alone imperative, and sometimes as an embedded imperative. Our paraphrase of agentives into normal form clarifies that situation. A stit sentence, [a stit: Q], since it displays its agent and appropriate declarative complement publicly and obviously, is the appropriate picture of the underlying agentive partly because it remains recognizably the same in any and every context. A further important feature about the grammar of imperatives, aside from their embeddability, is that they display an agent. As Castaneda (1975) and others have urged, they have the deep grammatical form a to verb. In this respect an imperative is unlike a declarative in general, which may or may not express an agentive proposition and even when it does may not wear its agent on its surface, as the linguists say. But imperatives must show forth an agent, at least in the sense that to be understood, and (the point is crucial) to be used in larger contexts, the agent must be uniquely recoverable from the surface (for example, as the addressee of a stand-alone imperative). Those engaged in the descriptive grammar of English have and are entitled to different views on this matter. Perhaps the work most pertinent to our concerns is Badecker 1987. Badecker surveys some Chomsky theories, which by deriving 3 We are not sure that this use of overlapping grammatical categories is for the best. Let us emphasize again that the aim of applying stit notation is to clarify, not to analyze, and, in particular, not to provide a syntactical criterion for when a certain surface form must be considered an imperative rather than a declarative. Let us also emphasize that "force" versus "content" has no role here, since the topic is purely grammatical. 4 Note that this "see" is not indicative, but rather subjunctive, like "hand" in "Ahab demanded that Mr. Starbuck hand him yon top-maul."

1. Stit: A canonical form for agentives

17

all infinitive constructions by transformation of declaratives are deeply at variance with the spirit of the present book, though our aims are so different from his that it is hard to call the variance a conflict. In healthy contrast, the lexicalist theory that Badecker offers in his chapter 3 awards infinitive constructions independence from declaratives, and thereby more nearly shares our direction; however, there remains the question of whether in agentive infinitive constructions, such as Mr. Starbuck refused to hand Ahab yon top-maul, we should or should not take it that there is a "trace" of Mr. Starbuck heading the infinitive phrase. We certainly need Mr. Starbuck to get the semantics right, but that far from settles the syntactical question for English. In any event, Badecker supplies a helpful framework for addressing this and related questions. A consequence of the fact that a stit sentence is a declarative as well as an imperative is this: It can be embedded wherever a declarative or an imperative can be embedded. For example, with regard to the former, a stit sentence can be embedded under a negation. The result of such an embedding is on the face of it not itself a stit sentence; in the special (and, to a logician, prominent) case of negation, the result of embedding looks at least on the surface like

not like some instance of

But more deeply, for the special case of negation, the result of embedding is not always any kind of agentive; that is, by the paraphrase test for agentives suggested by Thesis 3, to which we hope you have agreed, the declarative it is false that a sees to it that Q is not invariably paraphrasable (or indeed equivalent in truth value with) a sees to it that it is false that a sees to it that Q. Thus, by Thesis 3, (19) is not itself an agentive. In more colorful language that speaks against taking a naive approach to inventing a "logic of imperatives," we may say that the negation of an imperative is not always an imperative. We later discuss the interaction of negation with stit in a number of places; see for example §2B.6 and §2B.8 on "refraining." Finally, let us recall that when concerned with the modalities of agency, it is similarly helpful to use stit sentences to keep track of the agentives. The English Starbuck could have handed the top-maul to Ahab is helpfully paraphrased as [Starbuck could-have-stit: Starbuck hands the top-maul to Ahab], while

18

Introduction to stit Starbuck is obligated to hand the top-maul to Ahab

is paraphrased as Oblg:[Starbuck stit: Starbuck hands the top-maul to Ahab]. In all these cases it is crucial that the content of the order and the obligation, the ability and the action, be all the very same, all captured by a single stit sentence.

1D

Mini-history of the modal logic of agency

Throughout this book we constantly refer to "the action," "the obligation," "the ability," "the order," and so on. This is because English, through its love of the subject-predicate construction, drives us to such usages. We do not believe, however, that English also drives us to wax ontological about actions, obligations, abilities, orders, and so on. We may wish to fashion theories about entities taken to fall under these common nouns. Stit theory is not to be taken as arguing against such a wish. On the other hand, we may not. Stit theory has the advantage that it permits us to postpone attempting to fashion an ontological theory, while still advancing our grasp of some important features of action, obligation, and so on. It does so by invoking a "modal" construction in place of a subject-predicate construction that requires an ontology. By a modal construction we mean an intensional grammatical construct having sentences as both inputs and outputs. A "modal logic of agency" intends that some such construct express agency (or action), as for example our own favored English construct "a sees to it that Q." The modal logic of agency is not popular. Perhaps largely due to the influence of Davidson (see the essays in Davidson 1980), but based also on the very different work of such as Goldman 1970 and Thomson 1977, the dominant logical template takes an agent as a wart on the skin of an action, and takes an action as a kind of event. This "actions as events" picture is all ontology, not modality, and indeed, in the case of Davidson, is driven by the sort of commitment to firstorder logic that counts modalities as Bad. The project has had some successes, all of which we shall ignore, and some failures, most of which we shall ignore. (See Bennett 1988 for an indispensable perspective.) Certain of its failures, however, are to be attributed to the neglect of the modal features of agency. We compare the ontological approach to action with our modal approach in §3E. Here we pursue the modal history. The earliest modal logic of agency of which we have learned was formulated around the year 1100 by a Dominican trained at a famous Norman institution, the abbey of Bec. (The date of c. 1100 is implied by the sources that were conveniently available to us without scholarly digging, namely, the discussion on p. 120 of Henry 1967 together with the "Anselm" entry in the Encyclopedia Britannica of 1968.) We refer to St. Anselm, who succeeded Lanfranc as prior and then abbot of Bec, and then later as archbishop of Canterbury. No mere pale

1. Stit: A canonical form for agentives

19

theoretician nor private saint, the archbishop was deeply involved in controversy with the tyrant William Rufus and later his brother Henry in regard to the matters of lay investiture and clerical homage; he vigorously opposed the former. These controversies were heavily freighted with the concepts of promising and commitment and agentive powers. In order to make clear that his authority in matters spiritual was not at the pleasure of the king, Anselm refused to accept the papal pallium from the hands of William Rufus. Partly in consequence, the archbishop was in effect exiled by the king. Anselm's brief notes on the modal logic of agency were, we think, composed during this bitter exile. In the document that Henry 1967 calls N, Anselm writes: Quidquid autem 'facere' dicitur, aut facit ut sit aliquid, aut facit ut non sit aliquid. Omne igitur 'facere' dici potest aut 'facere esse' aut 'facere non esse.' (p. 124; from N 29.8.10) Paraphrase by Henry: For all x, if 'x does' is true, then x does so that something either is so or is not so. Hence the analysis of 'doing' will in fact be an analysis of x's doing so that p, and of x's doing so that not-p [where 'p' is a clause describing a state of affairs, and 'not-p' is short for 'it is not the case that p']. (p. 124) Anselm goes on to describe a kind of square of opposition that clearly indicates he had in mind a modal logic of agency (to the extent to which that can be said without anachronism), but his work seems to have remained unnoticed until after the stirring of modal logic in this century. If you promise to accept our remark as merely helpful rather than authoritative, we will hazard that Henry 1953 is the first reference to Anselm that appreciates his work as modal, and that Chisholm 1964a (who cites Henry 1960) is the earliest reference to Anselm by an active researcher in this field (see §3B for further discussion of Chisholm). Other references to Anselm on this topic: Danto 1973, Humberstone 1976 (the reference does not appear in the published abstract, Humberstone 1977), the perceptive Walton 1976b, 1976a, and 1980 (which cites Dazeley and Gombocz 1979), and a sterling account in Segerberg 1989. It is amazing (and perhaps a little sad) that over eight hundred years elapsed between Anselm's invention or discovery and the next contributions to the modal logic of agency. The first modern desire for a modal agentive construction seems to have been felt by philosophers working their various ways through the embedding requirements of legal and deontic concepts. One can certainly see the need expressed in the pioneering work of Hohfeld 1919, who introduces locutions such as the following (p. 38). X has a right against Y that he shall stay off the former's land. This Hohfeldian construct plainly embeds the modal agency construction in a "has a right" context. This is true of all of Hohfeld's work: The modal agency construction is always embedded in additional constructions imputing legal rights, duties, powers, and so on. Agency has not yet received a separate modal treatment. (We discuss Hohfeld a bit in §5B.l.)

20

Introduction to stit

The next place we know a modal agentive construct to crop up, much more explicitly but still embedded in the context of a normative expression, is in Kanger 1957:

Ought(Y sees to it that F(X, Y)). (p. 42) Although the locution "sees to it that" is displayed only in a normative context and wholly without comment, it is clear from the general tenor of Kanger's methodology that he intended to be isolating a norm-free concept of agency. The explicit grammatical breakthrough for the logic of agentive modality comes in Anderson 1962, who, reflecting on Hohfeld, introduces for the first time a separate form of expression intended to disengage the concept of agency from normative considerations.5 When on p. 40 Anderson takes

M(x, p, y) to represent the case "when x executes what is regarded as an 'action' ... and y is the recipient or patient of the action executed by x," he suddenly gives us a clean target for some analytic questions that otherwise seem confusing. Anderson sometimes reads M(x, p, y), with perhaps too little attention to the connections between formal and English grammar, as "x does p to (for) y." Evidently here agency is, for better or worse, not separated from patiency. And certainly there is in Anderson no semantic theory of agency or patiency, and only a trace of a deductive calculus (e.g., Anderson points out that the implication between ~M(x, p, y) and M(x, ~p, y) goes only from right to left). That is, Anderson pioneers in isolating agency and patiency, but he does so only immediately to recombine them with deontic concepts. In 1963 each of two logicians, Fitch and von Wright, advanced modal theories of agency, each of them stressing syntactic developments. Fitch 1963 defines "does A" in terms of two other modalities, "striving for" and "causes," and offers a deductive calculus. The work has not been taken up by later logicians and is seldom cited in the published literature. Indeed, although NB was Fitch's admiring and fond student and colleague, he regrets to say that he had to be reminded of this essay by Segerberg 1989, which contains a maximally useful account. Von Wright, beginning in 1963 and continuing at least through 1981, was, we think, the other logician to be a first to treat agency (or action) as a specific modal or quasi-modal topic, always with that specially honest von Wrightian insistence on the lack of finality of the formulation in question, including attending to nonmodal formulations in which complements are taken as terms signifying specific or generic actions, rather than sentences. As in other cases, the work keeps a close eye on deontic logic, to which he contributed so much. We think von Wright did not succeed in disentangling agency from change, 5 Very likely the breakthrough for Anderson came about after correspondence or conversations with his friend Kanger. Somewhat later Anderson visited Manchester, where Henry was. Henry remarked in personal correspondence that during this year of 1965 there was a colloquium involving a number of persons interested in agency, including, e.g., Hare and Kenny.

1. Stit: A canonical form for agentives

21

evince interest in the general problem of embedding of agentives. For instance (but only "for instance"), von Wright 1963 took as a convenient primitive the notation

d(p/p), which is to be read as expressing some such idea as "the agent preserves the state described by p" (pp. 43, 57). Like Anderson, von Wright tends to leave to the reader the task of putting bits of logical grammar together with bits of English grammar. In contrast with Anderson, however, agency here has been separated out from patiency. (We further discuss von Wright in §3A.) Kanger and Kanger 1966 introduce as a separate locution X causes F, where F is supposed to be a sentence, but in a fashion like Anderson's, they logicize about it only by setting down that F may be replaced by its logical equivalents, and that the proposition that X causes F implies that F. Three influential lines of research began about the same time as that of the Kangers, each of which highlighted the separate existence of agentive modalities, namely, those initiated by Castaneda, by Kenny, and by Chisholm.6 Castaneda, whose views concerning deontic logic have informed both philosophers and logicians for many years (since at least Castaneda 1954), has much to say that is relevant to agency as a modality. Though his philosophical concerns led him to pursue goals other than the formulation of a modal logic of agency, he repeatedly urged the fundamental importance of the grammatical and logical distinction between "propositions" and "practitions" (a distinction put as clearly as anywhere in Castaneda 1981); but because there is no possibility of constructing a Castaneda "practition" from an arbitrary sentence, in the way for instance that Anderson's M(x, p, y] or von Wright's d(p/p) each permits an arbitrary sentence in place of p, Castaneda practitions cannot themselves serve as the foundation for such a modal logic of agency. (§3D expands our consideration of Castaneda.) Kenny 1963, in the course of initiating a rich literature on the verbal structure of our causal and agentive discourse, says that any "performance" in his technical sense is describable in the form bringing it about that p. And Chisholm 1964b takes the following as a basic locution on which to found an extensive series of definitions and explanations in the vicinity of agency: There is a state of affairs A and a state of affairs B, such that he makes B happen with an end to making A happen, 6 Of course other work on the theory of action has also influenced the modal logic of agency, but that literature is unsurveyably vast. We note as a passing example that there is hardly a one of our past or present departmental colleagues who has not contributed.

22

Introduction to stit

where the letters stand in for "propositional clauses," and where the subject of "makes happen" can be either a person or a state of affairs. The discussions of Kenny and Chisholm, though relevant to logical questions, are themselves not directed toward the formulation of either proof-theoretical or semantic principles governing their respective basic locutions. They are sufficiently closely connected to our project, however, that we return to Kenny and Chisholm respectively in §3C and §3B. This is as accurate a record as we can manage of the early history of the modal logic of agency. If this story is right, then the following gives its gist. History of the modal logic of agency prior to 1969 Anselm c. 1100

facere esse (x does so that p)

Hohfeld 1919

X has a right against Y that he shall stay off the former's land

Kanger 1957

Ought(Y sees to it that F ( X , Y ) )

Anderson 1962

M(x,p, y) (x does p to [or for] y)

Fitch 1963

Does A

von Wright 1963

d(p/p) (the agent preserves the state described by p)

Kanger & Kanger 1966

X causes F

Castaneda 1954ff, Kenny 1963, Chisholm 1964ff

relevant discussions

As we note in several places previously, in chapter 3 we extend our discussion of certain among these figures by more closely relating their thoughts to stit theory: von Wright, Chisholm, Kenny, Castaneda, and Davidson. Also in §5B there is a little more consideration of some Hohfeldian themes. But for now we leave this early part of the history. The first modal logic of agency with an explicit semantics is, we think, that of Chellas 1969. The primitive locution is to be read as "T sees to it that O," where T is an agent and O takes the place of a sentence (pp. 62-63). Chellas only deploys this locution in one context, namely, as the argument of an imperative operator.7 7

But Chellas does not restrict the complement of an imperative operator to sentences having the form A-TO as is required by our Thesis 4.

1. Stit: A canonical form for agentives

23

As for semantics, Chellas takes as a paradigm the technique made famous by Kripke not long before Chellas was writing; we mean deployment of a binary relation between "worlds" in order to clarify modal concepts. Chellas in particular gives a semantic clause for ATO: is true at the present world just in case O is true at all those worlds under the control of—or responsive to the action of—the individual which is the value of T at the present world. (p. 63) The language that Chellas uses in this pioneering explanation, like the "relative possibility" language of Kripke a few years earlier, is neither familiar in itself nor further clarified by Chellas. Perhaps this is the reason that, like his predecessors, Chellas in practice confines his agentive locution to the imperative context from which his need for it sprang, and does not pause to investigate its separate properties. 8 After Chellas there is a substantial group of logicians all of whom have deployed a binary relation or a pair of binary relations in an effort to generate a semantic understanding of an agentive modality that might be used as the complement of an imperative or of a deontic operator; we know of Porn 1970, 1971, 1974, 1977; Needham 1971; Aqvist 1972; Kanger 1972; Hilpinen 1973; Humberstone 1977; Lindahl 1977; and Talja 1980. For a critique of the line of research being described, with special reference to Porn 1970, see Walton 1975; also of note are Walton 1976b, 1976a, 1980, which develop some insights in an independent and more nonsemantic fashion. The earlier Porn articles and that of Aqvist use only a single binary relation; the idea of using two binary relations seems to be independently due to Needham 1971, Kanger 1972, and Hilpinen 1973. (Unless we have overlooked it, there is no cross-mention; we have not seen Needham's M.A. thesis, but make the inference from Porn 1977.) The reason for the second binary relation is given as this: Agency has not only a sufficient condition aspect but a necessary condition aspect (Kanger 1972, p. 109; Hilpinen 1973, p. 119), and one needs a separate relation for each. The later workers in this mini-tradition play variations on this theme. In our judgment this line of investigation, although initially promising, and although producing some useful insights, has not been much followed up for the following reason: It has remained obscure what one is to make of the binary relations that serve as the founding elements of the entire enterprise. Kanger 1972 says, for example, that one of the relations holds between a person and a couple of worlds or indices when everything the person does in the second world is the case in the first; and the other relation holds when the opposite of everything the person does in the second is the case in the first (p. 109). That is far from clear, and no one in the tradition is, in our judgment, any clearer than that. For a final example, we describe and quote at length from Porn 1977, which among those we mention is the most developed grammatical and semantic 8

We discuss Chellas a bit more in §4A, and we rely on his agentive operator, ATO, in several places—writing it, however, as [a cstit: A] in order to conform to our standard symbolism for the stit sentence.

24

Introduction to stit

treatment of agentive modality. (All page references are to Porn 1977 and all words not inside quotation marks are ours.) Dap is read "it is necessary for something which a does that p" (p. 4). It is said (pp. 7-8) that an equivalent concept is found as the definition of "a sees to it that p" in Chellas 1969, chapter 3, section 4, and in Porn 1971. "Consider all those hypothetical situations u' in which the agent does at least as much as he does in u. If v is such a situation, it may be said to be possible relative to what the agent does in u. ... if p is necessary for something that a does in u, then there cannot be a situation which is possible relative to what a does in u and which lacks the state of affairs that p. ... A natural minimal assumption is that the relation [of relative possibility] is reflexive and transitive." (PP. 4-5) D'ap is read "but for a's action it would not be the case that p" (p. 5), and also "p is dependent on a's action." (p. 7) "For the articulation of the truth of D' a p at u we require all hypothetical situations u' such that the opposite of everything that a does in u is the case in u' ... [the relation must be] irreflexive and serial." (pp. 5-6) Further, to connect the two modalities D and D', a condition is imposed that "requires that worlds which are alternatives to a given world under the relation [for Da] be treated as equals in contexts of counteraction conditionality." (p. 6) C' ap is read "p is not independent of a's action." (p. 7) Eap is defined as the conjunction of Dap and C'ap, and read "a brings it about (causes it to be the case that, effects that) p" (p. 7). Porn says (p. 8) that an equivalent concept is found in Needham 1971, p. 154, an essentially equivalent concept in Hilpinen 1973, section 6, and explicitly in Porn 1974, p. 96. Porn finds unacceptable (p. 7) an alternative E* a p, defined as the conjunction of Dap and D'ap, which Porn says is equivalent to a definition of Kanger 1972. There are three points to be made about this extract. The first is that given the available apparatus, Porn 1977 seems to us to offer the best explanations of and the most detailed working out of the modal logic of agency as based on abstract binary relational semantics. Second, even these best-possible explanations seem difficult. The conclusion one might draw is that one should doubt the likelihood that the abstract relational-semantic point of view itself can continue to serve in the way that was hoped. But third, however, and counting against this conclusion, is that Porn 1977 is evidently formulating, in the context of the relational semantics, the very combination of "negative" and "positive" conditions that much later were built into stit theory (see §2A.2 and §2A.3). Aqvist 1974, 1978 provide a much less abstract and more intuitive semantic setting; these articles are the first of which we know that make the fundamental

1. Stit: A canonical form for agentives

25

suggestion that agency is illuminated by seeing it in terms of a tree structure such as is familiar from the extensive form of a game as described in von Neumann and Morgenstern 1944. Aqvist's account of agency is in some respects akin to that described in this book, in some respects less flexible, and in some respects richer. His aim is not strictly to provide a modal logic of agency; for example, the primitive of Aqvist 1978 is "DO(a, Pa)" to be read "a does, or acts, in such a way that he Ps," and where "Pa" must be an atomic sentence (rather than an arbitrary sentence), and like von Wright, Aqvist wraps agency together with change. But his goal is close enough to warrant (i) a comparison (which is not attempted here) and (ii) a suggestion that the reader consult these sources. A notable relevant article is Mullock 1988. Of decisive importance is the uncommonly rich joint work Aqvist and Mullock 1989, which applies insights derived from the tree structure to serious questions in the law. Aqvist and Mullock 1989 is, as we later say also of Hamblin 1987, required reading. There is one later commentator on the tradition just described who is of special excellence and interest: Makinson 1986. In a series of more than a dozen articles beginning with Segerberg 1980, and including among others Segerberg 1981, 1982, 1984, 1985a, 1985b, 1987, 1988a, 1988b, and 1989, a distinguished modal logician develops a richly motivated and intuitively based formal approach to action by taking a routine as the guiding concept. Segerberg explicitly bases some of the intuitive and formal aspects of his work on studies that in computer science have come to be called "dynamic logic," the influence on Segerberg being primarily through Pratt. Consult Elgesem 1989 for a sympathetic yet critical and penetrating account of Segerberg's line of research. The work is not fully in the modal logic of agency, since it stresses a grammar of (i) terms (including complex terms) for naming "actions" and (ii) predicates for expressing properties of "actions," and thus self-consciously avoids a grammar of nesting connectives. But instead of a complaint this is intended only as a reason for limiting ourselves to a mere mention of what may indeed turn out to be not only valuable in itself but a useful link between the ontological and modal points of view on agency. Mention of Pratt calls attention to the existence of a large and interesting formal literature that we fail to cite as part of this mini-history except insofar as it has influenced Segerberg, namely the work on "dynamic logic" and its cousins that has been done by Floyd. Hoare, Pratt, and other computer scientists (see Segerberg 1989 for a brief entree via Pratt that is written especially with the logic of action in view, and see Pratt 1980 for an excellent fuller account). There are three reasons for excluding this line of investigation from the present survey: (i) we are very far from familiar with the literature, so that making it accessible is best left to someone else. Further, what we know of it (ii) stresses the ontological rather than the modal approach, whereas the latter is the topic of this mini-history, and (in) what we know of it is relevant to action only in the wide sense of "action" that encompasses mechanical action, that is, the sense of "action" that encompasses the action of programs and starter motors. In fact the present modal point of view makes it arguable that this literature is no more relevant to agency than is the literature of any other discipline that gives us

26

Introduction to stit

ways to fill out the sentential complement of "sees to it that": An agent can see to it that the starter engages and passes through various stages, or that a certain recursive program runs, or ... . But it seems best to make explicit our failure to more than barely mention such a large literature just because so many persons think that although it may be arguable, it certainly isn't plausible that it has no special relevance to agency. On the other hand, NB once asked a well-known computer scientist/mathematician after a lecture on parallel processing if he had meant his use of "actor" and "agent" to be anything but an idle metaphor; he was aghast that one should need to inquire. The articles Brown 1988, 1990, and 1992 represent a sustained and important investigation of the modal approach to ability and its connection with action. Brown initially proposed a modal operator that has something of the force of "can do." Horty 2001, which uses [a bstit: Q] for this Brown connective, explains Brown's ideas and relates them to stit theory. Penultimately there is von Kutschera 1986, which articulates in one form or another nearly all of the essential underlying ideas concerning agency on which we base the semantics offered in subseqent chapters. We can describe the extent of von Kutschera's priority only by using some phrases not defined until later. At the very least, one must credit von Kutschera 1986 with the no choice between undivided histories condition, with generalization beyond the discrete, with generalization to multiple independent agents (including the independence of agents condition), with attention to strategies, and with semantics for the "deliberative stit" that we study in Definition 2-5 and §8G.l. It also needs to be remarked that von Kutschera 1986 cites the earlier von Kutschera 1980. Finally there is Hamblin 1987, which in the context of a study of imperatives provides a rich source of formal, informal, and semi-formal ideas on the topic at hand, many of which have influenced the present work; in particular, collegial reflection on Hamblin's "action-state semantics" was the immediate context of the beginning of the research reported in this book. Our own recommendation is that no one ought to try to move deeply into any part of the theory of agency without reading Hamblin.

1E

Conclusion and summary

Although not so grand as Ishmael's, we think our story well worth the telling, and though we have not sailed far, our mainsail has been unfurled and we have caught the first breeze. What we propose is an augmentation of our current linguistic resources with a linguistic form, the stit sentence [a stit: Q], that (i) leads us carefully to attend to the agent of an action, (ii) is capable of taking any English declarative as its complement, (iii) is recoverable as the same stit sentence either as a declarative or an imperative, and (iv) is grammatically suitable for embedding within wider contexts. Among its other virtues, the stit sentence sheds light on refraining and helps to clarify some of the agentive modalities. This linguistic addition, attentive to grammatical form and semantic

1. Stit: A canonical form for agentives

27

structure, promotes greater clarity in the way we talk and think about the phenomena of our world, and thus justifies its added complexity. In succeeding chapters we strive to deepen our understanding of agency by providing stit sentences with careful and well-motivated semantic analyses, and we apply and generalize on it in a variety of ways.

2

Stit: Introductory theory, semantics, and applications In chapter 1 we followed and extended the idea, going back at least to Anselm, of treating agency as a modality—a modality that represents through an intensional operator the agency, or action, of some individual in bringing about a particular state of affairs.* We proposed that using the stit construction as a normal form, when we are confused, is a happy way to clarify some aspects of action and agency. In this chapter we sharpen our understanding of stit—and thereby, if we are right, of agency. The central idea is that the concept of action must be understood in relation to an open future, and we formulate a rigorous theory that tries to understand how action is compatible with and indeed requires irideterminism. In this way we essay a contribution to what Kane 1998 calls ''the intelligibility question" (p. 105). We often label the chosen approach stit theory, because it concentrates on the linguistic form "a (an agent) sees to it that A," which we abbreviate simply as [a stit: A]. Part of the theory, however, has nothing to do with language. This nonlinguistic part instead purports to articulate in a general way how agency fits into the overall causal structure of our world. For reasons that will emerge, we often ponderously refer to this theory as "the theory of agents and choices in branching time,'' or, more briefly but less memorably, as BT + AC theory— or, when endowed with instants or times, as BT + I + AC theory. Stit theory, including its nonlinguistic part, provides a precise and intuitively compelling semantic account of the stit operator within an overall logical framework of indeterminism; the account is then used as a springboard for investigating a number of topics from the general logic of agency, such as the proper treatment of certain concepts naturally thought of as involving iterations of the agency "This chapter draws on several sources. We thank John Horty for co-authoring §2A, which, with the permission of Kluwer Academic Publishers, is drawn from Horty and Belnap 1995. §2B, except for §2B.10, finds its source in a portion of Belnap 1991, for the use of which we thank the International Phenomenological Society, while §2B.10 is based on part of Belnap 1996a, with the permission of Kluwer Academic Publishers.

28

2. Stit: Introductory theory, semantics, and applications

29

operator, as well as interactions of this operator with other truth-functional and modal connectives. The theory of agents and choices in branching time supports more than one semantic candidate for stit. We mention several of these briefly in various later chapters, but only two are primary in this book. One of the two derives from the early work of two of the authors, NB and MP. The other, which is simpler and for certain purposes easier to use, first appeared in von Kutschera 1986, prior to the work of the authors of this book; later one of the authors of §2A of this chapter independently suggested it, in Horty 1989, explicitly as an alternative to the account of stit put forth by NB and MP. The first major purpose of the present chapter, which we carry out in §2A, is to describe in §2A.l the underlying theory of branching time and the tense logic that is appropriate to that theory, and then in §2A.2 and §2A.3 to describe the individual semantics of each of the two stits, together with the theory of agents and choices on which they rest. These discussions are all explicitly intended to be preliminary; later chapters treat B T + I + A C theory and each of the two stit operators with more rigor and in more detail. A second purpose of this chapter, accomplished in §2B, is to offer an equally preliminary exploration, with many pictures, of some applications of the first, more complicated of the two stit operators. In order to distinguish between the two agency operators under discussion, and for other reasons that will soon become apparent, we describe the operator featured in the early work of NB and MP as the achievement stit, represented in this section as "astit"; and we describe the alternative suggested by von Kutschera and Horty as the deliberative stit, represented here as "dstit." When in this book we speak simply of a stit operator—or use "stit" alone as an operator in some sentence—we often mean to generalize over both the deliberative and achievement stit operators, and perhaps others of the same family. Other times, however, we use plain "stit" as meaning one of astit or dstit, context making it clear which stit is at issue.

2A 2A.1

Theory and semantics: The two stits Background: Branching time

Stit theory is cast against the background of an indeterministic temporal framework, in particular, the theory of branching time due originally to Prior 1967, pp. 126-127, and developed in more detail in Thomason 1970 and Thomason 1984.l 1 From time to time in this book we insert the reminder that although we use the phrase "branching time" because of its fixed place in the literature, we never, ever mean to suggest that time itself—which is presumably best thought of as linear—ever, ever "branches." The less misleading phrase, which we occasionally use, is "branching histories," with an essential plural to convey that it is the entire assemblage of histories that has a branching structure. A single history cannot branch, but two histories can branch from each other. As a further occasional reminder, we observe that although in this book we idealize each history as linear, there is a more adequate treatment of branching histories in Belnap 1992. There each history in the branching assemblage is idealized not as linear, but instead as a four-dimensional spacetime. For similar ideas see also McCall 1994, Rakic 1997, and Placek 2000.

Introduction to stit

30

Figure 2.1: Branching time: Moments and histories

The theory.is based on a picture of moments as ordered into a treelike structure, with forward branching representing the openness or indeterminacy of the future and the absence of backward branching representing the determinacy of the past. Such a picture leads, formally, to a notion of branching temporal structures as structures (we sometimes say BT structures) of the form (Tree, = {h: (Mnh) = O}, so that H<M> is the set of histories that pass through at least one member of the set of moments, M. For suitable M, this is the set of histories in which M comes to be.

7. Agents and choices in branching time with instants

193

H(m) represents the contingent proposition that m timelessly exists, a proposition that is true in all and only those histories that m inhabits. H[M] does not always make useful sense as a proposition. When, however, / is an initial event, H[M] is well interpreted as the (timeless) proposition that / is "completed," "finishes," or in Aristotle's phrase, "passes away." If you are thinking about experimental preparations or agonizing deliberations, this is a sensible thing to mean by "exists": A preparation-event or deliberation-event does not "exist" in a history unless it does so entirely. If some history, h, splits off in the middle of a deliberation, we decline to say that the deliberative event "exists" in h. (One would need explicitly to tackle the present progressive in order to have a rigorous account of a deliberation-event "existing" in histories in which it is not completed.) Note a comfortable interaction: The proposition that / is completed depends not at all on the causal "shape" of / in its nether region. Any initial cofinal with / toward the future will determine the same set of containing histories as does /. For example, if / is taken as a representation of the setting-up of an indeterministic experimental situation, whether or not the set-up is completed is insensitive to everything except the causal locus of its being completed. When O is an outcome event, the meaning of "exists" needs to be quite different. In that case, the proposition H is the right sense of "exists," since outcomes exist by beginning (not ending), and H is true in all and only those histories in which O begins, commences, or in Aristotle's language, comes to be. Suppose a coin is flipped. If we look only at the bare causal structure, it is natural (but not necessary) to think of the flipping as lasting as long as there is indetermination as to the outcomes heads or tails, no matter how that flipping may be related to the way that the coin dances in the air. In the same way, it is natural (but not necessary) to think of the heads outcome beginning whenever the possibility of tails is excluded. In other words, the heads outcome (not must but) may be taken to begin whenever that outcome is determinately settled, and the proposition that the heads outcome exists should be true in just those histories in which settled-heads begins to be. For purposes of causal analysis, it doesn't matter when the heads outcome ends. Consistency. Without belaboring the point, we note that the interplay between propositions and events in branching time generates a firmly based family of consistency/inconsistency concepts. We may take from possible-worlds theory the idea that when propositions are represented as sets of histories, consistency between them is definable as having some history in common, a history in which both propositions are true. We add that it is helpful to define the consistency of two initial events I1 and I2 by the consistency/inconsistency of the propositions H[I1] and -H[I 2 ], so that the question is whether or not there is a history in which both preparations are (not just started but) completed. Dually, two outcome events O1 and O2 are consistent/inconsistent iff the propositions H and H are consistent/inconsistent, so that the question is whether or not there is a history in which both outcomes begin to be. Even more delicately,

194

Foundations of indeterminism

one may ask whether a certain initial is consistent with a certain outcome by way of well-chosen propositions. Finally, one has the right definition of what makes a transition "contingent." CONTINGENT TRANSITION. (Definition. Reference: Def. 8) is a contingent transition iff is a transition, and if some history is dropped in passing from the completion of / to the beginning of O: H[I] -H = O. These remarks are intended to indicate with almost excessive brevity that branching time permits and indeed suggests a rigorous and modestly enlightening theory of the interrelations of (possible) concrete events and propositions. We mention that some additional ideas on initials and outcomes are offered in Szabo and Belnap 1996 and in Belnap 1996c, and in the unpublished set of notes Belnap 1995.

7A.5

Theory of instants

If "histories" are a way of making a sort of vertical division of Tree, then Instant, whose members are instants, is a kind of horizontal counterpart. In branching time, the doctrine of instants harkens back to the Newtonian doctrine of absolute time—and therefore is suspect. We use it, but we don't trust it. For that reason if for no other we try to be as clear about it as we can. Instants are perhaps not fully "times" because we are not in this study relying on measures or distances, but it is intuitively correct to think of i( m ) as the set of alternative possibilities for "filling" the time of TO. We need instants because we think that for the achievement sense of stit, in considering whether Autumn Jane stit she was muddy at a certain moment, it is relevant to consider what else might have been at the instant inhabited by that moment. Evidently our uses of "moment" (in which we follow Thomason) and "instant" are jargon not sanctioned in ordinary speech, although the distinction is certainly there to be drawn. Not all parts of stit theory rely on the theory of instants. Only the semantics for the achievement stit has need of these horizontal comparisons. For example, the theory of dstit, §8G.l, and the theory of strategies of chapter 13 are developed quite apart from the idea carried by Instant. In this important sense, the theory of instants is not a deep presupposition of stit theory. There is a contrast at this point with TxW theories as described in §7A.6. We nevertheless develop the theory of instants to the extent required by the semantics of the achievement stit. There are three postulates. Instant AND INSTANTS. (BT +I + AC postulate. Reference: Post. 5) i. Partition. Instant is a partition of Tree into equivalence classes; that is, Instant is a set of nonempty sets of moments such that each moment in Tree belongs to exactly one member of Instant. ii. Unique intersection. Each instant intersects each history in a unique moment; that is, for each instant i and history h, inh has exactly one member.

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195

iii. Order preservation. Instants never distort historical order: Given two instants i1 and i2 and two histories h and h', if the moment at which i1 intersects h precedes, or is the same as, or comes after the moment at which i2 intersects h, then the same relation holds between the moment at which i1 intersects h' and the moment at which i2 intersects h'. We next offer some convenient definitions and simple facts, after which we comment on the postulates.

INSTANTS. (Definition. Reference: Def. 9) • The members of Instant are called instants, i ranges over instants. • i(m) is the uniquely determined instant to which the moment m belongs, the instant at which m "occurs." •

m

(i,h) is the moment in which instant i cuts across (intersects with) history h; that is

• Order preservation can conveniently be stated in the symbols just introduced: implies • Fact: m(i ( m 0 ) , h 0 ) , a function of mo and ho, is the moment on history h0 that occurs at the same instant as does mo: • i|>m = {m0: m < mo & mo E i}. We say that i|> m is the horizon from moment m at instant i. • Where i1 and i2 are instants, we may induce a linear time order (not a causal order!) by defining i1 < i2 iff m1 < m2 for some moment m1 in i1 and some moment m2 in i2. Instants can also be temporally (not causally) compared with moments, m: i1 < m iff m1 < m for some moment m1 in i1 and m < i2 iff m < m2 for some moment m2 in i2 . Post. 5(i) encodes that making same-time comparisons between histories is objectively sound. We do not pretend to understand the conceptual problems involved in making such comparisons. The problem becomes ironically clearer when it is made more difficult by transference to branching space-time, where it is same-place-time comparisons between inconsistent point events that is at issue. All we can add is a conviction that it will not be possible to make suitable advances without consideration of the work of Bressan 1972, Bressan 1974, Zampieri 1982, and Zampieri 1982-1983, for they are the only persons we know who have worked within what seems to us the only reasonable position, that identifying place-times across possible situations is neither trivially easy (perhaps Kripke thinks this) nor a matter of partial constraint and partial stipulation (perhaps Lewis thinks this) nor empirically insignificant (perhaps this is van Fraassen's view), but a matter of serious physics.

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Postulates Post. 5(ii) and Post. 5(iii) on Instant are very likely too strong (too oversimplifying); our justification is that agency is already hard to understand, so that it won't hurt to try to see what it comes to in circumstances that are not altogether realistic—as long as we keep track of what we are doing so that later we can try to move closer to reality. Thus, which most reality-oriented persons think not so plausible, but which greatly simplifies our picture of time, all histories are said to have exactly the same temporal structure. It follows that all histories are isomorphic with each other, and with Instant, which justifies the ordering on instants defined in Def. 9. On the other hand, no assumption whatsoever is made about the order type that all histories share with each other and with Instant. For this reason the present theory of agency is immediately applicable regardless of whether we picture succession as discrete, dense, continuous, well-ordered, some mixture of these, or whatever; and regardless of whether histories are finite or infinite in one direction or the other. The theory of Instant is not, as we have said, as fundamental as that of S theory allows that histories can "overlap" in the sense of sharing exactly the same states of affairs at exactly the same times, and perhaps, though we are unsure, this is genuine overlap. Still, on h:T—> S theories, states of affairs are repeatables that can in principle occur at different times; they are therefore quite different from moments, each of which belongs to but a single instant, as is appropriate to a concrete event. Perhaps one could obtain a useful h:T—> S theory from branching time by locating an interesting sense of "state of affairs" as partially characterizing moments. Otherwise the h:T—>S theory seems faced with the double demerit of needing to provide both an account of times, and an account of states of affairs, on pain of being without application. Perhaps, for instance, belief in "states of affairs" is wild-eyed metaphysics (we hope not). Thomason 1984 discusses the notion of a "Kamp structure" which is a variation on the TxW idea. (The literature uses the word "frame" more often than "structure"; there is no difference in concept.) In a Kamp structure, each world is provided with its own temporal ordering. No real ground is gained, however, since using these structures to simulate branching still requires the concept expressed by saying that two worlds "perfectly match up through" a given time. Perhaps ground is even lost, since the ontology of Kamp structures requires making sense of the possibility that the times of two worlds could be dramatically different in their ordering, while nevertheless sharing some particular entity ontologically classified as a "time." For example, the respective times of two worlds might each contain both 4:00 P.M. and 5:00 P.M. The first world, however, could put 4:00 P.M. and 5:00 P.M. in their natural order, while the second world reverses their order. In considering what this could mean, we seem to lose our grip. In any case, all of our philosophical objections to Tx W as a theory of the structure of our world apply equally to Kamp structures. (This discussion is indebted to correspondence with A. Zanardo.) Both TxW structures and Kamp structures can be defined with first-order conditions. Yet another such representation of branching times is the notion of an "Ockhamist structure" (X, h2 E H. So for Tree to be deterministic at mo is for there to be but a single immediate possibility at mo. Sounds right. Obviously Tree may be deterministic at m but not so at either earlier or later moments. On the present account of determinism, one can coherently believe that our world used to be deterministic but is not so now, although it may become so once again. We go on to say that Tree itself is deterministic if it is deterministic at every past. In that case there is obviously but a single history. A determinist is someone who believes that the tree (world) of which our moment is a part is deterministic. It would appear that many philosophers believe that anyone who is not a determinist is softheaded and probably needs therapy. Others believe that anyhow all respectable philosophical theories, including theories of agency, should at least be consistent with determinism. Determinism, however, is an extremely strong theory, going far beyond determinism of the present moment. In any event, we are not determinists, even though the denial of determinism is not a postulate of this book. But more than that, on the theory here offered, if anyone could ever see to anything, then determinism is false. So even though we do not lay down indeterminism as a postulate, since we believe that sometimes people have choices, we are indeterminists. Accordingly we think that any theory (of anything) should be compatible with at least a little indeterminism. We are "compatibilists" in the best sense. We agree with Kane 1998 that in particular the question "whether a kind of freedom that requires indeterminism can be made intelligible" (p. 105) deserves, instead of a superficial negative, our most serious attention, and indeed we intend that this book contribute to what Kane calls "the intelligibility question." Note, incidentally, that some situations in our world could, for all we know, be governed by indeterminist laws that are nonprobabilistic. Such a law might describe the sorts of possible outcomes for some type of initial event, but without carrying information concerning the relative probability of those outcomes. Agency may or may not be like that.

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It is abstractly interesting to observe that our world could be strongly antidetermmistic in the sense that there is splitting of histories at every moment, so that H( m1 ) = H( m2 ) implies m1 = m2 for every m1 and m2 in Tree. It seems to us entirely possible that strong antideterminism is true—but maybe not. Certainly it would be mathematically attractive to be able sometimes to think of a moment as a set of histories (those that pass through it), just as we can think of a history as a set of moments (those that it passes through). With equal certainty, however, this is not a good place at which to argue from beauty to truth. As a final word on this topic, we note that even "strong" antideterminism is not all that strong. Even "strong" antideterminism is consistent with the truth of numerous and important determinist theories about "systems" and "states" and such. Indeterminism is not disorder.

7B.2

Arguments against branching

In a passage that we also quote on p. 170, Lewis 1986 advances persuasive arguments against branching time. The trouble with branching exactly is that it conflicts with our ordinary presupposition that we have a single future. If two futures are equally mine, one with a sea fight tomorrow and one without, it is nonsense to wonder which way it will be—it will be both ways—and yet I do wonder. The theory of branching suits those who think this wondering is nonsense. Or those who think the wondering makes sense only if reconstrued: You have leave to wonder about the sea fight, provided that really you wonder not about what tomorrow will bring but about what today predetermines. (pp. 207—208 of Lewis 1986; we quote the remainder of this passage later, on p. 208) In addition to our earlier comments, we respond to this argument in two quite different ways. The first response is that it takes our ordinary ways of thinking too seriously. We draw an analogy, due to Burgess 1978 (p. 165), between (i) objections to "some of our ordinary ways of speaking" based on standard relativity theory and (ii) objections to some of those ways based on branching time. In fact we sharpen the Burgess analogy by emphasizing the similarity of the roles of "many histories" in indeterminism and of "many frames of reference" in the theory of relativity. When taken as a piece of argumentation, the analogy could go like this. If the foundation of the Lewis argument in our ordinary ways were solid, then the following would be an easy reduction to absurdity of the theory of relativity. The trouble with the theory of relativity exactly is that it conflicts with our ordinary presupposition that we have a single present or "now." If two presents or "nows" are equally mine, one with a sea fight at Neptune's north pole and one without, it is nonsense to wonder which way it now is—it is now both ways—and yet we do wonder.

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That is a bad argument. Certainly many of us have a tendency to wonder what is going on at Neptune's north pole (or on the far side of Sagittarius) right "now." The presupposition underlying this wondering is "ordinary" and perhaps even "natural" for clock-aware and scientifically sophisticated persons born in the last few hundred years. But nevertheless it is false, as we are taught by relativity. We urge as apt the analogy suggested by this rewording of the quoted argument against branching time: Relativity insists that our world provides us with no uniquely natural spatially extended "now," although it may permit us to consider, if we wish, the limited family of all nows ("hyperplanes") to which a given utterance-event belongs. And indeterminism insists that our world provides us with no uniquely natural spatiotemporally extended actual history, although it permits us to consider, if we wish, the limited family of all histories to which a given utterance-event belongs. To the extent that common sense asks for a unique naturally given "now" to which a given utterance-event belongs, or for a unique naturally given "actual history" to which a given utterance-event belongs, to that extent, common sense is asking for something it cannot have. Nevertheless, the Prior-Thomason semantics, which explicitly recognizes the relativity of many statements to histories as well as moments, can give common sense a large amount of what it wants and can correct some parts of the Lewis formulation that are too hasty. "We have a single future." If this means that it is settled what will happen, for example, that either it is settled that there will be a sea battle or settled that there will not, it is false. If it means that it is settled that incompatible events will never happen, it is true. If it means that there is a single future history following upon this utterance, it is false. If it means there is a single future of possibilities, it is true. Branching time indeed claims that it can happen that "two futures are equally mine, one with a sea fight tomorrow and one without." But what does it mean to say that a future is "mine"? Branching time says that it means that it is among the futures now possible, where the "now" is indexically mine. To avoid making branching time look silly in a way that it surely isn't silly, the quoted description should be amended by insertion as follows: "two possible futures are equally mine, one with a sea fight tomorrow and one without." Lewis misdescribes the theory of branching time in saying of such a situation that "it will be both ways." Branching time is entirely clear that "Tomorrow there will be a sea fight and tomorrow there will not be a sea fight" is a contradiction. What is true and not surprising is that "It is possible that tomorrow there will be a sea fight and it is possible that tomorrow there will not be a sea fight" is eminently consistent. Nor is this merely a matter of formal tense logic. It seems to us deeply realistic to take it that if the captain is faced with two possibilities, sea battle tomorrow or no sea battle tomorrow, then those are possibilities for him, on that occasion. They are equally his, not one more than the other, exactly in accord with Lewis's account of (not his own theory but) branching time. Suppose the sea battle comes to pass. Then (after the sea battle) the two possibilities were his, and were equally his. In particular, branching time rejects the Lewis

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shadow-theory according to which the captain himself is to be found in the world of the sea battle, whereas merely one of his "counterparts" can be located in worlds without sea battles. The point goes back at least to Burgess 1978, who reflects as follows on his earlier decision to go to the office instead of the seashore: "Tense logic insists, pace Lewis, that I am the very same person who could have gone to the shore; it's not just someone like me who could have gone" (p. 173). Suppose we "step outside of branching time." To do this is to confine ourselves to language that has no trace of indexicality, a perfectly proper thing to do (see note 14 on p. 162). Histories and moments and persons are then linguistically accessible only via (rigid) naming or quantification. Branching time then says that for suitable moment mo and histories h1 and h2, the captain lives through a sea battle the day after m0 on h1, and lives through no-sea-battle the day after mo on h2. Here seems a premiss for a reductio of branching time, for branching time then seems to say that the captain has it both ways, both living through a sea battle and living through no-sea-battle. The reductio is, however, an illusion. Omitting the relativization to histories is intolerable. What branching time says is that the captain "has it both ways" in the entirely innocuous sense that he lives through a sea battle on history h1 and lives through no-sea-battle on history h2. That just says that there are at mo two possibilities for him, a fact about our world that we must keep. It does not say that the two possibilities will each be realized, an absurdity that branching time denies. It does not say that these possibilities remain possibilities at moments after the sea battle has commenced. It only says that in the past of such moments the two possibilities were available. Current possibilities drop off (McCall 1994) with passage into the future, but not the fact that they once were possibilities. Once was-possible, always was-possible. What about "wondering" whether or not there will be a sea battle? Evidently our wondering is history-independent: The fact that we wonder is dependent on the moment but independent of the history parameter. So what sense can we make of wondering about a history-dependent complement such as "there will be a sea battle tomorrow"? Lewis points out one alternative, which he rightly presents as not very ordinary, namely, that the complement of the wondering is the history-independent question, whether or not it is now settled that there will be a sea battle tomorrow. We proposed in §6E a two-point understanding of wondering that lets its complement remain open in the history parameter. First, it seems natural to construe "wondering q," where q is an indirect question (e.g., "whether there will be a sea battle tomorrow"), as "wanting to know" (or perhaps "wanting to have") a true answer to the question of q. Second, it seems obvious that our wants are not normally satisfied immediately; we must in general wait. Just so, in order to obtain satisfaction of the want expressed in our wondering, we shall need to wait until tomorrow; for only tomorrow is it possible for us to come to know whether or not a sea battle comes to pass. We next comment on some ideas in the remainder of the passage the beginning of which we quoted once on p. 170 and again on p. 205. Lewis continues in the following way.

208

Foundations of indeterminism But a modal realist who thinks in the ordinary way that it makes sense to wonder what the future will bring, and who distinguishes this from wondering what is already predetermined, will reject branching in favour of divergence. In divergence also there are many futures; that is, there are many later segments of worlds that begin by duplicating initial segments of our world. But in divergence, only one of these futures is truly ours. The rest belong not to us but to our other-worldly counterparts. Our future is the one that is part of the same world as ourselves. It alone is connected to us by the relations—the (strictly or analogically) spatiotemporal relations, or perhaps natural external relations generally—that unify a world. It alone is influenced causally by what we do and how we are in the present. We wonder which one is the future that has a special relation to ourselves. We care about it in a way that we do not care about all the other-worldly futures. Branching, and the limited overlap it requires, are to be rejected as making nonsense of the way we take ourselves to be related to our futures; and divergence without overlap is to be preferred. (p. 208)

The first sentence refers to "the future," the one that "is truly ours." Branching time says that only the future of possibilities is uniquely determined by the moment of utterance, so that "the future" either refers to this, or else is not history-independent (is open in the history parameter). Branching time says that if indeterminism be true, then there is no more sense to "the actual future" than there is to "the actual distant instantaneous present" or to "the odd prime number." But what about the future that "is part of the same world as ourselves"? Assuming indeterminism, there is the following dilemma. • If we read "world" as "history," then it makes no sense to speak of "the world of which we are part." There are many such possible histories to which this utterance-event equally belongs. All of them are "connected to us by the ... spatiotemporal relations ... that unify a world," for there is, in our opinion, no more fundamental "natural external relation" than the causal ordering itself. It is to be borne in mind that even wholly incompatible moments are mediately connected by ." It is striking that each point contains two independent references to moments: the moment of use, mc, and the moment of evaluation, m. They play of course entirely different roles: The moment of use is immobile (unshifted by any operation), and may be used to fix indexical expressions, whereas the moment of evaluation is paradigmatically mobile, being shifted by a variety of tense constructions. In addition to their uses in connection with various special indexical expressions, some context parameters have another role to play (Kaplan, p. 595; see our overlapping discussion in §6B.4.2). In the present case the following, which restates Policy 6-2, is critical. 8-2 OBSERVATION. (Starting evaluation of stand-alone sentences) The moment-of-use parameter is used to start the evaluation of any stand-alone sentence considered as being uttered to some purpose involving the semantics of the sentence, for example, uttered as an assertion. We expand the discussion of §6B.4.2. If you want to evaluate Themistocles was surprised,

(6)

and you understand that "was" moves evaluation into the past, you need a place to start that motion. When (6) is itself being considered as stand-alone, the moment of use gives us that starting point. It works in a special way: The (paradigmatically mobile) moment of evaluation is fixed, to begin with, as the very moment of use. It is only when we come to the sentence Themistocles is surprised,

(7)

which is implicitly embedded in (6) by past-tensing, that there is a divergence between moment of use (which remains the same as for (6) taken as stand-alone) and moment of evaluation (which needs shifting, existentially, toward the past). 3 Elsewhere in this book, in order to minimize complexity of exposition, we not only keep the model, m, implicit; we also pass over recognition of either the assignment-to-variable parameters or the context parameters. In short, in spite of thinking of just one target language, we relativize truth in different ways depending on which problems we are attacking. For example, when we are not concentrating on quantifiers, we omit relativization to the assignment parameters.

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You will note that in expressing Observation 8-2, which is derived from Kaplan, we use loose language. It is difficult to do otherwise for the following reason: Although we have in mind that a sentence can be considered either as stand-alone or as embeddable, the symbolic language (unlike, e.g., English) makes no such distinction. In the symbolic language, there is no syntactic mark (such as initial capitalization and final period in written English, or intonation in the spoken language) that distinguishes sentences taken as stand-alone from those taken as embeddable. This lack of match between English and the symbolic language makes analysis more difficult. Here is the best we know how to do (without describing a new kind of symbolic language) by means of a definition and a policy. (See Green 1998 for a study of "illocutionary-force-indicating devices," including Frege's sign of assertion.) First the definition, which is a BT +1 + AC-specific version of the more general concept indicated in Policy 6-2. CONTEXT-INITIALIZED POINT. (Definition. Reference: Def. 16) We say that a BT + 1 + AC point, <m, mc, a, m/h>, is context initialized iff the moment of use, mc, is identical to the moment of evaluation, m. That is, context-initialized BT + I + AC points have always the form <m, mc, a, mc/h>. The idea is that in a context-initialized BT + I + AC point, the mobile moment of evaluation is "initialized by" the moment of the context of use. Next the policy. 8-3 POLICY. (Differential

treatment of stand-alone and embeddable sentences)

• We recommend and urge, on pain of confusion, that sentences considered as stand-alone may usefully have their evaluation restricted to contextinitialized points. • We recommend and urge, on pain of confusion, that sentences considered as embeddable shall not have their evaluation so restricted, but that they shall be evaluated also at points in which the moment of use, mc, and the moment of evaluation, m, diverge. The rationale for the first part of this policy is that each utterance should be conceived as tied to a concrete context, and that such a context determines a unique causal position, with a definite past and a definite future of possibilities. We idealize such a position with the moment of use, mc. This moment of use is the very moment at which we wish to evaluate a stand-alone sentence. That is why, for stand-alone sentences, we initialize the moment of evaluation with the moment of use. Keep in mind, however, that the moment of evaluation is mobile, and can be shifted by tense constructions ingredient in the stand-alone sentence. And that is the very reason for the second part of Policy 8-3. Since there is in the symbolic language (and indeed often in the English examples of philosophers) no syntactic difference between stand-alone and embeddable sentences, the definition of "context-initialized point" (Def. 16) and

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Policy 8-3 represent the best that we can do. The definition and policy are, we think, exceptionally useful in discussing tenses and indeterminism, for in those ventures the failure to observe the distinction between stand-alone and embeddable sentences is especially harmful. Here is one way the definition and policy offer immediate progress. Kaplan (pp. 505-506) asks that in thinking about the "character" or meaning of a sentence, we first fix context, and then ask for the content of the sentence in that context. There should, however, be' two notions of content-in-a-context, depending on whether we are thinking of the sentence as stand-alone or embeddable. If we are thinking of it as stand-alone, then the moment of evaluation is initialized by the moment of the context. Since one cannot reasonably treat a sentence with free variables as stand-alone (Assertability thesis 6-7), it is obvious that for stand-alone sentences the history is the only mobile parameter that is left to vary when considering a stand-alone sentence. If one correlates time to moment and world to history, this explains the otherwise puzzling phenomenon noted by Kaplan on p. 546: ... the truth of a proposition is not usually thought of as dependent on time as well as a possible world. The time is thought of as fixed by the context. That is right for stand-alone sentences: Time (or moment) is fixed, while world (or history) is not. If, however, we are thinking of the sentence as embeddable by means of translocal connectives such as tense operators, then for "content in context" we must let (i) the moment of evaluation diverge from and vary independently of ( i i ) the moment of the context. This explains why Kaplan permits content to vary over times as well as "worlds." We shall remain unclear as to the point of our semantic constructions unless we bear this in mind. We note that Kaplan is working in a Tx W framework (see §7A.6). It is, we think, an indication of the relative helpfulness of the moment-history framework that it explains a phenomenon that from the point of view of Tx W seems just puzzling. 8D.1

History of the context?

Policy 8-3 arises partly in virtue of the fact that there is a pairing of two parameters that have the same range of variation, namely, the moment of context and the moment of evaluation. This is sometimes called "double indexing." The phenomenon in general is of no special interest; after all, each xj parameter in quantification theory has exactly the same range as any other, so that, for example, x1 and x2 exhibit "double indexing" of the domain. When, however, as in the case of moments, one of the paired parameters is a context parameter and the other a mobile parameter, we may speak more particularly of "context-mobile pairing." We deepen our appreciation for Policy 8-3 if we ask the following two "context-mobile pairing" questions, one about histories, and one about assignments to variables. (The following discussion expands on that of S6B.5.)

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8-4 QUESTION. (History of the context?) Because we shall have "modal" connectives that are translocal with respect to histories, it follows that there must be a mobile "history of evaluation" parameter. Furthermore, the mobility of the history of evaluation plays an essential role in our account of assertion (Semantic account 6-12); but why is there not also a "history of the context," to be a paired with the mobile history of evaluation, as context parameter? If there were, for stand-alone sentences we could initialize the history of evaluation with "the history of the context." It would clearly make technical sense to provide a context parameter ranging over histories, to be interpreted as "the history of the context of use." And if we had one, we could enlarge our definition of "context-initialized point" to recognize this pairing. Further, there is plausibility in the Kaplan intuition that with "a little ingenuity" one can always make sense out of pairing a context parameter with a content parameter (p. 511). Indeterminism, however, compels a view absolutely contrary to this: There is no "history of the context." When you utter something, you do not thereby uniquely determine the entire future course of history. Your utterance has many choices and chances ahead of it, and so belongs to many histories. The context of use determines a unique moment, but not a unique history. Just to be explicit, we mean to be challenging principles such as that suggested by Kaplan on p. 597: "Any difference in world history, no matter how remote, requires a difference in context." Turning this around says that the identity of the context of use is enough to fix the course of world history, both past and future. That, if indeterminism be true, holds well enough for past history: The past, though largely unknown, is fixed. But it fails for the future: A single, well-identified context of use is typically part of a large variety of possible future courses of history. There is no unique "future of the context." The note on pp. 334-335 of Salmon 1989 is similar to Kaplan, though more convoluted. Salmon specifies the "quasi-technical notion of the context of an utterance," to be distinguished from the "utterance" itself, by saying that if any facts had been different in any way, even if they are only facts entirely independent of and isolated from the utterance itself, then the context of the utterance would, ipso facto, be a different context, even if the utterance itself remains exactly the same. Salmon concludes that ... although a single utterance occurs in indefinitely many different possible worlds, any particular possible context of an utterance occurs in one and only one possible world. If, however, "facts" are what is fixed at the moment of utterance, whether they are "isolated" or not, they cannot fix what the future brings—if indeterminism be true. Distinguishing "utterance" from a quasi-technical notion of "context of utterance" cannot make it otherwise. From this perspective, it looks as if Salmon is wrong—if indeterminism be true. It is, however, a delicate matter to label the

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difference between Kaplan-Salmon and ourselves as a difference of opinion. Here is a place in which there is a dramatic difference between the ideas of "world" and of "history." Kaplan and Salmon and, for example, Lewis seem to picture a "world" as something like a single space-time. We, in contrast, picture a world as something like an indeterministic tree. What is common to these two pictures is the use of "world" for a system of events connected externally by some sort of causal-type relation. It is for this reason that on either picture, it is plausible that a given utterance, or context of utterance, will determine its one and only "world." This seems close to common ground. If, however, histories can branch indeterministically in the way that we suggest, then a single utterance, together with all the most distant "facts," belongs to many histories, no one of which is specially determined as "actual" by the moment of utterance. Of course one could define "context of utterance" as a pair consisting of the moment of utterance together with a particular future history, the "actual" future history, and Salmon's note seems to suggest that he does in fact rely on the notion of an "actual" future history. He seems to rely, as do others, on the view that among all possible futures, one is marked out as a "Thin Red Line" in exactly the sense that we decried in chapter 6.

8D.2

Assignment-to-variables of the context?

As we discussed in §6B.5, there is still another family of mobile parameters, the assignment parameters, for which we do not provide a matching context parameter. We expand on that discussion. 8-5 QUESTION. (Assignment-to-x\ of the context?) Because we shall have "quantifier" connectives that are translocal with respect to the assignment-to-X1 (for example) parameter, there must be an "assignment-to-:ri of evaluation" to be something like a "content" parameter, in which the quantifier is translocal. But why is there not also an "assignment-to-rri of the context" to be a paired context parameter? If there were, we could initialize the mobile assignmentto-xi parameter with the context assignment-to-x1 parameter for stand-alone sentences. It would clearly make technical sense to provide a matching assignment-to-X1 as a context parameter. In some passages, indeed, Kaplan suggests that image (see especially p. 592), although he does not go so far as to provide both an assignment-to-x1 context parameter and an assignment-to-x1 mobile parameter in which the quantifiers can be translocal: There is only one assignment-to-X1 parameter, not two. Here the explanation lies not in the nature of things, but rather in (presumably universal) linguistic practice. Assignments to x1 are not anchored in the context in any serious way. As Kaplan clearly says (p. 593), there is no "factof-the-matter" that contextually determines a unique assignment to x1. So the symbolic language we are describing fails to provide an assignment-to-x1 as a context parameter because there is nothing in our language (or in any language we know) to which such a technical device would correspond.

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It is good to recognize, however, that we speakers of English (supplemented with variables) could adopt a convention that supplied each variable with a value in each context of use. We could then go on to insist that, for stand-alone sentences, the mobile assignment used for evaluation is initialized by the context assignment. Suppose, for example, we require that the context determine that x1 and x2 are both assigned zero. That seems as good a convention as any, since there is (as Kaplan says) no "fact of the matter." Then if we took something like

as a stand-alone sentence, and accordingly used Policy 8-3 to restrict ourselves to context-initialized points of evaluation, we should find out that (8) is automatically true. Ugh. Having realized that we could adopt such a convention, we are glad that we speakers of English haven't done so. And the symbolic language we describe is, in this respect, just like ours. All of its context parameters are "fact of the matter" parameters really and objectively determined by the context of use; none are subject to some doubtful convention manufactured by a logician.

8D.3

Points and policies summary

(i) The special nature of context-initialized points is recognized. (ii) Policy 8-3 is firmly in place for differential consideration of stand-alone and embeddable sentences, (iii) That a point of evaluation, <m, mc, a, m/h>, contains a context parameter for the moment of use, but none for "history of the context of use" and none for the "assignment of the context of use," is no accident or idle logician-imposed convention, (iv) Nor is the fact that there is no analogy to Policy 8-3 for histories or assignments.

8E

Generic semantic ideas

In the following section we detail recursive semantic clauses for the various operators that we treat. Here we outline generic features of the semantic concepts that are defined by that recursion. Except for the "in-context" concepts, these are all standard. We are supposing two kinds of categorematic expressions, terms and sentences. The semantic value of a term is always an entity in Domain, while the semantic value of a sentence is always one of the two truth values, T or F. Since we are going to include operations that are translocal in assignments, moments, and histories, we know that we shall have to relativize semantic value to these parameters. We further explicitly relativize semantic value to the moment of context, even though it is immobile, partly for explicit indexicality, partly to exploit the idea of context-initialized points and the attendant Policy 8-3, but fundamentally because we are thinking of evaluating terms and sentences in many different contexts. And we explicitly relativize semantic value to

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the structure and interpretation parameters because we wish to have not only "truth," but also "equivalence," "implication," and "validity." We want a way of referring to the semantic value of any categorematic expression, be it term or sentence. Once we have "semantic value" for sentences, we automatically have truth, falsity, and the idea so important for indeterminism, settled truth. SEMANTIC VALUE, DENOTATION, AND TRUTH. (Definition.

