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Content IFFY CONFIDENCE by Johannes V. Schmitt A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHILOSOPHY) August 2012 Copyright 2012 Johannes V. Schmitt Acknowledgments I would like to especially thank Kenny Easwaran, Malte Willer and Shyam Nair for their very helpful comments both concerning the technical implementation of the major ideas in this work and the philosophical motivation of these ideas. I would also like to thank Karen Lewis, Dorothy Edgington, Seth Yalcin, Jim Higginbotham, Elena Guerzoni, Barry Schein, Viola Schmitt, Manuel K riz, Stefan Kaufmann, Jim Higginbotham, George Wilson and the members of the philosophy dissertation work group at USC for discussion of various aspects and themes of the project. I am espe- cially indebted to my doctoral advisor Mark Schroeder, who has given me incredibly detailed comments and extremely helpful (strategic) advice at every single stage of this project. Finally, I would like to thank two Frenchmen, Mika el Cozic and Paul Egr e, for spurning my interest in these issues with their excellent mini- seminar at ESSLLI 2008 in Hamburg, Germany. ii Table of Contents Acknowledgments ii List of Figures vi Abstract vii 1 Introduction 1 1.1 Conditionals and Hypothetical Reasoning . . . . . . . . . . . . . . . . 1 1.2 Two kinds of hypothetical reasoning . . . . . . . . . . . . . . . . . . . 3 1.3 The scope and the structure of this project . . . . . . . . . . . . . . . 6 1.4 The Horseshoe Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 The Ramsey Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.1 Possible-Worlds Views . . . . . . . . . . . . . . . . . . . . . . 22 1.5.2 The AGM framework . . . . . . . . . . . . . . . . . . . . . . . 27 1.6 Non-descriptivism about conditionals . . . . . . . . . . . . . . . . . . 34 2 Simple Credal Expressivism 47 2.1 Expressing Credal States . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Probability and Credences . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 The Axioms of Probability . . . . . . . . . . . . . . . . . . . . 51 2.2.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 55 2.2.3 Conditionalization . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Adams' Thesis and Adams' Logic . . . . . . . . . . . . . . . . . . . . 64 2.3.1 Assertibility and Conditional Belief . . . . . . . . . . . . . . . 64 2.3.2 Probabilistic Validity . . . . . . . . . . . . . . . . . . . . . . . 66 2.3.3 Adams' logic and uncertainty-unfriendly reasoning . . . . . . . 69 2.3.4 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3.5 Upshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4 SCE and Triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 iii 2.4.1 The Bombshell . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4.2 How robust are these results? . . . . . . . . . . . . . . . . . . 82 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3 The dynamic view 91 3.1 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2 Epistemic Commitment and Commitment States . . . . . . . . . . . . 94 3.3 The Recursive Denition of the Update-Function . . . . . . . . . . . 100 3.4 How well does the dynamic system do? . . . . . . . . . . . . . . . . . 110 3.4.1 The Or-If inference . . . . . . . . . . . . . . . . . . . . . . . . 113 3.5 The Dynamic System vs. SCE . . . . . . . . . . . . . . . . . . . . . . 120 3.5.1 Does the theory account for conditionals' important role in uncertain reasoning? . . . . . . . . . . . . . . . . . . . . . . . 122 4 Conditionals and Uncertain Reasoning 126 4.1 The Plan for this Chapter . . . . . . . . . . . . . . . . . . . . . . . . 126 4.2 Why does condence matter? . . . . . . . . . . . . . . . . . . . . . . 129 4.3 A rst proposal - Generalizing the Update-operation . . . . . . . . . 131 4.4 Rethinking CUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.1 Condence and Graded Modals . . . . . . . . . . . . . . . . . 136 4.4.2 Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.4.3 CUS with single probability functions . . . . . . . . . . . . . . 150 5 A probabilistic dynamic approach 168 5.1 Representing probabilistic information by sets . . . . . . . . . . . . . 168 5.1.1 Bayesian Closures . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2 The positive account . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.2.1 Fixing Negation requires rejecting exhaustivity . . . . . . . . . 181 5.2.2 CUS + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3 Alternate Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.4 Non-extendibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.5 Appendix to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.5.1 Construction of Q-closures . . . . . . . . . . . . . . . . . . . . 195 5.5.2 Proofs of Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 Can CUS + deliver on the promises? 201 6.1 In search of the right theory . . . . . . . . . . . . . . . . . . . . . . . 201 6.2 Uncertainty-unfriendly arguments in CUS + . . . . . . . . . . . . . . . 204 6.3 Embedded conditionals inL ! . . . . . . . . . . . . . . . . . . . . . . 209 iv 6.3.1 The second desideratum . . . . . . . . . . . . . . . . . . . . . 209 6.3.2 Right-nested conditionals . . . . . . . . . . . . . . . . . . . . . 210 6.3.3 Conjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.3.4 Disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.3.5 Left-nested conditionals: . . . . . . . . . . . . . . . . . . . . . 215 6.4 Remarks on the logic of CUS + . . . . . . . . . . . . . . . . . . . . . . 222 6.4.1 Transmission Principles . . . . . . . . . . . . . . . . . . . . . . 223 6.4.2 Expressive limitations . . . . . . . . . . . . . . . . . . . . . . 224 6.5 Appendix to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Bibliography 233 v List of Figures 2.1 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2 Conditionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Ordering Worlds - Why does (3*) hold? . . . . . . . . . . . . . . . . . 149 5.1 Gratuitous Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Bayesian closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.3 Combining functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 vi Abstract In Iy Condence, I investigate indicative conditional sentences and assimilate two non-descriptivist theories of the meaning of these sentences, a simple expressivist theory and theory using the resources of dynamic semantics. This assimilation is made possible through the construction of a non-standard dynamic semantic frame- work. The dynamic semantic framework relies on a more ne-grained representation of information, a representation in terms of sets of probability functions, and enables us to vindicate central commitments of both theories while extending the range of sentences involving conditionals that each of the theories makes predictions about. I argue that defenders of non- descriptivism about conditionals should abandon the simple expressivist view and adopt an extended dynamic theory that relies on my construction. vii Chapter 1 Introduction 1.1 Conditionals and Hypothetical Reasoning This dissertation investigates hypothetical reasoning, i.e. reasoning in the face of uncertainty or less than full condence. Hypothetical reasoning is reasoning starting from various hypotheses or assumptions about the state of the world. Forming hy- potheses and reasoning from those hypotheses to certain conclusions is an important part of the intellectual life of rational beings. Given the signicance of this kind of reasoning for decision-making, planning and various forms of rational inquiry it is not surprising that it has had an impact on the grammatical structure of natural languages like English. 1 When we utter (well-formed) sentences like `Suppose there was a major earth- quake on Tuesday...' we seem to succeed at entertaining an hypothesis that describes some kind of scenario, which then allows us to wonder what else is true in the envis- aged scenario. English has many expressive resources that allow (or invite ) us to talk about hypotheses (e.g. `Suppose that...'), but there seems to be one sentence type that functions as if it were designed to help us articulate reasoning from hypotheses.The sentence type in question is the conditional sentence (or conditional for short), exem- plied by sentences like `If the train is late again, she is going to miss the board meet- ing', `If you don't quit smoking, you'll get lung cancer' or even (like in Hardcastle's example) `If I call a horse's tail a leg, a horse has ve legs'. The term `conditionals' as it is going to be used refers to a grammatical category of sentences instantiated by most natural languages. Generally speaking, the if-clause or antecedent seems to enshrine one or more hypotheses about the world and the consequent contains a statement about what is the case or, sometimes, what is to be done in the light of these hypotheses. Hypothetical reasoning does not require certainty with respect to the inferences that are being drawn just like thinking about the future does not require certainty in the predictions that are being made. For example, we can all speculate about what 2 would happen if Singularity (i.e. greater-than-human intelligence through techno- logical progress) occurred and think that certain scenarios are more plausible than others but that none are plausible enough to bet large amounts of money on them. Accordingly, if conditionals are really linguistic devices for conducting hypothet- ical reasoning, we would expect there to be at least some conditionals that can be accepted in situations in which the speaker is less than fully condent that the con- sequent holds given the antecedent. Throughout most of this dissertation I will give particular attention to conditionals that exhibit this very feature of being uncertain. The main reason why these conditionals are of special interest is that they lie at the intersection of two important and well studied modes of reasoning: Uncertain reasoning and hypothetical reasoning. 1.2 Two kinds of hypothetical reasoning We can entertain hypotheses that we know and presuppose to be false and we can reason from those hypotheses, as in the sentence `If she had been born in Chicago she would have attended Illinois State University' . But we can also entertain hypotheses that we do not know to be false (and that we do not know to be true either) as in `If she was born in Chicago, she attended Illinois 3 State University'. 1 This form of hypothetical reasoning plays an important role in guiding our decisions on what to do when the hypothesis is no longer a hypothesis, but something that we believe or accept to be true about the world. Philosophers have often thought that these two dierent ways of engaging in hy- pothetical reasoning correspond to dierent grammatical features of the sentences that are used to express the thoughts that gure in our reasoning. Counterfactual reasoning has been grouped with conditionals that are at least partly in the subjunc- tive mood and the kind of hypothetical reasoning that departs from hypotheses that have not been ruled out has been linked with conditionals in the indicative mood. This neat and clear-cut correspondence is actually not quite tenable, but I shall not worry about various issues arising from categorizing conditionals in dierent ways. 2 In this dissertation I shall be almost exclusively concerned with the kind of hy- pothetical reasoning that is epistemically open. In the case of an epistemically open conditional the agent is typically not opinionated with respect to the question posed 1 We can also entertain hypotheses that we know to be in con ict with some law of nature or even some law of mathematics. In this dissertation, I won't be concerned with so-called counterlegals like `If the gravitational constant had been less than 9.81 m s 2 , the record for pole vaulting would be more than 6.15 meters'. I shall also not consider counterpossibles and their subclass,counteridenticals, e.g. `If I were Julius Ceasar, I would not have invaded Gaul'. 2 An example for a subjunctive that is not a counterfactual is the following conditional: `If the patient had mono(nucleosis) he would be showing the very symptoms that he is actually showing.' See Edgington (1995) 4 by the antecedent of the conditional, either because she has not considered it or because she has considered it and suspended judgment on it. 3 At any rate, she does not disbelieve the antecedent. It follows that in order to entertain the hypothesis that is encoded in the antecedent, the agent does not have to engage in the kind of belief revision that involves a contraction of her belief set. This is good news in so far as modeling belief contraction involves a number of challenges. 4 The focus of this dissertation is on sentences in the English sub-category of the category `conditionals', but most of the discussion occurs at a level of abstraction far removed from specic features of natural languages and so could be applied, mu- tatis mutandis, to Bulgarian, Yiddish or Latin. 5 More specically, I take it that the conditional that I am going to be interested in is customarily expressed in English by `If... (then)'- clauses in the indicative mood (with optional resumptive pronoun `then'). However, some apparent `non-conditionals' seem to be conditional in spirit, 3 By bracketing other types of hypothetical reasoning I do not wish to endorse a so-called `Apartheid-view' of the conditional (a very poor choice of a metaphor), on which the dierent kinds of reasoning have nothing in common. 4 Ernest Adams' famous example `If Oswald didn't kill Kennedy, someone else did' by way of which he tried to shed light on the important distinction between indicative conditionals and subjunctive conditionals shows that accepting or even evaluating indicative conditionals may at least sometimes require substantial amount of belief-revision, at least for those of us who are Warren-ites. 5 Conditionals have acquired proverbial status in many languages. A famous Latin proverb says Si tacuisses, philosophus mansisses (`If you had been silent you would have remained a philosopher') and a famous Yiddish proverb says (translated) `If my grandmother had grown a beard, she would have been my grandfather'. 5 or rather content. Subordinate clauses starting with locutions like `Given the as- sumption that...' `Supposing that...' or `In the event that...' signal conditional content. 6 I shall call deliberately fudge on this admittedly very dicult and messy issue and extend my use of the grammatical kind term to cover these types of sen- tences as well. When I use `conditional' I refer to any sentences that may be used to express something that can also be expressed by the more canonical `If...then'. 1.3 The scope and the structure of this project Now that we have a better sense of what conditionals are I can say a few more words about the scope and the structure of this project. My goal is to identify a theory of conditionals that says something interesting (and true!) about the meaning of the If-then-clauses, the nature of conditional belief (both full and partial) and that, even more importantly, has certain explanatory virtues that will be discussed throughout the course of the rst two chapters. Chapter 1 will introduce several concepts, questions and `themes' that subsequent chapters presuppose. Chapter 2 introduces a rst theory of conditionals, so-called Simple Credal Expressivism (SCE) along with the probabilistic apparatus that is needed to appreciate the way in which 6 Slogans such as `No risk, no fun', `No entity without identity' or idioms such as `No harm, no foul' seem to express conditional content. And even conjunctive quasi-warnings like `You make this call and you won't see me again' seem to express conditional content. 6 the theory works. Chapter 3 is going to introduce a dynamic view of the conditional. Chapter 4 focuses on the fact that the dynamic view lacks resources for representing partial belief or condence and suggests some preliminary strategies for lling the gap. Chapter 5 presents a technically more challenging extension of the dynamic view that allows us to represent partial belief in the extended dynamic framework. Chapter 6 looks at the explanatory virtues of this new view and compares them to other competing views that try to do good by the same explanatory standards. One of the central guiding and motivating ideas of the project is that the resulting theory incorporates virtues of both theories presented in earlier chapters while getting around some of their limitations. Views that try to vindicate various other views that precede them have sometimes been labeled `ecumenical' but I shy away from such grand ecclesiastical metaphors. In order to understand what counts as a virtue of a theory, it makes sense to look at a foil, the so-called `horseshoe theory', i.e. the view according to which `If then ' just means `: or '. As we shall see, its would-be virtues { simplicity and a slim semantic apparatus ( = classical truth-functional semantics) can't compensate for the inability to explain some core data. 7 1.4 The Horseshoe Theory Every undergraduate student who learns about the truth table for `If....then' is puz- zled by the fact that the truth table tells us that a conditional is true whenever the antecedent is false. Aren't there at least some conditionals with (contingently) false antecedents that seem less acceptable than others? Suppose you are, for whatever reason, speculating about Magic Johnson's height: (1) If Magic Johnson is over seven feet tall, he is over two meters tall. (2) If Magic Johnson is over seven feet tall, he is less than two yards tall. 7 As it turns out, Magic Johnson stand 6 ft 9 in, so the common antecedent is false. Yet is seems that the rst conditional is clearly acceptable, whereas the second is clearly not acceptable since it betrays confusion about metric units. However, the view according to which the truth table for the horseshoe captures the meaning of the English connective `If....then' predicts that both conditionals are true. It is not clear whether it can account for the intuitive dierence between (1) and (2). But it is not just failing to capture the dierence between (1) and (2) that should worry us. Whoever defends the horseshoe view is at least prima facie committed to 7 A similar example is used in Nute and Cross (2001), p.4. 8 treating many conditionals as true that we wouldn't normally treat as true, namely all those conditionals whose antecedents we rationally expect to be false, even though we do not know them to be false. Similarly, we are committed to treating all those conditionals whose consequents we expect to be true as true. I shall call these two commitments or rather the rules that express such commitments `The Antecedent Rule' and `The Consequent Rule' respectively. Given the truth-table for the horseshoe, the two rules are sound. This is because they are guaranteed to preserve truth and in so far as they are guaranteed to preserve truth, they act as a safeguard against fallacious reasoning, i.e. reasoning from true premises to false conclusions. In other words, the horseshoe theory together with the assumption that sound reasoning is good reasoning naturally suggests at least two rules that tell us how to reason with conditionals. i.e. which inferences to draw. Unfortunately both rules fail to capture how we actually reason with conditionals, let alone how we should rationally reason with them. 8 The antecedent rule is completely misguided as a rule for reasoning. This becomes apparent when we look at examples of reasoning from uncertain premises. Just because we are very condent that there is no life in other galaxies, we aren't committed to being condent that if there is life in other galaxies, 8 I am assuming here, optimistically, that the gap between our actual powers of reasoning and those of a suciently rational agent is not too wide. 9 aliens in their UFOs have visited Earth. A reason to believe or be condent in the negation of the antecedent of a conditional is not a (sucient) reason to believe or be condent in the conditional itself. The consequent rule is not as blatantly absurd as the antecedent rule, but it is still not a safe technique for reasoning. While believing a consequent of a conditional is a reason for believing the conditional itself, that reason is defeasible in the sense that it can be defeated by the information carried by the antecedent. Once we suppose that the antecedent holds, we may no longer have any reason (or at least any strong reason) to believe the consequent. I can be condent that I am going to go on a bike ride tonight but fail to be condent that I am going on a bike ride if I break both my legs in the next 5 minutes while walking down the stairs to get my bike. Both rules lead to the so-called paradoxes of material implication: The Paradox of material implication (antecedent rule, schema): : ) (! ) The Paradox of material implication (consequent rule, schema): ) ( !) The Paradox of material implication (rst half, example): There is no life in other galaxies.) If there is life in other galaxies, UFO-s have landed on Earth. 10 The Paradox of material implication (second half, example): There is going to be a concert at Disney hall tomorrow night) If there is a major earth- quake in L.A. tonight, there is going to be a concert at Disney Hall tomorrow night. But the antecedent rule and the consequent rule are not the only bona de sound rules of inference that are intuitively questionable. At least three additional rules of inference are all validated by the horseshoe theory and yet there seem to be natural counterexamples to each of them. Antecedent Restriction is a rule that allows us add a conjunct to the antecedent of any conditional that we already believe or accept. Hypothetical Syllogism has also been referred to as `transitivity' of the right arrow, i.e. the relation that holds between an antecedent and a consequent of a conditional. According to that rule, we are committed to accepting conditionals that follow, by transitivity of that relation, from what we already accept. 9 Finally, Contraposition is the rule that commits us to accepting the `dual' of the converse of every conditional we accept. 9 Note that if Antecedent Restriction is invalid, Hypothetical Syllogism is ipso facto invalid: Suppose you accept (! ) but not (^)! . Then since we always have (^)! we already have identied a schema for a counterexample to hypothetical syllogism. 11 Notice that all these rules are valid for the horseshoe, i.e. the `articial' connective used and introduced in textbooks of (mathematical) logic. The contentious claim is that the English conditional connective is that connective. Antecedent Restriction, schema (! )) (^) ). Hypothetical Syllogism, schema (! ), ( !)) (!). Contraposition, schema (! )) (: !:) Antecedent Restriction, example If you add pepper to the dressing, the salad is going to taste even better. ) If you add pepper to the dressing and you add rat poison to the dressing, the salad is going to taste even better. Hypothetical Syllogism, example If Obama dies in October 2012, Romney is going to win the 2012 presidential election. If Romney is going to win the 2012 presidential election. Obama is going to retire from politics in January 2013. ) If Obama dies in October 2012, he is going to retire from politics in January 2013. Contraposition, example If it rains in Amsterdam, it doesn't rain hard.) If it rains hard in Amsterdam, it doesn't rain (in Amsterdam). Presenting this list of schematic inferences and instantiations to the horseshoe theorists is not going to end the discussion. The horseshoe theorist is not going to 12 simply admit to being mistaken and give up. Here is where the actual philosophical debate starts. Ultimately, the question `Which theory should we accept?' translates into the question of which inferences involving conditionals are valid and which aren't. This is the place at which our theorizing heavily depends on data, by which I mean speakers' aggregated judgments as to whether or not a given inference is ok or not. Some philosophers have suggested that the data on which our theories rest do not constitute a very rm foundation. As Chris Gauker puts it (Gauker (2005)), `there are few plain data on which our theories can rest'. Gauker goes on to argue that when it comes to building a theory of the conditional, giving up on any particular inference is fair game. No single piece of data is non-negotiable. While a conditional, as uttered in a given conversation in a given situ- ation, might appear true or acceptable, we might be persuaded that it is not so given a persuasive explanation of why it appears so. While an argument containing conditionals might at rst appear valid we might be persuaded that it is not valid given a persuasive account of the appear- ance of validity. 13 Take Modus Ponens as an example: The validity of Modus Ponens has been taken to be undeniable. In fact, some philosophers 10 have argued that one cannot fail to be justied (or at least blameless, a somewhat weaker notion) in inferring by Modus Po- nens because to grasp a conditional's meaning is to reason using its introduction rule (conditional proof) and its elimination rule (Modus Ponens). Yet other philosophers like Van Mc Gee ( McGee (1985)), William Lycan (Lycan (2001)) and MacFarlane and Kolodny (MacFarlane and Kolodny (2010)) have famously denied that Modus Ponens is valid, sometimes even without explaining why it appears to be valid. 11 But if the validity of Modus Ponens can be questioned, it seems that the validity of just any rule of inference can be questioned. But not only can every piece of data be dismissed, it can also be explained away. And this is what the horseshoe theorist is likely to do vis-a-vis the above list. She can just argue that whereas all of the above inferences are strictly speaking valid, they exhibit some other feature, call it F, that makes these inferences look bad. Now, 10 e.g. P. Boghossian in Boghossian (2003). 11 McGee denies Modus Ponens for compound conditionals in order to be able to get the validity of importation and exportation. See McGee (1989). Lycan seems to think that Modus Ponens is not consistent with the fact that reasoning with conditionals is non-monotonic, e.g. in the case of so-called Sobel-sequences. If I believe that `If Jon and Susie come, the party is going to be terrible' but also believe that `if Jon comes, the party is going to be fun' and if I learn that Jon and Susie are actually coming, I cannot infer that the party is going to be fun. But if I know that Jon and Susie are coming, I also know that Jon is coming, yet I can't infer by Modus Ponens that the party is going to be fun. I am not convinced that this is a counterexample to Modus Ponens. 14 if you are a resourceful representative of the horseshoe-camp, it is your job to look for an actual pragmatic mechanism (e.g. an implicature) that can be lled in en lieu of the placeholder `F'. This is what Frank Jackson (Jackson (1979)) and his followers (among them David Lewis) have done. Jackson argues that what matters in a discourse containing conditionals is not so much the validity of the inferences, but the felicity of the inference. And unlike validity, felicity turns on the assertibility of a conditional that is dened in terms of probabilities and not the truth-table. 12 I won't have time to respond to this more sophisticated response on behalf of the horseshoe theory. The point of mentioning it is more to illustrate that data are an important source of evidence for a theory's correctness, but that this particular source of evidence can be weighted in dierent ways. However, even if we think that it is ok to not explain some piece of data or other, there is something deeply problematic about a theory that fails to be able to account for most of the core data and postulates a complicated mechanism to explain its own predictions away. There is, however, a second shortcoming of the horseshoe theory that is largely independent of the problem of predicting the wrong inferences to be reasonable. The problem concerns iterated conditionals, or conditionals embedded in other construc- tions. In order to see that the second problem is not just supercially dierent from 12 Jackson thinks that conditionals are assertible to the extent that our accepting them is robust with respect to possible evidence that the antecedent is false. 15 the rst problem, suppose for a moment that the above agenda of identifying some feature that `explains' away the data can in fact be carried out. Even if we iden- tify some extra-semantic, for example pragmatic feature F that explains why certain conditionals occurring in premises and conclusions of valid inferences sound odd in these inferential contexts, and that, as a consequence, the entire inference sounds invalid, that feature may not suce to explain why certain inferences involving em- bedded conditionals `sound' invalid as well. Just because producing a sentence `p' in a certain context results in triggering some implicature i it doesn't follow that pro- ducing a compound sentence that contains `p' as a constituent results in triggering that same implicature. 13 It suces to look at simple truth-functional (extensional) constructions contain- ing conditionals to illustrate the problem. Negated conditionals are a case in point. Suppose we play Clue and I am fairly condent but not certain that Mrs. White did it (with the candlestick). However, I believe that both Colonel Mustard and Reverend Green are also among the suspects, but I am more condent that Colonel 13 The reason is related to what has become well-known in the literature as the `Frege point' (or `Frege-Geach problem'). If asserting that p carries pragmatic feature F p , asserting that not p or that if p then q usually does not carry feature F p as well. Yet, the defender of the horseshoe-view is at least prima facie committed to the view that carrying pragmatic feature F p projects. But most pragmatic features fail to project in that way. For example, an assertion that p comes with a commitment to defend the claim that p. But clearly, an assertion that if p that q does not come with any commitment to defend the claim that p. E.g. asserting `If the skeptic is right, we know next to nothing' does not commit me to skepticism. 16 Mustard did it than that Reverend Green did it. In this situation, I assent to `It is not the case that if Mrs. White didn't do it, Reverend Green did it'. I assent to that conditional because I accept that if Mrs. White didn't do it, Colonel Mustard might have done it. But according to the horseshoe theory I thereby accept that Mrs. White did it and Reverend Green didn't do it. But that's certainly not what I accept and believe, since Reverend Green is an obvious suspect { he just isn't the prime suspect. An even more ludicrous example has been given by Dorothy Egington. Her exam- ple involves an iterated conditional that is also embedded under negation. Horseshoe theorists seem to have very little resources to block the following argument, which proves the existence of God from two premises that are both clearly acceptable for atheists. Unless horseshoe theorists somehow deny that the logical form of the argu- ment is as stated, they will have to accept the argument as valid or admit that they have not succeeded in making sense of embedded conditionals: Premise 1: If God doesn't exist, then it is not the case that if I pray to God my prayers will be answered (::( )) Premise 2: I do not pray to God (: ) Conclusion: God exists () 17 I believe that two basic desiderata for a theory of the conditional ow from these two shortcomings. The rst desideratum is that a theory of the conditional explain how we reason with conditionals when we are less than certain of the premises. Hypothetical reasoning occurs at dierent levels of certainty. The second desideratum is that a theory give an account of how the conditional embeds at least in some simple truth-functional constructions, mainly negation, the consequent of a conditional and disjunction. As we shall see in subsequent chapters, there is a certain tension between the two desiderata: It is hard to satisfy both. The two theories that are going to be presented and discussed in the course of this work will each fail to satisfy at least one of the desiderata. The theory that I defend combines both views' virtues. Both theories that will be discussed rely on a historically very important account of what it is to accept a conditional, namely the Ramsey-test. The Ramsey-test has been a point of departure for most contemporary theories of the conditional. In a way this multitude of interpretations suggests that the Ramsey-test is not a very powerful constraint on a view of the conditional. We will quickly survey views informed by the Ramsey test in the next section. 18 1.5 The Ramsey Test The so-called Ramsey test can be traced back to a footnote in the work of Frank Ramsey (making it probably one of the most important footnotes in the history of logic and/or epistemology): If two people are arguing 'if p will q' and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q: so that, in a sense, `If p, q' and `If p, : q' are contradictories. We can say they are xing their degrees of belief in q, given p. If p turns out false, these degrees of belief are rendered void. If either party believes: p for certain, the question ceases to mean anything to him except as a question about what follows from certain laws or hypotheses. Ramsey (1965), p. 247 Many if not most contemporary theories of the conditional can be understood as interpretations of the procedure (or at least certain aspects of the procedure) suggested by Frank Ramsey in this footnote. At a very fundamental level, The Ramsey test tells us what it is to accept or come to believe a conditional. There are at least three major frameworks of conditional belief that incorpo- rate the Ramsey test: Ernest Adams' probabilistic interpretation, Robert Stal- 19 naker's semantic theory Stalnaker (1969) and Peter G ardenfors' doxastic interpreta- tion G ardenfors (1986), based on the AGM framework for belief revision developed in the early 1980s. 14 I will take a closer look at Adams' probabilistic interpretation in Chapter 2. The next two subsections contain my gloss on the two views that won't receive further attention in subsequent chapters, namely Stalnaker's view and the AGM view. But before I discuss these views, I shall remind the reader of two important facts about the Ramsey-test. 1. The procedure specied by the Ramsey-test is very closely related to what logicians do when they want to derive a conditional in a deductive system. They make use of a deduction theorem, provided that they have proved it for the system in question (via induction on the length of the derivation) { So if we wonder whether a material conditional of the form is derivable from a premise set , we simply show that the consequent is derivable from the set [fg. Then, by one application of the deduction theorem, we have derived from the premise set and hence shown that the conditional is derivable. Since deduction theorems are rather standard even for the material implication 14 There has been some controversy over whether or not proponents of these three major views misinterpreted the gist of Ramsey's actual proposal. Slater Slater (2004) believes that all of these views misconstrue Ramsey's proposal. This largely interpretative issue cannot be settled here. 20 this may be taken to suggest that the Ramsey-test is a rather weak constraint on theories of the conditional. However, the Ramsey-test goes beyond a mere deduction theorem for the indicative conditional by inviting us to revise our existing beliefs in case the antecedent is inconsistent with them. Unlike in the case of (classical) logic we do not simply assume the antecedent and happily derive arbitrary conclusions from our newly inconsistent belief set. 2. A correct interpretation of the Ramsey-test forces a somewhat more sophis- ticated understanding of what it is to `hypothetically add beliefs to a stock of knowledge' on us. This procedure may not involve adding any belief, but rather supposing that the antecedent is the case and operate, as it were, in a suppositional mode. This is important because there are some conditionals that are acceptable for agents even though they shouldn't be acceptable if the agents applied the Ramsey-test via simply pretending, imagining or even sup- posing that they believed the antecedent. Conditionals like `If the CIA is spying on me, I don't know it' or `If I am a brain in the vat, I don't believe it' would never be acceptable to any agent, unless they were in a state in which they be- lieved the antecedent but also believed that they didn't believe the antecedent. This shows that supposing that p is dierent from imagining or pretending that one believe that p (which is a model of supposition defended by Levi). In 21 particular supposing that p does not commit us to supposing that we believe that p. Some LOT-imagery may help drive the point home: We have to keep track of which items are being added to and deleted from the belief- box and keep those separate from the items that are added to and deleted from the supposition-box. This being said, I will largely ignore this rather subtle issue in the remainder of the discussion of the Ramsey-test. It is, however, good news that the view that I am ultimately going to defend does have sucient resources to model the dierences between hypothetically believing that p and supposing that p, even though I shall present a simplied version of it that does not include the extra resources needed to mark this dierence. 