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Logics for Conditionals LOGICS FOR CONDITIONALS LOGICS FOR CONDITIONALS ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor in de Wijsbegeerte aan de Universiteit van Amsterdam, op gezag van de Rector Magnificus dr. D.W.Bresters, hoogleraar in de Faculteit der Wiskunde en Natuurwetenschappen, in het openbaar te verdedigen in de Aula van de Universiteit (tijdelijk Roetersstraat 15), op vrijdag 4 oktober 1985 te 14.30 uur door FRANCISCUS JACOBUS MARIA MARTINUS VELTMAN geboren te Eindhoven Pnomoioneb: prof.dr. J.F.A.K. van Benthem,Rijksuniversiteit, Groningen prof.dr. J.A.W. Kamp,University of Texas, Austin ACKNOWLEDGEMENTS I do not know to whomI am most indebted, to Johan van Benthem or to Hans Kamp. What I do know is that this would have been an altogether different thesis if I had not had the privilege of having mHfi.ofthem as supervisors. With his gift for anticipation Hansis the ideal person to air ones views to - even long before they deserve it. Muchof what is new in the following was inspired by discussions with him. But where Hans was the one who most encouraged me to follow up my own ideas, Johan saw to it that I did not get too wild about them. He has always urged me to put things in a broader perspective and his commentsand questions determined the final shape of this book. I amgrateful also to myteachers at the University of Utrecht, my former colleagues at the Erasmus University in Rotterdam, and mypresent colleagues at the University of Amsterdam.In this connection I want to mention in particular Jeroen van Rijen, Dick de Jongh, Roel de Vrijer, Jeroen Groenendijk, Martin Stokhof and Fred Landman: Jeroen van Rijen's ideas on the philosophy of logic greatly influenced mine; the proofs in this book have benefited much from Dick de Jongh's mathematical taste; Roel de Vrijer carefully read the manuscript and suggested several important improvements on the presentation; Jeroen Groenendijk's and Martin Stokhof's preoccupation with the role of background information in matters of semantics and pragmatics has been of great importance for the development of my own views; and Fred Landmangets the prize for asking the nastiest questions. As will become clear from the notes many other people have by their suggestions and criticism contributed to the contents of this book. I cannot here mention each of them by name, but I amgrateful to all of them. As for editional matters, I amgreatly indebted to Michael Morreau who tried his utmost to turn my double Dutch into readable English. Marjorie Pigge typed this book and I want to thank her for the great care with which she did so. This book has been nearing completion for quite some time. Nowthat it is finally finished, I wouldlike to dedicate it to all those whoare relieved that it is. Amsterdam Frank Veltman August, 1985 CONTENTS Acknowledgements PART I: THE PROBLEM OF CONDITIONALS 1 I.1. METHODOLOGICAL REMARKS 1 I.1.1. The case of the marbles 2 I.1.2. Logic as a descriptive science 5 I.1.2.1. Therationalist tradition 7 1.1.2.2. The empirical approach 9 I.1.3. -Amore pragmatic View 14 I.2. EXPLANATORYSTRATEGIES 22 1.2.1. Logical validity 23 I.2.1.1. The standard explanation 23 I.2.1.2. Truth and evidence 29 1.2.1.3. Probability semantics 32 I.2.1.4. Relevance Logici 38 I.2.2. Pragmatic correctness 40 I.2.3. Logical form 44 49 Notes to Part I PART II: POSSIBLE WORLDS SEMANTICS 52 (mainly on counterfactual conditionals) II.1. PRELIMINARIES §? II.1.1. Worlds and propositions 52 II.1.2. Logical notions 61 II.2. DELINEATIONS Bé II.2.1. Constraints on consequents 58 II.2.2. Constraints on antecedents 78 II.3. ALTERNATIVES gb II.3.1. Ramsey's suggestion and Stalnaker‘s theory 90 II.3.2. Comparative similarity according to Lewis 99 II.3.3. Premise semantics 103“ II.3.4. Comparative similarity induced 113 II.4. MODELTHEORETIC RESULTS 118 II.4.1. Comparingordering functions and selection 119 functions II.4.2. The completeness of 9 126 II.4.3. Extensions of P 132 Notes to Part II 143 PARTIII: DATA SEMANTICS &48 (mainly on indicative conditionals) III.1. SEMANTIC STABILITY AND INSTABILITY 149 III.1.1. Information models 149 III.1.2. Betweenassertability and truth 155 III.1.3. Data logic. Preliminaries 170 180 III .2. PRAGMATIC CORRECTNESS AND INCORRECTNESS III.2.1. Gricean constraints 181 III.2.2. Oddconditionals 183 III.2.3; A test for pragmatic correctness 188 III.3. DATA LOGIC 192 III.3.1. Deduction principles 192 III.3.2. Completeness and decidability 195 III.4. PROBLEMS