16)

Reference: Def.

• Semantic value. For any categorematic expression, E, be it term or sentence, Val m , m c , a , m / h ( E ) , is "the semantic value of E at the point (937, mc, a, m/h>." Val m,mc a m / h ( E ) is defined recursively by clauses given in §8F and §8G. Note that by the earlier clause of Def. 16 that appeared on p. 229, we may write Valtt (E) in place of • Denotation. Where t is any term, Val m , m c , a , m / h ( t ) E Domain. Valtt(t), or V a l m , m c , a , m / h (t) is "the denotation of t at the point <m, mc, a, m/h>." Also, as before, Valtt(t) stands in for Val m, m c , o , m / h ( t ) . • Truth. Val m,mc,a,m//i(A) is "the truth value of A at <m, mc. a, m/h>." Where A is any sentence, Alternate much-used notation for truth and falsity: - m, mc, a, m/h tt= A iff Va/m, m c , 0 ,m/h(A) = is true at point <m, mc, a, m/h>."

T.

Either is read "A

- m, mc, a, m/h ¥ A iff Va/gjt,m c ,a,m/h(-2, that witness does not stand in the past of mo. So at mo it is true that a might have seen to it that Q, but it is false that a could have seen to it that Q. In contrast, at mo it is true that a could have seen to it that R, since w0 witnesses [a stit: R] at each of mi-m3. Here is an example that fleshes out this abstract description of Figure 9.1. EXAMPLE. Don Quixote attacks the windmill. Commending himself most devoutly to his lady, Dulcinea, whom he begged to help him in this peril, he covered himself with his buckler, couched his lance, charged at Rozinante's full gallop, and rammed the first mill in his way. At the moment, WQ, that ends his commending, the Knight of the Mournful Countenance had the choice either to stand down or ride on. Once having begun his charge, however, there was a slightly later moment, Wi, at which Rozinante might by chance have collapsed. In the case of no collapse, there

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Figure 9.1: Don Quixote attacks the windmill

was then a later moment, W2, at which Quixote had the choice either to swerve toward or to swerve away from the disastrous confrontation. Let my, as shown in Figure 9.1 be the moment at which he rammed the windmill. Consider the history on which, as he finished his commending, the Knight regained his wits and stood down, and follow that history to a moment, mo, that is co-instantial with the moment, 7713, at which he rammed the windmill. At that moment, mo, it would be true to say that he might have attacked the windmill, but false to say that he could have. What decides the matter is that there is nothing he could have chosen at the end of his commending (WQ—which is the only choice point in the past of the moment of non-attack under consideration) that would guarantee his attack. Both chance (at wi) and the uncertainty of the outcome of a future choice (at w%) stand in the way of such a guarantee.1

9C

Might have been otherwise

To appreciate this next conjecture, consider a stit with a non-agentive state of affairs as complement, and let the anaphor, "otherwise," refer to just that non-agentive complement rather than to the entire stit sentence. CONJECTURE. If yon fellow sees to some state of affairs, then it might have been that the state of affairs not obtain—at that very instant. The final phrase accomplishes a task more easily than idiomatic English: Make sure that the "might" means that the absence of the state of affairs obtains 1 Permit us yet another restatement of methodology: Although we think the distinctions we are drawing are important for, e.g., moral analysis, we by no means fancy that our chosen expressions have a perfect fit with ordinary English.

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Applications of the achievement stit

as a co-instantial alternative to the very moment in question. In notation it is unambiguous: STIT VERSION 1. [Q stit: Q] implies Might-have-been:~Q. By the semantics given in §8F.6, this is evidently equivalent to STIT VERSION 2.

[a stit: Q] is inconsistent with

Was-always-inevitable:Q.

These stit versions say that if [a stit: Q], then it has not been inevitable (determined) from all eternity that Q should obtain at the instant in question. UPSHOT. The conjecture is, in its stit versions, true. It is an obvious consequence of the negative condition (§8G). EXAMPLE. If it was inevitable from all eternity that the hog gelder's reed flageolet sounded four times while Don Quixote was at his meal, then the hog gelder did not see to it that his reed flageolet sounded four times while Don Quixote was at his meal. A hard determinist valiantly endorses the consequent; a soft determinist becomes cross, changes the topic, and exits the lists.

9D

Might not have done it

Next an important conjecture with a straightforward disposition. CONJECTURE. If a does something, then it might have been otherwise; that is, a might not have done it. Here let the "otherwise" refer anaphorically to the entire stit sentence, not just to its complement. STIT VERSION,

[a stit: Q} implies Might-have-been:~[a stit: Q}.

UPSHOT. True. We belabor the obvious by offering two proofs. First, since [a stit: Q] implies Q (by the positive condition), so that ~Q implies ~[a stit: Q], this is an immediate consequence of Upshot §9C. The second and more important proof is this: The consequent is a truth of logic, so that the implication is vacuous! This is related to the Triponodo principle of Makinson 1986, except here instead of the "trivial (legal) power not to do," we have the "trivial possibility of not doing." The argument that it is logically true is an easy reductio. If [a stit: Q] were settled true throughout an instant, then by the positive condition, Q would be settled true throughout that same instant—which would leave no room for satisfaction of the negative condition.

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EXAMPLE. The literature contains cases that stand as putative counterexamples to "that a did it implies that a might not have done it." The following gives the flavor of how stit theory might respond to a portion of these "counterexamples." Consider the apparently agentive statement: Don Quixote "armed himself cap-a-pie, mounted Rozinante, placed his ill-constructed helmet on his head, braced on his buckler, grasped his lance, and through the door of his back yard sallied forth into the open country." If the statement is taken at face value as agentive, then it must possibly be false. For example, if one looks to the description of Don Quixote as "having lost his wits completely" in order to judge the agentive statement false (on the grounds that a man without wits cannot see to anything), one sees that its possible falsity is trivial. If, however, one uses the fact that Don Quixote's brains have dried up as an excuse to reinterpret the apparently agentive content of the statement as really non-agentive, perhaps something like a metaphor, so that Quixote mounted Rozinante as the storm mounts a distant hill, then the description is not agentive and we concede that stit theory does not pretend to have a strategem—which would doubtless involve intentional elements—for probing the statement in question. (At this point we might have included a discussion of "might not have refrained from preventing," but we refrained, noting only that even though "refrain from preventing," in the sense of [a stit: ~[a stit: ~Q]], is a distinct agentive modality, "might not have refrained from preventing" does not create distinct analytical problems).

9E

Could not have avoided doing

This conjecture, inserted here because its disposition involves an application of the preceding result, is a proposal for a sufficient condition of doing. CONJECTURE. "The fact that a person could not have avoided doing something is a sufficient condition of his having done it" (Frankfurt 1969, p. 150). This appears to be an instance of "necessity implies truth"; but analysis reveals that the conjecture is plausible and interesting only because it is ambiguous. Its status depends on whether "avoided doing it" means just "didn't do it," and so is non-agentive; or whether, agentively, it means "refrained from doing it," that is, "saw to it that he or she didn't do it." This ambiguity is difficult to detect in ordinary English; but when it is revealed by using stit normal forms, either the plausibility or the interest of the conjecture disappears, as we see by considering the following two versions. STIT VERSION 1. ~Might-have-been:-~[a stit: Q] implies [a stit: Q}. STIT VERSION 2. ~[a could-have-sht: ~[a stit: Q]} implies [a stit: Q}.

262

Applications of the achievement stit

UPSHOT. Recall that the antecedent of Stit version 1 comes to Was-alwaysmevitable:[a stit: Q], which makes that version of Frankfurt's conjecture sound like an interesting truth. Stit version 1 is indeed true, but only vacuously so, since by the trivial possibility of not doing principle of §9D, its antecedent is a logical falsehood. Stit version 2 is evidently false; for a counterexample, choose Q as any tautology. It is then past doubt that the antecedent of version 2 is trivially true and the consequent trivially false. Thus, in spite of the plausibility derived from thick and interesting stories, attention to austere form suggests that there is no reading of Frankfurt's conjecture on which it is both interesting and true. 2 EXAMPLE. Since the first version is an easy application of §9D, we illustrate only stit version 2. On the side of the antecedent, it is evident that not even the great Mameluke of Persia, either before or after his nine-hundred-year enchantment, could have refrained from seeing to it that if the golden helmet of Mambrino was made of brass, then it was made of brass; but on the side of the consequent, that dignitary certainly did not in fact see to that, nor to any tautology.

9F

Could have prevented

The following conjecture is confusing in ordinary language but easy to settle correctly when expressed symbolically. CONJECTURE. That we are responsible for some state of affairs implies that it must have been possible for us to have been responsible for its absence. STIT VERSION. [a stit: Q] implies [a could-have-stit: ~Q}. Examples of this conjecture sound plausible in English: It appears to follow from the fact that if Sancho Panza remained at rest beneath the cork tree, then he could have seen to it that he moved (Hume). UPSHOT. But as all contemporary logicians of action know, the most elementary story tells us that the conjecture is false. In stit theory, the relevant point emerges through the negative condition, which requires only that the falsity of Q be risked, not that its falsity be guaranteed. EXAMPLE. La Tolosa, the fair cobbler's daughter from Toledo, saw to it that Don Quixote was girded with his sword; but given the rough company of carriers, to say nothing of La Molinera, that poor wench was evidently in no position to see to it that the knight failed to be girded. 2 Perhaps the evident plausibility of the Frankfurt examples derives from the fact that so many verbs give rise to both agentive and non-agentive readings, a matter that we have suggested can be at least partly resolved by applying the stit paraphrase thesis, Thesis 3.

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Figure 9.2: Journey to the ducal castle

9G

Could have refrained

A subtler question is the following: QUESTION. If a saw to something, could a have refrained from seeing to it? • Some think yes. Chisholm 1964a, for instance, says that if some varlet loosed his firelock, then "there was a moment at which it was true, both that he could have fired the shot and also that he could have refrained from firing it." (Observe that Chisholm's "also" is not "could have seen to it that the shot was not fired"; he does not make the superficial mistake of supposing that the Conjecture of §9F is true.) • Some think no. Frankfurt 1969 supposes it possible that there should be such a thing as "the fact that a person who has done something could not have done otherwise." On our view this question is not to be happily represented without taking into consideration the stit analysis of refraining, so that we are not surprised to find that on those few occasions that the literature notices the existence of the question, it seems to resort to sheer postulation. STIT VERSION. Does [a stit: Q] imply [a could-have-stit: ~[a stit: Q}}? We interpret the question as asking whether or not the fact that [a stit: Q] is true at m implies that there is a moment in the past of m that stands as witness to [a stit: ~[a stit: Q}} at a moment co-instantial with TO. That is, is there some single choice point in the past of TO that, had a different choice been made, would have guaranteed the agent's failure to stit Ql UPSHOT FOR STIT. The implication fails, with an easy example, though not quite so easy as the counterexample to the Conjecture of §9F. In Figure 9.2, Q is settled true at TOO and TOI, and settled false at 7712, all of which are coinstantial.

264

Applications of the achievement stit

Abstractly put, each of WA and WB picture a choice for a; you can see from the diagram that Choice0^,A(mn} — {TOO}, so that WA witnesses the truth of [a stit: Q] at mo (looking to m^ for satisfaction of the negative condition). On the other hand, Choice^,A(mi) = {mi, 7712}, so that since [a stit: Q] is true at mi, WA does not satisfy the positive condition for witnessing that [a stit: ~[a stit: Q}\ is true at m^. The choice point WA is "too soon." (The choice point, WB, however, does that job.) Therefore, [a could-have-stit: ~[a stit: Q}} fails at m0, and therewith the implication stated in the stit version. Here is a concrete example based on Figure 9.2. EXAMPLE. We take some literary license in the following idealization, recounted in such a way that Figure 9.2 serves as both spatial map and a model of choices in branching time. One afternoon the Knight of the Lions and his squire, Sancho Panza, journeyed to the castle of the duke and duchess.3 In order to process our English tenses, station yourself at some moment later than m0- At a certain point A the knight and his squire chose the north path, which (ideally) guaranteed their arrival at the castle by sunset. We suppose that their only other choice at point A was the northeast path, which itself split, after a few minutes, at point B. At point B they could have either elected the north path from B, which also would (ideally) have guaranteed their arrival by sunset, or they could have chosen the northeast path from B, which would have led them astray with no possibility of arrival by sunset. Thus, when Don Quixote and Sancho actually arrived at the castle by sunset, there was no choice point in the past of their entrance to the fortress at which they could have guaranteed refraining from arriving by sunset. That is, no choice in the past of their arrival could have positively prevented them from choosing to arrive by sunset. Take notice that the moment of departure from B, at which indeed they could have chosen to refrain from arriving by sunset, is not in the past of the moment of their actual arrival. Heed also that the example is purely structural—the desires, beliefs, and intentions of the agents are irrelevant. We add that for this conjecture, it makes a difference whether one considers stit or dstit. DSTIT VERSION. Does [a dstit: Q] imply Can:[a dstit: ~[a dstit: Q}}7 Here the dstit-suitable Can: is taken from §8F.4. UPSHOT FOR DSTIT. In contrast to the stit version, the implication holds. As in §8F.4, we take Can:[a dstit: Q] as simply Poss:[a dstit: Q], noting that the formal countenance of "can" for dstit can be less wrinkled than that of "can" with the achievement stit, because one need not worry about a double temporal reference. The implication then comes to this: [a dstit: Q] implies Poss:[a dstit: ~[a dstit: Q}}. A proof can be found in Horty 2001. 3 We follow Pellicer in identifying the unnamed duke and duchess with Don Carlos de Borja and Maria Luisa de Aragon, whose ducal descendant celebrated the third centenary of Qmxote in Pedrola in 1905.

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EXAMPLE. We redescribe the journey to the castle in terms of dstit, changing only the Q, which is now to the effect that ( WilLDon Quixote and Sancho Panza arrive at the castle by sunset), which we evaluate at WA and the history on which they choose to go left. On this construal the later moment WB is irrelevant: At WA, given the choice they did make, they dstit (WilLthey arrive by sunset); also at WA they could have chosen to dstit they did not dstit (Will:they arrive by sunset). Moment WB is relevant in considering whether or not at WA they can dstit (Will:they fail to arrive at sunset). They can't.

9H

Might have refrained

We pray your close attention to a question whose answer depends on whether an infinite number of choices is made in a finite time (busy choosers, Def. 14). QUESTION. Suppose that a sees to it that Q; does it follow that a might have refrained from seeing to it that Q in the sense that there is a co-instantial alternative at which a refrains from seeing to it that Q? STIT VERSION. Does [a stit: Q] imply Might-have-been:[a stit: ~[a stit: Q}}? This is a tricky question. Its answer depends, of all things, on whether or not there are busy choosers, §7C.5. UPSHOT WITHOUT BUSY CHOOSERS. If there are no busy choosers, the implication is valid. PROOF. Lettered steps are keyed to Figure 9.3. (a) Grant [a stit: Q] true at mo /ho, and let WQ be the witness in question, (b) Let mi be some moment in i(m0) a* which [a stit: Q] is settled true, and which has the further feature that it "has a closest witness" in the sense that there is a witness, w\, to [a stit: Q] at mi such that between wi and i(mo) there are no further witnesses to the settled truth of [a stit: Q} at any moment in i(mo)- Because there are no busy choosers, m1 must exist. By the negative condition, (c) there is a moment, m2, lying in i(mo) and above wi at which Q is not settled true. We claim that w\ is a witness to [a stit: ~ a stit: Q}} at 7712- The negative condition is easy: [a stit: Q] at mi is just what is required. Suppose, for reductio, that the positive condition failed; that is, (d) suppose that [a stit: Q] were true at some moment m3 £ Choice^im?.}, with witness at, say, w2. Where could w2 be? Because both wi and w2 precede ma, by no backward branching, either wi < w2 or w2 ^ wi. ( e i ) The first alternative is impossible, because w\ is a closest witness. (e2) The second alternative is equally impossible, because then the positive condition of w2 witnessing [a stit: Q] at ma would conflict with the failure of Q to be settled true at m2. (/) It cannot therefore be gainsaid that [a stit: ~[a stit: Q}} is settled true at m2, which is a co-instantial alternative to m0. Therefore, Might-have-been:[a stit: ~[a stit: Q}] holds at mo/h0.

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Applications of the achievement stit

Figure 9.3: Implication without busy choosers

EXAMPLE WITHOUT BUSY CHOOSERS. The journey to the ducal couple's castle portrayed earlier in Figure 9.2 illustrates the subtle difference between "could have refrained" and "might have refrained." In that adventure the travelers might have refrained from arriving before sunset, though it was false that they could have refrained from doing so, since they were not busy choosers. And thus it is: The moment at which the wayfarers might have departed from point B is an excellent witness to the truth of [a stit: ~[a stit: Q}} at a moment co-instantial with the one in question, thus verifying "might have refrained." Since, however, that moment is not in the past of the moment of their actual arrival at the castle, it does not help verify "could have refrained." UPSHOT WITH BUSY CHOOSERS. In the presence of busy choosers and witness by chains, the implication fails. This is a near consequence of Theorem 18-8, which is based on a strong lemma, and in the vicinity of which there is more information. Here we offer a proof of the upshot with busy choosers that is based on a picture. PROOF. We turn to the busy picture of Figure 9.4 for a counterexample to the implication from [a stit: Q] to Might-have-been:[a stit: ~[a stit: Q}]. The abstract facts are these. We are ultimately interested only in a certain moment, mo, but we consider also (i) moments in i( mo ), which is represented by the dashed horizontal, and (ii) choice points of two sorts: those for a, represented by rectangles, and those for /3, represented by circles. Each choice point for a is binary, with the right possible choice containing a single history on which Q is assigned settled true where it intersects i( mo ). The left possible choice for a at that same choice point leads immediately to a choice point for (3, at which there

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Figure 9.4: The Mirror Game

are two possible choices, left and right, each of which leads immediately to a choice point for a of exactly the same kind as before. We suppose the temporal distance between choices for a is halving, and that each entire denumerable historical series of choice points for a and (3 approaches a unique member of i(mo)'i and we assign Q settled false at such members of i(ma)- The moment, mo, is the one lying above the right side of the first choice for a. We claim first for Figure 9.4 that each choice point, wi, for a witnesses the settled truth of [a stit: Q] at the moment, mi, in which the history belonging to the righthand possible choice for a at w1 intersects i(mo)- The positive condition is easy, since we assigned Q settled true at mi and since there is but a single history contained in that possible choice. The negative condition is satisfied by our having assigned Q settled false at those members of i(mo) approached by a denumerable historical series of choice points; one (and indeed many) of those members of i(mo) must be properly later than w\. As a special case, [a stit: Q] is true at mo- We claim next that nowhere in i(mo} is it settled true that [a stit: ~[a stit: Q}}. This is obvious for the members of i(mo), such as mo itself, that lie above some right-hand possible choice for a. Now suppose for reductio that [a stit: ~[a stit: Q}] is settled true at some moment m-2 in Z( m o ) that is approached by a denumerable series of choice points. There must then be a witness, and since we are allowing witness by chains, §8G.4, let the chain be 02, as indicated in Figure 9.4. The positive condition implies that ~[a stit: Q] be settled true at every moment in i(mo) that is choice equivalent to 7712 for a at C2. Choose some member, W2, of 02. Properly between W2 and m2 there will be a choice point, w3, for (3, and properly after w3, there will be a choice point, 104, for a that is not in the past of m,2 (w^ -f. m-i). Let 777.4 be the member of i(mo) lying above the right-hand possible choice for a at 104. The critical point is that 771,4 is choice equivalent to 7712 for a at 02, since—by the no choice between undivided histories principle—no choice for a in c? distinguishes 7774 from 7712So, since [a stit: Q] is certainly settled true at 7714, we have a contradiction.

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Applications of the achievement stit

Thus [a stit: Q] is settled true at mo but Might-have-been:[a stit: ~[a stit: Q}} is not, so that the implication fails. 9-1 EXAMPLE WITH BUSY CHOOSERS. (The mirror game.) The Knight of the Mournful Countenance and the Knight of the Mirrors engage in serious combat. (We thank S. Sterrett for supplying a basic idea of this game.) They play a busy game that begins at noon and ends at sunset—at which time the vanquished is to remain entirely at the mercy of the victor. (Thomsen 1990 reminds us that "there is nothing more serious than play," p. 171.) Some plays of the game consist of infinitely many moves, which our knights-errant manage by halving the time spent on each successive move. Though busy, it is still a simple game. Don Quixote has the first move. On his turn the Manchegan has the following choice: Press on or retire. If he retires then at sunset he is the vanquished. If he decides to press on, it is the turn of the Knight of the Mirrors, whose move always consists in selecting a phantasmic replica of one of two giants for the Manchegan to face: either Pandafilando of the Malignant Eye, or Briareus with many arms, each later phantasm being, in appearance, half as tall as its predecessor. Whatever he of the Mirrors selects, the next turn belongs again to the Knight of the Mournful Countenance. At sunset there are but two relevant possibilities: Either Don Quixote has retired, in which case he is the vanquished, or he has succeeded in facing some denumerable sequence of phantasms, in which case he is the victor. The curious fact to be illustrated is this: It is possible for Don Quixote to retire from the contest, but it is not possible for him to refrain from retiring. Contrary to our untutored intuitions, not even an entire chain of choices to press on, right up to sunset, can witness that Don Quixote refrains from retiring; for such a chain of choices does not establish that it was entirely up to him that he persevere. The choices that in fact were made by the Knight of the Mournful Countenance bestow no hard information about "what he would have chosen" had the Knight of the Mirrors confronted him with an unrelenting sequence of replicas of Pandafilando of the Malignant Eye. Quixote's famous victory, however, does allow him to wrest from the fallen Knight of the Mirrors the confession that "the torn and dirty shoe of Lady Dulcinea of El Toboso is better than the ill-combed though clean beard of Casildea." Trust the concreteness of this fable, we beseech you, only to the extent that you understand its structural properties. Symmetrically, if you think our chronicle is wrong, please try to find an alternative rigorous account of witnessing, refraining, and so on, and not just another picturesque story without a precisely described structure.

91

Had available a strategy for not doing

In §9D we observed that by the negative condition, doing something implies that it might have been that it was not done ([a stit: Q] implies Might-have-been:~[a stit: Q}). In §9H we showed, in contrast, that in the presence of busy choosers, that a does something does not imply that it might have been that a refrained

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from doing it ([a stit: Q] does not imply Might-have-been:[a stit: ~[a stit: Q}]}If you look at any picture, however, it certainly seems as if whenever there is a witness, w, for [a stit: Q], at that witness a has available a strategy he could follow that, provided a never deviated from that strategy, would guarantee his not-doing. That is, [a stit: Q] plausibly implies that there was (in the past) available to a a strategy guaranteeing ~[a stit: Q}. The intuitive idea is that a could avoid seeing to it that Q by shooting for a "counter," as called for by the negative condition, at which Q will not be settled true. This intuition works; but rigorously establishing the fact (Theorem 13-28) turns out to be less trivial than one might suppose, requiring as it does much of chapter 13. In the end we are led to see the sharp difference between (i) something coming about for which there was a guaranteeing choice by a, and (ii) something coming about for which there was (in the past) available to a a guaranteeing strategy. Since past choices are a matter of fact, in case (i) we are entitled to a "because": The thing came about because of a choice by a. In case (iz), however, we are not entitled to a "because." since we have said that the strategy was available to a and that a chose in accord with the strategy, without saying anything about the agent following the strategy. Even though this book sometimes lapses into the language of "following," one must recognize that the concepts of the theory of agents and choices in branching time are too austere to support such loose talk. But more of this in chapter 13.

9J

Summary

The proposition If an agent is morally responsible for doing something, then the agent could have done otherwise

, ,

conceals complex connections among actions, moral responsibility, and alternatives open to an agent. We simplify by dividing the proposition in two: If an agent is morally responsible for doing something, then the agent did it.

,_,

If an agent did something, then the agent could have done otherwise.

/o\

This division isolates the idea of doing something, validating the use of a logic of agency. We use stit theory to clarify proposition (3). CONJECTURE. Proposition (3) can be disambiguated by means of a logic of agents who make choices against a background of branching time. UPSHOT. The conjecture is true. If we interpret "could have done otherwise" as "might have been otherwise," the implication holds; if we interpret it as "might not have done it," the implication still holds, but vacuously. If we read it as "could have prevented," the implication fails. If "could have done otherwise"

270

Applications of the achievement stit

means "could have refrained" then it fails for the achievement stit, but holds for dstit. If "could have done otherwise" is taken as "might have refrained," then without busy choosers the implication holds, but with busy choosers it fails. If "could have done otherwise" is taken as "had available a strategy for not doing," then busy choosers or not, the implication holds.

10

Multiple and joint agency The goal of this chapter is to inspect some structural aspects of multiple and joint agency, a task sufficiently complex to give pause to the three inseparables, Aramis, Athos, and Porthos.* First, in §10B, we treat complex nestings of stits involving distinct agents. The discussion is driven by the logical impossibility of "a sees to it that 0 sees to it that Q" in the technical sense, even though that makes sense in everyday language. Of special utility are the concepts of "forced choice," of the creation of deontic states, and of probabilities. Second, in §10C, joint agency, both plain and strict (every participant is essential), is given a rigorous treatment in BT + I + AC theory. A central theorem is that strict joint agency is itself agentive. In the final section, §10D, we combine these two perspectives, looking briefly at other-agent joint agentives. Throughout this chapter we use "stit" for the achievement stit, §8G.3. As elsewhere in this book, even minimal progress toward the goal of this chapter has required a variety of simplifications, (i) As in stit theory generally, we totally avoid the reification of actions, and (ii) we minimize reference to intention in order to concentrate on causal structure, (in) Of relevant notions, we omit stits that are based on "witness by chains," §8G.4, and (iv) the deliberative stit, §8G.l. (v) Also, we omit concepts requiring the chapter 13 notion of strategies, and (vi) we do not consider the evident fact that agents interact in space-time, a topic yet to be studied. Finally, (vii) for the scope of this chapter, for the sake of expository simplicity we assume no busy choosers in the sense of Def. 14. In exchange, although we directly employ only the achievement stit, we have in this chapter limited ourselves to ideas and applications that, we think, work equally well for either the achievement or the deliberative dstit. We refer to Tuomela 1989a and 1989b for an alternative methodology that, in contrast to stit theory, freely permits one to (i) reify actions and (ii) refer *With the kind permission of Baltzer Science Publishers, this chapter draws on Belnap and Perloff 1993 As in the case of chapter 9, you will observe that we sometimes let the source of our examples influence our mode of speech.

271

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Applications of the achievement stit

to intentions. Those articles also provide access to some earlier studies of joint agency.

10A

Preliminary facts

We will need the following stit facts. (See Def. 12 for the notation mi =%, m?.) 10-1 FACT. (Some facts about the achievement stit) i. Backward monotony,

and

m2 imply

ii. Witness-identity lemma (Chellas 1992). Suppose that Qi implies Q-2, that m, Wi, and io2 are moments, and that a.\ and a2 are (possibly identical, possibly distinct) agents. Suppose further that w\ is a witness to [a\ stit: Qi] at m, and that wi be the witness for this. From (e), (/), and the witness-identity lemma, we infer (g) w\ < w. So (d) and (g) imply, by backward monotony (Fact 10-1), that (h) ms =£t mg. But then the second witness-identity lemma with (/) and (h) gives that [0 stit: Q] must be settled true at m^; which contradicts (b) and completes the proof. For a picture, think of Figure 10.1 as a diagram of a witnessing moment, mo, in which columns represent possible choices for a, and rows picture the choices for 0 (see §10C.2 for the attribution of this picture to von Neumann). If [a stit: [0 stit: Q}}, then, where A = [0 stit: Q], A must fill some "choice-column" for a in

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mo. But then, because A represents a stit by /?, whenever A appears anywhere in a "choice-row" for /? inTOO,it must fill that row. So A must fill the entire diagram ofTOO-If so, then A is by definition settled true atTOQ.This contradicts the principle that, by the negative condition, stit statements are never settled true at their witness. So [a stit: [(3 stit: Q}] won't do as a representation of anything consistent. Before further considering (3)-(6), we turn to the examples (7)-(8), whose complements appear to be negations of agentives. We represent (7) straightforwardly with the form [a stit: ~[/3 stit: Q]], inasmuch as the count makes it true by kidnapping Mme Bonacieux. We observe that although kidnapping is unusual, the form [a stit: ~[/3 stit: Q}}, in contrast to the impossible [a stit: [j3 stit: Q}}, depicts a common enough occurrence. Indeed, everything we do tends to limit i] p choices of others, and much that we do guarantees such limitations. Although (8) appears to be similar to (7), appearances can be deceiving. We cannot represent (8) with [a stit: ~[/3 stit: Q]}: Mme Bonacieux is in no position to guarantee that d'Artagnan fails to follow her. Observe that among (3)-(8), we have so far provided a representation only for (7). In particular, we have pointed out that neither [a stit: [(3 stit: Q]] nor [a stit: ~[/3 stit: Q}] is appropriate to represent any of the others. Does this indicate a weakness of stit theory? We think not; we think it indicates a strength. The failures of [a stit: [0 stit: Q}} and [a stit: ~[/3 stit: Q}} encourage us to find more adequate stit-involving interpretations of (3)-(6) and (8). Chellas 1992 says that it would be "bizarre to deny that an agent should be able to see to it that another agent sees to something." Our acceptance of this view for everyday language is exactly what drives our search. We will look at four interpretations involving other-agent nested stits: deontic, disjunctive, probabilistic, and strategic.

10B.1

Deontic reading of other-agent nested stits

Let us suppose that the facts of (3) are as follows: The queen calls d'Artagnan to her chambers and says, Retrieve my diamond tags.

(9)

D'Artagnan retrieves the tags for the queen. He succeeds in the task she assigned. Shall we represent the situation by [a stit: \/3 stit: Q}} (with a = Queen Anne, /3 = d'Artagnan, and Q •*-> d'Artagnan retrieves the diamond tags)? No, that would be a mistake. It would be a mistake because, as we have seen, [a stit: \f) stit: Q}} claims that the queen guaranteed the truth of [/? stit: Q]. Since her issuance of an imperative does not guarantee that d'Artagnan retrieves the diamond tags, we must be more realistic in our analysis. We begin by reminding ourselves of three things, (i) As Austin 1975 taught us, issuing an order is doing something with words; it is, as everyone now says, a speech act; (ii) speech acts are acts; and (Hi) stit encourages us to understand action by asking what it is that an agent sees to. Let us therefore ask, what did the queen see to when she uttered (9)? Our answer is that she created an obligation. She saw to it,

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by her pronouncement, that an obligation existed where there was none before. Specifically, she saw to it that d'Artagnan was obligated to retrieve the diamond tags. We mean, incidentally, that really giving an order really does create an obligation, not just that the speaker so intends. In making this distinction with absolute sharpness, we separate ourselves from Searle 1965, who appears indifferently to use the language of intention in his "essential condition" but not to use it in his "essential rule" for promises. Having read Hamblin 1987, we also reject the claim of Searle and Vanderveken 1985 that "the point of orders and commands is to try to get people to do things" (p. 14). Instead the point of an order is to create an obligation. Nor does advice have the "causing of action" as its point. Aramis is plainly right that "as a general rule, people ask for advice only in order not to follow it; or, if they do follow it, in order to have someone to blame for giving it." What needs telling is a better story of what deontic states agents really see to when they use not only orders and commands, but also advice, requests, invitations, promises, ..., and indeed assertions and questions. Structural features of stit theory accordingly lead us to the following as a preliminary interpretation of (3).

That, however, won't do, since (3) is a "success" locution. It implies its complement, "d'Artagnan retrieves her diamond tags," whereas the form (10) does not imply [/3 stit: Q}. In a way that is precisely the point: Anne can stit the obligation, but not that d'Artagnan carries it out. We need to add that as a separate conjunct:

Now (11) shows on its face that it involves agency by Anne (as well as agency by d'Artagnan) without being agentive for Anne. 2 Such cases are many and important. When in (4) Jussac orders Biscarat to surrender and Biscarat replies, "You're my commander and I must obey you," he recognizes that his commander has seen to an obligation, [a stit: [/3 stit: Q}} is not the appropriate reading here because Jussac did not guarantee Biscarat's surrender. What did Jussac accomplish with his order? Jussac saw to the creation of an obligation: [Jussac stit: Oblg:[Biscarat stit: Biscarat surrenders]].

(12)

Jussac created the obligation to surrender, but it was Biscarat who surrendered. Regarding (8), [a stit: ~[/3 stit: Q]\ is not appropriate because when Mme Bonacieux sees to it that d'Artagnan does not follow her she does not prevent him from following her; rather she sees to it that a prohibition exists where none existed previously. The form 2

If it were possible for us to use the theory of agents and choices in branching time to represent that d'Artagnan retrieved the tags because of Anne's order, we would do so. We cannot, since the "because" in question seems to be an intentional matter, and so falls among the many aspects of agency for which we offer no theory.

10. Multiple and joint agency [Mme Bonacieux stit: Fr&n: [d'Artagnan stit: d'Artagnan follows]]

277 (13)

is therefore preferable as a representation of Mme Bonacieux' agency in the matter. When d'Artagnan obeys, he is an agent: He refrains from following her. It therefore takes the conjunction of the two agentives in order to reflect these features of the situation. To summarize the results of this section, we make a conceptual advance when we represent the character of the narrow-scope agentive in examples like (3) and (4) with the deontic form

and (8) with

One final point before leaving the topic. Notice that if the equivalence, Frbn:[a stit: Q] Oblg:[a stit: ~[a stit: Q]], is correct, then by substitution in the complement, (13) is equivalent to [Mme Bonacieux stit: 06/(?:[d'Artagnan stit: ~[d'Artagnan stit: d'Artagnan follows]]].

(16)

We think this is right. If so, there is confirmation of the deontic equivalences worked out in §2B.9.

10B.2

Disjunctive readings of other-agent nested stits

Not all apparent other-agent nested agentives can be illuminated by using deontic-complemented stits. Some require use of a disjunctive complement. For example, while we cannot use the form (14) to represent (5), we can use an other-agent nested agentive provided that it has a disjunctive complement. That is, though the cardinal does not obligate M. Bonacieux to agree to spy on his wife, the cardinal does see to it that Either M. Bonacieux agrees to spy on his wife or M. Bonacieux returns to his cell.

/,--,

The normal form [The cardinal stit: ([M. Bonacieux stit: M. Bonacieux agrees to spy on his wife] V [M. Bonacieux stit: M. Bonacieux returns to his cell])],

(18)

helpfully articulates the situation in which a principal agent imposes a "forced choice" on another agent. Figure 10.2 provides a picture: Open to a at w? is a choice that forces /3 into the position at w\ of having to choose between Qi and Q2- This observation, that a form such as

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Figure 10.2: Simple forced choice

can successfully represent the wide-scope agentive of certain apparent otheragent nested agentives whereas [a stit: [(3 stit: Q]} always fails, seems surprising until one reflects that, if one considers future choices, it is common coin that every choice opens up some future possibilities and closes off others. As Calvin once said to his tiger, Hobbes, "each decision we make determines the range of choices we'll face next." Subtly different and somewhat more complicated is the following, for which Figure 10.3 gives a picture. Suppose that Cardinal Richelieu is determined to entangle M. Bonacieux more deeply in his plot. For this purpose, the cardinal sees to it that the unfortunate draper is given a "forced choice." By the act of the cardinal, the draper is compelled to choose whether or not to put himself into the situation of forced choice already pictured in Figure 10.2. For example, Cardinal Richelieu might see to it that M. Bonacieux has a way in which the latter can avoid the forced choice between (i) earning his freedom by agreeing to spy on his wife and (ii) returning to his cell. He can do so by instead choosing to face the executioner. That is, the cardinal is sufficiently powerful to arrange matters so that M. Bonacieux must himself choose between an awful alternative—facing the executioner—and putting himself in a position of forced choice—spying or returning to his cell. When looking at such complicated interactions, it is easy to lose sight of the fact that the principal agent is responsible for seeing to it that the other agent is forced into this terrible predicament. Stit theory has the resources to describe the dimensions of the situation in order to help us understand the relations between the choices of different agents.3 Even so, we hasten to advertise that reliance on stit as one's only linguistic tool makes it awkward to articulate what is obvious from Figure 10.3: The cardinal and the 3 Figure 10.2 approximates the structure of the young man's predicament in Frank Stockton's The Lady or the Tiger. The more complex Figure 10.3 approximates the structure of Sophie's situation in William Styron's Sophie's Choice.

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Figure 10.3: Complex forced choice

draper both had a role in the transition to the outcome, l 'M. Bonacieux spies for the cardinal."

10B.3

Probabilistic reading of other-agent nested stits

The next reading that we shall consider for other-agent nested stits is probabilistic. As background we observe the following: In this world we seldom guarantee the outcomes that everyday language expresses, even when these outcomes are not agentive; this is one of the principal reasons that we think of stit as only an approximation to "sees to it that." Observe that we do not intend this as a "concession." In analogy, it is no "concession" to remark that the law of the lever is only an approximation to what happens on a playground teeter-totter occupied by a couple of screaming, bouncing children whose parents are all too likely to intervene. The importance of the sociology of the teeter-tooter is no excuse for casting aspersions on the law of the lever in its teeter-totter application. The importance of the way in which outcomes are expressed in everyday language is no excuse for failing to appreciate the idealization of agency embodied in stit theory. In any event, our "nonconcession" is not meant to suggest that we never guarantee anything expressible in ordinary language! One of the things we can guarantee (we think) is the high probability of some outcome expressed in ordinary language. When Aramis sees to it that Lord de Winter learns of Milady's iniquity,

(19)

his choice by no means guarantees that de Winter is informed, but the following is fine:

280

Applications of the achievement stit [Aramis stit: it is highly probable that de Winter learns of Milady's iniquity.]

In designing a language to help us understand this matter, there are two choices: (i) permanently build the probabilistic element into the stit construction, so that stit itself indicates high probability instead of guarantee; or instead (if) represent the idea of probability as a separate linguistic element to be combined with stit as wanted. We think (i) should be avoided. In practice it makes it more difficult, not easier, to analyze problems. We recommend (ii). There is, however, a difficulty: The concealed double time reference of the achievement stit makes it at the least confusing to think through the interactions of probabilities and stit. We could reduce confusion by using dstit (see (iv) at the beginning of the chapter), but not having dstit available in this chapter, we cannot now profitably carry out the analytical work required. Furthermore there really isn't much sense in localizing probabilities in outcomes of moments. Moments are just too big: One ought to be suspicious of the intelligibility of saying that moments have "outcomes" that might or might not be probable. (We mean to refer to objective probabilities. If the probabilities are "epistemic," then the history of analytic philosophy testifies that anything goes.) Instead, outcomes and therefore probabilities of outcomes should attach to small, local events. (A relativistic foundation for this notion is given in Belnap 1992.) For these reasons we only indicate by an example how we think ( i i ) applies to other-agent nested agentives. Consider (6) on p. 273. Surely it is not literally true that Kitty guarantees that d'Artagnan seduces her by the provocative course of behavior she chooses. [Kitty stit: [d'Artagnan stit: d'Artagnan seduces Kitty]] is false. But the following, which introduces the required element of high probability, is true: [Kitty stit: it is highly probable that [d'Artagnan stit: d'Artagnan seduces Kitty]].

, ,

It is also true that d'Artagnan seduces Kitty.

(21)

So we agree with Chellas and common sense that (6) can sometimes be true, and we add a suggestion that in such cases a conjunction of (20) with (21) represents the matter correctly. The underlying point is that if in fact Kitty guarantees that (21), then it cannot be that d'Artagnan is agentive in the matter, and if he is agentive, then Kitty can guarantee the probability of the seduction, but not the seduction itself.

10B.4

Strategic reading of other-agent nested stits

In §5C we suggested at some length that the restricted complement thesis could and should be modified so that a strategy (instead of a stit) could be taken as

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the content of a promise. We mean "strategy" in the austere sense of chapter 13. It seems equally plausible to invoke a strategy as the content of a widescope operation of an apparent other-agent nested stit, perhaps along with, for example, the probabilistic reading. Consider (5). What the cardinal sees to may well be best expressed as the carrying out of a strategy whose desired issue is that M. Bonacieux spies on his wife. Perhaps, however, this thought is too superficial. Whether or not it proves helpful remains an open question.

10C Joint agency: Plain and strict In the previous section we explored some other-agent nested constructions with singular agents as subjects. In this section we will study agentives with jointagent subjects.

10C.1

Preliminaries

We start with English grammar. Constituent imperatives (see §1C) are embedded imperatives, analogous to embedded declaratives or embedded interrogatives. Their content, like that of agentive declaratives, can always be represented by stit sentences. An imperative, whether stand-alone or constituent, can have a collective term in subject position, as can an agentive declarative: M. de Treville announces: "I won't have my musketeers going to low taverns." The four friends scraped together nine or ten pistoles.

f^\ (23)

Example (22) might well be taken "distributively," and as analyzable in terms of stit sentences with subjects taken to denote a single agent (we call these "singular stits"), perhaps the subjects being individual variables bound by a quantifier. On a plausible reading, M. de Treville requires each musketeer to see to it that he does not go to low taverns. Examples like (23), however, drive us to widen the grammar of the language of agency. Here it is evidently the four friends "taken collectively" who succeeded in raising nine or ten pistoles; it is not something that each of them does. We cannot usefully represent (23) with only singular stits. We need to add to our formal grammar of singular stits the category of a "joint stit." Collectives can be represented by mereological sums as in Massey 1976; here we choose to represent collectives by sets. This choice limits applicability; the proposed apparatus cannot treat cases in which collectives change their membership over time (see Parks 1972), nor cases in which their membership is history-dependent. The limitation is for expository convenience only, and could be removed by using the language ML" of Bressan 1972 that we mentioned in §7C.l. We would realize Bressan's "cases" as moment-history pairs. In that language we would first represent Agent as an absolute concept, so that Agente would by definition be the extensionalization of Agent. Collectives of agents would be represented as properties F (possibly extensional, possibly not; possibly contingent, possibly not) such that F C

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Agente (this makes being a collective of agents a contingent property). Then for such F, [F stit: Q] would be denned as equivalent to the following: [Agent OF6 stit: Q}. The point is that at every moment-history m/h there is a unique subproperty T of Agent whose bearers are precisely those agents contingently identical at m/h to some individual concept falling under F. We can then use F in order to trace the same group of agents through the vicissitudes of time and history, since it must be a "substance concept"; see Gupta 1980 for a discussion of the use of substance concepts as principles of identity. The formal clothing of our decision is this, (a) We let F range over nonempty subsets of Agent and (&) we count [F stit: Q] as grammatical. Thus, we propose to represent (23) by [The four friends stit: the four friends have nine or ten pistoles],

(24)

where we let the four friends = {Athos, Porthos, Aramis, d'Artagnan} C Agent. (Note: We give this form for simplicity of illustration. The form [F stit: P]V[F stit: Q] seems more apt for (23) than the form of (24), namely, [F stit: P V Q]. They are certainly not equivalent, neither intuitively nor in stit theory.) There is more than one thing that one might mean by [F stit: Q}. First, one might mean that the bearers of F, without any outside help, guarantee that Q, on the basis of a prior simultaneous real choice by each of them. There is also a second, stronger, account. In this version the bearers of F, without any outside help, and with the essential input of each of them, guarantee that Q. Each account is useful and is worth a notation of its own. Since, however, we can give only one meaning to [F stit: Q], we choose the first. Later we introduce [F sstit: Q] as notation for the second account. We postpone to another occasion treatment of cases where Q is best seen as due to sequential efforts of the members of F. Our thought is that one must first be clear on sequential choices by a single agent, a topic on which we lightly touch in §8G.5 and in chapter 13. A consequence is that in this chapter we will often treat cases that in reality represent sequential choices as if they were simultaneous, provided the sequencing seems not important and the reconstrual as simultaneous seems enlightening.

10C.2

Plain joint stits

The key concept is the extension of choice equivalence to sets of agents. 10-3 DEFINITION. (Choice equivalence for sets of agents) For F a nonempty set of agents, we let m-i =£, m^ be defined as Va(a € F —> m\ =£, 7712). Choice equivalence for a set F of agents at a moment w is technically easy, but it is conceptually so important that we offer some further words. Let us go back to the idea that at bottom we are representing "possible choices" at a moment. The deepest idea of a possible choice for a single agent a at w is contained in its representation as a set of histories. The deepest idea of a possible choice for a set F of agents is also contained in its representation as a set of histories. We obtain

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one from the other as follows: Given a possible choice for each member of F, we define a possible choice for F as a whole to be the intersection or "combination" of all the individual possible choices. The "independence of agents" condition guarantees that such a combination always exists. The image we have in mind is due to von Neumann. Let an outcome be dependent on the choices of two agents a and (3. Von Neumann represents this graphically as follows. All the outcomes are arranged in a rectangular grid. Agent a can pick the row and agent (3 can pick the column. What happens is indicated in the intersection of the row picked by a and the column picked by B. "Independence of agents" just says that some outcome is indicated at each intersection of a row and a column. For example, if there are three rows (choices for a) and four columns (choices for /?), then there are twelve possible outcomes for their combined choice. With the help of the concept of choice equivalence for sets of agents, we can state the truth conditions for [F stit: Q}. We say that 10-4 DEFINITION. (Joint stit) [F stit: Q] is true at m/h iff there is a choice point w—a "witness to [F stit: Q] at TO"—satisfying the following conditions (compare the definition of the achievement stit in §8G.3): Agency: Agent. Priority: w < m. Positive: Q is settled true at each m\ such that m Negative: Q is not settled true at some moment—a "counter"—on the horizon from w at i(m). If we apply this definition to (24), it tells us that the raising of the pistoles was due to a simultaneous antecedent choice of the four friends. By so much the definition makes (24) a good approximation to (23). Furthermore, Definition 10-4 is good logic: 10-5 FACT. (Carryover from singular to joint stits) Results or analyses concerning singular agents established without the use of the postulate of the independence of agents also hold for joint agents. •Results not transferring include those expressed with the help of "a ^ /3" when these rely on the independence of agents. The point is that the possible choices for a and {3 at w will be independent if a ^ /?, but this is by no means true of the possible choices for FI and F2 when FI ^ F2. The obvious reason is that non-identity between the two collectives does not prohibit their having members in common. By Fact 10-5 we mean for example that any implication or non-implication that holds between singular stit sentences with just a also holds between the joint stit sentences that result when F is substituted throughout for a. For instance, [F stit: [F stit: Q}\ is equivalent to [F stit: Q], and ~[F stit: Q] is not in general agentive for F (i.e., it is not in general equivalent to [F stit: ~[F stit:

Q}})-

We can use joint agentives to express the independence of agents, provided we have the help of the following version of "ability" (this generalizes to a set of agents the ability concept of §8G.3).

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Applications of the achievement stit

10-6 DEFINITION. (Can stit) Let [T can i -stit: Q] be true at a moment-history pair w/h iff there is a moment m lying on the horizon from w at i such that w witnesses the settled truth of [F stit: Q] at m. So [T carn -stit: Q] says that T can see to it that Q is true at i. Then we have the following. 10-7 FACT. (Joint ability) Provided cant-stit: P& Q]. That is, if at w a can see to it that P at z and ß can see to it that Q at i, then at w they can jointly see to the conjunction P& Q at i. Someone might think that the following is a counterexample. Porthos can see to it that the pistoles are used to repay a debt. Athos can see to it that the pistoles are used to purchase meals. But the pistoles being so few, the conjunction is impossible even with their best joint effort.

(25)

If you take the situation described seriously, however, especially with regard to fixing the time references, you will find that it is impossible. Of course if Porthos chooses first, then what Athos can see to is not independent of Porthos's choice, and vice versa. But fix their choices as absolutely simultaneous, as required for our principle of the independence of agents. And fix the "can" not sloppily, but as Austin's all-in, no-holds-barred "can" (Austin 1961, p. 177). Suppose that there are only a few pistoles. Award Porthos the ability to see to it that the debt is repaid. You have by so much restricted the power to be ascribed to Athos; there is in this situation nothing Athos can do by his choice alone that guarantees that the pistoles are used to purchase meals. For unless you either supply more pistoles or weaken Porthos's ability, you must allow that no matter what choice Athos makes, it is not enough by itself to guarantee the availability of the pistoles. Since you have given Porthos the ability to use the pistoles to repay the debt, you have described a situation in which for Athos to use the pistoles for meals requires the de facto cooperation of Porthos. The second sentence of (25) is therefore not satisfied, so that (25) is not a counterexample to the principle of independence of agents as expressed in Fact 10-7. We believe that any conceivable counterexample to the principle of Fact 107 will be equi-peculiar with the quantum-mechanical phenomenon discovered by Einstein-Podolsky-Rosen, for in fact it would need to have the same form; namely, spatially separated events that are each absolutely indeterministic and perfectly correlated. (This formulation comes from Belnap 1992.) Ordinary language can easily fool us about this by permitting (normally useful) waffly readings of "can"; here is a place where theory helps.

10C.3

Strict joint stits

But [T stit: Q] still does not tell us all that we may wish to know. For instance, (24) does not imply that each of the four friends was involved. It might have

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been, for example, that d' Artagnan was not essential in raising the pistoles in the sense that [The three musketeers stit: the four friends have nine or ten pistoles],

(26)

where, as everyone knows, d' Artagnan E the three musketeers = {Athos, Porthos, Aramis}. Here is the easy fact about [F stit: Q] that informs us of this possibility: 10-8 FACT. (Weakening of ]oint stits) Given F1 C F2 C Agent: if [F1 stit: Q] then [F2 stit: Q}. That is, joint stits are closed under "weakening" by the addition of further agents. We need to define some related properties of agents in two versions before we can go further. The first relativizes the concepts to F and Q. The second drops the F, relativizing only to Q. The point is to be careful as to which concept is at stake. (We remark that although the terminology to be introduced seems apt in context, one needs to be sensitive to the considerations mentioned in §10C.l concerning sequential choices.) 10-9 DEFINITION. (Essential and inessential for stits) a is essential for [F a is inessential for 10-10 DEFINITION. (Essential and bystander for outcomes) a is essential [inessential, a mere bystander, not a mere bystander] for Q 3F[F stit: Q] & for every [not all, no, some] F such that [F stit: Q], a is essential for [F stit: Q]. Thus (26) says that d'Artagnan is inessential for (24), but (26) does not say that d'Artagnan is a mere bystander for the four friends having nine or ten pistoles. What then about the idea that F sees to it that Q, with the added provision that each of its members is essential? We think the intuitive concept thus described is rigorously definable (up to an approximation) just by saying that every member of F is essential for [F stit: Q] (but without requiring that every member is essential for Q). For preference we adopt an equivalent way of adding that there are no inessential members: F sees to it that Q, but no proper subset of F does so:

We will soon define "strictly stit" (an expression we introduce for joint agency when each of the agents is essential) by just this sentence, but first we must face a difficulty: Only the first part of (27) has an agentive form; the second conjunct is instead a denial of agency. So the whole may not itself be agentive! The difficulty is, however, easily overcome. In fact (27) is equivalent to each of the following.

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Applications of the achievement stit

The equivalence of (27) and (29) establishes that (27) is agentive for F in spite of the fact that a conjunct of (27) is a denial of agency. We state this as a 10-11 THEOREM. (No inessential members) The sentences (27), (28), and (29) are mutually equivalent. In other words, where we let

the following are equivalent:

"NIM" is an acronym for "no inessential members." The proof of Theorem 10-11 goes in a circle as follows. (27) —> (28). Suppose (27) true at m1/h with prior witness w, so that in particular (a1) [F stit: Q] and (a 2 ) NIM are each true at m1/h. From (01) we have that (b) Q is settled true at all m2 such that m1 =£, m2, and there is a counter m3 on the horizon from w at i(m1) at which (c) Q is not settled true. We show that the same witness and counter will serve for (28). The part about the counter is evident, since if (c) Q is not settled true at m3 then neither is its conjunction with NIM. What we need to show for the positive condition, since we already have (6), is that supposing (d) m1 =£, m2, we have (x) NIM is settled true at rn 2 . Suppose for reductio that (x) fails, i.e., that (e) and (/) [F1 stit: Q] true at m 2 with witness w1. By (a1) and (d) and the second witness-identity lemma we know that [F stit: Q] is true at m2 with witness w, and therefore by (/) and the witness-identity lemma we know that (g) w = w\ (any two stits to Q at m2 must have the same witness). But m1 = F1 m2 from (d) and (e), and hence (A) m1 = F1 m2 by (g). Now (h) with (/) and the second witness-identity lemma puts [F1 stit: Q] true at m1. But (a 2 ) and (e) imply that [F1 stit: Q] is false at m1, contradiction. (28) —> (29). Suppose (a) (28) true at m1 with prior witness w. By the positive condition, each of (b) Q and (c) NIM is settled true at every m2 such that m1 =F, m2, and by the negative condition there is a counter m3 such that at m3/h3 for some h3 to which m3 belongs, (d) (Q & NIM) is false, that is, either (e1) Q is false or (e 2 ) [F1 stit: Q] is true, for some nonempty proper subset F1 of F. We need to establish the positive and negative conditions for (29). The part about the negative condition is easy; since [F stit: Q] implies Q, (d) implies that ([F stit: Q] & NIM) is false at m3/h3. Choose m2 such that m1 = F m2. We need to show that ([F stit: Q] & NIM) is settled true there. Now (c) already tells us that NIM is settled true at m2, and indeed from (6) we know that Q is settled true at every moment m2/ such that m2

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which gives us the positive condition for w to witness [F stit: Q] at m2. We therefore are missing only the negative condition for w to witness [F stit: Q] at m2, namely (x) there is a moment on the horizon from w at i ( m 2 ) at which Q is not settled true. In case (e1) we obviously have that (x); we need to show that case ( e 2 ) also implies (x). But1 this is a consequence of ( e 2 ) with (c) and the sufficient condition for unsettledness of Fact 10-1. (29) -> (27). This is trivial, having the form that [F stit: P] implies P. We therefore enter the following definition, where "sstit" is to be read "strictly sees to it that," and connotes the absence of inessential members. 10-12 DEFINITION. (Joint strict stit) (T sstit: Q] «-> ([r stit: Q] & VTi[0 ^

10-13 DEFINITION. (Strict agency) sstit: Q}.

Q is strictly agentive for r iff Q NMB, but not conversely. Thus the proposition that ([F stit: Q] & NMB) stands as follows: [F sstit: Q] -> ([F stit: Q] & NMB) -> [F stit: Q}. (ii) There is the statement OMB that outside of F there are only mere bystanders for Q. The proposition ([F sstit: Q] & OMB) says that F is the one and only joint agent for Q; it is evidently not agentive (in the sense of stit). It is, however, something that could be seen to.

10D

Other-agent nested joint stits

It is evident that the investigations of §10B on other-agent nested stits and §10C on joint stits need combining. In this area there is much to be considered. Here we offer only a single illustration, which is that the apparatus developed can distinguish the content of the following in an illuminating way: The four friends required of one of Planchet and Fourreau that he see to it that Brisement has a proper burial.

(34)

The four friends required of Planchet and Fourreau that they see to it that Brisement has a proper burial.

(35)

The four friends required of Planchet and Fourreau that one of them see to it that Brisement has a proper burial.

(36)

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Letting a = Planchet, B = Fourreau, and Q Brisement has a proper burial, the normal forms are respectively [The four friends stit: [The four friends stit: and

[The four friends stit:

(37)

In (36) the obligation is jointly on Planchet and Fourreau as a pair, but the execution is supposed to be by one of them as an individual. This complex content, so subtly different from that of (34) and (35), can be clearly expressed by an other-agent nested joint stit as in (37). The lesson, easy to miss if you take these examples as little puzzles or tricks, is that seriously applicable deontic logic needs other-agent nested joint agentives; and it therefore needs to include a theory with at least the expressive power of joint stits.

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Part IV

Applications of the deliberative stit

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11 Conditional obligation, deontic paradoxes, and stit Consider the following argument, which is adapted from Castaneda 1981.* (a) Alchourron is obligated to do the following: If Bulygin sends him the draft of their latest joint paper, revise it. (b) Bulygin has sent Alchourron the draft of their latest joint paper. Therefore,

(1)

(c) Alchourron is obligated to revise the draft. Castaneda points out that this straightforward reasoning cannot be accommodated by most existing deontic calculi. To see why, let A stand for "Alchourron revises the draft" and B for "Bulygin sends Alchourron the draft." Then we can represent the argument (1) symbolically:

Here, Oblg: stands for the obligation operator. The proposition which would allow detachment of the obligation Oblg:A in (2), does not belong to the standard system of deontic logic. Consequently, the argument is invalid.1 *Paul Bartha is the author of this chapter. Publishers, it is based on Bartha 1993. 1 F011esdal and Hilpinen 1971 sets out the logic," known as KD, in which Oblg:(B>A) D is usually rejected in any system which is based Hintikka 1971.

With the kind permission of Baltzer Science axioms for the "standard system of deontic (BD Oblg:A) does not hold. This proposition on "deontically perfect world" semantics; see

295

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Applications of the deliberative stit

The argument becomes valid if we replace Oblg:(BDA) by BD Oblg:A in (2)(a). No corresponding change is required in English. The English version of the argument seems acceptable as stated. This suggests that English is less sensitive than standard deontic logics to the difference between the two forms Oblg:(BD A) and BD Oblg:A. These two forms will be referred to as OD-statements and D 0-statements respectively. Castaneda has developed one approach that brings the reasoning of deontic logic closer to that of English. He suggested in Castaneda 1981 that the two forms of obligation are equivalent under one assumption. Specifically, he proposed that the equivalence

is valid whenever B is a "circumstance" or "condition" and A is "an action considered deontically as a deontic focus" (p. 46). He develops a calculus which takes the circumstance/action-as-focus distinction as primitive. This chapter takes a different approach, starting from the equally important distinction between agentive and non-agentive sentences. Earlier chapters formalize this distinction by providing two varieties of semantics for the agentive construction "a sees to it that A" (written as [a stit: A]]: the achievement stit, §2A.2, and the deliberative stit, §2A.3. In this chapter, a simple semantics of obligation is developed as an extension of the logic of the deliberative stit, written [a dstit: A]. The basic idea is inspired by the reduction of deontic logic to alethic modal logic in Anderson 1956. After the technical preliminaries of §11A, in §11B we explain our adaptation of Anderson's semantics of obligation. The resulting concept is compared to other systems of deontic logic. A partial completeness result is described in §11C. The semantical system is then used to analyze conditional obligation. In particular, as §11D and §11E show, it provides a precise way to define a "circumstance" such that the argument (2) becomes valid under the assumption that B is a circumstance. In the remainder of the chapter, the semantics is used to shed light on two paradoxes of deontic logic. The point is not to give a final solution to any paradox or problem. Rather, we hope to show that many of the problems of deontic logic are essentially problems about agency rather than obligation. We also hope that readers will see that stit theory can be a useful tool in thinking about such problems, a theme that we further develop in chapter 12.

11A

Technical preliminaries

For this chapter we limit our target grammar as follows. Sentences in our language are constructed from propositional variables by truth-functional connectives ~ and &, as well as modal operators Universally: and Sett:, deontic operators Frbn: and Perm:, and the agentive operator, [a dstit: ]. There will also be tense operators Will: and Was:. As usual, V, D, =, T and _L are introduced as abbreviations, and we will also introduce by abbreviation the Chellas

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agentive operator [a cstit: A]. We use A, B, and so on to range over sentences. We recall that the semantics of [a dstit: A], where a is an agent and A a sentence, is based on the fundamental notions moment, history, agent, choice set, and possible choice, as reflected in a BT + AC model, m, which is a pair consisting of a BT + AC structure © = (Tree, \ Oblg:[a dstit: ~[a dstit: Pax]]: a is obligated to see to it that a does not park on highway x. Sett:(~[a dstit: ~[a dstit: Pax]] D Sa): It is settled true that if a does not see to it that a does not park on highway x, then Sa. That's a lot of words. One can see further into the proposal by a combination of cases and pictures. Take a particular concrete moment, TO, of concern to a and to the rules. There are three major cases, the first two of which are throwaways. No-good-choice case. Since §11B.1, in contrast to Anderson, has put no constraints on the sanction, there is the case in which Sa is settled true at m. In the no-good-choice case all is obligatory for a, but also all is forbidden. There is no available choice for a that avoids the sanction. Nothing is permitted, where, as in §3A, Perm:[a dstit: Q] ~Frbn:[a dstit: Q]. Figure 12.1 provides a picture. What to make of the different decisions, Anderson's requiring that the sanction not be universal and ours making no such requirement? We think that both decisions are plausible, and that the point of view of Marcus 1980 explains why. Anderson's sanction applies to entire worlds or perhaps histories. It would be strange indeed if every world or every history were deemed unacceptable. In this case there would clearly be nothing left for deontic logic to worry over. As Marcus puts what we take to be the same point, if a set of rules is inconsistent in this strong sense, then it "provides no guide to action under any circumstances" (p. 129). This makes reasonable Anderson's postulate that the sanction is not inevitable. On the other hand, when we mark as "sanctioned for a" every history through a particular moment, that sad situation is an extremely local affair. It represents only that in that particular circumstance, there is nothing acceptable to do.

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Applications of the deliberative stit

Figure 12.1: No good choices for a at m

Figure 12.2: Two proper no-can-do cases

This is what Marcus urges to be quite another thing. To deny that such a circumstance could obtain is, as she says, to deny something real, namely, that a moral dilemma can arise in a particular unfortunate circumstance. So, we conclude, the two choices are both in accord with Marcus's account, and they are both right. When all histories are at issue, it is right to assume with Anderson that the sanction is avoidable, but when we consider only the histories that are possible historical continuations of a particular moment, then it is right to allow with Marcus that in some circumstances there may be no way to avoid the sanction. No-can-do case. In the other throwaway case, at m, a cannot see to it that Q: It is settled true that a does not. Nothing a can choose at m counts as choosing to see to it that Q. No such choice is in fact open to a at m. There are two subcases. In one of them the previously discussed no-good-choice case obtains, and there is nothing more to say: All is obligatory and all is forbidden for a, regardless of what the agent can do. In the other subcase, which is new, Sa is false at m/h for at least one history h through m. We will call such a case a proper no-can-do case. Figure 12.2 conjures up two of them.