1.5.1 Possible-Worlds Views Robert Stalnaker oered the following reading of the test: Add the antecedent hypothetically to your stock of knowledge (or beliefs) and then consider whether or not the consequent is true. Your belief about the conditional should be the same as your hypothetical belief, under this condition, about the consequent. Stalnaker suggests exploiting the (then, i.e. in the 1960s) excitingly new machin- ery of possible world semantics to spell out the above procedure for the test. So 22 instead of starting with a `stock of knowledge' or beliefs we look at a set of possible worlds, intuitively the set of worlds that corresponds to the agent's beliefs (Let us call that set `S'). For a conditional to be accepted by an agent corresponds to the conditional's being true at every world in S. And the conditional ! is accepted at a world w just in case is true at the world f(, w), where f is a function that for each pair of possible worlds and propositions (sets of worlds) selects a world { intuitively the world at which is true that is closest to w. 15 Stalnaker translates Ramsey's instruction into a rigorous theoretical framework that has been defended by many philosophers to date. But even though many philoso- phers remain loyal to the possible-worlds treatment of indicative conditionals, they follow Lewis in rejecting some key assumptions that Stalnaker made. 16 Lewis' re- sulting view, which assumes the same model-theoretic framework, has mostly been 15 For a more contemporary defense of a possible-worlds account of indicative conditionals based on a `nearness'-relation, see Nolan (2003). Weatherson (2001) adds another layer of sophistication to this picture by using a two-dimensional modal semantics, allowing him to explain the oddness of `If Warren Buett becomes the next President, things will be dierent from the way they actually will be.' Weatherson (2001), p. 202. 16 Mainly the Uniqueness assumption, the claim that for every world, there is at most one world that is closest to it and the Limit Assumption, the claim that for each possible world, there is at least some world that is such that no other world is closer to it. In oering an alternate account without either assumption, Lewis put his independently motivated theory of comparative similarity to work: A conditional ! is true at w 0 if there is at least one world w 1 at which and are both true that is closer or more similar to w 0 than any worlds at which and: are true. See Lewis (1973) 23 oered as a theory of subjunctive conditionals but not indicative conditionals and will not be discussed here. Let us instead take a closer look at Stalnaker's implementation of the Ramsey- test. Stalnaker's view tells us what it is for a conditional to be true at a world. What it is to believe or accept a conditional can then be explained in terms of formally explicated notion of truth-at-a-world. Moreover, given the model-theory of possible worlds cum interpretation of the selection-function and a denition of truth and logical consequence we can determine a system of derivation for a propositional language with a conditional connective that is complete and sound with respect to the possible-worlds semantics. The resulting logic, VCS or C2, has some attractive features. C2 is a logic for a propositional language with one conditional connective `>', (`the corner') and the usual modal operators box and diamond (`2', `3') for necessity and possibility. It has two rules of inference, Modus Ponens and Necessitation. 1. If` , ! , then` 2. If` then`2 24 Stalnaker chooses the following axioms, but other axioms can be shown to charac- terize C2 equally well: 17 1. Any tautological well-formed formula 2. 2( ) (22 ) 3. 2( ) ( > ) 4. 3 ((> ):(>: )) 5. > ( _) (> )_ (>) 6. (> ) ( ) 7. (? ) (>) ( >) This logic tells us that both halves of the paradox of material implication are invalid and that the remaining patterns of inference discussed above are also invalid. Take Antecedent Restriction as an example: From the fact that, at some world w, it is true that > , it does not follow that it is true at w that (^) > . This is guaranteed by the structure of the Stalnaker models. Whereas, is, ex hypothesi true at f(, w), it need not be true at f(^, w) since there are models in which 17 For a short exposition see for example Lewis (Lewis (1973) or Arl o-Costa's SEP entry. A completeness proof for C2 with respect to these axioms is oered by Lewis (Lewis (1971)). 25 f(, w) =f(^:, w)6=f(^, w), i.e. the closest-world is not a^-world. 18 But the choice of the logic is not the only benet of Stalnaker's system. Given the fact that Stalnaker constructs a truth-conditional semantics, we can use its full compositional force to assign meanings to compounds of conditionals. Unfor- tunately, Stalnaker's system makes some odd predictions for embedded conditionals (just like the horseshoe theory, but for dierent reasons). According to Stalnaker's Axiom (4) above, denying a conditional is the same as accepting the conditional with the negated consequent, at least as long as the antecedent is possible. But when I say `It is not the case that if Mrs. White didn't do it, Reverend Green did it' I am not thereby saying that if Mrs. White didn't do it, Reverend Green didn't do it. What I am saying is weaker than what that last conditional says. Similarly, Stalnaker's model theory allows the construction of counterexamples to the law of exportation and importation. According to that law, we can always move from a conditional with an additional conjunct in the antecedent to a nested conditional and vice versa: Principle 1 (Exportation { Importation) (^ > ) ( > ( > )) 18 Stalnaker's four conditions on the functionf guarantee that the last inequality holds, at least if both and^ are possible. For indicative conditionals (as opposed to subjunctive conditionals) Stalnaker poses the further restriction that the range of f be the context set, i.e. the set of all and only those worlds at which everything the participants of a conversation take for granted is true. 26 For example, if you accept that if Julie has a GPA of 3.95 and majors in philosophy, she should apply to law school you ought to also accept that if Julie has a GPA of 3.95 then if she majors in philosophy, then she ought to apply to law school. This simple example is supposed to pump the intuition that the law is in fact valid. As we shall see in Chapter 6 it is one of the few principles governing our reasoning with embedded conditionals that we should accept. It is all the more important that we can account for its intuitive validity. 19 1.5.2 The AGM framework The AGM framework (named after its three founding fathers Alchourr on, Makinson and G ardenfors) is a powerful tool for analyzing the dynamics of belief, i.e. belief 19 Notice that accepting it poses at least one problem, which I won't address until Chapter 6. If we accept that whenever entails , the conditional ! is valid (a principle that I shall call Conditional Upper Bound) and if we accept Exportation, we have to reintroduce the paradoxes of material implication. The argument is the following: 1. (^ ) Simplication 2. (^ )! from 1 by (Conditional Upper Bound) 3. ! ( !) from 2 by (Exportation) 4. ( !) from 3 by (Conditional Upper Bound) The argument works mutatis mutandis for the other half of the paradox of material implication (instead of starting with Simplication of Conjunction, we start with Ex Falso Quodlibet). This illustrates the tension between the desideratum of explaining how conditionals embed in compound sentences on the one hand and making sense of reasoning from uncertain premises { for which the paradoxes of material implication are unreasonable { on the other hand. 27 change in the context of classical logic. 20 The basic idea is that we can add beliefs to a corpus of beliefs and see how the addition aects the entire corpus. We can distinguish three kinds of belief change. We may simply expand our corpus by incorporating a new belief. This is the default (non-probabilistic) way of modeling learning. We can also contract our corpus by some item, for example if we have come to have doubts about a particular issue and wish to remain agnostic about it. Lastly, we can revise our corpus by a sentence or proposition, which means that we either expand the corpus or rst contract it and then expand it (or perform a nite sequence of such operations). The core of AGM consists in formulating postulates for these three modes of belief change, with revision being the most complex and challenging form of belief change. Peter G ardenfors (G ardenfors (1986)) was the rst author working in the AGM tradition to investigate the addition of conditional belief to a corpus. His project is guided by the Ramsey-test as a principle of belief revision. 21 20 For an in uential non-classical system of belief revision, see Lewis (1982). 21 Upper case Roman letters will stand for elements of belief sets, either propositions or sentences in a language of thought or some computer code. 28 Accept a proposition of the form `if A then C' in a state (Ramsey-test) of belief K if and only if the minimal change of K needed to accept A also requires accepting C G ardenfors argues that next to the Ramsey-test the following principle of `doxastic economy' should govern any system of belief change. The underlying idea is that gathering evidence for a belief is a cumbersome process and beliefs should not be thrown away unless new evidence absolutely calls for it: If a proposition B is accepted in a given state of belief K and (Preservation) A is consistent with the beliefs in K, then B is still accepted in the minimal change of K needed to accept A 29 G ardenfors goes on to argue that these two principles do not sit well with a num- ber of other plausible assumptions about belief change. Suppose we have specied a syntax for a propositional language L that includes a conditional connective `!'. G ardenfors assumes a belief revision structurehK, Fi, where K is a set of belief sets or corpora and F is a function from K L to K, taking the pairhK, Ai into K A . A logical consequence relation (``') over the language L is dened in the standard way and the conditional cum consequence relation satises a deduction theorem. Moreover, every belief set is closed under logical consequence. Hence, the set of all sentences of the language L is also a limiting case of a belief set. Let us look at the following 8 principles: 1. (Conditionals) Belief sets include propositions containing the conditional con- nective as elements 2. (Success) A2 K A 3. (Minimal Change) If K6= K ? and K A = K ? , then`:A 4. (Ramsey-test) A! B2 K if and only if B2 K A 22 5. (Monotonicity) For all belief sets K and K and all propositions A, if K K , then K A K A 22 (Ramsey-test) presupposes that (Conditionals) is true. 30 6. (Preservation) If:A = 2 K and B2 K, then B2 K A 7. (Closure under Expansion) Let K be a belief set. Then K + A =fBj A! B 2 Kg, the expansion of K by A, is also a belief set. 8. (Non-triviality) There are at least three pairwise disjoint propositions A, B and C and some set K in K which is consistent with all three propositions, i.e.:A = 2 K,:B = 2 K, and:C = 2 K. 23 G ardenfors (G ardenfors (1986)) shows that principles (1)-(8) are jointly unsatis- able. 24 The details of the derivation need not concern us here. It is, however, an extremely interesting result that adding the Ramsey-test to a system of principles governing belief revisions forces us to reconsider some of the very principles that have seemed completely innocuous in other contexts. It turns out that the result holds for logics that are much weaker than the classical system adopted by G ardenfors (Hansson (1992), p. 524). The most interesting and famous response to the triviality result is probably Isaac Levi's rejection of (2) and 23 Two propositions A and B are pairwise disjoint iffA,Bg`?. 24 G ardenfors omits (Closure under Expansion) and does not make all his assumptions explicit. The above list of assumptions is based on Hansson (Hansson (1992)). Both Hannson and G ardenfors note that (Monotonicity) follows from (Ramsey-test). The proof is straightforward. Assume the antecedent of (Monotonicity) and suppose that C2 K A . By (Ramsey-test), A! C2 K and, by our assumption, A! C2 K . But then we have that C2 K A (by (Ramsey-test)) and hence we have shown the consequent of (Monotonicity). 31 the corresponding weakening of (Ramsey-test)(Levi (1988)). It is beyond the scope of this section to fully explore the commitments that arise from such a move. Two immediate consequences deserve to be mentioned though. The rst consequence is that conditionals are no longer possible members of belief sets and hence no longer fully respectable members of the object language L. In fact, Levi regards them as quasi-metalinguistic instructions to manage non-conditional sentences or beliefs in a certain way (e.g. to reason by Modus Ponens and Conditional Proof). The second consequence is that once conditionals lose this status, it will, again, become extremely dicult to oer an account of how they membed under other kind of constructions, for example negation. Levi acknowledges this problem and oers a more or less ad hoc way of determining under which conditions these compounds are acceptable. I shall not pursue this strategy because even a revised AGM framework on which a weakened version of the Ramsey-test is consistent with other principles of belief change does not seem to satisfy our two desiderata. 25 But there is one aspect of Levi's proposal that is very appealing even if one moves beyond the AGM framework, and that is the view according to which conditional beliefs are no longer beliefs like any others. There is a sense in which conditionals are 25 Without additional resources,.e.g. some comparative measure of `being at least as entrenched as' dened over the members of a belief set K the AGM framework is not able to represent reasoning from uncertain premises. 32 not sentences made for describing the world, but devices that allow us to organize our beliefs and knowledge about the world in certain ways: Conditional statements are not objects of belief which bear truth values and are subject to appraisal with respect to epistemic possibility and probability. (...) They are appraisals of serious or epistemic possibility relative to transformations of the current corpus of knowledge or belief set.(Levi (1988), p. 49-50) 26 Let us call the family of views according to which conditionals are not in the busi- ness of describing the world or carving up the beast at its joints non-descriptivism about conditionals. It is a picture that I am extremely sympathetic to. I shall close this introductory chapter with reviewing two important considerations for non- descriptivism about conditionals. These considerations are in no way knock-down arguments for non-descriptivism, but they suggest that the view is very plausible. All subsequent theories presented in this dissertation are committed to some version of the view. My project of building a theory that satises both desiderata can, however, be appreciated by someone who does not wish to be committed to non-descriptivism. Descriptivists need not get o the boat just yet. 26 According to Levi, a corpus of knowledge is primarily a resource for deliberation and its chief function is a standard for serious possibility. If h is in K, its negation is not a serious possibility. If h is not in K, its negation is a serious possibility (Levi (1988), p.54). 33 1.6 Non-descriptivism about conditionals When one says that conditionals do not describe reality one speaks metaphorically. Many declarative sentences do not describe reality and it is not clear that language and thought altogether have the function of `describing reality'. 27 Still, there is some probably very complicated relation between what we think and say and the structure of reality, at least if we think there is such a thing as reality. The Quinean web of beliefs is somehow attached to the way things are, even if it is only at the fringes. The negative claim made by non-descriptivists is that conditionals play no direct role in telling us how things are. They play a role in thinking about how things are, but they do not themselves state facts. If you check the inventory of the world there are fact about particulars and { according to your favorite ontological theory { maybe facts about events, properties, numbers and minds, but none of these facts are iy. There are no iy facts. 27 There are many other reasons why one has to be careful with the claim that conditionals do not describe reality. If I say `If you drop the vase, it will break' or `If you throw the mattress into the water, it won't oat' I seem to be saying something that is very much about the way the world turned out to be. These conditionals are assertible because certain mechanical laws hold (and because we all accept those laws). But asserting a conditional that is about a regularity is not necessarily stating a fact about the world as Humeans about causality will readily acknowledge. Conditionals allow me to talk about many regularities and pseudo-regularities that are not covered by some kind of natural law. That was one of the many reasons for the downfall of the account of conditionals that tries to derive them from laws or lawlike statements (For the problems with that account that cannot be reiterated here see Goodman (1947)). According to SCE, whoever states `If you drop the vase, it will break' is expressing a mental state and there is no reason why such an attitude couldn't be the endorsement of a law. 34 It is best to not spend too much time worrying about these very general state- ments and look at some concrete evidence for non-descriptivism. So I want to look at two classes of examples that are supposed to show that conditionals exhibit features that we wouldn't expect factual language or thought to exhibit. Non-descriptivism is then supposed to explain these phenomena better than descriptivism. I shall start with a locus classicus of the discussion of conditionals: Riverboat Sly Pete and Mr. Stone are playing poker on the Mississippi riverboat. It is now up to Pete to call or fold. Xs henchman Zack sees Stone's hand, which is quite good, and signals its content to Pete. X's henchman Jack sees both hands and sees that Pete's hand is rather low, so that Stones is the winning hand. At this point the room is cleared. A few minutes later Zack slips X a note which says `If Pete called, he won' and Jack slips X a note which says `If Pete called, he lost'. X concludes that Pete folded. This famous scenario, rst described by Gibbard (Gibbard (1981)) raises a lot of questions. For now, it suces to note that Jack and Zack accept conditionals that seem to be incompatible with each other but that both conditionals are (highly) assertible or acceptable for them, given their respective evidence. Moreover, as Lycan (Lycan (2001)) argues, neither Jack nor Zack seems to have any justied false (non- conditional) beliefs about the situation which shows that the case is dierent from a 35 standard case of factual disagreement in which at least one party has a false (possibly justied) belief. I can disagree with you about the height of the Eiel tower but one of us is going to be mistaken at the end of the day, however good our evidence is. However, the example has met with criticism, leading to the so-called `hard line', according to which Zack's belief is ultimately unwarranted and his conditional is simply false. According to that line, if Pete calls, he is not going to win. How could he? After all, his hand is worse than his opponent's hand. Pace Santos (Santos (2008))it is perhaps best to look for stand-o scenarios that are epistemically symmetric. Bennett's (Bennett (2003)) Top Gate example exhibits such symmetry { by design: Top Gate In a dam, a main channel (controlled by Top Gate) bifurcates into two sub-channels going East and West. These, in turn, are controlled by East Gate and West Gate respectively. If East gate is opened and Top Gate opens, all the water will run eastwards; if West Gate is opened and Top Gate opens, all the water will run westwards. If both the East and West gates are open nothing happens as Top Gate will then be closed { the irrigation system does not allow for the three gates to be open at the same time. Observer X sees the East Gate open and is therefore justied in saying `If Top Gate opens, all the water will run eastwards' whereas Y sees the West Gate opened and is 36 therefore vindicated in asserting: `If Top Gate opens, all the water will run westwards' In this case, it seems that X's conditional should be accepted by anyone who is in X's position and Y's conditional should be accepted by anyone who is in Y's position. So the conditionals are both warranted and since the scenario is symmetrical they are both equally warranted. 28 But they cannot both be true, at least if we accept the following theorem of C2: 29 Principle 2 (Conditional Non-Contradiction) :((> )^ (>: )) This is where the controversy starts. Gibbard, Edgington (Edgington (1986)) and their followers conclude that conditionals lack truth-values and reject C2's Axiom 4. 28 Notice also that an observer would be able to conclude from the two reports that Top Gate cannot be open (or that it is not going to open). The information conveyed by both conditionals is somehow pooled to derive the negation of the antecedent, even though the conditionals are not co- assertible. The use of both conditionals as premises in a sound C2 argument suggests that neither agent said something false. 1. If Top Gate opens, all the water will run eastwards. 2. If Top Gate opens, all the water will run westwards. 3. If Top Gate opens, all the water will run eastwards and westwards (follows from (1) and (2) by the C2 Axioms 4 and 5) 4. It is not the case that the water will run eastwards and westwards (analytical consequence of a tautology) 5. Top Gate will not open (from (4) by Modus Tollens) It is puzzling how the rst two premises can be co-tenable. However, we seem to sometimes accept conditionals for the sake of deriving consequences from them. Stalnaker's `One if by land, two if by sea' -example is a case in point Stalnaker (1984). 29 By symmetry, they cannot both be false. 37 That move in itself is not yet sucient to esh out any particular positive view since many views are compatible with such a negative claim. However, the move commits its defenders to non-descriptivism about conditionals. Other philosophers observe that not all pairs of assertions that share the surface- structure and: are subject to the law of non-contradiction. Pairs of sentences exhibiting context-sensitivity are generally not admissible counterexamples to non- contradiction. A prominent class of examples are sentences containing indexicals: If Ingrid says `I am a lawyer' and Peter says `I am not a lawyer' they have not contradicted each other (and both have possibly spoken truly). Another (overlapping) class is constituted by judgments of taste: If Ingrid says `Fish sticks are delicious' and Peter says `Fish sticks are not delicious' they disagree, but it doesn't follow from that disagreement that one party is wrong. Ingrid can be right relative to her culinary standard and Peter can be right relative to his and there is even a (not very robust) sense in which both assertions can be true. Appealing to the pervasiveness of context-sensitivity in natural languages like English, philosophers have tried to assimilate conditionals that allow for Gibbardian standos to one of those paradigm cases of context-sensitive sentences. One option is to defend a contextualist account of conditionals { an account according to which the agent's subjective state of mind or her information in a context makes a contribution 38 at the level of content, i.e. to what the conditional says. The basic idea is that the content of a conditional like `If Top Gate open, the water will ow eastwards' as uttered by S in a context C depends directly on the S's subjective situation in C, e.g. her information. 30 Alternatively, one can model conditionals on judgments of taste or perspectival statements and defend a relativist account. On such an account, the propositional content of the conditional is invariant across contexts, but its truth-value is assessor- relative. So `If , then can be true with respect to X's point of view and yet, `If , then ' can be true with respect to Y's point of view, thus allowing for the possibility of a Gibbardian stando. 31 Gibbard's stand-o cases thus lead to a dialectical stand-o between dierent explanatory strategies. On one family of views, conditionals do not have truth-values. All the views that deny that conditionals have truth-values are committed to non- descriptivism. This is because according to these views the reason why conditionals lack truth-values is that there are no suitable bearers of truth-values. There are no propositions corresponding to sentences of the form `If....then'. I have just argued that there are at least two alternative views to a `No-truth-values-view' (NTV). 30 Such an indexical contextualism is defended by De Rose and Santos (Santos (2008)). 31 John MacFarlane MacFarlane (2007) argues that a relativist analysis of context-sensitivity is superior to contextualism, inter alia because it avoids the problem of lost disagreement. 39 Neither one of the two alternative views, i.e. contextualism and relativism are non- descriptivist because both are compatible with their being propositions corresponding to conditionals. In particular, according to relativism conditionals do express a propositional content that is invariant across contexts. However, according to Jacob Ross and Mark Schroeder, relativist accounts have a hard time making sense of another important phenomenon that conditionals exhibit: Holmes' Reversal (Ross and Schroeder (2011)) Moriarty, Milverton and Moran are the three possible suspects in a murder that was committed with an air-ri e. Believing that only one of the suspects had an air-ri e at the time of the murder, Watson says: `Now all we need to nd out is who had an air- ri e. For each of the suspects, if he had an air-ri e, then he is the murderer. Holmes, however, has deduced that in fact two of the suspects had air-ri es. And so he says: `My dear Watson, it is not the case that if Moriarty had an air-ri e, then he is the murderer and it is not the case that if Milverton had an air-ri e then he is the murderer and it is not the case that if Moran had an air-ri e, then he is the murderer.' (Call this conjunctive sentence r) Holmes also knows that he will soon learn the identity of the innocent air-ri e owner and that he will subsequently know, of each of the remaining suspects, that if 40 he had an air-ri e, he is the murderer. And so Holmes predicts that he will assertively utter:r. Before we discuss the signicance of this phenomenon, it is worth noting that epistemic modals like `It might be that ' , `It may be that ', `it is possible that (with focus on `possible') and `It must be that ' are another type of sentence that exhibits both reversibility and standos. 32 In fact, on some views these striking parallels between conditionals and epistemic modals are hardly a coincidence: Con- ditionals are but a species of epistemic modals restricted by the antecedent and the history of analyzing `If...then'-sentences as sentences involving a binary operator is the `history of a syntactical mistake'. 33 But let us get back to reversibility. As Ross and Schroeder argue, reversals constitute prima facie violations of the principle of re ection, which is often taken as a principle that is constitutive of rationality. 34 A version of the principle says that if you currently have some positive credence c in and if you know that you will soon receive conclusive (or at least good enough) evidence that will change your credence 32 The point about standos was noted by Kratzer in her seminal 1986 paper. We both see a man approaching, but I am further away from him than you. I can be warranted in saying `The person approaching might be Fred' given my evidence and you can be warranted in asserting `The person approaching cannot be Fred' given your evidence ( Kratzer (1986), p.9). 33 Kratzer (1986) and more recently Rothschild (2011). 34 The principle is originally due to Bas van Fraassen, van Fraassen (1984), p. 244. 41 to c , then you should, other things being equal, now set your credence in to c . Holmes, however, does not seem to violate any principle of rationality - his reasoning is impeccable. Ross and Schroeder argue that so-called invariantists about conditionals (and epistemic modals), i.e. anyone who believes that a given conditional has the same content across contexts, cannot make sense of reversibility without giving up on Re- ection. It is irrational for Holmes to believe a proposition r and yet know of that same proposition r that he is going to disbelieve it. The only way to maintain invari- antism is to go for a relativist as opposed to absolutist invariantism (as advocated by Egan in Egan (2007)). But Ross and Schroeder think that any version of relativism that also explains disagreement, i.e. that for any contested sentence , two dierent speakers in two dierent contexts can disagree when one of them utters and the other of them utters : run afoul of making room for reversibility. This doesn't show that relativism has to be given up, put it puts pressure on a relativist view that does not deny that there can be Holmes-style reversals. So if we want to take reversibility and standos seriously and believe that rel- ativism struggles to make room for the former, we have reason to go for either a 42 contextualist or a non-descriptivist view. 35 When looking at the options that are available to us, it is important to remember an important constraint on our theo- rizing: Since indicative conditionals and epistemic modals exhibit the same kind of information-sensitivity, any explanation of these phenomena for conditionals should build on an explanation (or be consonant with an explanation) for epistemic modals. If we want to be contextualists or non-descriptivists, we want to be contextualists or non-descriptivists across the board { `mixed' views in the face of the same body of evidence for both kinds of sentences would be odd. I won't have time to discuss the details of a possible contextualist view either for epistemic modals or conditionals. On a very straightforward contextualist account, epistemic modals are simply statements about what the speaker knows or rather about what is compatible with her knowledge. According to such accounts a sen- tences like `It might be that' is true at a context C i is compatible with what the speaker or anyone in her group of relevant peers knows at C. On such accounts epis- temic modals are second-order claims about one's own (rst-order) epistemic states. And notice that reversibility is not at all surprising on this account. I can know that 35 We have to be cautious though. As we shall see, at least some non-descriptivist views have a hard time accounting for embedding conditionals. But Holmes' reversal turns on negating a conjunction of negated conditionals. Fortunately, some views have the resources to account for such embeddings. 43 something is not going to be compatible with my knowledge next week though it is compatible now, simply because I know that my knowledge is going to expand. Contextualism about epistemic modals has been subject to a series of criticisms. Qua descriptivist view it is subject to an important challenge formulated by Seth Yalcin. Yalcin argues that a descriptivist view is unable to explain epistemic con- tradictions, i.e. the fact that ` and it might be that:' is not only unassertible, but not even satisable: There is no epistemic state of any subject that satises the conjunction. 36 If the semantic contribution of `It might rain' in context C is { for example { that there is a world in which it rains that is among the worlds compatible with what speaker S knows, then it is hard to see what would make the conjunction ` and it might be that:' semantically unsatisable. For surely there is a world in which it actually rains but another world compatible with what we know in which it does not rain. Another frequently noted problem for contextualism if not for any account of (epistemic) modals is that no particular version of the view seems to be tenable. 36 Yalcin argues that these conjunctions are not merely Moore-paradooxical like the conjunction ` and I don't believe that' but actually unsatisable, i.e. epistemic contradictions. The fact that the contradictory feel persists when the conjunctions are embedded in contexts that seem to single out hypothetical scenarios, e.g. antecedent of conditionals and complements of `Suppose that...' - attitude statements is evidence for this stronger claim: 1. *Suppose it rains and it might not rain 2. *If it rains and it might not rain, then... These data can of course be challenged - and they have been challenged. See for example ?. 44 Take the contextualist view according to which `It might be that' is true in context C i is compatible with X's knowledge. For every choice of a particular value for the variable X, there will be some counterexample. That counterexamples are readily available is shown by cases that have been called `eavesdropping cases'. Typically, in these cases an agent A who knows that: is eavesdropping on someone B who wonders whether. If B says `It might be that' her statement is either true, namely in the case in which X does not include A, or it is false (if X does include A). In the rst case what B says is inconsistent with what A knows and so, paradoxically, it cannot be true. In the second case, it is unclear why an epistemic modal should ever get asserted (Egan et al. (2005)). Another related problem that arises mostly for contextualism about deontic modals is the inability to account for (the possibility of) advice. It is very customary to use `oughts' in deliberative situation, but also when advising deliberators on the best course of action. Like any other view, contextualism has to be able to make sense of the practice of using ought-statements in deliberative situations, but it seems that it is unable to do so. If I wonder what I ought to do next week and say to myself `I ought to read Brandom's `Making it Explicit" what I say in a context is roughly that reading Brandom is the best course of action in light of my (probably limited) information 45 about the world and my goals. If my dissertation adviser then tells me `No, Johannes, you ought to nish Chapter 2 of Iy Condence' he is saying that nishing Chapter 2 of `Iy Condence' is part of the best course of evidence in light of his information about the world and my goals. It is hard to see how my adviser's advice can be generally helpful and informative (which it usually is) without us talking past each other. 37 Given these problems with contextualism, I am going to focus on the non- descriptivist response to the two puzzles. 38 The particular view that I shall develop has been around for a while, but many theorists have neglected to work out the details of the view. 39 In conclusion, I take it that the foregoing discussion licenses some optimism that non-descriptivism has the right structure to explain the two phenomena. 37 This point is made in the course of MacFarlane and Kolodny's discussion of the Miners' Paradox (MacFarlane and Kolodny (2010). 38 That the non-descriptivist solution of the puzzles is ultimately superior to the solutions pre- sented by contextualism or relativism cannot be conclusively established here. However, I do believe that the discussion shows that both major alternatives come with considerable commitments. 39 There are other versions of non-descriptivism that are very well developed though. Seth Yalcin (Yalcin (2011), Yalcin (2007)) has presented a rather sophisticated non-descriptivist view in his two papers on epistemic modals. 46 Chapter 2 Simple Credal Expressivism 2.1 Expressing Credal States Simple Credal Expressivism (SCE) is the view that we express a credal state when we assert a conditional. The view is called `simple' because it says that we express one such state (and not more). What does it mean for a conditional to `express' a mental state? `Expressing' is not to be understood in a performative sense, i.e. in which a gesture, a dance or a poem express certain feelings, moods or states of mind. Mark Schroeder gives a helpful analysis of the notion in his paper `Expression for Expressivists' (Schroeder (2008b)) and further builds on the notion in `Being For' (Schroeder (2008a)). `Expressing' can be understood in a technical way as referring 47 to the relation that holds between a declarative sentence and a corresponding belief, i.e. between `grass is green' and the belief that grass is green. An utterance of the sentence `Grass is green' is (semantically) permissible if and only if one has the belief that grass is green. In the same sense, conditionals express beliefs or belief-like states of mind. The only dierence between conditionals and declarative sentences is that the beliefs that conditionals express are structurally dierent from beliefs that normal declarative sentences express. On a very simple model, believing is a relation be- tween a believer and a proposition, possibly mediated by a mode of presentation (I shall ignore the question of what logical form the belief-relation has. See Crimmins (1992) for a discussion of this question.). Believing a conditional is dierent. It is not to be analyzed as a relation between a believer and a proposition { for there are no propositions corresponding to conditionals 1 { but rather as a relation between a be- liever and a pair of propositions, i.e. the proposition expressed by the antecedent and the proposition expressed by the consequent (provided the antecedent and the con- 1 At least in the sense in which the structure of propositions is somehow isomorphic to the structure of reality, i.e. in the sense in which true propositions can be identied with facts. Schroeder (Schroeder (2011b)) argues that propositions play two separate theoretical roles, the role of being the objects of the attitudes and the bearers of truth and falsity on the one hand (Role 1) and the role of being appropriate objects of Excluded Middle and vehicles of metaphysical commitment on the other hand (Role 2). Schroeder advises expressivists about some domain D to grant that there are occupants of Role 1 in D but to deny that there are occupants of Role 2 in D. He then goes on to develop an account of propositions that are apt to play Role 1 but not Role 2. 