AND PROSPECTS 202 III.4.1. Data lattices 202 III.4.2. Complexconditionals 207 III.4.3. Counterfactuals 210 Notes to Part III 213 References 218 Samenvatting (summary in Dutch) 223 PART I THE PWROBLEM OF CONDITIONALS I.1. METHODOLOGICAL REMARKS To those whobelieve that there is such a thing as the logic of conditionals this dissertation mayappear to be yet another attempt to unravel its secrets - which it is not. As a logician, you can do no more than devise a logic 501 conditionals and try to persuade your readers to adopt it. Youmay succeed in doing so if you are able to demonstrate that the one you propose is a better logic for conditionals than the ones proposed so far. The phrase ‘better for conditionals‘ should, however, not be misunderstood. In particular, it should not be interpreted as meaning ‘morelike the real one’. The best logic for conditionals one might propose is not that which they actually possess. It cannot be, not because this actual logic would be not good enough, but simply because there is no such thing. Whethera given logic is better than somealternative has little to do with its better fitting the facts; it is more a question of efficacy. This is one of the theses defended in the following introductory pages. It is put forward whenthe question is discussed as to howone should choose between rival logical theories. That is a very natural question to ask in an introduction to the problem of conditionals. If only because so manytheories have been put forward, all purporting to solve the problem, that putting yet another one on the market might seemto only add to the difficulties. I.1.1. The case of the marbles Here is an example which will return regularly in muchof the following. There are three marbles: one red, one blue, and one yellow. They are known to be distributed among two matchboxes, called 1 and-2. The only other thing which you are told is that there is at least one marble in each box. The various possibilities which this leaves open can be summarized as follows: box 1. box 2 I blue yellow, red II yellow ' blue, red III red blue, yellow IV yellow, red blue V red, blue yellow VI blue, yellow red Bearing these possibilities in mind, you will have to agree that (1) The yeflflow manbfie LA in box 1, £5 both 05 the othenb ahe to be fiound in box 2' ' ' Nowsuppose someone no better informed than yourself were to claim that (2) The ye££ow*manb£e.tAtn box 7 t5 the btue manbte to tn box 2 Youwill disagree. Youmay even go as far as to assert (3) It tbiuflibo that t5 the btue manbteto tn box 2, the yettow one to tn box 1 After all, the yellow marble could just as well be in box 2 together with the blue one, as long as the red one might yet be in box 1. Things are different, however, if it is excluded that the red marble is in box 1. i In that case, (2) holds. That is, (4) 15 the ted manbte to tn box 2, then t5 the btue manbte to tn box 2 at wett, the ye££ow'manbte to tn box 1 For those whodo accept the last two statements there is a surprise. in . store. Using . standard logical . notation . 1 ), we see that sentences (3) and (4) are respectively of the form (3') -(b£ue tn 2-» yettow tn 1) (4‘) ned tn 2 a-(btue tn 2-» ye£tqw.tn 1) So we can apply the principle of ModusTollens to (3') and (4') and conclude (5') ~ned tn 2 That is, (5) Ititt not the cate that the ted manbteto tn box 2 But this conclusion will be a lot less acceptable than the premises (3) and (4) might have seemed.2) Is ModusTollens playing up here, or should (3) or (4) be rejected after all? Anywhohave been confused by the above will feel obliged to choose sides. Somemaychoose to stay with their initial intuitions about (3) and (4), regarding the example as evidence that ModusTollens sometimes fails. But what kind of evidence is this? Whyshould intuitions be treated with this kind of respect, especially where others do not share them? Choosing instead to do something about (3) and (4), and thus save the principle of ModusTollens, would seem a lot safer: all3) existing theories of conditionals will support this. Even so, one would not have an easy time working things out. For although everyone is agreed that something must be done, there is no consensus to be found in the literature as to precisely what it should be. Sometheories will advise you to reject (3), other ones to reject (4). Often, however, the response is more sophisticated, amounting to a denial that we are dealing with a proper instantiation of ModusTollens here.
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