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325

Figure 12.3: The normal case

You can easily calculate that in a proper no-can-do case, Definition 12-2 neither obligates nor forbids the agent to see to it that Q, which sounds right when the agent no can do. Of course it follows that in the proper no-can-do case the agent is permitted to see to it that Q, and also permitted to refrain from seeing to it that Q, which sounds peculiar given that the agent is powerless with respect to Q. But it sounds peculiar, we think, only because in ordinary English we confuse permission and power. (The effort to straighten us out on this goes back at least to Hohfeld 1919; see especially Makinson 1986.) For example, we often use the same modal verb, "may," for both. When we reflect, however, we easily see that in fact there are numerous situations in which an agent is permitted or authorized to do what in fact is not in their power to do. The agent a might be permitted to park at any meter in the entire state of Connecticut, even when a is enjoying Oregon. The proper no-can-do case highlights this. The normal case. In the central case of interest, the sanction Sa is false on at least one history through m, and there is at least one choice on which a sees to it that Q. We will call this the normal case, even though it certainly does not happen with much frequency. In the normal case, a can see to it that Q, and in the normal case, the sanction is not inevitable; see Figure 12.3. It is a calculation of dstit theory (as opposed to the achievement stit) that in the normal case, though not at all in general, "seeing to it that" and "refraining from seeing to it that" are contradictory. (The sanction is irrelevant to this calculation; one needs only can-do.)

12E

Generalized prohibitions

Let us pass on to the first of the two generalities ingredient in (1). It is not just one highway on which a is forbidden to park. In fact,

326

Applications of the deliberative stit Each highway is one on which a is forbidden to park. (Or "each highway within a given jurisdiction." We shall suppress this subtlety.)

(7)

In other words, we might say, a is forbidden to park on a highway. Or, put more ponderously in order to nail down a narrower scope for the quantifier: It is forbidden that a park on a highway.

(8)

So, obviously, to express that parking is forbidden on highways we need a quantifier. But which quantifier, and where should it go? The universally quantified prohibition (7) and the prohibition of an existential (8) sound more or less equivalent, but certainly the last hundred years of logic suggest that it is not very plausible that they should be. It will be instructive to observe a proposed train of reasoning that would show them to be so. This train contains some routine transitions and some interesting ones. Here is what will strike you: None of the interesting transitions concerns the deontic modalities themselves. All are concerned with either the background theory of branching time, or the theory of agency. In the end we shall find out that (7) and (8) are not equivalent, though nearly so. And coming to see that sort of thing is part of what logic is for. Marcus, in the shadow of some intolerant and irrational decades that had foolishly put modal considerations on the defensive, said that "establishing the foundations of mathematics is not the only purpose of logic" (Marcus 1960, p. 58). This should count as a tiny example of what she had in mind. Let us begin with the henscratches for (7) and (8), respectively. Symbols for (7): VxFrbn:[a dstit: Pax].

(9)

Symbols for (8): Frbn:[a dstit: 3xPax].

(10)

It would be all right to shorten [a dstit: Pax] in (9) to Pax, just because, according to Postulate 12-1, the latter is agentive, that is, [a dstit: Pax] and Pax are equivalent. But it is probably better to use the longer "normal form" in order to remind us that the complement of Frbn: must always be a dstit or something equivalent. This is in accord with the restricted complement thesis, Thesis 5. Let us add here that by the restricted complement thesis a third candidate, Frbn:Ex[a dstit: Pax], is not on the face of it grammatical. We should suspect that this third proposed complement of Frbn: might, in the language of Marcus 1966, describe a mere state of affairs that could not meaningfully be on a's list of prohibitions. And in fact the coming analysis of the 3x/dstit transition confirms this suspicion. Here is a list of plausible equivalences, which we call the "Main Calculation," that would lead us between (9) and (10). We have indicated on the right how each entry is tied to the one above it. In gently moving quantifiers from the outside to the inside we will run into a number of particular transitions: "QTF" means "quantifier-and-truth-function equivalence." Other transitions

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are applications of previous postulates, as indicated. And the three key transitions "Vx/Sett," "Vx/dstit," and "Ex/dstit" are based on hypothesized quantifier/modality relationships. These three we explicitly discuss, with an eye to either confirmatopn or disconfirmation.

Main Calculation

We take up the undiscussed quantifier/modality transitions Vx/Sett, Vx/dstit, and Ex/dstit in turn. Let us first underline that they have nothing to do with deontic logic per se. They are all either about branching time or else about pure agency. They are about the world and our doings, independently of whether we are getting ready to make some rules or set some standards.

12E.1

Barcan formula for Sett:

The first transition does not even concern agency, much less obligation. It is a matter for the general theory of branching time, or perhaps the general theory of highways. What are we to say of the principle Vx/Sett that permutes the Sett: modality with a universal quantifier over highways? In symbols: 12-3 POSTULATE. (Barcan formula for Sett:) VxSett:Qx Sett:VxQx. Not to put too fine a point on it, we think Postulate 12-3 is good. Here is why. Sett: at a moment means, close enough, "true in all histories" through that moment. You can therefore plainly see that Sett: has the intent of an S5 necessity. That means that the permutation of the universal quantifier with Sett: is precisely the Barcan formula in one of its guises. (From Barcan 1946. So named in Prior 1957.) The Barcan formula is not only time honored and of

328

Applications of the deliberative stit

exceedingly noble parentage, but of tremendous logical utility. Proof systems without the Barcan formula are awkward and difficult for us to use. Adding the Barcan formula simplifies and thereby increases our logical acumen. It makes us smarter. That's the benefit. But, you may ask, at what cost? If we use the Barcan formula we must suppose that the range of the quantifier does not vary with the history; so much is clear from the analysis of Kripke 1963. Here is a general point, and we mean to be echoing one made by Marcus, for example in Marcus 1972: Although English means what it means independently of what we say, our technical quantifiers mean exactly what we say they mean. Their meaning is up to us, and it is, as Carnap emphasized, a practical matter. In applying Carnap's advice, it makes a difference whether the quantifiers we are considering are for general use, or for a case at hand. For general use it seems clearly best to simplify our logical life by interpreting our technical quantifiers—the ones with which we calculate—so that their ranges do not depend on the particular historical continuation.2 As Marcus argues in Marcus 1961, for example, we shall still be able to find appropriate and indeed illuminating symbolic formulations for every one of those English sentences that might have led us to abandon the Barcan formula, had we been so weak on our philosophical pins as to be bowled over by derisive counterexamples hurled down the alley in an effort to obtain a crowd-pleasing strike. Need we add that such presumed counterexamples will be found on inspection to be made from a nonmetallic composite material, and to contain a hole the size of your thumb? Proceeding to the local case at hand, observe that the particular example involves quantification over highways. That makes it sound as if in this locality the Barcan formula should fail, since surely what highways there will be depends on what happens and thus on particular historical continuations. Well, yes, the "since" clause is true, but the argument is not. The argument derives from the paradigm of Kripke or Lewis total worlds, each with its worldwide quota of highways. That paradigm, however, is not wanted here. We are looking at a particular momentary event of forbidden parking. Whether or not something is a highway at a particular moment does not depend on historical continuations of that moment. The domain of highways at a moment (as we might say) is indeed independent of what historically happens next. It is, in the phrase of Castaneda, a "circumstance" of the parking (see the beginning of chapter 11). Thus we should accept the permutation labeled "Vx/Sett" in general, and therefore in the particular case. 2 This still leaves numerous options, of which the following are salient: (i) Marcus quantifiers ranging over actualia; (ii) quantifiers ranging over possibilia; (iii) Bressan "individual concept" quantifiers with intensional predication; and (iv) substitutional quantifiers. Arguments that some alternative is better or worse than one of these have tended to be based on mockery involving the double modal, "How could you possibly say such a thing?" That's amusing, but also sad. For many practical purposes of serious formalization, it is appropriate to use the scandalously neglected Bressan individual concept semantics with intensional predication, of which we say just a little more in §12F.

12. Marcus and the problem of nested deontic modalities

12E.2

329

Universal quantifier over dstit

The next transition we discuss, namely Vx/dstit, is also Barcan-like:

Here, however, appearances deceive: Dstit is not really a straightforward S5like necessity modality. As you can see from §8G.l, the semantics for [a dstit: Q] involves not only a "positive condition" saying that the (perhaps vacuous) choice that a makes necessitates Q, but also a "negative condition" saying that "it might have been otherwise than Q" (from which it follows that the choice was not after all vacuous). As indicated in §11A, we can separate the positive from the negative. For the positive we can adapt the Chellas stit, [a cstit: Q], defined in §8G.2, just as we did in §11A. For the negative we can take from §8F.4 the standard branching-time dual of Sett:, namely, Poss:, so Poss:Q m1 E U s ( m0 )]• Admm(s) = {m: s admits m}. • s excludes a history or a moment if it does not admit it. The ideas of exclusion and admission are central to the theory of strategies. As for exclusion, one may start out along an excluded history, but if one chooses in accord with s, then one cannot forever remain on an excluded history. Eventually one reaches a moment at which the strategy goes one way and the history goes another. Nor can one ever reach an excluded moment if one follows a strategy. Admission is in one sense a weak notion; for example, any history that contains no member of Dom(s) is vacuously admitted. Apart from this trace of vacuity, which we remove on a case-by-case basis when we apply the notion—for example, in the next definition—an admitted history is one that might happen given that the strategy is faithfully followed. An admitted moment is one that might be reached provided that the strategy is followed without exception. Admissibility represents what is possible, given that the strategy is followed. It is therefore natural to define what a strategy "guarantees" in terms of admissibility: 13-5 DEFINITION. (Guarantee)

• s guarantees HO iff • s really guarantees HQ iff s guarantees HQ, and furthermor

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Strategies

Think of H0 as a proposition expressed by a moment-independent sentence Q, that is, as the set of all histories in which such a Q is true. Then for SQ to guarantee H0 is for s to guarantee Q in the sense that Q is true in all those histories not forbidden by s. The reference to H(D 0 m(s)) insures that we don't have to worry, however, about histories that branch off who-knows-where long before the arrival of the domain of s. The definition of "real guarantee" adds a "negative condition" in accord with the general tenor of stit theory, §8G. For a strategy to be said to really guarantee H0, its choices must do at least some work of excluding non-H0 histories. The real sense of "guarantee" is exactly what we want for our main result, Corollary 13-29, to the effect that a certain strategy "guarantees" inaction. The following is obvious, but worth recording. 13-6 FACT. (Admissibility and extension) Extending or strengthening a strategy can only reduce admissibility, never increase it. Contrapositively, extending or strengthening can increase exclusion, but never reduce it. And what is guaranteed persists under extension or strengthening. Among strategies there is a sharp difference between those that do and those that do not tell you what to do after you have failed to follow their earlier advice. We call the former "secondary" and the latter "primary." (Our earlier work on strategies used the pair of words "redundant" and "irredundant." Horty 2001 uses "lean" where we say "primary." Although the ideas are stable, the best choice of vocabulary is unsettled.) The thought is that advice given by a strategy after you have already violated the strategy may be useful, but it is surely "secondary" to the primary portion of the strategy. So we call the strategy itself "secondary" or "primary" depending on whether it does or does not contain any secondary advice. 13-7 DEFINITION. (Secondary versus primary) s is secondary iff some moment that it excludes is anyhow in Dom(s); and s is otherwise primary. The strategies (for games in extended form) of von Neumann and Morgenstern 1944 are primary, or mostly so. A typical game theoretical strategy does not tell you what to do if you do not follow the strategy. Suppose a strategy for chess begins with advising king's pawn to KP4 as first move. That very strategy would typically not tell you what to do on your second move if instead you moved your king's rook pawn on your first move. Why should it? Such advice would be redundant (it would be secondary advice). One finds a contrast in the "deontic kinematics" of Thomason 1984, p. 155, which we discuss in chapter 14. In that scheme, obligation is laid on at every moment. If we think of these obligations as together making up a strategy (Thomason speaks of plans), the strategy would be secondary. And for good reason; as Thomason notes, secondary strategies enable consideration of "reparational" obligations, telling us what to do after we have done wrong. Secondariness can be important.

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Secondariness of a strategy should be distinguished from a form of strategy that at a certain moment gives you a "second-best" choice besides its first choice. One might in English describe both a secondary strategy and a strategy that gives a second-best choice in the same language: Each tells you what to do "if" you don't do what it primarily advises. But the cases are really different. If a strategy offers a second-best choice, then the second-best must be labeled so: "PQ4 is best, but PK4 is second best." See Horty 2001 for ideas that involve such grading of the choices on which stits are based. In this chapter, however, we do not deal at all with strategic concepts that reach outside the BT +1 + AC structure, (Tree, ^, Instant, Agent, Choice), in order to label some choices as second-best: The "second-best" of a secondary strategy needs no extra labeling. What makes it second-best is already definable. Namely (and roughly), it is the choice (and the only choice) that the strategy advises at a moment that is excluded by an earlier choice dictated by the strategy. It is the exclusion of a moment that renders advice at that moment automatically secondary. Note, incidentally, that when there are no busy choosers, we can always count our way down from a forbidden moment through a finite series of "fresh mistakes" to the main line of the strategy. This gives an already-definable account of "n-ary strategy" as well as "secondary strategy." On the other hand, if there are busy choosers, it is not so clear that such gradations make sense. 13C.2

Simple strategies

We shall be dealing largely with strategies that are both strict and primary, partly because they are in some respects the simplest. In fact we shall cause these two properties to contribute to the upcoming definition of "simplicity." There is, however, another ingredient. To introduce it, we need to think ahead a little ways to the completeness of strategies. When we turn to "completeness," we need to have in mind some population of moments M that is typically larger than the domain of the strategy but much smaller than the entire universe. These are the moments that we care about. Typically they will not extend backward all the way to the Bang, nor forward past the heat death of the sun.2 When M is given, we may call it a "field" for the strategy. Always a field M will include the domain of a strategy, that is, the set of moments at which the strategy gives advice. We use the phrase "in M" to carry this information: 13-8 DEFINITION, (strategy in M; field for s) • s is a strategy for a in M iff s is a strategy for a such that Dom(s) C M. • When s is a strategy for a in M, we call M a field for s. 2

Less pictorially, a "field" M for a strategy might have the property that in each history h with which M has nonempty intersection, there are upper and lower bounds for h n M. Or it might be that all of the nonempty hdM have a common past or even contain a common lower bound. Or it might be that each nonempty h n M is a maximal chain in M. But even though typically interesting strategies will be bounded or even have one of the stronger properties, our austere refusal to impose such conditions seems to clarify the concepts and results that this chapter considers.

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Strategies

It could be that a strategy gives advice for every moment in M, or even for every moment in all of Tree. Such a strategy is said to be "total": 13-9 DEFINITION. (Total strategy) A strategy s is total [in M\ iff Dom(s) = Tree [= M\. We shall not be interested in total strategies until chapter 14. We are instead presently interested in notions of "completeness," which are altogether different from totality. A strategy complete in a set of moments M will tell us what to do at appropriate moments within a field M—but by no means all. For instance, a "complete" strategy need not tell us what to do at moments in the field that the strategy itself forbids. That is precisely why a field for a strategy must be distinguished from its domain. Nor is a field uniquely determined by the domain; that is why we say "field for s" instead of "field of s." EXAMPLE. (Domain versus field) Picture Tree as representing a game of chess in extended form (von Neumann), that is, a tree, rooted in some concrete occasion, of all plays in accord with the rules of chess. That would be an appropriate "field" for a strategy. A particular strategy itself would be represented by a small subtree of Tree; for example, a tree that starts with "P to Q4." This strategy can be altogether "complete" in Tree, while giving no advice at all to the player who ignores it in favor of starting with KKt to KB3. As a small step toward completeness, we consider backward closure. Note that we confine ourselves to M partly so as not to have to go back to the beginning of the universe. 13-10 DEFINITION. (Backward closure) Suppose that s is a strategy for a in M. s is backward closed in M iff Vmo,mi[(mi € Dom(s) & mo < mi & mo 6 M) —> m0 G Dom(s)}. A "conditional" strategy might well not be backward closed: "If you reach Station A, get off the subway; but for earlier stations, there is no advice." So strategies that are not backward closed are important. But if a strategy is not backward closed, then that seems enough to give it a kind of not-yet-available flavor. Such a strategy gives us advice at a certain moment without giving us any advice whatsoever as to how to get there. It would in that respect be similar to an end-game strategy for chess as given in a book on end games, provided we chose an odd field for it. Namely, suppose we took as field not just the subtree where the book starts giving us advice. Suppose instead we oddly took as field the entire tree for chess that starts with the first move. Then that strategy would not be backward closed in that field. If we want to confine ourselves to the simple case in which a strategy is available from the very beginning of its field, we must think about backward closed strategies. This notion is therefore the third and final part of the concept of simplicity: 13-11 DEFINITION. (Simplicity) s is simple for a in M iff s is a strategy for a in M that is strict for a, primary, and backward closed in M.

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So a strategy that is not simple might fail of simplicity in one of three ways, (i) It might be loose (sometimes giving disjunctive advice), (ii) It might be secondary (sometimes giving advice at moments it forbids). (Hi) It might not be backward closed (some parts of its domain might be isolated high up in M). Simple strategies are not like that.

13C.3

Pre-simplicity

Permit us to say something parenthetical about strategies that are not simple in virtue of being not backward closed in M—parenthetical in the sense that later sections do not build on this one. Suppose such a strategy is in fact strict and primary, and fix the set M of relevant moments. The strategy might have a simple extension in M, or it might not. There is only one case in which it does not: There are two moments TOI and m^ in its domain, and they have a lowerbounding moment m0 in M (but not in Dom(s)) such that every choice for a atTOOexcludes one of m1 and m2. Were this case to occur, we should not know how to specify s(mo) while retaining both strictness (at TOO) and primariness (with respect to each of m1 and m 2 ). This observation points to an alternative concept of "simplicity" for use when we wish to consider strategies that are not backward closed. To remind us that the alternative is helpful only when we are looking forward to simplicity, let us call it "pre-simplicity." 13-12 DEFINITION. (Pre-simplicity) s is pre-simple for a in M iff s is a strategy for a in M that is strict for a, primary, and such that each two members of its domain are inseparable for a in M. EXAMPLE. (Pre-simplicity) Let s be a strategy for you that might be described in this way: "If you reach Station A or Station B, get off the subway; but for the earlier moments at which you have a choice about reaching Station A or Station B, there is no advice." You may not care about completeness at all. If you do, you may not care about it with respect to a field M for s that includes those earlier moments. In these cases you will not care about pre-simplicity in M (though you may still care about it with respect to a smaller field). Suppose, however, that a field M for s includes at least one of these earlier moments, say TOO, at which you can choose between reaching Station A and Station B; and suppose that you care about strengthenings of s that are complete in M. You may or may not be interested in primariness. If you are not, you can think about extending the strategy to mo in any way you like. You could then think of some or all of the "if you reach Station A or Station B" advice as "reparational." You could think of it as "redundantly" advising you what to do after you have fallen off the strategy. In this case—see chapter 14—pre-simplicity would be of no interest. Suppose, however, that you are interested in primariness. Even then it makes sense to extend the strategy toTOO,provided the advice there is not strict but loose: Head either for Station A or for Station B, whichever you choose. That would be a primary (though loose) extension. Nothing wrong with that. But if you are interested in a strategy only if it can be backward closed to a strict and primary extension, you need to think about pre-simplicity.

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Here are key facts about pre-simplicity. They are put here for help in keeping our bearings, and are not otherwise used in this chapter. 13-13 FACT. (Pre-simplicity) i. s is pre-simple for a in M iff s has a simple extension for a in M. ii. s is pre-simple for a in M iff s is a strategy for a in M, and

iii. In application to a strategy that is backward closed in M, there is no difference between simplicity and pre-simplicity. The smallest simple extension as guaranteed by (i) might be called the "backward closure" of s. But we won't need the idea, and in any event that phrase could well be used in other ways when we are interested in loose or inconsistent or separable strategies. Part ( i i ) gives an equivalent account of pre-simplicity so as to exhibit that pre-simplicity is a unified idea, not a mere conjunction: Strictness, primariness, and inseparability are all part of the same package. And (Hi) tells us that when we know a strategy is backward closed in M, we just don't have to worry about simplicity versus pre-simplicity. In any event, we shall nearly always be dealing with simple strategies.

PROOF. Ad Fact 13-13 (i). Suppose SQ is pre-simple for a in M. Define si so that Dom(si) = Dom(s0)U{m0: mo £ M & 3mi[m0 < mi & mi 6 Dom(s0)}}; and for m define Evidently si is backward closed in M, and si is primary if SQ is primary. Also pre-simplicity of S0 guarantees that s1 is strict. So s1 is a simple extension of s0. Conversely, if s0 is not pre-simple, then if it is either not strict or secondary, no extension can remove that defect. And if its failure of pre-simplicity comes from having two members m1 and m2 of its domain that are not choice equivalent at some moment mo in M that lies in the past of each of m1 and m2, then there is no way to extend s0 backward to m0 without violating either strictness (at mo) or primariness: If Si(mo) is a single member of Choicean0, then either m1 or m2 will be excluded, causing s1 to be secondary. Ad Fact 13-13 (ii), left-to-right. Argue by contraposition. Suppose that s is a strategy for a in M, but that the right-hand side is false: mi, m2 G Dom(s), hi S s (mi), hi e s (mi), m0 e M, m0 ^ mi, mo ^ m2, hi _L^ 0 hi. Argue by cases. If mo = mi = mi, then there is nonstrictness at mo since non-choice equivalent histories belong to s(m 0 ). If mo = mi and mo < mi, then mo excludes m2, witnessing that s is secondary. Similarly if m0 < mi and mo = m2. And if m0 < mi and m0 <mi, then there is a violation of inseparability. So in either case s is not pre-simple. Ad Fact 13-13 (ii), right-to-left. Argue by contraposition. If s is not presimple, it is either not strict, or secondary, or separable. If it is not strict at mo then we can find a counterexample to the right-hand side with mo = mi

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= m-2- If it is secondary in virtue of the exclusion of m\ at mo, we can find a counterexample to the right-hand side with mo < m\ andTOO= m^. And if s fails inseparability in virtue of mi,m,2 E Dom(s) being separable at mo & M, then we can find a counterexample to the right-hand side with mo < mi and m0 < TO2. Ad Fact 13-13 (Hi). This is a way of saying that simplicity implies inseparability. The argument is that a witness to separability that by backward closure lay in the domain of s would also witness either nonstrictness or secondariness.

13C.4

Ideas of completeness

We can now define "completeness." For convenience, we shall do this under the presupposition that we are dealing only with simple strategies; otherwise the concepts might need refining.3 13-14 DEFINITION. (Completeness) Suppose that s is a simple strategy for a in M. • s is complete along h in M iff M(~}h C Dom(s). • s completely admits h in M iff s admits h and is complete along h in M. • s is H-complete in M iff s is complete in M along every history in H that s admits. • s is complete in M iff s is complete in M along every history that s admits. • s is simply complete for a in M iff s is simple for a in M and s is complete in M. We offer the fourth of these five concepts as a conceptual analysis of the essential idea of strategic completeness. In contrast, the first three concepts are introduced as technically useful, while the last gives us a rhetorical variant of "simple and complete." (Strictly speaking, "simply" here is overdefined, but we nevertheless try to be consistent about including it.) Completeness is an easy idea: A complete strategy in M provides advice at each moment in the field M at which you can arrive without violating the dictates of the strategy itself. No matter which advised choice you make, no matter the outcomes of decisions by other agents or by nature, as long as you are within the field M, a complete strategy always tells you what to do. Not even a complete strategy in M, however, either gives or pretends to give any advice outside the confines of the specified field M. A strategy that is incomplete in a field M gives you some 3 For instance, the deontic trees defined in Definition 14-1 are "complete" in a sense almost unrelated to Definition 13-14: Deontic trees obligate at each and every moment. Strategies derived from that variety of deontic kinematics are not part of our present topic because they are not simple.

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advice, and then sometimes leaves you adviceless even when you follow it, even when you are still in M. Complete strategies are analogous to the complete theories known to modeltheoretic logic in the following respect: Although interesting specimens of either are few and far between, complete strategies, like complete theories, are important objects of study because of the limiting role they play. One can see that—and be a little surprised by the fact that—for strategies in M, completeness is converse to primariness: 13-15 FACT. (Completeness and primariness) Suppose that s is a simple strategy for a in M. Then the following give good accounts of completeness in M and primariness. • Completeness: (Admm(s)nM) • Primariness: Dom(s) C

C Dom(s).

(Admm(s)r\M).

Let us record that extending a strategy can never make it less complete than it was. 13-16 FACT. (Persistence of completeness) If s0 is a simple strategy for a in M, then if SQ has one of the following properties, so does any simple strategy si for a in M that extends SQ: (i) is complete along h in M; (ii) completely admits h in M; (Hi) is //-complete in M; (iv) is complete in M; (v) is simply complete for a in M. PROOF. Because extensions only add to the domain of a strategy, completeness evidently begets completeness. We need, however, to check the "admission" part of ( i i ) . This part says that if SQ and s\ are simple strategies for a in M such that si extends s0, and if s0 completely admits h, then sj admits h. So let mo € Dom(si)C\h; we need to show that h € SI(TOO). Because si is a strategy in M, mo € Mfl/i. So, since SQ is complete along h in M, mo 6 Dom(so), and so mo € (Dom(so)C\h). Therefore, since SQ admits h, h € So( m o)- But since mo £ (Dom(sQ)(~}Dom(si)), the definition of extension says that SI(TOO) = so( m o)So /i 6 si(mo), as required. The next definition and fact articulate the technically useful idea of extending a strategy by an entire admitted history (more accurately, by that portion of it in M). 13-17 DEFINITION. (Extension along h) Let s0 be a strategy for a in M. Define "si is the extension of S0 along h for a in ' as follows:

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The purpose of this construction is the following. In general admission of a history is not preserved under extension. We can get around this by a wholesale extension that always points along h, and that totally uses up the portion of h in M. Then there will be no leftover moments at which a further extension could exclude h. The following small fact is thus critical. 13-18 FACT. (Extension along h) If (i) s0 is a simple strategy for a in M and (M) s0 admits h, then, where sj is the extension of s0 along h for a in M, • S1 is itself a simple strategy for a in M; • s1icompletely admits h for a in M; and • any extension of S1 in M continues to admit h. PROOF. If s1 is the extension of s0 along h for a in M in the sense of Definition 13-17, then s1 is on the face of it strict and backward closed for a in M. Suppose that s1 were secondary, with mo, m1 S Dom(s1) and mo < m1 but m1 If m1 did not belong to h it would belong to 0), which would make SQ secondary, contrary to hypothesis. So mi € h, and since histories are closed backward (by the postulate of no backward branching, Post. 3), mo h. But then, since mi ^ U s i( m o)i h £ si(mo). Definition 13-17 then implies that mo G Dom(so) and that «o( m o) = SI(TTIQ). So after all SQ does not admit h, contrary to hypothesis. So s\ is simple for a in M. The "complete" part of "completely admits" is on the surface of Definition 13-17, and the "admits" part is not far below. Were h excluded by si, one would have mo with h £ si(mo). But a glance at Definition 13-17 suffices for this case implying that mo £ Dom(so), whence s0 would itself exclude h, contrary to hypothesis. That extensions of Si continue to admit h comes from Fact 13-16. We shall also need some near-the-surface facts about joins of chains of strategies in the sense of the "weaker than" relation of Fact 13-3. 13-19 FACT. (Joins of chains) Let S be a nonempty chain of strategies for a in M. Then the join sg (Fact 13-3) is itself a strategy for a in M. Furthermore, if every member of S has one of the following properties, then so does s-^: (i) strict for a, (ii) primary, (m) backward closed in M, (iv) simple for a in M, (v] admits h, and (m) completely admits h. PROOF. Use Zorn's lemma. A violation of any of (i)-(vi) for SE would be finite, and hence would be a violation of the same property for some member of This concludes our introduction of the most basic strategy concepts. We go on to introduce "favoring," an important but somewhat less obvious idea.

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13D

Favoring

Let us return to the problem mentioned at the beginning of the chapter: As promised in §91, find a strategy the following of which will guarantee not seeing to it that Q. The core of an intuitive solution is easy: Shoot for ~Q whenever given a choice. To make clear sense out of this suggestion, we introduce the concept of a strategy "favoring" a set of histories. The root idea comes from Hamblin 1987, p. 157, where the topic is carrying out (or "extensionally" satisfying) an imperative: ... the addressee of an imperative would be expected, at least, to act in such a way as to keep extensional satisfaction within the bounds of possibility. This means that he must not do any deed d that would infringe—that is, that would ensure dissatisfaction of—the imperative. The next definition aims to capture exactly this idea. In reading it, think of HO as the set of histories on which Hamblin's iniDerative is satisfied. 13-20 DEFINITION. (Favoring)

Let s be a simple strategy for a in M.

• s favors H0 iff • s completely favors HQ in M iff s favors HQ and is flo-complete in M. If s favors HO, then whenever any moment in the domain of s enables a choice that keeps HQ as a live possibility, then s advises such a choice. Furthermore— and critically—s does so by means of a history such that the same is true of any further moments in the domain of s that lie on that history. Let us relate this critical conceptual point to the notation. At a key point in the definiens of "s favors H" we used "Admh(s)," which refers to the set of histories that can be reached if the strategy is followed everywhere. If instead we had used "s(m)," we would have represented a weaker notion of "favoring." For example, consult Figure 14.1. Let HQ be the set of all the histories, {/iw, hi, h^, ...}. Then the depicted strategy, s, would favor H0 in the weaker sense, since each sm is nonempty. It would not, however, favor HQ in the stronger sense, since, as declared by Fact 14-9, Admh(s) = 0. The property of favoring need not persist under extension; but the property of completely favoring does: 13-21 FACT. (Persistence of completely favoring) Any simple extension si of a strategy s0 for Q in M that completely favors HQ also completely favors HQ. PROOF. Suppose h € HO and si admits h. Then s0 admits h by Fact 13-6, so SQ is complete along h in M by hypothesis, so si is complete along h in M by Fact 13-16. So si is Ho-complete. For the "favoring" part, remark that HoH Admh(so) C Hor\Admh(si) by Fact 13-16. Now suppose mi € Dom(s\) and

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and By the remark, r\Admh(so) = 0, so that to avoid violating the hypothesis that SQ completely favors HQ, mi ^ Dom(s0). Choose h 6 H(mi)r\H0; evidently /i ^ ^4dmft(s 0 ). Let the excluding occur at m0: m0 £ Dom(so)nh, h £ s0(m0). Since m0, TOI both belong to h, they are comparable; but mi ^ mo would put mi in Dom(so) by backward closure, so that mo < mj. So SQ excludes mi according to Definition 13-4, so that si excludes mi by Fact 13-6. So the supposal that mi € Dom(si) leads to the secondariness of si, contrary to hypothesis. We have developed just enough of the theory of strategies to prove a useful lemma: A favoring strategy can always be extended to a completely favoring strategy. This lemma will have important corollaries. 13-22 LEMMA. (Favoring extension lemma) Suppose s is simple for a in M, and favors H0. Then s has an extension s' that is simple for a in M and that completely favors HO. If furthermore m0 is such that m0 €. M and H^^CiHo then s' can be chosen to be available at m0. PROOF. Well-order H0 with the ordinals less than a limit ordinal A'. (We omit adapting the argument for finite H0.) For 7 < A', let /i7 be the history in H0 that is indexed by ordinal 7. Furthermore, when the "furthermore" clause of the lemma is wanted, in the special case 7 = 0, as the "furthermore" hypothesis we let h-y G H(mQ). In short, every history in HO admitted by s\> was "completed" in its turn. Furthermore, suppose the "furthermore" clause is operative. We know by the "furthermore" hypothesis that the privileged moment mo belongs to ho(~\M, where ho is the history in HQ that is indexed by the ordinal 0. Since ho is admitted by SQ = s, we know by definition that «i is complete along /i0 in M. This guarantees that mo S Dom(si). Since s\ extends si, it is sure that mo € Dom(s\). That is, the strategy s\' is available at mo, as desired. 13-23 COROLLARY. (Complete and favoring extensions) • If s0 is a simple strategy for a in M then s0 can be extended to a simply complete strategy s\ for a in M. • Suppose m G M and H^m^r\H0 ^ 0. Then there is available at m a simply complete strategy for a in M that favors HOPROOF. For the first part, supposing SQ to be a simple strategy for a in M, apply Lemma 13-22, choosing H0 as Admh(so)- Evidently SQ vacuously satisfies the requirement that it favor Admh(s0). Let si be the promised extension of SQ that, besides being simple for a in M, completely favors Admh(so)- We need to

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show that Si is complete. Suppose si admits h. Then h € Admh(so) by Fact 136, so by complete favoring, si is complete along h, as required. For the second part, suppose m £ M and H(m)f~}H0 ^ 0. Start with the empty strategy, which vacuously is a simple strategy for a in M that favors HQ. This gives us the main hypothesis of Lemma 13-22, and we have also the "furthermore" hypothesis since the empty strategy admits all histories. So there is a strategy So available at m that is simple for a in M and completely favors HQ. This strategy, however, may not be complete. No matter, by the already-established first part of this corollary, SQ itself has an extension si that is simply complete for a in M. Clearly si remains available at m. Lastly, by Fact 13-21, si also completely favors HQ.

13E

Application to finding a strategy for inaction

As we said at the beginning of this chapter, we provide an elementary application of the theory of strategies to the logic of "seeing to it that." Previous work in stit theory, §9G, made it clear that the ability to see to some result, Q, does not imply the ability to refrain from seeing to Q. (Since two stits were put in play in §8G, we note that this failure applies, delicately, to the achievement stit, but not to the deliberative stit. We are here concerned only with the former.) In the course of developing stit theory, it became plausible to suggest, however, that whenever an agent, a, can see to it that Q, in spite of being unable to refrain from seeing to Q, there is always available to the agent a strategy guaranteeing that, if followed, the result would be that the agent not see to it that Q. Attempts to demonstrate this intuitively, especially in the presence of infinities, always ran up against the need for rigor, and therefore had to await the development of the austere theory of strategies. Chiefly because we now have the favoring extension lemma in hand, we are ready for the application of the theory of strategies to the problem of proving the existence of a "strategy for inaction." (We certainly hope, however, that the favoring extension lemma finds more surprising applications.)

13E.1

Semantics for stit

At this point we need an exact statement of the semantics of [a stit: Q] as the achievement stit, which we restate. Recall from §2A and elsewhere that indeterminism requires that truth with respect to a BT + I+AC model, m, must be relative to moment-history pairs m/h, so that we write "971, m/h t= Q" for "Q is true in the model m at the pair m/h." The semantics for [a stit: Q] relies on the apparatus of "instants" described in §13B, which was not needed or wanted for the theory of strategies. Reminder (Def. 4): H(M) 'ls the set °f histories having nonempty intersection with M; i(mi) is the instant of time on

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which mi lies; m( J(m ) t h ) is the moment in which the instant i(mi) intersects the history h. The following is the concept of the truth of [a stit: Q] based on witness by a chain (instead of witness by only a moment); it is a merely verbal variant of §8G.4, with which it may be compared. 13-24 DEFINITION. (Truth of [a stit: Q]) Given that mi £ hi, we define that 97t, mi/hi 1= [a stit: Q] iff there is a chain of moments CQ, called a witness, such that the following conditions are satisfied. • Priority-nonemptiness. All moments in CQ must be properly earlier than mi, and CQ must be nonempty: CQ ^ 0 and Vmo[mo G CQ —>TOO< mi]. • Positive condition. Q must be settled true at every moment in i( mi ) that lies on a history intersecting CQ that is inseparable from hi for a in CQ: • Negative condition. Every moment, mo, in CQ "risks" the falsity of Q in the sense that there is above m0 some moment in i(mi) at which Q is not settled true. In other words, for every moment, m0, in CQ, it is not settled at m0 that Q be true at

13E.2

Abstraction of stit witness by chains

The rigorous result we are after goes something like this. Suppose that [a stit: Q] is true at a moment-history pair mi/hi, witnessed by a chain CQ. Fix mi, hi, and CQ. Then at every moment m0 in the witnessing chain CQ there is available a strategy such that following it really guarantees that [a stit: Q] fails at m( i(m j./io)/^) f°r suitable /i0. To prove this result, we need to provide a suitable interface between stit and strategy. From the stit side, we need much less than what we have. We surely need the agent, a, and the moment-history pair mi/h\. But we do not need all of the syntactic information in Q, nor a general semantic story. We need only whether the particular Q holds at each m(i( ),h)lh. This we can represent by a set of histories depending only on mi and Q: We also need a field of choice points for a. On the "stit" side, it must be large enough to include all the chains that could possibly witness any of the stits considered in the desired result. On the "strategy" side, since the strategy is to be "available" at mo 6 CQ, the field must include mo. The field, however, must not include any moments earlier than mo, which are evidently no longer "available" at mo. Let us take the set of all moments from mo onward that are temporally prior to mi, that is, that precede some moment co-instantial with mi: MI — {m: mo ^ m & m < i( mi )}. We record these two abstractions for later reference.

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13-25 STIPULATION. (Two critical sets) Suppose that TOO, mi, and Q have been fixed, and thatTOO< TOI. • Hi = {h: m, m ( l ( m i ) i h ) //i N Q}. • MI = {TO:TOO^ TO &TO< i(mi)}. Next notice that /ii doesn't really come into the Definition 13-24 account of truth for stit. Consequently we can take TOI to come in only as a peg on which to hang the definitions of H\ and M\. If we resolve the general nonemptiness condition on the chain witness c in favor of a particular member TOO of c, we can state the remainder of the truth condition for stit as a relation between TOO, HI, MI, a, and an arbitrary history ho. We codify this "stand-in" for stit as a general definition of a five-place relation, stit-stand-in(mo, HQ, M0, a, ho). 13-26 DEFINITION. (Stit-stand-in) For any moment TOO, set of histories HO, set of moments M0, agent a and history ho, stit-stand-in(m0, H0, M0, a, ho) iff there is a c such that the following all hold. • Priority-nonemptiness stand-in.TOO • Positive stand-in. • Negative stand-in. The positive stand-in says this. Let h be any history intersecting c such that c makes no choice distinction between ho and /i. Then h G HO- That is why we can say that the choices made in c by a are such as to "guarantee" being in HQ. Or to put it contrapositively, if h G (H(c) —Ho) then 3m[m G cD/ioPI/i & /IQ _L^j /i]. The negative stand-in says that every moment m in c keeps departure from HQ as a live possibility. That makes c nonvacuous as a series—though it does not and should not follow that a has a nonvacuous choice at each member of c. The condition says that at every moment TO in c there is a "live possibility" of finishing outside of HQ. It does not say that there is a currently available "live option." It is important to see that stit-stand-in is a concept entirely in the language of the theory of strategies. For the upcoming result about stit-stand-in to have its intended significance, however, we must verify that the concept is satisfactory on the "stit" side, as follows. 13-27 FACT. (Stit and stit-stand-in) Fix a and TOI, and h\ such that TOI € h\. Assume that 9JT, mi/h\ N [a stit: Q], and let CQ be a witness (Definition 13-24). Fix TOO G CQ. Define HI and MI as in Stipulation 13-25. Choose any history ho G H(mo). Then 971, m,(i(m },hQ)/ho ^ [ but it does not say, as required for stit-stand-in(mo, HI, MI, a, ho), that mo 6 c. Nevertheless, since mo is assumed to be part of a witness for [a stit: Q] at m\jhi, we can be sure that if c witnesses [a stit: Q] at TO(i ,,h 0 )//iO) then so does cUJmo}. The reason is this. We know by the negative condition for the witnessing of [a stit: Q] at mi jh\ that Q fails at m(j ( m /i)//i for some h 6 H(moy Therefore, the chain c cannot be entirely earlier than mo on pain of violating the positive condition for the witnessing of [a stit: Q] at m(i( }tho)/ho- So some member of c must come properly after mo. But then adding mo to c keeps the negative condition true, and does not disturb the positive condition, so that cU{mo} is also a witness to [a stit: Q] at m(j ( m ),h0)/ho- And of course that set is bound to contain mo, as required for stit-stand-in(mo, HI, MI, a, h0). (There is a discussion in Belnap 1996a of what can be added to or subtracted from a chain witness.)

13E.3

Main result

We use the stit-stand-in abstraction from the concept of seeing-to-it-that as follows. We prove generally that regardless of H0 and M0, whether as just defined or not, we can always find available at mo a strategy for a such that if a follows it faithfully at the choice points in MQ, then for any history /IQ that could possibly eventuate, stit-stand-in(mo, HO, MO, a, ho) must fail. This is a result in the pure theory of strategies. Then we use Fact 13-27 to argue that there is always a strategy the following of which guarantees that [a stit: Q] is false at the moment-history pairs in question. 13-28 THEOREM. (Strategy for inaction) Fix mo, MO, HO, and a such that mo Then there is available at mo a simply complete strategy s for a in MQ such that stit-stand-in(mo, HQ, MO, a, ho) fails for every history ho admitted by s. PROOF. By the second part of Corollary 13-23, let s be a simple strategy for a in MQ available at mo that is complete in MO and that favors -H(M 0 ) ~Ho m MQ. Suppose s admits ho (and is therefore complete along /i0 in MO), and, for contradiction, that stit-stand-in(m0, H0, MO, a, h0) holds (Definition 13-26). By priority-nonemptiness let mo G c C /i 0 nM 0 . By the negative condition, so that by favoring, we know that we may let So choose By the positive condition, hi (hence in h0r\Mo, therefore in Dom(s)) such that hi ±^ ll ho. Here is the contradiction: ho and hi are both admitted by s, so must both belong to s(mi). But then s is not after all strict. Our main result is a direct corollary.

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13-29 COROLLARY. (Main result) Suppose that 9Jt, mi/hi 1= [a stit: Q], with witness CQ. Then at each moment mo G CQ, there is available a simply complete strategy s for a in that really guarantees PROOF. Start with the hypothesis, and choose m0 € CQ. Define MI, #1 in accord with Stipulation 13-25. Since m0 e MiHljHi, by Theorem 13-28 there is available atTOOa simply complete strategy s for a in MI such that stit-standOT(TOQ, -Hi, MI, a, ho) fails for every history /i0 admitted by s. By Fact 13-27, that amounts to saying that s guarantees And the hypothesis allows us to calculate that s is not vacuous for the set of histories

since we can promote "guarantee" to "really guarantee."

For this reason,

14

Deontic kinematics and austere strategics Deontic concepts are naturally linked with strategic concepts.* Surely any obligation can be viewed as an obligation to follow a certain strategy in the austere sense of chapter 13, always making such choices, depending on circumstances, as conduce to satisfying the obligation. Conversely, to follow a strategy can be viewed as very like living up to a set of deontic requirements, "doing what the strategy requires." We do not here directly discuss such common-speech linkages. Instead we report a specific and detailed theoretical linkage that allows reciprocal illumination between certain deontic concepts and certain strategic concepts. An unexpected result is the connection between the no choice between undivided histories condition from the austere theory of strategies, and a new deontic kinematic condition adduced by Thomason to help describe how oughts fit into branching time. Deontic kinematics I. On the deontic side we invoke Thomason 1984, a fundamental essay that describes a family of structures carrying information about (z) oughts-at-moments in (ii) branching time. The branching mentioned in (ii) represents that some things might happen that are not guaranteed to happen. In other words, there are inconsistent possibilities for the future. In particular, an existing obligation might be fulfilled, or might not. The purpose of relativizing oughts to moments as in (z) is simple and pervasive: One knows that oughts come to be and pass away. For example, an obligation can be created by a promise; for another example, a "reparational obligation" can be created by breaking a promise. One asks for general principles governing how such event-located oughts fit together in our world. This is the topic of "deontic kinematics." (Thomason 1984 calls the topic "ought-kinematics.") Deontic kinematics is an enrichment of the "static" or "timeless" versions of the deontic concepts that are prevalent in the literature. 'With the kind permission of Filosophia, this chapter draws on Belnap 1996b.

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Austere strategics I. The strategic concepts we use arise in the theory of agents making choices in branching time, BT + AC theory, §3. In earlier chapters we explored stit theory, which presupposes BT + AC or BT + I+AC theory; and then in chapter 13 we considered the austere theory of strategies. Austere strategics sits on the same foundation as stit theory, which is to say, upon the theory of choices in branching time; but this theory of strategies is not semantic and does not itself depend on the semantics of stit. We describe the theory as "austere" because it involves none of the psychological or value concepts typically taken as underpinnings for the concept of strategy. In fact, the austere theory of chapter 13 simply identifies a strategy as a pattern of choices in branching time, neglecting entirely the mental work that goes into adopting a strategy, and equally neglecting what a strategy is "for." It is surprising how rich and also clarifying such a minimal theory can be. Deontic kinematics and austere strategics both "take time seriously": As we see, each has a key postulate that in the former vertically integrates oughts across time, and in the latter vertically integrates choices. Our most striking result exhibits the interplay between these two apparently disconnected ideas.

14A

Basic concepts

First we review the underlying concepts.

14A.I

Branching time

We begin by noting with maximum brevity what is common to the two theories, namely, branching time. Deontic kinematics and austere strategics follow the lead of Prior 1957 and Thomason 1970 in representing the indeterminism of our world by means of a BT structure (Tree, ^}. That is, deontic kinematics and austere strategics have the Prior-Thomason theory of branching time as a common part. In this book the postulates and definitions of this theory are briefly described in §2A, more extensively discussed in chapter 7, and summarily presented in §3. See the latter for help with the notation.

14A.2

Deontic kinematics II

Thomason 1984 arrives at the theory of deontic kinematics by adding a deontic primitive "0" to the primitives Tree and ^ of branching time. He calls the resulting structures "treelike frames for deliberative deontic tense logic." Here, for our immediate purposes, we substitute the phrase "deontic tree." 14-1 DEFINITION. (Deontic tree) A deontic tree is a triple (Tree, ^, 0), where (Tree, Choice'^(h) C s(m}\. • s is available at m iff m € Dom(s). • For m S Dom(s), s is consistent at m iff s(m) ^ 0, and s is consistent iff s consistent throughout its domain. • For m 6 Dom(s), s is vacuous at m iff s(m) = H(my

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• A strategy s is total [in M] iff Dom(s) •=• Tree [= M]. These ideas are explained at more leisure in §13C. Recall that we represent a choice as a set of histories. The key idea is that a strategy tells you which of your available choices count as following the strategy—and which do not. A strategy that is inconsistent at m explicitly tells you that no history available at m counts as following the strategy (there is nothing good to do). At the other end of the spectrum, a strategy that is vacuous at m says that everything goes at m. A strategy is total if it gives you advice at every moment in the entire tree. 14-3 DEFINITION. (Strategic concepts: admission and exclusion) • s admits h iff Vrn0[m0 G Dom(s)r\h —> h G s(m0)}. • Admh(s) — {h: s admits h}. • s admits mi iff Vm0[(m0 G Dom(s) & mo < mi) —> mi e U s ( TO o)]• Admm(s) = {m: s admits m}. • s excludes a history or a moment if it does not admit it. The admission and exclusion concepts on this list, drawn from Definition 13-4, are of critical importance for this chapter. Let us start with exclusion, which is defined (with inescapable confusion) for both moments and histories. A moment is excluded by a strategy if never violating the strategy implies that you will never reach that moment. A history is excluded by a strategy if never violating the strategy implies that eventually you will fall off that history. Since admission is simply the contradictory of exclusion, a moment is admitted by a strategy if it is possible to arrive there without violating the strategy, and a history is admitted by a strategy if staying on that history never requires that you violate the strategy. 14-4 DEFINITION. (Strategic concepts: simplicity, completeness, etc.) • For m G Dom(s), s is strict for a at m iff s(m) G Choice^, and s is strict for a iff s is strict for a throughout its domain. • s is secondary iff some moment that it excludes is nevertheless in Dom(s); and s is otherwise primary. (This dichotomy arises from the following: The dictates of s are primary if they concern moments that are admitted by s, but secondary when they concern moments that are excluded by s.) • s is a strategy for a in M iff s is a strategy for a such that Dom(s) C M. • When s is a strategy for a in M, we call M a field for s. • Suppose that s is a strategy for a in M. s is backward closed in M iff

368

Strategies • s is simple for a in M iff s is a strategy for a in M that is strict for a, primary, and backward closed in M.

• s is complete along h in M iff Mfl/i C Dom(s). • s completely admits h in M iff s admits /i and is complete along h in M. • s is H-complete in M iff s is complete in M along every history in H that s admits. • s is complete in M iff s is complete in M along every history that s admits. • s is simply complete for a in M iff s is simple for a in M and s is complete in M. This last group of definitions supports a rich theory of simple strategies. For example, a major fact proved in chapter 13 is that every strategy for a in M that is simple can be extended to a strategy that is simply complete for a in m (Corollary 13-23). We mention this here, however, only for contrast, since we will not be dealing with simple strategies at all. And for the same reason, we refer to chapter 13 for explanations of these concepts: In this chapter, we occasionally mention some of these concepts for contrast, but we do not really use them.

14B

From Thomason's deontic kinematics to austere strategics

To think about oughts and strategies together, zero in on a single agent, a, and think of Tree as both supporting 0(m] representing what ought to be at m and also supporting a choice-function Choice^ defining what a can do at m. What, if anything, would then be wrong with thinking of the deontic tree itself as a candidate account of a strategy for a? That is, what would be wrong with taking 0 as a strategy? A great deal. From the agency point of view, the deontic tree would have the defect that it omits consideration of what a can do at m. This defect seems to us so substantial that it calls into question the propriety of using the adjective "deliberative" in connection with a deontic tree. Such a tree could not be a strategy for a, since there is no requirement that 0(m) be closed under choice equivalence for a at m. 0(m) picks out histories containing m, but not choices available to a at m. Strategies ("deliberative" strategies) tell you what to do; 0 does not—at least not directly. Perhaps there is an indirect route from 0 to strategic advice. Let us then ask: How to make a strategic choice at m given a particular 0(m)? Consider a possible choice for a at m, say H (H will be a subset of all the histories H(m) to which m belongs). If H contains only histories lying outside of 0(m), certainly the strategy should steer away from H. And if H contains only histories acceptable to 0(m), then with equal certainty the strategy should

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permit the choice H. But what if H is "mixed," containing both some histories acceptable to 0(m) and others that are unacceptable to 0(m)l There are then evidently two salient strategic policies available, neither happy. The "weak" policy says that choosing H, even though mixed, is in accord with the strategy. The "strong" policy says that choosing a mixed H violates the strategy. There is also a third policy, itself not entirely happy. The "Hamblin" policy, suggested in Hamblin 1987, makes it depend upon what other options there are, as follows. Let H be mixed. If there is at least one possible choice that contains only histories acceptable to 0(m), then the Hamblin strategy forbids choosing such a mixed H (like the strong policy). But if there is no possible choice containing only histories acceptable to 0(m), then the Hamblin strategy permits choosing mixed H. There is no point moralizing about this here; we simply put the matter in terms of clear definitions about which something definite can be said.2 14-5 DEFINITION. (Deontic trees as strategies: sweak,a,o and sstrong,a,o and SHambhn,a, o) Suppose that we are given a deontic tree (Tree, ^, 0) together with a choice partition Choice1^ for a at every moment m. We define the weak, the strong, and the Hamblin strategies for a determined by this deontic tree, written respectively as sweak^a, o, sstTong,c>, o, and sHam,bhn,a, o, by specifying the domain of each function as the same as the domain of 0—namely, the entire tree—and continuing as follows. • sweak,a,o(fn)

=

• sstrong,a,o(m) = • SHambhn,a, o("i): If there is a possible choice H for a at m (that is, H such that then otherwise, sHambhn,a, o(m) We can now intelligibly speak of a deontic tree as in effect one of the strategies {sweak,a, o, sstrong,a, o, sHamblin,a, o}, and ask what properties it may have. 14-6 FACTS. (Deontic trees as strategies) • For S € {Sweak,a, O, Strong,a, O, SHamblin,a, o}, s(m)

is a Subset of H(m)

that is closed under choice equivalence for a at m. That is, each of {sweak,ce,o, strong,a, o, sHamUin,a, o} is a strategy for a. None of the conditions (i)-(iii) of Definition 14-1 play a role here: The form of Definition 14-5 suffices.

• Since 0 is total on Tree, so is each of the three strategies sweak,a, o, sstrong,a,o, and Sffambim,a, o- Each is available at every m in Tree. Each strategy is ferociously "complete." 2

Horty 2001 shows that there is much more to be said in the context of an apparatus involving utilitarian degrees-of-value comparisons. Here, however, we have only the deontic O.

370

Strategies • The consistency condition (z) of Definition 14-1 implies that sweak, a, o and SHambiin,a,o are consistent. In contrast, the strong cousin s s tr<mg,a,o may well be inconsistent even though the consistency condition on 0 is satisfied. Suppose, for example, that the consistency condition is satisfied at m by means of just a single history h through m that is acceptable to 0(m). If the possible choice for a at m that contains h also contains an unacceptable history, then according to sstr0ng,a, o, that choice is not open to a. The strategy s s t r ong,a,o tells a that at m every option is forbidden. That is, sstr0ng,a,o 1S inconsistent. • Any member of {sweak,a,o, sstrong,a,o, sHa.mbiin,a,o}, except in trivial cases, is bound to be secondary. This feature distinguishes any strategy based on a deontic tree from any simple strategy. The positive role of secondariness in the deontic-kinematic framework is to permit consideration of reparational obligations, obligations that come into force after some earlier obligation has been violated. We underline that the feature is neither good nor bad; but one should bear it in mind. Primariness/secondariness is the most significant difference between simple strategies and strategies that are derived from deontic trees. • If S G {sweak,a, O, Sstrong.a, O, SHamblin,a, o}, S is total, 3S W6 Said. So it

is natural to expect that at some moments m, s is close to being undefined; i.e., s lays on a at m only a vacuous obligation, the set of all histories containing m: s(m) = H(my This is not primarily a technical point: Any strategic advice that covers every moment in Tree must be largely vacuous.

• A strategy s € {sweak,a,o, sstrong,a,o, SHambiin,a, o} derived from a deontic tree is therefore never simple: Although s is bound to be backward closed (in any M), it will never be both strict for a and primary. Notably missing from this account is any reference to the kinematic condition (in) of Definition 14-1. What does the kinematic condition mean? Thomason 1984 connects the notion with the idea of a "coherent plan." But there seems nothing incoherent about a strategy derived from a deontic tree, whether or not condition (Hi) is satisfied. This suggests that we have not found the best way of relating austere strategics and deontic kinematics.

14C

From austere strategics to Thomason's deontic kinematics

We now start from the other end. We start with austere strategics as given, and ask what we can make of deontic kinematics in this context. It turns out— unexpectedly—that here is where we find the most interesting results. These are the results that seem to shed the most light both on austere strategics and on deontic kinematics.

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So suppose we start at the other end, with a strategy s. One might think of the histories S(TO) as what a "ought to do" at TO. Suppose we were then to turn things around and define 0(m) as s(m). We should learn that conditions (i) and (ii) of Definition 14-1 would be satisfied at least when s is consistent. We should also find out that condition (lii) is something that does not readily spring to mind. Mutual illumination seems again absent.

14C.1

Admission onward and Os

There is, however, another idea that begins with strategies. This idea pays attention to a recurring theme in the deontic literature: The histories defining obligation should in some sense be "ideal." At least they should be "ideal" to the extent that in these histories no one ever violates an obligation. If we take the given strategy s as the source of obligations, then we have already defined these histories. They are the admitted histories (Definition 14-3). In an admitted history a never departs from the strategy. This never-failing agreement with the strategy is evidently "ideal." Therefore admission, referring as it does to a generality of moments, would appear as a concept of deontic kinematics. We might therefore think about defining 0(m) as Admh(s)r\H(mj so as to include the element of ideality. Observe that since it involves quantifying over moments, admission is essentially a "global" idea, not bound to a particular moment. Admitted histories are acceptable to the strategy, and are "ideal" as far as the strategy goes. None of them can be improved upon, as far as strategy-following goes. So we can talk deontically about what is true in all "ideal" histories, that is, the admitted ones, those in which the strategy is never violated.3 The idea of defining obligation in terms of admission will not do, however, for every case of a deontic tree. And that is because some of these trees are designed to allow intelligible "reparational obligations." In such a tree there can be things that you should do at a certain moment m even though those very things were excluded by earlier obligations. (You only reached TO because you did what you ought not to have done.) The suggestion to define 0(m) as Admh(s)C\ H(m) might be all right if we were only concerned with simple strategies. In the reparational case at hand, however, we should have 0(m) = Admh(s)r\ H(m) = 0, which would represent an inconsistency: The consistency condition, Definition 14-1 (i), on 0 would not be satisfied. This definition cannot, then, offer advice to those who have ever, even once, made a strategic mistake. If this seems too abstract, any of the old tales will do to make the point. You ought atTOOto visit your mother—perhaps because of a promise—and you have adopted that as your strategy. But atTOOyou choose not to catch the plane to visit your mother. You find yourself at TOI, not having caught the plane. At TOI you must choose whether or not to call your mother to tell her that you will not be coming. Both of these histories through TOI, however, are such that on them you have not visited your mother. They are therefore both excluded 3 We say that admitted histories are "acceptable" rather than "permitted" because we think that permission is an essentially agentive concept and therefore not applicable to individual histories. Only choices for a can be permitted—or not—for a.

372

Strategies

(not admitted) by your strategy to visit your mother. Your strategy therefore excludes all alternatives through mi. Your strategy is inconsistent at mi. What has gone wrong? This: One wants for the reparational or secondary situation a concept that gives a a "fresh start." That is why taking admission as an absolute character of histories did not work. It is better to invoke the idea of a history that is "admitted by s from m onward." In other words, the history may have been excluded earlier, but don't look back: From m onward it forever after counts as ideal. In still other words, past failures don't count. As one might expect of a purely deliberative idea, only the future of possibilities is relevant. Here is the definition. 14-7 DEFINITION. (Admission onward; Admh(s, m)) let mo € Dom(s). • s admits h from mo onward iff mo

Let s be a strategy, and and

• For any strategy s and moment mo, Admh(s, mo) is the set of histories admitted by s from mo onward. In applying this definition, we run into the contrast that strategies can be partial while deontic trees are always total. For present purposes we therefore choose to concentrate on total strategies. It is easy to see that from the point of view of admissibility, nothing is lost. Given any strategy SQ we can always consider instead the total strategy si obtained by vacuously defining Si(m) = H(m) for every m & (Tree —Dom(so)). (It cannot be inferred, however, that nothing is changed tout court; obviously total strategies are not simple at all, so that exclusive concentration on them would put potholes in the road to inquiring about the properties of simple strategies.) The upshot is that in the context of reparational obligations, given a total strategy s, Admh(s, m) seems a good candidate for the concept 0(m) of deontic kinematics. These are the histories that, having reached m, are and will remain "ideal" as far as the strategy goes. They are the histories through m on which (although obligations may have been violated earlier) no more violations occur. Sounds "ideal," doesn't it? Let us record the suggestion in a definition. 14-8 DEFINITION. (Strategies as candidate deontic trees: Os) Suppose we have a choice tree (Tree, ^, Choice). Given a total strategy s for a, we define the candidate deontic tree determined by s as (Tree, ^, Os), where for m e Tree, Os(m) = Admh(s, m). Here is a decisive observation: Os is not a strategy and should not be confused with a strategy. A strategy s picks out choices at m, whereas Os can pick out some individual history within a possible choice for a at m, while excluding other histories in that same choice. Such ideal histories are important; but one must also keep in mind that no agent can choose the set of such histories. What can we say about the so-induced candidate deontic tree, especially with respect to conditions (i)-(iii) that Definition 14-1 laid on deontic trees?

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The following sections report that all of these conditions—(i) consistency, (ii) locality, and (Hi) kinematic—fall into place given Definition 14-8. There you have it: Think of deontic kinematics as generated by austere strategics, and each of the deontic-kinematic conditions is illuminated. This is especially helpful for appreciating the least obvious of these conditions, the kinematic condition (Hi). We start at the first of the deontic-kinematic conditions.