48 sequent do not themselves contain conditionals). It is, as it were, a bi-propositional attitude. The basic idea behind SCE is rather simple, but very powerful. It looks like it allows the proponent of SCE to get by without being committed to the existence of conditional propositions. This is why SCE and non-descriptivism about conditionals are natural partners, as it were. If one is attracted to the metaphysical picture of non- descriptivism about conditionals one has ample reason to go for a credal expressivism, mostly because there aren't many other views compatible with non-descriptivism which are nearly as worked out as expressivism about conditionals. And if one is, for whatever reason, already committed to expressivism about conditionals, it would be odd if one ended up rejecting non-decsriptivism. For what is the point of allowing objects such as `conditional facts' to sneak into one's ontology but then insisting that conditionals stand in no direct semantic relation to these would-be facts? The next question facing the expressivist is of course the question of exactly what mental states get expressed by a conditional. Here (SCE) has a short answer, but giving the short answer may prompt questions that call for a longer answer. The short answer is that the state of mind that gets expressed is an agent's conditional 49 credence. 2 The long(er) answer is a philosophical account of that attitude, an account that tells us when and how we should ascribe it to agents and maybe also how it interacts with other attitudes. Outlining such an account will keep us busy for most of the remainder of this chapter. Two short remarks about conditional credence are in order. First, conditional credence is a gradable attitude. One can have greater or less conditional credence in something. But not only is it gradable, we can talk about (and quantify over) degrees of that attitude. Notice that not all gradable attitudes are such that they allow us to talk about degrees. For example, one can be more or less grateful that something is the case, but it is odd to say things like `Benny was 3.5 times as grateful that the Easter bunny brought him chocolates as Mary'. Second, conditional credences can be dened in terms of credences, namely as ratios of simple credences, if we help ourselves to classical probability theory. 3 This is what I am going to do in the next section. 2 It is possible to defend a more sophisticated expressivist view according to which several mental states are expressed. Wolfgang Spohn gives a ranking-theoretic account of dierent beliefs that may be expressed by a conditional in a forthcoming paper (personal communication). 3 If the reader nds the notion of conditional belief to be illuminating one can characterize conditional credence by saying that it is to credence as conditional belief is to belief. 50 2.2 Probability and Credences 2.2.1 The Axioms of Probability The proponent of SCE is not committed to Bayesianism, but Bayesianism is ar- guably the most eective background view to complement the expressivist picture. Bayesianism can be characterized by two core assumptions, one about static belief and one about the dynamics of belief. Bayesianism 1 The belief state of a rational agent at any given time is represented by a probability measure which obeys the laws of probability. Bayesianism 2 When a rational agent acquires new information, she changes her belief state by the so-called rule of conditionalization. The next thing we need to do is say more about probabilities. According to Bayesianism, the mathematical functions that we study in probability theory and that we call `probabilities' just are properties or features of doxastic attitudes, just like lengths are features of middle-size dry goods. Hence, there is no mystery as to how to interpret the probability-calculus. Every suciently rational agent has 51 credences and in virtue of the agent's being rational all credences satisfy some basic properties. Denition 1 (Probability) LethW, F, Pri be a probability system. Let W be a set of worlds and let F be a eld of subsets of W, i.e. a non-empty family of subsets on W containing W itself and closed under complementation, union and intersection. F selects the sets of worlds corresponding to some proposition of the language L. Pr is a function from F into the unit interval that satises the following conditions: 1.8 e2 F: Pr(e)> 0 2. Pr(W ) = 1 3.8 k8 e 1 ...8 e k such that if i6= j, e i and e j are disjoint : Pr (e 1 [e 2 [:::[e k ) = P 0ik Pr(e i ) Various theorems follow from these axioms and I shall state some of them that will be relevant for the presentation of the main results: 4 Theorem 1 Pr(:) = 1 Pr(). 4 See Howson and Urbach (2006) for a thorough discussion of these theorems. The proofs of all these theorems are straightforward and are omitted here. 52 Theorem 2 Pr(?) = 0 Theorem 3 If then Pr() = Pr( ). Theorem 4 If entails then Pr()6 Pr( ). Theorem 5 06 Pr()6 1 for all in the domain of Pr. Theorem 6 (Theorem of Total Probability) If Pr( 1 _ 2 _:::::_ n ) = 1 and i entails: j for i6= j, then Pr( ) = Pr( ^ 1 ) +Pr( ^ 2 )....+Pr( ^ n ), for any proposition . One of the strategies that aim to show the truth of the rst principle of Bayesian- ism, i.e. (Bayesianism 1), is the so-called Dutch-Book argument. 5 Suppose an agent A's degree of belief in is r (for some real number r). Dutch Book arguments assume that A is willing to buy a bet on that pays $ X if won for any amount less than $(rX) and willing to sell the bet for any amount more than $(rX). If X=1, $r is the very price point at which the agent is indierent between buying and selling a bet on . Now, the Dutch Book Theorem states that agents with betting quotients whose degrees of belief violate one of the axioms of probability are susceptible to a net loss 5 Another broad strategy is proving a representation-theorem, i.e. showing that if the agents preference ordering satises certain requirements, her credences can be represented by a probability- function that obeys the axioms of probability. For an interesting discussion of this route see Maher (Maher (1993)). 53 come what may. This theorem, together with the assumption that the attitude of being prepared to lose money (or other goods that have positive utility) come what may is a symptom of irrationality { delivers the desired result. Let us look at an example in order to illustrate the general strategy. Suppose my credences violate theorem 3 because, e.g. my degree of belief in (^ ) is (for some real number with 0<< 1) and my degree of belief in:(:_: ) is ( +), for > 0. Now, some evil-meaning bookie can sell me a package of bets consisting of a bet on the falsity of (^ ) and a bet on the truth of:(:_: ) . Since, ex hypothesi, my degrees of belief satisfy the axioms, I will pay $ ( +) + (1) for the package, which amounts to $ 1 +. So I would end up paying $ 1 + (which is strictly greater than 1 given that > 0) for a package of bets that never pays more than $1, facing a guaranteed loss of $ . 6 6 As Patrick Maher (Maher (1993), p.96) has pointed out, one problematic assumption is the assumption that an agent who is willing to buy a bet B 1 for x dollars and another bet B 2 for y dollars will always buy the package consisting of both bets for (x + y) dollars. I may rationally be prepared to spend $ 3,000,000 on a house in Malibu or on a house on the Canary islands, but not $ 6,000,000 on both (even if I had enough money I might not be prepared to spend 15 hours traveling to get from one house to the other house). Formally this just means that utilities are not additive. 54 2.2.2 Conditional Probability With simple probability functions in place, we can proceed to discuss conditional probability functions. The conditional probability Pr( j) is often taken to be de- ned by the ratio Pr(^ ) Pr() if Pr()> 0, undened otherwise. Hence: Denition 2 (Conditional Probability) Let Pr be as in Denition 1. We dene Pr( j) = def Pr(^ ) Pr() if Pr()> 0 This denition together with the theorem of total probability allows us to imme- diately derive an important consequence: Theorem 7 If Pr( 1 _ 2 _:::::_ n ) = 1 and i entails: j for i6= j and for all i Pr( i ) > 0;then, Pr( ) = Pr( j 1 )Pr( 1 ) +Pr( j 2 )Pr( 2 )....+Pr( j n )Pr( n ), for any proposition . 7 A few comments on the conceptual status of (Conditional Probability) are in order. The denition says that some conditional probabilities are ratios of simple probabil- ities (in the same sense in which rational numbers are ratios of natural numbers) but it is silent on the status of conditional probabilities Pr(j ) such that Pr() = 0. 7 Again, this is a simple consequence of the theorem of total probability and the denition of conditional probability. 55             A  A&B  A  A&B  A  A& B  Figure 2.1: Conditional Probability Intuitively, the conditional probability answers the question, once an agent has xed her credence in and her credence in , what credence should she assign to , on the hypothesis that holds? The basic idea is that we look at a partition of logical space into three cells, the : situations, the^: situations and the^ situations. Now, in order to deter- mine the value of Pr(j ) we focus exclusively on the two -scenarios (temporarily ignoring all:-situations) and look at the ratio of ^ situations among the situations. The ratio may be the same even though the size of the and ^ -cells in the partition vary widely (See Figure 2.1). The interpretation of conditional probability raises no new or special issues. Con- ditional probabilities characterize doxastic states for agents, namely conditional cre- 56 dences. These states are dierent from simple credences. Yet, there is an interesting correspondence: If two agents share a distribution of simple credences, they are guar- anteed to also share a distribution of those conditional credences that are denable in terms of simple credences. And if two agents' `agree' in terms of all their conditional credences denable in terms of simple credences they also agree in terms of their sim- ple credences. 8 Still, agents may share enough of each others' conditional credences without sharing the `underlying' (i.e. denitionally more basic) simple credences. The fact that conditional probability is undened if Pr () = 0 would not be an is- sue if numerical probability functions tracked judgments of what is sometimes called epistemic possibility, assigning 0 to all and only epistemically impossible proposi- tions. But some epistemically possible propositions get assigned a probability of 0 and the corresponding conditional probabilities will be undened even though think- ing about them is not thinking about abstruse impossibilities. To give an oft-cited example: Suppose some suitably restricted version of the principle of indierence holds. Suppose moreover that you know for sure that a meteor will hit Earth tomorrow (your friend, the generally reliable astronomer, has told you) but you are completely in the dark as to where it will impact. Your 8 Proof: If two agents share all their conditional credences, they are in particular going to assign the same value to allPr(j ) for which is of the form:_ or:? if? is part of the language. But it is a consequence of the denition of conditional probability plus axiom 1 that Pr(j:_) = Pr() for all . 57 credence that the center of the meteor (once it will be at rest) is going to be in the Western hemisphere given that it will be on the equator should intuitively be 0.5 { but any conditional probability function satisfying (Conditional Probability) will be undened becausePr (the center of the meteor will be located on the equator) = 0, given that there is an (uncountably) innite number of circumferences on the globe. 9 This is one of the reasons why theorists like Karl Popper have oered a set of axioms characterizing conditional probabilities without purporting to dene them. If conditional probabilities are taken as primitive, unconditional probabilities can be dened in terms of conditional probabilities. 10 In Popper's system, conditional probabilities will obey (Conditional Probability) whenever Pr() > 0. If Pr() = 0, the conditional probability is still dened, but in cannot be derived from the value of the unconditional probabilities. The simple credal expressivist can go either way: She can take conditional cre- dences to be dened by simple credences as (Conditional Probability) has it. Or she can, pace Popper, take them to be constrained by simple credences in the case in 9 H ajek oers an example that involves the undened probability of a dart's landing in C given that it is landing in C { where C is a probability gap for a Kolmogorov probability function Pr, i.e. Pr(C) is undened. For another interesting rationale for a Popper-style approach see McGee (1994) pp. 179-181. 10 Pr() can be understood as shorthand forC(;_:), whereC( , ) is a Popper-style conditional probability function.The dening axiom (Conditional Probability) follows from Popper's Axiom C(^ , ) = C(, ^) C( , ) once we set = _:. 58 which the agent has positive credence in the antecedent and unconstrained by simple credences in all other cases. As far as (SCE) is concerned, nothing turns on whether our sympathies are with interpreting (Conditional Probability) as a denition or as a theorem that follows from Popper's axioms. 2.2.3 Conditionalization Let us now look at the dynamic bit of Bayesianism, (B2), which is concerned with learning from experience and responding to evidence in the right way by updating credences. Conditionalization is a probabilistic change rule that takes us from an old (or prior) probability function and a proposition (a set of worlds) to update on to a new (or posterior) probability function. Let PR stand for a set of priobability functions. Then this transition can be formally represented as a function from the setPR L into the setPR. Suppose the subject S learns that is true but nothing else, other than what is entailed by the new information when taken together with the old beliefs. How should this impact her credence function Pr old ? Bayesianism is informed by two crucial assumptions on belief-change, which are (i) the assumption, shared with the AGM approach, that the change has to be minimal (or maximally conservative) and (ii) the assumption that the change has to be successful { the so-called principle of certainty { Pr () = 1. 59 This is where conditional probabilities come in: When determining the condi- tional probability Pr( j) we operate on the hypothesis that is certain { that is why we always have that Pr(j) = 1. So Bayesians simply identify the new prob- ability function with the old conditional probability function, in other words they assume that Pr ( ) = Pr( j). Principle 3 (Conditionalization) If Pr new is the result of updating Pr old by , then Pr new () = Pr old (j ) Bayesians think they can show that failing to update by (Conditionalization) makes a subject vulnerable to a diachronic Dutch Book. 11 This means that learning that is true consists in simplifying our partitions by permanently deleting (not only temporally ignoring) the:-cells and expand- ing the -cells accordingly (The underlying renormalization process is illustrated in Fig. 2.2). 12 Notice that the ratios of probabilities within the -cells remain in- variant, which entails that conditional probabilities on are not aected by this 11 See Teller (1976) pp 209-212. As Teller points out the argument relies on the assumption that the probability function is dened over a set of mutually exclusive and collectively exhausting set of hypotheses on the course of the subjects experiences between the time of the bet and the time of the belief-change. 12 In fact, the deletion is somewhat of a problem. Once we go from one belief-state to another by conditionalization, we cannot revert to the old state { the belief-change is irreversible and destructive, i.e. we can never `unlearn' anything we have learned before. 60             φ&ψ  φ  φ    ∼φ  Figure 2.2: Conditionalization kind of belief-change. 13 In fact, this re ects another important feature of the rule of conditionalization, namely invariance of conditional probabilities. Now, suppose you accept (Conditionalization) and want to extend the Ramsey- test to the domain of partial belief, i.e. you want to nd out what it means for you { as a Bayesian { to believe an indicative conditional ! to a certain degree. Remember from Chapter 1 that when performing the Ramsey-test, you hypo- thetically undergo a belief-change { you hypothetically move from your current probability-function to a new probability function. The only dierence between the two belief-states, represented by the two probability-functions, is that in your new belief-state you hypothetically accept the antecedent of the conditional and 13 Conditional probabilities of the formPr(j) wherePr(j)6= 1 may vary because theirPr(^ :)-share will be reduced to 0. 61 let that new state of certainty in uence your other beliefs in the most conservative way. So the new belief-state is just like the state measured by Pr { what you do when you perform the Ramsey-test is wonder what the value of Pr ( ) is { but of course you haven't really learned and so the probability-function that actually still best represents your credences is Pr old . Now, the one and only state in your old belief system that corresponds to the hypothetical `Ramsey-state' Pr ( ) is, by (Conditionalization), Pr old (j ). Hence your credence in a conditional should equal your conditional credence. 1. Pr(! ) = Pr ( ) Ramsey Test, as applied to partial belief (Ramsey) 2. Pr ( ) = Pr(j ) Conditionalization 3. Pr(! ) = Pr(j ) (1), (2), logic of `=' I have just sketched one possible route { within the Bayesian framework { from the Ramsey-test as a rule for sorting out credences in conditionals to one of the central claims made by defenders of SCE, namely the claim that one's degree of belief or credence in a conditional reduces to one's conditional degree of belief in the consequent given the antecedent. 62 Whereas anyone who accepts (B1), (B2) and (Ramsey) is committed to endorsing this reasoning, some theorists actually reject (Conditionalization) as an update-rule and endorse Jerey conditionalization (of which standard conditionalization is a lim- iting case) or some other update-rule instead). Let us call philosophers who accept (B1) and reject (B2), i.e. updating by (Condi- tionalization), Quasi-Bayesians. Quasi-Bayesians who are attracted to Simple Credal Expressivism can still accept line (3) of the derivation. Accepting (SCE) does cer- tainly not require any theoretical commitment to (B2), since there are alternate ways of arguing for the truth of line (3). 14 But notice that theorists who do accept both claim (3) and (Ramsey) are committed to at least some instances of (Conditional- ization). 14 Here is one such way: Assume (i) the independence of the conditional from its antecedent, i.e. Pr[^ (! )] = Pr() Pr(! ) and (ii) Probabilistic Centering, i.e. the claim that Pr(^ ) = Pr (^ (! )). Putting these two claims together, we get (3). One problem with this is that we dont really have strong reasons to go for the independence of the conditional form its antecedent unless we are already committed to (3). See H ajek and Hall (1994), pp 82-83. 63 2.3 Adams' Thesis and Adams' Logic 2.3.1 Assertibility and Conditional Belief Claim (3) is often called Adams' Thesis in the literature because Ernest Adams 15 rst suggested it as a hypothesis concerning the probability of conditionals. Philosophers like Stalnaker (Stalnaker (1970)) happily endorsed the hypothesis until Lewis showed that the thesis has triviality consequences (see next section). As I have just pointed out, philosophers thinking about indicative conditionals and claim (3) (Adams' Thesis) need not be Bayesians or even Quasi-Bayesians. They may just think of probabilities as a numerical tool that tells an agent how safe it is to assert or believe something (rather similar to the sense in which a credit rating by Standard & Poor tells an investor how safe it is to invest in a given bond). In the wake of Adams' hypothesis and Lewis' results, philosophers writing on conditionals have very often taken the very concept of probability to not measure probability of truth, but rather some notion of ideal assertibility and { consequently Adams' Thesis itself has often been understood as a claim about assertibility: The conditional probability of given measures the probability of the conditional 15 In his 1975 book (Adams (1975)). 64 ! , where that means nothing but the extent to which we are prepared to assert that ! . But whereas Adams' Thesis is compatible with denying Bayesianism, it is of course also compatible with accepting Bayesianism. When discussing SCE, I shall assume that the follower of SCE accepts both Bayesianism (in the sense of (B1) and (B2)) and Adams' Thesis. They understand probabilities to measure the strengths of certain doxastic states, i.e. credences and they understand credences to be the very states that are expressed by most of our (sincere) utterances of declarative sentences. However, when we say that followers of (SCE) accept Adams' Thesis it is im- portant to be clear on how they understand the thesis they accept. They do not interpret the claim as saying that there are two functions that coincide on a certain section of their graphs. This is because according to the expressivist there really is no simple probability function that is dened for an expression like `! '. Rather, expressivists interpret Adams' Thesis as an explication of terms like `Pr(! )'. These terms do not strictly speaking refer to probabilities, but to ratios of probabilities. They are, as it were, remnants of a time when our use of formal language was informed by mistaken beliefs about conditionals. So ultimately, instead of talking about probabilities of conditionals we should only be talking about conditional probabilities, for there are no such things as probabilities of conditionals. 65 `Pr(! )' is just a somewhat misleading label for the result of applying a condi- tional probability function to two propositions, the antecedent and the consequent of the conditional. 2.3.2 Probabilistic Validity Adams' Thesis has been advanced in combination with a probabilistic notion of validity that does not rely on the notion of truth. There is a sense in which it is a package deal: It comes with a number of assumptions about how we reason with conditionals that are rigorously formulated in Adams' Logic. Of course the thesis can be attacked independently of the logic that it comes with, and it has been. Kaufmann argues that our assessment of certain scenarios reveals that we sometimes assign a probability to a conditional that does not equal its conditional probabilityKaufmann (2004). Adams himself was primarily interested in formal notions such as entailment and consistency that can be dened in his logic. Adams works with a conditional exten- sion over a factual propositional basisL containing the truth-functional connectives, including the horseshoe (). For every pair of formulae and of the factual basis 66 L, there is a formula ) in the conditional extension ofL ) . 16 Truth values are only assigned to formulae of the factual language, but probabilities are assigned to both formulae of the factual language and formulae in the conditional extension. Adams' Thesis then boils down to the stipulation that the probabilityPr assigned to an expression of the form ) equals the conditional probability Pr( j) if Pr() > 0. Adams also originally stipulated that the value of Pr( j) should be 1 if Pr() = 0, but philosophers like McGee have criticized this awkward stipulation and argued that Adams' logic sits better with a Popperian treatment of conditional probability functions. 17 But regardless of how we dene Pr( j) for the case in which has zero proba- bility, we can note an interesting corollary: Corollary 1 (Upper Bound) For for all probability functions Pr, if Pr() > 0, then Pr( )> Pr( j), i.e. the probability of the material implication is an upper bound to the corresponding conditional probability. 16 However, there is no formula containing ) as a proper subformula. Conditionals do not embed in any truth-functional contexts. To put it metaphorically, a sentence of the form ) is hovering somewhere between the object-language and the meta-language. 17 McGee (McGee (1994)) shows that if Adams denes the probability of ) to be measured by a Popper function C(; ), he can say that an inference is probabilistically valid if and only if it is strictly valid, where an inference is strictly valid i its conclusion has probability 1 under any assignments of probability under which its premises each have probability 1. 67 Moreover, we have Pr( ) = Pr( j) in the special case in which Pr() = 1. 18 With this apparatus in place, we can proceed to dene a notion of probabilistic validity. Adams' probabilistic semantics builds on the notion of uncertainty. Intu- itively, valid deductive arguments are those arguments that preserve certainty and fail to increase the amount of uncertainty { this is just another way of saying that they are not ampliative. An agent who is certain of the premises may also be cer- tain of the conclusion (and in fact, she must be certain of the conclusion). Adams generalized this epistemic interpretation of validity to arbitrary levels of condence: Valid inferences are those inferences that preserve not just certainty, the highest possible level of condence, but arbitrary levels of condence. Applying this idea to single-premise arguments is straightforward. Applying it to multi-premise arguments is a little more tricky. Adams stipulated that in the case of a valid multi-premise 18 For the proof, assume for reductio that Pr( j) > Pr( ), i.e. Pr( j) > Pr(:_ ) and that 0 < Pr() < 1. By theorem 7 and the assumption about Pr() that allows us to discharge the consequent of the theorem, this is equivalent to Pr( j) > Pr(:_ j)Pr() + Pr(:_ j:)Pr(:). This last inequality reduces to Pr( j) > Pr( j)Pr() + 1 Pr(:), since Pr(:_ j:) = 1 and Pr(:_ j) = Pr( j). However, since Pr() and Pr(:) sum to 1 the left hand side can be rewritten as Pr( j)Pr() +Pr( j)Pr(:). We can now subtract the term Pr( j)Pr() from both sides of the inequality. Hence, we get Pr( j)Pr(:)>Pr(:) which is always false, since Pr( j) cannot be greater than 1. Contradiction. What remains to be shown is that Pr( j) = Pr( ) if Pr() = 1 (This `if' cannot be strengthened to an `i'): So assume that Pr() = 1 and assume that Pr( j)> Pr(:_ ) holds. The relational expression Pr( j)> Pr(:_ ) reduces to Pr( )> Pr(:_ ), but the right hand side reduces to Pr( ) (since:^ ?) but the relation Pr( )> Pr( ) can only hold if the two terms are equal, so the original terms must be equal as well. 68 argument our uncertainty in the conclusion must not be greater than our overall uncertainty in all the premises, understood as the sum of our uncertainty in each individual premise. We can state these ideas a little more formally. Let the uncertainty of relative to a function Pr, U Pr (), be dened as 1{ Pr(). Moerover, call a function Pr proper for a set of formulae ofL ) if there is no conditional formula ) in such that Pr() = 0. Adams' denition of probabilistic entailment ensures that if a set of formulae probabilistically entails a formula , then the uncertainty of the conclusion,U Pr () is no greater than the sum of the uncertainties U Pr () of the premises, for all formulae (premises) in and all probability functions Pr that are proper for . 19 One of Adams' theorems is the claim that any classically valid inference that can be formulated in the languageL ) is probabilistically valid. 2.3.3 Adams' logic and uncertainty-unfriendly reasoning Notice that some probabilistically valid arguments are such that it is reasonable to be highly condent in each of their premises and yet completely uncondent in the 19 Adams denes entailment as certiability: Let be a formula ofL ) , the extension of the factual language by the connective), and let be a set of formulae ofL ) . p-entails i for all > 0 there exists > 0 such that all-probability-assignments Pr forL ) which are proper for and , if Pr( )> 1 for all in , then Pr()> 1 . Intuitively, the idea is that we should be able to be arbitrarily close to certain of the conclusion of a (probabilistically) valid inference without being fully certain of any of the premises. As we get more and more condent in the conclusion, we should get more and more condent in the premises. 69 conclusion. For example, the argument with 99 premises of the form `Ticket n is not a winner' to the conclusion `No ticket in the lottery is a winner' is probabilistically valid (of course this argument is also classically valid). Probabilistic validity does not rule out that an agent is much more condent in every individual premise than she is in the conclusion. What it does rule out is that the sum of her uncertainties in the premises is greater than her uncertainty in the conclusion. Yet, as I suggested above, Adams' notion of probabilistic validity is based on an important epistemic conception of validity. I now want to introduce one more notion, namely the notion of uncertainty-friendliness. Call an argument uncertainty-friendly i it is safe to reason from its premises to its conclusion not only if we are certain of the premises, but also if we are less than certain of the premises. Uncertainty- friendly arguments preserve arbitrary levels of condence. If we think of probabilities as degrees of belief or credences (as Bayesians recommend) this feature of arguments simply falls out of the axioms of probability stated above. Yet, one of the reasons why conditionals are so puzzling is that they seem to lack this feature. If the argument from the premise that `2 + 2 = 4' to the conclusion `If 2 + 2 = 3, I am the Dalai Lama' is really valid, conditionals are not uncertainty-friendly. They are uncertainty- unfriendly in the following sense: Some valid inferences involving them preserve 70 certainty but fail to preserve levels of condence that fall short of certainty. This is an interesting anomaly that only conditionals seem to exhibit. Looking at the examples of the paradoxes presented along with the schematic ar- guments in Chapter 1, we notice that we seem to be able to be very condent in their premises and yet uncondent in their conclusions (they are, after all, `paradoxes'). So they must all be uncertainty-unfriendly. But that just means that they are also probabilistically invalid, i.e. invalid Adams . If we identify probabilistic validity with validity tout court as Adams did, the argument schemata listed in Chapter 1 are all invalid, conforming to our intuitions. On the other hand, if we want to leave it open whether the arguments are ulti- mately valid but think that Adams' notion of probabilistic validity is an adequate formal analysis of the informal notion of uncertainty-unfriendliness we can at least notice that the arguments are all uncertainty-unfriendly. So regardless of whether we identify validity tout court with probabilistic validity or if we take probabilistic validity to provide an analysis of uncertainty-friendliness but not of validity tout court, we have used Adams' logic and more specically his notion of probabilistic validity to explain why the examples of the paradoxes of material implication seem bad. 71 These arguments are uncertainty-unfriendly and hence also probabilistically in- valid. Per Adams' denition of probabilistic validity, there will be choices of proba- bility functions that assign the premises arbitrarily high values (and arbitrarily small uncertainties) and the conclusion an uncertainty that is higher than the sum of the uncertainties of the premise. Per (B1) there will be agents who have high credences in the premises and yet a low credence in the conclusion. Hence, Adams' Logic pre- dicts that we get intuitive counterexamples to the argument schemata discussed in Chapter 1. That Adams is on to something and that he is right about how we (ought to) reason with conditionals is also suggested by an impressive convergence result due to Gibbard (1981): Gibbard shows that Adams' Logic coincides with Stalnaker's C2 Logic on their common domain. Any argument of Stalnaker's language that can be expressed in Adams' language consisting of factual base and conditional extension is C2-valid (see Chapter 1) if and only if it is probabilistically valid. 20 20 The proof of this result is fairly technical and cannot be discussed here. 72 2.3.4 Embeddings Due to the restrictions imposed on the language, Adams' logic does not have anything to say about conditionals that are embedded under disjunctions, conjunctions or other conditionals. Not making any predictions about a given domain may be thought to be better than making patently wrong predictions about the domain. While the charge of not telling us enough about how conditionals behave when they are constituents of com- pound sentences is not necessarily fatal for someone interested in formal properties of a language like Adams, it is deeply problematic for accounts like SCE that have looked to Adams' logic in order to come up with some story about how we can talk about validity of inferences involving conditionals without thereby being committed to conditionals' expressing propositions or having truth-values. So what is the champion of SCE going to say about occurrences of embedded conditionals? The most popular strategy consists in breaking the problem down into more manageable and smaller pieces and devising an ad hoc response to each individual type of bona de embedding. Edgington distinguishes four cases: 1. (If 1 , then 1 ) or (if 2 , then 2 ) 2. Not (If , then ) 73 3. If then (if then ) 4. If (If , then ), then Of these cases, (1) and (4) simply do not exist in the language, for example Adams' formal language, while (2) and (3) have to be reinterpreted or rewritten in a way that does not involve embedded conditionals. Let us rst investigate case (1), the example of a conditional embedded in a disjunction. The defender of SCE can claim that whenever we have conditionals embedded under disjunctions the disjunctions are but pseudo-disjunctions that are ultimately equivalent to sequences of statements containing no embeddings of condi- tionals. 21 There is actually some empirical evidence that this may not be a desperate move. The evidence concerns disjunctions (or pseudo-disjunctions) with two condi- tional disjuncts, i.e. sentences like `Either he will stay in America if he is oered tenure or he will return to Europe if he isn't'. Woods, who defends a view just like SCE, notices that this sentence seems to entail the disjunction of its consequents. On Michael Woods' view a pseudo-disjunction like `If 1 then 1 or if 2 , then 2 is 21 The strategy for conjunctions is very straightforward. We cannot conjoin conditionals, but we can conjoin what is expressed by them, namely our attitudes. Believing that if p, then q and that if s then r is the case is just believing that if p, then q is the case and believing that if s is the case, then r is the case { which according to SCE amounts to having two high-ish conditional credences, namely a credence in q given p and a credence in s given r. 74 a telescoped version of a whole series of statements: 1 or 2 : If 1 , then 1 ; If 2 , then 2 . 22 Further evidence for a conclusion along these lines is provided by the so-called Johnson-Laird and Savary fallacy: Subjects are presented with the following task: Suppose you have the following information about a specic hand of cards: 23 1. If there is a king in the hand then there is an ace in the hand, or else if there isnt a king in the hand then there is an ace in the hand. 2. There is a king in the hand. Suppose you know nothing else about the case. What, if anything, follows? All of Johnson-Laird and Savary's subjects conclude that there is an ace in the hand. But this inference is of course not valid on any view on which English disjunction is truth-functional and conditionals embed under that disjunction without interfering with them in funny ways. A defender of SCE can of course vindicate the `fallacious' judgment of the laymen: The information that the subjects really have and that they have because it is conveyed by these sentences is that there is an ace in the hand: 22 Woods (1997), p.63. Edgington discusses these cases in her commentary to Woods' posthumous monograph. 23 Johnson-Laird and Savary (1999), p. 219. 75 If there is a king in the hand, there is an ace in the hand. But also, if there isn't a king in the hand, there is an ace in the hand. The fact that embedding conditionals under disjunctions gives rise to examples that have been considered infelicitous may look like a reason to be optimistic about the prospect of tackling the other three items on Edgington's list. K olbel (2000) 24 argues that the expressivist about conditionals is indeed in a much better position than the expressivist about moral language when it comes to tackling their respective version of the Frege-Geach problem. It is undeniable that `rape is wrong' can be embedded to the same extent to which `rape is painful' can be embedded. But conditionals form their own syntactical category and it is at least not prima facie absurd to deny that such embeddings are always felicitous. Maybe `He is in Rome if he is in Italy or he is in Bordeaux if he is in France' is really not a sentence of English but just a convenient and sloppy way of saying multiple things at once. But if I am right at least the second and the third item on Edgington's to-do- list present serious problems. Negating a conditional is usually not just negating the consequent (I will say more about this in the next chapter). The third item 24 K olbel (2000), pp 98-99. K olbel shows that whatever ad hoc manoever expressivists choose, cases of (existentially) quantifying into a conditional construction cannot be paraphrased in any way that does not involve embedding conditionals. Since I will largely be concerned with a language containing expressive resources that can be analyzed at the level of the propositional calculus, I will not discuss K olbel's cases. 