14C.2

The consistency condition on deontic trees

Of course if s is inconsistent, Admh(s, m) can be empty. That's trivial. The interesting question concerns the consistency (nonemptiness) of Admh(s, m) given that s is consistent. We shall find that the consistency of Admh(s, m) is by no means guaranteed. But we shall also locate an unexpected condition under which consistency of Admh(s, m) is restored: Namely, it makes a difference whether or not there are busy choosers, Def. 14. 14C.2.1

Inconsistency of Os given busy choosers

The first relevant fact is that Admh(s, m) can be empty even if 5 is consistent, if one allows a to be a busy chooser. 14-9 FACT. (Consistency condition on Os, Definition 14-1, with busy choosers) There is a total consistent strategy s for a such that Os is nevertheless inconsistent. PROOF. Here is a (peculiar) case in which a particular Admh(s, mu) is empty, even though each s(m) is nonempty. We need to look at an example of "feathering" of histories (see Figure 14.1). Suppose after some moment mw there is an infinite chain of moments mo > mi > m^ > ... > ml > ml + i > ... descending toward mw (mw is not part of the sequence, but mw is the greatest lower bound of the sequence). We suppose that s offers a vacuous choice for a at TOO and also at every other moment other than TOI, 7712, ..., mu. At mu and at each member ml of the chain, i > 0, there is, however, a binary choice for a. (Since there are infinitely many nonvacuous choices for a between m^ and mo, a is by Def. 14 a busy chooser.) Going left stays in the chain, going right leaves it. The strategy at mw says: Go left (stay in the chain), that is, s(mu) = the set of all histories passing through some mz, 0 < i. But after m^ and before TOO, s always prescribes going right ("leave the chain"). So Admh(s, mu) is empty even though s(m) is nonempty for every m. To verify this, pick any "right" history, hi, intersecting the chain atTOZ,i > 0. You will find that ht is excluded by some (indeed every) moment that is strictly between TOW and TO;. For example, you can see from Figure 14.1 that h% is excluded at ma. Furthermore, the "leftmost" history hu, which containsTOO,is excluded by every m-i that is earlier than mo and later than mu. So every history through mw is excluded at some point at or after mu. None of the histories through ma is admitted from mu onward. None of them is "ideal." So Admh(s, mu) = 0. Which is to say, Os is inconsistent at mw, even though s is everywhere consistent. D

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Strategies

Figure 14.1: Consistent s does not imply consistent Os

It does not appear that the strategy pictured in Figure 14.1 is any more incoherent than any other more "finite" reparational strategy. The appearance is more that of a merely set-theoretic construction without intuitive content. But perhaps that appearance is only due to our reluctance to think about the possibility of busy choosers; it's hard to be sure. 14C.2.2

Consistency of Os given no busy choosers

The strange counterexample is not, however, the end of the story about how consistency of s connects with consistency of Os. We may add the following: Absence of busy choosers is quite enough to provide the link between consistency of s and consistency of Os. 14-10 FACT. (Consistency condition on Os (Definition 14-1 (i)), given no busy choosers) Let s be a total consistent strategy for a. Then Os (m) is nonempty— and the consistency condition (i) of Definition 14-1 is satisfied—provided a is not a busy chooser, Def. 14. PROOF. Suppose that we have a choice tree (Tree, w F A iff 371, M 1= [a stit: A] and 371, i\>w ¥ A. Hence, when w witnesses [a stit: A] at m, w also witnesses [a stit: A] and [a stit: [a stit: A]] at any m' choice equivalent to m for a at w. Let & = (Tree, \ w ^ w' < m], forward denseness, that is, Vw[w e BC —* 3w'[w' 6 5(7 & w < w'}}, and nonvacuous choices, that is, Vtu[w 6 BC —> Choice^ ^ {H(wj}}.2 A &MSJ/ choice sequence in & (or 971) is a busy a-choice sequence in & (or 971) for some agent a in ©. It is easy to see that & is finite only if there is no busy choice sequence in &, but the converse does not hold. Let K be the set of all sentences A such that & 1= A for each BT + I + AC + nbc structure ©, and let Lai + rr be the axiomatic system that takes as axioms all substitution instances of truth-functional tautologies, refref, and the following schemata:

2 The idea of a busy choice sequence is listed with a some somewhat more general definition in Def. 14. A formal study of busy choice sequences can be found in Xu 1995a. Note that the forward denseness condition is only a particular way in which a chain can be busy. We only consider forward denseness because when considering the truth values of sentences in our language, only forward denseness is relevant. One can find, however, other issues in which backward denseness is more crucial. See, for example, Fact 14-9.

384

Proofs and models

and takes as rules of inference modus ponens and RE From A = B to infer [a stit: A] = (a stit: B\. The definitions of theorems (I-), consistency and maximal consistent sets are as usual, keeping in mind that in this chapter all these concepts refer to Lai + rr. Let A be any sentence and E any set of sentences. A is a deductive consequence of E, written E h A, if h (A1 &...&;Ak D A) for some >4i, ..., .Afc e E. E is deductively closed if j4 € E whenever E h .A. The deductive closure of ,4 will be denoted by DC(A). Identifying Lai + rr with the set of all its theorems, we show in the following sections that K = Lai + rr, and that Lai + rr has the finite model property: For every sentence A, V- A only if SOT ¥ A for some finite model SDt. This property will be obtained simultaneously with the completeness of Lai + rr. For the convenience of our upcoming discussions, we present some theorems and derived rules in Lai + rr. Note that all the derived rules and all the theorems TT1-19 can be proved without applying refref.

15. Decidability of one-agent achievement-stit theory with refref

15B

385

Companions

In this section, we will consider some basic semantic features of so-called companions to stit formulas, which we will not define until we obtain the companion theorem (Theorem 15-11). But it will be helpful to present a description of it at the beginning so that the reader may have some idea of where our discussion is going. Let 9m be any BT + I + AC model such that m, m t= [a stit: A] with witness w, and let M = {m1: m' € i(m) & m' =° m}. Clearly, m, M N [a stit: A}. We will show in the companion theorem that for each sentence C, either m, M N [a stit: A&Ca] or m, M ~[a stit: A,Ca\. Whichever is true together with [a stit: A] in M will be called a "companion" to [a stit: A}. Note that in this section, "structures" and "models" always mean BT + I+AC structures and BT + I + AC models, and we do not require that these structures and models satisfy the property of no busy choice sequences. Let us start with some simple lemmas.

386

Proofs and models

15-4 LEMMA. (A consequence of no choice between undivided histories) Let & be any structure in which w < m and w' < m, and let a be any agent, i = i(m), M - {m1: m1 6 i & m' =awm} and M1 = {m'\ m! € i & m' = , m}. Then

PROOF. Suppose that w < w'. Consider any TO' 6 i\>w>, that is, m' £ i and w' < m'. Assume that m' ^ m. Then by the axiom of choice there are two histories h and h' such that {w, w', m} C h and {w, w', m'} C h', and hence h' =w h by definition. This implies, by no choice between undivided histories, h1 =aw,h and hence m' =£, m, that is, m' 6 M. It follows that i| >UJ ' C M. (ii) Suppose that w' ^ w. By (%), if w' < w, then i\>w C M', and hence M C M'. If to' = w, then m' =£,, m iff m' =£, m for every m' € i, and hence M = M'.

n

15-5 LEMMA. (Some sufficient conditions for stit) Let SOT be any model in which w < m, i = i( m j and M = {m": m" e z & m" = ° m}, and let >1 be any sentence. Then

PROOF, (i) Suppose that SOT, M N A and for some m' € M, SOT, m' N [a stit: .A] with witness w'. Setting M' = {m": m" 6 i & m" =£,/ m'}, we then have

Since m' € M, w < m', and hence by no backward branching, either w < w' or w' ^ u/. If w < w', then by Lemma 15-4(i), i\>w> C M, and hence SOT, i|>u/ N .A, contrary to (1). Hence it must be true that w' ^ w, and then by Lemma 15-4(ii), M C M'. It follows from (1) that 971, M N [a stit: A], (ii) Suppose that !Ut, i\>w t= yl and for some m' € i\>w, 3H, m' N [a stit: A] with witness w'. Similarly, setting M' = {m": m" € i & m" =^, m'}, we also have (1), and applying no backward branching, we have either w ^ w' or w' < w. By the transitivity of ^, w ^ w' implies i\>w' C i\>w, which in turn impliesm,i\>w' t= A since SOT; i\>w N A . It follows from (1) that w' < w, and hence by Lemma 15-4(%), i|>«, C M'. Then DOT, i|>tu N [a stit: A] by (1). D 15-6 LEMMA. (Concerning [a stit: A&B]) Let 9H,TON [a stzt: A] with witness w, and let i = i(m). Suppose that 9JI, M 1= 5, where M = {m': TO' 6 i & m' = ^ m}. Then SOT, m 1= [a stit: A&B] with the same witness. PROOF. Suppose that 271, M 1= B. Then by hypothesis, Jt, M *= a&.B and 9Jt, i|>UI ^ 4, and hence mOT, i|>m J^ ^&5. It follows that £01, m N [a stit: A&B] with witness w. D

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15-7 LEMMA. (Concerning Aa) Let m be any model in which w < m, i = i(m) and M = { m / : m ' E i & m ' s ^ m } . Suppose that A is any sentence such that m, M F= A and M, M F [a stit: A]. Then m, i >w F= Aa. In particular, if m, m 1= [a stit: A&B] with witness w, and if M, M F [a stit: B], then Wl, i>w F= Ba and M, M F= [a stit: /4&Ba]. PROOF. First, if m, i\>w F A, then, since M, M F= A, M,F 1= [a stit: A], contrary to our hypothesis. It follows that M, i|>w, F= A. Next, suppose for reductio that M, i|>w, F ~[a stit: A]. Then there is an m" € i|>«, such that M, m" F= [a stit: A]. Since we have shown that M, i\>w F=A, M, i|>w, F= [a stit: A] by Lemma 15-5 (ii), contrary to the hypothesis. From this reductio we conclude that M, i\>w 1= ~[a stit: A] and hence M, i|>w F= Aa. In particular, if M, m F= [a stit: A&B] with witness w, and if M, M F [a stit: 5], then M, i|>w F= Ba and hence M, M F [a stit: A&;Ba] by Lemma 15-6. 15-8 LEMMA. (01 refraining lemma) Let M be any model, and let A and B be any sentences such that M, m F= [a stit: A&~[a stit: B]] with witness w, and let i = i( m ). Suppose that for some m' £ i|> w , M, m' F= [a stit: B] with witness w'. Then w < w'. PROOF. Let M' = {m": m" 6 i & m" =£, m'}. Then M, M' F= [a stit: B]. Since m' E i|>w,, w < m' and w' < m'. Then by no backward branching, either w' < w or w < w'. Suppose for reductio that w' < w. Then i|>w C M' by Lemma I5-4.(i). It follows that M, m F [a stit: B] since m € i\>w. But by hypothesis we have M, m F= ~[a stit: B], a contradiction. From this reductio we conclude that w < w'. 15-9 LEMMA. (Concerning Aa and stit) Let M be any model, and let A and B be any sentences. Then the following hold: i. If M, m F [a stit: A&B a ] with witness w, then M, m F= [a stit: A] with the same witness; ii. If M, m F= [a stit: A&~[a stit: A&B 0 ]] with witness w, then M, m F [a stit: A] with the same witness. PROOF. (i) Suppose that M, m F= [a stit: A&.Ba] with witness w. Then, setting i = i(m) and M = {m': m' € i & m' =aw m}, we have M, M F A&5a and M, i|>w ¥ A&Ba. Hence by Lemma 15-7, M, i\>w F= Ba, and hence M, i|>w, F A. It follows that M, m F= [a stit: A] with witness w. (ii) Suppose that M, m F= [a stit: A&~[a stit: A&B a ]] with witness w, and let i = i(m) and M = {m': m' € i & m' = awm}. Then

and either M, i|>w F A, or M, i\>w ¥ ~[a stit: A&Ba}. Consider the latter case. There must be an m' € i\>w such that M, m' t= [a stit: A&Ba} with witness w'. By applying Lemma 15-8, we have w < w', and hence i\>w' C i\>w.

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Since by (i), M, m' F= [a stit: A] with witness w', it follows that M, i\>w ¥ A. So in both cases, we have M, i\>w ¥ A. It follows from (2) that M, m F= [a stit: A] with the same witness w. 15-10 COROLLARY. (Validity of A7 For any model M, M F= A7. PROOF. Suppose that M, m F [a stit: ~[a stit: A&.B}&.Ba] with witness w, and let i = i(m) and M = {m": m" € i & m" =aw m}. By Lemma 15-7, we have that

Then there must be an m' £ i\>w such that M, m' F= [a stit: A&B] with witness w'. By Lemma 15-8, w < w', and hence, i\>w> C i\>w. Let M' = {m": m" € i & m" = aw, m'}. Then M' C i|>u, and hence by Lemma 15-6, M, m' F [a stit: A&.Ba] with witness w', and hence by Lemma 15-9(i), M, m' F= [a stit: A] with witness w', it follows that

We show as follows that M, M F ~[a sizt: A]. Suppose for reductio that there is an m* C M such that M, m* F [a stit: A] with witness w*. Set M* = {m": m" C i & m" = aw. m*}. Then M, M* F [astit:A]. Since m* € M, w < m*, and hence either w < w* or w* < w by no backward branching. Now if w < w*, then by Lemma 15-4(ii), M* i\>w, and hence by (3) and Lemma 15-6, M, M* F= [a stit: A & B ] , contrary to (3). If w* < w, then we have i\>w C M* by Lemma 15-4(i), and hence, since M, M* F [a stit: A], M, i\>w F= [a stit: A], contrary to (4). This reductio shows that M, M F= ~[a stit: A]. It follows from (4), (3) and Lemma 15-6 that M, M F= [a stit: ~[a stit: A]&Ba}. Now we are ready to prove the companion theorem. 15-11 THEOREM. (Companion theorem) Let M, m F [a stit: A] with witness w, and let i = i(m) and M — {m': m' € i & m' =aw, m}. Then for every sentence 5, either M, M F= [a stit: A&Ba] or M, M F ~[a soit: A&Ba], and consequently, either M, M F= [a stit: A&Ba] or M, M F= [astit:A&~[a stit: A & B a ] ] . PROOF. Suppose that M, M ¥ ~[a stit: A&Ba}. Then there is an m' € M such that M, m' F [a stit: A&Ba] with witness w'. Hence, setting M' = {m": m" € i & m" =aw, m'}, we have

By Lemma 15-9(i), we have M, m' F [a stait: A] with the same witness w'. This and our hypothesis imply

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Since m' E M, w < m'. Hence by no backward branching, either w < w', or w' < w, or w = w'. If w < w', then by Lemma 15-4(i), i\>w> C M, and hence by (6), m, i\>w> F A, contrary to (7). Similarly, if w' < w, then by Lemma 15-4(i), i\>w C M', and hence by (6), M, i\>w F= A, contrary to (7). We then conclude that w = w'. Since m'' =aw, m (i.e., m' € M), it follows that M = M', and hence by (5), M, M F [a stit: A&B"]. By the foregoing proof, we know that if M, M F [a stit: A&Ba], then M, M F ~[a stit: A & B a ] , and hence m, M F= [a stit: a&~[a stit: A&Ba}} by Lemma 15-6. Let M, m 1= [a stit: .A] with witness tu. For each sentence C, if M, m F= [a stit: A&Ca], we call [a stit: A&Ca] a (semantic] positive companion (poscompanion) to [a stit: A] at m (w.r.t. M), and C a pos-companion root of [a stit: A] at m ( w . r . t . M ) . Similarly, if M, m F ~[a stit: A&Ca], we call ~[a stit: A&(Ca] a negative companion (neg-companion) to [a stit: A] at m, and we call C a neg-companion root of [a stit: A] at m. Pos-companions and negcompanions to [a stit: A] at m are called companions to [a stit: A] at m. Note that every sentence must be either a pos-companion root or a neg-companion root of [a stit: A] at m. Let M be any model in which M = {m': m' € i(m) & m' = aw m} and suppose that m, m F= [a stit: A] with witness w. The companion theorem guarantees that for each sentence C, if M, m F= [a stit: A&. Ca], then M, M F [a stit: A &ca]; and That is to say,[a stit; A] must be true together with all its companions through all of

M, not only with its consequences. It can be seen from our further discussions that it does not suffice to consider only consequences of [a stit: A}; the notion of companions is essential to our proof of the completeness theorem. Of particular interest is that companions help us to compare witnesses to stit sentences, which we show as follows. 15-12 LEMMA. (Companions and witness) Let M be any model in which w < m, i = i(m) and m' e i >w, and let M, m F= [a stit: A&Ca\ with witness w, and M, m' (= [a stit: B&~[a stit: B&(Ca]] with witness w'. Then w' < w. PROOF. Since w < m' and w' < m', either w' < w or w < w' by no backward branching. Suppose for reductio that w < w'. Then, setting M = {m": m" € i & TO" =aw, m'}, we know that M C i >w. Since m, i\>w F= Ca by hypothesis and Lemma 15-7, it follows that m, M F Ca. By Lemma 15-9 (ii) we have that M, m' F= [a stit: B] with witness w', and hence by Lemma 15-6, M, M F= [a stit: B&C"]. But we know by hypothesis that M, M F ~[a stit: B&Ca], a contradiction. It follows from this reductio that w' < w. Among other things, what Lemma 15-12 tells us is this: If a sentence is a pos-companion root of [a stit: A] at m but a neg-companion root of [a stit: B] at m, then the witness to [a stit: A] at m must be strictly later than the witness to [a stit: B] at m, as formulated in the following corollary.

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15-13 COROLLARY. (Companions and witness) Let 97T, m 1= [a stit: A&cCa] with witness w, and 971, m 1= [a stit; B&~[a stit; B&C"*]] with witness w'. Then w' < w. 15-14 COROLLARY. (Validity of A9)

For any model 971, 97T N A9.

PROOF. Suppose that

and let i = i (m) and M = {m': m' £ i & m' =* m}. Then m, M \= Ca, and hence by Lemma 15-7, 971, i >w 1= tu ^ ~[a stit; ^4&[a stit; 5& ~[a stit; 5&C"*]]]. Hence there must be an m' 6 i| >U) such that

and by Lemma 15-9 (ii), 971, m' t= [a stit; B] with witness w'. Set M' = {m": m" 6 i & m" =°, m'}. Then 971, M' 1= [a stit; 5]. Applying Lemma 15-12 to (8) and (9), we have w' < w, and hence by Lemma 15-4^, i\>w C M'. It follows that 971, m E [a stit; B]. D

15C

Soundness: Validity of refref equivalence

In the last section, we showed the validity of all axioms A6-A9 of Lai + rr for all BT + I + AC structures (Lemma 15-7, Corollary 15-10, Theorem 15-11, Corollary 15-14). It is easy to see that for all BT + I + AC structures, all axioms A1-A5 are valid (see §15A) and that the rules modus ponens and RE are "validity-preserving." Thus, in order to establish the soundness of La1 + rr, the only thing left to show is the validity of refref for all BT + I+AC + nbc structures. 15-15 LEMMA. (Doing implies refraining from refraining) Let M be any BT + I + AC + nbc model. Suppose that M, m F= [a stit; A}. Then M, m F [a stit: ~[a stit; ~[a stit; A}}}. PROOF. Let M, m \= [a stit: A] with witness w and counter m', and let i = Since

Suppose for reductio that M, m¥ [a stit: ~[a stit; ~[a stit; A]]]. Then we have by (10) and Lemma 15-7 that

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We claim that there must be an mo € M' such that M, mo F [a stit: A] with witness WQ and counter mo; for if M, M' F= ~[a stit: A], then by (11), M, M' N= [a stit: ~[a stai: A]], contrary to (12). Furthermore, since mo € M', M' = {m": m" € i & m" =° m0}; and since w < mo and w0 < m0, either m < wo or Wo < w by no backward branching. If WQ < w, then, setting MO = {m": m" E i & mO" =aw0 M}, M' C M0 by Lemma 15-4, and hence, since M, MO F= [a stit: A], M, M' F [a stit: A], contrary to (11). So it must be true that w < Wo. Let M'o = {m": m" E i & TO" =awo mo'}. In general, we can inductively define two sequences m0, mi, ... and w0, w1, ... such that for each k > 0, mk + 1 E M''k = {m": m" € i & m"=awkmk} and M, mk + 1 F [a stit: A] with witness Wk + 1 > wk > w and counter m'fe + 1. The proof of this is similar to the earlier one. But {w, WQ, w\, ...} is obviously a busy choice sequence, contrary to our hypothesis. 15-16 LEMMA. (Refraining from refraining implies doing) Let 971 be any BT + I+AC + nbc model. Suppose that 971, m t= [a stit: ~[a stit: ~[a stai: 4]]]. Then 971, m N [a sizi; 4]. PROOF. Let 971, m 1= [a stit: ~[a stai: ~[a stit: A]}] with witness w and counter m*, and let i = i (m ) and M = {m": m" € i & TO" =£ TO}. Then 97t, m* (= [a siit' ~[a stit: A]] with the witness w* ^ w by Lemma 15-8, and hence

Suppose for reductio that 971, m K [a siii: /!]. Since

there must be an mo G M such that 97i, mo N [a sfoi: /IJ with witness WQ and counter m0; for if 071, M 1= ~[a stit: A], then 971, M N (~[a stt .4])Q by (14), and hence by Lemma 15-7, 971, i\>w t= (~[a stit: A ] ) a , contrary to (13). Since m0 € M, M = {m": m" E i & m" =™TOO};and since w < mQ and w0 < TOO, either w < w0 or w0 ^ w by no backward branching. If w0 ^ w, then, setting M0 = {m": m" G i & m" =aw0 mo}, we have M C M0 by Lemma 15-4, and hence, since 971, Mo1= [a stit: A], M, M F= [a stit: A], contrary to the assumption of our reductio. It follows that w < WQ. Let M0 = {m": m" € i & m" =aw0 In general, we can inductively define two sequences mo, m1, ... and w0, mo}. w1 ... such that for each k > 0, mK + 1 € M'k = {m": m" E i & m" =awk mk} and M, mk + 1 F= [a stit: A] with witness wk + 1 > Wk > w and counter m'k+1. The proof of this is similar to the earlier proof. But {w, WQ, w\, ...} is a busy choice sequence, contrary to our hypothesis. D These two lemmas enable us to establish the soundness of Lai + rr. 15-17 THEOREM. (Soundness theorem) For every sentence A, and for every BT + I + AC + nbc structure 6, h- A in La1 + rr only if 6 f= A.

392

15D

Proofs and models

Companion sets

In §15B we studied some basic features of semantic companions to stit sentences. From now on we study the syntactic features of companions. Our proof of the finite model property and the completeness of Lai + rr needs to consider, for each consistent sentence A, a finite set F of sentences including all subsentences of A. From now on, whenever F appears, we presuppose that F is finite and nonempty. Corresponding to §15B, we define (syntactic) companions to stit sentences as follows. Let F be any finite set of sentences, and let $ be any maximal consistent set (MCS) containing [a stit: A}. For any C € F, if [a stit: A&Ca] E $, we call [a stit: A&Ca] a (syntactic) T-pos-companion to [a stit: A] in 0 and m > 0:v

Then for any sentence C, PROOF. As in our proof in the previous lemma, we obtain by applying T7 k times and T15 m times. 15-21 LEMMA. (Properties of companion sets) Let E be any F-companion set for [a stit: A], and let [a stit: B] be the characteristic sentence of E w.r.t. A. Then the following hold: i. For every sentence

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Proofs and models

and consequently E h [a stit: B&C"*] or E F ~[a stit: 5&C"]; ii. If C"i, ..., Ck, and DI, ..., Dm are, respectively, all the F-pos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E, then

and consequently

PROOF, (i) By Lemma 15-20 and A2, we need only to show one direction of each equivalence. Suppose that C £ F and £ I- [a stit: A&Ca]. Since F [a stit: B] D Ca by A2, it follows from R3 that E F [a stit: B&C"]. Suppose next that E E ~[a stit: A & C a ] . Then E T [a stit: A&~[a stit: A&Ca}] by A8 since E T [a stit: A]. We show as follows that E F [a stit: B&~[o; stit: B & C a ] ] . By Lemma 15-20, F [a stit: B & < C a ] D [a stit: A&C a] . So E T ~[a stit: B & C a ] , and hence, T [a stit: B] D ~[a stit: B&C"*]. It follows from R3 that E h [a stit: B&~[a stit: B & C a ] ] . (li) By (i), we have that F [a stit: B] D [a stit: B&Cai] for all i with 1 < i < K, and that F [a stit: 5] D [a stit: B &~[a stit: 5 &£>"]] for all j with 1 <j < m. Then we can complete the proof by applying T7, T15, and A4. A direct consequence of Lemma 15-19 and Lemma 15-21 is the following. Let E be the F-companion set for [a stit: A] in an MCS $ with C\, ..., Ck, and DI, ..., Dm to be, respectively, all the F-pos-companion roots and all the Fneg-companion roots of [a stit: A] w.r.t. $. Let BQ, BI, B%, ... be defined as follows:

and let £„ = DC([a stit: Bn + i\) for all n > Q. Then E = E0 = EI = .... That is to say, for each n > 0, E is the F-companion set for [a stit: Bn] in $ (with [a stit: Bn + i] to be the characteristic sentence of E w.r.t. Bn), and [a stit: Bn] has the same F-pos-companion roots and the same F-neg-companion roots w.r.t. E as [a stit: A] has. 15-22 LEMMA. (Properties of companion sets) Let E be a F-companion set for [a stit: A]. Then for every sentence C 6 F,

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PROOF, (i) By A2, it suffices to suppose that E h Ca and show that E h [a stit: A&Ca]. Let [a stit: B] be the characteristic sentence of E w.r.t. .A. Since h [a stit: B] D Ca*, it follows that E h [a stit: B & < C a ] by R3, and hence E 1- [a stit: A & C a } by Lemma 15-21 (i). (ii) Suppose that E \- C and E F [a stit: C]. We know that Eu{~[a stit: C}} is consistent, and hence it must be included in some MCS &[a stit: D]. Then by Lemma 15-28(Hi), n P Da and II h Ca. Since II h A& C by A2, II h- £>&[« stit: D] by Lemma 15-22(ii). It follows that II h [a stit: A&C}&~[a stit: C], and hence II h [a

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stit: j4& Ca] by A6. The other direction of the proof is similar, (ii) Let $ be any MCS including E, and let £' be the F-companion set for [a stit: A] in $. By Lemma 15-24, E contains all the F-companions to [a stit: A] in $. Setting ^ to be any MCS including II, we know for the same reason that Ii contains all F-companions to [a stit: A] in ty. According to (i), each F-companion to [a stit: A] in $ must be in \? and vice versa. It follows that E' C EnII. D 15-30 LEMMA. (An inclusion relation among companion sets concerning a particular stit sentence) Let E and E' be F-companion sets in an MCS <J>, let II be a F-alternative to E such that E' C EnII, E h [a stit: A], II h [a stit: A] and £' V- [a stit: A], and let [a stit: B] be any characteristic sentence of £'. Suppose that F is equivalently closed under conjunction, A € F, and E" is the F-companion set for [a stit: A&[a stit: B}} in $. Then £' C E" C Enll. PROOF. Since E" h [a stit: B] by A2, it is then clear that E' C E". To show that E" C Enll, we need only to show that E and II contain all the F-companions to [a stit: A&c[a stit: B]} in $. To that end, we first note that since A € F and E h [a stit: A] and II h [a stit: A], it follows from Lemma 15-24 and Lemma 15-29 that

Consider any F-pos-companion root C of [a stit: A] w.r.t. $. Since £' C Enll, E h [a stit: AkCa]k[a stit: B} and II h [a stit: A&Ca}&[a stit: B} by (17). It then follows from A3 and A4 that E h [a stit: A&[a stit: B}&Ca] and II h [a stit: A&[a stit: B}&Ca}. Hence by Lemma 15-26 (substituting E' for E in that lemma), E and II contain all the F-pos-companions for [a stit: A & [a stit: B]] in $. Consider any F-neg-companion root D of [a stit: A] w.r.t. $. By (17), we have

It follows from E' C Enll that E' P [a stit: A] D [a stit: A&Da] (otherwise we would have E h ~[a stit: A] and II h ~[a stit: A]), and hence by Lemma 15-25, E' y [a stit: B&Da}. Then £' h ~[a stit: B&Da] by Lemma 15-21 ft), and hence by A5, E' h ~[a stit: [a stit: B]&Da]. It follows from Rl and T4 (by contraposition) that

Hence by (18), we have E h ~[Q stit: A&[a stit: B]&Da] and H h ~[a stit: A &[a stit: 5]&Z) a ]. It follows from Lemma 15-26 that E and H contain all the F-neg-companions—and hence all the F-companions—to [a stit: A&[a stit: B]} in $, that is, E" C EnH. This completes the proof. D II is a F-counter to £ relative to [a stit: A] iff H is a F-alternative to E, E h [a stit: A], and IIV- A. Assume that F is equivalently closed under conjunction. H is a T-semi-ref-counter to E relative to [a stit: A] iff

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Proofs and models

II is a T-ref-counter to E relative to [a stit: A] if II is a F-counter to E relative to [a stit: A] and II h [a stit: ~[a stit: A}]. It is easy to see that ref-counters are semi-ref-counters. Note that when II is a F-counter (F-semi-ref-counter, F-refcounter) to E relative to [a stit: A], E does not have to be a F-companion set for [a stit: A}; but when F is equivalently closed under conjunction, by Lemma 1524, S does contain all the F-companions to [a stit: A] w.r.t. any MCS including S. As we said in the last section, a F-companion set E may represent a possible choice for a at a moment. In the same way, II, as a F-semi-ref-counter to E, may represent another possible choice for a at that moment. In the structure that we will construct in the proof of completeness and the finite model property, if S represents one choice for a at a moment, then II will represent the other choice for a at that moment (i.e., a has only binary choice). The semi-refcounter relation between F-companion sets is so important for our proof that we introduce the following notation for it. Let F be equivalently closed under conjunction. R£ (S, FT) iff the following conditions hold.

Note that whenever we use R£(E, II) in a discussion, we presuppose that F is equivalently closed under conjunction. Comparing (19), (20), and (21) earlier and (22), (23), and (24), the definition of -Rp(E, II) may not appear to give us the semi-ref-counter relation. But Lemma 15-31 guarantees that RT (E, II) iff [a stit: A] 6 F and II is a F-semi-ref-counter to S relative to [a stit: A], provided that F satisfies the following condition: F is closed under negated stit sentences if for every sentence A, [a stit: A] £ F implies ~[a stit: A] e F.3 3

Note that the notion Rp (E, II) is sufficient for our proof of the completeness and the finite model property. flp(E, II) does not imply that II is a F-semi-ref-counter to E relative to [a stit: A] if F does not have the property of being closed under negated stit sentences. That is to say, F's property of being closed under negated stit sentences is not essential for our proof. We give F this property only because it makes fip (E, II) match the notion of F-semi-ref-counters, while the ideas of F-semi-ref-counters and F-ref-counters are easier to follow than the idea of fl£(E, n).

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15-31 LEMMA. (Serm-ref-counter relation and RA ) Let F be equivalently closed under conjunction. Then

and if for every and if also

is a

pos-companion root of

is closed under negated stit sentences, then and is a semi-ref-counter to relative to

PROOF, (i) Assume that RA(£, II). We first show that I I P [a stit: A], which, by definition, is true when there is no F-neg-companion to [a stit: A] in E. Assume that D\, ..., Dm (m ^ 1) be all the F-neg-companion roots to [a stit: A] w.r.t. E. Suppose for reductio that E h [a stit: A}. Then, on the one hand, we know by definition that II I- V i< »«s m [a stit: A&D?]. On the other hand, Lemma 15-29 provides H t- &i^ is^m ~[a stit: A&cD?}. Hence II is inconsistent, contrary to our assumption of consistency of H. It follows from this reductio that H Y- [a stit: A}. We next show that H Y- A. Suppose for reductio that H h- A. Then by Lemma 15-22(ii) and II Y- [a stit: A], Ft h v4&~[a stit: A], and hence E h ~[a stit: A] by Lemma 15-28 (in), contrary to our assumption of consistency on E. It follows from this reductio that E V- A. (ii) Suppose that RA(S, II), ~[a stit: A] £ F and that every C 6 F is a F-pos-companion root of [a stit: A] w.r.t. E. Then by RA(E, II), II h ~[a stit: A}. Suppose for reductio that II Y- [a stit: ~[a stit: A}}. Then II h ~[a stit: ~[a siii: 4]] by Lemma 15-22^, and hence by Lemma 15-28(Hi), E h ~[a stit: A], contrary to our assumption of consistency on E. It follows from this reductio that II h [a stit: ~[a stit: A]]. It is easy to see that (Hi) follows from (i), (ii) and related definitions.

15F

Semi-ref-counters

From our discussion in the last two sections, one can see that F's property of being equivalently closed under conjunction is important for our proof. We now show that for every sentence A, we can associate a finite set F that has this property. Let A be any sentence, and let I" be the set of all subsentences of A, and let F* = F'U{~[a stit: C]: [a stit: C] e F'}. We define FA = {5j&...&5 n : BI, ..., Bn are distinct sentences in F*}. In the following lemma, for any set X = {Bi, ..., Bn} of sentences, &X = Bi&...&B n . 15-32 LEMMA. (Finding a set that has the desired properties) Let A be any sentence. Then TA is a finite set that is closed under subsentences, closed under negated stit sentences, and equivalently closed under conjunction. PROOF. It is easy to see by definition that TA is a finite set that is both closed under subsentences and closed under negated stit sentences. To show that TA

402

Proofs and models

is equivalently closed under conjunction, let B = £?i&...&5 m and C = C"i& ...&C n be any sentences in YA such that I ^ m, n ^ |F*|, and BI, ..., Bm are distinct sentences in F*, and C\, ..., C n are distinct sentences in F*. We show as follows that there is a D e FA such that 1- BhC = D. Let Conj(5) = {#!, ..., 5m} and Conj(C) = {C1; ..., Cn}. Clearly, if Conj(5) = Conj(C7), we can set D = B. Suppose that Conj(5) 7^ Conj(C'). Let us set X = Conj(5) -Conj((7), Y = Coiy(C)-Coiy(5), Z = Conj(B)nConj(C r ), B' = &X, C" = &Y, and E = &Z. It is easy to see that X, Y, and Z are disjoint subsets of F* and h 5'& C'hE = £& C. We can thus set D = £'& C"&£ to complete the proof. In this section we focus on F-semi-ref-counters to F-companion sets. From now on, whenever F appears in our discussion, we presuppose that F is a finite set that is closed under subsentences, closed under negated stit sentences, and equivalently closed under conjunction.4 15-33 LEMMA. (A property of semi-ref-counters corresponding to the absence of busy choice sequences) Let S, II and E' be any F-companion sets such that ,Rr(S, II), n C E' and E' h [a stit: A}. Suppose that there are n + l F-negcompanions to [a stit: A] in E. Then there are at most n F-neg-companions to [a stit: A] in S'. PROOF. We first show that

Consider any C € F. Suppose that E h [a stit: A&, Ca}. Then E h Ca by A2, and hence H h Ca by Lemma 15-28(Hi). Because H C S' and E' h [a stit: A], it is then clear that H F ~[a stit: A}. Since -Rp(E, H), II Y- [a stit: A] by Lemma 15-31 (i). Hence H h [a stit: A] D [a stit: A&Ca} by Lemma 15-25, and hence E' h [a stit: A&Ca}. It follows that (25) holds, which implies by Lemma 15-24 that

Setting DI, ..., Dn + i to be all the F-neg-companion roots of [a stit: A] w.r.t. E, we have by definition of R£ (E, II) that II h [a stit: A] D V i«s z< n + i [a stit: AkD?}. Hence E' H V i < t $ n + i [a stit: AkD?]. Since, by (26), S' contains all the F-companions to [a stit: A] in any MCS including S', it follows that there is a k such that 1 ^ k ^ n + l and E' h [a stit: A8z.D%}. Hence by Lemma 15-24, there are at most n F-neg-companions to [a stit: A] in S'. This completes the proof. D 4 As we said in note 3, the property of F of being closed under negated stit sentences is not essential. This property has no effect in our upcoming discussions, except that it helps one to see some of the ideas of our proof (by referring to Lemma 15-31).

15. Decidability of one-agent achievement-stit theory with refref

403

A direct consequence of Lemma 15-33 is the following: Let n > 0, and let EQ, • • • > E n and HO, ..., IIn be F-companion sets such that for each i with 0 ^ i ^ n, .R r (£j, II,), and for each i with 0 ^ i ^ n —1, IIj C EI+ I. Suppose that there are n F-neg-companions to [a stit: A] in EQ. Then Hn must be a F-ref-counter to E n relative to [a stit: A], and there can be no E and II such that -Rp(E, LT) and nra C E. This feature of F-semi-ref-counters corresponds to a semantic feature of Lai + rr—there is no busy choice sequence in its models. The following three lemmas establish some sufficient conditions for F-semiref-counters. 15-34 LEMMA. (A sufficient condition that semi-ref-counters exist) Let E be a F-companion set for [a stit: A] with [a stit: B] to be the characteristic sentence of E w.r.t. A. Let Ci, ..., Ck and D I , ..., Dm be, respectively, all the Fpos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E. Suppose that Ft = DC([a stit: ~[a stit: J5J&C 1 ]) is consistent, where C = &i< z < k C™. Then II is a F-companion set for (a stit: ~[a stit: B}} and is a Falternative to S, and II h- fa stit: A] D V i ^ j < m la s^: A&D"]. In particular, then PROOF. We first show that H is a F-companion set for [a stit: ~[a stit: B}} and is a F-alternative to £. To that end, we first note that by T7, II h [a stit: ~[a stit: B]}. Consider any D = D} with 1 ^ j < m. We know by Lemma 15-21 (ii) and T16 that

Hence h ~[a stit: ~[a stit: B]&Da} by Rl. It follows from hypothesis and Rl that

Hence by Lemma 15-19, II is a F-companion set for [a stit: ~[a stit: B}}. It is easy to check that H is a F-alternative to E. Next, since II h [a stit: ~[a stit: B]bC], and since h ~[a stit: C} by T6, it follows from Rl that H h [a stit: ~[a stit: B&~[a stit: C]]&.Ca]. Because B is the sentence v4&(7&;(&i^j^ m ~[a stit: A k D f } ) , setting B* = ^ & ( & i < j < m ~[a stit: A&DJ*]), we have II h [a stti: ~[a siii: B'&^j&C 1 0 ], and hence by T14, U\-[a stit: ~[a stti: 5*] & (7°]. It follows by A2 that II h ~[a sfit: B*], and hence by Lemma 15-27, II h [a sttt: v4] D V i 5 S j = £ m ta s^; ^&-0"]- If there is no F-neg-companion to [a stit: A] in E, then B = A& C, and hence by T6, II h [a stit: ~[a stai: ^4& Q ]; and applying (30) and Rl to Til, we have I- ~[a stit: [a stit: £]&£>"]. It follows from T4 and Rl that I- ~[a stit: ~[a stit: B']&[a stit: 5]&D a ]. This completes our proof of (28). Hence II' is a F-companion set for [a stit: ~[a stit: B'} &[a stit: B]] and a F-alternative to E'. It is easy to see that E C E'nll' since E' h [a stit: B} and II' h [a siii: 5]. We show next that #r( s / > n / )- Since A e F and E' h A by A2, £' h [a stai: A] by Lemma 15-22 (h). Let B* and C1* be the following sentences:

Then by Rl, h [a stit: B1} = [a stit: 5*&[a stit: 5]&C*], and hence by hypothesis and A2, H' h ~[o; stit: 5*&[a siii; 5]&C7*]&[o! siif; B]. On the one hand, by applying A4 (by contraposition), we have II' h ~[a stit: B*]V ~[a stit: [a stit: B]&C""]. On the other hand, by applying Lemma \5-2l (ii) and T10, we have II' h [a stit: [a stit: 5]& C1?] for all i with 1 < i ^ fc (since IT h [a stit: B } ) , and hence by A4, II' h [a stit: [a stit: B}&C*}. It follows that II' I- ~[a stit: B*} (when there is no F-neg-companion to [a stit: A] in E', the same argument here, replacing B* by A, yields II' I- ~[a stit: A}). Hence by Lemma 15-27 (substituting A&[Q stit: B] for A in the lemma), we know that

15. Decidability of one-agent achievement-stit theory with refref

405

Since II' h [a stit: B] and h [a siz'i: 4]&[a sizi: B\ D [a s£zi: A&[a stit: B}} by A3 and A4, it follows that

We also know by Rl, T4, and A5 that for each j with 1 ^ j ^ TO,

But for each such j, £ h ~[a s£z't: 5&.D"] by hypothesis and Lemma 15-21 (ii), and hence IT h ~[a siif: 5&D°] by £ C n'. It follows from (31), (32), and (33) that

We have shown that £' h [a s£z't: .4], and hence by Lemma 15-24, £' contains all the F-companions to [a stit: A] in $. Since £ .K [a stit: A], we know by Lemma 15-26 that D I , ..., Dm are all the F-neg-companion roots of [a stit: A] w.r.t. E'. Hence by definition, R$(Z', IT). D 15-36 LEMMA. (A sufficient condition for extending a companion set to another and its semi-ref-counter) Let £ be a F-companion set in an MCS $ such that £F [a stit: A] and [a stit: A] e 3>nF, let E' be the F-companion set for [a stit: j4&[a stit: B}] in $, where [a stit: B] is a characteristic sentence of E, and let [a stit: B'} be the characteristic sentence of £' w.r.t. A&[o; stit: B]. Suppose that there is a C 6 F such that

and II' = DC([a stit: ~[a stit: 5']&C"]) is consistent, where C' = fci^j^fc C? and Ci, ..., Ck are all the F-pos-companion roots of [a stit: A&[a stit: B\] w.r.t. $. Then #r(E', n') and E C E'nn'. PROOF. Since A e F and E' h A by A2, it follows from Lemma 15-22(ii) that £' h [a sizi: yl]. By Lemma 15-34 we know that II' is a F-companion set for [a stit: ~[a stit: B'}} and is a F-alternative to E', and also, setting D I , ..., Dm to be all the F-neg-companion roots of [a stit: A&.[a stit: B}] w.r.t.

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FACING THE FUTURE Agents and Choices in Our Indeterminist World

NUEL BELNAP MICHAEL PERLOFF M I N G XU

With Contributions by Paul Bartha Mitchell Green John Horty

OXPORD

UNIVERSITY PRESS

2001

OXFORD U N I V E R S I T Y PRESS Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Pans Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan

Copyright © 2001 by Oxford University Press, Inc. Published by Oxford University Press, Inc 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press Library of Congress Cataloging-in-Publication Data Belnap, Nuel D , 1930Facing the future • agents and choices in our indeterminist world / Nuel Belnap, Michael Perloff, Ming Xu, with contributions by Paul Bartha, Mitchell Green, John Horty. P cm Includes bibliographical references and index. ISBN 0-19-513878-3 1 Agent (Philosophy) 2. Choice (Psychology) 3 Free will and determinism I Perloff, Michael II Xu, Ming III Title B105 A35 2001 128'4—dc21 00-064995

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

Preface This is a book about the causal structure of agency and action. It frames a rigorous theory by using techniques and ideas from philosophical logic, philosophy of language, and metaphysics with a small "m." This theory, which we sometimes call "the theory of agents and choices in branching time," describes agents as facing a future replete with real possibilities, some of which various agents realize by making choices. It is central to our theory that choices and the actions that they ground are radically indeterministic: Before an event of choosing, there are multiple alternatives open to the agent. Furthermore, since the choice is real, so must be the alternatives, and each alternative must be as real as any other. All we can say before the moment of choice is that the agent will make one of the open choices, leaving behind the unchosen alternatives. After the choice, it is correct to say that they were once possible, but are no longer possible. None of the possible choices is a mental or linguistic figment, nor is any a mere ghost image of "the actual choice." Given that the possibilities relevant for action are always possibilities for our future, the theory also refrains from appealing to "possible worlds" other than the one and only world that we all inhabit. These ideas are in some part rooted in common sense. Without help, however, common sense cannot seem to pull them together into a coherent whole. One of our principal aims is to carry out that job by articulating them in a completely intelligible exact theory. The resultant theory of agents and choices in branching time pictures the causal structure of our world as made up of alternative courses of events branching tree-like toward the future. Each branch point represents a choice event or chance event. On the one hand, each continuation from a branch point is individually possible; on the other hand, it is impossible that more than one of these continuations should be realized. If that sounds obscure, we agree: It is, we think, almost impossible to speak clearly and accurately about indeterminism except in the framework of a rigorously fashioned theory such as the one we propose. The theory of agents and choices in branching time is of real but limited interest without its application to understanding the language of action. We propose a certain linguistic form, a "modal connective," as being unusually helpful to anyone who wishes to think deeply about agents and their actions.1 1

Our usage here is common but not universal. A connective is any grammatical construe-

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Preface

The form is "a sees to it that Q," where a names an agent and Q holds the place of a sentence; for example, we think of "Ahab sailed the seven seas" as "Ahab saw to it that he sailed the seven seas." The form is so important to our enterprise—all but three of our eighteen chapters are devoted to its study or use—that we give it an abbreviation, "[a stit: Q]" with "stit" as an acronym for "sees to it that." Stit theory explains the meaning of [a stit: Q] in the idiom of philosophical logic. In doing so, stit theory invokes a certain melding of the Prior-Thomason indeterministic semantics with Kaplan's indexical semantics. The combined semantics make [a stit: Q] roughly equivalent to "a prior choice by a guaranteed that Q." The stit idea is many-sided. We explore its grammar, semantics, and proof theory as logicians do. We delight in the fact that stit does not treat actions primarily as "things" to be counted or named. We explore some of the linguistics of stit (especially how and why its status as a modal connective lends itself to usefully complicated constructions), we consider some applications to difficult conceptual problems, and we argue the ability of stit to illuminate agency in a variety of ways. We look at some ways in which stit might be modified or generalized. In all of this, however, we try never to forget the central constraining thought: There is neither action nor agency nor doings without real choices, choices that find their place not merely in the agent's mind, but within the (indeterminist) causal order of our world. To see to it that Q, an agent must make a real choice among objectively incompatible future alternatives. When we say that an event may have many possible but incompatible outcomes, we thereby come down on the side of "hard" indeterminism as against determinism. There is no consensus on these ideas. Since the eighteenth century became understandably awed by the success of Newtonian science, the presumption of determinism has guided most of the philosophical and scientific explorations of both agency and nature. Taking determinism to be delivered by science as an unquestioned "fact," philosophers since Hume and Kant have worked at developing "compatibilist" theories that hold agency, in the guise of moral responsibility, to be compatible with what James called the block universe. Such theories have often taken possibilities as unreal: as arising from the mind, or from social practices, or from language—for example from consistency with the bits of language called "scientific laws." Our contrasting indeterministic presumption is eloquently expressed by the eminent paleontologist Stephen Jay Gould. I don't think that any deeper or more important principle pervades nature, and lies at the heart of all historical sequences, than this central but underappreciated notion of "contingency"—the great and tion that maps one sentence (or several sentences) into another sentence. For example, when you put sentences (i) "Ahab is captain" and (ii) "Ishmael is not captain" into the blanks of " and ," the result is another sentence, (iii) "Ahab is captain and Ishmael is not captain." That makes the "and" construction a connective. A truth functional connective is one like " and ": If for example you know whether each of (i) and (ii) is true, then you automatically know whether (iii) is true. A modal connective is defined negatively as being one that is not truth functional.

Preface

vii

liberating truth that tiny inputs, virtually invisible and risibly impotent in appearance at the outset, can cause history to cascade down any route in a vast array of entirely different pathways. (Gould 1999, p. 30) This book neither argues for indeterminism nor tries to pick holes in arguments for compatibilism. Our project assumes the indeterminism of the causal order in which agency is embedded, it assumes that actions are based on real choices, and it assumes that choices are therefore not predetermined. Our goal is not to persuade, but to make these ideas intelligible. Although numerous philosophers share our general point of view, not many exact theories share these assumptions and aims. Our strategy is to concentrate almost exclusively on the objectively causal side of indeterminism and agency, which already presents enough difficulties without bringing in noncausal concepts. We therefore lay aside many deeply important aspects of agency and choice that involve intentions, propositional attitudes, or other mental phenomena. We look for ways in which applications of stit theory can engender a better understanding of agency. Seven examples: (i) an analysis of refraining that clarifies how it can be both a doing and a not-doing; (ii) an analysis of imperatives that emphasizes their agentive content; (iii) an extended treatment of deontic logic that insists that obligations and permissions (a) are directed to agents capable of making choices, and (b) are embedded in the indeterministic causal order of our world; (iv) fresh analyses of promising and of assertion, analyses that argue the unwisdom of the doctrine that among all the objective possibilities, a unique course of events constitutes the one and only "actual future"; (v) an exploration of the causal side of the requirement on action that the agent "could have done otherwise"; (vi) the causal structure of joint agency; and (vii) a generalization of stit theory to strategies considered from a causal point of view. In sum: Holding the extra-mental and extra-linguistic status of incompatible possibilities as given, and supposing that the future sometimes depends upon an agent's choices among incompatible options, we offer a tense-modal theory intended to describe some causal aspects of agency in our indeterministic world. Guidance on reading this book. In the spirit of Carnap, each chapter begins with some introductory remarks in order to permit easy skipping of topics not of present interest to the reader. We append the following largescale structural notes as additional guidance. The book divides into six parts of varying degrees of technicality, followed by an appendix. Each of these may be characterized as follows. Part I introduces stit theory. We intend the chapters in this part to be accessible to all those with an interest in our topic. Portions of chapter 2 and chapter 5 do, however, involve willingness to put up with some elementary logical constructions, and chapter 2 offers a brief explanation of the theory of agents and choices in branching time that underlies stit theory. In this part we introduce key grammatical and semantic features of stit, including the distinction

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of the deliberative stit from the achievement stit. Largely concentrating on the achievement stit, the various chapters of part I suggest applications with the help of many pictures, make comparisons with some other work on agency, and, beginning with Anselm's work in 1100, give a little history of the modal logic of agency. This part also contains applications of stit theory to imperatives and to promising. Part II supplies precisely and in detail the nuts and bolts—or, more aptly, roots and branches—of the theoretical structure that supports our account of agents, actions, and our indeterminist world. The three chapters of this part are foundational in character, and involve substantially more rigor, though not much more mathematics. They stress conceptual analysis rather than theoremproving. This part more than any other focuses on the problems faced by any indeterministic theory. Here we argue against the beguiling but harmful doctrine of "the actual future," which says that among the many courses of events that might come to pass, there now exists a privileged such course that will actually do so. This foundational part examines, postulate by postulate, the theory of agents and choices in branching time, and explains in detail the semantic subtleties required of a language spoken in an indeterminist world. Part III offers two applications of the achievement stit: One chapter aims to illuminate the dark idea of "could have done otherwise," and another considers the causal aspects of joint agency. These chapters are a little more technical. Part IV is of the same level of technicality as part III: It offers applications of the deliberative stit, chiefly to help in elaborating such deontic concepts as obligation and permission, which, we believe, are in much need of a theory of agency. Part V uses the already-established theoretical structure of agents and choices in branching time in order to develop an austere (causal, not normative) account of strategies as a kind of generalization of stit. One chapter in this part connects our theory of strategies to Thomason's deontic kinematics. Part V proves a theorem or two, though much of it is, again, conceptual analysis. Part VI provides the technical backbone of stit theory, including proofs of decidability, soundness, and completeness. The chapters of this part are required reading for those who wish to investigate or develop the logical and mathematical properties of theories of agency similar to ours. The appendix gathers, for easy reference, most of the various theses, structures, postulates, definitions, semantic ideas, and systems that are introduced elsewhere and are employed throughout the book. We use boldface to refer to certain of these items: Look in the appendix for its sections §l-§9, for stit theses Thesis 1-Thesis 6, for postulates Post. 1-Post. 10, for definitions Def. 1-Def. 20, and for axiomatic concepts Ax. Conc. 1-Ax. Conc. 3. Although in each section of this book we feel free to refer to any other section, it may be useful to indicate the following dependency-structure among the various parts. Parts I (introduction to stit), II (foundations of indeterminism), and VI (proofs and models) are almost entirely independent. The reader's primary interests may therefore be allowed to determine with which of parts I, II, or VI it is best to begin. Part III (applications of the achievement stit) and part IV

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(applications of the deliberative stit) are mutually independent, whereas each presupposes familiarity with part I. Part V (strategies) requires chapter 2 of part I as minimum background.

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Acknowledgments Our contributors deserve special thanks for giving us permission to use their materials: P. Bartha for contributing chapter 11 and for co-authoring chapter 12; M. Green for co-authoring chapter 6; and J. Horty for co-authoring §2A of chapter 2, which has its source in Horty and Belnap 1995. This essay is also used in a substantially different way in Horty 2001, a book elaborating the normative aspect of choice. Horty 2001 should be consulted by every reader of this book. We add that each contributor has also helped in other ways too numerous to count. Many other people have helped in the research reported in this book. We apologize for the simple alphabetical listings that we offer as sincere but inadequate thanks. The following assisted with the development of ideas and with the preparation of the manuscript during their respective terms as Alan Ross Anderson fellows: M. Allen, D. Bruckner, C. Campbell (who caught many dozens of errors and infelicities during a final proof-reading), K. Davey, U. Ergun (who created many of the figures), C. Hitchcock, C. Jones, J. MacFarlane (who produced the initial draft of the index), J. Roberts, L. Shapiro, A. Staub, V. Venkatachalam (who was responsible for the initial draft of the bibliography), and M. Weiner. In addition, a number of others have helped over the years, especially by commenting on or offering us fresh information in connection with the separate articles on which this book is based: G. Antonelli, L. Aqvist, C. Bicchieri, R. Brandom, M. Brown, D. Davidson, D. Elgesem, R. Gale, A. Gupta, D. Henry, I. Humberstone, D. Kaplan, M. Lange, W. Lycan, D. Makinson, J. Moore, R. Neta, I. Porn, K. Schlechta, K. Segerberg, S. Sterrett, J. Thomson, D. Turner, D. Vanderveken, F. von Kutschera, D. Walton, H. Wansing, and anonymous referees for several journals. Finally, we thank two anonymous OUP referees for encouraging us to turn a heap of articles into a book.

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Contents I

Introduction to stit 1 Stit: A canonical form for agentives 1A Agentives 1B Stit: Simple cases 1C Grammar of the modal logic of agency 1D Mini-history of the modal logic of agency 1E Conclusion and summary

. . . . .

3 5 9 14 18 26

2 Stit: Introductory theory, semantics, and applications 2A Theory and semantics: The two stits 2B Applications of stit, with many pictures

. .

28 29 39

3 Small yet important differences from earlier proposals 3A Von Wright 3B Chisholm 3C Kenny 3D Castaneda 3E Davidson 3F Conclusion

. . . . . .

59 60 65 68 74 78 81

4 Stit and the imperative 4A The theory of fiats 4B Ross's paradox and stit 4C Chellas's theory 4D Agentive constructions 4E Negations of imperatives 4F The many varieties of imperatives 4G Embedding imperatives 4H Conclusion

. . . . . . . .

5 Promising: Stits, claims, and strategies 5A From stit to promising 5B From RR to promising 5C Strategic content of promises and word-givings

97 . 98 . 107 . 116

82

82 84 85 87 89 92 94 96

xiv

II

Contents

Foundations of indeterminism

6 Indeterminism and the Thin Red Line 6A Preliminary considerations 6B Parameters of truth 6C The assertion problem 6D The Thin Red Line 6E Time's winged chariot hurries near

. . . . . . , .

133 134 141 156 160 170

7 Agents and choices in branching time with instants 7A Theory of branching time 7B Theoretical reflections on indeterminism 7C Theory of agents and choices 7D Domain

. . . .

. . . .

177 177 203 210 219

8 Indexical semantics under indeterminism 8A Sources 8B Structure parameters: The "world" of the speakers 8C Interpretation and model: The "language" of the speakers . 8D Points of evaluation, and policies 8E Generic semantic ideas 8F Semantics for stit-free locutions 8G Clauses for stit functors

, . . . . . .

220 . 221 . 226 . 227 . 228 . 234 . 239 . 247

9 Could have done otherwise 9A Could have been and might have been 9B Could have done and might have done 9C Might have been otherwise 9D Might not have done it 9E Could not have avoided doing 9F Could have prevented 9G Could have refrained 9H Might have refrained 9I Had available a strategy for not doing 9J Summary

. . . . . . . . . .

255 . 257 . 257 . 259 . 260 . 261 . 262 . 263 . 265 . 268 . 269

10 Multiple and joint agency 10A Preliminary facts 10B Other-agent nested stits 10C Joint agency: Plain and strict 10D Other-agent nested joint stits

. . . .

271 . 272 . 273 . 281 . 290

III

Applications of the achievement stit

Contents

IV

xv

Applications of the deliberative stit

11 Conditional obligation, deontic paradoxes, and stit 11A Technical preliminaries 11B Semantics of obligation 11C Completeness 11D Conditional obligation 11E Oc-statements versus cO-statements 11F The Good Samaritan 11G Contrary-to-duty obligations 11H Problems with the proposed semantics of obligation . . . .

295

12 Marcus and the problem of nested deontic modalities 12A The parking problem 12B The form of obligations 12C The Anderson/dstit simplification 12D The form of prohibitions 12E Generalized prohibitions 12F Generalization on agents 12G Temporal generalization 12H The outer ought

318 318 319 321 322 325 333 335 335

V

296 298 303 304 306 309 312 314

Strategies

13 An austere theory of strategies 13A Nature of austere strategics 13B Review of choices in branching histories 13C Elementary theory of strategies 13D Favoring 13E Application to finding a strategy for inaction

. . . . .

14 Deontic kinematics and austere strategics 14A Basic concepts 14B From Thomason's deontic kinematics to austere strategics 14C From austere strategics to Thomason's deontic kinematics 14D Remarks

. . . .

VI

341 342 344 345 356 359 364

365 368 370 376

Proofs and models

15 Decidability of one-agent achievement-stit theory with refref 15A Preliminaries 15B Companions 15C Soundness: Validity of refref equivalence 15D Companion sets

381

. . . .

382 385 390 392

xvi

Contents 15E Alternatives and counters 15F Semi-ref-counters 15G Completeness and finite model property

. . 397

. . 401 . . 408

16 On the basic one-agent achievement-stit theory 16A Preliminaries 16B Soundness 16C Companion sets and their alternatives 16D Construction of preliminary structures 16E Completeness

. . . . .

415 . 415 . 417 . 419 . 424 . 428

17 Decidability of many-agent deliberative-stit theories 17A Preliminaries 17B Soundness 17C Completeness and compactness 17D Finite model property

. . . .

. . . .

18 Doing and refraining from refraining 18A Preliminaries 18B Main results

451 . . 452 . . 454

Appendix: Lists for reference 1 Stit theses: Thesis 1-Thesis 6 2 Structures 3 BT + I + AC postulates: Post. 1-Post. 10 4 Branching-time-with-instants definitions: Def. 1-Def. 9 . . 5 Agent-choice definitions: Def. 10-Def. 14 6 Basic semantic definitions: Def. 15-Def. 16 7 Derivative semantic definitions: Def. 17-Def. 20 8 Grammar 9 Axiomatics concepts: Ax. Conc. 1, Ax. Conc. 2, and Ax. Conc. 3

435 435 439 441 445

459

. . . . . . . .

. 459 . 460 . 461 . 463 . 466 . 468 . 469 . 471

. . 472

Bibliography

475

Index

483

Part I

Introduction to stit

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1 Stit: A canonical form for agent ives Among the topics of discussion in this world, none are more common than those concerning the achievements and refrainings, obligations and prohibitions, successes and failures of the agents with whom we share a common space.* "What happened when so-and-so did that?" we ask, or "What should have been done?" Biographies and narrative histories, which form a sizable segment of our reading materials, have as their central concern the doings of agents, their obligations and prohibitions, the outcomes of their choices, and the range of things from which they refrain. Philosophers have long sought to find a distinction between those sentences which attribute agency and those that do not in the verbal configurations we commonly use to talk about such matters. If such a distinction existed it should be a relatively simple matter to uncover some general principle in everyday speech to differentiate Joshua fit the battle of Jericho from Joshua survived the battle of Jericho. We find, to our dismay, that we are no closer now to a linguistic litmus test for agency than was Aristotle. After all, it would have been of some importance for Aristotle's peers to decide between an agentive interpretation of Alexander succeeded to the throne of Macedonia and a non-agentive interpretation. For if Alexander was agentive in the matter of his becoming king, then there was a prior choice of his which led directly to * With the kind permission of Theoma, Belnap and Perloff 1988 is the basis of each section of this chapter except §1D. That section is drawn from Belnap 1991, for the use of which we thank the International Phenomenological Society.

3

4

Introduction to stit

that outcome, and he was likely guilty of regicide; while if he was not agentive in the matter of becoming king, then there was no choice of his that guaranteed his succession to the throne. J. L. Austin said that "The beginning of sense, not to say wisdom, is to realize that 'doing an action', as used in philosophy, is a highly abstract expression—it is a stand-in used in the place of any (or almost any) verb with a personal subject ..." (Austin 1961, p. 126). In his essay he tried to throw light on the question of "doing an action" by looking at the range of cases in which excuses are offered both in everyday usage and in the law, and to arrive at a proper vocabulary for action by "induction" on the proper uses of words. Many years have passed, the lesson has been learned, and it is time for philosophy to go beyond the mere beginnings of sense and progress toward a deeper understanding of an agent doing an action. How then should we proceed? The Austin legacy is of course a particularly rich one that is being carried on in a variety of ways. One way attempts to explain certain facts about language in terms of propositional content and illocutionary forces: "speech act theory" as represented for example by Searle and Vanderveken. That is not our program, as we make clear in many places hereafter, suggesting that a theory of "speech acts" should begin with a prior theory of acts. Accordingly we follow in the wake of Austin's suggestion that when faced with the question of the meaning of a word or a term such as "doing an action," we should "reply by explaining its syntactics and demonstrating its semantics" (Austin 1961, p. 28). A suggestion of this book is that the next step in the progression toward greater sense and wisdom is to have available the sort of clean and well-honed linguistic resource that Austin, and other philosophers, have realized to be necessary. We think that the most promising path to a deeper understanding of an agent making a choice among alternatives that lead to action is to augment our philosophical language with a class of sentences whose fundamental syntactic and semantic structures are so well designed and easily understood that they illuminate not only their own operations but the nature and structure of the linguistic settings in which they function. An example of a doing-an-action sentence that Austin might have had in mind is Ahab sailed in search of Moby Dick.

(1)

It has a personal subject, "Ahab," and an action-like verb, "sailed," and seems to be describing an action in Austin's deliberately wide sense. We take (1) not only to be true, but to be agentive for Ahab, for Ahab's sailing in search of Moby Dick was a direct result of a choice he made among alternatives available to him. On the other hand, although the perfectly ordinary sentence, The Pequod sailed in search of Moby Dick

(2)

is surely true, and though we may be hard pressed to say exactly why, we are not hard pressed to say that it is not agentive: It does not even have a personal subject. Consider now

1. Stit: A canonical form for agentives Ishmael sailed in search of Moby Dick.

5 (3)

Ishmael signed on as a member of the ship's company in total ignorance of Ahab's vengeful purpose; are we then to say that (3) is agentive for Ishmael, but false, on the consideration that both "Ahab" and "Ishmael" are appropriately "personal" subjects? Or is (3) true on analogy with the example about the Pequod? English is not to be trusted in these waters. Since English fails to serve us as an adequate pilot, what we want is a resource sensitive to the difference between those cases using an action-like verb in which agency is ascribed, and those cases in which there is merely the appearance of agency. With decent Austinian respect and regard for the structure of the language, we propose, insofar as is possible, to locate such a resource within English itself, a resource that will also allow us to become clearer about the relation between agentive sentences in their declarative uses and agentives in their imperative uses; for surely Mr. Starbuck, hand me yon top-maul, and

(4)

Mr. Starbuck handed Ahab the top-maul

(5)

are, in context, more than accidentally related. In fact, except for contextdetermined indexicals, they are the same agentive sentence in two different appearances: (4) is an imperative issued to Mr. Starbuck by Ahab, while (5) is an agentive declarative whose truth or falsity is intimately connected to the satisfaction or failure of satisfaction of the imperative.