76 on Edgington's list, `right-nested' conditionals are another very real phenomenon that cannot be explained away that easily. The expressivist is unable to predict the import-export principle (see Chapter 1). She is forced to interpret it as a `meaning postulate', a principle that tells her how to interpret a strictly speaking ill-formed conditional of the form (! ( ! ). This is similar to her reading of Adams' Thesis, but the dierence is that Adams' Thesis was always intended as a postulate or a hypothesis, whereas the export-import principle should be expected to be a logical theorem that should fall out of whatever axioms we assume for conditionals. Finally, expressivists deny that constructions of type (4) are felicitous. I actually believe that many of them are felicitous, but I admit that making sense of them is very tricky. I will discuss these conditionals when I assess the merits of my own view in Chapter 6. 2.3.5 Upshot In this section on Adams' Logic I have reviewed a logical system that seems tailor made for the defender of SCE. According to SCE, it is ok to say `If , then ' for an agent S (in some context C) i S has a suciently high conditional credence in given . The availability of Adams' Logic has shown us that the defender of SCE can go beyond the stage of merely telling us what it is to utter a conditional: she can 77 actually oer a full-blown account of what it is to reason with conditionals, a story that ts comfortably with her commitment to Bayesianism and the probabilistic version of the Ramsey-test. Moreover, Adams' Logic allows her to spell out such an account sans having to say anything interesting about what it is for a conditional to be true. But there is a point at which her ability to tell a coherent non-descriptivist story about the conditional comes to an abrupt end and that point is the point at which we decide to embed conditionals in other truth-functional constructions. It is just before we reach this nal grade of conditional involvement, as it were, that she quits the game. 2.4 SCE and Triviality 2.4.1 The Bombshell SCE's problem with embeddings is a liability, but in the context of discussions of so-called triviality results it is often perceived as an asset. Philosophers like Anthony Appiah (Appiah (1986)) have argued that at least certain triviality results give us good reason to reject the view that conditionals have truth values or express propo- sitions. It was Appiah's idea to turn what might be thought of as a limitation of the view into a dialectical weapon. 78 But what are triviality results anyway? At their most ambitious, triviality re- sults show that Adams' Thesis plus various other assumptions impose constraints on the interpretation of the probability measure that can only be satised by trivial interpretations. A trivial interpretation of the probability calculus is any interpretation according to which the range of the probability function contains a lesser number of distinct values than would be needed to adequately represent actual dierences in credences or dierences in likelihood. That means that what counts as trivial depends on our expectations of how ne-grained we would like partial beliefs or likelihoods to be represented. So `triviality' is always to be understood relative to an expectation{ more specically, relative to the expectation of how ne-grained non-trivial measures of partial belief or credences are. For example, any probability functionPr that has at most k distinct values may be trivial with respect to credences concerning a lottery that contains at least k+1 tickets. Consider the sequence of statements: `One of the rst j tickets is going to be the winner', with j ranging from 0 to k +1 { the function Pr will have to assign the same value to at least two members of the sequence that intuitively deserve dierent credibility. So it will miss out on some relevant distinctions, making it inadequate at best. 79 But a scarcity of values of a probability function is not the only way for an interpretation of the probability-calculus to be trivial or simply inadequate. Recall from Chapter 1 that G ardenfors showed that a certain construal of the Ramsey-test forces an interpretation of the AGM-calculus on us that is inadequate. In fact, the literature on indicative conditionals teems with triviality proofs, using an array of dierent assumptions H ajek and Hall (1994). All the proofs look to the `bombshell', Lewis' surprising result in his 1976 paper, as their common ancestor Lewis (1976). The rst of Lewis two original results runs as follows. Let Pr be an arbitrary probability function dened over a class of propositions, understood as sets of worlds, that satises the probability axioms. LetPr(j ) be a conditional probability function dened in terms of Pr in the usual way (i.e. it satises (Conditional Probability)). The main lemma is very straightforward, but it requires an auxiliariy lemma that relies on a closure-assumption. Then we have, for all probability functions Pr in a class C that is closed under conditionalization (i.e. if the class C contains a function Pr, it also contains Pr , for an arbitrary sentence of the language) the following derivation: 1. Pr(! ) = Pr((! )^ ) + Pr((! )^: ) by theorem 6 2. Pr(! ) =Pr((! j )Pr( ) +Pr((! j: )Pr(: ) by (1) and th. 7 80 3. Pr(! ) = Pr( j^ )Pr( ) + Pr( j^: )Pr(: ) by the auxiliary lemma and (2) 4. Pr(! ) =Pr( ) 1 +Pr(: ) 0 by simplication of (3) 5. Pr(! ) =Pr( ) by simplication of (4) 6. Pr( j) = Pr( ) by (5) and (AT) (6) is a triviality result because it shows that arbitrary conditional probability functions collapse into unconditional probability function. As Bennett (Bennett (2003)) puts it, if (6) were to characterize an agent's belief state (which we may safely assume to be non-trivial), that agents credence in `I am going to die tomorrow' should equal her credence in `I am going to die tomorrow, given that I have terminal cancer'. Our expectation is that any probability functions that are candidates for representing a rational agent's credences should be able to discriminate between these belief-states (such discriminatory power is simply called for). Since the expectations 81 are not met, the interpretation of a belief-state by a functionPr() that satises Lewis conditions is a trivializing interpretation. 25 SCE can bypass Lewis' conclusion by blocking the derivation of the main lemma at the very beginning. The term Pr((! )^ ) does not pick out a value of a probability function: We have no rule for assigning probabilities to the result of conjunctively combining a conditional with some non-conditional formula. We have to keep in mind that according to SCE, there is no such thing as `the probability of a conditional': Even though the term `Pr(! )' does refer, via Adams' Thesis, to a real number, it does not refer to the result of applying a probability function Pr to a proposition denoted by a sentence of the language. 2.4.2 How robust are these results? Lewis' result might have been due to his specic package of assumptions. What if some of these assumptions are dropped? As almost forty years of research in philosophical logic have shown, triviality results are surprisingly robust. They persist even if dierent assumptions are made or if some of the core assumptions, including Adams' Thesis, are weakened. This is of course a fascinating (meta-)result in its own 25 Lewis also shows in his second triviality result that any member Pr of a class of probability functions that satises Adams' Thesis and is closed under conditionalization has at most four dierent values. Lewis (1976), p. 139. 82 right, but the really interesting question for the current dialectic is whether SCE is in as good as a position to bypass these results as it was in the case of Lewis' original proof. Let us start with H ajek's 1989 result (H ajek (1989)). H ajek can actually show that we do not need to assume that the class of probability functions is closed under any operation like conditionalization. Moreover, he cites Stalnaker and van Fraassen, who both identied another (according to them) objectionable assumption in Lewis' original proof, namely `metaphysical realism', the view that the proposition expressed by a conditional is independent of the probability assigned to it. 26 H ajek's proof does not rely on this second assumption either. H ajek shows that as long as we suppose that ! expresses a proposition (i.e. a set of possible worlds) whenever and do (never mind that it expresses the same proposition under every probability 26 Van Fraassen criticizes the idea that credences wax and wane without any changes to the way an agent represents the world. Here is what he says: A certain person, who has a certain amount of information, and a commitment to certain theories, represents the world to himself by means of a model structure with probability measure P on its set of possible worlds (...) Now a revision occurs; this persons information and/or theoretical commitment changes in a certain way. His ideas about what the world is like change; and also the degrees of belief he attaches to the sentences in his language. So he revises his model structure cum probability measure. And here Lewis introduces the requirement that it should be possible to make this revision by changing the probability measure alone and not the constitution of the possible worlds and the nearness relation on them.van Fraassen (1976), p. 264 This idea could be (tentatively) exploited by contextualists about conditionals who want to deny that they are subject to Lewis' result. However, since H ajek's result is independent of the assumption of `metaphysical realism' it looks like this contextualist move is not going to be robust. 83 function), we can show that there are strictly more distinct values of conditional probability functions than there are values of un-conditional probability functions. 27 This shows that the value of at least one conditional probability function cannot have `a match', i.e. the value of an unconditional probability function that shares a value with it. Again, champions of SCE happily avoid this result: There are no propositions expressed by sentences of the form! , so the fact that there are more values for conditional probabilities than there are values for unconditional probabilities need not concern them. But triviality rears its ugly head even when Adams' Thesis itself is weakened. Morgan (Morgan (1999) looks at a theory of comparative probability and shows that an assumption that is weaker Adams' original assumption leads to triviality. Morgan denes a four-place relation of comparative probability on a standard propositional language. The relational expression `A, LE B, ' is to be read as `A given the set is less likely than or equally likely as B given the set .' The sets (representing sets of background beliefs or evidence) and can be empty or identical to each 27 H ajek assumes that we work in a nite model, i.e. a model with nitely many worlds. While this is clearly a substantive assumption, it is not implausible that there are only nitely many ways things could have been. If we use possible worlds primarily as a tool for modal semantics (for a propositional language), we do not need more than nitely many worlds. If a set of sentences is consistent, it is satisable in a nite model.Burgess (2009), p.61. 84 other, but they may also be distinct. Notice that theories of comparative or qual- itative probability on which those orderings are re exive and transitive are weaker than standard numerical probability in the sense that one ordering of sentences is compatible with multiple distinct assignments of probabilities to those sentences. 28 Morgan then shows that the `Adams assumption', i.e. the two assumptions 1. If A then B, LE B, [fAg 2. B, [fAg LE If A then B, lead to triviality when added to a number of other defensible assumptions about the relation LE. This may be a worrisome result, but simple credal expressivists are fortunate enough to be able to avoid it. Morgan allows the sentence `If A then B' to occur in any argument place of the relation LE. In other words, Morgan's language allows me to express belief states such as `I am at least as condent that A given that if C then D holds as I am that B given that if C then D holds'. Such a permissiveness should raise suspicions. If there are no numerically sharp belief states corresponding to expressions like Pr(j ! ), there is no reason to believe that there should be `blunt' or merely comparative belief states corresponding 28 A preordering like Morgan's relation is re exive and transitive, but not total. That means that some items may not be ordered with respect to each other, i.e. for some and , neither is at least ast likely as , nor is at least as likely as . For an axiomatic account of qualitative probability that does not assume that the relation of being at least as condent in than in is (doubly) relativized to bodies of evidence, see Hawthorne (2009). 85 to relational expressions like `A,fIf C then D, ...g LE B,fIf C then D, ...g'. The use of these expressions relies on an ability to treat conditionals themselves as antecedents or as hypotheses from which other claims are derived, something that simple credal expressivists agree we cannot do. Hence, the expressivist can just roll her eyes at Morgan and say: `No wonder you generate these results { you misunderstand the nature of conditional thought!' Interestingly, Morgan himself concluded that in the presence of Adams' Thesis or a weakened version thereof the conditional is not apt to be included in the object- language. Interestingly, this is far more pessimistic than the conclusion that Adams himself drew, since Adams granted the conditional at least a limited role in the object language. Morgan's result shows that there is no need to assign (numerical) probabilities to conditionals in order to generate triviality results. But suppose we do want to assign numerical probabilities to propositions and believe in sharp credences: A result triggered by Bradley's so-called preservation condition suggests that something much (much!) weaker than Adams' Thesis suces to force trivial interpretations on us Bradley (2000). All we have to assume is that if Pr(A) > 0 and Pr(B) = 0, then Pr(A! B) = 0. This condition on probability assignments is entailed by Adams' 86 Thesis but does itself not entail it since it is much weaker. There is no question that the condition is extremely plausible. As Bradley puts it: You cannot, for instance, hold that we might go to the beach, but that we certainly won't go swimming and at the same time consider it possible that if we go to the beach we will go swimming! To do so would reveal a misunderstanding of the indicative conditional (or just plain inconsis- tency).Bradley (2000), p. 220. Given this principle, Bradley shows that if we treat conditional as belonging to the same Boolean algebra as the sentences from which it is compounded, all interpretations are such that either A or A! B is going to imply B. 29 But this is of course the point at which the defender of SCE gets o the boat. So, again, expressivists can argue that triviality results are a symptom that calls for (only) one diagnosis: Conditionals are not the kind of sentences that express thoughts or propositions, i.e. items apt to be constituents of more complex thoughts, and accordingly they are not to be embedded under truth functions or conditionals. 29 There are of course some conditionals that are such that either their antecedents or the con- ditionals themselves imply their consequents, e.g. `If Anna is at the wedding and Peter is at the wedding, then Peter is at the wedding' but a conditional connective for which either of these con- ditions is always satised is of course pointless. 87 2.5 Conclusion SCE is a strange animal. In the previous section I concluded that it says too little about the English `If....then'. Conditionals are sentences of English that occur as constituents of more complex sentences. In n.10 of the dissertation Iy Condence I use the following sentence: `If I believe that If Jon and Susie come, the party is going to be terrible but also believe that if Jon comes, the party is going to be fun and if I learn that Jon and Susie are actually coming, I cannot infer that the party is going to be fun.' This sentence may be hard to parse, but I am pretty sure that it is grammatical. Explaining embeddings is certainly a challenge for any view and so it isn't sur- prising that many views appeal to some kind of partners-in-crime consideration in order to excuse their own inability to do so. Yet, SCE just atly denies that there are any real (as opposed to pseudo-) embeddings. This is one of the greatest weaknesses of SCE and yet, ironically, it guards the view against some nasty consequences that descriptivist views like contextualism or relativism have to face, namely the triviality results. Or so it seems. If we revisit my hasty discussion of some of the results we will notice that most of the work is done by the claim that conditionals do not express propositions (and are hence to be treated dierently than sentences expressing propositions) and not by the 88 claim that they never embed in other sentences. It is one thing for the conditional to be able to form compounds with other sentences of English. And it is quite another thing for these compounds to be appropriate objects of propositional attitudes like credences. I think there is any important lesson to be drawn from this chapter, even though the `lesson' is probably speculative enough to merely qualify as a hypothesis. The hypothesis is that we may have reason to allow conditionals to behave just like other sentences in most respects (see SCE's problems with embeddings) but that we also have reason (see the triviality results) to think that assigning credences to conditionals is dierent from assigning credences to other sentences of the language. Maybe conditionals behave like ordinary descriptive sentences as far as their syntactic role is concerned but fail to behave like ordinary descriptive sentences as far as their epistemic role is concerned. These two aspects of their behavior would seem to militate against each other, leading to a peculiar tension that can be, depending on our philosophical temper- ament, either puzzling or fascinating or both. 30 In the next chapter I shall discuss a view that is true to SCE's non-descriptivist spirit but overcomes its awkward re- 30 A tension that is by the way not unrelated to the tension between our two desiderata formulated in Chapter 1. 89 jection of allowing the conditional to embed freely in truth-functional (as well as non-truth-functional) contexts. 90 Chapter 3 The dynamic view 3.1 Recap In Chapter 1 I discussed a simple, but mistaken view of the conditional, namely the horseshoe view. The two major shortcomings of the view led me to formulate two desiderata for any views of the conditional. The rst desideratum is that a theory of the conditional explain how we reason with conditionals when we are less than certain of the premises. The second desideratum is that a theory give an account of how the conditional embeds at least in some simple truth-functional constructions. I then surveyed a number of views on which conditionals conform to the Ramsey test but fail to satisfy at least one of the desiderata. I also argued that the family of 91 views that are committed to non-descriptivism about conditionals promise to capture Gibbardian standos and reversibility phenomena and deserve our attention. In Chapter 2 I also looked at Simple Credal Expressivism. According to SCE, what it is to be in the state of getting a positive result on the Ramsey test is to have a suciently high conditional credence in the consequent, given the antecedent. That means that there is not just one such state of passing the Ramsey test, but many dierent states. Moreover, saying that p if q is to express that attitude, not to assert that one has that attitude. I observed that SCE's commitment to Adams' Thesis has two roots: Acceptance of a probabilistic version of the Ramsey test and acceptance of conditionalization as a rule for belief change, a view called (Bayesianism 2) in Chapter 2. In this chapter I will look at a major alternative to SCE, namely a dynamic semantics for the conditional. The view that such a dynamic semantics captures the meaning of the conditional can be characterized as a view that is still non- descriptivist in spirit, but that oers a more elaborate picture of what the point of asserting conditionals is. As I will show, the view oers a model of successful communication involving the conditional. I believe the view has three important and theoretically fruitful features: 92 The theory predicts that the assertion of a conditional can make a real contri- bution to a conversation. The theory substantiates the intuition that the indicative conditional diers in meaning from the material implication, while explaining why the inference from ` or ' to `If:, then ' seems trivially valid. The theory is consistent with the non-descriptivist idea that conditionals are not in the business of describing the world. In the remainder of this chapter, I will argue that a certain standard version of a dynamic semantics for the indicative conditional that interprets the indicative conditional as a Ramsey conditional exhibits these features. However, as I will argue in subsequent chapters, all is not well with the dynamic account: It needs to be revised and extended if it is to meet the second main desideratum (i.e. explain the conditional's role in uncertain reasoning), which is a desideratum that SCE seems to fully satisfy. Dynamic Semantics comes with a package of assumptions about what it is for a term to have a certain meaning (and a fortiori what it is for that term to have the same meaning as some other term). In Section I, I will look at the most important of these assumptions. In Section 3.2, I will introduce a sample theory. Section 3.3 contains a discussion of what makes such a theory special (and what made it so 93 innovative, at least at the time when it was rst presented). Section 3.4. introduces the account of conditionals that falls out of the theory. In Section 3.5 I will revisit the above list of positive features and argue that the theory has them all. 3.2 Epistemic Commitment and Commitment States Donald Davidson famously (and controversially) argued that a theory of truth can do duty for a theory of meaning. 1 According to Davidson we dont need to introduce meanings as theoretical entities in order to obtain a theory that qualies as a theory of meaning. Now, the dynamic view that I shall develop is hardly Davidsonian, but it shares the methodological assumption that we do not need to introduce new theoretical entities in order to get a theory of meaning. All we need is a space of epistemic possibilities or epistemic scenarios and a binary relation dened on it. In the current version the binary relation is going to be interpreted as the relation of epistemic or doxastic commitment. Intuitively, acquiring certain beliefs commits an agent or a group of agents to holding other beliefs (that they may or may not actually have). For example, if I believe that all Cretans are schmoozers and I learn that Epimenides is a Cretan I am committed to believing that Epimenides is a schmoozer. 1 In his famous paper `Truth and Meaning', Davidson (1967). 94 A rational agent can foresee the impact that the (undefeated) information that Epimenides is Cretan is going to have on their belief system { the `impact' of a sentence is measured by the extent to which the agent's overall commitment state changes after coming to accept the sentence. Notice that beliefs are not the only intentional states that are related to each other via commitment relations. Intentional states like supposing that and hoping that also carry commitments, even though they may be distinct from believing that. The idea that what a sentence means can be captured by looking at the ways it aects a certain conversational (not doxastic) context is at the center of the approach to meaning called Dynamic Semantics. Proponents of dynamic semantics understand the meaning of a sentence as something that can only be measured by looking at a sentence's context change potential or CCP. Along with this view goes the assumption that no static feature of a sentence can fully capture its meaning. Frank Veltman (Veltman (1996)) has described this semantic conception with a famous slogan: `You know the meaning of a sentence if you know the change it brings about in the information state of anyone who accepts the news conveyed by it'. My interpretation of dynamic semantics departs from the standard interpreta- tion in at least one respect: Unlike `standard' dynamic semantics (as presented in Heim (1992), Groenendijk and Stokhof (1991) and Beaver (1999)) the current ver- 95 sion is not supposed to model how a conversation or an inquiry actually evolves over time. Rather, it shows how accepting new information would aect the overall com- mitments taken by a group of agents. Just like in the case of Stalnaker's common ground, the commitment states recorded by the system are publicly accessible in the sense that agents have common knowledge of the fact that they are committed to whenever holds at the information state. Since the commitment state does not track a conversation, there is also a sense in which the system does not aspire to tell us what agents actually are committed to after they accept the new piece of information, but rather what they would be committed to at time t if they were to accept the new piece of information, holding their remaining commitments at t xed. This is somewhat similar to assessing outcomes of actions in decision theory: Decision theory does not tell us what course of action an agent or a group of agents actually settle on, nor what they have most reason to settle on, but merely how an action aects the agent's expected utilities or payos, given the agent's preferences at a certain time t. Since the dynamic system that we will be studying is based on classical logic and since classical logic validates the principle Ex Falso Quodlibet, a state in which a group of agents take on mutually inconsistent commitments (e.g. a commitment to 96 believing thatp paired with a commitment to believing that:p) will be characterized as a state in which the agent has arbitrary commitments. Given my understanding of the point of dynamic semantics, this shouldn't be interpreted as saying that a group of agents who are about to accept something that con icts with their overall commitments as a group will eventually come to be committed to believing that there are round squares. It only means that accepting the new piece of information would result in a state of being overcommitted, in which our notion of epistemic commitment ceases to do any theoretically interesting work. When faced with a situation in which the agents have received con icting informa- tion they can always avoid the threat of being overcommitted, either by discounting one of the `incriminating' pieces of evidence or by re-examining and adjusting their beliefs. But the fact that there are many techniques for belief revision is not to say that states of being epistemically overcommitted are always irrational. It is plausible to think that agents can have inconsistent practical and epistemic commitments while still being rational. If I inadvertently believe that p and that q and if q entails:p I am committed to both p and:p. 2 2 Epistemic commitment is a normative notion, but `X is committed to believing that p' is not quite as strong as `X ought to believe that p' { The question of whether there are genuine ought- dilemmas, i.e. acts that are such that someone ought to do them and someone ought to not do them (at the same time) is fairly controversial { commitment-dilemmas on the other hand should be 97 While the notion of epistemic commitment is the conceptual basis of the system, we will not treat it as primitive for the purposes of the technical reconstruction. Instead, we will start with the notion of a commitment state, which is the state that records a group's collective commitments at a certain time. Just like individual agents can have overall commitments, groups of agents can have overall commitments some of which may not be commitments of any individual in that group. In dynamic semantics commitments states (Veltman uses the term `information state' but the two terms are interchangeable) are conventionally characterized by sets of epistemically possible worlds { the worlds in the sets are the worlds that have not been ruled out or dismissed. A sentence's change potential itself is then simply modeled as a function from commitment states to commitment states. Given the notion of a commitment state and some recursive change or update function" (to be dened in the next section), we can model the intuitive notion of epistemic commitment in the following way: Suppose that a group of agents is, at time t, in a state s. If at t, s" = s, i.e. if the update by the formula is redundant, then the agents are epistemically regarded as rather common. For a very compelling account of epistemic and practical commitment, see Chapter 5 of Shpall (2011). 98 committed to . And vice versa i at t, the agents are committed to , s" = s. 3 The basic idea is that an update is redundant only if the information added by the update is already accounted for by the commitment state. Principle 4 (Redundancy) An agent in commitment state s is committed to believing that i the update of s by is redundant, i.e. s" = s. 4 (Redundancy) is not a denition of what it is to be committed to something, but something like a postulate for the meaning of `commitment'. Any adequate update-operation has to satisfy (Redundancy): To see that (Redundancy) is at least prima facie plausible, pick a simple example: If all scenarios in my commitment state represent Kansas as being east of Colorado, 3 This means that the states that commit agents to believing that are the very states that are xed points of the" - operation. Belief is the limit of the inquiry (into the question whether ) { the very state that makes further inquiry pointless. This is a substantive assumption, since one might think that believing is the kind of state is unbounded in the sense that one can come to believe something more and more strongly, without limit. 4 Both directions are very straightforward. Left-to-right: If the agent is already committed to believing that, coming to accept should not alter their overall commitments: If it did, accepting would commit them to believing something new, over and above the commitment to believing that . But accepting that does not commit them to anything over and above believing that . Right-to-left: Suppose a given update by is redundant, i.e. it means that the commitment state hasnt changed. That means that either carries no information at all, in which case the agent is already (vacuously) committed to believing that or it carries substantive information that is however not new to the agent and so the agent is already committed to believing that . Either way, the agent will be committed to believing that . 99 adding the accepted information that Kansas is east of Colorado will not result in any change. Conversely, if accepting the information that Kansas is east of Colorado makes no dierence as far as my commitment state is concerned, it must be because I was already committed to believing that Kansas is east of Colorado. 3.3 The Recursive Denition of the Update-Function Before we can appreciate how the dynamic account explains the meaning of con- ditionals while exhibiting the features listed in Section I we will have to give an overview of how the semantics works. The propositional language that I am using is to be understood as a regimented and expressively impoverished version of English. Greek letters will be used mainly as metavariables for actual formulae of the language. On occasion, I will use Greek letters when mentioning (as opposed to using) the formulae they range over. While the basic formal properties of the system have been rst explored and presented by Frank Veltman, I shall introduce my own terminology and diverge in my denitions from Veltman's. The formal properties of the current system are the same as those of Gillies' systemGillies (2004). As mentioned in the previous section, the update-function that will be dened operates on commitment states. Now, let W be a non-empty set of worlds or ersatz 100 items (situations, etc.) and let I be a family of subsets of W. Then the members of I are commitment states of some individual agent. We will let the variables s,s 1 , s 2 range over members of I. If s = W, the agent has not come to accept anything no world has been ruled out as misrepresenting reality. If, on the other hand, s is the singleton of some member of W, the agent has narrowed the space of possibilities down to one world, i.e. the agent is fully opinion- ated. A process in the course of which an agent accepts more and more sentences (e.g. a scientic inquiry or a criminal investigation) can be modeled as a sequence of possible commitment stateshs 0 ... s n i where s 0 is the initial commitment state and s n is the endpoint of the process. Such a process is fully successful if it terminates in a state that is the singleton of not just some world, but the singleton of the actual world. Finally, the absurd acceptance state, to be equated with the empty set, is the state in which inconsistent pieces of information would be accepted. Rather than being a state of epistemic anarchy and unruliness (`anything goes'), the absurd state should be understood as dead end for some particular conversation, inquiry or other cognitive-investigative process { a new starting point has to be identied and the process of gathering information can start afresh, premised on a slightly dierent 101 commitment state, but the dynamic system does not provide or even suggest any particular repair strategy. 5 Let b A be the set of propositional variables of a language of propositional logicL, which we imagine to be an impoverished version of English. LetL ^ be the closure of b A under the usual truth-functional connectives. An atomic sentencep can be said to be true or false at worldw depending on whether some two-place valuation function V assigns 1 to the pair consisting of w and p. 6 Denition 3 (Updates for the languageL ! of CUS) LetL ! be the closure of b A under the truth-functional connectives plus2,3 and!. Consider any w2W, s 2 I, p2 b A, 2L ! . The update function": I! I is dened by the following recursion: 1. s" p =fw2 s : p is true at wg 2. s": = sn(s") 3. s" (^ ) = (s")" 7 5 There is no reason though why repair strategies discussed in the context of the AGM-framework shouldn't be applied to dynamic semantics. 6 Truth for the non-atomic members ofL ^ can be dened in terms of their intensions (see below), i.e. we can say that is true at w i w2 [] 7 Disjunction (_ ) will be dened as:(:^: ) and the material implication ( ) is dened as (:_ ) 102 4. s"2 =fw2 s : s" = sg 5. s"3 =fw2 s : s":6= sg 6. For all 2L ^ : s" (! ) =fw2 s : (s")" = s"g 8 Some comments and explanations are in order. Denition (1) shows that updates come in at least two dierent kinds: Intersective Updates It can be argued that the update semantics for the truth- functional fragmentL ^ of the language is a notational variant of Stalnaker's 1978 viewStalnaker (1978), according to which asserting that removes the possibilities incompatible with (the information that) from the common ground. 9 Put a little more rigorously, we can show that for all in the truth-functional fragment and all information states s an update of s by will be intersective in the following sense: s" = s\ [] (3.1) 8 This restriction, i.e. the restriction on formulae in the antecedent of a conditional is not imposed by Gillies in Gillies (2004) 9 There is a very general result that shows that any bona de dynamic structurehI,"i that satises two conditions, namely eliminativity and distributivity, reduces to a static structure. 103 The set [] is the intension of , intuitively the set of worlds that are characterized by the information carried by { The intension of for modal-free sentences can be implicitly dened by equation (1). 10 The novel and exciting aspect of dynamic semantics comes to the fore with con- nectives or operators that do not simply eliminate possibilities, i.e. add information but perform a dierent operation. Certain commands in computer programs oer good examples of operations that do something other than adding to a body of infor- mation. For example, a rather primitive command would check if certain syntactic rules are being violated and return the original input if there aren't violations and an error symbol if there is at least one violation. Similarly for a command that searches a database for a key term or a letter etc. Updates that are tests What makes dynamic semantics an interesting and worthwhile project is that it allows us to introduce sentences whose meanings cannot be charac- terized by intensions and hence not straightforwardly explained by static semantics. These sentences fail to characterize sets of worlds. For our current purposes these sentences will be limited to conditionals and epistemic modals. The basic idea is that updates by epistemic modals are non- intersective, i.e. they run tests on an agent's information state, tests that either 10 Modals and conditionals lack intensions that can be dened via the"-operation. 104 return the original state or crash, i.e. they result in the empty set. Notice that updating by `might' either returns the absurd acceptance state; or the original acceptance states { there is no middle ground. The reason for this is that the world- variablew does not occur in the condition on set membership in the class abstraction symbol and so whether or not the condition holds does not depend on any particular world. If the condition holds, every world ins will be in the set and if it fails to hold no world in s will be in the set. 11 It follows that updating by an epistemic modal will never result in a net increase of information. On Veltman's view epistemic modals do not knock out some possi- bilities while retaining others. They run tests on acceptance states. This is where the dynamic system parts company with Stalnakerian common ground-semantics, because in Stalnalker's system every sentence plays the same role, i.e. the role of narrowing down the context. By saying `Anne might be in Paris' an agent is not committing herself to Anne's being in Paris or her not being in Paris, but she is making sure that her and her addressees' common information state allows for the possibility of Anne's being in 11 If we think of surviving an update on as a property of worlds, then surviving an update by `It must be that ' is a property that a world in s has or fails to have vacuously, in a similar sense in which Peter has or fails to have the property that all men are mortal vacuously. But just like Peter's having that property depends on his mortality (he wouldn't have it if he were immortal), a worlds' having that property depends on the world's being a certain way, namely being a world where is true. Intuitively, the property of worlds that can be characterized as surviving an update by is nothing but the property of 's being true at w. 105 Paris. By saying `Ann must be in Paris' the agent is checking that she (and her addressees) are committed to Ann's being in Paris, without explicitly informing them that she is in Paris. Conditionals, too, behave semantically like tests { just like epistemic modals they can be conceptualized as tests on acceptance states. The only open question is: What kind of test do they run? { what is an acceptance state being tested for by someone's assertion that if p, then q? Plausibly, a conditional runs the following test. Take the current acceptance state { update it (via a hypothetical update-operation) by the antecedent and ask yourself: Does the resulting state support the consequent? In other words, a group of agents' being committed to a conditional means that their acceptance state passes the test just described. Using the notion of epistemic commitment, we can formulate the following adequacy condition for the dynamic test performed by a conditional. (It should not be surprising that this adequacy condition corresponds to a deduction theorem for the conditional that captures the gist of the Ramsey Test): Principle 5 (Dynamic Ramsey test) Ramsey Test s (! ) if and only if s + The formulas + denotes the acceptance state that results from s after is being hypothetically accepted. Now, if we interpret hypothetical updating + simply as 106 enriching an information state via the"-operation (the straightforward updating operation that has been introduced), the right hand side of Ramsey Test tells us how to dene an update for the conditional that has the desired features. Of course hypothetically updating may still turn out to be dierent from (merely) supposing that one really learns something (the received view made popular by Levi). Should the received view about hypothetical updating be false, the simple dynamic conditional would fail to satisfy the Ramsey Test and we would have to construct a more elaborate dynamic function in order to capture the meaning of the conditional But for now the simplifying assumption about hypothetical updating will do no harm. 12 This clause describes a test because s" (! ) either returns s or the empty set. Like in the clause for `might' and `must' the condition for class membership does not depend on the variable w, which means that the set will contain all worlds in s as members if the condition holds and none if it fails to hold. But the condition for class membership is just passing the Ramsey test. Hence, if the states passes the Ramsey test, i.e. if the result of updating s by the antecedent supports the consequent, the 12 There are in fact some doubts about whether Levis view is correct. If we think { as I do { that at least some so-called Thomason conditionals are true and assertible (`If p, then I don't know that p') we have strong reasons to reject this story about hypothetical updating, as Willer (2011a) argues convincingly.The good news is that dynamic semantics can be rened to allow for a separate hypothetical update operation, as Willer shows. 107 test procedure returns the original commitment state. On the other hand, if s fails the Ramsey test, s" (! ) will not be equal to s" (! ) { consequently, the update crashes and the empty state is returned. Of course, an update can be a hybrid of test and an intersective update, as for example it will be in the case of (3^:). As any other system, CUS has to tell us which inferences are valid and which aren't, so we need to quickly discuss validity within the update semantics. Glossing over many complications, we can oer the following denition of acceptance and entail- ment. Both re ect important ideas: The clause for acceptance re ects the idea that a (group of) agent(s) accept(s) something if and only if her (their) overall informa- tion does not change when the accepted item is added. The clause for entailment is based on the idea that entailment is a meta-linguistic rephrasing of the conditional: ought to entail if and only if the conditional ! is a tautology. Denition 4 (Acceptance and Entailment) Let s be an information state, and let , be formulas ofL ! 1. An agent in s accepts , s i s" = s; 2. commits an agent to or entails , i for all s: s" 3. and are epistemically equivalent ( = ) i for all s: s" = s" 108 Some comments are in order. An information state supports a sentence i up- dating the state by that sentence has a null-impact. (Redundancy) tells us that a set s supports a sentence if the agent whose commitment state s is committed to . Now, as long as I contains only sets that are commitment states of some agent we will also get the other direction of the entailment. Notice that s could be the absurd state;. The absurd state supports any sentence, which should not be surprising, given that the absurd state is equated with any commitment state in which an agent has inconsistent commitments. 13 The denition of entailment guarantees that ` entails ' is co-extensional with ` commits an agent to '. One could try to generalize this relation to sets of sentences, i.e. to the relation ` entails ' in a natural way. Being a set of formulae, has at most denumerably many members 1 , 2 ,.. , n , so we could say that entails i the conjunction 1 ^ 2 ^...^ n entails . However, this strategy for extending the denition of logical consequence does not work. Conjunction fails to be commutative in the system, i.e. we do not have that ^ entails ^ for arbitrary formulae and . 14 13 That the absurd state supports every sentence can be shown by induction on the length of the formula . 14 A counterexample to the commutativity of conjunction is given by the formula 3^:, a formula that does not entail:^3. This can be easily seen by noticing that updating a state 109 We can now say that a formula is valid in the system i it is entailed by any formula whatsoever { or equivalently, i is supported by the state W , the maximally unopinionated state. 3.4 How well does the dynamic system do? It is time to verify if dynamic semantics actually has the features listed in Section 3.1. Let us start with (1): The theory predicts that asserting a conditional can be a real contribution to a conversation At rst sight, it may not be clear that the system exhibits this feature. And this is because the fact that the conditional update is a dynamic test means that the update will never result in a net increase of information. How can the assertion of a conditional make a real contribution to a conversation if it never results in a net increase of information? In order to appreciate how the dynamic account makes room for assertions of conditionals to be real contributions one has to realize two things: First, asserting a conditional can make a real contribution even without resulting, dynamically, in a net increase of information. And second, asserting a that contains at least one world at which is true by the conjunction will not result in a non-empty set s , but updating s by the second conjunction will result in a crash, i.e.;. 110 conditional can be (intuitively) informative even without resulting, dynamically, in a net increase of information. While the rst point is my own, the second point relies on some comments made by Gillies. An assertion of a conditional can make a real contribution to a conversation Conditionals are devices that `review' commitment states { they check if the commit- ment state is compatible with the information contributed by the material implication { clearly such `housekeeping' of a body of information can be very important since it will point a conversation in a new direction (in the case in which there is an immedi- ate threat of a crash) or it can reassure participants that they are still on board with what is being discussed. Also, remember the rough analogy of a computer program that checks a body of information or a text for certain syntactic violations or certain search terms and returns the original state or an error symbol { clearly, having the computational means to set up such a program can be enormously helpful and make a real contribution to settling a question. An assertion of a conditional can be informative There are two ways for a conditional assertion to result in an information gain: First, an assertion of a conditional can indirectly increase the available information. The reason for this is that conditionals satisfy various `transmission principles' { MP is just the most famous one. So if you already have the information that p and learn 111 that if p then q, you are ipso facto entitled to claim an extra bit of (non-conditional) information, namely the information that q. Similarly, if you are almost certain that p (or think that it is likely that p) and learn that If p then q, you can be almost certain that q (or think that it is likely that q). Conditionals have been called `inference tickets' because of their use in these transmission principles. In fact, according to some views, conditionals are nothing but explicitly verbalized commitments to infer by these principles. 15 Second, we can gain information though a conditional by attending to the fact that the conditional has been uttered (by some party to the conversation). If the an- tecedent were accepted across the shared acceptance state, the conditional would not have been uttered (since uttering it would be pointless) - hence the information state either does not support the antecedent or, if it does, it also supports the consequent (assuming that the other party got it right). Hence, the material implication holds throughout the information state. We have gained a genuine piece of information by appealing to pragmatic rules and some basic semantic features of the conditional. 15 In this work I am not defending any such view. Suce it to say that such a view goes some way towards explaining why the logic of conditionals is so perplexing and why embedding them in certain ways leads to triviality. If we recall Lewis Carroll's (Carrol (1895)) story of what the tortoise said to Achilles treating a rule of inference such as MP like an explicit premise and adding it to the set of premises leads to an innite regress{ This shows that endorsing a rule (in the sense of inferring by it) must at least in some cases be dierent from believing something factual. And if conditionals are but explicit endorsements of rules it would show why treating them as items of the object-language may cause trouble. On the other hand the view runs into trouble due to its inability to explain any kind of embedding. 112 We can now turn to the next philosophically interesting and theoretically fruitful feature of the view: 3.4.1 The Or-If inference Can the dynamic theory substantiate the intuition that the indicative conditional diers in meaning from the material implication, while explaining why the inference from `or' to `if' seems trivially valid? 16 Showing that the dynamic theory is up to the task is going to take us a little longer. I will start by laying out a strategy that will include solving a puzzle about epistemic modals along the way. After that, I will prove a corollary and four theorems that will suce to show that the conditional and the material implication dier in meaning. Let us start with the puzzle about epistemic modals, namely the puzzle of how { if at all { `p' and `it must be that p' dier in meaning. What is puzzling is that `p' and `it must be that p' appear to have the very same epistemic signicance. That is, anyone who is epistemically committed to `p' is epistemically committed to `it must 16 There are many reasons why explaining the inference is a real achievement for the view. For one, there is a famous argument marshaled by McGee that shows that any connective that validates Modus Ponens, Exportation, the Or-If-inference and that is at most as strong as logical implication and at least as strong as material implication will be exactly as strong as the material implication. There is a loophole in the argument which the dynamic account exploits: The logical consequence relation in dynamic semantics is non-classical. In particular, it is not persistent and re exive. For a thorough discussion of McGee's argument see Gillies (2004) and Gillies (2009). 113 be that p', and vice-versa. But `it is not the case that p' and `it is not the case that it must be that p' have very dierent epistemic signicances. The rst commits one to:p whereas the second one appears to commit one to `it might be that:p' { at least if we assume that `might' and `must' are duals. Now, I think the puzzle about the dierence in meaning between indicative con- ditional and material implication can be regarded as a mirror image of the puzzle about epistemic modals. Notice rst that in terms of epistemic signicance, there does not seem to be any dierence between ! and , i.e. :_ . In fact, the inference from _ to:! has a taste of triviality: Anyone who is committed to believing that either the gardener or the butler did it is committed to believing that if the gardener didnt do it, the butler did it. The apparent validity of the or-if inference shows that there is strong reason to think that the indicative conditional and the material implication are equivalent at some level. But the negated sentences:( ! ) and:( ) dier in meaning quite dramatically, as the following example shows: 17 Suppose there are three candidates competing for a seat in the Chamber of Deputies: Salazar, Sarrazin and Santandin { with polls showing Santandin trailing the other two contenders. Jones, who knows about Sarrazin's track record of losing run-os after closely missing the majority of 17 What is essentially the same point was illustrated by the `Clue'-example in Chapter 1. 114 votes in the rst poll, points out: `If Sarrazin doesn't get over 50 % of the votes in rst round, Salazar will be elected' {You disagree: It may well be that Sarrazin withdraws from the race if he fails to get a majority of votes, leaving Santandin, who runs on a similar platform, with realistic chances to win the run-o ballot. So you say: `It is not so that if Sarrazin doesn't get over 50% of the votes, Salazar will be elected'. According to Epistemic Equivalence, what you have just committed yourself to is that Sarrazin will not get over 50 % of the votes and that Salazar will be elected (i.e. the negation of the material implication) But that is clearly implausible. What do the two puzzles have in common? In both cases we have two sentential components with very similar meaning that cannot be substituted for one another in larger truth-functional contexts, viz. negated contexts, without a change in epis- temic commitments. A plausible principle of substitutivity of sentential constituents with the same epistemic signicance seems to fail. Now, as we have seen, the dy- namic system allows us to dene a very ne-grained equivalence relation, epistemic equivalence, that is such `it must be that p and `p' can be shown to not stand in that relation. On the other hand, `It must be that p' entails `p' and vice versa, i.e. for any commitment state, whoever is committed to the one is committed to the other. Thus we have both (i) and (ii): (i):(2 =) 115 (ii) 2 and2 Given (i),2 and cannot be substituted for one another in a context without a change of meaning (mutatis mutandis this holds for the conditional and the material implication). But the remarkably beautiful feature of the dynamic solution of the two puzzles about meaning is that it shows us how the two puzzles can be understood to be at core dierent manifestations of one and the same puzzle. And this is because, on the dynamic view theorems (iii) and (iv), fall out of (i) and (ii): (iii):(! = ) (iv) ! and ! The reason why (iii) and (iv) fall out of (i) and (ii) is because on the dynamic view, indicative conditionals are just instances of epistemic modals. But the parallels between (i) and (ii) on the one hand and (iii) and (iv) on the other hand are also theoretically important because they vindicate the observation that there is a very close conceptual connection between epistemic modals and indicative conditionals. In several papers paper Kratzer famously argued that (unembedded) conditionals are but restricted epistemic modals (Kratzer 1981, 1986). A number of reasons have been oered in support of this view { too many to survey here. In Chapter 1 we already saw conditionals and epistemic modals share two important features: They can both be subject to Gibbardian Standos and Reversibility phenomena . 116 Intuitively, what we commit ourselves to when asserting that it might be that p is the same as what we commit ourselves to when denying a conditional like `No matter what, not p' (I take these locutions to be conditional in nature: `No matter what, p' seems to be the natural language version of `If:? then p'). On the other hand, what we commit ourselves to when we say `it must be that p' seems to be identical to that which we commit ourselves to when we say `No matter what, p'. These intuitions can be explained by the following corollary { they are logical consequences of it: Corollary 2 For all and : (! ) =2(:_ ) Corollary 2 follows from Denition (1) { the proof is straightforward but some- what laborious. 18 The result is very signicant because it tells us that the conditional 18 One shows that the setfw2sjs":_ = sg equals the setfw2sj(s" )" = (s")g. More precisely, one shows that whenever the rst set equals; the second set equals; and vice versa and that if the rst set does not equal; the second set does not equal; either. But if neither set equals ;, they both equal s. First case, i.e.fw2sjs":_ = sg =; Thus the crashing condition sn (s"j (s")" )6= s is fullled Hence the set s"n (s")" 6=; Hence s"6= (s")" But that means that the test for the conditional will not go through, and sofw2sjs":_ = (s")g =; Second Case, i.e.fw2sjs":_ = sg6=; Thus the crashing condition is not fullled, i.e. sn (s"j (s")" ) = s Hence the set s"n (s")" =; Hence s" = (s")" But that means that the test for the conditional will go through, and sofw2sj(s")" = (s")g6=; This completes the proof. 117 is a special case or instance of an epistemic modal and that the test it performs is a species of a test performed by the epistemic modal. We can see that just like the epistemic modal tests whether holds throughout the acceptance state s, the conditional tests whether the material implication holds throughout the state s. But this naturally explains how the puzzle about the dierence in meaning be- tween `it must be that p' and `p' and the very similar puzzle about the dierence in meaning between the indicative conditional and the epistemic modal are related: The latter is but a special case of the former. Moreover, the identication (i) vindicates the independent prediction that conditionals and epistemic modals are very closely related, e.g. by exhibiting Standos and Reversibility. Let us now nally establish (i) through (iv): Proof 1 (Proof of (i)-(iv)) Proof of (i): we can pick any state with a proper substate throughout which holds (pick one such state and call its - call its maximal substate throughout which holdss ) notice that since the worlds ins at which holds form a proper subset of s , some worlds (namely all the members of s ns ) will fail to make true. Accordingly, updatings by will yield the absurd state, while updatings by will yield the substate s - so by clause (3) of Denition 2 and are not epistemically 118 equivalent. Proof of (iii): By Corollary 1, (iii) follows from (i) Proof of (ii): This follows by induction on the complexity of . I will prove (ii) for the induction base, i.e. the case in which is atomic. The induction step is straightforward. First conjunct: We have to show that for arbitrary s, (s"2p)" p = s"2p. We distinguish two cases. First case: s"2p crashes, hence s"2p =; and we get (s"2p)"p since the absurd state support every formula. Second case: s"2p = s, i.e. s =fw2sjs"p = sg, i.e. s"p = s, hence by substituting equals for equals (s"2p)"p = s"2p. Hence, by proof by cases, (s"2p)"p = s"2p. Second conjunct: We have to show that for arbitrary s, (s"2p)"p=s"2p. Again, we distinguish two cases. First case: s"p crashes, hences"p =; and we get (s"2p)"p =s"2p because the absurd state supports every formula. Second case: s"p =s , for some non-empty s s. But we also have that p holds throughout s , hence s " p = s , hence s = fw2s js "p = s g, i.e. s "p = s , hence by substitution (s"2p)"p= s"2p. Hence, by proof by cases, (s"2p)" p= s"2p. 119 Proof of (iv): (iv) follows immediately from (ii), by Corollary 1 So to conclude, theorem (iii), the failure of epistemic equivalence, explains the intuitive dierence in meaning between conditionals and material implication while (iv) explains the validity if the Or-If-inference { On the dynamic view, disjunctions dynamically entail negated conditionals and vice versa, so it is hardly surprising that the or-if-inference feels valid. The theory satises desideratum (2) and not only that, it earns bonus points, as it were, for making sense of the puzzle about epistemic modals. 3.5 The Dynamic System vs. SCE It is now time to wonder whether the theory vindicates the non-descriptivist idea that conditionals are not in the business of describing the world. Nothing about the technical apparatus of dynamic semantics forces us to interpret it as being non- cognitivist about conditionals. But on the interpretation according to which the semantics operates on (epistemic) commitment states, it is plausible to think that the role of conditionals in conversation (and inquiry) diers more or less dramati- cally from the role of non-conditional sentences (sans epistemic modals) The point of asserting a conditional is not just adding a new item to the aggregate of infor- 120 mation that one is committed to, but reviewing the information that one is already committed to. Unfortunately, though, this dierence in roles does not yet establish that (3) has been met because it does not establish that conditional thought will not ultimately aim at describing the world. In order to show that there is room for a proponent of dynamic semantics to argue that conditional thought is unlike descrip- tive thought I will quickly discuss the example of belief. I havent said much about what it is to believe something dynamically, but I think that we can notice that there is a dierence between believing the material implication and the indicative conditional, even though believing the one commits an agent to believing the other and vice versa (our theorem (iv)). But that only shows that 'being committed to' is not a very ne-grained notion and that we need a notion of believing that is more ne-grained. Believing that (for modals-free formulae) could be characterized as being in a commitment state and ascribing certain properties to the worlds in the state. It also involves ascribing a property to the actual world, namely the property of being one of those worlds that conform to the disjunctive information that has been gained. A world w conforms to a piece of information i it is not eliminated by an update on the piece of information. No such property of conforming to a piece of information is being ascribed when someone believes a conditional: If anything, one ascribes a 121 property not to the world, but to one's commitment state. A plausible candidate for the property that is being ascribed to the commitment state is the property of passing the Ramsey test. But that property is only derivatively a property of the worlds in the commitment state { it is a property of those worlds in the sense in which being such that humans are mortal is a property of any human being (or none). This, then, seems to constitute some positive reason for saying that the dynamic account can be robustly non-descriptivist. On the dynamic view that I have sketched conditionals play a distinctive role in communication and inquiry: They can be regarded as tools that help us manage and control information. And even though their role in thought is less clear, conditional thought can be conceptualized as being directed at one's overall information or commitments { and only indirectly at the world. I would now like to conclude this chapter by looking at one of the two main desiderata. 3.5.1 Does the theory account for conditionals' important role in uncertain reasoning? Now, in order to assess the chances that the dynamic view meets our desideratum, it is worth discussing the way in which SCE captures uncertain reasoning involving conditionals. Recall that the very goal of SCE is to explain what is it to be more or 122 less condent in a conditional { it takes condence to be the central propositional attitude directed at conditionals and the main analysandum. An account of all the other attitudes is somehow expected to fall out of the account of (degrees of) condence. SCE's verdict is that condence in a conditional diers quite a bit from condence in the truth of other sentences. Condence in a conditional is a sui generis form of condence. In fact, according to SCE there is no `iy' proposition that plays the content-role for condence and other doxastic attitudes. Even if it comes with a lot of philosophical baggage, SCE's story about condence in a conditional is very compelling. Unlike SCE, the dynamic view is completely silent on the important question of what it is to be condent in a conditional. Actually, it is not only silent on that question, it presupposes that the only mode of coming to accept a conditional is full condence. This can be seen when we look at the clause for conjunction: Principle 6 (Conjunction) s" (^ ) = (s")" This clause does not go through when the uparrow is interpreted as dynamically coming to be in a state of less than full condence, just like the equation x x = x does not have solutions on which x6=1. It is simply not true that becoming x% 123 condent in a conjunction is the same as becoming x% condent in the rst conjunct and then x% condent in the second conjunct. Degrees of condence behave like probabilities and the only probability that sat- ises the truth-conditional rule for conjunction is the probability 1. I will discuss the signicance of this observation at greater length in the next chapter. The dynamic framework does a great job of characterizing commitment states in which agents are never less than fully condent of that which they (provisionally) accept. But it is a serious limitation of the view that it stops at full condence. If SCE has taught us anything, it has taught us that in order to adequately capture reasoning with conditionals, we need to capture uncertain reasoning involving conditionals. And the net that the dynamic account casts is just not wide enough to capture that (cognitive) phenomenon. Looking at how SCE treats condence in a conditional can prove benecial for defenders of the dynamic picture, even rough integrating condence into the dy- namic system may be a highly non-trivial task. But there is reason for trying hard to amend dynamic semantics: For if the dynamic view were properly extended to allow for the full range of uncertain reasoning, it would get very close to reaping all the benets of SCE while avoiding all of its pitfalls: The view could vindicate the important intuition that conditionals are non-descriptive linguistic devices, tools 124 for inquiry but not for representation, without having to deny that the meanings of complex constructions containing conditionals can be derived compositionally from the meanings of their parts. If the dynamic system were upgraded in the right way, it would hold quite some promise to become an extremely attractive theory. 125 Chapter 4 Conditionals and Uncertain Reasoning 4.1 The Plan for this Chapter In the previous chapter I introduced a system of dynamic semantics { a system that is built around the notion of epistemic commitment. The meaning of a sentence was informally explained as the sentence's CCP or context change potential. The main rationale for introducing this new approach was the hope to latch onto the meaning of the conditional { and indeed, I argued that the dynamic approach oers a number of clear benets or improvements over SCE. 126 But SCE is not without its merits. What made it a prime facie attractive choice was that it oers an elegant account of condence in a conditional. Being x% con- dent in a conditional is simply having a conditional credence that equals x 100 . And believing a conditional is being suciently condent in the conditional. 1 So whereas analyzing condence in a conditional is an asset to SCE, not providing any analysis of it is a severe liability of the dynamic view. Of course I haven't yet shown that the dynamic system does not leave any room for dening condence in a conditional. So my rst goal for this chapter is showing that in the current version of the dynamic framework there is indeed no room for accommodating degrees of condence. Acceptance states lack resources for allowing us to say that an agent in state s is x% condent that p or that she is more condent that p than that q and more condent that q than that r. 2 My second goal, which is subsidiary to the rst goal, is to establish that any viable theory of conditionals that is a theory of meaning has to leave room for distinguishing between dierent degrees of condence in the conditional, especially if it is going to 1 Defenders of SCE can say that what counts as suciently condent depends on a threshold value, a value that is determined by a parameter of the context, e.g. the stakes of committing oneself to something. 2 With a framework that allows us to model full condence and nothing but full condence we can still explain the extension of `being more condent that ... than that ...' with respect to an acceptance state s. This extension will be trivial, though, in the sense that it is simply the Cartesian Product of the set of sentences that are accepted at s and the set of sentences whose negations are accepted at s. 127 tell us something about the meaning of the conditional. Missing out on condence in conditionals is missing out on an important aspect of their meaning. The argument that I present builds on some of the points made about uncertainty-friendliness in Chapter 2. Finally, my third goal is to show that certain initially plausible extensions of the current framework will not do. In particular, generalizing update functions, introducing orderings on worlds each leave us with what seem to be insurmountable problems. What is required is probability measures, i.e. sets of probability functions and model acceptance states in terms of them. But as I will show probability functions introduce new problems of their own. { I will present a problem which I will call `the negation problem': The problem is, in a nutshell, that any simulation of dynamic negation through probability functions cannot simply proceed via complementation. The presentation of these diculties and obstacles is supposed to drive home the point that designing the right dynamic theory is highly non-trivial. The neg- ative points that will be made in this chapter can be regarded as groundwork or `prolegomena' for the positive part of the account in the next chapter. The positive part consists in the presentation and discussion of a correct probabilistic system in Chapter 5 and 6. 128 4.2 Why does condence matter? My claim at the end of Chapter 3 was that the dynamic system is somehow in- complete without an account of condence. But one might think that the dynamic system accomplishes everything it is supposed to accomplish without any need of explaining condence in a conditional. This thought relies on a certain de ationist understanding of what it is to believe a conditional as well as the dynamic semantic clause for the conditional as introduced in Chapter 3. It may help to present this line of reasoning as a formal argument: 1. What a conditional means is explained by what it is to believe the conditional 2. Believing a conditional is nothing but getting a positive result on the Ramsey Test 3. Getting a positive result on the Ramsey test is being in an acceptance state that satises the dynamic clause for the conditional (see Chapter 3) 4. (from 1,2 & 3) What a conditional means is explained by what it is to be in an acceptance state that satises the dynamic clause for the conditional Now, this argument seems to show that the dynamic theory as is captures the mean- ing of the conditional { in other words, if the argument is sound, there is no need to worry about condence in a conditional. And the argument seems at least prima facie 129 plausible. As we have seen in the last chapter, premise (3) is very compelling. The dynamic clause for the conditional runs a test that is tailored to track the Ramsey test. Premise (2) seems plausible too, at least if we understand `believing' as `being fully condent in'. But the problem is that as soon as we interpret the occurrence of `believing' in premise (2) as being fully condent in we have to also interpret the occurrence of `believing' in premise (1) as being fully condent in { for otherwise the argument would not be valid { it would equivocate on the meaning of `believe'. But unfortunately the required `full condence' - reading of `believing' in premise (1) makes the premise come out false or so I shall argue. Hence, if the argument is valid, it is not sound. And if all its premises are true, it is not valid. So the status of the argument turns on premise (1) { why do I think that premise (1) is false if `believing' is understood as `being fully condent in'? The reason is very simple, to the point that it may almost sound naive: One cannot fully un- derstand the meaning of a conditional unless one understands what role it plays in uncertain reasoning. And one cannot fully understand what role a conditional plays in uncertain reasoning unless one also understands what it is to be uncertain of the conditional. 3 3 Another way of making the same point is the following: The conditional's meaning is constituted by its inferential role. Hence, if the conditional's inferential role is dierent when we reason from uncertain premises, at least some aspects of its meaning can only be grasped if we investigate how conditionals behave in uncertain reasoning. 130 The main rationale for this claim is that many arguments involving conditionals are uncertainty-unfriendly, as I have argued extensively in Chapter 1 and 2. An account that does not say anything about condence or partial belief (in conditionals) fails to predict which inferences involving conditionals are reasonable. 