1A

Agentives

Let us accordingly begin with the following convention: The agentive form that we are about to introduce shall be set off with square brackets [ ... ]. It shall have two open places as indicated, the first to take an agent term, the second to take a declarative sentence (the declarative complement of the new form). The point about the second open place is nontrivial: Having noted that declarative sentences can either ascribe agency or not, we specifically include as possible declarative complements for the second open place both those sentences that do ascribe agency and those that do not ascribe agency. The resulting squarebracket sentence is to say that the proposition expressed by the declarative complement is guaranteed true by a prior choice of the agent. So [the carpenter ... Ahab has a new snow white ivory leg] is to be agentive for the carpenter, and is to say that he is the agent in the matter of Ahab's having a new snow white ivory leg. With what verb or verb phrase shall we replace the ellipsis in that sentential form (it was after all only elliptical)?1 Among the candidate English verb phrases that history suggests are the following: 1

Numerous other philosophers have considered this or similar questions, including at least Anderson, Aqvist, Bennett, Chellas, Chisholm, Danto, Davidson, Fitch, Hamblin, Hilpinen,

6

Introduction to stit i. brings it about that ii. makes it the case that Hi. causes it to be the case that iv. is responsible for the fact that v. lets it be the case that vi. allows it to be the case that vii. takes steps in order that

viii. behaves so that in consequence ix. sees to it that As you can see, these are all grammatically acceptable; but items (i)-(iii) suggest that causal processes are either at work or hovering in the background, whereas we wish to de-emphasize this suggestion. Items (iv)-(vi) give the impression of moral judgment and ethical responsibility, and whereas those are important ideas (none more so), it is inappropriate to build them into the foundation of this enterprise. Items (vii) and (viii) are closer to the mark in their straightforward association of an agent and something made true by the agent, but even these candidates might suggest to some ears that we want you to concentrate on a second, prior action performed by an agent. The English verb form (ix), sees to it that, has, to our ears at least, fewer of the obvious defects of the others, and sees to it that has the definite advantage of suggesting alternatives and choices. So our preferred sentence form is [a sees to it that Q], which for logical emphasis we abbreviate as [a stit: Q], referring to such a sentence as a stit sentence. We intend that "Q" here take up a sentential position, as is proper for a complement of "sees to it that." (Note: In early chapters we use "Q," whereas in later chapters we use "A" or "5," and so on, with exactly the same intention. The slight difference in notation derives from the diversity of the origins of the various chapters, and in order to avoid fresh mistakes, we have thought it best not to enforce uniformity in this particular regard.) It is a consequence of what has just been said that a sentence [a stit: Q] with a as its subject is always agentive for a. Let us enter this important idea as the "agentiveness of stit thesis." Hohfeld, Humberstone, the Kangers, Kenny, Lindahl, Makinson, Melden, Needham, Porn, Talja, Thalberg, and von Wright. Bibliographic access to the literature with which we are familiar can be gained through Thalberg 1972, Aqvist 1984, Bennett 1988, Makinson 1986, and the bibliography of this book. Although we do not undertake to discuss all of these contributions, we do discuss some of them, especially in §1D and chapter 3. Above all, we should be remiss if we failed to note that NB and MP were each introduced to this family of ideas by our teacher, Alan Ross Anderson—see Anderson 1962 and Anderson 1970.

7

1. Stit: A canonical form for agentives AGENTIVENESS OF STIT THESIS. (Stit thesis. Reference: Thesis 1) Q] is always agentive for a.

[a stit:

This is the first of six "stit theses"; you can find a list in §1 of the appendix. By this thesis we intend the descriptive claim that the English sentence "a sees to it that Q," which we abbreviate as [a stit: Q], always describes a as an agent carrying out an action. The complement, Q, may be agentive for a or not. This almost equally important claim is our "stit complement thesis." STIT COMPLEMENT THESIS. (Stit thesis. Reference: Thesis 2) grammatical and meaningful for any arbitrary sentence Q.

[a stit: Q] is

The two theses together emphasize the double aspect of stit: Q in [a stit: Q] may be any sentence at all, but [a stit: Q] is always agentive for a—perhaps difficult to interpret, perhaps contradictory, perhaps merely false, but always agentive for a. So [Ishmael stit: Ishmael sailed on board the Pequod] is agentive for Ishmael and a true sentence, while [Ishmael stit: Ishmael refused to share a room at the Spouter Inn] is equally agentive for Ishmael, but false. The stit sentence [Queequeg stit: the Pequod is fitted out for its voyage], which has a non-agentive as its declarative complement, is agentive for Queequeg but false, for it is not due to Queequeg, but to Peleg, that the Pequod is fitted out for its voyage. At this point you may fairly ask how stit treats other agentive sentences, that is, sentences other than stit sentences that ascribe an action to an agent. Our answer is that if such a sentence is truly agentive, then it can appropriately be paraphrased as a stit sentence. Furthermore, if it is not truly agentive, then to attempt such a paraphrase will tend to reveal this fact. We are therefore led to advance the "stit paraphrase thesis." STIT PARAPHRASE THESIS. (Stit thesis. Reference: Thesis 3) Q is agentive for a just in case Q may usefully be paraphrased as [a stit: Q]. Therefore, up to an approximation, Q is agentive for a whenever Q [a stit: Q}. Given this thesis, we may use the equivalence Q [a stit: Q] as a helpful test. Contrariwise, the helpfulness of the test tends to support the thesis, rough as it is. In any case, we intend this strategy, clarification by paraphrase, as neither definitional nor reductive. It is rather an attempt to isolate, by way of a canonical form, a particular set of English sentences in order to study them more closely as they interact with each other and with other parts of language

8

Introduction to stit

in different linguistic environments, just as a biologist might tag a particular organism to follow its activities as it interacts with members of its own species and with other species in various physical environments. Analogically, most of us are comfortable in saying that an English sentence is conditional if it can be paraphrased into the canonical form "if Q1 then Q2," and then revealing that the sentence The Prophet will tell Queequeg and Ishmael about Ahab provided they stop a minute is in fact conditional by paraphrasing it as If Queequeg and Ishmael stop a minute then the Prophet will tell them about Ahab. EXAMPLE. The sentence (1) is agentive since it can be paraphrased by prefixing an "Ahab saw to it that": [Ahab stit: (1)].

(6)

By Thesis 3, then, since (1) "Ahab sailed in search of Moby Dick" is correctly paraphrased as (6) [Ahab stit: Ahab sailed in search of Moby Dick], both the original and its longer transform are not only true but agentive for Ahab. EXAMPLE. It seems a travesty to paraphrase the sentence about the Pequod, (2), by [Pequod stit: (2)]. Indeed, on their ordinary readings, (2) seems true, while its attempted stit paraphrase seems false. Furthermore, it appears to us that this difference helps us see the non-agentiveness of (2)—helps us to see, that is, that (2) does not ascribe an action to an agent and is therefore non-agentive. EXAMPLE. It strikes us that (3) has two quite ordinary readings, an agentive reading in analogy with the Ahab example, (1), and a non-agentive reading in analogy with the Pequod example, (2). For many purposes the difference is irrelevant or at least of negligible importance; but if one is discussing serious matters of, for example, ethical responsibility, then the difference counts. It matters in such cases whether or not (3), as employed in a particular discussion, can be paraphrased by [Ishmael stit: (3)]. If the non-agentive reading of (3) seems forced, take others. Ishmael sailed over the seas

(7)

is presumably agentive, while Ishmael sailed over the side of the Pequod,

(8)

in its most natural use, would not be agentive. Our agreement on these verdicts tracks our agreement on the effort to paraphrase with stit: It seems right to claim that (7) [Ishmael stit: (7)], and it seems wrong to claim that (8) [Ishmael stit: (8)]. Observe also that there are certainly circumstances under which one would take

1. Stit: A canonical form for agentives [Ishmael stit: (8)]

9 (9)

as true (perhaps he deliberately saw to it that he went flying over the side). Nevertheless, this possibility should not be confused with passing the stit paraphrase test for agentiveness. The truth of the stit sentence (9) does not imply that its complement, (8), is agentive, nor that the stit sentence is equivalent to its complement. Only when (8) is taken as meaning the same as (9) is it plausible to take (8) as ascribing agency to Ishmael.

1B

Stit: Simple cases

We now use the strategy of clarification by paraphrase to demonstrate how stit sentences can help us better to know our way around in some areas in which agency counts. Consider for example Ahab found the White Whale.

(10)

Certainly it is true, but is it agentive? In order to answer we have to address ourselves specifically to the question: Did Ahab see to it that Ahab found the White Whale? As we have seen already, Ahab was agentive in his search for Moby Dick, but was he agentive in finding it? We think not. Although Ahab was a participant in guiding the Pequod to the ultimate outcome, and the major participant, his actually finding the White Whale was due in large part to chance, to natural forces beyond his control. "Time and tide flow wide," remarks Mr. Starbuck. "The hated fish has the round watery world to swim in." Thus, though (10) is true, it is false when paraphrased as [Ahab stit: Ahab found the White Whale]. Therefore, insofar as (10) is true, it is non-agentive; and insofar as it is agentive, it is not true. Consider the following pair: Queequeg struck home with his harpoon.

(11)

Queequeg's harpoon struck home.

(12)

Some careful speakers of English might always use (11) as an agentive and (12) as a non-agentive, whereas most of us are liable to use them interchangeably, sometimes in one way and sometimes the other. But [Queequeg stit: Queequeg struck home with his harpoon]

(13)

[Queequeg stit: Queequeg's harpoon struck home]

(14)

and

come to much the same thing. Such differences between (11) and (12) as there are disappear when they are embedded into stit contexts; the stit sentences (13) and (14), regardless of the uncertainties of their complements, are transparent with respect to agency.

10

Introduction to stit

1B.1

Imperatives

"Clear away the boats! Luff!" cried Ahab. Treatment of such imperatives is by no means tangential to our concerns. We endorse the view that C. H. Hamblin expresses in his masterful study: Imperatives are not only among the most frequent of utterances; they are also, surely, among the most important. If the human race had to choose between being barred from uttering imperatives and being barred from uttering anything else, there is no doubt which it should prefer. (Hamblin 1987, p. 2) Hamblin additionally reports that a full twenty percent of Shakespeare is in the imperative mood. (We happily acknowledge that a study of Hamblin's book set us under way and helped us avoid some threatening reefs.) Imperatives, in our usage, constitute a grammatical category. Following an established tradition, however, we think of each use of an imperative as having a force and a content. With regard to force, Ahab's imperatives may have been orders or commands, which many think the only possibilities; but Ahab might instead have been inviting, requesting, suggesting, advising, ... the helmsman to luff. Putting force to one side, however, we are after the content, about which stit theory has a definite and (as it happens) helpful opinion that we sum up in the "imperative content thesis." IMPERATIVE CONTENT THESIS. (Stit thesis. Reference: Thesis 4) Regardless of its force on an occasion of use, the content of every imperative is agentive. For example, Luff! can have its content represented as The helmsman luffs, which in turn, since it is agentive, can be paraphrased as The helmsman sees to it that he luffs. Thus, Luff! can be paraphrased as Helmsman, see to it that you luff! In this case the application of our thesis is easy because The helmsman luffs, which looks to be the most plausible content for the imperative, is already agentive. Still, there is more to be learned. To luff is to see to it that the bow of the boat is heading directly into the wind. Accordingly, the content of the imperative Luff! can be put into the canonical form [Helmsman stit: the helmsman luffs]

(15)

or equivalently, [Helmsman stit: the boat is headed into the wind].

(16)

The two stit sentences are equivalent, but while the complement of the former is agentive, the complement of the latter doesn't mention the agent at all.2 Unlike Luff! the imperative 2

"Complement" versus "content" sometimes sounds confusing. The (grammatical) complement of [a stit: Q] is Q. Its (semantic) content is an agentive proposition.

1. Stit: A canonical form for agentives Be on deck at dawn!

11 (17)

does not show its content so obviously; the stit apparatus, however, increases in value as problems become more complex. Consider (17) as addressed to the helmsman. The obviously non-agentive The helmsman is on deck by dawn

(18)

cannot, by our thesis, represent its content; we need an agentive form, and fortunately there is already something at hand: the stit sentence, [The helmsman stit: the helmsman is on deck at dawn]. This, in turn, can be transformed back into the imperative Helmsman, see to it that you are on deck at dawn, which we take to be an accurate paraphrase of the original imperative (17). This is important because you might be tempted to think that (18), inasmuch as it displays an agent as its subject, is the content of the imperative (17); but there are many ways in which it might be true that the helmsman is on deck at dawn, while false that he sees to it that he is. Because the content of every imperative must be agentive, a helpful paraphrase will not merely have the name of the addressee as its subject: That name will appear as subject of a sentence of the form [a stit: Q}. We therefore have the following: Since every imperative has agentive content, and since every agentive can be paraphrased as a stit sentence, it must be that every imperative will have a stit paraphrase of the form [a stit: Q]—sometimes with Q an agentive (hence a possible content of an imperative) and sometimes not. These brief remarks do not suffice. We return twice to the topic of imperatives, once in §1C, and again in chapter 4.

1B.2

Deontic contexts

The traditional account (see, e.g., Aqvist 1984) has it that deontic statements have one of the forms

Oblg:Q,i.e. it is obligatory that Q, Frbn:Q, i.e. it is forbidden that Q, or

Perm:Q, i.e. it is Permitted that Q where Q is a declarative. 1-1 REMARK. (On notation) The standard deontic notation is Op, Fp, and Pp. We depart from the standard for good reason: We shall in the course of this book deal with many other modalities, several of which would have an equal right to single letters such as 0, F, and P. We therefore use abbreviations for deontic and other modalities that we hope are short enough for the eye to take

12

Introduction to stit

in at a glance and long enough for the mind to remember. (It helps that we do not often use these longer expressions in the course of complicated calculations in which extreme brevity is a decided advantage.) Among its many virtues, this approach allows such great latitude that it can be connected to every possible action by any agent. On the other hand, agents have in this grammar no distinguished place; if invoked at all, the agent's name is only an accidental feature of the declarative complement. It is an easy mistake to think that Oblg:Q says more than it actually does say. It does not say, for example, who, if anyone, is obliged to see to it that Q. Consider It is forbidden that cooks be on the bridge. It is tempting to read this as though it imposes a prohibition on the cook, but it doesn't, as we can see from the analogous It is forbidden that dogs be on the bridge; in either case the sentence form is not fit to tell us who is to see to it that no cooks (or dogs) are on the bridge. While a deontic language of declarativesentence complements may be satisfactory for impersonal oughts, many within the tradition have seen that agents need to be treated with more care. Perhaps foremost among those who have argued that these standard alternatives are inadequate is Castaneda, for example, in Castaneda 1974 (see additional discussion in §3D). Some deontic logicians have suggested the step of changing the grammar so that Oblg:, Perm:, and Frbn: are taken to be adjectives modifying actionnominals. The fundamental forms are then taken as Oblg:a, i.e., a is obligatory, Frbn:a, i.e., a is forbidden, or Perm:a, i.e., a is permitted, where a stands in place of a term designating an action. So Oblg:a might be instanced by Sailing is obligatory, or Frbn:a by Bringing the boat into the wind is forbidden. Since actions are always the actions of agents, this step is in the right direction, but it still fails appropriately to recognize the agent. Furthermore, there is a considerable loss in expressive power, since clearly understood declarative sentences, Q, are so much easier to come by than are clearly understood action-nominals, a. In order to regain the headway lost in the move from sentences to action-nominals, some have tried adding negative doings such as not-sailings, or disjunctive doings such as luffing-or-flensing; but even with these additions, there remain two features missing from the overhauled model: first, the flexibility that follows from permitting an arbitrary declarative within the scope of a deontic statement, and second, the grammatical means to identify and keep track of the agent. We propose a combination having the strengths of both the declarativesentence complement plan, Oblg:Q, and the action-nominal complement plan,

1. Stit: A canonical form for agentives

13

Oblg:a. We propose to focus on deontic statements that have only agentives for their complements and thus can always be paraphrased into one of the following forms. Oblg:[a stit: Q], i.e., a is obligated to see to it that Q Frbn:[a stit: Q}, i.e., a is forbidden to see to it that Q Perm:[a stit: Q}, i.e., a is permitted to see to it that Q Though Perm:[a stit: Q], for example, is not agentive (in our technical sense), we intend it as quasi-agentive in the loose sense that it involves both an agent and an agentive, and in the stricter sense that like an agentive, it has the agent itself as a recoverable part of its intension. Our proposal calls, then, for a deontic language enriched by the restriction that the complements of obligation, permission, and prohibition be limited to stit sentences, a proposal that forms part of the "restricted complement thesis." RESTRICTED COMPLEMENT THESIS. (Stit thesis. Reference: Thesis 5) A variety of constructions concerned with agents and agency—including deontic statements, imperatives, and statements of intention, among others—must take agentives as their complements. Recall that (i) stit sentences always express an action, (ii) stit sentences never lose or misplace the agent, and (TO) there are no grammatical or metaphysical or semantic restrictions on the declaratives that may be put in place of Q. With these features of stit in mind, applying Thesis 5 to deontic constructions retains, we think, the most valuable features of both the declarative-sentence complement account, Oblg:Q, and the action-nominal complement account, Oblg:a. For example, in context the burden of the prohibition of cooks (or dogs) from the bridge might be Oblg:[the Third Mate stit: no cooks (or dogs) are on the bridge]. Or we might have Frbn:[the cook stit: the boats are lowered]. Question: Does this imply Oblg:[the cook stit: ~(the boats are lowered)]? The application of stit theory to deontic concerns is a topic deserving treatment at length. We return to it explicitly in §2B.9, chapter 11, chapter 12, and chapter 14, as well as more indirectly in chapter 4 (via imperatives) and in chapter 5 (via the idea of promising).

14

1C

Introduction to stit

Grammar of the modal logic of agency

The formal grammar of stit is easy: A singular term goes in one blank, and a sentence in the other. The grammar of English agentives, especially that of imperatives, is more delicate. The first topic is embedding. We all know that declarative sentences can either occur stand-alone or be embedded in a larger context. One may encounter the following standing alone: The Prophet will tell Queequeg and Ishmael about Ahab. One may also find the selfsame declarative as the consequent of a conditional: If they will stop a minute, then the Prophet will tell Queequeg and Ishmael about Ahab. One should bear in mind that exactly the same possibilities exist for the two other "moods" of English, the interrogative and the imperative. With respect to interrogatives, one has for example the two stand-alone interrogatives Did Queequeg and Ishmael give their informed consent to sailing after the White Whale? and

Whom did the Prophet tell about Ahab? One also has these selfsame interrogatives embedded in a (declarative) "dependence" construction: Whether they gave their informed consent to sailing after the White Whale depends on whom the Prophet told about Ahab. When interrogatives are embedded, we tend to call them "indirect questions," a terminology that distracts from the chief point: The purely grammatical phenomenon of the very same interrogative sentence occurring now as stand-alone and now embedded is precisely analogous to declaratives occurring in these ways. Finally, and crucially for our purposes, the English grammar of imperative sentences exhibits precisely the same behavior. For example, start with the stand-alone imperative Mr. Starbuck, hand me yon top-maul. (Of course the "Mr. Starbuck" may be dropped when context makes clear the addressee, and "me" must be supplanted by "Ahab" when the imperative is used by some third party.) Then each of the following is a way to embed this selfsame imperative in a larger context. Mr. Flask, request Mr. Starbuck to hand me yon top-maul.

1. Stit: A canonical form for agentives

15

Ahab ordered Mr. Starbuck to hand him yon top-maul. Mr. Starbuck carried out Ahab's order to hand him yon top-maul; or at least Mr. Starbuck handed Ahab yon top-maul. Did Ahab advise Mr. Starbuck to hand him yon top-maul? Ahab demanded that Mr. Starbuck hand him yon top-maul. Ahab demanded that yon top-maul be handed to him by Mr. Starbuck. Mr. Starbuck refused to hand Ahab yon top-maul. Mr. Starbuck refused Ahab's request (order, advice) to hand him yon top-maul. Mr. Starbuck is obligated (permitted, forbidden) to hand Ahab yon topmaul. There is clearly no doubt, then, that imperatives can be either stand-alone or embedded. In English, embedding any of declaratives, interrogatives, or imperatives is a complicated matter requiring at least inversion and context-driven switching of pronouns. Our deepest comment is this: Embedded imperatives are in truth embedded imperatives. That is, they are constituent or embeddable grammatical transforms of the very same imperative sentences. In a logically perspicuous language, they would be the very same sign designs. Furthermore, agentive declarative sentences are, we think, also syntactic variants of imperatives, whether stand-alone or embedded. Both Mr. Starbuck handed Ahab yon top-maul and the clause embedded in That the top-maul was handed to Ahab by Mr. Starbuck was of no concern to Mr. Flask represent variants on exactly the same theme. The intimidating variety of syntactic structures is a good reason for thinking about stit. In all these cases the paraphrase into canonical form allows us to see that whatever the complications of natural language, we have one and the same agentive sentence, its surface grammatical form varying from context to context. We may put this as a thesis, our last: the "stit normal form thesis." STIT NORMAL FORM THESIS. (Stit thesis. Reference: Thesis 6) In investigations of those constructions that take agentives as complements, nothing but confusion is lost if the complements are taken to be all and only stit sentences. Our recommendation is that both imperatives and agentive declaratives be normal-formed as stits. What, then, of the grammar of stit itself? We think that the best thing to say is that a stit sentence itself is both a declarative and

16

Introduction to stit

an imperative.3 A stit sentence can be embedded wherever a declarative or an imperative can be embedded. We have seen the variety of English sentences that are helped by paraphrase as stits. Let us consider the other direction. Given a stit sentence, [a stit: Q], we should expect to read this piece of notation differently in English, depending on how it is used. Here are some paradigmatic examples of readings of [a stit:

Q}.

• As a stand-alone imperative: a, see to it that Q! • As a stand-alone declarative: a sees to it that Q, a is seeing to it that Q, a saw to it that Q. • As an embedded imperative: a to see to it that Q, for a to see to it that Q, that a see to it that Q, a's seeing to it that Q. • As an embedded declarative: a sees to it that Q, that a see to it that Q.4 These subtle alterations required by English grammar are really required; but they tend to obscure rather than reveal the fact that we can locate the selfsame agentive in a variety of contexts: sometimes as a stand-alone declarative, sometimes as an embedded declarative, sometimes as a stand-alone imperative, and sometimes as an embedded imperative. Our paraphrase of agentives into normal form clarifies that situation. A stit sentence, [a stit: Q], since it displays its agent and appropriate declarative complement publicly and obviously, is the appropriate picture of the underlying agentive partly because it remains recognizably the same in any and every context. A further important feature about the grammar of imperatives, aside from their embeddability, is that they display an agent. As Castaneda (1975) and others have urged, they have the deep grammatical form a to verb. In this respect an imperative is unlike a declarative in general, which may or may not express an agentive proposition and even when it does may not wear its agent on its surface, as the linguists say. But imperatives must show forth an agent, at least in the sense that to be understood, and (the point is crucial) to be used in larger contexts, the agent must be uniquely recoverable from the surface (for example, as the addressee of a stand-alone imperative). Those engaged in the descriptive grammar of English have and are entitled to different views on this matter. Perhaps the work most pertinent to our concerns is Badecker 1987. Badecker surveys some Chomsky theories, which by deriving 3 We are not sure that this use of overlapping grammatical categories is for the best. Let us emphasize again that the aim of applying stit notation is to clarify, not to analyze, and, in particular, not to provide a syntactical criterion for when a certain surface form must be considered an imperative rather than a declarative. Let us also emphasize that "force" versus "content" has no role here, since the topic is purely grammatical. 4 Note that this "see" is not indicative, but rather subjunctive, like "hand" in "Ahab demanded that Mr. Starbuck hand him yon top-maul."

1. Stit: A canonical form for agentives

17

all infinitive constructions by transformation of declaratives are deeply at variance with the spirit of the present book, though our aims are so different from his that it is hard to call the variance a conflict. In healthy contrast, the lexicalist theory that Badecker offers in his chapter 3 awards infinitive constructions independence from declaratives, and thereby more nearly shares our direction; however, there remains the question of whether in agentive infinitive constructions, such as Mr. Starbuck refused to hand Ahab yon top-maul, we should or should not take it that there is a "trace" of Mr. Starbuck heading the infinitive phrase. We certainly need Mr. Starbuck to get the semantics right, but that far from settles the syntactical question for English. In any event, Badecker supplies a helpful framework for addressing this and related questions. A consequence of the fact that a stit sentence is a declarative as well as an imperative is this: It can be embedded wherever a declarative or an imperative can be embedded. For example, with regard to the former, a stit sentence can be embedded under a negation. The result of such an embedding is on the face of it not itself a stit sentence; in the special (and, to a logician, prominent) case of negation, the result of embedding looks at least on the surface like

not like some instance of

But more deeply, for the special case of negation, the result of embedding is not always any kind of agentive; that is, by the paraphrase test for agentives suggested by Thesis 3, to which we hope you have agreed, the declarative it is false that a sees to it that Q is not invariably paraphrasable (or indeed equivalent in truth value with) a sees to it that it is false that a sees to it that Q. Thus, by Thesis 3, (19) is not itself an agentive. In more colorful language that speaks against taking a naive approach to inventing a "logic of imperatives," we may say that the negation of an imperative is not always an imperative. We later discuss the interaction of negation with stit in a number of places; see for example §2B.6 and §2B.8 on "refraining." Finally, let us recall that when concerned with the modalities of agency, it is similarly helpful to use stit sentences to keep track of the agentives. The English Starbuck could have handed the top-maul to Ahab is helpfully paraphrased as [Starbuck could-have-stit: Starbuck hands the top-maul to Ahab], while

18

Introduction to stit Starbuck is obligated to hand the top-maul to Ahab

is paraphrased as Oblg:[Starbuck stit: Starbuck hands the top-maul to Ahab]. In all these cases it is crucial that the content of the order and the obligation, the ability and the action, be all the very same, all captured by a single stit sentence.

1D

Mini-history of the modal logic of agency

Throughout this book we constantly refer to "the action," "the obligation," "the ability," "the order," and so on. This is because English, through its love of the subject-predicate construction, drives us to such usages. We do not believe, however, that English also drives us to wax ontological about actions, obligations, abilities, orders, and so on. We may wish to fashion theories about entities taken to fall under these common nouns. Stit theory is not to be taken as arguing against such a wish. On the other hand, we may not. Stit theory has the advantage that it permits us to postpone attempting to fashion an ontological theory, while still advancing our grasp of some important features of action, obligation, and so on. It does so by invoking a "modal" construction in place of a subject-predicate construction that requires an ontology. By a modal construction we mean an intensional grammatical construct having sentences as both inputs and outputs. A "modal logic of agency" intends that some such construct express agency (or action), as for example our own favored English construct "a sees to it that Q." The modal logic of agency is not popular. Perhaps largely due to the influence of Davidson (see the essays in Davidson 1980), but based also on the very different work of such as Goldman 1970 and Thomson 1977, the dominant logical template takes an agent as a wart on the skin of an action, and takes an action as a kind of event. This "actions as events" picture is all ontology, not modality, and indeed, in the case of Davidson, is driven by the sort of commitment to firstorder logic that counts modalities as Bad. The project has had some successes, all of which we shall ignore, and some failures, most of which we shall ignore. (See Bennett 1988 for an indispensable perspective.) Certain of its failures, however, are to be attributed to the neglect of the modal features of agency. We compare the ontological approach to action with our modal approach in §3E. Here we pursue the modal history. The earliest modal logic of agency of which we have learned was formulated around the year 1100 by a Dominican trained at a famous Norman institution, the abbey of Bec. (The date of c. 1100 is implied by the sources that were conveniently available to us without scholarly digging, namely, the discussion on p. 120 of Henry 1967 together with the "Anselm" entry in the Encyclopedia Britannica of 1968.) We refer to St. Anselm, who succeeded Lanfranc as prior and then abbot of Bec, and then later as archbishop of Canterbury. No mere pale

1. Stit: A canonical form for agentives

19

theoretician nor private saint, the archbishop was deeply involved in controversy with the tyrant William Rufus and later his brother Henry in regard to the matters of lay investiture and clerical homage; he vigorously opposed the former. These controversies were heavily freighted with the concepts of promising and commitment and agentive powers. In order to make clear that his authority in matters spiritual was not at the pleasure of the king, Anselm refused to accept the papal pallium from the hands of William Rufus. Partly in consequence, the archbishop was in effect exiled by the king. Anselm's brief notes on the modal logic of agency were, we think, composed during this bitter exile. In the document that Henry 1967 calls N, Anselm writes: Quidquid autem 'facere' dicitur, aut facit ut sit aliquid, aut facit ut non sit aliquid. Omne igitur 'facere' dici potest aut 'facere esse' aut 'facere non esse.' (p. 124; from N 29.8.10) Paraphrase by Henry: For all x, if 'x does' is true, then x does so that something either is so or is not so. Hence the analysis of 'doing' will in fact be an analysis of x's doing so that p, and of x's doing so that not-p [where 'p' is a clause describing a state of affairs, and 'not-p' is short for 'it is not the case that p']. (p. 124) Anselm goes on to describe a kind of square of opposition that clearly indicates he had in mind a modal logic of agency (to the extent to which that can be said without anachronism), but his work seems to have remained unnoticed until after the stirring of modal logic in this century. If you promise to accept our remark as merely helpful rather than authoritative, we will hazard that Henry 1953 is the first reference to Anselm that appreciates his work as modal, and that Chisholm 1964a (who cites Henry 1960) is the earliest reference to Anselm by an active researcher in this field (see §3B for further discussion of Chisholm). Other references to Anselm on this topic: Danto 1973, Humberstone 1976 (the reference does not appear in the published abstract, Humberstone 1977), the perceptive Walton 1976b, 1976a, and 1980 (which cites Dazeley and Gombocz 1979), and a sterling account in Segerberg 1989. It is amazing (and perhaps a little sad) that over eight hundred years elapsed between Anselm's invention or discovery and the next contributions to the modal logic of agency. The first modern desire for a modal agentive construction seems to have been felt by philosophers working their various ways through the embedding requirements of legal and deontic concepts. One can certainly see the need expressed in the pioneering work of Hohfeld 1919, who introduces locutions such as the following (p. 38). X has a right against Y that he shall stay off the former's land. This Hohfeldian construct plainly embeds the modal agency construction in a "has a right" context. This is true of all of Hohfeld's work: The modal agency construction is always embedded in additional constructions imputing legal rights, duties, powers, and so on. Agency has not yet received a separate modal treatment. (We discuss Hohfeld a bit in §5B.l.)

20

Introduction to stit

The next place we know a modal agentive construct to crop up, much more explicitly but still embedded in the context of a normative expression, is in Kanger 1957:

Ought(Y sees to it that F(X, Y)). (p. 42) Although the locution "sees to it that" is displayed only in a normative context and wholly without comment, it is clear from the general tenor of Kanger's methodology that he intended to be isolating a norm-free concept of agency. The explicit grammatical breakthrough for the logic of agentive modality comes in Anderson 1962, who, reflecting on Hohfeld, introduces for the first time a separate form of expression intended to disengage the concept of agency from normative considerations.5 When on p. 40 Anderson takes

M(x, p, y) to represent the case "when x executes what is regarded as an 'action' ... and y is the recipient or patient of the action executed by x," he suddenly gives us a clean target for some analytic questions that otherwise seem confusing. Anderson sometimes reads M(x, p, y), with perhaps too little attention to the connections between formal and English grammar, as "x does p to (for) y." Evidently here agency is, for better or worse, not separated from patiency. And certainly there is in Anderson no semantic theory of agency or patiency, and only a trace of a deductive calculus (e.g., Anderson points out that the implication between ~M(x, p, y) and M(x, ~p, y) goes only from right to left). That is, Anderson pioneers in isolating agency and patiency, but he does so only immediately to recombine them with deontic concepts. In 1963 each of two logicians, Fitch and von Wright, advanced modal theories of agency, each of them stressing syntactic developments. Fitch 1963 defines "does A" in terms of two other modalities, "striving for" and "causes," and offers a deductive calculus. The work has not been taken up by later logicians and is seldom cited in the published literature. Indeed, although NB was Fitch's admiring and fond student and colleague, he regrets to say that he had to be reminded of this essay by Segerberg 1989, which contains a maximally useful account. Von Wright, beginning in 1963 and continuing at least through 1981, was, we think, the other logician to be a first to treat agency (or action) as a specific modal or quasi-modal topic, always with that specially honest von Wrightian insistence on the lack of finality of the formulation in question, including attending to nonmodal formulations in which complements are taken as terms signifying specific or generic actions, rather than sentences. As in other cases, the work keeps a close eye on deontic logic, to which he contributed so much. We think von Wright did not succeed in disentangling agency from change, 5 Very likely the breakthrough for Anderson came about after correspondence or conversations with his friend Kanger. Somewhat later Anderson visited Manchester, where Henry was. Henry remarked in personal correspondence that during this year of 1965 there was a colloquium involving a number of persons interested in agency, including, e.g., Hare and Kenny.

1. Stit: A canonical form for agentives

21

evince interest in the general problem of embedding of agentives. For instance (but only "for instance"), von Wright 1963 took as a convenient primitive the notation

d(p/p), which is to be read as expressing some such idea as "the agent preserves the state described by p" (pp. 43, 57). Like Anderson, von Wright tends to leave to the reader the task of putting bits of logical grammar together with bits of English grammar. In contrast with Anderson, however, agency here has been separated out from patiency. (We further discuss von Wright in §3A.) Kanger and Kanger 1966 introduce as a separate locution X causes F, where F is supposed to be a sentence, but in a fashion like Anderson's, they logicize about it only by setting down that F may be replaced by its logical equivalents, and that the proposition that X causes F implies that F. Three influential lines of research began about the same time as that of the Kangers, each of which highlighted the separate existence of agentive modalities, namely, those initiated by Castaneda, by Kenny, and by Chisholm.6 Castaneda, whose views concerning deontic logic have informed both philosophers and logicians for many years (since at least Castaneda 1954), has much to say that is relevant to agency as a modality. Though his philosophical concerns led him to pursue goals other than the formulation of a modal logic of agency, he repeatedly urged the fundamental importance of the grammatical and logical distinction between "propositions" and "practitions" (a distinction put as clearly as anywhere in Castaneda 1981); but because there is no possibility of constructing a Castaneda "practition" from an arbitrary sentence, in the way for instance that Anderson's M(x, p, y] or von Wright's d(p/p) each permits an arbitrary sentence in place of p, Castaneda practitions cannot themselves serve as the foundation for such a modal logic of agency. (§3D expands our consideration of Castaneda.) Kenny 1963, in the course of initiating a rich literature on the verbal structure of our causal and agentive discourse, says that any "performance" in his technical sense is describable in the form bringing it about that p. And Chisholm 1964b takes the following as a basic locution on which to found an extensive series of definitions and explanations in the vicinity of agency: There is a state of affairs A and a state of affairs B, such that he makes B happen with an end to making A happen, 6 Of course other work on the theory of action has also influenced the modal logic of agency, but that literature is unsurveyably vast. We note as a passing example that there is hardly a one of our past or present departmental colleagues who has not contributed.

22

Introduction to stit

where the letters stand in for "propositional clauses," and where the subject of "makes happen" can be either a person or a state of affairs. The discussions of Kenny and Chisholm, though relevant to logical questions, are themselves not directed toward the formulation of either proof-theoretical or semantic principles governing their respective basic locutions. They are sufficiently closely connected to our project, however, that we return to Kenny and Chisholm respectively in §3C and §3B. This is as accurate a record as we can manage of the early history of the modal logic of agency. If this story is right, then the following gives its gist. History of the modal logic of agency prior to 1969 Anselm c. 1100

facere esse (x does so that p)

Hohfeld 1919

X has a right against Y that he shall stay off the former's land

Kanger 1957

Ought(Y sees to it that F ( X , Y ) )

Anderson 1962

M(x,p, y) (x does p to [or for] y)

Fitch 1963

Does A

von Wright 1963

d(p/p) (the agent preserves the state described by p)

Kanger & Kanger 1966

X causes F

Castaneda 1954ff, Kenny 1963, Chisholm 1964ff

relevant discussions

As we note in several places previously, in chapter 3 we extend our discussion of certain among these figures by more closely relating their thoughts to stit theory: von Wright, Chisholm, Kenny, Castaneda, and Davidson. Also in §5B there is a little more consideration of some Hohfeldian themes. But for now we leave this early part of the history. The first modal logic of agency with an explicit semantics is, we think, that of Chellas 1969. The primitive locution is to be read as "T sees to it that O," where T is an agent and O takes the place of a sentence (pp. 62-63). Chellas only deploys this locution in one context, namely, as the argument of an imperative operator.7 7

But Chellas does not restrict the complement of an imperative operator to sentences having the form A-TO as is required by our Thesis 4.

1. Stit: A canonical form for agentives

23

As for semantics, Chellas takes as a paradigm the technique made famous by Kripke not long before Chellas was writing; we mean deployment of a binary relation between "worlds" in order to clarify modal concepts. Chellas in particular gives a semantic clause for ATO: is true at the present world just in case O is true at all those worlds under the control of—or responsive to the action of—the individual which is the value of T at the present world. (p. 63) The language that Chellas uses in this pioneering explanation, like the "relative possibility" language of Kripke a few years earlier, is neither familiar in itself nor further clarified by Chellas. Perhaps this is the reason that, like his predecessors, Chellas in practice confines his agentive locution to the imperative context from which his need for it sprang, and does not pause to investigate its separate properties. 8 After Chellas there is a substantial group of logicians all of whom have deployed a binary relation or a pair of binary relations in an effort to generate a semantic understanding of an agentive modality that might be used as the complement of an imperative or of a deontic operator; we know of Porn 1970, 1971, 1974, 1977; Needham 1971; Aqvist 1972; Kanger 1972; Hilpinen 1973; Humberstone 1977; Lindahl 1977; and Talja 1980. For a critique of the line of research being described, with special reference to Porn 1970, see Walton 1975; also of note are Walton 1976b, 1976a, 1980, which develop some insights in an independent and more nonsemantic fashion. The earlier Porn articles and that of Aqvist use only a single binary relation; the idea of using two binary relations seems to be independently due to Needham 1971, Kanger 1972, and Hilpinen 1973. (Unless we have overlooked it, there is no cross-mention; we have not seen Needham's M.A. thesis, but make the inference from Porn 1977.) The reason for the second binary relation is given as this: Agency has not only a sufficient condition aspect but a necessary condition aspect (Kanger 1972, p. 109; Hilpinen 1973, p. 119), and one needs a separate relation for each. The later workers in this mini-tradition play variations on this theme. In our judgment this line of investigation, although initially promising, and although producing some useful insights, has not been much followed up for the following reason: It has remained obscure what one is to make of the binary relations that serve as the founding elements of the entire enterprise. Kanger 1972 says, for example, that one of the relations holds between a person and a couple of worlds or indices when everything the person does in the second world is the case in the first; and the other relation holds when the opposite of everything the person does in the second is the case in the first (p. 109). That is far from clear, and no one in the tradition is, in our judgment, any clearer than that. For a final example, we describe and quote at length from Porn 1977, which among those we mention is the most developed grammatical and semantic 8

We discuss Chellas a bit more in §4A, and we rely on his agentive operator, ATO, in several places—writing it, however, as [a cstit: A] in order to conform to our standard symbolism for the stit sentence.

24

Introduction to stit

treatment of agentive modality. (All page references are to Porn 1977 and all words not inside quotation marks are ours.) Dap is read "it is necessary for something which a does that p" (p. 4). It is said (pp. 7-8) that an equivalent concept is found as the definition of "a sees to it that p" in Chellas 1969, chapter 3, section 4, and in Porn 1971. "Consider all those hypothetical situations u' in which the agent does at least as much as he does in u. If v is such a situation, it may be said to be possible relative to what the agent does in u. ... if p is necessary for something that a does in u, then there cannot be a situation which is possible relative to what a does in u and which lacks the state of affairs that p. ... A natural minimal assumption is that the relation [of relative possibility] is reflexive and transitive." (PP. 4-5) D'ap is read "but for a's action it would not be the case that p" (p. 5), and also "p is dependent on a's action." (p. 7) "For the articulation of the truth of D' a p at u we require all hypothetical situations u' such that the opposite of everything that a does in u is the case in u' ... [the relation must be] irreflexive and serial." (pp. 5-6) Further, to connect the two modalities D and D', a condition is imposed that "requires that worlds which are alternatives to a given world under the relation [for Da] be treated as equals in contexts of counteraction conditionality." (p. 6) C' ap is read "p is not independent of a's action." (p. 7) Eap is defined as the conjunction of Dap and C'ap, and read "a brings it about (causes it to be the case that, effects that) p" (p. 7). Porn says (p. 8) that an equivalent concept is found in Needham 1971, p. 154, an essentially equivalent concept in Hilpinen 1973, section 6, and explicitly in Porn 1974, p. 96. Porn finds unacceptable (p. 7) an alternative E* a p, defined as the conjunction of Dap and D'ap, which Porn says is equivalent to a definition of Kanger 1972. There are three points to be made about this extract. The first is that given the available apparatus, Porn 1977 seems to us to offer the best explanations of and the most detailed working out of the modal logic of agency as based on abstract binary relational semantics. Second, even these best-possible explanations seem difficult. The conclusion one might draw is that one should doubt the likelihood that the abstract relational-semantic point of view itself can continue to serve in the way that was hoped. But third, however, and counting against this conclusion, is that Porn 1977 is evidently formulating, in the context of the relational semantics, the very combination of "negative" and "positive" conditions that much later were built into stit theory (see §2A.2 and §2A.3). Aqvist 1974, 1978 provide a much less abstract and more intuitive semantic setting; these articles are the first of which we know that make the fundamental

1. Stit: A canonical form for agentives

25

suggestion that agency is illuminated by seeing it in terms of a tree structure such as is familiar from the extensive form of a game as described in von Neumann and Morgenstern 1944. Aqvist's account of agency is in some respects akin to that described in this book, in some respects less flexible, and in some respects richer. His aim is not strictly to provide a modal logic of agency; for example, the primitive of Aqvist 1978 is "DO(a, Pa)" to be read "a does, or acts, in such a way that he Ps," and where "Pa" must be an atomic sentence (rather than an arbitrary sentence), and like von Wright, Aqvist wraps agency together with change. But his goal is close enough to warrant (i) a comparison (which is not attempted here) and (ii) a suggestion that the reader consult these sources. A notable relevant article is Mullock 1988. Of decisive importance is the uncommonly rich joint work Aqvist and Mullock 1989, which applies insights derived from the tree structure to serious questions in the law. Aqvist and Mullock 1989 is, as we later say also of Hamblin 1987, required reading. There is one later commentator on the tradition just described who is of special excellence and interest: Makinson 1986. In a series of more than a dozen articles beginning with Segerberg 1980, and including among others Segerberg 1981, 1982, 1984, 1985a, 1985b, 1987, 1988a, 1988b, and 1989, a distinguished modal logician develops a richly motivated and intuitively based formal approach to action by taking a routine as the guiding concept. Segerberg explicitly bases some of the intuitive and formal aspects of his work on studies that in computer science have come to be called "dynamic logic," the influence on Segerberg being primarily through Pratt. Consult Elgesem 1989 for a sympathetic yet critical and penetrating account of Segerberg's line of research. The work is not fully in the modal logic of agency, since it stresses a grammar of (i) terms (including complex terms) for naming "actions" and (ii) predicates for expressing properties of "actions," and thus self-consciously avoids a grammar of nesting connectives. But instead of a complaint this is intended only as a reason for limiting ourselves to a mere mention of what may indeed turn out to be not only valuable in itself but a useful link between the ontological and modal points of view on agency. Mention of Pratt calls attention to the existence of a large and interesting formal literature that we fail to cite as part of this mini-history except insofar as it has influenced Segerberg, namely the work on "dynamic logic" and its cousins that has been done by Floyd. Hoare, Pratt, and other computer scientists (see Segerberg 1989 for a brief entree via Pratt that is written especially with the logic of action in view, and see Pratt 1980 for an excellent fuller account). There are three reasons for excluding this line of investigation from the present survey: (i) we are very far from familiar with the literature, so that making it accessible is best left to someone else. Further, what we know of it (ii) stresses the ontological rather than the modal approach, whereas the latter is the topic of this mini-history, and (in) what we know of it is relevant to action only in the wide sense of "action" that encompasses mechanical action, that is, the sense of "action" that encompasses the action of programs and starter motors. In fact the present modal point of view makes it arguable that this literature is no more relevant to agency than is the literature of any other discipline that gives us

26

Introduction to stit

ways to fill out the sentential complement of "sees to it that": An agent can see to it that the starter engages and passes through various stages, or that a certain recursive program runs, or ... . But it seems best to make explicit our failure to more than barely mention such a large literature just because so many persons think that although it may be arguable, it certainly isn't plausible that it has no special relevance to agency. On the other hand, NB once asked a well-known computer scientist/mathematician after a lecture on parallel processing if he had meant his use of "actor" and "agent" to be anything but an idle metaphor; he was aghast that one should need to inquire. The articles Brown 1988, 1990, and 1992 represent a sustained and important investigation of the modal approach to ability and its connection with action. Brown initially proposed a modal operator that has something of the force of "can do." Horty 2001, which uses [a bstit: Q] for this Brown connective, explains Brown's ideas and relates them to stit theory. Penultimately there is von Kutschera 1986, which articulates in one form or another nearly all of the essential underlying ideas concerning agency on which we base the semantics offered in subseqent chapters. We can describe the extent of von Kutschera's priority only by using some phrases not defined until later. At the very least, one must credit von Kutschera 1986 with the no choice between undivided histories condition, with generalization beyond the discrete, with generalization to multiple independent agents (including the independence of agents condition), with attention to strategies, and with semantics for the "deliberative stit" that we study in Definition 2-5 and §8G.l. It also needs to be remarked that von Kutschera 1986 cites the earlier von Kutschera 1980. Finally there is Hamblin 1987, which in the context of a study of imperatives provides a rich source of formal, informal, and semi-formal ideas on the topic at hand, many of which have influenced the present work; in particular, collegial reflection on Hamblin's "action-state semantics" was the immediate context of the beginning of the research reported in this book. Our own recommendation is that no one ought to try to move deeply into any part of the theory of agency without reading Hamblin.

1E

Conclusion and summary

Although not so grand as Ishmael's, we think our story well worth the telling, and though we have not sailed far, our mainsail has been unfurled and we have caught the first breeze. What we propose is an augmentation of our current linguistic resources with a linguistic form, the stit sentence [a stit: Q], that (i) leads us carefully to attend to the agent of an action, (ii) is capable of taking any English declarative as its complement, (iii) is recoverable as the same stit sentence either as a declarative or an imperative, and (iv) is grammatically suitable for embedding within wider contexts. Among its other virtues, the stit sentence sheds light on refraining and helps to clarify some of the agentive modalities. This linguistic addition, attentive to grammatical form and semantic

1. Stit: A canonical form for agentives

27

structure, promotes greater clarity in the way we talk and think about the phenomena of our world, and thus justifies its added complexity. In succeeding chapters we strive to deepen our understanding of agency by providing stit sentences with careful and well-motivated semantic analyses, and we apply and generalize on it in a variety of ways.

2

Stit: Introductory theory, semantics, and applications In chapter 1 we followed and extended the idea, going back at least to Anselm, of treating agency as a modality—a modality that represents through an intensional operator the agency, or action, of some individual in bringing about a particular state of affairs.* We proposed that using the stit construction as a normal form, when we are confused, is a happy way to clarify some aspects of action and agency. In this chapter we sharpen our understanding of stit—and thereby, if we are right, of agency. The central idea is that the concept of action must be understood in relation to an open future, and we formulate a rigorous theory that tries to understand how action is compatible with and indeed requires irideterminism. In this way we essay a contribution to what Kane 1998 calls ''the intelligibility question" (p. 105). We often label the chosen approach stit theory, because it concentrates on the linguistic form "a (an agent) sees to it that A," which we abbreviate simply as [a stit: A]. Part of the theory, however, has nothing to do with language. This nonlinguistic part instead purports to articulate in a general way how agency fits into the overall causal structure of our world. For reasons that will emerge, we often ponderously refer to this theory as "the theory of agents and choices in branching time,'' or, more briefly but less memorably, as BT + AC theory— or, when endowed with instants or times, as BT + I + AC theory. Stit theory, including its nonlinguistic part, provides a precise and intuitively compelling semantic account of the stit operator within an overall logical framework of indeterminism; the account is then used as a springboard for investigating a number of topics from the general logic of agency, such as the proper treatment of certain concepts naturally thought of as involving iterations of the agency "This chapter draws on several sources. We thank John Horty for co-authoring §2A, which, with the permission of Kluwer Academic Publishers, is drawn from Horty and Belnap 1995. §2B, except for §2B.10, finds its source in a portion of Belnap 1991, for the use of which we thank the International Phenomenological Society, while §2B.10 is based on part of Belnap 1996a, with the permission of Kluwer Academic Publishers.

28

2. Stit: Introductory theory, semantics, and applications

29

operator, as well as interactions of this operator with other truth-functional and modal connectives. The theory of agents and choices in branching time supports more than one semantic candidate for stit. We mention several of these briefly in various later chapters, but only two are primary in this book. One of the two derives from the early work of two of the authors, NB and MP. The other, which is simpler and for certain purposes easier to use, first appeared in von Kutschera 1986, prior to the work of the authors of this book; later one of the authors of §2A of this chapter independently suggested it, in Horty 1989, explicitly as an alternative to the account of stit put forth by NB and MP. The first major purpose of the present chapter, which we carry out in §2A, is to describe in §2A.l the underlying theory of branching time and the tense logic that is appropriate to that theory, and then in §2A.2 and §2A.3 to describe the individual semantics of each of the two stits, together with the theory of agents and choices on which they rest. These discussions are all explicitly intended to be preliminary; later chapters treat B T + I + A C theory and each of the two stit operators with more rigor and in more detail. A second purpose of this chapter, accomplished in §2B, is to offer an equally preliminary exploration, with many pictures, of some applications of the first, more complicated of the two stit operators. In order to distinguish between the two agency operators under discussion, and for other reasons that will soon become apparent, we describe the operator featured in the early work of NB and MP as the achievement stit, represented in this section as "astit"; and we describe the alternative suggested by von Kutschera and Horty as the deliberative stit, represented here as "dstit." When in this book we speak simply of a stit operator—or use "stit" alone as an operator in some sentence—we often mean to generalize over both the deliberative and achievement stit operators, and perhaps others of the same family. Other times, however, we use plain "stit" as meaning one of astit or dstit, context making it clear which stit is at issue.

2A 2A.1

Theory and semantics: The two stits Background: Branching time

Stit theory is cast against the background of an indeterministic temporal framework, in particular, the theory of branching time due originally to Prior 1967, pp. 126-127, and developed in more detail in Thomason 1970 and Thomason 1984.l 1 From time to time in this book we insert the reminder that although we use the phrase "branching time" because of its fixed place in the literature, we never, ever mean to suggest that time itself—which is presumably best thought of as linear—ever, ever "branches." The less misleading phrase, which we occasionally use, is "branching histories," with an essential plural to convey that it is the entire assemblage of histories that has a branching structure. A single history cannot branch, but two histories can branch from each other. As a further occasional reminder, we observe that although in this book we idealize each history as linear, there is a more adequate treatment of branching histories in Belnap 1992. There each history in the branching assemblage is idealized not as linear, but instead as a four-dimensional spacetime. For similar ideas see also McCall 1994, Rakic 1997, and Placek 2000.

Introduction to stit

30

Figure 2.1: Branching time: Moments and histories

The theory.is based on a picture of moments as ordered into a treelike structure, with forward branching representing the openness or indeterminacy of the future and the absence of backward branching representing the determinacy of the past. Such a picture leads, formally, to a notion of branching temporal structures as structures (we sometimes say BT structures) of the form (Tree, = {h: (Mnh) = O}, so that H<M> is the set of histories that pass through at least one member of the set of moments, M. For suitable M, this is the set of histories in which M comes to be.

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H(m) represents the contingent proposition that m timelessly exists, a proposition that is true in all and only those histories that m inhabits. H[M] does not always make useful sense as a proposition. When, however, / is an initial event, H[M] is well interpreted as the (timeless) proposition that / is "completed," "finishes," or in Aristotle's phrase, "passes away." If you are thinking about experimental preparations or agonizing deliberations, this is a sensible thing to mean by "exists": A preparation-event or deliberation-event does not "exist" in a history unless it does so entirely. If some history, h, splits off in the middle of a deliberation, we decline to say that the deliberative event "exists" in h. (One would need explicitly to tackle the present progressive in order to have a rigorous account of a deliberation-event "existing" in histories in which it is not completed.) Note a comfortable interaction: The proposition that / is completed depends not at all on the causal "shape" of / in its nether region. Any initial cofinal with / toward the future will determine the same set of containing histories as does /. For example, if / is taken as a representation of the setting-up of an indeterministic experimental situation, whether or not the set-up is completed is insensitive to everything except the causal locus of its being completed. When O is an outcome event, the meaning of "exists" needs to be quite different. In that case, the proposition H is the right sense of "exists," since outcomes exist by beginning (not ending), and H is true in all and only those histories in which O begins, commences, or in Aristotle's language, comes to be. Suppose a coin is flipped. If we look only at the bare causal structure, it is natural (but not necessary) to think of the flipping as lasting as long as there is indetermination as to the outcomes heads or tails, no matter how that flipping may be related to the way that the coin dances in the air. In the same way, it is natural (but not necessary) to think of the heads outcome beginning whenever the possibility of tails is excluded. In other words, the heads outcome (not must but) may be taken to begin whenever that outcome is determinately settled, and the proposition that the heads outcome exists should be true in just those histories in which settled-heads begins to be. For purposes of causal analysis, it doesn't matter when the heads outcome ends. Consistency. Without belaboring the point, we note that the interplay between propositions and events in branching time generates a firmly based family of consistency/inconsistency concepts. We may take from possible-worlds theory the idea that when propositions are represented as sets of histories, consistency between them is definable as having some history in common, a history in which both propositions are true. We add that it is helpful to define the consistency of two initial events I1 and I2 by the consistency/inconsistency of the propositions H[I1] and -H[I 2 ], so that the question is whether or not there is a history in which both preparations are (not just started but) completed. Dually, two outcome events O1 and O2 are consistent/inconsistent iff the propositions H and H are consistent/inconsistent, so that the question is whether or not there is a history in which both outcomes begin to be. Even more delicately,

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one may ask whether a certain initial is consistent with a certain outcome by way of well-chosen propositions. Finally, one has the right definition of what makes a transition "contingent." CONTINGENT TRANSITION. (Definition. Reference: Def. 8) is a contingent transition iff is a transition, and if some history is dropped in passing from the completion of / to the beginning of O: H[I] -H = O. These remarks are intended to indicate with almost excessive brevity that branching time permits and indeed suggests a rigorous and modestly enlightening theory of the interrelations of (possible) concrete events and propositions. We mention that some additional ideas on initials and outcomes are offered in Szabo and Belnap 1996 and in Belnap 1996c, and in the unpublished set of notes Belnap 1995.

7A.5

Theory of instants

If "histories" are a way of making a sort of vertical division of Tree, then Instant, whose members are instants, is a kind of horizontal counterpart. In branching time, the doctrine of instants harkens back to the Newtonian doctrine of absolute time—and therefore is suspect. We use it, but we don't trust it. For that reason if for no other we try to be as clear about it as we can. Instants are perhaps not fully "times" because we are not in this study relying on measures or distances, but it is intuitively correct to think of i( m ) as the set of alternative possibilities for "filling" the time of TO. We need instants because we think that for the achievement sense of stit, in considering whether Autumn Jane stit she was muddy at a certain moment, it is relevant to consider what else might have been at the instant inhabited by that moment. Evidently our uses of "moment" (in which we follow Thomason) and "instant" are jargon not sanctioned in ordinary speech, although the distinction is certainly there to be drawn. Not all parts of stit theory rely on the theory of instants. Only the semantics for the achievement stit has need of these horizontal comparisons. For example, the theory of dstit, §8G.l, and the theory of strategies of chapter 13 are developed quite apart from the idea carried by Instant. In this important sense, the theory of instants is not a deep presupposition of stit theory. There is a contrast at this point with TxW theories as described in §7A.6. We nevertheless develop the theory of instants to the extent required by the semantics of the achievement stit. There are three postulates. Instant AND INSTANTS. (BT +I + AC postulate. Reference: Post. 5) i. Partition. Instant is a partition of Tree into equivalence classes; that is, Instant is a set of nonempty sets of moments such that each moment in Tree belongs to exactly one member of Instant. ii. Unique intersection. Each instant intersects each history in a unique moment; that is, for each instant i and history h, inh has exactly one member.

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iii. Order preservation. Instants never distort historical order: Given two instants i1 and i2 and two histories h and h', if the moment at which i1 intersects h precedes, or is the same as, or comes after the moment at which i2 intersects h, then the same relation holds between the moment at which i1 intersects h' and the moment at which i2 intersects h'. We next offer some convenient definitions and simple facts, after which we comment on the postulates.

INSTANTS. (Definition. Reference: Def. 9) • The members of Instant are called instants, i ranges over instants. • i(m) is the uniquely determined instant to which the moment m belongs, the instant at which m "occurs." •

m

(i,h) is the moment in which instant i cuts across (intersects with) history h; that is

• Order preservation can conveniently be stated in the symbols just introduced: implies • Fact: m(i ( m 0 ) , h 0 ) , a function of mo and ho, is the moment on history h0 that occurs at the same instant as does mo: • i|>m = {m0: m < mo & mo E i}. We say that i|> m is the horizon from moment m at instant i. • Where i1 and i2 are instants, we may induce a linear time order (not a causal order!) by defining i1 < i2 iff m1 < m2 for some moment m1 in i1 and some moment m2 in i2. Instants can also be temporally (not causally) compared with moments, m: i1 < m iff m1 < m for some moment m1 in i1 and m < i2 iff m < m2 for some moment m2 in i2 . Post. 5(i) encodes that making same-time comparisons between histories is objectively sound. We do not pretend to understand the conceptual problems involved in making such comparisons. The problem becomes ironically clearer when it is made more difficult by transference to branching space-time, where it is same-place-time comparisons between inconsistent point events that is at issue. All we can add is a conviction that it will not be possible to make suitable advances without consideration of the work of Bressan 1972, Bressan 1974, Zampieri 1982, and Zampieri 1982-1983, for they are the only persons we know who have worked within what seems to us the only reasonable position, that identifying place-times across possible situations is neither trivially easy (perhaps Kripke thinks this) nor a matter of partial constraint and partial stipulation (perhaps Lewis thinks this) nor empirically insignificant (perhaps this is van Fraassen's view), but a matter of serious physics.

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Postulates Post. 5(ii) and Post. 5(iii) on Instant are very likely too strong (too oversimplifying); our justification is that agency is already hard to understand, so that it won't hurt to try to see what it comes to in circumstances that are not altogether realistic—as long as we keep track of what we are doing so that later we can try to move closer to reality. Thus, which most reality-oriented persons think not so plausible, but which greatly simplifies our picture of time, all histories are said to have exactly the same temporal structure. It follows that all histories are isomorphic with each other, and with Instant, which justifies the ordering on instants defined in Def. 9. On the other hand, no assumption whatsoever is made about the order type that all histories share with each other and with Instant. For this reason the present theory of agency is immediately applicable regardless of whether we picture succession as discrete, dense, continuous, well-ordered, some mixture of these, or whatever; and regardless of whether histories are finite or infinite in one direction or the other. The theory of Instant is not, as we have said, as fundamental as that of S theory allows that histories can "overlap" in the sense of sharing exactly the same states of affairs at exactly the same times, and perhaps, though we are unsure, this is genuine overlap. Still, on h:T—> S theories, states of affairs are repeatables that can in principle occur at different times; they are therefore quite different from moments, each of which belongs to but a single instant, as is appropriate to a concrete event. Perhaps one could obtain a useful h:T—> S theory from branching time by locating an interesting sense of "state of affairs" as partially characterizing moments. Otherwise the h:T—>S theory seems faced with the double demerit of needing to provide both an account of times, and an account of states of affairs, on pain of being without application. Perhaps, for instance, belief in "states of affairs" is wild-eyed metaphysics (we hope not). Thomason 1984 discusses the notion of a "Kamp structure" which is a variation on the TxW idea. (The literature uses the word "frame" more often than "structure"; there is no difference in concept.) In a Kamp structure, each world is provided with its own temporal ordering. No real ground is gained, however, since using these structures to simulate branching still requires the concept expressed by saying that two worlds "perfectly match up through" a given time. Perhaps ground is even lost, since the ontology of Kamp structures requires making sense of the possibility that the times of two worlds could be dramatically different in their ordering, while nevertheless sharing some particular entity ontologically classified as a "time." For example, the respective times of two worlds might each contain both 4:00 P.M. and 5:00 P.M. The first world, however, could put 4:00 P.M. and 5:00 P.M. in their natural order, while the second world reverses their order. In considering what this could mean, we seem to lose our grip. In any case, all of our philosophical objections to Tx W as a theory of the structure of our world apply equally to Kamp structures. (This discussion is indebted to correspondence with A. Zanardo.) Both TxW structures and Kamp structures can be defined with first-order conditions. Yet another such representation of branching times is the notion of an "Ockhamist structure" (X, h2 E H. So for Tree to be deterministic at mo is for there to be but a single immediate possibility at mo. Sounds right. Obviously Tree may be deterministic at m but not so at either earlier or later moments. On the present account of determinism, one can coherently believe that our world used to be deterministic but is not so now, although it may become so once again. We go on to say that Tree itself is deterministic if it is deterministic at every past. In that case there is obviously but a single history. A determinist is someone who believes that the tree (world) of which our moment is a part is deterministic. It would appear that many philosophers believe that anyone who is not a determinist is softheaded and probably needs therapy. Others believe that anyhow all respectable philosophical theories, including theories of agency, should at least be consistent with determinism. Determinism, however, is an extremely strong theory, going far beyond determinism of the present moment. In any event, we are not determinists, even though the denial of determinism is not a postulate of this book. But more than that, on the theory here offered, if anyone could ever see to anything, then determinism is false. So even though we do not lay down indeterminism as a postulate, since we believe that sometimes people have choices, we are indeterminists. Accordingly we think that any theory (of anything) should be compatible with at least a little indeterminism. We are "compatibilists" in the best sense. We agree with Kane 1998 that in particular the question "whether a kind of freedom that requires indeterminism can be made intelligible" (p. 105) deserves, instead of a superficial negative, our most serious attention, and indeed we intend that this book contribute to what Kane calls "the intelligibility question." Note, incidentally, that some situations in our world could, for all we know, be governed by indeterminist laws that are nonprobabilistic. Such a law might describe the sorts of possible outcomes for some type of initial event, but without carrying information concerning the relative probability of those outcomes. Agency may or may not be like that.