4 But if the meaning of the conditional cannot be captured unless we make sense of what it is to be condent in a conditional we should at least try to model condence in a dynamic framework. And this is a project that I will now turn to. 4.3 A rst proposal - Generalizing the Update- operation If information states are understood to represent what an agent actually believes, as opposed to what they are merely committed to believing, the operation" will correspond to the process of becoming fully certain that . Whenever an agent is fully certain or condent that , it will be true that there is a commitment state s 4 It also fails to predict which (non-doxastic) attitudes are reasonable. Partial belief plays a dierent role in our cognitive lives than all-out belief. Unfortunately I do not have time to argue for this rather sweeping claim. It seems to me that standing in certain attitudes, attitudes like hoping that notp or being worried thatp rationally requires that one be at least somewhat condent that p. But it also seems to me that standing in those same attitudes does not rationally require that one believe that p. 131 that makes the update by redundant. And whenever they are not fully condent, the update by will not be redundant. But maybe there could be dierent update-operations, operations that take an agent from a commitment state to a commitment state in which the agent is less than maximally condent that what they updated by is the case. These operations would re ect more hesitant ways of drawing out ones' commitments. We could measure the `strength' of these operations by looking at their redun- dancy conditions. For example, if an agent is initially 80% condent that and an update by results in a state in which she is still 80% condent that , i.e. if the update is redundant, then the update must be one that would make the agent 80% condent, regardless of what her initial condence is like. There are examples of updating one's commitments in such a way, even though they may seem somewhat contrived. Suppose I am reading a local newspaper. But suppose that I know that the newspaper has a track record of being satirical (and hence not truthful) with respect to one fth of the `news' that are being covered. Now imagine I read that Pima County has voted to secede from Arizona and that I have no evidence as to whether this piece is satirical or factual. Given my knowledge of the (lack of) reliability of my source and the absence of additional evidence, I should be 80% condent that Pima County will secede from Arizona. Another way 132 of saying this is that my commitments have shifted to a state in which I am strongly, but not fully committed to Pima County wanting to split from Arizona and that the strength of my commitment corresponds to 8 on a scale from 1-10. But what goes for 80% goes for 79%, 81% , 78%, etc. So one could dene a denitional schema for an update operation," x from which update operations for every level of certainty corresponding to a real number between 0 and 1 could be derived. Each of these would, when applied to a commitment state, result in a commitment state in which the subject is x% condent in . The original"-operation could be modeled as limiting case of becoming maximally condent and the other operations representing the dierent ways of becoming par- tially condent. Just like someone for whom there is an acceptance state s such that s" 100 = s can be said to dynamically believe that or to be fully condent that , someone for whom there is an s with s" x = s (with x < 100) could be said to be x% condent that . This would seem like a natural extension of what has already been said about belief and epistemic commitment. But there is a problem: No update operation that works like the dynamic semantic system discussed in chapter 4 can model x% condence, for 0 < x < 100. And this is for reasons that I have brie y discussed at the end of Chapter 4. We just have to look at the clause for conjunction in order to remind ourselves of the problem: 133 Principle 7 (Conjunction*) s" x ^ = (s" x )" x for all 0 < x < 1. It is very easy to come by counterexamples to this clause. Becoming 80% condent that Lindsey is at the party and Dennis is at the party does not correspond to rst becoming 80% condent that Lindsey is at the party and then 80% condent that Dennis is at the party. The problem seems to lie in the assumption that being x% condent that and can always be `decomposed' into two constituent events, becoming x% condent that and becoming x% condent that (given ). While it is very easy to say what it is intuitively wrong with (Conjunction*), it is a little harder to give more principled reasons for thinking that (Conjunction*) cannot work. Let us see what has to be the case in order for (Conjunction*) to work. There needs to be some function f that (i) species the degree of condence in the con- junction in terms of the degree of condence in its conjuncts, or more precisely in terms of the degree of condence in its second conjunct given the same degree of condence in its rst conjunct and (ii), moreover, that function needs to have the property f(, ) = for all degrees of condence that lie between 0 and 1. The good news is that there is a function that satises (Conjunction*) for the two extreme values 0 and 1, namely multiplication (provided the conjuncts are logically independent) { the bad news, however, is that multiplication has certain mathemati- 134 cal properties that entail that (Conjunction*) cannot be satised for arbitrary values between 0 and 1. Needless to say, if there is no function that will do the job of satisfying (Conjunc- tion*) for all values between 0 and 1, there will not be a recursive function either. But we need a recursive function in order to get a clean semantics for conjunctive formulae of arbitrary complexity { remember that compositionality is supposed to be one of the great selling points of the dynamic picture over SCE, so it shouldn't be thrown out of the window that early on. If there is any `deep' lesson that is brought out by the failure of (Conjunction*), it is the lesson that Boolean operations on set of worlds or sets of any items whatsoever behave dierently from Boolean functions on degrees of beliefs or other measures that represent information numerically. Hence, the upshot of this discussion is that degrees of belief behave dierently and that there cannot be a standard Boolean semantics for degrees of belief. And since a proper treatment of condence is going to rely of some account of degrees of belief, this is bad news for the prospects of analyzing condence within a dynamic framework. 135 4.4 Rethinking CUS 4.4.1 Condence and Graded Modals How can we incorporate talk about degrees of condence into CUS? In this section I argue that enriching our object language with some expressions that can be used by a speaker to convey uncertainty is sucient for an account of condence that ultimately explains Adams' Thesis. 5 So the basic idea is to enrich the object languageL ! with operators that will then be dynamically interpreted as conveying the agent's uncertainty. The most plausible choice for these operators are graded modals like `it is probably the case' and `it is x% likely', for the simple reason that these modals are conventionally used (in English) to convey less than full condence in their prejacents. 6 In static semantics for attitude ascriptions, we can explain the attitude of con- dence in a proposition very elegantly by appealing to an agents credence in that proposition, measured by some probability function. We can explain what `John is more condent that Ann passes than her mom is' means by quantifying over cre- 5 I also think that this extension of the object language is necessary in order for CUS to explain dierent degrees of condence, but I am not very condent that there is a forceful argument showing that no other way of extending CUS will allow us to make sense of degrees of condence. 6 See for example Swanson (2011), p.251. 136 dence functions and saying that John's credence function assigns a higher value to the sentence (as uttered in the right context) `Ann passes [the exam]' than her mom's credence function. So talk about condence can be cashed out in terms of talk about agents' credence functions. Now, defenders of SCE make a key move that the dynamic account can take advantage of. According to SCE, we can talk about condence (i.e. an agents credence function) and quantify over people's credence functions but we can also use certain sentences of English to express attitudes of uncertainty like condence. Certain sentences of English are semantically appropriate for a speaker in a context if and only if the speaker has the relevant attitude in that context. Sentences like (5) and (6) are semantically appropriate for a speaker in a context i the speaker has sucient condence in , viz. her credence function assigns a high enough value to : (1) It is likely that (2) It is probably the case that So according to SCE the very same (subjective) state of aairs, namely S's credence function, does two things: It makes it the case that it is appropriate for S to assent 137 to (5) or (6) and it also makes it the case that S counts as being condent that to a certain (high-ish) degree. 7 This fact can be exploited by our dynamic framework. If the underlying states that explain uncertain attitudes and provide semantic appropriateness conditions for the language of subjective uncertainty are the same, we can simulate the work that attitudes like condence do by looking at the sentences that are used to express them. What it is to be condent in something can be captured by looking at what it is to update by a sentence like (5) or (6). But we have to be cautious. Just like we haven't dened the attitude of believing something within framework of dynamic semantics, we are not going to purport to dene the attitude of condence within that framework either. Belief and condence belong to the realm of thought and we are not after the structure of thought, but only after meanings as far as they are explained by the way agents trade information. So when I say that we can `simulate' the work that the attitude of condence does dynamically I mean that we can measure or record in the semantic system what taking that attitude towards a sentence commits an agent or a group of agents to. 7 One of the commitments of SCE is that `I believe that it is x% likely that p' and `I am x% condent that p' can be used to express the very same attitude, viz. condence to degree x. Expressivist theories with semantic appropriateness conditions that map two sentences with intuitively dierent semantic content to the same attitude are not always robust. As Schroeder (Schroeder (2010)) shows, a related problem proves all but fatal for an early expressivist view advocated by Simon Blackburn. 138 And the working hypothesis is that the epistemic commitments that are being had by someone who is x% condent in are the same as the commitments taken on by someone who updates by a sentence like `It is x% likely that . This is what the principle (Condence) states: Principle 8 (Condence) Becoming (x %) condent that and updating by the information that is x % likely result in the same information states. (Condence) does not purport to provide a philosophical analysis of the attitude of being condent, or the many logically similar attitudes of being `x % condent for all choices of x. The actual attitude(s) may be more ne-grained than the state of having certain epistemic commitments. Dynamic Semantics leaves room for a more ne-grained approach to dening attitudes, but its main focus is not on attitudes, their phenomenology or their functional role etc. but on the epistemic commitments that having those attitudes typically generate. The points I have made in this section about condence apply straightforwardly to condence in a conditional. Being condent in a conditional is `commitment- equivalent' to updating one's commitment state by the sentence that the conditional in question is likely. And just like I argued in Chapter 1 that believing a conditional is being in a state that passes the Ramsey test, being committed to believing that the 139 conditional is likely is being in a state that passes an analogous test, a `Likely-Ramsey test' if you will. Not surprisingly, the schema for the test that gets the right result has great structural similarities with the Ramsey test. Instead of using the awkward name `Likely Ramsey test' I will call it the `Generalized Ramsey Test' (GR), suggesting that the test procedure is somehow a generalization of the Ramsey test procedure. Here is the adequacy condition for the test: Principle 9 (GR-test) A state s supports `it is x % likely that ! if and only if s + supports `it is x % likely that And in fact, calling the test `generalized' isnt promising too much: The hypothesis that the GR-test tracks the state of believing a conditional to be likely has the nice features of allowing us to derive the Ramsey test if we set x = 100%. So the Ramsey test becomes a limiting case of the GR-test. But this is not only a nice result, it actually makes a lot of sense. Just like believing a conditional is a special case of believing a conditional to be x% likely, the Ramsey test is a special case of the Generalized Ramsey test. I think for those of us who accept the Ramsey test there is a lot of intuitive pull to also accept the GR-Test, even though there is no argument that shows that 140 accepting the Ramsey test as an analysis of belief in a conditional commits one to accepting the GR-test as an analysis of the belief that a conditional is x% likely. What (Condence) and (GR-test) together entail is something very intuitive: Whoever accepts `Probably ' and understands the meaning of the English words is condent, but not fully condent, in and whoever is condent, but not fully condent in and understands the meaning of the English words accepts `Probably '. Similarly, whoever accepts `If, probably ' and understands the meaning of the English words is condent in `If then ' and whoever is condent in `If then ' and understands the meaning of the English words accepts `If then probably '. Since the foregoing principle re ects a basic fact about the meaning of `Probably', adding probability operators to the language of CUS will suce to represent condence in conditionals as well as in non-conditional sentences. So letO stand for `it is probably the case that ' and letO x stand for the schematic modal `it is at least x% likely that ', a stand-in for arbitrarily many modals that refer to a numerical degree of condence (and let our languageL ! be extended by the probably-operator and innitely many likelihood-operators). The task of representing condence in the framework of CUS reduces to showing that Conjecture 1 is true: 141 Conjecture 1 (Dynamic Condence) Leth I ," i be a structure for a language L ! [fO;O x g, where" is an update function on the set of information statesI .Then h I ," i will allow us to dene recursive clauses" O and" O x such that the following are true: Information state s represents agent S as being condent in i s acceptsO Information state s represents agent S as being condent in ! i s accepts !O Information state s represents agent S as being at least x% condent in i s accepts O x Information state s represents agent S as being at least x% condent in ! i s accepts !O x ( ) In order to show that Conjecture 1 is true, we have to show that (i) the semantic condition that an information state s has to meet in order for its agent to acceptO guarantees that whoever is in s will be condent in and vice versa and (ii) that the semantic condition that an information state s has to meet in order for its agent to accept (!O ) guarantees that whoever is in s will be condent in (! ) and vice versa. It will turn out that the task of showing that the modals can be given the right kind of denition is highly non-trivial. 142 The reason for this lies in the structure of CUS. Recall that the semantic frame- work of CUS is very coarse-grained, since information is represented as set of worlds. But as we shall see, this is not just an artifact of the model { the representation is essential for Denition 1 to go through. This just means that Conjecture 1 can only be true for a structure that does not conform to Denition 1. 4.4.2 Orderings All that Denition 1 gives us is I, a set of subsets of worlds. It is challenging to represent dierent degrees of condence by looking at worlds and nothing but worlds. The natural reaction, on seeing that sets of worlds don't capture degrees of condence, is to replace them with probability functions. And indeed I believe that this is what we ultimately must do. But because we will see later that it is quite dicult to adjust the dynamic framework in order to work with probabilities, it is worth taking the time to see that a more conservative way of enriching the dynamic framework will not suce. The idea I want to discuss is adding structure to sets of worlds by imposing an ordering on the worlds in the sets. A very similar move, i.e. dening an ordering on worlds has helped us make decisive progress in other areas of semantics. A good case in point is deontic language. Dening an ordering on sets of worlds has allowed 143 us to give truth conditions for sentences like `You ought to write three papers but you absolutely must write two', truth conditions that we would not have been able to give without orderings (McNamara (1996)). Angelika Kratzer (Kratzer (1981)) was the rst semanticist to implement this idea for the language of probability. 8 We start with a likelihood ordering, a partial ordering on worlds. The source for the likelihood ordering will be a set of hypotheses on regularities that worlds conform to: If the set of regularities thatw 1 conforms to is a subset of the set of regularities thatw 2 conforms to,w 2 is at least as highly ranked as w 1 . For example, if I know that ravens are black and that I am not infallible, a world in which I just saw a white raven, will { other things being equal { be lower ranked than a world in which I saw a white bird that looked like a raven. Rankings in turn allow us dene the likelihood relation between intensions. For a sentence to be at least as likely as a sentence (> s ) is for no world in the intension of to be higher ranked than any of the worlds in the intension of . 9 Finally, let `Probably ' orO be an abbreviation for > s : and:(:6 s ), i.e. an abbreviation for ` is more likely than:', i.e. (< s :). Finally, we could revise 8 While Kratzer used a static semantic framework, Frank VeltmanVeltman (1996) sought to give a dynamic semantic account of `presumably ' by using orderings. 9 Since the relation `x is at least as likely as y' (> L )is being modeled as a relation between intensions, it only relates sentences in the truth-functional fragment of the language. 144 Denition 1 of CUS by adding the following simple clauses for the modal `probably ': For all 2L ^ : s"O =fw2 sj < s :g 10 (4.1) For all and 2L ^ : s"O(! ) = s" (!O ) (4.2) What is remarkable about this strategy is that, if successful, it would allow us to dene a conservative extension of CUS. We could simply add a clause forO to Denition 1, without having to alter any of the other clauses. Hence, the `meaning' of the non-probabilistic fragment of the language would remain intact and no `new' truths formulated in the languageL ! could be shown to follow from the denitions. In this section I shall show that no such relational denition of likelihood is going to allow us to introduce a probability operator that satises the conjecture of the previous subsection. A `probably'-operator based on a qualitative likelihood relation fails to represent condence. This limitative result is interesting in its own right, but it gives us strong reason to make more signicant revisions to the framework of CUS. 11 10 Notice that this approach does not allow us to dene the generalized graded modalO x because we do not have any means to dene numerical degrees of condence in terms of a qualitative likelihood-relation. 11 For further discussion of the limitations of the ordering approach for dealing with `probably' see Yalcin Yalcin (2010). 145 To show the negative point, let us look at the following sequence of sentences: 1. The rst time the coin comes up heads is on the rst toss. 2. The rst time the coin comes up heads is on the second toss. 3. The rst time the coin comes up heads is on the third toss. (........) 100. The rst time the coin comes up heads is on the 100 th toss. Plausibly, a rational agent who has no further information about the sequence of coin tosses and does not think the coin is biased will set her condence equal to the objective chances of the events that are described by the sentences in the sequence. 12 So she will be 50% condent in (i), 25% condent in (ii), 12.5% condent in (iii), etc. Given this probability distribution over the members of the sequence, the agent will not assent to (S1) but she will assent to (S2). This is so because `probably' is used to express condence that is greater than 50%: (S1) # Probably, the rst time the coin comes up heads is on the rst toss. (S2) Probably, the rst time the coin comes up heads is on the rst toss or the second toss. 12 That this is the rational way to set one's credence is entailed by Lewis' Principal Principle and possibly other bridge principles. See Lewis (1980). 146 Notice that on the suggested picture, on which the semantics of `probably depends on some qualitative likelihood relation, the reason why (S1) is not acceptable and (S2) is acceptable is that the translation (S1*) of (S1) into the semantic metalanguage fails to hold, but the translation (S2*) of (S2) into the semantic metalanguage does hold: (S1*) `The rst time the coin comes up heads is on the rst toss' > L `It is not the case that the rst time the coin comes up heads is not on the rst toss' (S2*) `The rst time the coin comes up heads is on the rst or the second toss' > L `It is not the case that the rst time the coin comes up heads is on the rst or the second toss' Let us introduce some abbreviations that will make it easier to follow the dis- cussion. Let Let `F' stand for the sentence `The rst time the coin comes up heads is on the rst toss', and let `S' stand for `The rst time the coin comes up heads is on the second time'. Now, it is easy to show that the negation of (1*) and the sentence (2*) together with the assumption that the> L ordering respects the axioms of probability 13 , entail (3*): 13 This assumption says that we can show > L whenever the axioms of probability dictate that Pr() > Pr( ). 147 (3*) `The rst time the coin comes up heads is on the rst or second toss'>L `The rst time the coin comes up heads is on the rst toss' 14 But in order for worlds to be ordered in this way, there has to be at least one world in the intension of (F_S) that outranks every world in the intension of F. But which world could that be? All the worlds in the intension of (F_S) are either worlds in which the coin comes up heads for the rst time on the rst toss or in which the coin comes up heads for the rst time on the second toss. Another way of saying this is that the intension of (F_S) is the union of the intension of F and the intension of S. So if there is some world in the intension of (F_S) that outranks every world in the intension of F, that world has to be either in the intension of F or the intension of S. But clearly, no world in the intension of F outranks every world in the intension of F { for no world outranks itself. So the world that outranks every world in the intension of F has to be in the intension of S. Now, remember that in order for a world w to outrank another world v, the world w has to satisfy strictly more hypotheses or regularities than the world v. So suppose there is some world 14 Assume that (3*) does not hold, hence (F_ S)6 L F . Assume furthermore that6 L respects the axioms of probability. By (1*) we have:(F > L :F). Similar considerations suggest that:(: F > L F) and hence, since a sentence and its negation are always connected through > L, we get F= L F. But since (2*) holds, we have (F_ S) > L :(F_S) and so we get the following likelihood ordering::(F_ S) < L (F_ S)6 L F = L :F. Since on any assignment of probabilities to F, Pr(F) = Pr(:F) = 0.5 we get that on any assignment of probabilities, Pr(:(F_ S)) + Pr(F_ S) <1, contradicting the axioms of probability. Hence (F_ S) < L F 148 w in the intension of S that conforms to more regularities than any world in the intension of F { what are those extra regularities that w conforms to? Image.pdf > ? Figure 4.1: Ordering Worlds - Why does (3*) hold? Of course we can stipulate ad hoc regularities like `When tossed twice, a coin comes up heads for the rst time on the second toss' but there is absolutely no reason to think that any world would conform to those kind of regularities about chancy events. Moreover, postulating these kinds of regularities will not even be coherent, since the problem can be iterated. When we notice that the account is committed to (F_ S_ T) > (F_ S) we can show by an analogous argument that some world in the intension of T will have to outrank all the worlds in the intension of S (and F) { but the very ad hoc regularity that gives worlds in the intension of S the boost will prevent them from being outranked by worlds in the intension of T. So postulating ad hoc regularities does not only fail to solve the problem, it creates new problems. So it seems like 149 making sense of (1) and (2) commits the defender of the qualitative account of `probably ' to (3*), but they cannot give a reason for why (3*) should hold. If the data expressed by (1) and (2) are correct, `probably' cannot be analyzed as meaning ` is more likely than:', where `is more likely than' picks out a Kratzer-style ordering of worlds. But the fact that Kratzer's idea is awed also means that any CUS-style semantics that incorporates an ordering of worlds will be equally awed. So if we want to make sense of `probably' and other modals of its ilk we cannot do this in a possible-worlds framework cum ordering source. 4.4.3 CUS with single probability functions All along we have been assuming that an agent's information is best represented in terms of epistemically possible worlds and epistemically possible worlds only. But as we have seen, there are at least some cases in which an agent's information may be too ne-grained to be represented by sets of worlds or sets of items that play the world-role. When the agent has information that is re ected in the distribution of her credences possible worlds by themselves will not suce { credences have a mathematical structure that cannot be mimicked by possible worlds. Let us quickly remind ourselves of why this observation is relevant for dynamic semantics. So why can dynamic semantics not just ignore this added complication 150 and idealize the probabilistic `dimension' of a given body of information away? The reason is that we are interested in dynamic semantics in so far as it allows us to say something interesting about the meaning of the conditional. As I have argued, making sense of condence in a conditional is an important constraint on this project, i.e. the project of elucidating the meaning of the conditional. To make sense of condence in a conditional is to dynamically explain the meaning of locutions that convey the speaker's condence in a conditional, e.g. `If most of the proofs are awed, the dissertation will probably not be accepted'. And in order to dynamically explain these locutions, we need to pinpoint the contributions of the epistemic modals like `probably' in them. But unless we have a richer notion of what information states are we will not be able to pin down how these modals aect a commitment or information state. This is, then, the reason why we need a richer notion of information state and cannot get away with idealizing the probabilistic dimension of information away. This realization is not new. The fact that epistemic probability operators are unlike other epistemic modals in that they model probabilistic information and that treating them adequately requires `extra-modal' resources has been recognized and discussed at great length by Seth Yalcin. 15 15 Yalcin (2010) 151 Yalcin's reaction vis- a-vis this problem is a very natural one: He suggests enrich- ing the notion of an information state to include a probability distribution or some other function that measures uncertainty. 16 According to Yalcin, information states proper are really not just sets of worlds, or more accurately, sets of cells (i.e. sub- sets of partitions of sets of worlds), they are sets of cells plus probability functions ranging over the cells in such a set. Enriching information states in such a way allows Yalcin to derive semantic clauses for probability operators without being committed to assigning sentences containing these operators conventional truth-values (they are true with respect to an information state and a world, not true with respect to a world simpliciter). In what follows, I shall pursue a dierent strategy, dierent from what I call Yalcin's enrichment strategy. The most fundamental dierence is of course that Yalcin operates within a static semantic framework { even though he assigns a special place in his theory to information states he does not dene meaning in terms of functions from information states to information states. 17 16 Yalcin (2011) 17 Through personal communication I have learned that Yalcin has { by March 2012 { implemented his idea in a dynamic system. Unlike myself, he also oers a semantic treatment for attitude- ascriptions like `S knows that p'. 152 But there is another noteworthy dierence that will become obvious when com- paring the two approaches across semantic frameworks. Instead of enriching sets of worlds with probability measures, I shall replace them with probability measures . Probability functions will play the role of information states. So, again, to the extent to which comparisons across dierent theoretical frameworks make sense, the current strategy constitutes a signicant departure from Yalcin's strategy. I think the current approach is promising because, no matter what the outcome is, there is positive heuristic value in sorting out whether probability functions can play the same (functional) role as sets of worlds even though they have a dierent mathematical structure. Probabilities will play the role of that which is updated by update-functions but also the role of that which allows us to read o what an agent is committed to. Whether or not we can fully capture the meaning of an expression by attending to the way it impacts our existing credal commitments is an interesting question. But besides helping tackle or at least get clear on these foundational, meta- semantic issues, the strategy also allows us to establish a direct link between the two theories of meaning for conditionals that have been discussed in this dissertation, the two theories being SCE and the Gillies-Veltman dynamic view. Recall that SCE's semantics is minimal in the sense that it posits attitudes, i.e. credences and 153 nothing else. If dynamic semantics operates on probabilities and on probabilities only and if these probabilities can be interpreted as credences, there is a sense in which dynamic semantics vindicates a core intuition behind SCE since both views operate on the same domain. The full- edged dynamic view can then be regarded as oering a more sophisticated, mature picture of the point of conditionals than SCE does. Conditionals are not linguistic devices for expressing credences, but devices that perform checks on our credences. Let us try to esh out the basic idea of a probability-based dynamic semantic system. Recall from Chapter 2 that a probability function is any function that satises the axioms introduced and discussed there, even an ordinary truth- function that assigns 0 and 1 and nothing else. But what shall we take to be the domain of the functions? The domain of the function will not be a set of sentences, but a set of intensions, i.e. sets of worlds. Sentences will be assigned probabilities derivatively, in so far as they are semantically correlated with the intension. This correlation may not always be very tight. This is because in the dynamic framework intensions are at best partial representations of a sentence's meaning. If they were fully accurate representations of a sentence's meaning, dynamic semantics would be a pointless endeavor and we could settle for a static possible-worlds semantics. 154 Recall that for any that has an intension, the intension of is the set of worlds w at which is true, regardless of whether some of these worlds belong to some commitment states for which an update by crashes. I am appealing to an intuitive, but unanalyzed notion of truth at a world, as it is used in textbooks of modal logic. Now, since we haven't specied what it is for epistemic modals and conditionals to be true at a world, we cannot really say what it is for them to have intensions and hence their intensions cannot be the argument of a probability function. This is yet another sense in which intensions are inept to be identied with meanings { certain sentences in English and other languages are assertible but lack intensions { all they have is dynamic meaning. A short comment on truth at a world is in order. We have specied the semantic contribution of epistemic modals at the level of commitment states, but dodged the question of what is for them to be true at a world. And indeed, there is no readily available sense in which epistemic modals and conditionals are true at a world. Maybe certain theoretical reasons could lead us to say that conditionals and epistemic modals can, after all, be `true at a world' in a dierent sense, for example qua being true at a world-information state pair. Such a truth-predicate may be thought to be decient and pointless, but it is at least disquotational and would 155 go some way towards explaining speakers' widespread use of `true' as applying to conditionals and epistemic modals. 18 The current point is that regardless of whether such a project is worthwhile, not assigning probabilities to epistemic modals and conditionals does not once and for all prevent us from assigning truth-values to these sentences. At any rate, not being able to dene probability for epistemic modals and condi- tionals is not an unwelcome limitation. We do not really need probability functions to measure values for sentences like `it might be that ' because we can just declare the sentences that would result from prexing probability-operators e.g. `it is likely that it might be Julius Caesar who is crossing the Rubicon' to be ill-formed. The analogous move for conditionals is a little more subtle. We do of course want to say that one can be condent in a conditional and hence, via (Condence) that one can think that a conditional is likely, so prima facie it would seem like we need to say what it is for a conditional sentence to get assigned such and such probability. But per (GR-Test) providing a dynamic semantic analysis of `It is likely that if then ' does not require us to ever assign probabilities to a conditional, but only to its consequent. So it looks like we can do without getting into the business of assign- 18 See Cappelen and Hawthorne (2009), p.11. 156 ing probabilities to epistemic modals. Instead we can limit ourselves to assigning probabilities to the prejacents of epistemic modals. Let us start with the simplest possible idea. An information state will be replaced by a single probability functionPr. Pr will assign probabilities to members of}(W), corresponding to the intensions correlated with the formulae of the target language. We will be sloppy and writePr(), which stands forPr (jj). If contains epistemic modals, Pr() will not be dened, as it lacks an argument { as we will see, even though a functionPr will not assign any value to an epistemic modal, it can still be updated by it. This re ects the subtle, yet important distinction between checking one's commitments and changing them. Also, I shall use the variable `Pr' without blank space as a function-symbol, designating a probability function. Since functions are thought to be gappy or unsaturated entities, they are usually designated by leaving a blank argument space to the right of the main function symbol. Hence, in the following the reader may encounter both the term `Pr' and the term `Pr(p)' the latter being a variable designating a real number and the former being a variable designating a function into real numbers. Even though it is more intuitive and suggestive to speak of probability-functions, strictly speaking the update-relation has to be dened on sets of quasi-probability functions. A quasi-probability function is a function that is either a probability 157 function or #, the absurd function that assigns 1 to any intension. This detour via `quasi-probability-functions' may be compromising the aesthetic value of the dynamic system, but it is no cause for concern. A very natural idea is to replace the entire information state with a probability function. Just like a set of worlds is said to represent the information that Anna sings karaoke by containing only worlds in which Anna sings karaoke, a probability function can be said to represent the information that Anna sings karaoke by assigning the intension of `Anna sings karaoke' a real number between 0 and 1 that exceeds some threshold value. But the probability function { unlike the sets of worlds { also represents probabilistic information, since it tells us how much more condent the agent is that Anna sings karaoke than that Maria sings karaoke. Two remarks are in order. First, notice that we will not be able to assign prob- abilities to sentences that are not characterized by intensions. Hence, we will not be able to apply probability functions to2,3 or ! . But since probability functions now play the role of information states we can still update a probabil- ity function by sentences like epistemic modals even though we cannot apply the probability function that represents an agent's information to the (intension) of the sentence. 158 Second, updating sets of worlds occasionally results in the empty set of worlds, an information state that is interpreted as the absurd state, since it can be shown to support every sentence. If we want probability functions to play the role of an information state, we need to specify a function that behaves analogously. So let # ( ) be the absurd function, assigning the maximum value 1 to any intension. 19 Let us call any function that is either a probability function or an absurd function, i.e. a function assigning 1 to just any intension, a quasi-probability function and let us write `QPr' for an arbitrary quasi-probability function. The good news is that it is easy to see how most clauses of Denition 1 can be altered if the set of worlds is replaced by a probability function. Once we have a plausible choice for how updating works in the atomic case, almost all the remaining clauses fall into place. The most plausible choice for atomic updates is Bayesian conditionalization (see Chapter 2): To be informed through conversation or other forms of linguistic communication that p is the case is to learn that p holds. Remember that atomic updates in the old system CUS crash just in case the infor- mational contribution of the item that a state is updated by is incompatible with the remaining commitment(s) of the agent(s) in question. Accordingly, atomic updates in the new system crash just in case the result of conditionalization is undened. 19 #( ) cannot be classied as a probability function, since no probability function satisfying Denition 3 assigns 1 to every intension. 159 We can now give the following recursive characterization of the update operation " QPr" p = 8 > > > < > > > : QPr p if QPr p is dened # otherwise (4.3) The rules for most connectives are very straightforward. The clause forO shows why it can only be prexed to sentences in ofL ^ : QPr" (^ ) = (QPr")" (4.4) QPr"3 = QPr where 8 > > > < > > > : QPr = QPr if QPr":6= QPr QPr* = # otherwise (4.5) QPr"2 = QPr where 8 > > > < > > > : QPr = QPr if QPr" = QPr QPr = # otherwise (4.6) 160 QPr" (! ) = QPr where 8 > > > < > > > : QPr = QPr if (QPr")" = QPr" QPr = # otherwise (4.7) QPr"O = QPr where 8 > > > < > > > : QPr = QPr if QPr(jj) > 0.5 QPr = # otherwise (4.8) QPr"O(! ) = QPr" (!O ) (4.9) No surprises here: An update by3 crashes i updating the original function by: is redundant and an update byO crashes i the value of the probability function as applied to the intension of the prejacent of the modal is greater than 0.5. If does not have a (standard) intension, the value ofO will be irrelevant. The only embedding of non-descriptive vocabulary under the `probably'-operator is the conditional. The last clause stipulates that in the case of such an embedding, the conditional is always understood to take wide scope. This implementation strategy looks promising, since it shows that almost all updates rules can be dened on single probability functions. 