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It is abstractly interesting to observe that our world could be strongly antidetermmistic in the sense that there is splitting of histories at every moment, so that H( m1 ) = H( m2 ) implies m1 = m2 for every m1 and m2 in Tree. It seems to us entirely possible that strong antideterminism is true—but maybe not. Certainly it would be mathematically attractive to be able sometimes to think of a moment as a set of histories (those that pass through it), just as we can think of a history as a set of moments (those that it passes through). With equal certainty, however, this is not a good place at which to argue from beauty to truth. As a final word on this topic, we note that even "strong" antideterminism is not all that strong. Even "strong" antideterminism is consistent with the truth of numerous and important determinist theories about "systems" and "states" and such. Indeterminism is not disorder.

7B.2

Arguments against branching

In a passage that we also quote on p. 170, Lewis 1986 advances persuasive arguments against branching time. The trouble with branching exactly is that it conflicts with our ordinary presupposition that we have a single future. If two futures are equally mine, one with a sea fight tomorrow and one without, it is nonsense to wonder which way it will be—it will be both ways—and yet I do wonder. The theory of branching suits those who think this wondering is nonsense. Or those who think the wondering makes sense only if reconstrued: You have leave to wonder about the sea fight, provided that really you wonder not about what tomorrow will bring but about what today predetermines. (pp. 207—208 of Lewis 1986; we quote the remainder of this passage later, on p. 208) In addition to our earlier comments, we respond to this argument in two quite different ways. The first response is that it takes our ordinary ways of thinking too seriously. We draw an analogy, due to Burgess 1978 (p. 165), between (i) objections to "some of our ordinary ways of speaking" based on standard relativity theory and (ii) objections to some of those ways based on branching time. In fact we sharpen the Burgess analogy by emphasizing the similarity of the roles of "many histories" in indeterminism and of "many frames of reference" in the theory of relativity. When taken as a piece of argumentation, the analogy could go like this. If the foundation of the Lewis argument in our ordinary ways were solid, then the following would be an easy reduction to absurdity of the theory of relativity. The trouble with the theory of relativity exactly is that it conflicts with our ordinary presupposition that we have a single present or "now." If two presents or "nows" are equally mine, one with a sea fight at Neptune's north pole and one without, it is nonsense to wonder which way it now is—it is now both ways—and yet we do wonder.

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That is a bad argument. Certainly many of us have a tendency to wonder what is going on at Neptune's north pole (or on the far side of Sagittarius) right "now." The presupposition underlying this wondering is "ordinary" and perhaps even "natural" for clock-aware and scientifically sophisticated persons born in the last few hundred years. But nevertheless it is false, as we are taught by relativity. We urge as apt the analogy suggested by this rewording of the quoted argument against branching time: Relativity insists that our world provides us with no uniquely natural spatially extended "now," although it may permit us to consider, if we wish, the limited family of all nows ("hyperplanes") to which a given utterance-event belongs. And indeterminism insists that our world provides us with no uniquely natural spatiotemporally extended actual history, although it permits us to consider, if we wish, the limited family of all histories to which a given utterance-event belongs. To the extent that common sense asks for a unique naturally given "now" to which a given utterance-event belongs, or for a unique naturally given "actual history" to which a given utterance-event belongs, to that extent, common sense is asking for something it cannot have. Nevertheless, the Prior-Thomason semantics, which explicitly recognizes the relativity of many statements to histories as well as moments, can give common sense a large amount of what it wants and can correct some parts of the Lewis formulation that are too hasty. "We have a single future." If this means that it is settled what will happen, for example, that either it is settled that there will be a sea battle or settled that there will not, it is false. If it means that it is settled that incompatible events will never happen, it is true. If it means that there is a single future history following upon this utterance, it is false. If it means there is a single future of possibilities, it is true. Branching time indeed claims that it can happen that "two futures are equally mine, one with a sea fight tomorrow and one without." But what does it mean to say that a future is "mine"? Branching time says that it means that it is among the futures now possible, where the "now" is indexically mine. To avoid making branching time look silly in a way that it surely isn't silly, the quoted description should be amended by insertion as follows: "two possible futures are equally mine, one with a sea fight tomorrow and one without." Lewis misdescribes the theory of branching time in saying of such a situation that "it will be both ways." Branching time is entirely clear that "Tomorrow there will be a sea fight and tomorrow there will not be a sea fight" is a contradiction. What is true and not surprising is that "It is possible that tomorrow there will be a sea fight and it is possible that tomorrow there will not be a sea fight" is eminently consistent. Nor is this merely a matter of formal tense logic. It seems to us deeply realistic to take it that if the captain is faced with two possibilities, sea battle tomorrow or no sea battle tomorrow, then those are possibilities for him, on that occasion. They are equally his, not one more than the other, exactly in accord with Lewis's account of (not his own theory but) branching time. Suppose the sea battle comes to pass. Then (after the sea battle) the two possibilities were his, and were equally his. In particular, branching time rejects the Lewis

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shadow-theory according to which the captain himself is to be found in the world of the sea battle, whereas merely one of his "counterparts" can be located in worlds without sea battles. The point goes back at least to Burgess 1978, who reflects as follows on his earlier decision to go to the office instead of the seashore: "Tense logic insists, pace Lewis, that I am the very same person who could have gone to the shore; it's not just someone like me who could have gone" (p. 173). Suppose we "step outside of branching time." To do this is to confine ourselves to language that has no trace of indexicality, a perfectly proper thing to do (see note 14 on p. 162). Histories and moments and persons are then linguistically accessible only via (rigid) naming or quantification. Branching time then says that for suitable moment mo and histories h1 and h2, the captain lives through a sea battle the day after m0 on h1, and lives through no-sea-battle the day after mo on h2. Here seems a premiss for a reductio of branching time, for branching time then seems to say that the captain has it both ways, both living through a sea battle and living through no-sea-battle. The reductio is, however, an illusion. Omitting the relativization to histories is intolerable. What branching time says is that the captain "has it both ways" in the entirely innocuous sense that he lives through a sea battle on history h1 and lives through no-sea-battle on history h2. That just says that there are at mo two possibilities for him, a fact about our world that we must keep. It does not say that the two possibilities will each be realized, an absurdity that branching time denies. It does not say that these possibilities remain possibilities at moments after the sea battle has commenced. It only says that in the past of such moments the two possibilities were available. Current possibilities drop off (McCall 1994) with passage into the future, but not the fact that they once were possibilities. Once was-possible, always was-possible. What about "wondering" whether or not there will be a sea battle? Evidently our wondering is history-independent: The fact that we wonder is dependent on the moment but independent of the history parameter. So what sense can we make of wondering about a history-dependent complement such as "there will be a sea battle tomorrow"? Lewis points out one alternative, which he rightly presents as not very ordinary, namely, that the complement of the wondering is the history-independent question, whether or not it is now settled that there will be a sea battle tomorrow. We proposed in §6E a two-point understanding of wondering that lets its complement remain open in the history parameter. First, it seems natural to construe "wondering q," where q is an indirect question (e.g., "whether there will be a sea battle tomorrow"), as "wanting to know" (or perhaps "wanting to have") a true answer to the question of q. Second, it seems obvious that our wants are not normally satisfied immediately; we must in general wait. Just so, in order to obtain satisfaction of the want expressed in our wondering, we shall need to wait until tomorrow; for only tomorrow is it possible for us to come to know whether or not a sea battle comes to pass. We next comment on some ideas in the remainder of the passage the beginning of which we quoted once on p. 170 and again on p. 205. Lewis continues in the following way.

208

Foundations of indeterminism But a modal realist who thinks in the ordinary way that it makes sense to wonder what the future will bring, and who distinguishes this from wondering what is already predetermined, will reject branching in favour of divergence. In divergence also there are many futures; that is, there are many later segments of worlds that begin by duplicating initial segments of our world. But in divergence, only one of these futures is truly ours. The rest belong not to us but to our other-worldly counterparts. Our future is the one that is part of the same world as ourselves. It alone is connected to us by the relations—the (strictly or analogically) spatiotemporal relations, or perhaps natural external relations generally—that unify a world. It alone is influenced causally by what we do and how we are in the present. We wonder which one is the future that has a special relation to ourselves. We care about it in a way that we do not care about all the other-worldly futures. Branching, and the limited overlap it requires, are to be rejected as making nonsense of the way we take ourselves to be related to our futures; and divergence without overlap is to be preferred. (p. 208)

The first sentence refers to "the future," the one that "is truly ours." Branching time says that only the future of possibilities is uniquely determined by the moment of utterance, so that "the future" either refers to this, or else is not history-independent (is open in the history parameter). Branching time says that if indeterminism be true, then there is no more sense to "the actual future" than there is to "the actual distant instantaneous present" or to "the odd prime number." But what about the future that "is part of the same world as ourselves"? Assuming indeterminism, there is the following dilemma. • If we read "world" as "history," then it makes no sense to speak of "the world of which we are part." There are many such possible histories to which this utterance-event equally belongs. All of them are "connected to us by the ... spatiotemporal relations ... that unify a world," for there is, in our opinion, no more fundamental "natural external relation" than the causal ordering itself. It is to be borne in mind that even wholly incompatible moments are mediately connected by ." It is striking that each point contains two independent references to moments: the moment of use, mc, and the moment of evaluation, m. They play of course entirely different roles: The moment of use is immobile (unshifted by any operation), and may be used to fix indexical expressions, whereas the moment of evaluation is paradigmatically mobile, being shifted by a variety of tense constructions. In addition to their uses in connection with various special indexical expressions, some context parameters have another role to play (Kaplan, p. 595; see our overlapping discussion in §6B.4.2). In the present case the following, which restates Policy 6-2, is critical. 8-2 OBSERVATION. (Starting evaluation of stand-alone sentences) The moment-of-use parameter is used to start the evaluation of any stand-alone sentence considered as being uttered to some purpose involving the semantics of the sentence, for example, uttered as an assertion. We expand the discussion of §6B.4.2. If you want to evaluate Themistocles was surprised,

(6)

and you understand that "was" moves evaluation into the past, you need a place to start that motion. When (6) is itself being considered as stand-alone, the moment of use gives us that starting point. It works in a special way: The (paradigmatically mobile) moment of evaluation is fixed, to begin with, as the very moment of use. It is only when we come to the sentence Themistocles is surprised,

(7)

which is implicitly embedded in (6) by past-tensing, that there is a divergence between moment of use (which remains the same as for (6) taken as stand-alone) and moment of evaluation (which needs shifting, existentially, toward the past). 3 Elsewhere in this book, in order to minimize complexity of exposition, we not only keep the model, m, implicit; we also pass over recognition of either the assignment-to-variable parameters or the context parameters. In short, in spite of thinking of just one target language, we relativize truth in different ways depending on which problems we are attacking. For example, when we are not concentrating on quantifiers, we omit relativization to the assignment parameters.

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You will note that in expressing Observation 8-2, which is derived from Kaplan, we use loose language. It is difficult to do otherwise for the following reason: Although we have in mind that a sentence can be considered either as stand-alone or as embeddable, the symbolic language (unlike, e.g., English) makes no such distinction. In the symbolic language, there is no syntactic mark (such as initial capitalization and final period in written English, or intonation in the spoken language) that distinguishes sentences taken as stand-alone from those taken as embeddable. This lack of match between English and the symbolic language makes analysis more difficult. Here is the best we know how to do (without describing a new kind of symbolic language) by means of a definition and a policy. (See Green 1998 for a study of "illocutionary-force-indicating devices," including Frege's sign of assertion.) First the definition, which is a BT +1 + AC-specific version of the more general concept indicated in Policy 6-2. CONTEXT-INITIALIZED POINT. (Definition. Reference: Def. 16) We say that a BT + 1 + AC point, <m, mc, a, m/h>, is context initialized iff the moment of use, mc, is identical to the moment of evaluation, m. That is, context-initialized BT + I + AC points have always the form <m, mc, a, mc/h>. The idea is that in a context-initialized BT + I + AC point, the mobile moment of evaluation is "initialized by" the moment of the context of use. Next the policy. 8-3 POLICY. (Differential

treatment of stand-alone and embeddable sentences)

• We recommend and urge, on pain of confusion, that sentences considered as stand-alone may usefully have their evaluation restricted to contextinitialized points. • We recommend and urge, on pain of confusion, that sentences considered as embeddable shall not have their evaluation so restricted, but that they shall be evaluated also at points in which the moment of use, mc, and the moment of evaluation, m, diverge. The rationale for the first part of this policy is that each utterance should be conceived as tied to a concrete context, and that such a context determines a unique causal position, with a definite past and a definite future of possibilities. We idealize such a position with the moment of use, mc. This moment of use is the very moment at which we wish to evaluate a stand-alone sentence. That is why, for stand-alone sentences, we initialize the moment of evaluation with the moment of use. Keep in mind, however, that the moment of evaluation is mobile, and can be shifted by tense constructions ingredient in the stand-alone sentence. And that is the very reason for the second part of Policy 8-3. Since there is in the symbolic language (and indeed often in the English examples of philosophers) no syntactic difference between stand-alone and embeddable sentences, the definition of "context-initialized point" (Def. 16) and

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Policy 8-3 represent the best that we can do. The definition and policy are, we think, exceptionally useful in discussing tenses and indeterminism, for in those ventures the failure to observe the distinction between stand-alone and embeddable sentences is especially harmful. Here is one way the definition and policy offer immediate progress. Kaplan (pp. 505-506) asks that in thinking about the "character" or meaning of a sentence, we first fix context, and then ask for the content of the sentence in that context. There should, however, be' two notions of content-in-a-context, depending on whether we are thinking of the sentence as stand-alone or embeddable. If we are thinking of it as stand-alone, then the moment of evaluation is initialized by the moment of the context. Since one cannot reasonably treat a sentence with free variables as stand-alone (Assertability thesis 6-7), it is obvious that for stand-alone sentences the history is the only mobile parameter that is left to vary when considering a stand-alone sentence. If one correlates time to moment and world to history, this explains the otherwise puzzling phenomenon noted by Kaplan on p. 546: ... the truth of a proposition is not usually thought of as dependent on time as well as a possible world. The time is thought of as fixed by the context. That is right for stand-alone sentences: Time (or moment) is fixed, while world (or history) is not. If, however, we are thinking of the sentence as embeddable by means of translocal connectives such as tense operators, then for "content in context" we must let (i) the moment of evaluation diverge from and vary independently of ( i i ) the moment of the context. This explains why Kaplan permits content to vary over times as well as "worlds." We shall remain unclear as to the point of our semantic constructions unless we bear this in mind. We note that Kaplan is working in a Tx W framework (see §7A.6). It is, we think, an indication of the relative helpfulness of the moment-history framework that it explains a phenomenon that from the point of view of Tx W seems just puzzling. 8D.1

History of the context?

Policy 8-3 arises partly in virtue of the fact that there is a pairing of two parameters that have the same range of variation, namely, the moment of context and the moment of evaluation. This is sometimes called "double indexing." The phenomenon in general is of no special interest; after all, each xj parameter in quantification theory has exactly the same range as any other, so that, for example, x1 and x2 exhibit "double indexing" of the domain. When, however, as in the case of moments, one of the paired parameters is a context parameter and the other a mobile parameter, we may speak more particularly of "context-mobile pairing." We deepen our appreciation for Policy 8-3 if we ask the following two "context-mobile pairing" questions, one about histories, and one about assignments to variables. (The following discussion expands on that of S6B.5.)

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8-4 QUESTION. (History of the context?) Because we shall have "modal" connectives that are translocal with respect to histories, it follows that there must be a mobile "history of evaluation" parameter. Furthermore, the mobility of the history of evaluation plays an essential role in our account of assertion (Semantic account 6-12); but why is there not also a "history of the context," to be a paired with the mobile history of evaluation, as context parameter? If there were, for stand-alone sentences we could initialize the history of evaluation with "the history of the context." It would clearly make technical sense to provide a context parameter ranging over histories, to be interpreted as "the history of the context of use." And if we had one, we could enlarge our definition of "context-initialized point" to recognize this pairing. Further, there is plausibility in the Kaplan intuition that with "a little ingenuity" one can always make sense out of pairing a context parameter with a content parameter (p. 511). Indeterminism, however, compels a view absolutely contrary to this: There is no "history of the context." When you utter something, you do not thereby uniquely determine the entire future course of history. Your utterance has many choices and chances ahead of it, and so belongs to many histories. The context of use determines a unique moment, but not a unique history. Just to be explicit, we mean to be challenging principles such as that suggested by Kaplan on p. 597: "Any difference in world history, no matter how remote, requires a difference in context." Turning this around says that the identity of the context of use is enough to fix the course of world history, both past and future. That, if indeterminism be true, holds well enough for past history: The past, though largely unknown, is fixed. But it fails for the future: A single, well-identified context of use is typically part of a large variety of possible future courses of history. There is no unique "future of the context." The note on pp. 334-335 of Salmon 1989 is similar to Kaplan, though more convoluted. Salmon specifies the "quasi-technical notion of the context of an utterance," to be distinguished from the "utterance" itself, by saying that if any facts had been different in any way, even if they are only facts entirely independent of and isolated from the utterance itself, then the context of the utterance would, ipso facto, be a different context, even if the utterance itself remains exactly the same. Salmon concludes that ... although a single utterance occurs in indefinitely many different possible worlds, any particular possible context of an utterance occurs in one and only one possible world. If, however, "facts" are what is fixed at the moment of utterance, whether they are "isolated" or not, they cannot fix what the future brings—if indeterminism be true. Distinguishing "utterance" from a quasi-technical notion of "context of utterance" cannot make it otherwise. From this perspective, it looks as if Salmon is wrong—if indeterminism be true. It is, however, a delicate matter to label the

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difference between Kaplan-Salmon and ourselves as a difference of opinion. Here is a place in which there is a dramatic difference between the ideas of "world" and of "history." Kaplan and Salmon and, for example, Lewis seem to picture a "world" as something like a single space-time. We, in contrast, picture a world as something like an indeterministic tree. What is common to these two pictures is the use of "world" for a system of events connected externally by some sort of causal-type relation. It is for this reason that on either picture, it is plausible that a given utterance, or context of utterance, will determine its one and only "world." This seems close to common ground. If, however, histories can branch indeterministically in the way that we suggest, then a single utterance, together with all the most distant "facts," belongs to many histories, no one of which is specially determined as "actual" by the moment of utterance. Of course one could define "context of utterance" as a pair consisting of the moment of utterance together with a particular future history, the "actual" future history, and Salmon's note seems to suggest that he does in fact rely on the notion of an "actual" future history. He seems to rely, as do others, on the view that among all possible futures, one is marked out as a "Thin Red Line" in exactly the sense that we decried in chapter 6.

8D.2

Assignment-to-variables of the context?

As we discussed in §6B.5, there is still another family of mobile parameters, the assignment parameters, for which we do not provide a matching context parameter. We expand on that discussion. 8-5 QUESTION. (Assignment-to-x\ of the context?) Because we shall have "quantifier" connectives that are translocal with respect to the assignment-to-X1 (for example) parameter, there must be an "assignment-to-:ri of evaluation" to be something like a "content" parameter, in which the quantifier is translocal. But why is there not also an "assignment-to-rri of the context" to be a paired context parameter? If there were, we could initialize the mobile assignmentto-xi parameter with the context assignment-to-x1 parameter for stand-alone sentences. It would clearly make technical sense to provide a matching assignment-to-X1 as a context parameter. In some passages, indeed, Kaplan suggests that image (see especially p. 592), although he does not go so far as to provide both an assignment-to-x1 context parameter and an assignment-to-x1 mobile parameter in which the quantifiers can be translocal: There is only one assignment-to-X1 parameter, not two. Here the explanation lies not in the nature of things, but rather in (presumably universal) linguistic practice. Assignments to x1 are not anchored in the context in any serious way. As Kaplan clearly says (p. 593), there is no "factof-the-matter" that contextually determines a unique assignment to x1. So the symbolic language we are describing fails to provide an assignment-to-x1 as a context parameter because there is nothing in our language (or in any language we know) to which such a technical device would correspond.

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It is good to recognize, however, that we speakers of English (supplemented with variables) could adopt a convention that supplied each variable with a value in each context of use. We could then go on to insist that, for stand-alone sentences, the mobile assignment used for evaluation is initialized by the context assignment. Suppose, for example, we require that the context determine that x1 and x2 are both assigned zero. That seems as good a convention as any, since there is (as Kaplan says) no "fact of the matter." Then if we took something like

as a stand-alone sentence, and accordingly used Policy 8-3 to restrict ourselves to context-initialized points of evaluation, we should find out that (8) is automatically true. Ugh. Having realized that we could adopt such a convention, we are glad that we speakers of English haven't done so. And the symbolic language we describe is, in this respect, just like ours. All of its context parameters are "fact of the matter" parameters really and objectively determined by the context of use; none are subject to some doubtful convention manufactured by a logician.

8D.3

Points and policies summary

(i) The special nature of context-initialized points is recognized. (ii) Policy 8-3 is firmly in place for differential consideration of stand-alone and embeddable sentences, (iii) That a point of evaluation, <m, mc, a, m/h>, contains a context parameter for the moment of use, but none for "history of the context of use" and none for the "assignment of the context of use," is no accident or idle logician-imposed convention, (iv) Nor is the fact that there is no analogy to Policy 8-3 for histories or assignments.

8E

Generic semantic ideas

In the following section we detail recursive semantic clauses for the various operators that we treat. Here we outline generic features of the semantic concepts that are defined by that recursion. Except for the "in-context" concepts, these are all standard. We are supposing two kinds of categorematic expressions, terms and sentences. The semantic value of a term is always an entity in Domain, while the semantic value of a sentence is always one of the two truth values, T or F. Since we are going to include operations that are translocal in assignments, moments, and histories, we know that we shall have to relativize semantic value to these parameters. We further explicitly relativize semantic value to the moment of context, even though it is immobile, partly for explicit indexicality, partly to exploit the idea of context-initialized points and the attendant Policy 8-3, but fundamentally because we are thinking of evaluating terms and sentences in many different contexts. And we explicitly relativize semantic value to

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the structure and interpretation parameters because we wish to have not only "truth," but also "equivalence," "implication," and "validity." We want a way of referring to the semantic value of any categorematic expression, be it term or sentence. Once we have "semantic value" for sentences, we automatically have truth, falsity, and the idea so important for indeterminism, settled truth. SEMANTIC VALUE, DENOTATION, AND TRUTH. (Definition.

16)

Reference: Def.

• Semantic value. For any categorematic expression, E, be it term or sentence, Val m , m c , a , m / h ( E ) , is "the semantic value of E at the point (937, mc, a, m/h>." Val m,mc a m / h ( E ) is defined recursively by clauses given in §8F and §8G. Note that by the earlier clause of Def. 16 that appeared on p. 229, we may write Valtt (E) in place of • Denotation. Where t is any term, Val m , m c , a , m / h ( t ) E Domain. Valtt(t), or V a l m , m c , a , m / h (t) is "the denotation of t at the point <m, mc, a, m/h>." Also, as before, Valtt(t) stands in for Val m, m c , o , m / h ( t ) . • Truth. Val m,mc,a,m//i(A) is "the truth value of A at <m, mc. a, m/h>." Where A is any sentence, Alternate much-used notation for truth and falsity: - m, mc, a, m/h tt= A iff Va/m, m c , 0 ,m/h(A) = is true at point <m, mc, a, m/h>."

T.

Either is read "A

- m, mc, a, m/h ¥ A iff Va/gjt,m c ,a,m/h(-2, that witness does not stand in the past of mo. So at mo it is true that a might have seen to it that Q, but it is false that a could have seen to it that Q. In contrast, at mo it is true that a could have seen to it that R, since w0 witnesses [a stit: R] at each of mi-m3. Here is an example that fleshes out this abstract description of Figure 9.1. EXAMPLE. Don Quixote attacks the windmill. Commending himself most devoutly to his lady, Dulcinea, whom he begged to help him in this peril, he covered himself with his buckler, couched his lance, charged at Rozinante's full gallop, and rammed the first mill in his way. At the moment, WQ, that ends his commending, the Knight of the Mournful Countenance had the choice either to stand down or ride on. Once having begun his charge, however, there was a slightly later moment, Wi, at which Rozinante might by chance have collapsed. In the case of no collapse, there

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Figure 9.1: Don Quixote attacks the windmill

was then a later moment, W2, at which Quixote had the choice either to swerve toward or to swerve away from the disastrous confrontation. Let my, as shown in Figure 9.1 be the moment at which he rammed the windmill. Consider the history on which, as he finished his commending, the Knight regained his wits and stood down, and follow that history to a moment, mo, that is co-instantial with the moment, 7713, at which he rammed the windmill. At that moment, mo, it would be true to say that he might have attacked the windmill, but false to say that he could have. What decides the matter is that there is nothing he could have chosen at the end of his commending (WQ—which is the only choice point in the past of the moment of non-attack under consideration) that would guarantee his attack. Both chance (at wi) and the uncertainty of the outcome of a future choice (at w%) stand in the way of such a guarantee.1

9C

Might have been otherwise

To appreciate this next conjecture, consider a stit with a non-agentive state of affairs as complement, and let the anaphor, "otherwise," refer to just that non-agentive complement rather than to the entire stit sentence. CONJECTURE. If yon fellow sees to some state of affairs, then it might have been that the state of affairs not obtain—at that very instant. The final phrase accomplishes a task more easily than idiomatic English: Make sure that the "might" means that the absence of the state of affairs obtains 1 Permit us yet another restatement of methodology: Although we think the distinctions we are drawing are important for, e.g., moral analysis, we by no means fancy that our chosen expressions have a perfect fit with ordinary English.

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as a co-instantial alternative to the very moment in question. In notation it is unambiguous: STIT VERSION 1. [Q stit: Q] implies Might-have-been:~Q. By the semantics given in §8F.6, this is evidently equivalent to STIT VERSION 2.

[a stit: Q] is inconsistent with

Was-always-inevitable:Q.

These stit versions say that if [a stit: Q], then it has not been inevitable (determined) from all eternity that Q should obtain at the instant in question. UPSHOT. The conjecture is, in its stit versions, true. It is an obvious consequence of the negative condition (§8G). EXAMPLE. If it was inevitable from all eternity that the hog gelder's reed flageolet sounded four times while Don Quixote was at his meal, then the hog gelder did not see to it that his reed flageolet sounded four times while Don Quixote was at his meal. A hard determinist valiantly endorses the consequent; a soft determinist becomes cross, changes the topic, and exits the lists.

9D

Might not have done it

Next an important conjecture with a straightforward disposition. CONJECTURE. If a does something, then it might have been otherwise; that is, a might not have done it. Here let the "otherwise" refer anaphorically to the entire stit sentence, not just to its complement. STIT VERSION,

[a stit: Q} implies Might-have-been:~[a stit: Q}.

UPSHOT. True. We belabor the obvious by offering two proofs. First, since [a stit: Q] implies Q (by the positive condition), so that ~Q implies ~[a stit: Q], this is an immediate consequence of Upshot §9C. The second and more important proof is this: The consequent is a truth of logic, so that the implication is vacuous! This is related to the Triponodo principle of Makinson 1986, except here instead of the "trivial (legal) power not to do," we have the "trivial possibility of not doing." The argument that it is logically true is an easy reductio. If [a stit: Q] were settled true throughout an instant, then by the positive condition, Q would be settled true throughout that same instant—which would leave no room for satisfaction of the negative condition.

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EXAMPLE. The literature contains cases that stand as putative counterexamples to "that a did it implies that a might not have done it." The following gives the flavor of how stit theory might respond to a portion of these "counterexamples." Consider the apparently agentive statement: Don Quixote "armed himself cap-a-pie, mounted Rozinante, placed his ill-constructed helmet on his head, braced on his buckler, grasped his lance, and through the door of his back yard sallied forth into the open country." If the statement is taken at face value as agentive, then it must possibly be false. For example, if one looks to the description of Don Quixote as "having lost his wits completely" in order to judge the agentive statement false (on the grounds that a man without wits cannot see to anything), one sees that its possible falsity is trivial. If, however, one uses the fact that Don Quixote's brains have dried up as an excuse to reinterpret the apparently agentive content of the statement as really non-agentive, perhaps something like a metaphor, so that Quixote mounted Rozinante as the storm mounts a distant hill, then the description is not agentive and we concede that stit theory does not pretend to have a strategem—which would doubtless involve intentional elements—for probing the statement in question. (At this point we might have included a discussion of "might not have refrained from preventing," but we refrained, noting only that even though "refrain from preventing," in the sense of [a stit: ~[a stit: ~Q]], is a distinct agentive modality, "might not have refrained from preventing" does not create distinct analytical problems).

9E

Could not have avoided doing

This conjecture, inserted here because its disposition involves an application of the preceding result, is a proposal for a sufficient condition of doing. CONJECTURE. "The fact that a person could not have avoided doing something is a sufficient condition of his having done it" (Frankfurt 1969, p. 150). This appears to be an instance of "necessity implies truth"; but analysis reveals that the conjecture is plausible and interesting only because it is ambiguous. Its status depends on whether "avoided doing it" means just "didn't do it," and so is non-agentive; or whether, agentively, it means "refrained from doing it," that is, "saw to it that he or she didn't do it." This ambiguity is difficult to detect in ordinary English; but when it is revealed by using stit normal forms, either the plausibility or the interest of the conjecture disappears, as we see by considering the following two versions. STIT VERSION 1. ~Might-have-been:-~[a stit: Q] implies [a stit: Q}. STIT VERSION 2. ~[a could-have-sht: ~[a stit: Q]} implies [a stit: Q}.

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Applications of the achievement stit

UPSHOT. Recall that the antecedent of Stit version 1 comes to Was-alwaysmevitable:[a stit: Q], which makes that version of Frankfurt's conjecture sound like an interesting truth. Stit version 1 is indeed true, but only vacuously so, since by the trivial possibility of not doing principle of §9D, its antecedent is a logical falsehood. Stit version 2 is evidently false; for a counterexample, choose Q as any tautology. It is then past doubt that the antecedent of version 2 is trivially true and the consequent trivially false. Thus, in spite of the plausibility derived from thick and interesting stories, attention to austere form suggests that there is no reading of Frankfurt's conjecture on which it is both interesting and true. 2 EXAMPLE. Since the first version is an easy application of §9D, we illustrate only stit version 2. On the side of the antecedent, it is evident that not even the great Mameluke of Persia, either before or after his nine-hundred-year enchantment, could have refrained from seeing to it that if the golden helmet of Mambrino was made of brass, then it was made of brass; but on the side of the consequent, that dignitary certainly did not in fact see to that, nor to any tautology.

9F

Could have prevented

The following conjecture is confusing in ordinary language but easy to settle correctly when expressed symbolically. CONJECTURE. That we are responsible for some state of affairs implies that it must have been possible for us to have been responsible for its absence. STIT VERSION. [a stit: Q] implies [a could-have-stit: ~Q}. Examples of this conjecture sound plausible in English: It appears to follow from the fact that if Sancho Panza remained at rest beneath the cork tree, then he could have seen to it that he moved (Hume). UPSHOT. But as all contemporary logicians of action know, the most elementary story tells us that the conjecture is false. In stit theory, the relevant point emerges through the negative condition, which requires only that the falsity of Q be risked, not that its falsity be guaranteed. EXAMPLE. La Tolosa, the fair cobbler's daughter from Toledo, saw to it that Don Quixote was girded with his sword; but given the rough company of carriers, to say nothing of La Molinera, that poor wench was evidently in no position to see to it that the knight failed to be girded. 2 Perhaps the evident plausibility of the Frankfurt examples derives from the fact that so many verbs give rise to both agentive and non-agentive readings, a matter that we have suggested can be at least partly resolved by applying the stit paraphrase thesis, Thesis 3.

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Figure 9.2: Journey to the ducal castle

9G

Could have refrained

A subtler question is the following: QUESTION. If a saw to something, could a have refrained from seeing to it? • Some think yes. Chisholm 1964a, for instance, says that if some varlet loosed his firelock, then "there was a moment at which it was true, both that he could have fired the shot and also that he could have refrained from firing it." (Observe that Chisholm's "also" is not "could have seen to it that the shot was not fired"; he does not make the superficial mistake of supposing that the Conjecture of §9F is true.) • Some think no. Frankfurt 1969 supposes it possible that there should be such a thing as "the fact that a person who has done something could not have done otherwise." On our view this question is not to be happily represented without taking into consideration the stit analysis of refraining, so that we are not surprised to find that on those few occasions that the literature notices the existence of the question, it seems to resort to sheer postulation. STIT VERSION. Does [a stit: Q] imply [a could-have-stit: ~[a stit: Q}}? We interpret the question as asking whether or not the fact that [a stit: Q] is true at m implies that there is a moment in the past of m that stands as witness to [a stit: ~[a stit: Q}} at a moment co-instantial with TO. That is, is there some single choice point in the past of TO that, had a different choice been made, would have guaranteed the agent's failure to stit Ql UPSHOT FOR STIT. The implication fails, with an easy example, though not quite so easy as the counterexample to the Conjecture of §9F. In Figure 9.2, Q is settled true at TOO and TOI, and settled false at 7712, all of which are coinstantial.

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Applications of the achievement stit

Abstractly put, each of WA and WB picture a choice for a; you can see from the diagram that Choice0^,A(mn} — {TOO}, so that WA witnesses the truth of [a stit: Q] at mo (looking to m^ for satisfaction of the negative condition). On the other hand, Choice^,A(mi) = {mi, 7712}, so that since [a stit: Q] is true at mi, WA does not satisfy the positive condition for witnessing that [a stit: ~[a stit: Q}\ is true at m^. The choice point WA is "too soon." (The choice point, WB, however, does that job.) Therefore, [a could-have-stit: ~[a stit: Q}} fails at m0, and therewith the implication stated in the stit version. Here is a concrete example based on Figure 9.2. EXAMPLE. We take some literary license in the following idealization, recounted in such a way that Figure 9.2 serves as both spatial map and a model of choices in branching time. One afternoon the Knight of the Lions and his squire, Sancho Panza, journeyed to the castle of the duke and duchess.3 In order to process our English tenses, station yourself at some moment later than m0- At a certain point A the knight and his squire chose the north path, which (ideally) guaranteed their arrival at the castle by sunset. We suppose that their only other choice at point A was the northeast path, which itself split, after a few minutes, at point B. At point B they could have either elected the north path from B, which also would (ideally) have guaranteed their arrival by sunset, or they could have chosen the northeast path from B, which would have led them astray with no possibility of arrival by sunset. Thus, when Don Quixote and Sancho actually arrived at the castle by sunset, there was no choice point in the past of their entrance to the fortress at which they could have guaranteed refraining from arriving by sunset. That is, no choice in the past of their arrival could have positively prevented them from choosing to arrive by sunset. Take notice that the moment of departure from B, at which indeed they could have chosen to refrain from arriving by sunset, is not in the past of the moment of their actual arrival. Heed also that the example is purely structural—the desires, beliefs, and intentions of the agents are irrelevant. We add that for this conjecture, it makes a difference whether one considers stit or dstit. DSTIT VERSION. Does [a dstit: Q] imply Can:[a dstit: ~[a dstit: Q}}7 Here the dstit-suitable Can: is taken from §8F.4. UPSHOT FOR DSTIT. In contrast to the stit version, the implication holds. As in §8F.4, we take Can:[a dstit: Q] as simply Poss:[a dstit: Q], noting that the formal countenance of "can" for dstit can be less wrinkled than that of "can" with the achievement stit, because one need not worry about a double temporal reference. The implication then comes to this: [a dstit: Q] implies Poss:[a dstit: ~[a dstit: Q}}. A proof can be found in Horty 2001. 3 We follow Pellicer in identifying the unnamed duke and duchess with Don Carlos de Borja and Maria Luisa de Aragon, whose ducal descendant celebrated the third centenary of Qmxote in Pedrola in 1905.

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EXAMPLE. We redescribe the journey to the castle in terms of dstit, changing only the Q, which is now to the effect that ( WilLDon Quixote and Sancho Panza arrive at the castle by sunset), which we evaluate at WA and the history on which they choose to go left. On this construal the later moment WB is irrelevant: At WA, given the choice they did make, they dstit (WilLthey arrive by sunset); also at WA they could have chosen to dstit they did not dstit (Will:they arrive by sunset). Moment WB is relevant in considering whether or not at WA they can dstit (Will:they fail to arrive at sunset). They can't.

9H

Might have refrained

We pray your close attention to a question whose answer depends on whether an infinite number of choices is made in a finite time (busy choosers, Def. 14). QUESTION. Suppose that a sees to it that Q; does it follow that a might have refrained from seeing to it that Q in the sense that there is a co-instantial alternative at which a refrains from seeing to it that Q? STIT VERSION. Does [a stit: Q] imply Might-have-been:[a stit: ~[a stit: Q}}? This is a tricky question. Its answer depends, of all things, on whether or not there are busy choosers, §7C.5. UPSHOT WITHOUT BUSY CHOOSERS. If there are no busy choosers, the implication is valid. PROOF. Lettered steps are keyed to Figure 9.3. (a) Grant [a stit: Q] true at mo /ho, and let WQ be the witness in question, (b) Let mi be some moment in i(m0) a* which [a stit: Q] is settled true, and which has the further feature that it "has a closest witness" in the sense that there is a witness, w\, to [a stit: Q] at mi such that between wi and i(mo) there are no further witnesses to the settled truth of [a stit: Q} at any moment in i(mo)- Because there are no busy choosers, m1 must exist. By the negative condition, (c) there is a moment, m2, lying in i(mo) and above wi at which Q is not settled true. We claim that w\ is a witness to [a stit: ~ a stit: Q}} at 7712- The negative condition is easy: [a stit: Q] at mi is just what is required. Suppose, for reductio, that the positive condition failed; that is, (d) suppose that [a stit: Q] were true at some moment m3 £ Choice^im?.}, with witness at, say, w2. Where could w2 be? Because both wi and w2 precede ma, by no backward branching, either wi < w2 or w2 ^ wi. ( e i ) The first alternative is impossible, because w\ is a closest witness. (e2) The second alternative is equally impossible, because then the positive condition of w2 witnessing [a stit: Q] at ma would conflict with the failure of Q to be settled true at m2. (/) It cannot therefore be gainsaid that [a stit: ~[a stit: Q}} is settled true at m2, which is a co-instantial alternative to m0. Therefore, Might-have-been:[a stit: ~[a stit: Q}] holds at mo/h0.

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Figure 9.3: Implication without busy choosers

EXAMPLE WITHOUT BUSY CHOOSERS. The journey to the ducal couple's castle portrayed earlier in Figure 9.2 illustrates the subtle difference between "could have refrained" and "might have refrained." In that adventure the travelers might have refrained from arriving before sunset, though it was false that they could have refrained from doing so, since they were not busy choosers. And thus it is: The moment at which the wayfarers might have departed from point B is an excellent witness to the truth of [a stit: ~[a stit: Q}} at a moment co-instantial with the one in question, thus verifying "might have refrained." Since, however, that moment is not in the past of the moment of their actual arrival at the castle, it does not help verify "could have refrained." UPSHOT WITH BUSY CHOOSERS. In the presence of busy choosers and witness by chains, the implication fails. This is a near consequence of Theorem 18-8, which is based on a strong lemma, and in the vicinity of which there is more information. Here we offer a proof of the upshot with busy choosers that is based on a picture. PROOF. We turn to the busy picture of Figure 9.4 for a counterexample to the implication from [a stit: Q] to Might-have-been:[a stit: ~[a stit: Q}]. The abstract facts are these. We are ultimately interested only in a certain moment, mo, but we consider also (i) moments in i( mo ), which is represented by the dashed horizontal, and (ii) choice points of two sorts: those for a, represented by rectangles, and those for /3, represented by circles. Each choice point for a is binary, with the right possible choice containing a single history on which Q is assigned settled true where it intersects i( mo ). The left possible choice for a at that same choice point leads immediately to a choice point for (3, at which there

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Figure 9.4: The Mirror Game

are two possible choices, left and right, each of which leads immediately to a choice point for a of exactly the same kind as before. We suppose the temporal distance between choices for a is halving, and that each entire denumerable historical series of choice points for a and (3 approaches a unique member of i(mo)'i and we assign Q settled false at such members of i(ma)- The moment, mo, is the one lying above the right side of the first choice for a. We claim first for Figure 9.4 that each choice point, wi, for a witnesses the settled truth of [a stit: Q] at the moment, mi, in which the history belonging to the righthand possible choice for a at w1 intersects i(mo)- The positive condition is easy, since we assigned Q settled true at mi and since there is but a single history contained in that possible choice. The negative condition is satisfied by our having assigned Q settled false at those members of i(mo) approached by a denumerable historical series of choice points; one (and indeed many) of those members of i(mo) must be properly later than w\. As a special case, [a stit: Q] is true at mo- We claim next that nowhere in i(mo} is it settled true that [a stit: ~[a stit: Q}}. This is obvious for the members of i(mo), such as mo itself, that lie above some right-hand possible choice for a. Now suppose for reductio that [a stit: ~[a stit: Q}] is settled true at some moment m-2 in Z( m o ) that is approached by a denumerable series of choice points. There must then be a witness, and since we are allowing witness by chains, §8G.4, let the chain be 02, as indicated in Figure 9.4. The positive condition implies that ~[a stit: Q] be settled true at every moment in i(mo) that is choice equivalent to 7712 for a at C2. Choose some member, W2, of 02. Properly between W2 and m2 there will be a choice point, w3, for (3, and properly after w3, there will be a choice point, 104, for a that is not in the past of m,2 (w^ -f. m-i). Let 777.4 be the member of i(mo) lying above the right-hand possible choice for a at 104. The critical point is that 771,4 is choice equivalent to 7712 for a at 02, since—by the no choice between undivided histories principle—no choice for a in c? distinguishes 7774 from 7712So, since [a stit: Q] is certainly settled true at 7714, we have a contradiction.

268

Applications of the achievement stit

Thus [a stit: Q] is settled true at mo but Might-have-been:[a stit: ~[a stit: Q}} is not, so that the implication fails. 9-1 EXAMPLE WITH BUSY CHOOSERS. (The mirror game.) The Knight of the Mournful Countenance and the Knight of the Mirrors engage in serious combat. (We thank S. Sterrett for supplying a basic idea of this game.) They play a busy game that begins at noon and ends at sunset—at which time the vanquished is to remain entirely at the mercy of the victor. (Thomsen 1990 reminds us that "there is nothing more serious than play," p. 171.) Some plays of the game consist of infinitely many moves, which our knights-errant manage by halving the time spent on each successive move. Though busy, it is still a simple game. Don Quixote has the first move. On his turn the Manchegan has the following choice: Press on or retire. If he retires then at sunset he is the vanquished. If he decides to press on, it is the turn of the Knight of the Mirrors, whose move always consists in selecting a phantasmic replica of one of two giants for the Manchegan to face: either Pandafilando of the Malignant Eye, or Briareus with many arms, each later phantasm being, in appearance, half as tall as its predecessor. Whatever he of the Mirrors selects, the next turn belongs again to the Knight of the Mournful Countenance. At sunset there are but two relevant possibilities: Either Don Quixote has retired, in which case he is the vanquished, or he has succeeded in facing some denumerable sequence of phantasms, in which case he is the victor. The curious fact to be illustrated is this: It is possible for Don Quixote to retire from the contest, but it is not possible for him to refrain from retiring. Contrary to our untutored intuitions, not even an entire chain of choices to press on, right up to sunset, can witness that Don Quixote refrains from retiring; for such a chain of choices does not establish that it was entirely up to him that he persevere. The choices that in fact were made by the Knight of the Mournful Countenance bestow no hard information about "what he would have chosen" had the Knight of the Mirrors confronted him with an unrelenting sequence of replicas of Pandafilando of the Malignant Eye. Quixote's famous victory, however, does allow him to wrest from the fallen Knight of the Mirrors the confession that "the torn and dirty shoe of Lady Dulcinea of El Toboso is better than the ill-combed though clean beard of Casildea." Trust the concreteness of this fable, we beseech you, only to the extent that you understand its structural properties. Symmetrically, if you think our chronicle is wrong, please try to find an alternative rigorous account of witnessing, refraining, and so on, and not just another picturesque story without a precisely described structure.

91

Had available a strategy for not doing

In §9D we observed that by the negative condition, doing something implies that it might have been that it was not done ([a stit: Q] implies Might-have-been:~[a stit: Q}). In §9H we showed, in contrast, that in the presence of busy choosers, that a does something does not imply that it might have been that a refrained

9. Could have done otherwise

269

from doing it ([a stit: Q] does not imply Might-have-been:[a stit: ~[a stit: Q}]}If you look at any picture, however, it certainly seems as if whenever there is a witness, w, for [a stit: Q], at that witness a has available a strategy he could follow that, provided a never deviated from that strategy, would guarantee his not-doing. That is, [a stit: Q] plausibly implies that there was (in the past) available to a a strategy guaranteeing ~[a stit: Q}. The intuitive idea is that a could avoid seeing to it that Q by shooting for a "counter," as called for by the negative condition, at which Q will not be settled true. This intuition works; but rigorously establishing the fact (Theorem 13-28) turns out to be less trivial than one might suppose, requiring as it does much of chapter 13. In the end we are led to see the sharp difference between (i) something coming about for which there was a guaranteeing choice by a, and (ii) something coming about for which there was (in the past) available to a a guaranteeing strategy. Since past choices are a matter of fact, in case (i) we are entitled to a "because": The thing came about because of a choice by a. In case (iz), however, we are not entitled to a "because." since we have said that the strategy was available to a and that a chose in accord with the strategy, without saying anything about the agent following the strategy. Even though this book sometimes lapses into the language of "following," one must recognize that the concepts of the theory of agents and choices in branching time are too austere to support such loose talk. But more of this in chapter 13.

9J

Summary

The proposition If an agent is morally responsible for doing something, then the agent could have done otherwise

, ,

conceals complex connections among actions, moral responsibility, and alternatives open to an agent. We simplify by dividing the proposition in two: If an agent is morally responsible for doing something, then the agent did it.

,_,

If an agent did something, then the agent could have done otherwise.

/o\

This division isolates the idea of doing something, validating the use of a logic of agency. We use stit theory to clarify proposition (3). CONJECTURE. Proposition (3) can be disambiguated by means of a logic of agents who make choices against a background of branching time. UPSHOT. The conjecture is true. If we interpret "could have done otherwise" as "might have been otherwise," the implication holds; if we interpret it as "might not have done it," the implication still holds, but vacuously. If we read it as "could have prevented," the implication fails. If "could have done otherwise"

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Applications of the achievement stit

means "could have refrained" then it fails for the achievement stit, but holds for dstit. If "could have done otherwise" is taken as "might have refrained," then without busy choosers the implication holds, but with busy choosers it fails. If "could have done otherwise" is taken as "had available a strategy for not doing," then busy choosers or not, the implication holds.

10

Multiple and joint agency The goal of this chapter is to inspect some structural aspects of multiple and joint agency, a task sufficiently complex to give pause to the three inseparables, Aramis, Athos, and Porthos.* First, in §10B, we treat complex nestings of stits involving distinct agents. The discussion is driven by the logical impossibility of "a sees to it that 0 sees to it that Q" in the technical sense, even though that makes sense in everyday language. Of special utility are the concepts of "forced choice," of the creation of deontic states, and of probabilities. Second, in §10C, joint agency, both plain and strict (every participant is essential), is given a rigorous treatment in BT + I + AC theory. A central theorem is that strict joint agency is itself agentive. In the final section, §10D, we combine these two perspectives, looking briefly at other-agent joint agentives. Throughout this chapter we use "stit" for the achievement stit, §8G.3. As elsewhere in this book, even minimal progress toward the goal of this chapter has required a variety of simplifications, (i) As in stit theory generally, we totally avoid the reification of actions, and (ii) we minimize reference to intention in order to concentrate on causal structure, (in) Of relevant notions, we omit stits that are based on "witness by chains," §8G.4, and (iv) the deliberative stit, §8G.l. (v) Also, we omit concepts requiring the chapter 13 notion of strategies, and (vi) we do not consider the evident fact that agents interact in space-time, a topic yet to be studied. Finally, (vii) for the scope of this chapter, for the sake of expository simplicity we assume no busy choosers in the sense of Def. 14. In exchange, although we directly employ only the achievement stit, we have in this chapter limited ourselves to ideas and applications that, we think, work equally well for either the achievement or the deliberative dstit. We refer to Tuomela 1989a and 1989b for an alternative methodology that, in contrast to stit theory, freely permits one to (i) reify actions and (ii) refer *With the kind permission of Baltzer Science Publishers, this chapter draws on Belnap and Perloff 1993 As in the case of chapter 9, you will observe that we sometimes let the source of our examples influence our mode of speech.

271

272

Applications of the achievement stit

to intentions. Those articles also provide access to some earlier studies of joint agency.

10A

Preliminary facts

We will need the following stit facts. (See Def. 12 for the notation mi =%, m?.) 10-1 FACT. (Some facts about the achievement stit) i. Backward monotony,

and

m2 imply

ii. Witness-identity lemma (Chellas 1992). Suppose that Qi implies Q-2, that m, Wi, and io2 are moments, and that a.\ and a2 are (possibly identical, possibly distinct) agents. Suppose further that w\ is a witness to [a\ stit: Qi] at m, and that wi be the witness for this. From (e), (/), and the witness-identity lemma, we infer (g) w\ < w. So (d) and (g) imply, by backward monotony (Fact 10-1), that (h) ms =£t mg. But then the second witness-identity lemma with (/) and (h) gives that [0 stit: Q] must be settled true at m^; which contradicts (b) and completes the proof. For a picture, think of Figure 10.1 as a diagram of a witnessing moment, mo, in which columns represent possible choices for a, and rows picture the choices for 0 (see §10C.2 for the attribution of this picture to von Neumann). If [a stit: [0 stit: Q}}, then, where A = [0 stit: Q], A must fill some "choice-column" for a in

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mo. But then, because A represents a stit by /?, whenever A appears anywhere in a "choice-row" for /? inTOO,it must fill that row. So A must fill the entire diagram ofTOO-If so, then A is by definition settled true atTOQ.This contradicts the principle that, by the negative condition, stit statements are never settled true at their witness. So [a stit: [(3 stit: Q}] won't do as a representation of anything consistent. Before further considering (3)-(6), we turn to the examples (7)-(8), whose complements appear to be negations of agentives. We represent (7) straightforwardly with the form [a stit: ~[/3 stit: Q]], inasmuch as the count makes it true by kidnapping Mme Bonacieux. We observe that although kidnapping is unusual, the form [a stit: ~[/3 stit: Q}}, in contrast to the impossible [a stit: [j3 stit: Q}}, depicts a common enough occurrence. Indeed, everything we do tends to limit i] p choices of others, and much that we do guarantees such limitations. Although (8) appears to be similar to (7), appearances can be deceiving. We cannot represent (8) with [a stit: ~[/3 stit: Q]}: Mme Bonacieux is in no position to guarantee that d'Artagnan fails to follow her. Observe that among (3)-(8), we have so far provided a representation only for (7). In particular, we have pointed out that neither [a stit: [(3 stit: Q]] nor [a stit: ~[/3 stit: Q}] is appropriate to represent any of the others. Does this indicate a weakness of stit theory? We think not; we think it indicates a strength. The failures of [a stit: [0 stit: Q}} and [a stit: ~[/3 stit: Q}} encourage us to find more adequate stit-involving interpretations of (3)-(6) and (8). Chellas 1992 says that it would be "bizarre to deny that an agent should be able to see to it that another agent sees to something." Our acceptance of this view for everyday language is exactly what drives our search. We will look at four interpretations involving other-agent nested stits: deontic, disjunctive, probabilistic, and strategic.

10B.1

Deontic reading of other-agent nested stits

Let us suppose that the facts of (3) are as follows: The queen calls d'Artagnan to her chambers and says, Retrieve my diamond tags.

(9)

D'Artagnan retrieves the tags for the queen. He succeeds in the task she assigned. Shall we represent the situation by [a stit: \/3 stit: Q}} (with a = Queen Anne, /3 = d'Artagnan, and Q •*-> d'Artagnan retrieves the diamond tags)? No, that would be a mistake. It would be a mistake because, as we have seen, [a stit: \f) stit: Q}} claims that the queen guaranteed the truth of [/? stit: Q]. Since her issuance of an imperative does not guarantee that d'Artagnan retrieves the diamond tags, we must be more realistic in our analysis. We begin by reminding ourselves of three things, (i) As Austin 1975 taught us, issuing an order is doing something with words; it is, as everyone now says, a speech act; (ii) speech acts are acts; and (Hi) stit encourages us to understand action by asking what it is that an agent sees to. Let us therefore ask, what did the queen see to when she uttered (9)? Our answer is that she created an obligation. She saw to it,

276

Applications of the achievement stit

by her pronouncement, that an obligation existed where there was none before. Specifically, she saw to it that d'Artagnan was obligated to retrieve the diamond tags. We mean, incidentally, that really giving an order really does create an obligation, not just that the speaker so intends. In making this distinction with absolute sharpness, we separate ourselves from Searle 1965, who appears indifferently to use the language of intention in his "essential condition" but not to use it in his "essential rule" for promises. Having read Hamblin 1987, we also reject the claim of Searle and Vanderveken 1985 that "the point of orders and commands is to try to get people to do things" (p. 14). Instead the point of an order is to create an obligation. Nor does advice have the "causing of action" as its point. Aramis is plainly right that "as a general rule, people ask for advice only in order not to follow it; or, if they do follow it, in order to have someone to blame for giving it." What needs telling is a better story of what deontic states agents really see to when they use not only orders and commands, but also advice, requests, invitations, promises, ..., and indeed assertions and questions. Structural features of stit theory accordingly lead us to the following as a preliminary interpretation of (3).

That, however, won't do, since (3) is a "success" locution. It implies its complement, "d'Artagnan retrieves her diamond tags," whereas the form (10) does not imply [/3 stit: Q}. In a way that is precisely the point: Anne can stit the obligation, but not that d'Artagnan carries it out. We need to add that as a separate conjunct:

Now (11) shows on its face that it involves agency by Anne (as well as agency by d'Artagnan) without being agentive for Anne. 2 Such cases are many and important. When in (4) Jussac orders Biscarat to surrender and Biscarat replies, "You're my commander and I must obey you," he recognizes that his commander has seen to an obligation, [a stit: [/3 stit: Q}} is not the appropriate reading here because Jussac did not guarantee Biscarat's surrender. What did Jussac accomplish with his order? Jussac saw to the creation of an obligation: [Jussac stit: Oblg:[Biscarat stit: Biscarat surrenders]].

(12)

Jussac created the obligation to surrender, but it was Biscarat who surrendered. Regarding (8), [a stit: ~[/3 stit: Q]\ is not appropriate because when Mme Bonacieux sees to it that d'Artagnan does not follow her she does not prevent him from following her; rather she sees to it that a prohibition exists where none existed previously. The form 2

If it were possible for us to use the theory of agents and choices in branching time to represent that d'Artagnan retrieved the tags because of Anne's order, we would do so. We cannot, since the "because" in question seems to be an intentional matter, and so falls among the many aspects of agency for which we offer no theory.

10. Multiple and joint agency [Mme Bonacieux stit: Fr&n: [d'Artagnan stit: d'Artagnan follows]]

277 (13)

is therefore preferable as a representation of Mme Bonacieux' agency in the matter. When d'Artagnan obeys, he is an agent: He refrains from following her. It therefore takes the conjunction of the two agentives in order to reflect these features of the situation. To summarize the results of this section, we make a conceptual advance when we represent the character of the narrow-scope agentive in examples like (3) and (4) with the deontic form

and (8) with

One final point before leaving the topic. Notice that if the equivalence, Frbn:[a stit: Q] Oblg:[a stit: ~[a stit: Q]], is correct, then by substitution in the complement, (13) is equivalent to [Mme Bonacieux stit: 06/(?:[d'Artagnan stit: ~[d'Artagnan stit: d'Artagnan follows]]].

(16)

We think this is right. If so, there is confirmation of the deontic equivalences worked out in §2B.9.

10B.2

Disjunctive readings of other-agent nested stits

Not all apparent other-agent nested agentives can be illuminated by using deontic-complemented stits. Some require use of a disjunctive complement. For example, while we cannot use the form (14) to represent (5), we can use an other-agent nested agentive provided that it has a disjunctive complement. That is, though the cardinal does not obligate M. Bonacieux to agree to spy on his wife, the cardinal does see to it that Either M. Bonacieux agrees to spy on his wife or M. Bonacieux returns to his cell.

/,--,

The normal form [The cardinal stit: ([M. Bonacieux stit: M. Bonacieux agrees to spy on his wife] V [M. Bonacieux stit: M. Bonacieux returns to his cell])],

(18)

helpfully articulates the situation in which a principal agent imposes a "forced choice" on another agent. Figure 10.2 provides a picture: Open to a at w? is a choice that forces /3 into the position at w\ of having to choose between Qi and Q2- This observation, that a form such as

278

Applications of the achievement stit

Figure 10.2: Simple forced choice

can successfully represent the wide-scope agentive of certain apparent otheragent nested agentives whereas [a stit: [(3 stit: Q]} always fails, seems surprising until one reflects that, if one considers future choices, it is common coin that every choice opens up some future possibilities and closes off others. As Calvin once said to his tiger, Hobbes, "each decision we make determines the range of choices we'll face next." Subtly different and somewhat more complicated is the following, for which Figure 10.3 gives a picture. Suppose that Cardinal Richelieu is determined to entangle M. Bonacieux more deeply in his plot. For this purpose, the cardinal sees to it that the unfortunate draper is given a "forced choice." By the act of the cardinal, the draper is compelled to choose whether or not to put himself into the situation of forced choice already pictured in Figure 10.2. For example, Cardinal Richelieu might see to it that M. Bonacieux has a way in which the latter can avoid the forced choice between (i) earning his freedom by agreeing to spy on his wife and (ii) returning to his cell. He can do so by instead choosing to face the executioner. That is, the cardinal is sufficiently powerful to arrange matters so that M. Bonacieux must himself choose between an awful alternative—facing the executioner—and putting himself in a position of forced choice—spying or returning to his cell. When looking at such complicated interactions, it is easy to lose sight of the fact that the principal agent is responsible for seeing to it that the other agent is forced into this terrible predicament. Stit theory has the resources to describe the dimensions of the situation in order to help us understand the relations between the choices of different agents.3 Even so, we hasten to advertise that reliance on stit as one's only linguistic tool makes it awkward to articulate what is obvious from Figure 10.3: The cardinal and the 3 Figure 10.2 approximates the structure of the young man's predicament in Frank Stockton's The Lady or the Tiger. The more complex Figure 10.3 approximates the structure of Sophie's situation in William Styron's Sophie's Choice.

10. Multiple and joint agency

279

Figure 10.3: Complex forced choice

draper both had a role in the transition to the outcome, l 'M. Bonacieux spies for the cardinal."

10B.3

Probabilistic reading of other-agent nested stits

The next reading that we shall consider for other-agent nested stits is probabilistic. As background we observe the following: In this world we seldom guarantee the outcomes that everyday language expresses, even when these outcomes are not agentive; this is one of the principal reasons that we think of stit as only an approximation to "sees to it that." Observe that we do not intend this as a "concession." In analogy, it is no "concession" to remark that the law of the lever is only an approximation to what happens on a playground teeter-totter occupied by a couple of screaming, bouncing children whose parents are all too likely to intervene. The importance of the sociology of the teeter-tooter is no excuse for casting aspersions on the law of the lever in its teeter-totter application. The importance of the way in which outcomes are expressed in everyday language is no excuse for failing to appreciate the idealization of agency embodied in stit theory. In any event, our "nonconcession" is not meant to suggest that we never guarantee anything expressible in ordinary language! One of the things we can guarantee (we think) is the high probability of some outcome expressed in ordinary language. When Aramis sees to it that Lord de Winter learns of Milady's iniquity,

(19)

his choice by no means guarantees that de Winter is informed, but the following is fine:

280

Applications of the achievement stit [Aramis stit: it is highly probable that de Winter learns of Milady's iniquity.]

In designing a language to help us understand this matter, there are two choices: (i) permanently build the probabilistic element into the stit construction, so that stit itself indicates high probability instead of guarantee; or instead (if) represent the idea of probability as a separate linguistic element to be combined with stit as wanted. We think (i) should be avoided. In practice it makes it more difficult, not easier, to analyze problems. We recommend (ii). There is, however, a difficulty: The concealed double time reference of the achievement stit makes it at the least confusing to think through the interactions of probabilities and stit. We could reduce confusion by using dstit (see (iv) at the beginning of the chapter), but not having dstit available in this chapter, we cannot now profitably carry out the analytical work required. Furthermore there really isn't much sense in localizing probabilities in outcomes of moments. Moments are just too big: One ought to be suspicious of the intelligibility of saying that moments have "outcomes" that might or might not be probable. (We mean to refer to objective probabilities. If the probabilities are "epistemic," then the history of analytic philosophy testifies that anything goes.) Instead, outcomes and therefore probabilities of outcomes should attach to small, local events. (A relativistic foundation for this notion is given in Belnap 1992.) For these reasons we only indicate by an example how we think ( i i ) applies to other-agent nested agentives. Consider (6) on p. 273. Surely it is not literally true that Kitty guarantees that d'Artagnan seduces her by the provocative course of behavior she chooses. [Kitty stit: [d'Artagnan stit: d'Artagnan seduces Kitty]] is false. But the following, which introduces the required element of high probability, is true: [Kitty stit: it is highly probable that [d'Artagnan stit: d'Artagnan seduces Kitty]].

, ,

It is also true that d'Artagnan seduces Kitty.

(21)

So we agree with Chellas and common sense that (6) can sometimes be true, and we add a suggestion that in such cases a conjunction of (20) with (21) represents the matter correctly. The underlying point is that if in fact Kitty guarantees that (21), then it cannot be that d'Artagnan is agentive in the matter, and if he is agentive, then Kitty can guarantee the probability of the seduction, but not the seduction itself.