161 Notice in particular that this account has all the resources to handle an update by the `probable conditional' (!O ), just like it handles (!3 ). But what does the current clause predict for the update `!O ' ? Given our current denition, this update crashes whenever QPr 6= #, since Pr( ! ) is never dened and hence a fortiori not greater than 0.5. So the update either crashes or doesnt crash { and if it doesnt crash that it is only because the update is an update on the absurd function. At any rate, this is a bad result, mostly because we wantO(! ) to carry meaning. It would be great to be able to semantically derive all instances of the schema: Principle 10 (Wide Scope) PR" (! ) = PR"(! ) where stands for an epistemic modal operator (either `must' or `might' or `prob- ably'). If we were able to derive the instances of (Wide Scope), we would be in a position to prove that the new system satises the adequacy-condition that comes with the GR-test (the Generalized Ramsey test), namely that for a conditional to be likely is for its consequent to be likely on the supposition that its antecedent is true. But since an update by `O(! )' is currently undened, we cannot derive anything about the state that such an update results in. So It looks like we will have to declare (Wide Scope) true by stipulation, for lack of being able to derive it. The 162 fact that such a stipulation is called for would point to a blind spot in our semantic system. The nature of this stipulation resembles the stipulation that probabilities of con- ditionals are to be treated as or equated with conditional probabilities. That stip- ulation is nothing but our old friend, Adams' Thesis. Recall from Chapter 2 that (AT) is a hypothesis that may be supported by various theoretical reasons, but that is not itself derived from axioms or general principles within Adams logic. This is why I shall call the present phenomenon SCEs revenge. SCEs revenge refers to the fact that we have to stipulate that `O(! ) just means `(!O )' without being able to derive that semantic fact compositionally from the individual meanings of the conditional and the probability-operator. There is a lot more to be said about SCEs revenge but I think that the stipulation that it requires is not a huge impediment to progress. In fact, I know of no account that is able to derive (Wide Scope) { as opposed to stipulating its truth. The orthodox approach defended by Kratzer and Lewis and their followers is no exception. 20 Kratzer maintains that If-clauses always restrict covert modals and that these covert modals default to epistemic necessity. She explains the truth of (Wide 20 Lewis (1975) advocated a view that is structurally similar to Kratzer's for adverbs of quantica- tion like `always' `mostly `usually' { for these reasons the orthodoxy is often called the `Lewis-Kratzer view' 163 Scope) by appealing to some kind of displacement of modals: Since the if-clause always restricts modals or quantiers, the overt modal has to take narrow scope with respect to `If'. So far so good: But how does the overt modal interact with the covert modal? Kratzerians stipulate that the overt modal somehow takes precedence and displaces the covert modal. 21 Notice that such a displacement mechanism is being hypothesized and not derived. 22 I agree with Kratzer that modals always take narrow scope when they embed if-clauses. I dont have an explanation for this peculiar scoping behavior but neither has Kratzer (nor do Veltman and Gillies & Von Fintel in their respective discussions of `Presumably + If' ). So there is a sense in which I can justify my stipulation that (Wide Scope) holds by appealing to a `partners in crime' consideration { if the standard account of conditionals in formal semantics cannot do better, why should the present account 21 This is not the only possibility. Geurts (Geurts (2005)) endorses a view on which there is an ambiguity that shows up when certain modals are embedded under `If'. One reading is the reading under which the overt modal displaces the covert modal as the modal that is being restricted, another reading is the reading under which the covert modal embeds under the restricted overt modal. According to Geurts, the following sentence has these two readings: (*) If Brown is depressed, he has to recite one of his poems To quote Geurts: `This may be understood as saying that Brown has to recite one his poems whenever he feels down, or as saying that his being depressed indicates that Brown has to recite one of his poems. On the rst construal, the sentence contains one modal operator; on the second construal, it contains two modals, one of which is overt while the other is covert'. 22 This is a `key' strategic move that Kratzerians make. As Rothschild (Rothschild (2011)) shows, it allows them to avoid triviality results { Unlike champions of SCE, they cannot just retreat to the view that conditionals do not express propositions in order to avoid the triviality results. 164 be expected to solve the problem? Explaining scoping phenomena was not on the list of desiderata presented in Chapter 4, however desirable it may be. Despite possible reservations about the status of (Wide Scope), probabilistic dy- namic semantics or probabilistic update semantics looks like a promising starting point - provided that we come up with a clause for negation the last outstanding task for this chapter. Here is a prima facie plausible candidate for the clause for negation: QPr": = QPr* where 8 > > > < > > > : QPr* = QPr : if QPr : is dened QPr* = # otherwise (4.10) But there is a problem with (ii). Whereas (ii) invites us to conditionalize on the negated sentence, it will not always be the case that such a conditionalization is dened, especially if we are conditionalizing on a sentence containing an epistemic modal or a conditional. If epistemic modals are not in the domain of the probability function, conditionalization which is dened in terms of conditional probabilities and ultimately in terms of ratios of simple probabilities { will not be dened for any formula containing them either. So if (ii) were right, we would only get an explanation of what it is to update an acceptance state by certain types of sentences, 165 but not by others, leading to an incomplete picture of what English sentences mean. So how should we extend this denition of negation? Answering this question will be fairly complicated and requires some stage setting, which is why it will be tackled in Chapter 5. But as we will see in just a moment negation poses a serious and, in fact, insur- mountable problem. Going back to Denition (1) of CUS, we can observe that one of the really nice features of the clause for negation was the fact that it gave us the right results both when applied to sentences inL ^ and when applied to sentences containing modals or conditionals, i.e. members ofL O nL ^ the update by: always gives us the complement, within s, of the update by regardless of whether the update by is a test, an intersective update or a hybrid update. In the present probabilistic framework, we are unable to dene a recursive clause for negation that has this `one size ts all' quality to it. However, we can oer a clause for sentences inL ^ by specifying that updating a quasi-probability function QPr by: amounts to conditionalizing QPr on: as long as this conditionalization is dened, but such a clause is not going to tell us what it is to negate an arbitrary sentence inL O , i.e. a sentence containing epistemic modals or conditionals, since no intensions are dened for these types of sentences. 166 The reason why negation worked in Denition 1 is that it was highly specic to the underlying representation of information as sets of worlds. The result of an update by a negated sentences": was always dened because it always happened to coincide with the complement of s" . As soon as we give up this simplistic picture of how information change occurs and go in for a genuinely probabilistic representation of information, we lose the formal prerequisites for giving the clause for negation its full recursive power. 167 Chapter 5 A probabilistic dynamic approach 5.1 Representing probabilistic information by sets We have just seen that the clause for negation does not get o the ground if infor- mation states are conceptualized as (single) probability functions. I would now like to revisit the original clause for negation in Denition (1) and see if we can derive constraints on the structure of probabilistic information states that will allow us to dene an update for negation that is true to the clause in Denition (1). s": = sn(s") (5.1) There are three formal features of (10) that are worth noting: 168 1. Negation operates on sets of worlds. 2. It can be shown that all the compositional rules in Denition 1 are eliminative, in the sense that s" is always a subset of s. If the rules weren't eliminative, s" would not necessarily be a subset of s, which might result in scenarios in which sn(s") would collapse into s even though it shouldn't. An example of such a gratuitous collapse will be discussed below. 3. It can be shown that the set s is always exhausted by the set s" and the set s":, i.e. pairs of incompatible updates are exhaustive. If updates weren't exhaustive in this sense, sn(s") might contain worlds that intuitively would not belong in s":. The point of stating these constraints is that they give us a blueprint for con- structing information states that will allow us to dene an update for negation. So the heuristic for making negation work is to mimic clause (10) as closely as possi- ble, which requires at least trying to make sure that probabilistic information states satisfy conditions (1)-(3). Much of the material in the following technical sections follows the thread of this general heuristic. Getting information states to satisfy the rst constraint is straightforward: In- formation states will simply be dened as sets of probability functions, for there is no complementation without sethood. 169 But not only is it easy to implement this new way of modeling information states, it is also a representation that is independently plausible because philosophers have defended accounts on which credences are not sharp, but blunt. On one conception, the members of the set could be taken to represent all the permissible ways in which an agent could come to adjust her credences consistently, given their information at time t. For example, suppose a bag contains 100 marbles, 30 of which are known to be red and 70 of which are known to be either blue or yellow, but the exact proportion is unknown. It seems permissible to place a credence of 0.4 in `The next marble taken out of the bag is blue' and credence of 0.3. in `The next marble taken out of the bag is yellow' but not 0.4 in both of these propositions (Halpern (2003), p.25). But of course it is not (rationally) required or mandatory to precisify one's credences in any of these ways. 1 On an alternate conception of what sets of probability functions correspond to the intervals obtained by looking at the outer (maximal and minimal) values of the functions can be understood as vague credences. According to this conception 1 Halpern discusses a prima facie problem for this view. Experiments seem to show that agents invariably prefer a wager on `The marble will be red' to both the wager on `The marble will be yellow' and `The marble will be blue' even though they are indierent between the last two wagers. The problem is that any way of precisfying credences should lead an agent to prefer at least one of the two last wagers to the wager on `The marble will be red'. One might want to conclude that the fact that a function Pr is a permissible precisication is not sucient for rational decision-making. What is required is that a function Pr is a permissibe precisication and no other function is also a permissible precisication. 170 credences cannot always be represented as points on a line but sometimes merely as regions of a line, i.e. open or closed intervals on the real line. Since points are limits of intervals, this view does not have to clumsily hypothesize two kinds of credences. It can say that credences in certain sentences like tautologies and contradictions are naturally sharp 2 while credences in many empirical propositions are naturally blunt. Philosophers have long felt that single probability functions are overly precise tools of representation, especially in the face of unspecic evidence. 3 After all, a typical subject's credence in claims such as `There will be a nuclear attack on an American city during this century' is likely not to have a numerically precise value. Accordingly, it has been argued that the agent's credence in the sentence is best represented as a set of all those probability functions that she is indierent between { or, alternatively, those functions that have not been eliminated by her evidence. 4 We have just seen that there are excellent reasons for conceptualizing information states as sets of probability functions and not as single functions. We now simply have to make the necessary adjustments at the technical level. 2 `Naturally' because it is always possible to espouse a philosophical view like dialetheism on which it is not a priori false that a given contradiction could turn out to be true. Holding such a view may supersede a natural inclination to have a credence of 0 in any contradiction. 3 The locus classicus for this idea is Jerey (1983) For a more recent framework in which this assumption is relaxed, see ?. 4 Cf. Elga (2010), who denies the view that credences are blunt. 171 I shall use the variable `[PR]' as variable ranging over sets of probability functions. Since we are still working in the framework of dynamic semantics, we have to identify a set that is going to play the role of the absurd state. Being in the absurd state is being in a state that would commit an agent to everything, so we should still get the result that all updates are redundant with respect to the absurd state, i.e. probabilistically updating the absurd state by any formula results in the absurd state. The empty set looks like a natural choice: Conditionalizing any member of the empty set on any intension whatsoever is going to result in the empty set. The denition of the"-operation changes accordingly, but for ease of presentation I will use the same uparrow to refer to it. If we make these changes, the rst clause of the denition of the operation will look as follows: [PR]" p = [PR ] where [PR ] =fPr*j9Pr2[PR]Pr*=Pr p g (5.2) Let us now turn to a prima facie plausible candidate for the negation clause in order to see if we have made progress: [PR]": = [PR]n [PR]" (5.3) 172 [PR] = [PR] ↑ ∼p ↑ p [PR] ↑ p ↑ ∼p Figure 5.1: Gratuitous Collapse This clause may look like it is in good shape, but it will not do, since it does not satisfy condition (2) or (3). To see why eliminativity matters, let us look at a scenario in which its failure leads to counterintuitive results. The diagram in Fig. 5.1 shows a scenario in which the two sets [PR] and [PR ] are disjoint. It should be clear that nothing in clauses (11) and (12) prevents these two sets from being disjoint. After all, [PR] is not closed under conditionalization and may not contain any functions that assign p the value 1. Let us suppose it contains at least one function that assigns p the value 0.5. But in the event that the two sets are disjoint, [PR]": p will simply collapse into the original set [PR]. And this is clearly the wrong result. Clearly, the act of denying something you were 50% condent in changes your information because it carries negative information (as illustrated in Fig. 5.1). 173 So the rst problem with clause (12) is that it does not prevent [PR]": from collapsing into [PR]. 5 This problem can be remedied by making the update elimi- native. The strategy that I now want to discuss consists in imposing a closure constraint on sets of probability functions. Updates could be dened only for those sets of probability functions that are closed under conditionalization, i.e. sets that satisfy the constraint that if they contain a function Pr, they are also guaranteed to contain the function Pr(jp), for arbitrary sentences p. It is relatively easy to see that for any set [PR], its closure under conditional- ization will contain the setfPr*j9 Pr2 [PR]Pr* = Pr p g as a proper subset, hence making the update for atomic sentences dened in (5.2) eliminative. This idea is illustrated in Fig. 5.2. But even if dening updates on closures of probability func- tions turns out to make all updates eliminative and ultimately allows us to dene a clause for negation, it raises an important conceptual issue: In what way can a set that obeys a closure constraint still be said to represent an information state? This important question shall be taken up in the next section. 5 The second problem, the failure of exhaustivity, will be discussed in a subsequent section. 174 CL[PR] CR[PR] ↑∼ϕ [PR] CR[PR] ↑ϕ Figure 5.2: Bayesian closures 5.1.1 Bayesian Closures Unlike the original sets, closures of sets of probability functions do not seem adequate for representing an agent's information state. They are inadequate representations for the simple reason that they contain too many functions. For example, for any credence function in the original set that assigns the intension of p a value strictly between 0 and 1, thereby re ecting an agent's uncertainty with respect to the ques- tion whetherp, there will be at least one other function in the closure of the set that assigns p the value 1 and one other function in the closure that assignsp the value 0. Hence, the agent's ne-grained distribution of condence as recorded in [PR] seems to get erased through its closure. 175 Fortunately, there is a response to the concern that closures fail to preserve an agent's actual distribution of condence. I shall show that given a strong assumption on the structure of a probabilistic information state [PR], we can always recover the set [PR] from its closure. The strong assumption will guarantee that the process by which we construct closures out of `base sets' is going to be reversible. This corresponds to a division of labor between base sets and their closures. Base sets continue to play the representational role of modeling information states and closures will be regarded as mere codes for the actual information states. However, updates and information change potentials will be dened on the codes or place- holders and not the actual information states. So an agent's or a group of agents' epistemic commitments are captured by a set [PR] of probability functions. This set can be nite (In fact, it can be a singleton) or innite. 6 In order to implement this idea, we need to give the term `closure' a precise meaning. A Bayesian closure of a set of probability functions is a set that contains the result of conditionalizing any of the original probability functions on any number of formulae arbitrarily many times. For our purposes the set of formulae will be limited to atomic formulae ofL ^ and their negations, which can all be enumerated 6 I have not imposed any constraints on the set. It may very well be reasonable to demand that the set be convex, i.e. for any p and any two functions Pr 1 andPr 2 in [PR] such that Pr 1 (p) = r 1 and Pr 1 (p) = r 2 any linear combination of the two functions is also in the set, i.e. for all a9 Pr 3 in [PR] such that Pr 3 (p) = a r 1 + (1a) r 2 .Kyburg and Teng (2001), p.107. 176 in a sequence { a sequence which we will refer to as `Q'. A detailed construction for a Bayesian closure is outlined in the appendix. I will use the superscript `CL' to refer to the closure-operation, hence [PR] CL is the Bayesian closure of [PR]. I will also occasionally refer to Bayesian closures (relative to the sentences in Q) as Q-closures. Given that we want closures to play the role of codes, looking at the closure of a set [PR] should allow us to read o the original information state, namely [PR]. The problem is that in the absence on any kind of constraint on [PR], there is no way back from the closure to the base set, so we are unable to recover an agent's original information state. Stated in technical terms, even if there is a function from a set [PR] to [PR] CL , there is generally no function back from [PR] CL to [PR]. For example, the setsfPr 1 ,Pr 2 ,Pr 3 g andfPr 1 ,Pr 2 g share the same closure as long as Pr 3 = Pr 2 (jp), for some sentence letter p. But these two sets share a feature that turns out to be relevant for the constraint that we need to impose. They contain pairs of probability functions that are Q-successors in the sense of the following denition: Denition 5 (Q-successor) Pr 2 is a Q-successor of Pr 1 i Pr 2 is the result of conditionalizing Pr 1 on some non-tautological member of the sequence Q. In order for there to be a function from [PR] CL `back' to [PR], the base set [PR] must not contain any pairs of functions that are Q-successors of each other. The 177 constraint that base sets be free of pairs of Q-successors seems to be the weakest constraint that suces for there being a function from [PR] CL back to [PR]. However, the `No Q-successors'-constraint is somewhat arbitrary. It is not easy to see how it could be philosophically motivated. But there is a somewhat stronger constraint that happens to be more plausible: It says that no members of the set should assign the maximal or minimal value, 1 and 0 respectively, to an intension unless all other members assign the same member. As soon as one probability func- tionPr in the set assigns 0 or 1 to some member of Q, all other probability functions are required to `follow suit' and agree with that function on the value of Pr. At the outer bounds of the space of uncertainty probabilistic information has to be univocal or `clustered'. Luckily the stronger constraint that I am suggesting we adopt is independently plausible, especially under the interpretation on which the dierent values in the sets correspond to the dierent ways in which an agent can permissibly set their credences, given their current information. It is possible that my information allows me to be between 90% and 95% condent that I turned o the stove before leaving my apartment. In that scenario, my information is compatible with both levels of condence and does not force me to choose. But if my information allows me to be 100% condent that I did not turn o the stove, then I should be fully certain that 178 I did not turn o the stove. That my information permits (probabilistic) certainty means that it also guarantees (probabilistic) certainty. Intuitively, any evidence that is good enough for certainty that p has to be already so substantial that it does not leave room for any evidence that points in the direction of: p. In other words, if, given their information, an agent has the option to be certain of a proposition or sentence, the agent is committed to being certain of that proposition or sentence. Formally, what the constraint says is that if there is a function in the set [PR] that assigns 1 to a sentence, there are no functions that do not assign 1 to the sentence, i.e. the agent is certain of the sentence. I shall call the principle that we need weak regularity: Principle 11 (Weak Regularity) For all agents S: Unless all functions in S's information state assign the value 1, no function in S's information state assigns the value 1. Weak regularity says that an agent who is not completely certain that a sentence is true should not assign that sentence the probability 1. 7 Weak regularity constrains the structure of sets of probability functions: If one function in the set assigns one 7 And by parity of reasoning she should not assign it the probability 0 either. For suppose she did assign 0 to even though she would not be completely certain that was false. Then, by the axioms of probability, she would assign 1 to:. But if S is not completely certain that is false she won't be completely certain that: is true either. 179 of the two extreme values (1 or 0) to some sentence (intension), all other functions have to `follow suit' and assign the same value to that sentence. Conversely, if some function does not assign an extreme value to some sentence, no function in the set will. 8 As a normative epistemic principle weak regularity is quite controversial, but an even stronger claim, namely Regularity, has been defended by a number of philoso- phers. 9 Such an appeal to authority is not supposed to show that it is true, but that it is a defensible assumption, especially if the assumption has the heuristic value of allowing us to extend CUS. 10 8 The principle is a substantive constraint on the structure of probabilistic information. If Anne at timet has no information as to whether the coin tossed by Jon at an earlier timet landed heads or tails but knows that it is a fair coin, the principle guarantees that it will not be permissible for her to have a credence of 1 in the sentence `The coin came up heads'. While I do not think this is implausible, it could be argued that it should at least be permissible for Anne to set her credence in heads to 1, e.g. because she would have a 50% chance of having full credence in something true. Given a state of uncertainty, how can it ever be impermissible to have a full credence in something that is as likely to be true as it is likely to be false? It seems that one cannot do better than having a 50% chance of getting it right and that it should be permissible to take that ind of gamble. My (short) answer to this question is that one should only have full credence in something if one's evidence allows it and that having no evidence is not (ever) enough evidence for full credence. Full credence is a serious commitment and serious commitments call for serious evidence. 9 for example.Shimony (1955) in his doctoral dissertation orAppiah (1986). 10 Some philosophers deny that assigning non-zero credences to is a necessary condition for 's being epistemically possible for the agent. Proponents of regularity allow that some epistemically possible propositions get assigned innitesimal values.Easwaran (2011)argues that the strategy is awed. 180 With weak regularity in place we can show that there is a function ( )* or x.x* { the `star' function that takes us from a Bayesian closure [PR] CL back to [PR] via stripping [PR] CL of all those functions that are Q-successors of some other function in [PR] CL . This is what Lemma 1 says: Lemma 1 (Starring) If a family of sets of probability functions F satises Weak Regularity, there are no two distinct sets [PR] and [PR ] such that [PR] CL = ([PR] ) CL . That means the function CL is invertible. Its inverse will be called the starring function ( ). We have: 8 [PR]2 F: ([PR] CL ) = [PR]. Proof: See Appendix. We now have all the tools in our toolbox that we need to dene an update semantics for the enriched languageL O . 5.2 The positive account 5.2.1 Fixing Negation requires rejecting exhaustivity Let us revisit the proposed clause for negation, this time amended by dening the update on closures of sets of probability functions: 181 [PR] CL ": = [PR] CL n [PR] CL " (5.4) Unfortunately this will (still) not do. It would do if were were still dealing with sets of worlds instead of probability functions: In CUS any information state s can be shown to satisfy the following principle of exhaustivity: Denition 6 (Exhaustivity) An update operation" satises the principle of exhaustivity i8s in the domain of the function: s = (s")[ (s":) In CUS the two sets, s" and s": exhaust the set s, but as we shall see soon, this does not hold for sets of probability functions. Let us ask again what we expect negation to do. Given the initial set [PR] CL , the set [PR] CL ": will contain only those members of [PR] CL that are intuitively in- compatible with the update by. In CUS this incompatibility requirement naturally translated into satisfying exhaustivity. If you can be either certain that Jon speaks Korean or certain that he does not speak Korean but nothing in between, then the only epistemic scenario that is incompatible with the possibility of Jon's speaking Korean is the one in which you are certain that he does not speak Korean. In the new system the incompatibility requirement will be satised, but it will not yield 182 anything as strong as exhaustivity. A negated formula (of arbitrary complexity) af- fects an agent's information by eliminating all permissible ways of coming to believe something { except those that are incompatible with an update by, i.e. those that are contained in a state that would crash if updated by . In other words, [PR] CL ": is a subset of [PR] CL that would crash if updated by. But it is not just any subset of [PR] CL that would crash if updated by , it is the union of all the subsets that would crash if updated by - making it the weakest revision of [PR] that is incompatible with . Denition 7 (Negation) [PR] CL ": =fPr2 [PR] CL j9 y [PR] CL : (Pr2 y^ [y] CL " =;)g But this means that we will not generally have exhaustivity. In fact, we can show that there are at least some `admissible' information states such that: Corollary 3 (Failure of Exhaustivity) 9[PR] ([PR] CL 6= [PR] CL ":[ [PR] CL ") It is very straightforward to identify witnesses for this corollary: Suppose [PR] = fPr 1 ;Pr 2 ;Pr 3 g and Pr 1 (p) = 0.8, Pr 2 (p) = 0.9 and Pr 3 (p) = 0.95. It is easy to see that Pr 1 is not in [PR] CL ": p and that Pr 1 is not in [PR] CL " p either, so 183 these two sets cannot exhaust [PR] CL , since the non-empty set [PR] is not a subset of their union. But even if negation does not allow us to derive a corollary that is analogous to exhaustivity, it still has the crucial feature that allows us to dene a recursive clause for negation: It tells us that the result of an update by a negated formula depends on the result of the update by the non-negated formula. This is the `hybrid' nature that we expect negation to have. If an update by behaves like a test, the update by: will be a test. If on the other hand the update by is an intersective update, the update by: will be an intersective update. And lastly, if the update by is { due to its truth-functionally complex structure { a hybrid between a test and an intersective update, the update by: will be a hybrid as well. 5.2.2 CUS + We can now look at the complete recursive denition of the update-function. s:. s" is a two-place function from pairs of formulae and simple Bayesian closures into simple Bayesian closures. LetPROB be a set of sets of simple probability functions that satisfy weak regularity. In accordance with our earlier discussion, these sets will play the conceptual roles of information states. However, the update function will be dened on their Q-closures. 184 That means that we will have to talk about the image of PROB under the Q- closure-operation. That set, to be calledCPROB, will contain the simple Bayesian closures of all the members of PROB. So for every set [PR] in PROB, [PR] CL will be a member of CPROB. I will introduce a notational convention for ease of presentation. The capital letter `C' will be used as a (meta)variable ranging over members ofCPROB. Before I oer the recursive clauses, it is important to remember that we always have; CL =;, i.e. the empty set is a member of CPROB: Closing the empty set under conditionalizing on members of Q yields nothing but the empty set. It is critical that the empty set is in the range (and domain) of the update-function since it plays the role of the absurd state. As before, updates may crash, i.e. result in the empty information state. The languageL O includes the languageL ! , but contains extra operators: `O', the generic likelihood operator and `O x ', the sharp likelihood operator. Denition 8 (Updates for the languageL O of CUS + ) Let the set [PR] be an arbitrary set of probability functions and let C = [PR] CL . " is the function that behaves as follows: 1. C" p =fPr*2 Cj9 Pr2 C: Pr* = Pr p g 2. C": =fPr2 Cj9y C (Pr2 y^ [y] CL " =;)g 185 3. C" (^ ) = (C")" ) 4. C"2 =fx2 Cj C"= Cg 5. C"3 =fx2 Cj C":6= Cg 6. C"O =fx2 Cj8 Pr2(C)* (Pr() > 0.5)g 11 7. C"O x =fx2 Cj8 Pr2(C)* Pr()> xg 8. For all 2L ^ : C"(! ) =fx2 Cj(C")" = C"g A few comments are in order. We have already discussed the atomic and negation clause, but we have not shown that updating Q-closures by an atomic formula or a negated formula results in a set of probability functions that is itself a Q-closure of some set of probability functions. But given the denition of the"-function, its output states had better be closures. Lemma 2 assures us that atomic updates and updates by a negated formula take us from closures to closures: Lemma 2 (Q-closures) 11 Clauses (6) and (7) are the only place in the recursion in which we apply probability functions to sentences or rather intensions. Since this requires that the sentences' intensions be dened and since modals and conditionals lack intensions, it follows that we cannot straightforwardly embed modals or conditionals under probability operators. But embedding conditionals under probability operators and thereby conveying condence in a conditional was of course one of the most important goals of our project. So have we just lost sight of this goal? The answer to this question lies in the stipulation (Wide Scope) made in Chapter 4. Conditionals always take wide scope with respect to epistemic modals. Hence, we stipulate that saying that it is probably the case that if then is just saying that If , then probably . 186 1.fPr*2 Cj9 Pr2 C: Pr* = Pr p g CL =fPr*2 Cj9 Pr2 C: Pr* = Pr p g 2.fPr2 Cj9y C (Pr2 y^ [y] CL " =;)g CL =fPr2 Cj9y C (Pr2 y^ [y] CL " =;)g Proof: See Appendix Notice moreover that the clause for conjunction is the same as the one oered in Denition 1 { again, its well-denedness depends on the input and output states both being Q-closures. It is straightforward to verify that the test clauses all take us from Q-closures to Q-closures. The modals `2' and `3' behave as expected, checking whether all probability functions assign the value 1 to (`must') or whether some function does not assign the value 1 to: (`might'). They return the original state if the state satises the test condition and the empty state if the state fails to satisfy the test condition. The two modals `Probably' (O) and `It is at least x % likely' are especially interesting. 12 They are the only places in our system in which we need to appeal to the original set [PR], which is the set designated by (C)*. Thus, the fact that the starring operation x.x* can be shown { via Weak Regularity { to be a function is critical for this clause. It would not be attainable without the special constraints on sets of probability functions. 12 Strictly speaking `O x ' is a variable standing for innitely many operators. 187 The two `probability-themed' clauses check if all the functions in (C)* assign a numerical value that is greater than 0.5 or a value that equals x, respectively. Notice also that a given state may fail both a test by `Probably ' and `Probably :'. Sometimes we do not have any kind of decisive evidence with respect to a proposition. In that case, an update both by `Probably ' and `Probably:' will crash. 13 The current system does a good job distinguishing between two distinct states of having inconclusive evidence. In the rst state the evidence does not commit an agent to , but it does commit the agent to `Probably ', since all probability func- tions assign a value that is greater than 0.5 or some other contextually determined threshold value. In the second state the evidence does not suce for either commit- ment. In that state, it is permissible to be more condent in than in: but it is also permissible to be more condent in: than in . The only commitment with respect to that an agent takes on is the epistemically `cheap' commitment to the claim that it might be that (and the corresponding commitment to the claim that it might be that:). I shall say more about the probability-operators and their 13 This is an important dierence between `probably' and `might'. It is never the case that both `It might be that ' and `It might be that:' crash. 188 interaction with the conditional in Chapter 6, when we turn to assessing whether CUS + has the right resources to satisfy the desiderata formulated in Chapter 1. 14 Given this denition of the update-operation, we can proceed to (re)dene the notion of acceptance and logical consequence. A nice feature of the system is that our commitment and logical consequence relation will behave exactly as they behaved before. The state that commits an agent to is still dened as the state that is the xed point of updating by and logical consequence is still dened as the relation that holds between two formulae i the corresponding conditional holds at every information state: Denition 9 (Acceptance and Logical Consequence in CUS + ) C commits an agent to i for all C: C"= C entails ( ) i for all C: (C")" = C" and are epistemically equivalent ( = ) i for all C: C" = C" 14 There might be a worry that a system that proves such an epistemic completeness is unt for predicting `objective' readings of `might' as in Keith De Rose's cancer test example (DeRose (1991)) in which the result of a complicated test provides the basis for asserting `The patient might have cancer'. Before the test, it is not appropriate to assent to that sentence. These objective readings could be accommodated by amending the test run by `It might be that ' to a test that xes a threshold, say 0.25 and requires that every function in C assigns a value that is greater than the threshold. This would allow for states that commit an agent neither to `it might be that ' nor to `it might be that:'. The duality with`must' would of course have to be given up. 189 This completes the rst part of extending CUS to represent agents as being in in- formation states that license arbitrary levels of condence in conditionals and other sentences. As we have seen, the extension is fairly laborious { to the extent that some philosophers may nd it hard to stomach. I do, however, believe that we shouldn't be intimidated by the unruliness of the apparatus of CUS + . Quite the contrary: Non-descriptivists about conditionals should be attracted to the current picture: It preserves all the supposed benets of SCE while actually providing a machinery that allows us to do compositional se- mantics with the conditional. As already announced, I shall walk the reader through some of the selling points of CUS + in the next chapter. Before we get to the grand nale of Iy Condence, I want to focus on two instructive formal limitations of the system. 