10B.4

Strategic reading of other-agent nested stits

In §5C we suggested at some length that the restricted complement thesis could and should be modified so that a strategy (instead of a stit) could be taken as

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281

the content of a promise. We mean "strategy" in the austere sense of chapter 13. It seems equally plausible to invoke a strategy as the content of a widescope operation of an apparent other-agent nested stit, perhaps along with, for example, the probabilistic reading. Consider (5). What the cardinal sees to may well be best expressed as the carrying out of a strategy whose desired issue is that M. Bonacieux spies on his wife. Perhaps, however, this thought is too superficial. Whether or not it proves helpful remains an open question.

10C Joint agency: Plain and strict In the previous section we explored some other-agent nested constructions with singular agents as subjects. In this section we will study agentives with jointagent subjects.

10C.1

Preliminaries

We start with English grammar. Constituent imperatives (see §1C) are embedded imperatives, analogous to embedded declaratives or embedded interrogatives. Their content, like that of agentive declaratives, can always be represented by stit sentences. An imperative, whether stand-alone or constituent, can have a collective term in subject position, as can an agentive declarative: M. de Treville announces: "I won't have my musketeers going to low taverns." The four friends scraped together nine or ten pistoles.

f^\ (23)

Example (22) might well be taken "distributively," and as analyzable in terms of stit sentences with subjects taken to denote a single agent (we call these "singular stits"), perhaps the subjects being individual variables bound by a quantifier. On a plausible reading, M. de Treville requires each musketeer to see to it that he does not go to low taverns. Examples like (23), however, drive us to widen the grammar of the language of agency. Here it is evidently the four friends "taken collectively" who succeeded in raising nine or ten pistoles; it is not something that each of them does. We cannot usefully represent (23) with only singular stits. We need to add to our formal grammar of singular stits the category of a "joint stit." Collectives can be represented by mereological sums as in Massey 1976; here we choose to represent collectives by sets. This choice limits applicability; the proposed apparatus cannot treat cases in which collectives change their membership over time (see Parks 1972), nor cases in which their membership is history-dependent. The limitation is for expository convenience only, and could be removed by using the language ML" of Bressan 1972 that we mentioned in §7C.l. We would realize Bressan's "cases" as moment-history pairs. In that language we would first represent Agent as an absolute concept, so that Agente would by definition be the extensionalization of Agent. Collectives of agents would be represented as properties F (possibly extensional, possibly not; possibly contingent, possibly not) such that F C

282

Applications of the achievement stit

Agente (this makes being a collective of agents a contingent property). Then for such F, [F stit: Q] would be denned as equivalent to the following: [Agent OF6 stit: Q}. The point is that at every moment-history m/h there is a unique subproperty T of Agent whose bearers are precisely those agents contingently identical at m/h to some individual concept falling under F. We can then use F in order to trace the same group of agents through the vicissitudes of time and history, since it must be a "substance concept"; see Gupta 1980 for a discussion of the use of substance concepts as principles of identity. The formal clothing of our decision is this, (a) We let F range over nonempty subsets of Agent and (&) we count [F stit: Q] as grammatical. Thus, we propose to represent (23) by [The four friends stit: the four friends have nine or ten pistoles],

(24)

where we let the four friends = {Athos, Porthos, Aramis, d'Artagnan} C Agent. (Note: We give this form for simplicity of illustration. The form [F stit: P]V[F stit: Q] seems more apt for (23) than the form of (24), namely, [F stit: P V Q]. They are certainly not equivalent, neither intuitively nor in stit theory.) There is more than one thing that one might mean by [F stit: Q}. First, one might mean that the bearers of F, without any outside help, guarantee that Q, on the basis of a prior simultaneous real choice by each of them. There is also a second, stronger, account. In this version the bearers of F, without any outside help, and with the essential input of each of them, guarantee that Q. Each account is useful and is worth a notation of its own. Since, however, we can give only one meaning to [F stit: Q], we choose the first. Later we introduce [F sstit: Q] as notation for the second account. We postpone to another occasion treatment of cases where Q is best seen as due to sequential efforts of the members of F. Our thought is that one must first be clear on sequential choices by a single agent, a topic on which we lightly touch in §8G.5 and in chapter 13. A consequence is that in this chapter we will often treat cases that in reality represent sequential choices as if they were simultaneous, provided the sequencing seems not important and the reconstrual as simultaneous seems enlightening.

10C.2

Plain joint stits

The key concept is the extension of choice equivalence to sets of agents. 10-3 DEFINITION. (Choice equivalence for sets of agents) For F a nonempty set of agents, we let m-i =£, m^ be defined as Va(a € F —> m\ =£, 7712). Choice equivalence for a set F of agents at a moment w is technically easy, but it is conceptually so important that we offer some further words. Let us go back to the idea that at bottom we are representing "possible choices" at a moment. The deepest idea of a possible choice for a single agent a at w is contained in its representation as a set of histories. The deepest idea of a possible choice for a set F of agents is also contained in its representation as a set of histories. We obtain

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one from the other as follows: Given a possible choice for each member of F, we define a possible choice for F as a whole to be the intersection or "combination" of all the individual possible choices. The "independence of agents" condition guarantees that such a combination always exists. The image we have in mind is due to von Neumann. Let an outcome be dependent on the choices of two agents a and (3. Von Neumann represents this graphically as follows. All the outcomes are arranged in a rectangular grid. Agent a can pick the row and agent (3 can pick the column. What happens is indicated in the intersection of the row picked by a and the column picked by B. "Independence of agents" just says that some outcome is indicated at each intersection of a row and a column. For example, if there are three rows (choices for a) and four columns (choices for /?), then there are twelve possible outcomes for their combined choice. With the help of the concept of choice equivalence for sets of agents, we can state the truth conditions for [F stit: Q}. We say that 10-4 DEFINITION. (Joint stit) [F stit: Q] is true at m/h iff there is a choice point w—a "witness to [F stit: Q] at TO"—satisfying the following conditions (compare the definition of the achievement stit in §8G.3): Agency: Agent. Priority: w < m. Positive: Q is settled true at each m\ such that m Negative: Q is not settled true at some moment—a "counter"—on the horizon from w at i(m). If we apply this definition to (24), it tells us that the raising of the pistoles was due to a simultaneous antecedent choice of the four friends. By so much the definition makes (24) a good approximation to (23). Furthermore, Definition 10-4 is good logic: 10-5 FACT. (Carryover from singular to joint stits) Results or analyses concerning singular agents established without the use of the postulate of the independence of agents also hold for joint agents. •Results not transferring include those expressed with the help of "a ^ /3" when these rely on the independence of agents. The point is that the possible choices for a and {3 at w will be independent if a ^ /?, but this is by no means true of the possible choices for FI and F2 when FI ^ F2. The obvious reason is that non-identity between the two collectives does not prohibit their having members in common. By Fact 10-5 we mean for example that any implication or non-implication that holds between singular stit sentences with just a also holds between the joint stit sentences that result when F is substituted throughout for a. For instance, [F stit: [F stit: Q}\ is equivalent to [F stit: Q], and ~[F stit: Q] is not in general agentive for F (i.e., it is not in general equivalent to [F stit: ~[F stit:

Q}})-

We can use joint agentives to express the independence of agents, provided we have the help of the following version of "ability" (this generalizes to a set of agents the ability concept of §8G.3).

284

Applications of the achievement stit

10-6 DEFINITION. (Can stit) Let [T can i -stit: Q] be true at a moment-history pair w/h iff there is a moment m lying on the horizon from w at i such that w witnesses the settled truth of [F stit: Q] at m. So [T carn -stit: Q] says that T can see to it that Q is true at i. Then we have the following. 10-7 FACT. (Joint ability) Provided cant-stit: P& Q]. That is, if at w a can see to it that P at z and ß can see to it that Q at i, then at w they can jointly see to the conjunction P& Q at i. Someone might think that the following is a counterexample. Porthos can see to it that the pistoles are used to repay a debt. Athos can see to it that the pistoles are used to purchase meals. But the pistoles being so few, the conjunction is impossible even with their best joint effort.

(25)

If you take the situation described seriously, however, especially with regard to fixing the time references, you will find that it is impossible. Of course if Porthos chooses first, then what Athos can see to is not independent of Porthos's choice, and vice versa. But fix their choices as absolutely simultaneous, as required for our principle of the independence of agents. And fix the "can" not sloppily, but as Austin's all-in, no-holds-barred "can" (Austin 1961, p. 177). Suppose that there are only a few pistoles. Award Porthos the ability to see to it that the debt is repaid. You have by so much restricted the power to be ascribed to Athos; there is in this situation nothing Athos can do by his choice alone that guarantees that the pistoles are used to purchase meals. For unless you either supply more pistoles or weaken Porthos's ability, you must allow that no matter what choice Athos makes, it is not enough by itself to guarantee the availability of the pistoles. Since you have given Porthos the ability to use the pistoles to repay the debt, you have described a situation in which for Athos to use the pistoles for meals requires the de facto cooperation of Porthos. The second sentence of (25) is therefore not satisfied, so that (25) is not a counterexample to the principle of independence of agents as expressed in Fact 10-7. We believe that any conceivable counterexample to the principle of Fact 107 will be equi-peculiar with the quantum-mechanical phenomenon discovered by Einstein-Podolsky-Rosen, for in fact it would need to have the same form; namely, spatially separated events that are each absolutely indeterministic and perfectly correlated. (This formulation comes from Belnap 1992.) Ordinary language can easily fool us about this by permitting (normally useful) waffly readings of "can"; here is a place where theory helps.

10C.3

Strict joint stits

But [T stit: Q] still does not tell us all that we may wish to know. For instance, (24) does not imply that each of the four friends was involved. It might have

10. Multiple and joint agency

285

been, for example, that d' Artagnan was not essential in raising the pistoles in the sense that [The three musketeers stit: the four friends have nine or ten pistoles],

(26)

where, as everyone knows, d' Artagnan E the three musketeers = {Athos, Porthos, Aramis}. Here is the easy fact about [F stit: Q] that informs us of this possibility: 10-8 FACT. (Weakening of ]oint stits) Given F1 C F2 C Agent: if [F1 stit: Q] then [F2 stit: Q}. That is, joint stits are closed under "weakening" by the addition of further agents. We need to define some related properties of agents in two versions before we can go further. The first relativizes the concepts to F and Q. The second drops the F, relativizing only to Q. The point is to be careful as to which concept is at stake. (We remark that although the terminology to be introduced seems apt in context, one needs to be sensitive to the considerations mentioned in §10C.l concerning sequential choices.) 10-9 DEFINITION. (Essential and inessential for stits) a is essential for [F a is inessential for 10-10 DEFINITION. (Essential and bystander for outcomes) a is essential [inessential, a mere bystander, not a mere bystander] for Q 3F[F stit: Q] & for every [not all, no, some] F such that [F stit: Q], a is essential for [F stit: Q]. Thus (26) says that d'Artagnan is inessential for (24), but (26) does not say that d'Artagnan is a mere bystander for the four friends having nine or ten pistoles. What then about the idea that F sees to it that Q, with the added provision that each of its members is essential? We think the intuitive concept thus described is rigorously definable (up to an approximation) just by saying that every member of F is essential for [F stit: Q] (but without requiring that every member is essential for Q). For preference we adopt an equivalent way of adding that there are no inessential members: F sees to it that Q, but no proper subset of F does so:

We will soon define "strictly stit" (an expression we introduce for joint agency when each of the agents is essential) by just this sentence, but first we must face a difficulty: Only the first part of (27) has an agentive form; the second conjunct is instead a denial of agency. So the whole may not itself be agentive! The difficulty is, however, easily overcome. In fact (27) is equivalent to each of the following.

286

Applications of the achievement stit

The equivalence of (27) and (29) establishes that (27) is agentive for F in spite of the fact that a conjunct of (27) is a denial of agency. We state this as a 10-11 THEOREM. (No inessential members) The sentences (27), (28), and (29) are mutually equivalent. In other words, where we let

the following are equivalent:

"NIM" is an acronym for "no inessential members." The proof of Theorem 10-11 goes in a circle as follows. (27) —> (28). Suppose (27) true at m1/h with prior witness w, so that in particular (a1) [F stit: Q] and (a 2 ) NIM are each true at m1/h. From (01) we have that (b) Q is settled true at all m2 such that m1 =£, m2, and there is a counter m3 on the horizon from w at i(m1) at which (c) Q is not settled true. We show that the same witness and counter will serve for (28). The part about the counter is evident, since if (c) Q is not settled true at m3 then neither is its conjunction with NIM. What we need to show for the positive condition, since we already have (6), is that supposing (d) m1 =£, m2, we have (x) NIM is settled true at rn 2 . Suppose for reductio that (x) fails, i.e., that (e) and (/) [F1 stit: Q] true at m 2 with witness w1. By (a1) and (d) and the second witness-identity lemma we know that [F stit: Q] is true at m2 with witness w, and therefore by (/) and the witness-identity lemma we know that (g) w = w\ (any two stits to Q at m2 must have the same witness). But m1 = F1 m2 from (d) and (e), and hence (A) m1 = F1 m2 by (g). Now (h) with (/) and the second witness-identity lemma puts [F1 stit: Q] true at m1. But (a 2 ) and (e) imply that [F1 stit: Q] is false at m1, contradiction. (28) —> (29). Suppose (a) (28) true at m1 with prior witness w. By the positive condition, each of (b) Q and (c) NIM is settled true at every m2 such that m1 =F, m2, and by the negative condition there is a counter m3 such that at m3/h3 for some h3 to which m3 belongs, (d) (Q & NIM) is false, that is, either (e1) Q is false or (e 2 ) [F1 stit: Q] is true, for some nonempty proper subset F1 of F. We need to establish the positive and negative conditions for (29). The part about the negative condition is easy; since [F stit: Q] implies Q, (d) implies that ([F stit: Q] & NIM) is false at m3/h3. Choose m2 such that m1 = F m2. We need to show that ([F stit: Q] & NIM) is settled true there. Now (c) already tells us that NIM is settled true at m2, and indeed from (6) we know that Q is settled true at every moment m2/ such that m2

287

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which gives us the positive condition for w to witness [F stit: Q] at m2. We therefore are missing only the negative condition for w to witness [F stit: Q] at m2, namely (x) there is a moment on the horizon from w at i ( m 2 ) at which Q is not settled true. In case (e1) we obviously have that (x); we need to show that case ( e 2 ) also implies (x). But1 this is a consequence of ( e 2 ) with (c) and the sufficient condition for unsettledness of Fact 10-1. (29) -> (27). This is trivial, having the form that [F stit: P] implies P. We therefore enter the following definition, where "sstit" is to be read "strictly sees to it that," and connotes the absence of inessential members. 10-12 DEFINITION. (Joint strict stit) (T sstit: Q] «-> ([r stit: Q] & VTi[0 ^

10-13 DEFINITION. (Strict agency) sstit: Q}.

Q is strictly agentive for r iff Q NMB, but not conversely. Thus the proposition that ([F stit: Q] & NMB) stands as follows: [F sstit: Q] -> ([F stit: Q] & NMB) -> [F stit: Q}. (ii) There is the statement OMB that outside of F there are only mere bystanders for Q. The proposition ([F sstit: Q] & OMB) says that F is the one and only joint agent for Q; it is evidently not agentive (in the sense of stit). It is, however, something that could be seen to.

10D

Other-agent nested joint stits

It is evident that the investigations of §10B on other-agent nested stits and §10C on joint stits need combining. In this area there is much to be considered. Here we offer only a single illustration, which is that the apparatus developed can distinguish the content of the following in an illuminating way: The four friends required of one of Planchet and Fourreau that he see to it that Brisement has a proper burial.

(34)

The four friends required of Planchet and Fourreau that they see to it that Brisement has a proper burial.

(35)

The four friends required of Planchet and Fourreau that one of them see to it that Brisement has a proper burial.

(36)

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291

Letting a = Planchet, B = Fourreau, and Q Brisement has a proper burial, the normal forms are respectively [The four friends stit: [The four friends stit: and

[The four friends stit:

(37)

In (36) the obligation is jointly on Planchet and Fourreau as a pair, but the execution is supposed to be by one of them as an individual. This complex content, so subtly different from that of (34) and (35), can be clearly expressed by an other-agent nested joint stit as in (37). The lesson, easy to miss if you take these examples as little puzzles or tricks, is that seriously applicable deontic logic needs other-agent nested joint agentives; and it therefore needs to include a theory with at least the expressive power of joint stits.

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Part IV

Applications of the deliberative stit

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11 Conditional obligation, deontic paradoxes, and stit Consider the following argument, which is adapted from Castaneda 1981.* (a) Alchourron is obligated to do the following: If Bulygin sends him the draft of their latest joint paper, revise it. (b) Bulygin has sent Alchourron the draft of their latest joint paper. Therefore,

(1)

(c) Alchourron is obligated to revise the draft. Castaneda points out that this straightforward reasoning cannot be accommodated by most existing deontic calculi. To see why, let A stand for "Alchourron revises the draft" and B for "Bulygin sends Alchourron the draft." Then we can represent the argument (1) symbolically:

Here, Oblg: stands for the obligation operator. The proposition which would allow detachment of the obligation Oblg:A in (2), does not belong to the standard system of deontic logic. Consequently, the argument is invalid.1 *Paul Bartha is the author of this chapter. Publishers, it is based on Bartha 1993. 1 F011esdal and Hilpinen 1971 sets out the logic," known as KD, in which Oblg:(B>A) D is usually rejected in any system which is based Hintikka 1971.

With the kind permission of Baltzer Science axioms for the "standard system of deontic (BD Oblg:A) does not hold. This proposition on "deontically perfect world" semantics; see

295

296

Applications of the deliberative stit

The argument becomes valid if we replace Oblg:(BDA) by BD Oblg:A in (2)(a). No corresponding change is required in English. The English version of the argument seems acceptable as stated. This suggests that English is less sensitive than standard deontic logics to the difference between the two forms Oblg:(BD A) and BD Oblg:A. These two forms will be referred to as OD-statements and D 0-statements respectively. Castaneda has developed one approach that brings the reasoning of deontic logic closer to that of English. He suggested in Castaneda 1981 that the two forms of obligation are equivalent under one assumption. Specifically, he proposed that the equivalence

is valid whenever B is a "circumstance" or "condition" and A is "an action considered deontically as a deontic focus" (p. 46). He develops a calculus which takes the circumstance/action-as-focus distinction as primitive. This chapter takes a different approach, starting from the equally important distinction between agentive and non-agentive sentences. Earlier chapters formalize this distinction by providing two varieties of semantics for the agentive construction "a sees to it that A" (written as [a stit: A]]: the achievement stit, §2A.2, and the deliberative stit, §2A.3. In this chapter, a simple semantics of obligation is developed as an extension of the logic of the deliberative stit, written [a dstit: A]. The basic idea is inspired by the reduction of deontic logic to alethic modal logic in Anderson 1956. After the technical preliminaries of §11A, in §11B we explain our adaptation of Anderson's semantics of obligation. The resulting concept is compared to other systems of deontic logic. A partial completeness result is described in §11C. The semantical system is then used to analyze conditional obligation. In particular, as §11D and §11E show, it provides a precise way to define a "circumstance" such that the argument (2) becomes valid under the assumption that B is a circumstance. In the remainder of the chapter, the semantics is used to shed light on two paradoxes of deontic logic. The point is not to give a final solution to any paradox or problem. Rather, we hope to show that many of the problems of deontic logic are essentially problems about agency rather than obligation. We also hope that readers will see that stit theory can be a useful tool in thinking about such problems, a theme that we further develop in chapter 12.

11A

Technical preliminaries

For this chapter we limit our target grammar as follows. Sentences in our language are constructed from propositional variables by truth-functional connectives ~ and &, as well as modal operators Universally: and Sett:, deontic operators Frbn: and Perm:, and the agentive operator, [a dstit: ]. There will also be tense operators Will: and Was:. As usual, V, D, =, T and _L are introduced as abbreviations, and we will also introduce by abbreviation the Chellas

11. Conditional obligation, deontic paradoxes, and stit

297

agentive operator [a cstit: A]. We use A, B, and so on to range over sentences. We recall that the semantics of [a dstit: A], where a is an agent and A a sentence, is based on the fundamental notions moment, history, agent, choice set, and possible choice, as reflected in a BT + AC model, m, which is a pair consisting of a BT + AC structure © = (Tree, \ Oblg:[a dstit: ~[a dstit: Pax]]: a is obligated to see to it that a does not park on highway x. Sett:(~[a dstit: ~[a dstit: Pax]] D Sa): It is settled true that if a does not see to it that a does not park on highway x, then Sa. That's a lot of words. One can see further into the proposal by a combination of cases and pictures. Take a particular concrete moment, TO, of concern to a and to the rules. There are three major cases, the first two of which are throwaways. No-good-choice case. Since §11B.1, in contrast to Anderson, has put no constraints on the sanction, there is the case in which Sa is settled true at m. In the no-good-choice case all is obligatory for a, but also all is forbidden. There is no available choice for a that avoids the sanction. Nothing is permitted, where, as in §3A, Perm:[a dstit: Q] ~Frbn:[a dstit: Q]. Figure 12.1 provides a picture. What to make of the different decisions, Anderson's requiring that the sanction not be universal and ours making no such requirement? We think that both decisions are plausible, and that the point of view of Marcus 1980 explains why. Anderson's sanction applies to entire worlds or perhaps histories. It would be strange indeed if every world or every history were deemed unacceptable. In this case there would clearly be nothing left for deontic logic to worry over. As Marcus puts what we take to be the same point, if a set of rules is inconsistent in this strong sense, then it "provides no guide to action under any circumstances" (p. 129). This makes reasonable Anderson's postulate that the sanction is not inevitable. On the other hand, when we mark as "sanctioned for a" every history through a particular moment, that sad situation is an extremely local affair. It represents only that in that particular circumstance, there is nothing acceptable to do.

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Applications of the deliberative stit

Figure 12.1: No good choices for a at m

Figure 12.2: Two proper no-can-do cases

This is what Marcus urges to be quite another thing. To deny that such a circumstance could obtain is, as she says, to deny something real, namely, that a moral dilemma can arise in a particular unfortunate circumstance. So, we conclude, the two choices are both in accord with Marcus's account, and they are both right. When all histories are at issue, it is right to assume with Anderson that the sanction is avoidable, but when we consider only the histories that are possible historical continuations of a particular moment, then it is right to allow with Marcus that in some circumstances there may be no way to avoid the sanction. No-can-do case. In the other throwaway case, at m, a cannot see to it that Q: It is settled true that a does not. Nothing a can choose at m counts as choosing to see to it that Q. No such choice is in fact open to a at m. There are two subcases. In one of them the previously discussed no-good-choice case obtains, and there is nothing more to say: All is obligatory and all is forbidden for a, regardless of what the agent can do. In the other subcase, which is new, Sa is false at m/h for at least one history h through m. We will call such a case a proper no-can-do case. Figure 12.2 conjures up two of them.

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Figure 12.3: The normal case

You can easily calculate that in a proper no-can-do case, Definition 12-2 neither obligates nor forbids the agent to see to it that Q, which sounds right when the agent no can do. Of course it follows that in the proper no-can-do case the agent is permitted to see to it that Q, and also permitted to refrain from seeing to it that Q, which sounds peculiar given that the agent is powerless with respect to Q. But it sounds peculiar, we think, only because in ordinary English we confuse permission and power. (The effort to straighten us out on this goes back at least to Hohfeld 1919; see especially Makinson 1986.) For example, we often use the same modal verb, "may," for both. When we reflect, however, we easily see that in fact there are numerous situations in which an agent is permitted or authorized to do what in fact is not in their power to do. The agent a might be permitted to park at any meter in the entire state of Connecticut, even when a is enjoying Oregon. The proper no-can-do case highlights this. The normal case. In the central case of interest, the sanction Sa is false on at least one history through m, and there is at least one choice on which a sees to it that Q. We will call this the normal case, even though it certainly does not happen with much frequency. In the normal case, a can see to it that Q, and in the normal case, the sanction is not inevitable; see Figure 12.3. It is a calculation of dstit theory (as opposed to the achievement stit) that in the normal case, though not at all in general, "seeing to it that" and "refraining from seeing to it that" are contradictory. (The sanction is irrelevant to this calculation; one needs only can-do.)

12E

Generalized prohibitions

Let us pass on to the first of the two generalities ingredient in (1). It is not just one highway on which a is forbidden to park. In fact,

326

Applications of the deliberative stit Each highway is one on which a is forbidden to park. (Or "each highway within a given jurisdiction." We shall suppress this subtlety.)

(7)

In other words, we might say, a is forbidden to park on a highway. Or, put more ponderously in order to nail down a narrower scope for the quantifier: It is forbidden that a park on a highway.

(8)

So, obviously, to express that parking is forbidden on highways we need a quantifier. But which quantifier, and where should it go? The universally quantified prohibition (7) and the prohibition of an existential (8) sound more or less equivalent, but certainly the last hundred years of logic suggest that it is not very plausible that they should be. It will be instructive to observe a proposed train of reasoning that would show them to be so. This train contains some routine transitions and some interesting ones. Here is what will strike you: None of the interesting transitions concerns the deontic modalities themselves. All are concerned with either the background theory of branching time, or the theory of agency. In the end we shall find out that (7) and (8) are not equivalent, though nearly so. And coming to see that sort of thing is part of what logic is for. Marcus, in the shadow of some intolerant and irrational decades that had foolishly put modal considerations on the defensive, said that "establishing the foundations of mathematics is not the only purpose of logic" (Marcus 1960, p. 58). This should count as a tiny example of what she had in mind. Let us begin with the henscratches for (7) and (8), respectively. Symbols for (7): VxFrbn:[a dstit: Pax].

(9)

Symbols for (8): Frbn:[a dstit: 3xPax].

(10)

It would be all right to shorten [a dstit: Pax] in (9) to Pax, just because, according to Postulate 12-1, the latter is agentive, that is, [a dstit: Pax] and Pax are equivalent. But it is probably better to use the longer "normal form" in order to remind us that the complement of Frbn: must always be a dstit or something equivalent. This is in accord with the restricted complement thesis, Thesis 5. Let us add here that by the restricted complement thesis a third candidate, Frbn:Ex[a dstit: Pax], is not on the face of it grammatical. We should suspect that this third proposed complement of Frbn: might, in the language of Marcus 1966, describe a mere state of affairs that could not meaningfully be on a's list of prohibitions. And in fact the coming analysis of the 3x/dstit transition confirms this suspicion. Here is a list of plausible equivalences, which we call the "Main Calculation," that would lead us between (9) and (10). We have indicated on the right how each entry is tied to the one above it. In gently moving quantifiers from the outside to the inside we will run into a number of particular transitions: "QTF" means "quantifier-and-truth-function equivalence." Other transitions

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are applications of previous postulates, as indicated. And the three key transitions "Vx/Sett," "Vx/dstit," and "Ex/dstit" are based on hypothesized quantifier/modality relationships. These three we explicitly discuss, with an eye to either confirmatopn or disconfirmation.

Main Calculation

We take up the undiscussed quantifier/modality transitions Vx/Sett, Vx/dstit, and Ex/dstit in turn. Let us first underline that they have nothing to do with deontic logic per se. They are all either about branching time or else about pure agency. They are about the world and our doings, independently of whether we are getting ready to make some rules or set some standards.

12E.1

Barcan formula for Sett:

The first transition does not even concern agency, much less obligation. It is a matter for the general theory of branching time, or perhaps the general theory of highways. What are we to say of the principle Vx/Sett that permutes the Sett: modality with a universal quantifier over highways? In symbols: 12-3 POSTULATE. (Barcan formula for Sett:) VxSett:Qx Sett:VxQx. Not to put too fine a point on it, we think Postulate 12-3 is good. Here is why. Sett: at a moment means, close enough, "true in all histories" through that moment. You can therefore plainly see that Sett: has the intent of an S5 necessity. That means that the permutation of the universal quantifier with Sett: is precisely the Barcan formula in one of its guises. (From Barcan 1946. So named in Prior 1957.) The Barcan formula is not only time honored and of

328

Applications of the deliberative stit

exceedingly noble parentage, but of tremendous logical utility. Proof systems without the Barcan formula are awkward and difficult for us to use. Adding the Barcan formula simplifies and thereby increases our logical acumen. It makes us smarter. That's the benefit. But, you may ask, at what cost? If we use the Barcan formula we must suppose that the range of the quantifier does not vary with the history; so much is clear from the analysis of Kripke 1963. Here is a general point, and we mean to be echoing one made by Marcus, for example in Marcus 1972: Although English means what it means independently of what we say, our technical quantifiers mean exactly what we say they mean. Their meaning is up to us, and it is, as Carnap emphasized, a practical matter. In applying Carnap's advice, it makes a difference whether the quantifiers we are considering are for general use, or for a case at hand. For general use it seems clearly best to simplify our logical life by interpreting our technical quantifiers—the ones with which we calculate—so that their ranges do not depend on the particular historical continuation.2 As Marcus argues in Marcus 1961, for example, we shall still be able to find appropriate and indeed illuminating symbolic formulations for every one of those English sentences that might have led us to abandon the Barcan formula, had we been so weak on our philosophical pins as to be bowled over by derisive counterexamples hurled down the alley in an effort to obtain a crowd-pleasing strike. Need we add that such presumed counterexamples will be found on inspection to be made from a nonmetallic composite material, and to contain a hole the size of your thumb? Proceeding to the local case at hand, observe that the particular example involves quantification over highways. That makes it sound as if in this locality the Barcan formula should fail, since surely what highways there will be depends on what happens and thus on particular historical continuations. Well, yes, the "since" clause is true, but the argument is not. The argument derives from the paradigm of Kripke or Lewis total worlds, each with its worldwide quota of highways. That paradigm, however, is not wanted here. We are looking at a particular momentary event of forbidden parking. Whether or not something is a highway at a particular moment does not depend on historical continuations of that moment. The domain of highways at a moment (as we might say) is indeed independent of what historically happens next. It is, in the phrase of Castaneda, a "circumstance" of the parking (see the beginning of chapter 11). Thus we should accept the permutation labeled "Vx/Sett" in general, and therefore in the particular case. 2 This still leaves numerous options, of which the following are salient: (i) Marcus quantifiers ranging over actualia; (ii) quantifiers ranging over possibilia; (iii) Bressan "individual concept" quantifiers with intensional predication; and (iv) substitutional quantifiers. Arguments that some alternative is better or worse than one of these have tended to be based on mockery involving the double modal, "How could you possibly say such a thing?" That's amusing, but also sad. For many practical purposes of serious formalization, it is appropriate to use the scandalously neglected Bressan individual concept semantics with intensional predication, of which we say just a little more in §12F.

12. Marcus and the problem of nested deontic modalities

12E.2

329

Universal quantifier over dstit

The next transition we discuss, namely Vx/dstit, is also Barcan-like:

Here, however, appearances deceive: Dstit is not really a straightforward S5like necessity modality. As you can see from §8G.l, the semantics for [a dstit: Q] involves not only a "positive condition" saying that the (perhaps vacuous) choice that a makes necessitates Q, but also a "negative condition" saying that "it might have been otherwise than Q" (from which it follows that the choice was not after all vacuous). As indicated in §11A, we can separate the positive from the negative. For the positive we can adapt the Chellas stit, [a cstit: Q], defined in §8G.2, just as we did in §11A. For the negative we can take from §8F.4 the standard branching-time dual of Sett:, namely, Poss:, so Poss:Q m1 E U s ( m0 )]• Admm(s) = {m: s admits m}. • s excludes a history or a moment if it does not admit it. The ideas of exclusion and admission are central to the theory of strategies. As for exclusion, one may start out along an excluded history, but if one chooses in accord with s, then one cannot forever remain on an excluded history. Eventually one reaches a moment at which the strategy goes one way and the history goes another. Nor can one ever reach an excluded moment if one follows a strategy. Admission is in one sense a weak notion; for example, any history that contains no member of Dom(s) is vacuously admitted. Apart from this trace of vacuity, which we remove on a case-by-case basis when we apply the notion—for example, in the next definition—an admitted history is one that might happen given that the strategy is faithfully followed. An admitted moment is one that might be reached provided that the strategy is followed without exception. Admissibility represents what is possible, given that the strategy is followed. It is therefore natural to define what a strategy "guarantees" in terms of admissibility: 13-5 DEFINITION. (Guarantee)

• s guarantees HO iff • s really guarantees HQ iff s guarantees HQ, and furthermor

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Think of H0 as a proposition expressed by a moment-independent sentence Q, that is, as the set of all histories in which such a Q is true. Then for SQ to guarantee H0 is for s to guarantee Q in the sense that Q is true in all those histories not forbidden by s. The reference to H(D 0 m(s)) insures that we don't have to worry, however, about histories that branch off who-knows-where long before the arrival of the domain of s. The definition of "real guarantee" adds a "negative condition" in accord with the general tenor of stit theory, §8G. For a strategy to be said to really guarantee H0, its choices must do at least some work of excluding non-H0 histories. The real sense of "guarantee" is exactly what we want for our main result, Corollary 13-29, to the effect that a certain strategy "guarantees" inaction. The following is obvious, but worth recording. 13-6 FACT. (Admissibility and extension) Extending or strengthening a strategy can only reduce admissibility, never increase it. Contrapositively, extending or strengthening can increase exclusion, but never reduce it. And what is guaranteed persists under extension or strengthening. Among strategies there is a sharp difference between those that do and those that do not tell you what to do after you have failed to follow their earlier advice. We call the former "secondary" and the latter "primary." (Our earlier work on strategies used the pair of words "redundant" and "irredundant." Horty 2001 uses "lean" where we say "primary." Although the ideas are stable, the best choice of vocabulary is unsettled.) The thought is that advice given by a strategy after you have already violated the strategy may be useful, but it is surely "secondary" to the primary portion of the strategy. So we call the strategy itself "secondary" or "primary" depending on whether it does or does not contain any secondary advice. 13-7 DEFINITION. (Secondary versus primary) s is secondary iff some moment that it excludes is anyhow in Dom(s); and s is otherwise primary. The strategies (for games in extended form) of von Neumann and Morgenstern 1944 are primary, or mostly so. A typical game theoretical strategy does not tell you what to do if you do not follow the strategy. Suppose a strategy for chess begins with advising king's pawn to KP4 as first move. That very strategy would typically not tell you what to do on your second move if instead you moved your king's rook pawn on your first move. Why should it? Such advice would be redundant (it would be secondary advice). One finds a contrast in the "deontic kinematics" of Thomason 1984, p. 155, which we discuss in chapter 14. In that scheme, obligation is laid on at every moment. If we think of these obligations as together making up a strategy (Thomason speaks of plans), the strategy would be secondary. And for good reason; as Thomason notes, secondary strategies enable consideration of "reparational" obligations, telling us what to do after we have done wrong. Secondariness can be important.

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Secondariness of a strategy should be distinguished from a form of strategy that at a certain moment gives you a "second-best" choice besides its first choice. One might in English describe both a secondary strategy and a strategy that gives a second-best choice in the same language: Each tells you what to do "if" you don't do what it primarily advises. But the cases are really different. If a strategy offers a second-best choice, then the second-best must be labeled so: "PQ4 is best, but PK4 is second best." See Horty 2001 for ideas that involve such grading of the choices on which stits are based. In this chapter, however, we do not deal at all with strategic concepts that reach outside the BT +1 + AC structure, (Tree, ^, Instant, Agent, Choice), in order to label some choices as second-best: The "second-best" of a secondary strategy needs no extra labeling. What makes it second-best is already definable. Namely (and roughly), it is the choice (and the only choice) that the strategy advises at a moment that is excluded by an earlier choice dictated by the strategy. It is the exclusion of a moment that renders advice at that moment automatically secondary. Note, incidentally, that when there are no busy choosers, we can always count our way down from a forbidden moment through a finite series of "fresh mistakes" to the main line of the strategy. This gives an already-definable account of "n-ary strategy" as well as "secondary strategy." On the other hand, if there are busy choosers, it is not so clear that such gradations make sense. 13C.2

Simple strategies

We shall be dealing largely with strategies that are both strict and primary, partly because they are in some respects the simplest. In fact we shall cause these two properties to contribute to the upcoming definition of "simplicity." There is, however, another ingredient. To introduce it, we need to think ahead a little ways to the completeness of strategies. When we turn to "completeness," we need to have in mind some population of moments M that is typically larger than the domain of the strategy but much smaller than the entire universe. These are the moments that we care about. Typically they will not extend backward all the way to the Bang, nor forward past the heat death of the sun.2 When M is given, we may call it a "field" for the strategy. Always a field M will include the domain of a strategy, that is, the set of moments at which the strategy gives advice. We use the phrase "in M" to carry this information: 13-8 DEFINITION, (strategy in M; field for s) • s is a strategy for a in M iff s is a strategy for a such that Dom(s) C M. • When s is a strategy for a in M, we call M a field for s. 2

Less pictorially, a "field" M for a strategy might have the property that in each history h with which M has nonempty intersection, there are upper and lower bounds for h n M. Or it might be that all of the nonempty hdM have a common past or even contain a common lower bound. Or it might be that each nonempty h n M is a maximal chain in M. But even though typically interesting strategies will be bounded or even have one of the stronger properties, our austere refusal to impose such conditions seems to clarify the concepts and results that this chapter considers.

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It could be that a strategy gives advice for every moment in M, or even for every moment in all of Tree. Such a strategy is said to be "total": 13-9 DEFINITION. (Total strategy) A strategy s is total [in M\ iff Dom(s) = Tree [= M\. We shall not be interested in total strategies until chapter 14. We are instead presently interested in notions of "completeness," which are altogether different from totality. A strategy complete in a set of moments M will tell us what to do at appropriate moments within a field M—but by no means all. For instance, a "complete" strategy need not tell us what to do at moments in the field that the strategy itself forbids. That is precisely why a field for a strategy must be distinguished from its domain. Nor is a field uniquely determined by the domain; that is why we say "field for s" instead of "field of s." EXAMPLE. (Domain versus field) Picture Tree as representing a game of chess in extended form (von Neumann), that is, a tree, rooted in some concrete occasion, of all plays in accord with the rules of chess. That would be an appropriate "field" for a strategy. A particular strategy itself would be represented by a small subtree of Tree; for example, a tree that starts with "P to Q4." This strategy can be altogether "complete" in Tree, while giving no advice at all to the player who ignores it in favor of starting with KKt to KB3. As a small step toward completeness, we consider backward closure. Note that we confine ourselves to M partly so as not to have to go back to the beginning of the universe. 13-10 DEFINITION. (Backward closure) Suppose that s is a strategy for a in M. s is backward closed in M iff Vmo,mi[(mi € Dom(s) & mo < mi & mo 6 M) —> m0 G Dom(s)}. A "conditional" strategy might well not be backward closed: "If you reach Station A, get off the subway; but for earlier stations, there is no advice." So strategies that are not backward closed are important. But if a strategy is not backward closed, then that seems enough to give it a kind of not-yet-available flavor. Such a strategy gives us advice at a certain moment without giving us any advice whatsoever as to how to get there. It would in that respect be similar to an end-game strategy for chess as given in a book on end games, provided we chose an odd field for it. Namely, suppose we took as field not just the subtree where the book starts giving us advice. Suppose instead we oddly took as field the entire tree for chess that starts with the first move. Then that strategy would not be backward closed in that field. If we want to confine ourselves to the simple case in which a strategy is available from the very beginning of its field, we must think about backward closed strategies. This notion is therefore the third and final part of the concept of simplicity: 13-11 DEFINITION. (Simplicity) s is simple for a in M iff s is a strategy for a in M that is strict for a, primary, and backward closed in M.

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So a strategy that is not simple might fail of simplicity in one of three ways, (i) It might be loose (sometimes giving disjunctive advice), (ii) It might be secondary (sometimes giving advice at moments it forbids). (Hi) It might not be backward closed (some parts of its domain might be isolated high up in M). Simple strategies are not like that.

13C.3

Pre-simplicity

Permit us to say something parenthetical about strategies that are not simple in virtue of being not backward closed in M—parenthetical in the sense that later sections do not build on this one. Suppose such a strategy is in fact strict and primary, and fix the set M of relevant moments. The strategy might have a simple extension in M, or it might not. There is only one case in which it does not: There are two moments TOI and m^ in its domain, and they have a lowerbounding moment m0 in M (but not in Dom(s)) such that every choice for a atTOOexcludes one of m1 and m2. Were this case to occur, we should not know how to specify s(mo) while retaining both strictness (at TOO) and primariness (with respect to each of m1 and m 2 ). This observation points to an alternative concept of "simplicity" for use when we wish to consider strategies that are not backward closed. To remind us that the alternative is helpful only when we are looking forward to simplicity, let us call it "pre-simplicity." 13-12 DEFINITION. (Pre-simplicity) s is pre-simple for a in M iff s is a strategy for a in M that is strict for a, primary, and such that each two members of its domain are inseparable for a in M. EXAMPLE. (Pre-simplicity) Let s be a strategy for you that might be described in this way: "If you reach Station A or Station B, get off the subway; but for the earlier moments at which you have a choice about reaching Station A or Station B, there is no advice." You may not care about completeness at all. If you do, you may not care about it with respect to a field M for s that includes those earlier moments. In these cases you will not care about pre-simplicity in M (though you may still care about it with respect to a smaller field). Suppose, however, that a field M for s includes at least one of these earlier moments, say TOO, at which you can choose between reaching Station A and Station B; and suppose that you care about strengthenings of s that are complete in M. You may or may not be interested in primariness. If you are not, you can think about extending the strategy to mo in any way you like. You could then think of some or all of the "if you reach Station A or Station B" advice as "reparational." You could think of it as "redundantly" advising you what to do after you have fallen off the strategy. In this case—see chapter 14—pre-simplicity would be of no interest. Suppose, however, that you are interested in primariness. Even then it makes sense to extend the strategy toTOO,provided the advice there is not strict but loose: Head either for Station A or for Station B, whichever you choose. That would be a primary (though loose) extension. Nothing wrong with that. But if you are interested in a strategy only if it can be backward closed to a strict and primary extension, you need to think about pre-simplicity.

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Here are key facts about pre-simplicity. They are put here for help in keeping our bearings, and are not otherwise used in this chapter. 13-13 FACT. (Pre-simplicity) i. s is pre-simple for a in M iff s has a simple extension for a in M. ii. s is pre-simple for a in M iff s is a strategy for a in M, and

iii. In application to a strategy that is backward closed in M, there is no difference between simplicity and pre-simplicity. The smallest simple extension as guaranteed by (i) might be called the "backward closure" of s. But we won't need the idea, and in any event that phrase could well be used in other ways when we are interested in loose or inconsistent or separable strategies. Part ( i i ) gives an equivalent account of pre-simplicity so as to exhibit that pre-simplicity is a unified idea, not a mere conjunction: Strictness, primariness, and inseparability are all part of the same package. And (Hi) tells us that when we know a strategy is backward closed in M, we just don't have to worry about simplicity versus pre-simplicity. In any event, we shall nearly always be dealing with simple strategies.

PROOF. Ad Fact 13-13 (i). Suppose SQ is pre-simple for a in M. Define si so that Dom(si) = Dom(s0)U{m0: mo £ M & 3mi[m0 < mi & mi 6 Dom(s0)}}; and for m define Evidently si is backward closed in M, and si is primary if SQ is primary. Also pre-simplicity of S0 guarantees that s1 is strict. So s1 is a simple extension of s0. Conversely, if s0 is not pre-simple, then if it is either not strict or secondary, no extension can remove that defect. And if its failure of pre-simplicity comes from having two members m1 and m2 of its domain that are not choice equivalent at some moment mo in M that lies in the past of each of m1 and m2, then there is no way to extend s0 backward to m0 without violating either strictness (at mo) or primariness: If Si(mo) is a single member of Choicean0, then either m1 or m2 will be excluded, causing s1 to be secondary. Ad Fact 13-13 (ii), left-to-right. Argue by contraposition. Suppose that s is a strategy for a in M, but that the right-hand side is false: mi, m2 G Dom(s), hi S s (mi), hi e s (mi), m0 e M, m0 ^ mi, mo ^ m2, hi _L^ 0 hi. Argue by cases. If mo = mi = mi, then there is nonstrictness at mo since non-choice equivalent histories belong to s(m 0 ). If mo = mi and mo < mi, then mo excludes m2, witnessing that s is secondary. Similarly if m0 < mi and mo = m2. And if m0 < mi and m0 <mi, then there is a violation of inseparability. So in either case s is not pre-simple. Ad Fact 13-13 (ii), right-to-left. Argue by contraposition. If s is not presimple, it is either not strict, or secondary, or separable. If it is not strict at mo then we can find a counterexample to the right-hand side with mo = mi

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= m-2- If it is secondary in virtue of the exclusion of m\ at mo, we can find a counterexample to the right-hand side with mo < m\ andTOO= m^. And if s fails inseparability in virtue of mi,m,2 E Dom(s) being separable at mo & M, then we can find a counterexample to the right-hand side with mo < mi and m0 < TO2. Ad Fact 13-13 (Hi). This is a way of saying that simplicity implies inseparability. The argument is that a witness to separability that by backward closure lay in the domain of s would also witness either nonstrictness or secondariness.

13C.4

Ideas of completeness

We can now define "completeness." For convenience, we shall do this under the presupposition that we are dealing only with simple strategies; otherwise the concepts might need refining.3 13-14 DEFINITION. (Completeness) Suppose that s is a simple strategy for a in M. • s is complete along h in M iff M(~}h C Dom(s). • s completely admits h in M iff s admits h and is complete along h in M. • s is H-complete in M iff s is complete in M along every history in H that s admits. • s is complete in M iff s is complete in M along every history that s admits. • s is simply complete for a in M iff s is simple for a in M and s is complete in M. We offer the fourth of these five concepts as a conceptual analysis of the essential idea of strategic completeness. In contrast, the first three concepts are introduced as technically useful, while the last gives us a rhetorical variant of "simple and complete." (Strictly speaking, "simply" here is overdefined, but we nevertheless try to be consistent about including it.) Completeness is an easy idea: A complete strategy in M provides advice at each moment in the field M at which you can arrive without violating the dictates of the strategy itself. No matter which advised choice you make, no matter the outcomes of decisions by other agents or by nature, as long as you are within the field M, a complete strategy always tells you what to do. Not even a complete strategy in M, however, either gives or pretends to give any advice outside the confines of the specified field M. A strategy that is incomplete in a field M gives you some 3 For instance, the deontic trees defined in Definition 14-1 are "complete" in a sense almost unrelated to Definition 13-14: Deontic trees obligate at each and every moment. Strategies derived from that variety of deontic kinematics are not part of our present topic because they are not simple.

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advice, and then sometimes leaves you adviceless even when you follow it, even when you are still in M. Complete strategies are analogous to the complete theories known to modeltheoretic logic in the following respect: Although interesting specimens of either are few and far between, complete strategies, like complete theories, are important objects of study because of the limiting role they play. One can see that—and be a little surprised by the fact that—for strategies in M, completeness is converse to primariness: 13-15 FACT. (Completeness and primariness) Suppose that s is a simple strategy for a in M. Then the following give good accounts of completeness in M and primariness. • Completeness: (Admm(s)nM) • Primariness: Dom(s) C

C Dom(s).

(Admm(s)r\M).

Let us record that extending a strategy can never make it less complete than it was. 13-16 FACT. (Persistence of completeness) If s0 is a simple strategy for a in M, then if SQ has one of the following properties, so does any simple strategy si for a in M that extends SQ: (i) is complete along h in M; (ii) completely admits h in M; (Hi) is //-complete in M; (iv) is complete in M; (v) is simply complete for a in M. PROOF. Because extensions only add to the domain of a strategy, completeness evidently begets completeness. We need, however, to check the "admission" part of ( i i ) . This part says that if SQ and s\ are simple strategies for a in M such that si extends s0, and if s0 completely admits h, then sj admits h. So let mo € Dom(si)C\h; we need to show that h € SI(TOO). Because si is a strategy in M, mo € Mfl/i. So, since SQ is complete along h in M, mo 6 Dom(so), and so mo € (Dom(so)C\h). Therefore, since SQ admits h, h € So( m o)- But since mo £ (Dom(sQ)(~}Dom(si)), the definition of extension says that SI(TOO) = so( m o)So /i 6 si(mo), as required. The next definition and fact articulate the technically useful idea of extending a strategy by an entire admitted history (more accurately, by that portion of it in M). 13-17 DEFINITION. (Extension along h) Let s0 be a strategy for a in M. Define "si is the extension of S0 along h for a in ' as follows:

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The purpose of this construction is the following. In general admission of a history is not preserved under extension. We can get around this by a wholesale extension that always points along h, and that totally uses up the portion of h in M. Then there will be no leftover moments at which a further extension could exclude h. The following small fact is thus critical. 13-18 FACT. (Extension along h) If (i) s0 is a simple strategy for a in M and (M) s0 admits h, then, where sj is the extension of s0 along h for a in M, • S1 is itself a simple strategy for a in M; • s1icompletely admits h for a in M; and • any extension of S1 in M continues to admit h. PROOF. If s1 is the extension of s0 along h for a in M in the sense of Definition 13-17, then s1 is on the face of it strict and backward closed for a in M. Suppose that s1 were secondary, with mo, m1 S Dom(s1) and mo < m1 but m1 If m1 did not belong to h it would belong to 0), which would make SQ secondary, contrary to hypothesis. So mi € h, and since histories are closed backward (by the postulate of no backward branching, Post. 3), mo h. But then, since mi ^ U s i( m o)i h £ si(mo). Definition 13-17 then implies that mo G Dom(so) and that «o( m o) = SI(TTIQ). So after all SQ does not admit h, contrary to hypothesis. So s\ is simple for a in M. The "complete" part of "completely admits" is on the surface of Definition 13-17, and the "admits" part is not far below. Were h excluded by si, one would have mo with h £ si(mo). But a glance at Definition 13-17 suffices for this case implying that mo £ Dom(so), whence s0 would itself exclude h, contrary to hypothesis. That extensions of Si continue to admit h comes from Fact 13-16. We shall also need some near-the-surface facts about joins of chains of strategies in the sense of the "weaker than" relation of Fact 13-3. 13-19 FACT. (Joins of chains) Let S be a nonempty chain of strategies for a in M. Then the join sg (Fact 13-3) is itself a strategy for a in M. Furthermore, if every member of S has one of the following properties, then so does s-^: (i) strict for a, (ii) primary, (m) backward closed in M, (iv) simple for a in M, (v] admits h, and (m) completely admits h. PROOF. Use Zorn's lemma. A violation of any of (i)-(vi) for SE would be finite, and hence would be a violation of the same property for some member of This concludes our introduction of the most basic strategy concepts. We go on to introduce "favoring," an important but somewhat less obvious idea.

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13D

Favoring

Let us return to the problem mentioned at the beginning of the chapter: As promised in §91, find a strategy the following of which will guarantee not seeing to it that Q. The core of an intuitive solution is easy: Shoot for ~Q whenever given a choice. To make clear sense out of this suggestion, we introduce the concept of a strategy "favoring" a set of histories. The root idea comes from Hamblin 1987, p. 157, where the topic is carrying out (or "extensionally" satisfying) an imperative: ... the addressee of an imperative would be expected, at least, to act in such a way as to keep extensional satisfaction within the bounds of possibility. This means that he must not do any deed d that would infringe—that is, that would ensure dissatisfaction of—the imperative. The next definition aims to capture exactly this idea. In reading it, think of HO as the set of histories on which Hamblin's iniDerative is satisfied. 13-20 DEFINITION. (Favoring)

Let s be a simple strategy for a in M.

• s favors H0 iff • s completely favors HQ in M iff s favors HQ and is flo-complete in M. If s favors HO, then whenever any moment in the domain of s enables a choice that keeps HQ as a live possibility, then s advises such a choice. Furthermore— and critically—s does so by means of a history such that the same is true of any further moments in the domain of s that lie on that history. Let us relate this critical conceptual point to the notation. At a key point in the definiens of "s favors H" we used "Admh(s)," which refers to the set of histories that can be reached if the strategy is followed everywhere. If instead we had used "s(m)," we would have represented a weaker notion of "favoring." For example, consult Figure 14.1. Let HQ be the set of all the histories, {/iw, hi, h^, ...}. Then the depicted strategy, s, would favor H0 in the weaker sense, since each sm is nonempty. It would not, however, favor HQ in the stronger sense, since, as declared by Fact 14-9, Admh(s) = 0. The property of favoring need not persist under extension; but the property of completely favoring does: 13-21 FACT. (Persistence of completely favoring) Any simple extension si of a strategy s0 for Q in M that completely favors HQ also completely favors HQ. PROOF. Suppose h € HO and si admits h. Then s0 admits h by Fact 13-6, so SQ is complete along h in M by hypothesis, so si is complete along h in M by Fact 13-16. So si is Ho-complete. For the "favoring" part, remark that HoH Admh(so) C Hor\Admh(si) by Fact 13-16. Now suppose mi € Dom(s\) and

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and By the remark, r\Admh(so) = 0, so that to avoid violating the hypothesis that SQ completely favors HQ, mi ^ Dom(s0). Choose h 6 H(mi)r\H0; evidently /i ^ ^4dmft(s 0 ). Let the excluding occur at m0: m0 £ Dom(so)nh, h £ s0(m0). Since m0, TOI both belong to h, they are comparable; but mi ^ mo would put mi in Dom(so) by backward closure, so that mo < mj. So SQ excludes mi according to Definition 13-4, so that si excludes mi by Fact 13-6. So the supposal that mi € Dom(si) leads to the secondariness of si, contrary to hypothesis. We have developed just enough of the theory of strategies to prove a useful lemma: A favoring strategy can always be extended to a completely favoring strategy. This lemma will have important corollaries. 13-22 LEMMA. (Favoring extension lemma) Suppose s is simple for a in M, and favors H0. Then s has an extension s' that is simple for a in M and that completely favors HO. If furthermore m0 is such that m0 €. M and H^^CiHo then s' can be chosen to be available at m0. PROOF. Well-order H0 with the ordinals less than a limit ordinal A'. (We omit adapting the argument for finite H0.) For 7 < A', let /i7 be the history in H0 that is indexed by ordinal 7. Furthermore, when the "furthermore" clause of the lemma is wanted, in the special case 7 = 0, as the "furthermore" hypothesis we let h-y G H(mQ). In short, every history in HO admitted by s\> was "completed" in its turn. Furthermore, suppose the "furthermore" clause is operative. We know by the "furthermore" hypothesis that the privileged moment mo belongs to ho(~\M, where ho is the history in HQ that is indexed by the ordinal 0. Since ho is admitted by SQ = s, we know by definition that «i is complete along /i0 in M. This guarantees that mo S Dom(si). Since s\ extends si, it is sure that mo € Dom(s\). That is, the strategy s\' is available at mo, as desired. 13-23 COROLLARY. (Complete and favoring extensions) • If s0 is a simple strategy for a in M then s0 can be extended to a simply complete strategy s\ for a in M. • Suppose m G M and H^m^r\H0 ^ 0. Then there is available at m a simply complete strategy for a in M that favors HOPROOF. For the first part, supposing SQ to be a simple strategy for a in M, apply Lemma 13-22, choosing H0 as Admh(so)- Evidently SQ vacuously satisfies the requirement that it favor Admh(s0). Let si be the promised extension of SQ that, besides being simple for a in M, completely favors Admh(so)- We need to

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show that Si is complete. Suppose si admits h. Then h € Admh(so) by Fact 136, so by complete favoring, si is complete along h, as required. For the second part, suppose m £ M and H(m)f~}H0 ^ 0. Start with the empty strategy, which vacuously is a simple strategy for a in M that favors HQ. This gives us the main hypothesis of Lemma 13-22, and we have also the "furthermore" hypothesis since the empty strategy admits all histories. So there is a strategy So available at m that is simple for a in M and completely favors HQ. This strategy, however, may not be complete. No matter, by the already-established first part of this corollary, SQ itself has an extension si that is simply complete for a in M. Clearly si remains available at m. Lastly, by Fact 13-21, si also completely favors HQ.

13E

Application to finding a strategy for inaction

As we said at the beginning of this chapter, we provide an elementary application of the theory of strategies to the logic of "seeing to it that." Previous work in stit theory, §9G, made it clear that the ability to see to some result, Q, does not imply the ability to refrain from seeing to Q. (Since two stits were put in play in §8G, we note that this failure applies, delicately, to the achievement stit, but not to the deliberative stit. We are here concerned only with the former.) In the course of developing stit theory, it became plausible to suggest, however, that whenever an agent, a, can see to it that Q, in spite of being unable to refrain from seeing to Q, there is always available to the agent a strategy guaranteeing that, if followed, the result would be that the agent not see to it that Q. Attempts to demonstrate this intuitively, especially in the presence of infinities, always ran up against the need for rigor, and therefore had to await the development of the austere theory of strategies. Chiefly because we now have the favoring extension lemma in hand, we are ready for the application of the theory of strategies to the problem of proving the existence of a "strategy for inaction." (We certainly hope, however, that the favoring extension lemma finds more surprising applications.)

13E.1

Semantics for stit

At this point we need an exact statement of the semantics of [a stit: Q] as the achievement stit, which we restate. Recall from §2A and elsewhere that indeterminism requires that truth with respect to a BT + I+AC model, m, must be relative to moment-history pairs m/h, so that we write "971, m/h t= Q" for "Q is true in the model m at the pair m/h." The semantics for [a stit: Q] relies on the apparatus of "instants" described in §13B, which was not needed or wanted for the theory of strategies. Reminder (Def. 4): H(M) 'ls the set °f histories having nonempty intersection with M; i(mi) is the instant of time on

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which mi lies; m( J(m ) t h ) is the moment in which the instant i(mi) intersects the history h. The following is the concept of the truth of [a stit: Q] based on witness by a chain (instead of witness by only a moment); it is a merely verbal variant of §8G.4, with which it may be compared. 13-24 DEFINITION. (Truth of [a stit: Q]) Given that mi £ hi, we define that 97t, mi/hi 1= [a stit: Q] iff there is a chain of moments CQ, called a witness, such that the following conditions are satisfied. • Priority-nonemptiness. All moments in CQ must be properly earlier than mi, and CQ must be nonempty: CQ ^ 0 and Vmo[mo G CQ —>TOO< mi]. • Positive condition. Q must be settled true at every moment in i( mi ) that lies on a history intersecting CQ that is inseparable from hi for a in CQ: • Negative condition. Every moment, mo, in CQ "risks" the falsity of Q in the sense that there is above m0 some moment in i(mi) at which Q is not settled true. In other words, for every moment, m0, in CQ, it is not settled at m0 that Q be true at

13E.2

Abstraction of stit witness by chains

The rigorous result we are after goes something like this. Suppose that [a stit: Q] is true at a moment-history pair mi/hi, witnessed by a chain CQ. Fix mi, hi, and CQ. Then at every moment m0 in the witnessing chain CQ there is available a strategy such that following it really guarantees that [a stit: Q] fails at m( i(m j./io)/^) f°r suitable /i0. To prove this result, we need to provide a suitable interface between stit and strategy. From the stit side, we need much less than what we have. We surely need the agent, a, and the moment-history pair mi/h\. But we do not need all of the syntactic information in Q, nor a general semantic story. We need only whether the particular Q holds at each m(i( ),h)lh. This we can represent by a set of histories depending only on mi and Q: We also need a field of choice points for a. On the "stit" side, it must be large enough to include all the chains that could possibly witness any of the stits considered in the desired result. On the "strategy" side, since the strategy is to be "available" at mo 6 CQ, the field must include mo. The field, however, must not include any moments earlier than mo, which are evidently no longer "available" at mo. Let us take the set of all moments from mo onward that are temporally prior to mi, that is, that precede some moment co-instantial with mi: MI — {m: mo ^ m & m < i( mi )}. We record these two abstractions for later reference.

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13-25 STIPULATION. (Two critical sets) Suppose that TOO, mi, and Q have been fixed, and thatTOO< TOI. • Hi = {h: m, m ( l ( m i ) i h ) //i N Q}. • MI = {TO:TOO^ TO &TO< i(mi)}. Next notice that /ii doesn't really come into the Definition 13-24 account of truth for stit. Consequently we can take TOI to come in only as a peg on which to hang the definitions of H\ and M\. If we resolve the general nonemptiness condition on the chain witness c in favor of a particular member TOO of c, we can state the remainder of the truth condition for stit as a relation between TOO, HI, MI, a, and an arbitrary history ho. We codify this "stand-in" for stit as a general definition of a five-place relation, stit-stand-in(mo, HQ, M0, a, ho). 13-26 DEFINITION. (Stit-stand-in) For any moment TOO, set of histories HO, set of moments M0, agent a and history ho, stit-stand-in(m0, H0, M0, a, ho) iff there is a c such that the following all hold. • Priority-nonemptiness stand-in.TOO • Positive stand-in. • Negative stand-in. The positive stand-in says this. Let h be any history intersecting c such that c makes no choice distinction between ho and /i. Then h G HO- That is why we can say that the choices made in c by a are such as to "guarantee" being in HQ. Or to put it contrapositively, if h G (H(c) —Ho) then 3m[m G cD/ioPI/i & /IQ _L^j /i]. The negative stand-in says that every moment m in c keeps departure from HQ as a live possibility. That makes c nonvacuous as a series—though it does not and should not follow that a has a nonvacuous choice at each member of c. The condition says that at every moment TO in c there is a "live possibility" of finishing outside of HQ. It does not say that there is a currently available "live option." It is important to see that stit-stand-in is a concept entirely in the language of the theory of strategies. For the upcoming result about stit-stand-in to have its intended significance, however, we must verify that the concept is satisfactory on the "stit" side, as follows. 13-27 FACT. (Stit and stit-stand-in) Fix a and TOI, and h\ such that TOI € h\. Assume that 9JT, mi/h\ N [a stit: Q], and let CQ be a witness (Definition 13-24). Fix TOO G CQ. Define HI and MI as in Stipulation 13-25. Choose any history ho G H(mo). Then 971, m,(i(m },hQ)/ho ^ [ but it does not say, as required for stit-stand-in(mo, HI, MI, a, ho), that mo 6 c. Nevertheless, since mo is assumed to be part of a witness for [a stit: Q] at m\jhi, we can be sure that if c witnesses [a stit: Q] at TO(i ,,h 0 )//iO) then so does cUJmo}. The reason is this. We know by the negative condition for the witnessing of [a stit: Q] at mi jh\ that Q fails at m(j ( m /i)//i for some h 6 H(moy Therefore, the chain c cannot be entirely earlier than mo on pain of violating the positive condition for the witnessing of [a stit: Q] at m(i( }tho)/ho- So some member of c must come properly after mo. But then adding mo to c keeps the negative condition true, and does not disturb the positive condition, so that cU{mo} is also a witness to [a stit: Q] at m(j ( m ),h0)/ho- And of course that set is bound to contain mo, as required for stit-stand-in(mo, HI, MI, a, h0). (There is a discussion in Belnap 1996a of what can be added to or subtracted from a chain witness.)

13E.3

Main result

We use the stit-stand-in abstraction from the concept of seeing-to-it-that as follows. We prove generally that regardless of H0 and M0, whether as just defined or not, we can always find available at mo a strategy for a such that if a follows it faithfully at the choice points in MQ, then for any history /IQ that could possibly eventuate, stit-stand-in(mo, HO, MO, a, ho) must fail. This is a result in the pure theory of strategies. Then we use Fact 13-27 to argue that there is always a strategy the following of which guarantees that [a stit: Q] is false at the moment-history pairs in question. 13-28 THEOREM. (Strategy for inaction) Fix mo, MO, HO, and a such that mo Then there is available at mo a simply complete strategy s for a in MQ such that stit-stand-in(mo, HQ, MO, a, ho) fails for every history ho admitted by s. PROOF. By the second part of Corollary 13-23, let s be a simple strategy for a in MQ available at mo that is complete in MO and that favors -H(M 0 ) ~Ho m MQ. Suppose s admits ho (and is therefore complete along /i0 in MO), and, for contradiction, that stit-stand-in(m0, H0, MO, a, h0) holds (Definition 13-26). By priority-nonemptiness let mo G c C /i 0 nM 0 . By the negative condition, so that by favoring, we know that we may let So choose By the positive condition, hi (hence in h0r\Mo, therefore in Dom(s)) such that hi ±^ ll ho. Here is the contradiction: ho and hi are both admitted by s, so must both belong to s(mi). But then s is not after all strict. Our main result is a direct corollary.

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13-29 COROLLARY. (Main result) Suppose that 9Jt, mi/hi 1= [a stit: Q], with witness CQ. Then at each moment mo G CQ, there is available a simply complete strategy s for a in that really guarantees PROOF. Start with the hypothesis, and choose m0 € CQ. Define MI, #1 in accord with Stipulation 13-25. Since m0 e MiHljHi, by Theorem 13-28 there is available atTOOa simply complete strategy s for a in MI such that stit-standOT(TOQ, -Hi, MI, a, ho) fails for every history /i0 admitted by s. By Fact 13-27, that amounts to saying that s guarantees And the hypothesis allows us to calculate that s is not vacuous for the set of histories

since we can promote "guarantee" to "really guarantee."

For this reason,

14

Deontic kinematics and austere strategics Deontic concepts are naturally linked with strategic concepts.* Surely any obligation can be viewed as an obligation to follow a certain strategy in the austere sense of chapter 13, always making such choices, depending on circumstances, as conduce to satisfying the obligation. Conversely, to follow a strategy can be viewed as very like living up to a set of deontic requirements, "doing what the strategy requires." We do not here directly discuss such common-speech linkages. Instead we report a specific and detailed theoretical linkage that allows reciprocal illumination between certain deontic concepts and certain strategic concepts. An unexpected result is the connection between the no choice between undivided histories condition from the austere theory of strategies, and a new deontic kinematic condition adduced by Thomason to help describe how oughts fit into branching time. Deontic kinematics I. On the deontic side we invoke Thomason 1984, a fundamental essay that describes a family of structures carrying information about (z) oughts-at-moments in (ii) branching time. The branching mentioned in (ii) represents that some things might happen that are not guaranteed to happen. In other words, there are inconsistent possibilities for the future. In particular, an existing obligation might be fulfilled, or might not. The purpose of relativizing oughts to moments as in (z) is simple and pervasive: One knows that oughts come to be and pass away. For example, an obligation can be created by a promise; for another example, a "reparational obligation" can be created by breaking a promise. One asks for general principles governing how such event-located oughts fit together in our world. This is the topic of "deontic kinematics." (Thomason 1984 calls the topic "ought-kinematics.") Deontic kinematics is an enrichment of the "static" or "timeless" versions of the deontic concepts that are prevalent in the literature. 'With the kind permission of Filosophia, this chapter draws on Belnap 1996b.

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Austere strategics I. The strategic concepts we use arise in the theory of agents making choices in branching time, BT + AC theory, §3. In earlier chapters we explored stit theory, which presupposes BT + AC or BT + I+AC theory; and then in chapter 13 we considered the austere theory of strategies. Austere strategics sits on the same foundation as stit theory, which is to say, upon the theory of choices in branching time; but this theory of strategies is not semantic and does not itself depend on the semantics of stit. We describe the theory as "austere" because it involves none of the psychological or value concepts typically taken as underpinnings for the concept of strategy. In fact, the austere theory of chapter 13 simply identifies a strategy as a pattern of choices in branching time, neglecting entirely the mental work that goes into adopting a strategy, and equally neglecting what a strategy is "for." It is surprising how rich and also clarifying such a minimal theory can be. Deontic kinematics and austere strategics both "take time seriously": As we see, each has a key postulate that in the former vertically integrates oughts across time, and in the latter vertically integrates choices. Our most striking result exhibits the interplay between these two apparently disconnected ideas.

14A

Basic concepts

First we review the underlying concepts.

14A.I

Branching time

We begin by noting with maximum brevity what is common to the two theories, namely, branching time. Deontic kinematics and austere strategics follow the lead of Prior 1957 and Thomason 1970 in representing the indeterminism of our world by means of a BT structure (Tree, ^}. That is, deontic kinematics and austere strategics have the Prior-Thomason theory of branching time as a common part. In this book the postulates and definitions of this theory are briefly described in §2A, more extensively discussed in chapter 7, and summarily presented in §3. See the latter for help with the notation.

14A.2

Deontic kinematics II

Thomason 1984 arrives at the theory of deontic kinematics by adding a deontic primitive "0" to the primitives Tree and ^ of branching time. He calls the resulting structures "treelike frames for deliberative deontic tense logic." Here, for our immediate purposes, we substitute the phrase "deontic tree." 14-1 DEFINITION. (Deontic tree) A deontic tree is a triple (Tree, ^, 0), where (Tree, Choice'^(h) C s(m}\. • s is available at m iff m € Dom(s). • For m S Dom(s), s is consistent at m iff s(m) ^ 0, and s is consistent iff s consistent throughout its domain. • For m 6 Dom(s), s is vacuous at m iff s(m) = H(my

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• A strategy s is total [in M] iff Dom(s) •=• Tree [= M]. These ideas are explained at more leisure in §13C. Recall that we represent a choice as a set of histories. The key idea is that a strategy tells you which of your available choices count as following the strategy—and which do not. A strategy that is inconsistent at m explicitly tells you that no history available at m counts as following the strategy (there is nothing good to do). At the other end of the spectrum, a strategy that is vacuous at m says that everything goes at m. A strategy is total if it gives you advice at every moment in the entire tree. 14-3 DEFINITION. (Strategic concepts: admission and exclusion) • s admits h iff Vrn0[m0 G Dom(s)r\h —> h G s(m0)}. • Admh(s) — {h: s admits h}. • s admits mi iff Vm0[(m0 G Dom(s) & mo < mi) —> mi e U s ( TO o)]• Admm(s) = {m: s admits m}. • s excludes a history or a moment if it does not admit it. The admission and exclusion concepts on this list, drawn from Definition 13-4, are of critical importance for this chapter. Let us start with exclusion, which is defined (with inescapable confusion) for both moments and histories. A moment is excluded by a strategy if never violating the strategy implies that you will never reach that moment. A history is excluded by a strategy if never violating the strategy implies that eventually you will fall off that history. Since admission is simply the contradictory of exclusion, a moment is admitted by a strategy if it is possible to arrive there without violating the strategy, and a history is admitted by a strategy if staying on that history never requires that you violate the strategy. 14-4 DEFINITION. (Strategic concepts: simplicity, completeness, etc.) • For m G Dom(s), s is strict for a at m iff s(m) G Choice^, and s is strict for a iff s is strict for a throughout its domain. • s is secondary iff some moment that it excludes is nevertheless in Dom(s); and s is otherwise primary. (This dichotomy arises from the following: The dictates of s are primary if they concern moments that are admitted by s, but secondary when they concern moments that are excluded by s.) • s is a strategy for a in M iff s is a strategy for a such that Dom(s) C M. • When s is a strategy for a in M, we call M a field for s. • Suppose that s is a strategy for a in M. s is backward closed in M iff

368

Strategies • s is simple for a in M iff s is a strategy for a in M that is strict for a, primary, and backward closed in M.

• s is complete along h in M iff Mfl/i C Dom(s). • s completely admits h in M iff s admits /i and is complete along h in M. • s is H-complete in M iff s is complete in M along every history in H that s admits. • s is complete in M iff s is complete in M along every history that s admits. • s is simply complete for a in M iff s is simple for a in M and s is complete in M. This last group of definitions supports a rich theory of simple strategies. For example, a major fact proved in chapter 13 is that every strategy for a in M that is simple can be extended to a strategy that is simply complete for a in m (Corollary 13-23). We mention this here, however, only for contrast, since we will not be dealing with simple strategies at all. And for the same reason, we refer to chapter 13 for explanations of these concepts: In this chapter, we occasionally mention some of these concepts for contrast, but we do not really use them.

14B

From Thomason's deontic kinematics to austere strategics

To think about oughts and strategies together, zero in on a single agent, a, and think of Tree as both supporting 0(m] representing what ought to be at m and also supporting a choice-function Choice^ defining what a can do at m. What, if anything, would then be wrong with thinking of the deontic tree itself as a candidate account of a strategy for a? That is, what would be wrong with taking 0 as a strategy? A great deal. From the agency point of view, the deontic tree would have the defect that it omits consideration of what a can do at m. This defect seems to us so substantial that it calls into question the propriety of using the adjective "deliberative" in connection with a deontic tree. Such a tree could not be a strategy for a, since there is no requirement that 0(m) be closed under choice equivalence for a at m. 0(m) picks out histories containing m, but not choices available to a at m. Strategies ("deliberative" strategies) tell you what to do; 0 does not—at least not directly. Perhaps there is an indirect route from 0 to strategic advice. Let us then ask: How to make a strategic choice at m given a particular 0(m)? Consider a possible choice for a at m, say H (H will be a subset of all the histories H(m) to which m belongs). If H contains only histories lying outside of 0(m), certainly the strategy should steer away from H. And if H contains only histories acceptable to 0(m), then with equal certainty the strategy should

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permit the choice H. But what if H is "mixed," containing both some histories acceptable to 0(m) and others that are unacceptable to 0(m)l There are then evidently two salient strategic policies available, neither happy. The "weak" policy says that choosing H, even though mixed, is in accord with the strategy. The "strong" policy says that choosing a mixed H violates the strategy. There is also a third policy, itself not entirely happy. The "Hamblin" policy, suggested in Hamblin 1987, makes it depend upon what other options there are, as follows. Let H be mixed. If there is at least one possible choice that contains only histories acceptable to 0(m), then the Hamblin strategy forbids choosing such a mixed H (like the strong policy). But if there is no possible choice containing only histories acceptable to 0(m), then the Hamblin strategy permits choosing mixed H. There is no point moralizing about this here; we simply put the matter in terms of clear definitions about which something definite can be said.2 14-5 DEFINITION. (Deontic trees as strategies: sweak,a,o and sstrong,a,o and SHambhn,a, o) Suppose that we are given a deontic tree (Tree, ^, 0) together with a choice partition Choice1^ for a at every moment m. We define the weak, the strong, and the Hamblin strategies for a determined by this deontic tree, written respectively as sweak^a, o, sstTong,c>, o, and sHam,bhn,a, o, by specifying the domain of each function as the same as the domain of 0—namely, the entire tree—and continuing as follows. • sweak,a,o(fn)

=

• sstrong,a,o(m) = • SHambhn,a, o("i): If there is a possible choice H for a at m (that is, H such that then otherwise, sHambhn,a, o(m) We can now intelligibly speak of a deontic tree as in effect one of the strategies {sweak,a, o, sstrong,a, o, sHamblin,a, o}, and ask what properties it may have. 14-6 FACTS. (Deontic trees as strategies) • For S € {Sweak,a, O, Strong,a, O, SHamblin,a, o}, s(m)

is a Subset of H(m)

that is closed under choice equivalence for a at m. That is, each of {sweak,ce,o, strong,a, o, sHamUin,a, o} is a strategy for a. None of the conditions (i)-(iii) of Definition 14-1 play a role here: The form of Definition 14-5 suffices.

• Since 0 is total on Tree, so is each of the three strategies sweak,a, o, sstrong,a,o, and Sffambim,a, o- Each is available at every m in Tree. Each strategy is ferociously "complete." 2

Horty 2001 shows that there is much more to be said in the context of an apparatus involving utilitarian degrees-of-value comparisons. Here, however, we have only the deontic O.

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Strategies • The consistency condition (z) of Definition 14-1 implies that sweak, a, o and SHambiin,a,o are consistent. In contrast, the strong cousin s s tr<mg,a,o may well be inconsistent even though the consistency condition on 0 is satisfied. Suppose, for example, that the consistency condition is satisfied at m by means of just a single history h through m that is acceptable to 0(m). If the possible choice for a at m that contains h also contains an unacceptable history, then according to sstr0ng,a, o, that choice is not open to a. The strategy s s t r ong,a,o tells a that at m every option is forbidden. That is, sstr0ng,a,o 1S inconsistent. • Any member of {sweak,a,o, sstrong,a,o, sHa.mbiin,a,o}, except in trivial cases, is bound to be secondary. This feature distinguishes any strategy based on a deontic tree from any simple strategy. The positive role of secondariness in the deontic-kinematic framework is to permit consideration of reparational obligations, obligations that come into force after some earlier obligation has been violated. We underline that the feature is neither good nor bad; but one should bear it in mind. Primariness/secondariness is the most significant difference between simple strategies and strategies that are derived from deontic trees. • If S G {sweak,a, O, Sstrong.a, O, SHamblin,a, o}, S is total, 3S W6 Said. So it

is natural to expect that at some moments m, s is close to being undefined; i.e., s lays on a at m only a vacuous obligation, the set of all histories containing m: s(m) = H(my This is not primarily a technical point: Any strategic advice that covers every moment in Tree must be largely vacuous.

• A strategy s € {sweak,a,o, sstrong,a,o, SHambiin,a, o} derived from a deontic tree is therefore never simple: Although s is bound to be backward closed (in any M), it will never be both strict for a and primary. Notably missing from this account is any reference to the kinematic condition (in) of Definition 14-1. What does the kinematic condition mean? Thomason 1984 connects the notion with the idea of a "coherent plan." But there seems nothing incoherent about a strategy derived from a deontic tree, whether or not condition (Hi) is satisfied. This suggests that we have not found the best way of relating austere strategics and deontic kinematics.

14C

From austere strategics to Thomason's deontic kinematics

We now start from the other end. We start with austere strategics as given, and ask what we can make of deontic kinematics in this context. It turns out— unexpectedly—that here is where we find the most interesting results. These are the results that seem to shed the most light both on austere strategics and on deontic kinematics.

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So suppose we start at the other end, with a strategy s. One might think of the histories S(TO) as what a "ought to do" at TO. Suppose we were then to turn things around and define 0(m) as s(m). We should learn that conditions (i) and (ii) of Definition 14-1 would be satisfied at least when s is consistent. We should also find out that condition (lii) is something that does not readily spring to mind. Mutual illumination seems again absent.

14C.1

Admission onward and Os

There is, however, another idea that begins with strategies. This idea pays attention to a recurring theme in the deontic literature: The histories defining obligation should in some sense be "ideal." At least they should be "ideal" to the extent that in these histories no one ever violates an obligation. If we take the given strategy s as the source of obligations, then we have already defined these histories. They are the admitted histories (Definition 14-3). In an admitted history a never departs from the strategy. This never-failing agreement with the strategy is evidently "ideal." Therefore admission, referring as it does to a generality of moments, would appear as a concept of deontic kinematics. We might therefore think about defining 0(m) as Admh(s)r\H(mj so as to include the element of ideality. Observe that since it involves quantifying over moments, admission is essentially a "global" idea, not bound to a particular moment. Admitted histories are acceptable to the strategy, and are "ideal" as far as the strategy goes. None of them can be improved upon, as far as strategy-following goes. So we can talk deontically about what is true in all "ideal" histories, that is, the admitted ones, those in which the strategy is never violated.3 The idea of defining obligation in terms of admission will not do, however, for every case of a deontic tree. And that is because some of these trees are designed to allow intelligible "reparational obligations." In such a tree there can be things that you should do at a certain moment m even though those very things were excluded by earlier obligations. (You only reached TO because you did what you ought not to have done.) The suggestion to define 0(m) as Admh(s)C\ H(m) might be all right if we were only concerned with simple strategies. In the reparational case at hand, however, we should have 0(m) = Admh(s)r\ H(m) = 0, which would represent an inconsistency: The consistency condition, Definition 14-1 (i), on 0 would not be satisfied. This definition cannot, then, offer advice to those who have ever, even once, made a strategic mistake. If this seems too abstract, any of the old tales will do to make the point. You ought atTOOto visit your mother—perhaps because of a promise—and you have adopted that as your strategy. But atTOOyou choose not to catch the plane to visit your mother. You find yourself at TOI, not having caught the plane. At TOI you must choose whether or not to call your mother to tell her that you will not be coming. Both of these histories through TOI, however, are such that on them you have not visited your mother. They are therefore both excluded 3 We say that admitted histories are "acceptable" rather than "permitted" because we think that permission is an essentially agentive concept and therefore not applicable to individual histories. Only choices for a can be permitted—or not—for a.

372

Strategies

(not admitted) by your strategy to visit your mother. Your strategy therefore excludes all alternatives through mi. Your strategy is inconsistent at mi. What has gone wrong? This: One wants for the reparational or secondary situation a concept that gives a a "fresh start." That is why taking admission as an absolute character of histories did not work. It is better to invoke the idea of a history that is "admitted by s from m onward." In other words, the history may have been excluded earlier, but don't look back: From m onward it forever after counts as ideal. In still other words, past failures don't count. As one might expect of a purely deliberative idea, only the future of possibilities is relevant. Here is the definition. 14-7 DEFINITION. (Admission onward; Admh(s, m)) let mo € Dom(s). • s admits h from mo onward iff mo

Let s be a strategy, and and

• For any strategy s and moment mo, Admh(s, mo) is the set of histories admitted by s from mo onward. In applying this definition, we run into the contrast that strategies can be partial while deontic trees are always total. For present purposes we therefore choose to concentrate on total strategies. It is easy to see that from the point of view of admissibility, nothing is lost. Given any strategy SQ we can always consider instead the total strategy si obtained by vacuously defining Si(m) = H(m) for every m & (Tree —Dom(so)). (It cannot be inferred, however, that nothing is changed tout court; obviously total strategies are not simple at all, so that exclusive concentration on them would put potholes in the road to inquiring about the properties of simple strategies.) The upshot is that in the context of reparational obligations, given a total strategy s, Admh(s, m) seems a good candidate for the concept 0(m) of deontic kinematics. These are the histories that, having reached m, are and will remain "ideal" as far as the strategy goes. They are the histories through m on which (although obligations may have been violated earlier) no more violations occur. Sounds "ideal," doesn't it? Let us record the suggestion in a definition. 14-8 DEFINITION. (Strategies as candidate deontic trees: Os) Suppose we have a choice tree (Tree, ^, Choice). Given a total strategy s for a, we define the candidate deontic tree determined by s as (Tree, ^, Os), where for m e Tree, Os(m) = Admh(s, m). Here is a decisive observation: Os is not a strategy and should not be confused with a strategy. A strategy s picks out choices at m, whereas Os can pick out some individual history within a possible choice for a at m, while excluding other histories in that same choice. Such ideal histories are important; but one must also keep in mind that no agent can choose the set of such histories. What can we say about the so-induced candidate deontic tree, especially with respect to conditions (i)-(iii) that Definition 14-1 laid on deontic trees?

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The following sections report that all of these conditions—(i) consistency, (ii) locality, and (Hi) kinematic—fall into place given Definition 14-8. There you have it: Think of deontic kinematics as generated by austere strategics, and each of the deontic-kinematic conditions is illuminated. This is especially helpful for appreciating the least obvious of these conditions, the kinematic condition (Hi). We start at the first of the deontic-kinematic conditions.

14C.2

The consistency condition on deontic trees

Of course if s is inconsistent, Admh(s, m) can be empty. That's trivial. The interesting question concerns the consistency (nonemptiness) of Admh(s, m) given that s is consistent. We shall find that the consistency of Admh(s, m) is by no means guaranteed. But we shall also locate an unexpected condition under which consistency of Admh(s, m) is restored: Namely, it makes a difference whether or not there are busy choosers, Def. 14. 14C.2.1

Inconsistency of Os given busy choosers

The first relevant fact is that Admh(s, m) can be empty even if 5 is consistent, if one allows a to be a busy chooser. 14-9 FACT. (Consistency condition on Os, Definition 14-1, with busy choosers) There is a total consistent strategy s for a such that Os is nevertheless inconsistent. PROOF. Here is a (peculiar) case in which a particular Admh(s, mu) is empty, even though each s(m) is nonempty. We need to look at an example of "feathering" of histories (see Figure 14.1). Suppose after some moment mw there is an infinite chain of moments mo > mi > m^ > ... > ml > ml + i > ... descending toward mw (mw is not part of the sequence, but mw is the greatest lower bound of the sequence). We suppose that s offers a vacuous choice for a at TOO and also at every other moment other than TOI, 7712, ..., mu. At mu and at each member ml of the chain, i > 0, there is, however, a binary choice for a. (Since there are infinitely many nonvacuous choices for a between m^ and mo, a is by Def. 14 a busy chooser.) Going left stays in the chain, going right leaves it. The strategy at mw says: Go left (stay in the chain), that is, s(mu) = the set of all histories passing through some mz, 0 < i. But after m^ and before TOO, s always prescribes going right ("leave the chain"). So Admh(s, mu) is empty even though s(m) is nonempty for every m. To verify this, pick any "right" history, hi, intersecting the chain atTOZ,i > 0. You will find that ht is excluded by some (indeed every) moment that is strictly between TOW and TO;. For example, you can see from Figure 14.1 that h% is excluded at ma. Furthermore, the "leftmost" history hu, which containsTOO,is excluded by every m-i that is earlier than mo and later than mu. So every history through mw is excluded at some point at or after mu. None of the histories through ma is admitted from mu onward. None of them is "ideal." So Admh(s, mu) = 0. Which is to say, Os is inconsistent at mw, even though s is everywhere consistent. D

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Strategies

Figure 14.1: Consistent s does not imply consistent Os

It does not appear that the strategy pictured in Figure 14.1 is any more incoherent than any other more "finite" reparational strategy. The appearance is more that of a merely set-theoretic construction without intuitive content. But perhaps that appearance is only due to our reluctance to think about the possibility of busy choosers; it's hard to be sure. 14C.2.2

Consistency of Os given no busy choosers

The strange counterexample is not, however, the end of the story about how consistency of s connects with consistency of Os. We may add the following: Absence of busy choosers is quite enough to provide the link between consistency of s and consistency of Os. 14-10 FACT. (Consistency condition on Os (Definition 14-1 (i)), given no busy choosers) Let s be a total consistent strategy for a. Then Os (m) is nonempty— and the consistency condition (i) of Definition 14-1 is satisfied—provided a is not a busy chooser, Def. 14. PROOF. Suppose that we have a choice tree (Tree, w F A iff 371, M 1= [a stit: A] and 371, i\>w ¥ A. Hence, when w witnesses [a stit: A] at m, w also witnesses [a stit: A] and [a stit: [a stit: A]] at any m' choice equivalent to m for a at w. Let & = (Tree, \ w ^ w' < m], forward denseness, that is, Vw[w e BC —* 3w'[w' 6 5(7 & w < w'}}, and nonvacuous choices, that is, Vtu[w 6 BC —> Choice^ ^ {H(wj}}.2 A &MSJ/ choice sequence in & (or 971) is a busy a-choice sequence in & (or 971) for some agent a in ©. It is easy to see that & is finite only if there is no busy choice sequence in &, but the converse does not hold. Let K be the set of all sentences A such that & 1= A for each BT + I + AC + nbc structure ©, and let Lai + rr be the axiomatic system that takes as axioms all substitution instances of truth-functional tautologies, refref, and the following schemata:

2 The idea of a busy choice sequence is listed with a some somewhat more general definition in Def. 14. A formal study of busy choice sequences can be found in Xu 1995a. Note that the forward denseness condition is only a particular way in which a chain can be busy. We only consider forward denseness because when considering the truth values of sentences in our language, only forward denseness is relevant. One can find, however, other issues in which backward denseness is more crucial. See, for example, Fact 14-9.

384

Proofs and models

and takes as rules of inference modus ponens and RE From A = B to infer [a stit: A] = (a stit: B\. The definitions of theorems (I-), consistency and maximal consistent sets are as usual, keeping in mind that in this chapter all these concepts refer to Lai + rr. Let A be any sentence and E any set of sentences. A is a deductive consequence of E, written E h A, if h (A1 &...&;Ak D A) for some >4i, ..., .Afc e E. E is deductively closed if j4 € E whenever E h .A. The deductive closure of ,4 will be denoted by DC(A). Identifying Lai + rr with the set of all its theorems, we show in the following sections that K = Lai + rr, and that Lai + rr has the finite model property: For every sentence A, V- A only if SOT ¥ A for some finite model SDt. This property will be obtained simultaneously with the completeness of Lai + rr. For the convenience of our upcoming discussions, we present some theorems and derived rules in Lai + rr. Note that all the derived rules and all the theorems TT1-19 can be proved without applying refref.

15. Decidability of one-agent achievement-stit theory with refref

15B

385

Companions

In this section, we will consider some basic semantic features of so-called companions to stit formulas, which we will not define until we obtain the companion theorem (Theorem 15-11). But it will be helpful to present a description of it at the beginning so that the reader may have some idea of where our discussion is going. Let 9m be any BT + I + AC model such that m, m t= [a stit: A] with witness w, and let M = {m1: m' € i(m) & m' =° m}. Clearly, m, M N [a stit: A}. We will show in the companion theorem that for each sentence C, either m, M N [a stit: A&Ca] or m, M ~[a stit: A,Ca\. Whichever is true together with [a stit: A] in M will be called a "companion" to [a stit: A}. Note that in this section, "structures" and "models" always mean BT + I+AC structures and BT + I + AC models, and we do not require that these structures and models satisfy the property of no busy choice sequences. Let us start with some simple lemmas.

386

Proofs and models

15-4 LEMMA. (A consequence of no choice between undivided histories) Let & be any structure in which w < m and w' < m, and let a be any agent, i = i(m), M - {m1: m1 6 i & m' =awm} and M1 = {m'\ m! € i & m' = , m}. Then

PROOF. Suppose that w < w'. Consider any TO' 6 i\>w>, that is, m' £ i and w' < m'. Assume that m' ^ m. Then by the axiom of choice there are two histories h and h' such that {w, w', m} C h and {w, w', m'} C h', and hence h' =w h by definition. This implies, by no choice between undivided histories, h1 =aw,h and hence m' =£, m, that is, m' 6 M. It follows that i| >UJ ' C M. (ii) Suppose that w' ^ w. By (%), if w' < w, then i\>w C M', and hence M C M'. If to' = w, then m' =£,, m iff m' =£, m for every m' € i, and hence M = M'.

n

15-5 LEMMA. (Some sufficient conditions for stit) Let SOT be any model in which w < m, i = i( m j and M = {m": m" e z & m" = ° m}, and let >1 be any sentence. Then

PROOF, (i) Suppose that SOT, M N A and for some m' € M, SOT, m' N [a stit: .A] with witness w'. Setting M' = {m": m" 6 i & m" =£,/ m'}, we then have

Since m' € M, w < m', and hence by no backward branching, either w < w' or w' ^ u/. If w < w', then by Lemma 15-4(i), i\>w> C M, and hence SOT, i|>u/ N .A, contrary to (1). Hence it must be true that w' ^ w, and then by Lemma 15-4(ii), M C M'. It follows from (1) that 971, M N [a stit: A], (ii) Suppose that !Ut, i\>w t= yl and for some m' € i\>w, 3H, m' N [a stit: A] with witness w'. Similarly, setting M' = {m": m" € i & m" =^, m'}, we also have (1), and applying no backward branching, we have either w ^ w' or w' < w. By the transitivity of ^, w ^ w' implies i\>w' C i\>w, which in turn impliesm,i\>w' t= A since SOT; i\>w N A . It follows from (1) that w' < w, and hence by Lemma 15-4(%), i|>«, C M'. Then DOT, i|>tu N [a stit: A] by (1). D 15-6 LEMMA. (Concerning [a stit: A&B]) Let 9H,TON [a stzt: A] with witness w, and let i = i(m). Suppose that 9JI, M 1= 5, where M = {m': TO' 6 i & m' = ^ m}. Then SOT, m 1= [a stit: A&B] with the same witness. PROOF. Suppose that 271, M 1= B. Then by hypothesis, Jt, M *= a&.B and 9Jt, i|>UI ^ 4, and hence mOT, i|>m J^ ^&5. It follows that £01, m N [a stit: A&B] with witness w. D

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387

15-7 LEMMA. (Concerning Aa) Let m be any model in which w < m, i = i(m) and M = { m / : m ' E i & m ' s ^ m } . Suppose that A is any sentence such that m, M F= A and M, M F [a stit: A]. Then m, i >w F= Aa. In particular, if m, m 1= [a stit: A&B] with witness w, and if M, M F [a stit: B], then Wl, i>w F= Ba and M, M F= [a stit: /4&Ba]. PROOF. First, if m, i\>w F A, then, since M, M F= A, M,F 1= [a stit: A], contrary to our hypothesis. It follows that M, i|>w, F= A. Next, suppose for reductio that M, i|>w, F ~[a stit: A]. Then there is an m" € i|>«, such that M, m" F= [a stit: A]. Since we have shown that M, i\>w F=A, M, i|>w, F= [a stit: A] by Lemma 15-5 (ii), contrary to the hypothesis. From this reductio we conclude that M, i\>w 1= ~[a stit: A] and hence M, i|>w F= Aa. In particular, if M, m F= [a stit: A&B] with witness w, and if M, M F [a stit: 5], then M, i|>w F= Ba and hence M, M F [a stit: A&;Ba] by Lemma 15-6. 15-8 LEMMA. (01 refraining lemma) Let M be any model, and let A and B be any sentences such that M, m F= [a stit: A&~[a stit: B]] with witness w, and let i = i( m ). Suppose that for some m' £ i|> w , M, m' F= [a stit: B] with witness w'. Then w < w'. PROOF. Let M' = {m": m" 6 i & m" =£, m'}. Then M, M' F= [a stit: B]. Since m' E i|>w,, w < m' and w' < m'. Then by no backward branching, either w' < w or w < w'. Suppose for reductio that w' < w. Then i|>w C M' by Lemma I5-4.(i). It follows that M, m F [a stit: B] since m € i\>w. But by hypothesis we have M, m F= ~[a stit: B], a contradiction. From this reductio we conclude that w < w'. 15-9 LEMMA. (Concerning Aa and stit) Let M be any model, and let A and B be any sentences. Then the following hold: i. If M, m F [a stit: A&B a ] with witness w, then M, m F= [a stit: A] with the same witness; ii. If M, m F= [a stit: A&~[a stit: A&B 0 ]] with witness w, then M, m F [a stit: A] with the same witness. PROOF. (i) Suppose that M, m F= [a stit: A&.Ba] with witness w. Then, setting i = i(m) and M = {m': m' € i & m' =aw m}, we have M, M F A&5a and M, i|>w ¥ A&Ba. Hence by Lemma 15-7, M, i\>w F= Ba, and hence M, i|>w, F A. It follows that M, m F= [a stit: A] with witness w. (ii) Suppose that M, m F= [a stit: A&~[a stit: A&B a ]] with witness w, and let i = i(m) and M = {m': m' € i & m' = awm}. Then

and either M, i|>w F A, or M, i\>w ¥ ~[a stit: A&Ba}. Consider the latter case. There must be an m' € i\>w such that M, m' t= [a stit: A&Ba} with witness w'. By applying Lemma 15-8, we have w < w', and hence i\>w' C i\>w.

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Proofs and models

Since by (i), M, m' F= [a stit: A] with witness w', it follows that M, i\>w ¥ A. So in both cases, we have M, i\>w ¥ A. It follows from (2) that M, m F= [a stit: A] with the same witness w. 15-10 COROLLARY. (Validity of A7 For any model M, M F= A7. PROOF. Suppose that M, m F [a stit: ~[a stit: A&.B}&.Ba] with witness w, and let i = i(m) and M = {m": m" € i & m" =aw m}. By Lemma 15-7, we have that

Then there must be an m' £ i\>w such that M, m' F= [a stit: A&B] with witness w'. By Lemma 15-8, w < w', and hence, i\>w> C i\>w. Let M' = {m": m" € i & m" = aw, m'}. Then M' C i|>u, and hence by Lemma 15-6, M, m' F [a stit: A&.Ba] with witness w', and hence by Lemma 15-9(i), M, m' F= [a stit: A] with witness w', it follows that

We show as follows that M, M F ~[a sizt: A]. Suppose for reductio that there is an m* C M such that M, m* F [a stit: A] with witness w*. Set M* = {m": m" C i & m" = aw. m*}. Then M, M* F [astit:A]. Since m* € M, w < m*, and hence either w < w* or w* < w by no backward branching. Now if w < w*, then by Lemma 15-4(ii), M* i\>w, and hence by (3) and Lemma 15-6, M, M* F= [a stit: A & B ] , contrary to (3). If w* < w, then we have i\>w C M* by Lemma 15-4(i), and hence, since M, M* F [a stit: A], M, i\>w F= [a stit: A], contrary to (4). This reductio shows that M, M F= ~[a stit: A]. It follows from (4), (3) and Lemma 15-6 that M, M F= [a stit: ~[a stit: A]&Ba}. Now we are ready to prove the companion theorem. 15-11 THEOREM. (Companion theorem) Let M, m F [a stit: A] with witness w, and let i = i(m) and M — {m': m' € i & m' =aw, m}. Then for every sentence 5, either M, M F= [a stit: A&Ba] or M, M F ~[a soit: A&Ba], and consequently, either M, M F= [a stit: A&Ba] or M, M F= [astit:A&~[a stit: A & B a ] ] . PROOF. Suppose that M, M ¥ ~[a stit: A&Ba}. Then there is an m' € M such that M, m' F [a stit: A&Ba] with witness w'. Hence, setting M' = {m": m" € i & m" =aw, m'}, we have

By Lemma 15-9(i), we have M, m' F [a stait: A] with the same witness w'. This and our hypothesis imply

15. Decidability of one-agent achievement-stit theory with refref

389

Since m' E M, w < m'. Hence by no backward branching, either w < w', or w' < w, or w = w'. If w < w', then by Lemma 15-4(i), i\>w> C M, and hence by (6), m, i\>w> F A, contrary to (7). Similarly, if w' < w, then by Lemma 15-4(i), i\>w C M', and hence by (6), M, i\>w F= A, contrary to (7). We then conclude that w = w'. Since m'' =aw, m (i.e., m' € M), it follows that M = M', and hence by (5), M, M F [a stit: A&B"]. By the foregoing proof, we know that if M, M F [a stit: A&Ba], then M, M F ~[a stit: A & B a ] , and hence m, M F= [a stit: a&~[a stit: A&Ba}} by Lemma 15-6. Let M, m 1= [a stit: .A] with witness tu. For each sentence C, if M, m F= [a stit: A&Ca], we call [a stit: A&Ca] a (semantic] positive companion (poscompanion) to [a stit: A] at m (w.r.t. M), and C a pos-companion root of [a stit: A] at m ( w . r . t . M ) . Similarly, if M, m F ~[a stit: A&Ca], we call ~[a stit: A&(Ca] a negative companion (neg-companion) to [a stit: A] at m, and we call C a neg-companion root of [a stit: A] at m. Pos-companions and negcompanions to [a stit: A] at m are called companions to [a stit: A] at m. Note that every sentence must be either a pos-companion root or a neg-companion root of [a stit: A] at m. Let M be any model in which M = {m': m' € i(m) & m' = aw m} and suppose that m, m F= [a stit: A] with witness w. The companion theorem guarantees that for each sentence C, if M, m F= [a stit: A&. Ca], then M, M F [a stit: A &ca]; and That is to say,[a stit; A] must be true together with all its companions through all of

M, not only with its consequences. It can be seen from our further discussions that it does not suffice to consider only consequences of [a stit: A}; the notion of companions is essential to our proof of the completeness theorem. Of particular interest is that companions help us to compare witnesses to stit sentences, which we show as follows. 15-12 LEMMA. (Companions and witness) Let M be any model in which w < m, i = i(m) and m' e i >w, and let M, m F= [a stit: A&Ca\ with witness w, and M, m' (= [a stit: B&~[a stit: B&(Ca]] with witness w'. Then w' < w. PROOF. Since w < m' and w' < m', either w' < w or w < w' by no backward branching. Suppose for reductio that w < w'. Then, setting M = {m": m" € i & TO" =aw, m'}, we know that M C i >w. Since m, i\>w F= Ca by hypothesis and Lemma 15-7, it follows that m, M F Ca. By Lemma 15-9 (ii) we have that M, m' F= [a stit: B] with witness w', and hence by Lemma 15-6, M, M F= [a stit: B&C"]. But we know by hypothesis that M, M F ~[a stit: B&Ca], a contradiction. It follows from this reductio that w' < w. Among other things, what Lemma 15-12 tells us is this: If a sentence is a pos-companion root of [a stit: A] at m but a neg-companion root of [a stit: B] at m, then the witness to [a stit: A] at m must be strictly later than the witness to [a stit: B] at m, as formulated in the following corollary.

390

Proofs and models

15-13 COROLLARY. (Companions and witness) Let 97T, m 1= [a stit: A&cCa] with witness w, and 971, m 1= [a stit; B&~[a stit; B&C"*]] with witness w'. Then w' < w. 15-14 COROLLARY. (Validity of A9)

For any model 971, 97T N A9.

PROOF. Suppose that

and let i = i (m) and M = {m': m' £ i & m' =* m}. Then m, M \= Ca, and hence by Lemma 15-7, 971, i >w 1= tu ^ ~[a stit; ^4&[a stit; 5& ~[a stit; 5&C"*]]]. Hence there must be an m' 6 i| >U) such that

and by Lemma 15-9 (ii), 971, m' t= [a stit; B] with witness w'. Set M' = {m": m" 6 i & m" =°, m'}. Then 971, M' 1= [a stit; 5]. Applying Lemma 15-12 to (8) and (9), we have w' < w, and hence by Lemma 15-4^, i\>w C M'. It follows that 971, m E [a stit; B]. D

15C

Soundness: Validity of refref equivalence

In the last section, we showed the validity of all axioms A6-A9 of Lai + rr for all BT + I + AC structures (Lemma 15-7, Corollary 15-10, Theorem 15-11, Corollary 15-14). It is easy to see that for all BT + I + AC structures, all axioms A1-A5 are valid (see §15A) and that the rules modus ponens and RE are "validity-preserving." Thus, in order to establish the soundness of La1 + rr, the only thing left to show is the validity of refref for all BT + I+AC + nbc structures. 15-15 LEMMA. (Doing implies refraining from refraining) Let M be any BT + I + AC + nbc model. Suppose that M, m F= [a stit; A}. Then M, m F [a stit: ~[a stit; ~[a stit; A}}}. PROOF. Let M, m \= [a stit: A] with witness w and counter m', and let i = Since

Suppose for reductio that M, m¥ [a stit: ~[a stit; ~[a stit; A]]]. Then we have by (10) and Lemma 15-7 that

15. Decidability of one-agent achievement-stit theory with refref

391

We claim that there must be an mo € M' such that M, mo F [a stit: A] with witness WQ and counter mo; for if M, M' F= ~[a stit: A], then by (11), M, M' N= [a stit: ~[a stai: A]], contrary to (12). Furthermore, since mo € M', M' = {m": m" € i & m" =° m0}; and since w < mo and w0 < m0, either m < wo or Wo < w by no backward branching. If WQ < w, then, setting MO = {m": m" E i & mO" =aw0 M}, M' C M0 by Lemma 15-4, and hence, since M, MO F= [a stit: A], M, M' F [a stit: A], contrary to (11). So it must be true that w < Wo. Let M'o = {m": m" E i & TO" =awo mo'}. In general, we can inductively define two sequences m0, mi, ... and w0, w1, ... such that for each k > 0, mk + 1 E M''k = {m": m" € i & m"=awkmk} and M, mk + 1 F [a stit: A] with witness Wk + 1 > wk > w and counter m'fe + 1. The proof of this is similar to the earlier one. But {w, WQ, w\, ...} is obviously a busy choice sequence, contrary to our hypothesis. 15-16 LEMMA. (Refraining from refraining implies doing) Let 971 be any BT + I+AC + nbc model. Suppose that 971, m t= [a stit: ~[a stit: ~[a stai: 4]]]. Then 971, m N [a sizi; 4]. PROOF. Let 971, m 1= [a stit: ~[a stai: ~[a stit: A]}] with witness w and counter m*, and let i = i (m ) and M = {m": m" € i & TO" =£ TO}. Then 97t, m* (= [a siit' ~[a stit: A]] with the witness w* ^ w by Lemma 15-8, and hence

Suppose for reductio that 971, m K [a siii: /!]. Since

there must be an mo G M such that 97i, mo N [a sfoi: /IJ with witness WQ and counter m0; for if 071, M 1= ~[a stit: A], then 971, M N (~[a stt .4])Q by (14), and hence by Lemma 15-7, 971, i\>w t= (~[a stit: A ] ) a , contrary to (13). Since m0 € M, M = {m": m" E i & m" =™TOO};and since w < mQ and w0 < TOO, either w < w0 or w0 ^ w by no backward branching. If w0 ^ w, then, setting M0 = {m": m" G i & m" =aw0 mo}, we have M C M0 by Lemma 15-4, and hence, since 971, Mo1= [a stit: A], M, M F= [a stit: A], contrary to the assumption of our reductio. It follows that w < WQ. Let M0 = {m": m" € i & m" =aw0 In general, we can inductively define two sequences mo, m1, ... and w0, mo}. w1 ... such that for each k > 0, mK + 1 € M'k = {m": m" E i & m" =awk mk} and M, mk + 1 F= [a stit: A] with witness wk + 1 > Wk > w and counter m'k+1. The proof of this is similar to the earlier proof. But {w, WQ, w\, ...} is a busy choice sequence, contrary to our hypothesis. D These two lemmas enable us to establish the soundness of Lai + rr. 15-17 THEOREM. (Soundness theorem) For every sentence A, and for every BT + I + AC + nbc structure 6, h- A in La1 + rr only if 6 f= A.

392

15D

Proofs and models

Companion sets

In §15B we studied some basic features of semantic companions to stit sentences. From now on we study the syntactic features of companions. Our proof of the finite model property and the completeness of Lai + rr needs to consider, for each consistent sentence A, a finite set F of sentences including all subsentences of A. From now on, whenever F appears, we presuppose that F is finite and nonempty. Corresponding to §15B, we define (syntactic) companions to stit sentences as follows. Let F be any finite set of sentences, and let $ be any maximal consistent set (MCS) containing [a stit: A}. For any C € F, if [a stit: A&Ca] E $, we call [a stit: A&Ca] a (syntactic) T-pos-companion to [a stit: A] in 0 and m > 0:v

Then for any sentence C, PROOF. As in our proof in the previous lemma, we obtain by applying T7 k times and T15 m times. 15-21 LEMMA. (Properties of companion sets) Let E be any F-companion set for [a stit: A], and let [a stit: B] be the characteristic sentence of E w.r.t. A. Then the following hold: i. For every sentence

394

Proofs and models

and consequently E h [a stit: B&C"*] or E F ~[a stit: 5&C"]; ii. If C"i, ..., Ck, and DI, ..., Dm are, respectively, all the F-pos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E, then

and consequently

PROOF, (i) By Lemma 15-20 and A2, we need only to show one direction of each equivalence. Suppose that C £ F and £ I- [a stit: A&Ca]. Since F [a stit: B] D Ca by A2, it follows from R3 that E F [a stit: B&C"]. Suppose next that E E ~[a stit: A & C a ] . Then E T [a stit: A&~[a stit: A&Ca}] by A8 since E T [a stit: A]. We show as follows that E F [a stit: B&~[o; stit: B & C a ] ] . By Lemma 15-20, F [a stit: B & < C a ] D [a stit: A&C a] . So E T ~[a stit: B & C a ] , and hence, T [a stit: B] D ~[a stit: B&C"*]. It follows from R3 that E h [a stit: B&~[a stit: B & C a ] ] . (li) By (i), we have that F [a stit: B] D [a stit: B&Cai] for all i with 1 < i < K, and that F [a stit: 5] D [a stit: B &~[a stit: 5 &£>"]] for all j with 1 <j < m. Then we can complete the proof by applying T7, T15, and A4. A direct consequence of Lemma 15-19 and Lemma 15-21 is the following. Let E be the F-companion set for [a stit: A] in an MCS $ with C\, ..., Ck, and DI, ..., Dm to be, respectively, all the F-pos-companion roots and all the Fneg-companion roots of [a stit: A] w.r.t. $. Let BQ, BI, B%, ... be defined as follows:

and let £„ = DC([a stit: Bn + i\) for all n > Q. Then E = E0 = EI = .... That is to say, for each n > 0, E is the F-companion set for [a stit: Bn] in $ (with [a stit: Bn + i] to be the characteristic sentence of E w.r.t. Bn), and [a stit: Bn] has the same F-pos-companion roots and the same F-neg-companion roots w.r.t. E as [a stit: A] has. 15-22 LEMMA. (Properties of companion sets) Let E be a F-companion set for [a stit: A]. Then for every sentence C 6 F,

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PROOF, (i) By A2, it suffices to suppose that E h Ca and show that E h [a stit: A&Ca]. Let [a stit: B] be the characteristic sentence of E w.r.t. .A. Since h [a stit: B] D Ca*, it follows that E h [a stit: B & < C a ] by R3, and hence E 1- [a stit: A & C a } by Lemma 15-21 (i). (ii) Suppose that E \- C and E F [a stit: C]. We know that Eu{~[a stit: C}} is consistent, and hence it must be included in some MCS &[a stit: D]. Then by Lemma 15-28(Hi), n P Da and II h Ca. Since II h A& C by A2, II h- £>&[« stit: D] by Lemma 15-22(ii). It follows that II h [a stit: A&C}&~[a stit: C], and hence II h [a

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stit: j4& Ca] by A6. The other direction of the proof is similar, (ii) Let $ be any MCS including E, and let £' be the F-companion set for [a stit: A] in $. By Lemma 15-24, E contains all the F-companions to [a stit: A] in $. Setting ^ to be any MCS including II, we know for the same reason that Ii contains all F-companions to [a stit: A] in ty. According to (i), each F-companion to [a stit: A] in $ must be in \? and vice versa. It follows that E' C EnII. D 15-30 LEMMA. (An inclusion relation among companion sets concerning a particular stit sentence) Let E and E' be F-companion sets in an MCS <J>, let II be a F-alternative to E such that E' C EnII, E h [a stit: A], II h [a stit: A] and £' V- [a stit: A], and let [a stit: B] be any characteristic sentence of £'. Suppose that F is equivalently closed under conjunction, A € F, and E" is the F-companion set for [a stit: A&[a stit: B}} in $. Then £' C E" C Enll. PROOF. Since E" h [a stit: B] by A2, it is then clear that E' C E". To show that E" C Enll, we need only to show that E and II contain all the F-companions to [a stit: A&c[a stit: B]} in $. To that end, we first note that since A € F and E h [a stit: A] and II h [a stit: A], it follows from Lemma 15-24 and Lemma 15-29 that

Consider any F-pos-companion root C of [a stit: A] w.r.t. $. Since £' C Enll, E h [a stit: AkCa]k[a stit: B} and II h [a stit: A&Ca}&[a stit: B} by (17). It then follows from A3 and A4 that E h [a stit: A&[a stit: B}&Ca] and II h [a stit: A&[a stit: B}&Ca}. Hence by Lemma 15-26 (substituting E' for E in that lemma), E and II contain all the F-pos-companions for [a stit: A & [a stit: B]] in $. Consider any F-neg-companion root D of [a stit: A] w.r.t. $. By (17), we have

It follows from E' C Enll that E' P [a stit: A] D [a stit: A&Da] (otherwise we would have E h ~[a stit: A] and II h ~[a stit: A]), and hence by Lemma 15-25, E' y [a stit: B&Da}. Then £' h ~[a stit: B&Da] by Lemma 15-21 ft), and hence by A5, E' h ~[a stit: [a stit: B]&Da]. It follows from Rl and T4 (by contraposition) that

Hence by (18), we have E h ~[Q stit: A&[a stit: B]&Da] and H h ~[a stit: A &[a stit: 5]&Z) a ]. It follows from Lemma 15-26 that E and H contain all the F-neg-companions—and hence all the F-companions—to [a stit: A&[a stit: B]} in $, that is, E" C EnH. This completes the proof. D II is a F-counter to £ relative to [a stit: A] iff H is a F-alternative to E, E h [a stit: A], and IIV- A. Assume that F is equivalently closed under conjunction. H is a T-semi-ref-counter to E relative to [a stit: A] iff

400

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II is a T-ref-counter to E relative to [a stit: A] if II is a F-counter to E relative to [a stit: A] and II h [a stit: ~[a stit: A}]. It is easy to see that ref-counters are semi-ref-counters. Note that when II is a F-counter (F-semi-ref-counter, F-refcounter) to E relative to [a stit: A], E does not have to be a F-companion set for [a stit: A}; but when F is equivalently closed under conjunction, by Lemma 1524, S does contain all the F-companions to [a stit: A] w.r.t. any MCS including S. As we said in the last section, a F-companion set E may represent a possible choice for a at a moment. In the same way, II, as a F-semi-ref-counter to E, may represent another possible choice for a at that moment. In the structure that we will construct in the proof of completeness and the finite model property, if S represents one choice for a at a moment, then II will represent the other choice for a at that moment (i.e., a has only binary choice). The semi-refcounter relation between F-companion sets is so important for our proof that we introduce the following notation for it. Let F be equivalently closed under conjunction. R£ (S, FT) iff the following conditions hold.

Note that whenever we use R£(E, II) in a discussion, we presuppose that F is equivalently closed under conjunction. Comparing (19), (20), and (21) earlier and (22), (23), and (24), the definition of -Rp(E, II) may not appear to give us the semi-ref-counter relation. But Lemma 15-31 guarantees that RT (E, II) iff [a stit: A] 6 F and II is a F-semi-ref-counter to S relative to [a stit: A], provided that F satisfies the following condition: F is closed under negated stit sentences if for every sentence A, [a stit: A] £ F implies ~[a stit: A] e F.3 3

Note that the notion Rp (E, II) is sufficient for our proof of the completeness and the finite model property. flp(E, II) does not imply that II is a F-semi-ref-counter to E relative to [a stit: A] if F does not have the property of being closed under negated stit sentences. That is to say, F's property of being closed under negated stit sentences is not essential for our proof. We give F this property only because it makes fip (E, II) match the notion of F-semi-ref-counters, while the ideas of F-semi-ref-counters and F-ref-counters are easier to follow than the idea of fl£(E, n).

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15-31 LEMMA. (Serm-ref-counter relation and RA ) Let F be equivalently closed under conjunction. Then

and if for every and if also

is a

pos-companion root of

is closed under negated stit sentences, then and is a semi-ref-counter to relative to

PROOF, (i) Assume that RA(£, II). We first show that I I P [a stit: A], which, by definition, is true when there is no F-neg-companion to [a stit: A] in E. Assume that D\, ..., Dm (m ^ 1) be all the F-neg-companion roots to [a stit: A] w.r.t. E. Suppose for reductio that E h [a stit: A}. Then, on the one hand, we know by definition that II I- V i< »«s m [a stit: A&D?]. On the other hand, Lemma 15-29 provides H t- &i^ is^m ~[a stit: A&cD?}. Hence II is inconsistent, contrary to our assumption of consistency of H. It follows from this reductio that H Y- [a stit: A}. We next show that H Y- A. Suppose for reductio that H h- A. Then by Lemma 15-22(ii) and II Y- [a stit: A], Ft h v4&~[a stit: A], and hence E h ~[a stit: A] by Lemma 15-28 (in), contrary to our assumption of consistency on E. It follows from this reductio that E V- A. (ii) Suppose that RA(S, II), ~[a stit: A] £ F and that every C 6 F is a F-pos-companion root of [a stit: A] w.r.t. E. Then by RA(E, II), II h ~[a stit: A}. Suppose for reductio that II Y- [a stit: ~[a stit: A}}. Then II h ~[a stit: ~[a siii: 4]] by Lemma 15-22^, and hence by Lemma 15-28(Hi), E h ~[a stit: A], contrary to our assumption of consistency on E. It follows from this reductio that II h [a stit: ~[a stit: A]]. It is easy to see that (Hi) follows from (i), (ii) and related definitions.

15F

Semi-ref-counters

From our discussion in the last two sections, one can see that F's property of being equivalently closed under conjunction is important for our proof. We now show that for every sentence A, we can associate a finite set F that has this property. Let A be any sentence, and let I" be the set of all subsentences of A, and let F* = F'U{~[a stit: C]: [a stit: C] e F'}. We define FA = {5j&...&5 n : BI, ..., Bn are distinct sentences in F*}. In the following lemma, for any set X = {Bi, ..., Bn} of sentences, &X = Bi&...&B n . 15-32 LEMMA. (Finding a set that has the desired properties) Let A be any sentence. Then TA is a finite set that is closed under subsentences, closed under negated stit sentences, and equivalently closed under conjunction. PROOF. It is easy to see by definition that TA is a finite set that is both closed under subsentences and closed under negated stit sentences. To show that TA

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is equivalently closed under conjunction, let B = £?i&...&5 m and C = C"i& ...&C n be any sentences in YA such that I ^ m, n ^ |F*|, and BI, ..., Bm are distinct sentences in F*, and C\, ..., C n are distinct sentences in F*. We show as follows that there is a D e FA such that 1- BhC = D. Let Conj(5) = {#!, ..., 5m} and Conj(C) = {C1; ..., Cn}. Clearly, if Conj(5) = Conj(C7), we can set D = B. Suppose that Conj(5) 7^ Conj(C'). Let us set X = Conj(5) -Conj((7), Y = Coiy(C)-Coiy(5), Z = Conj(B)nConj(C r ), B' = &X, C" = &Y, and E = &Z. It is easy to see that X, Y, and Z are disjoint subsets of F* and h 5'& C'hE = £& C. We can thus set D = £'& C"&£ to complete the proof. In this section we focus on F-semi-ref-counters to F-companion sets. From now on, whenever F appears in our discussion, we presuppose that F is a finite set that is closed under subsentences, closed under negated stit sentences, and equivalently closed under conjunction.4 15-33 LEMMA. (A property of semi-ref-counters corresponding to the absence of busy choice sequences) Let S, II and E' be any F-companion sets such that ,Rr(S, II), n C E' and E' h [a stit: A}. Suppose that there are n + l F-negcompanions to [a stit: A] in E. Then there are at most n F-neg-companions to [a stit: A] in S'. PROOF. We first show that

Consider any C € F. Suppose that E h [a stit: A&, Ca}. Then E h Ca by A2, and hence H h Ca by Lemma 15-28(Hi). Because H C S' and E' h [a stit: A], it is then clear that H F ~[a stit: A}. Since -Rp(E, H), II Y- [a stit: A] by Lemma 15-31 (i). Hence H h [a stit: A] D [a stit: A&Ca} by Lemma 15-25, and hence E' h [a stit: A&Ca}. It follows that (25) holds, which implies by Lemma 15-24 that

Setting DI, ..., Dn + i to be all the F-neg-companion roots of [a stit: A] w.r.t. E, we have by definition of R£ (E, II) that II h [a stit: A] D V i«s z< n + i [a stit: AkD?}. Hence E' H V i < t $ n + i [a stit: AkD?]. Since, by (26), S' contains all the F-companions to [a stit: A] in any MCS including S', it follows that there is a k such that 1 ^ k ^ n + l and E' h [a stit: A8z.D%}. Hence by Lemma 15-24, there are at most n F-neg-companions to [a stit: A] in S'. This completes the proof. D 4 As we said in note 3, the property of F of being closed under negated stit sentences is not essential. This property has no effect in our upcoming discussions, except that it helps one to see some of the ideas of our proof (by referring to Lemma 15-31).

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A direct consequence of Lemma 15-33 is the following: Let n > 0, and let EQ, • • • > E n and HO, ..., IIn be F-companion sets such that for each i with 0 ^ i ^ n, .R r (£j, II,), and for each i with 0 ^ i ^ n —1, IIj C EI+ I. Suppose that there are n F-neg-companions to [a stit: A] in EQ. Then Hn must be a F-ref-counter to E n relative to [a stit: A], and there can be no E and II such that -Rp(E, LT) and nra C E. This feature of F-semi-ref-counters corresponds to a semantic feature of Lai + rr—there is no busy choice sequence in its models. The following three lemmas establish some sufficient conditions for F-semiref-counters. 15-34 LEMMA. (A sufficient condition that semi-ref-counters exist) Let E be a F-companion set for [a stit: A] with [a stit: B] to be the characteristic sentence of E w.r.t. A. Let Ci, ..., Ck and D I , ..., Dm be, respectively, all the Fpos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E. Suppose that Ft = DC([a stit: ~[a stit: J5J&C 1 ]) is consistent, where C = &i< z < k C™. Then II is a F-companion set for (a stit: ~[a stit: B}} and is a Falternative to S, and II h- fa stit: A] D V i ^ j < m la s^: A&D"]. In particular, then PROOF. We first show that H is a F-companion set for [a stit: ~[a stit: B}} and is a F-alternative to £. To that end, we first note that by T7, II h [a stit: ~[a stit: B]}. Consider any D = D} with 1 ^ j < m. We know by Lemma 15-21 (ii) and T16 that

Hence h ~[a stit: ~[a stit: B]&Da} by Rl. It follows from hypothesis and Rl that

Hence by Lemma 15-19, II is a F-companion set for [a stit: ~[a stit: B}}. It is easy to check that H is a F-alternative to E. Next, since II h [a stit: ~[a stit: B]bC], and since h ~[a stit: C} by T6, it follows from Rl that H h [a stit: ~[a stit: B&~[a stit: C]]&.Ca]. Because B is the sentence v4&(7&;(&i^j^ m ~[a stit: A k D f } ) , setting B* = ^ & ( & i < j < m ~[a stit: A&DJ*]), we have II h [a stti: ~[a siii: B'&^j&C 1 0 ], and hence by T14, U\-[a stit: ~[a stti: 5*] & (7°]. It follows by A2 that II h ~[a sfit: B*], and hence by Lemma 15-27, II h [a sttt: v4] D V i 5 S j = £ m ta s^; ^&-0"]- If there is no F-neg-companion to [a stit: A] in E, then B = A& C, and hence by T6, II h [a stit: ~[a stai: ^4& Q ]; and applying (30) and Rl to Til, we have I- ~[a stit: [a stit: £]&£>"]. It follows from T4 and Rl that I- ~[a stit: ~[a stit: B']&[a stit: 5]&D a ]. This completes our proof of (28). Hence II' is a F-companion set for [a stit: ~[a stit: B'} &[a stit: B]] and a F-alternative to E'. It is easy to see that E C E'nll' since E' h [a stit: B} and II' h [a siii: 5]. We show next that #r( s / > n / )- Since A e F and E' h A by A2, £' h [a stai: A] by Lemma 15-22 (h). Let B* and C1* be the following sentences:

Then by Rl, h [a stit: B1} = [a stit: 5*&[a stit: 5]&C*], and hence by hypothesis and A2, H' h ~[o; stit: 5*&[a siii; 5]&C7*]&[o! siif; B]. On the one hand, by applying A4 (by contraposition), we have II' h ~[a stit: B*]V ~[a stit: [a stit: B]&C""]. On the other hand, by applying Lemma \5-2l (ii) and T10, we have II' h [a stit: [a stit: 5]& C1?] for all i with 1 < i ^ fc (since IT h [a stit: B } ) , and hence by A4, II' h [a stit: [a stit: B}&C*}. It follows that II' I- ~[a stit: B*} (when there is no F-neg-companion to [a stit: A] in E', the same argument here, replacing B* by A, yields II' I- ~[a stit: A}). Hence by Lemma 15-27 (substituting A&[Q stit: B] for A in the lemma), we know that

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Since II' h [a stit: B] and h [a siz'i: 4]&[a sizi: B\ D [a s£zi: A&[a stit: B}} by A3 and A4, it follows that

We also know by Rl, T4, and A5 that for each j with 1 ^ j ^ TO,

But for each such j, £ h ~[a s£z't: 5&.D"] by hypothesis and Lemma 15-21 (ii), and hence IT h ~[a siif: 5&D°] by £ C n'. It follows from (31), (32), and (33) that

We have shown that £' h [a s£z't: .4], and hence by Lemma 15-24, £' contains all the F-companions to [a stit: A] in $. Since £ .K [a stit: A], we know by Lemma 15-26 that D I , ..., Dm are all the F-neg-companion roots of [a stit: A] w.r.t. E'. Hence by definition, R$(Z', IT). D 15-36 LEMMA. (A sufficient condition for extending a companion set to another and its semi-ref-counter) Let £ be a F-companion set in an MCS $ such that £F [a stit: A] and [a stit: A] e 3>nF, let E' be the F-companion set for [a stit: j4&[a stit: B}] in $, where [a stit: B] is a characteristic sentence of E, and let [a stit: B'} be the characteristic sentence of £' w.r.t. A&[o; stit: B]. Suppose that there is a C 6 F such that

and II' = DC([a stit: ~[a stit: 5']&C"]) is consistent, where C' = fci^j^fc C? and Ci, ..., Ck are all the F-pos-companion roots of [a stit: A&[a stit: B\] w.r.t. $. Then #r(E', n') and E C E'nn'. PROOF. Since A e F and E' h A by A2, it follows from Lemma 15-22(ii) that £' h [a sizi: yl]. By Lemma 15-34 we know that II' is a F-companion set for [a stit: ~[a stit: B'}} and is a F-alternative to E', and also, setting D I , ..., Dm to be all the F-neg-companion roots of [a stit: A&.[a stit: B}] w.r.t.

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