5.3 Alternate Updates One reason why the system is so complicated is that updates are not dened on information states directly, but on Q-closures of information states. This detour may be cumbersome, but it is indispensable. Notice that for every application of the update-operation to an information state [PR], we go through several steps. First, we move from [PR] to its closure under CL , 190 C (= [PR] CL ) CL [PR] C ↑ ϕ ([PR] CL ↑ ϕ)* …↑ϕ (….)* ?? Figure 5.3: Combining functions C. Then we apply the function" to C. Finally we can apply the starring application to get back to an information state (C" ) . What happens if we combine these functions? (See Fig. 5.3) The combinationfgh of three functional operationsf, g, h should itself be a function. Let us call this operation*. The denition of the `shortcut' update operation is hence the following: Denition 10 [PR]* = ([PR] CL " ) However, there is no interesting way to characterize the* function recursively, due to the problem with negation that we encountered earlier. We can see that there is guaranteed to be such a function, but its value depend on the nature of the series of update(s) prescribed by { i.e. on whether these updates are all tests, all intersective update or both tests and intersective updates. If all updates were 191 either tests or updates, we could do with a relatively simple disjunctive clause. But updates can be hybrids of tests and intersective updates. And it is not clear to me that we can provide any denition of the*-operation at all. Not being able to recursively dene a function does not mean that we have to impose a ban on using functors referring to it. If it please the reader to update information states directly, sans detour, she can do so. In the remainder of this work I will stick with the update-function", as dened in Denition 3. 5.4 Non-extendibility The current system has a very peculiar feature: It never allows an agent to come to be less than certain of a proposition that is being asserted or communicated in other ways. It allows the agent to check her credence in a sentence that she is uncertain of but it does not allow her to adjust her credences to something that falls short of absolute certainty. Put another way, the system does not dierentiate between gaining information and becoming fully condent in something (i.e. the content of the information). Even though this idealization is compatible with Bayesianism as discussed in Chapter 2, it is not true to the spirit of our current endeavor, which is to investigate the role of conditionals in uncertain reasoning. The basic idea is 192 that we gain information in many ways and that some of them involve increments in condence without attaining full condence. There may be a remedy for this limitation: Instead of treating `O x ' as per- forming a test on an information state, why not simply let it perform a genuine eliminative update? You hear someone with sucient epistemic authority assert `Its 50% likely that there will be an earthquake in L.A.' and as a result, your credence in `There will be an earthquake in L.A.' goes up (or down, whatever it may be) to 50% { your credences aren't merely under review, they have shifted: You have gained genuine information about the world, information that is incompatible with some (probabilistic) scenarios that purport to represent your evidence. In addition to this epistemological reason there is another, methodological reason for dening `O x ' as a genuine eliminative update and not merely as a test. The methodological reason appeals to the value of expressive richness of the language. Given the presence of the box 2 in our language, we can always turn an operator or connective that performs a genuinely eliminative update into an operator that performs a test, but the reverse is not possible: We cannot turn a test-performing update into an eliminative update, so other things being equal introducing elimina- tive operators will yield an expressively richer language. If `O x ' were to be dened 193 to perform a genuine update, '2O x ' could be regarded as the corresponding test- performing modal that would merely check states. Suppose we think, as seems natural, that the update performed by `O x ' should proceed via Jerey-conditionalization. That means that our new credence in would equal x and our credence in an arbitrary sentence would be derivable via the weighted sum of the conditional probabilities: Principle 12 (Jerey Conditionalization) Pr JC () = (Pr old (j ) Pr JC ( )) + (Pr old (j: ) Pr JC (: )) Unfortunately Jerey-conditionalization cannot be implemented in the system. The output state of an update byO x would, for any x < 1, not be included in the input state since our information states are not currently dened to be closed under Jerey-conditionalization. It is not the case thatPr JC (...) is a member of the simple Bayesian closure whenever the unconditionalized functionPr(...) is. But even if we require information states to be closed under Jerey-conditionalization, we will not get what we need in order to dene a recursive update-operation like our current "-operation: We cannot retrieve original information states form Jerey-closures of information states. There is no way to look at a `Jerey-closure' of a set of probability functions and read the original information state o of it. Hence, we cannot dene a `starring' operation for Jerey closures of probabilistic information states. 194 This shows that the current approach does not function if we assume Jerey conditionalization as an update rule. It is not `Jerey-extendible' to coin a fancy term for the diagnosis. But it also shows that the acceptability of CUS is hostage to the truth of (Bayesianism 2): If updating by conditionalization is not the canonical method for belief change, CUS is not tenable either. In the next Chapter, I will take a look at the logic of CUS and focus on embed- dings and expressive limitations while assessing the value of CUS vis- a-vis its various syntactic limitations. Most importantly though, I will argue that the framework has all the resources to meet our desiderata. 5.5 Appendix to Chapter 5 5.5.1 Construction of Q-closures Suppose S is a set containing arbitrarily many probability functions. Let Q = hq 1 ;q 2 ;q 3 ;:::i be a sequence of intensions, e.g. sets of worlds corresponding to the literals ofL ! . (We are assuming that there are at most countably many literals). One can think of Q as providing the elementary sentences required for the construc- tion of a Carnap-style state description, i.e. a complete summary of the world. So the set Q contains sentences like `Grass is green' and `Coal is black' as well as their 195 negations. However, Q does not contain sentences whose logical structure is more complex. First, we can easily show that for every set [PR] there exists a Boolean Closure [PR] CL . To see this, notice that there are at most as many members of the sequence Q as there are integers. And for every singleton set S 0 =fPrg such thatS 0 [PR], there are at most as many members of the closure under conditionalization of that singleton as there are sets of integers. There is a mapping99K PR from }(@) into (S CL 0 ) for every singleton set S 0 . For every member of }(@), that mapping takes us to the probability function that is the original function Pr conditionalized on all the members of Q whose positions in the sequence correspond to the members of the set in }(@) if that function exists. For example, the mapping99K PR assigns the setf3, 724, 1099g the function that corresponds to Pr conditionalized on the intensions q 3 , q 724 and q 1099 , provided that function is dened. If the probability function is not dened, e.g. because it would require us to rst update on `grass is green' and subsequently on `grass is not green' the mapping will simply return the original function Pr. Since we are interested in information states that are supposed to model a certain aspect of `the cognitive life' of nite agents we will only be interested in a countable subset of }(@), namely the set of nite sets of integers. Each of these sets gets 196 assigned a probability function that is the result of applying conditionalization a nite amount of times to the original probability function Pr. The mapping from the nite fragment of }(@) into the corresponding class of probability functions can be summarized by the following inductive schema: Denition 11 (Pairing integers with sets) Let99K PR be the function that is characterized by the following recursion: ;99K PR Pr Iffk 1 ;k 2 ;:::k n g99K PR Pr(jq k 1 ^q k 2 ^::^q kn ) thenfk 1 ;k 2 ;:::k n ;mg99K PR Pr k 1 k 2 :::knm where 8 > > > < > > > : Pr k 1 k 2 :::knm = Pr(j q k 1 ^q k 2 ^::^q kn ^q m ) if that term is defined Pr k 1 k 2 :::knm = Pr(j q k 1 ^q k 2 ^::^q kn ) otherwise Now, let the range of99K PR be the collection S CL 0 . Since any subsets of }(@) are sets, S CL 0 is guaranteed to also be a set. Now, the closure of [PR], to be called [PR] CL , is the union of the sets S CL 0 , for all S 0 2 [PR], and hence it is also a set. 5.5.2 Proofs of Lemmas Proof 2 (Proof of Lemma 1) Let the star-operation be dened in the following way: ([PR]) =fPr2 [PR]j:9q2 Q9Pr # 2 [PR] (Pr = Pr # q )g 197 Now, let [PR] be an arbitrary set of probability functions that satises weak reg- ularity. It is easy to see that ([PR] CL ) is a function. it is much harder to see that ([PR] CL ) = [PR]. Now suppose this equation does not hold. There are two possibilities: The two sets overlap or they are disjoint. They cannot be disjoint, for assume they are: Let Pr be a member of [PR]. If Pr2[PR], it will also be the case that Pr2 [PR] CL (This is guaranteed by the construction of [PR] CL ). But since [PR] satises weak regularity, Pr will not be a Q-successor of any function in Q and hence Pr will satisfy the condition for membership in ([PR] CL ) . Now suppose they overlap. Then there are members of [PR] that are not in ([PR] CL ) or there are members of ([PR] CL ) that are not in [PR]. Suppose the rst disjunct holds and let Pr a be such a member of [PR]. Again, we have Pr a 2 [PR] CL . But since [PR] satises WR, Pr a is not a Q-successor of any function in [PR] and so it satises the condition for membership in ([PR] CL ) . Now suppose the second disjunct holds and letPr b be a member of ([PR] CL ) that is not in [PR]. ThenPr b is such that it is not a Q-successor of any function in ([PR] CL ) . But thenPr b must be in the original [PR] or in [PR] CL n[PR]. If it is in [PR] CL n[PR], it must have been added to [PR] at some (nite) stage of the construction, say k. Now, k codes some nite sequence of literalsfq 1 ;q 2 ;:::q m g butPr(jq 1 ^q 2 ^::^q m ) for some Pr in S either reduces to Pr (and so it is in [PR]) or it is a Q-successor of Pr. But, by our 198 assumption,Pr b cannot be a Q-successor of any function in [PR] CL . This completes our proof. Proof 3 (Proof of Lemma 2) Part 1 Let C be the set in question and dene (C) to be [PR]. We simply show that (i)fPr*2 Cj9 Pr2 C: Pr* =Pr p g [PR p ] CL and that (ii) [PR p ] CL fPr*2 Cj9 Pr2 C: Pr* = Pr p g, where [PR p ] =fPr p j Pr2 [PR]g. (i) IffPr*2 Cj9 Pr2 C: Pr* = Pr p g is empty we are done. So let it be non- empty and let Pr a 2fPr*2 Cj9 Pr2 C: Pr* = Pr p g. Then, by the condition on membership infPr*2 Cj9 Pr2 C: Pr* = Pr p g, Pr a must be the result of taking some function Pr(jq 1 ^q 2 ^::^q m ) in C and conditionalizing it on p. But since the order in which we conditionalize does not matter, we know thatPr(jq 1 ^q 2 ^::^q m ^ p)= Pr(jp^q 1 ^q 2 ^::^q m ) and the latter is in CL[PR p ], i.e. Pr a 2 [PR p ]. (ii) For the other direction of the inclusion, we assume [PR p ] CL to be non-empty and let Pr b 2 [PR p ] CL . Hence, we know that Pr b equals some Pr(jp^q 1 ^q 2 ^::^q m ). Again, this is equivalent to Pr(jq 1 ^q 2 ^::^q m ^ p) which is the result of taking some function in C and conditionalizing it on p. But that is just the condition for mem- bership infPr*2 Cj9 Pr2 C: Pr* =Pr p g, so Pr b 2fPr*2 Cj9 Pr2 C: Pr* =Pr p g. 199 Part 2 Again, we have to prove that the inclusion goes both ways. The direction fPr2 Cj9y C (Pr2 y^ [y] CL " =;)gfPr2 Cj9y C (Pr2 y^ [y] CL " =;)g CL is easy, given the denition of the closure operation. To make the proof of the other direction more perspicuous, I shall write `B' for the setfPr2 Cj9y C (Pr2 y^ [y] CL " =;)g and `B CL ' forfPr2 Cj9y C (Pr2 y^ [y] CL " = ;)g CL . Suppose some functionPr is in the set B CL , but not in B. ThenPr must be obtained from some memberPr of B via (possibly repeated) application(s) of conditionaliza- tion (by the construction of closures outlined above). But if any probability function is in B, it is also in some subset of B that crashes if updated by , by the condition on membership. For any function Pr in such a subset, the result of conditionalizing Pr n times on any intension (named by a member of Q) is also in the subset (because the subsets are closed under conditionalization) and therefore also in B. Hence, if Pr is in B, Pr is in B as well. But this contradicts our assumption. 200 Chapter 6 Can CUS + deliver on the promises? 6.1 In search of the right theory When discussing the reasons for the inadequacy of the horseshoe-view in Chapter 1 I introduced two important desiderata for any theory that tries to shed light on how we reason with conditionals. The rst desideratum was that the theory explain how we are to reason with conditionals when we are but uncertain of the premises we reason from. The second desideratum was that the theory explain how we reason with conditionals that are embedded in truth-functional contexts. I pointed out that 201 there is a tension between these two desiderata: It is hard to satisfy both and many theories surveyed in Chapter 1 fail at least one of them. The attempt to satisfy both desiderata has been one of the major themes, or leitmotivs of this philosophical exploration. In Chapter 2 I argued that Simple Credal Expressivism is in a great position to explain uncertain reasoning, at least to the extent to which its defenders take Adams' notion of probabilistic validity on board. As I also argued in Chapter 2, SCE falls short of explaining how conditionals embed in constructions like negation, disjunctions and the consequents and antecedents of conditionals. The dynamic theory that I turned to in Chapter 3 does much better in this regard. It allows us to embed conditionals truth-functionally, for example under negation and conjunction and intuitively gets the right results. As I shortly argue in a later section of this nal chapter, it predicts the validity of a number of plausible inferences involving embedded conditionals. However, the semantic framework that the theory uses treats the central notion of accepting a sentence as coming to be certain of a sentence. Not only is an update of an information states by explained as a function from an information state to the revised information state at which the agent(s) is certain that holds, it is also impossible to alter the operation so as to allow that updates can result in the agent's being merely 80% or 35% condent in. 202 This impossibility was one of the insights gleaned through the discussion in Chapter 4. The trick that got us around this impasse was to use graded modals like `It is likely that ' in order to represent an agent's doxastic attitude of being uncertain that . By introducing these modals directly into the object language we were able to hold onto the idea that the operation designated by `" ' takes us from an information state to a state at which the agent is certain that . In the presence of this assumption, enshrined in Conjecture 1 (Chapter 4), being uncertain of can then be represented as being certain that it is likely that . So far so good. However, introducing these new items, i.e. innitely many numerically sharp graded modals `It is at least x% likely that...' into our language forced some non- trivial changes to the underlying semantic framework on us. Instead of sets of worlds, information states had to be modeled as sets of probability functions. Moreover, due to a rather complicated problem with dening a recursive update for negation on sets of probability functions, we had to go on another digression and dene updates on Bayesian closures of information states instead of information states directly. The result of all this technical work was the system CUS + . 203 6.2 Uncertainty-unfriendly arguments in CUS + It is now time to step back and make sure that the long journey that I took my reader on was worth the (opportunity) cost. I set out to explain uncertain reasoning. Uncertain reasoning with conditionals is dierent from uncertain reasoning involving other declarative sentences due to the phenomenon that I christened uncertainty- unfriendliness in Chapter 2. In order to know whether it is ok to reason from some (conditional-free) declara- tive sentence, e.g. `Snow is white and grass is green' to another declarative sentence, e.g. `Snow is white' we have to make sure if the inference is valid. 1 Once we know that the inference is valid, it does not matter how condent we are in the premise(s). The reasoning will be ok (or not ok) no matter how condent we are in the premise(s). But this doesn't quite hold for conditionals. Some inferences involving condition- als are valid (at least vacuously) when we are certain of the premises, but they feel invalid when we are less than certain of the premises. The class of arguments that exhibit this behavior of uncertainty-unfriendliness was given in Chapter 1. The most prominent member of the class is the pair of inferences that constitute the paradox of material implication: 1 Inferences that aren't valid can of course still be ok if they are cogent or carry other features of good inductive arguments. 204 (1) :) ! (2) ) ! Now, can CUS + deal with these inferences? In Chapter 2 I argued that Adams' logic provides an intuitively correct model for uncertain reasoning. Adams' logic allows us to explain the uncertainty-unfriendliness of the above inferences. It allows us to vindicate the intuition that someone can be 80 % condent that there is no life in other galaxies and yet be only 10 % condent that if there is life in other galaxies, UFOs have landed on earth. The inference is invalid in Adams' system simply because there is a probability function that assigns a value of 0.8 to the premise while assigning a value of 0.1. to the conditional probability that is { qua Adams' Thesis { the probability of the conditional. The good news is that CUS + allows us to `recreate' this counterexample. More generally, it allows us to create counterexamples to every uncertainty-unfriendly ar- gument form identied by Adams. All we have to do is translate the argument as formulated in the language of Adams' Logic into a corresponding argument in the languageL O and add probability-operators of the right kind. That is what our next corollary tells us: 205 Corollary 4 (Probabilistic Counterpart Arguments) For any set of formula 1 , 2 , ..., n and ofL ) : The argument from the premises 1 , 2 ,..., n to the conclusion is valid in Adams' logic, i for all numbers 0 < x 1 , x 2 , .. ,x n < 1 and y there is probabilistic counterpart-argument inL O with premisesO x 1 1 ,O x 2 2 , ..., O xn n and conclusionO y that is valid in CUS + , where y = 8 > > > < > > > : 0 if 1 P 0<in (1x i )< 0 1 P 0<in (1x i ) otherwise Proof : See Appendix One direction of this corollary tells us that any argument that is invalid in Adams' Logic corresponds to a series of invalid arguments in CUS + , intuitively to be thought of as its translations intoL O or, more suggestively, its counterparts inL O . Hence, for any argument that seems to be an intuitive counterexample to the paradoxes of material implication there is at least one parallel argument in CUS + that is invalid. For example, the argument from the one premise `It is not the case that there is life in other galaxies' to the conclusion `If there is life in other galaxies, UFO-s have landed on Earth' can be `translated' into the following argument: 1.O 0:8 : There is life in other galaxies 206 2.O 0:8 (There is life in other galaxies! UFO-s have landed on Earth) This argument is one of the many invalid arguments that correspond to the original argument (i.e. the argument sans probability-operators). The example illustrates how CUS + allows us to capture the uncertainty-unfriendliness of arguments by ex- ploiting an important feature that all uncertain-unfriendly arguments share: At least some (in fact: all) of their counterparts in the sense of Corollary 4 are invalid. But even beyond analyzing uncertainty-unfriendliness, Corollary 4 tells us how to investigate and assess reasoning from uncertain premises inL O . Take any argu- ment we are interested in: We have to embed the premises (and conclusions) of the argument under probability-operators { but not just any old probability-operators. The corollary tells us that the numerical subscripts of the probability operators in the premises have to be related in a special way to the numerical subscript of the probability-operator in the conclusion. We should also remember from our discussion of implementing uncertainty in a dynamic system in Chapter 4 that uncertainty can never be expressed by a formula that does not contain a probability-operator as its outermost operator. That is a major dierence between a natural language like English andL O . In most contexts in which the stakes aren't very high we can say `She went to the grocery store' when 207 we are but 90 % condent that she went to the grocery store. InL O we have to use a probability-operator to express what has been said. There is another interesting feature of CUS + 's representation of uncertainty that is worth pointing out. We have just seen that it is possible to recreate all the important results of Adams' logic in CUS + . But Adams' logic depends on a core assumption, namely Adams' Thesis, the claim that the probability of a conditional just is the conditional probability of its consequent given its antecedent. The natural question is, then: What does CUS + have to say about Adams' Thesis? Strictly speaking there is no equivalent to Adams' Thesis in CUS + , for the simple reason that we do not assign probabilities to sentences (like conditionals) that fail to pick out intensions or sets of worlds. And where there are no probabilities there can be no equations involving those probabilities. But there is something weaker in the spirit of Adams' Thesis that we can say about our CUS + -models: A sentence like `! is at least 60% likely', or rather O 0:6 (! ) is accepted at C i there is some conditional probability function Pr ! which is not a member of C with Pr ! ( j)> 0.6 and the conditional probability functionPr ! can be dened as the ratio of two probability functions that are both in the set C. This is what our next principle says: 208 Principle 13 (Dynamic Version of Adams' Thesis) For all and 2L ^ : C O x ! i for all Pr2 C, Pr(^ ) Pr() > x (Dynamic Adams' Thesis) tell us that an agent's or a group of agents' condence in the conditional can be equated with the conditional probability of the consequent given the antecedent that results from their credences in and . (Dynamic Adams' Thesis) falls out of our denition of the clause for the condi- tional plus the clause for `O x '. To be condent in a conditional (whose antecedent and consequent picks out a set of worlds) is to accept, for some x,O x (! ) which is, by denition, to accept (!O x ) and whoever accepts that last sentence will do so only if once they update C by, their credence in is greater than x, i.e. only if their conditional probability in given is greater than x. This consideration lies at the heart of the proof of Principle 13, which will not be given here. 6.3 Embedded conditionals inL ! 6.3.1 The second desideratum When I introduced the dynamic view in Chapter 3, I claimed that the view would be able to satisfy the second desideratum, namely to explain how we reason with conditionals embedded in truth-functional contexts. Since my readers may have 209 wondered whether and how the view can actually deliver on this promise I want to give a quick overview of the arguments that the view validates or fails to validate. In Chapter 3 we already checked that the account of negation allows us to say that :(! ) diers in meaning from:( ). This is certainly a great selling point for CUS and its conservative extension CUS + . However, I didn't examine conditionals embedded in disjunctions and in the consequents or antecedents of other conditionals. 6.3.2 Right-nested conditionals Let us start with right-nested conditionals, which is a fancy way of saying condi- tionals with conditional consequents. Right-nested conditionals play an important role in the dialectic because they seem paradigmatically successful examples of em- bedded conditionals. That right-nested conditionals reduce to conditionals that do not contain any conditionals embedded in the consequent is one of the consequences of the principle of importation. The principle of exportation is valid in CUS and CUS + , and so is its converse, the principle of importation. Not only do they entail each other, but something stronger holds: They are epistemically equivalent: Principle 14 (Exportation { Importation) ^ ! = ! ( ! ) 210 That the epistemic equivalence holds follows from Denition 3 and is fairly easy to see (it will not be veried here). Recall from the argument sketched in Chapter 1 that any system that validates the principle of exportation and some further assumptions is ipso facto going to validate the paradoxes of material implication. So the paradoxes are valid in CUS and CUS + . 2 Notice, however, that they are not valid if epistemic modals are prexed to the formulae for which they are valid. Whereas CUS + validates the inference from: to ! , it doesn't validate the inference from3: to3(! ). But even more importantly, CUS + does not validate the inference fromO x : to O x (! ) either. 3 If an argument is valid in CUS , it does not follow that the modal counterparts of the argument are valid, i.e. it does not follow that the arguments obtained by prexing probability-operators in the way prescribed by (Corollary 4) are all valid. As we have seen in the preceding section, this observation is the key to understanding how CUS + can satisfy the rst desideratum of explaining reasoning from uncertain premises, i.e. explaining uncertainty-unfriendliness. 2 This was the tension between the desiderata that I noticed. I know of only two accounts that try to bypass it: McGee (1989) and Stalnaker and Jerey (1994). 3 This holds for any x such that x < 1. 211 6.3.3 Conjunctions Conjunctions of conditionals behave (almost) as we would expect them to behave, so no big surprises here. Interestingly, in CUS (and CUS + )^( !) is epistemically equivalent to ^ (( ^)! ). Yet, Conjunction Introduction and Conjunction Elimination both hold, i.e. , ! ^ ( ! ) and ^ ( ! ) and ^ ( !) ( !). The same goes for the conditional as rst conjunct. I omit the proofs, but it is easy to see that these arguments are valid. 6.3.4 Disjunctions Many theories of the conditional struggle to deal with the interaction between dis- junctions and conditionals. The dynamic view makes some independently interesting predictions about the meaning of disjunctions and not all of these predictions can be reviewed here. 4 There are three cases of potential interest: 1. Disjunctions whose rst disjunct is a conditional (! )_ 4 For example, disjunctive syllogism fails to be valid in CUS. Counterexamples to the principle contain modals in their second disjuncts: 1. Either her birthday is on July 4 or it must be on December 25. (She told me her birthday is always on a holiday.) 2. It is not the case that her birthday must be on December 25. 3. # Hence, her birthday is not on July 4. Similar cases can be constructed with `probably' in the second disjunct. SeeSchroeder (2011a) for a discussion of these cases. 212 2. Disjunctions whose second disjunct is a conditional _ (! ) 3. Disjunctions whose rst and second disjuncts are conditionals (! )_ (! ) Disjunctions in class (1) are tricky. When they occur in English, it is very hard to avoid reading the conditional as having scope over the disjunction. Disjunctions in class (2) are clearly assertible in English. (3) She either got her degree from Harvard or if she didn't get it from Columbia, she got it from NYU. It seems that by uttering a sentence like (3) the speaker commits herself to the disjunction (2): (4) Either she got her degree from Harvard or she got it from Columbia or she got it from NYU. It is therefore good news that according to CUS + (as well as CUS) the argument from _ (: !) to _ _ is valid (Proof : See Appendix). Interestingly, the converse argument fails to be valid in CUS + . Whereas the `or-if' inference is valid in CUS + (and CUS), the `embedded' or-if inference from_ _ to 213 _ (: !) is invalid. The critical case is the case in which we have a disjunction _ _ in which the rst disjunct, , entails the second disjunct, . For example, let : =:. The premise_:_ is a trivial consequence of the tautology_:, yet_ (!) is not supported by every information state (Proof : See Appendix). The disjunctions in class (1) exhibit behavior that is similar to the disjunctions in class (2). The inference from (:! )_ to_ _ is valid and the converse inference is invalid. Finally, disjunctions in class (3) have gained some notoriety in the literature. An infamous example of an argument that is valid according to the horseshoe theory but intuitively invalid is Adams' Switch Argument: 5 1. If switch A is thrown and switch B is thrown, the engine will start 2. If switch A is thrown the motor will start or if switch B is thrown the engine will start This argument is valid according to the horseshoe-view because the premise:_ (: _) trivially entails the conclusion (:_)_ (: _) (by associativity and 5 Adams (1975), p. 32. This inference is invalid on most other theories of the conditional that allow it to embed under disjunctions, e.g. Stalnaker's, (Stalnaker (1969)) Kratzer's (Kratzer (1981)) and Gauker's.(Gauker (2005)) 214 distributivity of `_'). 6 This argument had better be invalid in CUS + and, as we can easily verify, it is. 7 Another argument (form) that has received some attention is the argument from (! )_ (! ) to the disjunction of its consequents _. There is evidence that the overwhelming majority of subjects that are surveyed take this form or at least some instantiations to be valid (See my discussion in Chapter 2). The dynamic account nicely vindicates the folk opinion: (! )_ (!) entails _, but only if and do not entail each other (Proof : See Appendix). The argument does not go through if the disjuncts entail each other or are the same. Hence, the inference from (! )_ (! ) to _ , hence , which sounds like a clear non-sequitur, is invalid. Unless and exhaust logical space there is no reason to think that must hold whenever the two conditionals hold. 8 6.3.5 Left-nested conditionals: In many philosophical discussions of conditionals, left-nested conditionals are rele- gated to a footnote. Since I decided by at that in CUS these constructions are 6 That the horseshoe theory validates this inference is just another nail in its own con. 7 A simple parallel circuit design can serve as an intuitive counterexample. 8 In the example discussed in Chapter 2 (Johnson-Laird and Savary (1999)) the two disjuncts do exhaust logical space. 215 not part of the semantic denitions, I have joined the lot of philosophers who ei- ther bracket the issue or try to argue that left-nested conditionals are ultimately ill-formed, incoherent or otherwise decient. 9 Unlike many philosophers who deny that left-nested conditionals can make sense I actually believe that most left-nested conditionals are perfectly respectable members of the family (of conditional compounds): Witness `If he is not grateful if you watch his cat for him you shouldn't do it again' or `If the cup breaks if it is dropped, the material it is made of is fragile'. Philosophers who want to deny that conditionals can be embedded in antecedents and acknowledge the felicity of these sentences tend to gravitate towards two general explanatory `strategies'. The rst strategy is to explain how we as hearers can make sense of some conditionals by appealing to our cooperative and charitable nature. If our interlocutor in a context C utters `If (if A then B), then C' we can try to discern, within the context C, a `categorical' basis D for the conditional `If A then B' and reinterpret the left-nested conditional as `If D, then C'. For example: (5) If the light goes on if you press the switch, the electrician has called. 9 Gauker (2005), Bennett (2003), Edgington (1995), Gibbard (1981). 216 The basis for the rst conditional seems to be the realization that power is on. Hence, we can reinterpret the compound as saying: (6) If power is on, the electrician has called. If this were in fact the way we proceed, it would also allow us to explain failures to make sense of a conditional. That the sentences (7) If Kripke is there if Strawson is there, then Anscombe is there. is barely intelligible can be explained by the fact that in the relevant contexts we sim- ply do not succeed in identifying a (categorical) basis for the conditional embedded in the antecedent. Another reinterpretation strategy that is not incompatible with but rather com- plementary to the rst one tells us to interpret the antecedent of a left-nested con- ditional as containing a covert attitude verb like `accept'. A sentence of the form (8) If John should be punished if he took the money, then Mary should be pun- ished if she took the money. 217 is hence to be understood as elliptical version of `If you accept that John should be punished if he took the money, then you should accept that Mary should be punished if she took the money.' - a simple encouragement to be consistent in one's moral judgments. Needless to say, these reinterpretation strategies are both ad hoc. But for a defender of a theory that entails that sentences of type t are ungrammatical it is probably better to devise an ad hoc strategy for explaining away sentences that seem to be of type t than to not say anything at all about these sentences. Unlike SCE, CUS + is not committed to treating left-nested compounds as un- grammatical. But just like SCE, CUS + may not be in a position to say anything terribly illuminating about left-nested conditionals. This is because according to the dynamic view the conditional embedded in the antecedent amounts to just another test. Let us take a closer look at what the dynamic view would predict about these compounds if they were syntactically admissible. How would we update an infor- mation state by a formula (! )! . There seem to be two cases. First case: If (! ), the conditional embedded in the antecedent, passes the test, the entire conditional just amounts to2. Second case: In the event that the test crashes, the 218 entire conditional is going to (vacuously) pass the test and the compound conditional is accepted no matter what. Hence a conditional like (9) If a fair coin comes up heads if I throw it, I must be possessed by a demon. is going to be accepted no matter what (given that the embedded conditional is not going to be accepted). But that is not what we expect such a conditional to do. The antecedent is supposed to somehow restrict the information state (i.e. set of probabilities) that is relevant for deciding if the consequent holds. This requirement is in place even if the antecedent is itself a conditional. So in the above example what we try to intuitively get at is whether or not the consequent is accepted at those information states at which the antecedent holds, i.e. whether I am really possessed by a demon in those information states at which it is accepted that if I throw a fair coin it comes up heads. And this is where the dynamic account reaches its limits: It cannot explain how a conditional itself is able to restrict a set of worlds or probability functions in an interesting, i.e. non-trivial way. I will not try to convince my readers that there is an easy x for this bug. All I want to do is suggest that there is some strategy for making sense of left- nested conditionals within CUS (and CUS + ) that seems at least initially promising. 219 It requires defending a broader claim about the kind of semantic context that an antecedent of a conditional creates. We know that antecedents of conditional are dierent from standard sentential contexts in other ways. For example, they license negative polarity items like `ever' or `any'. So it wouldn't be completely bizarre to claim that these contexts have some further feature that sets them apart from regular sentential contexts. And the hy- pothesis could be that they require `factual' readings of epistemic modals and condi- tionals. In other words, whenever a conditional or an epistemic modal occurs inside the antecedent of a conditional, it has to be read as making a factual claim. 10 An- tecedents of conditionals would, in this sense, alter the semantic role of modals that are embedded in them. More concretely, the idea is that a conditional in the context 10 Geurts advances a similar hypothesis and provides the following circumstantial evidence (Geurts (2005), p.407): (i) If you will give me an apple, you may have a pear. (ii) If you may have a pear, you will give me an apple. (iii) If Brown is in Ouagadougou, Green may be with him. (iv) If Green may be in Ouagadougou, Brown is with her. To quote Geurts: Another observation that points towards a fundamental asymmetry between an- tecedents and consequents is that one environment restricts the interpretation of modals in a way the other does not. In particular, what we might call de se modality seems to be conned to conditional consequents: (...) The may' in (ii) cannot have the permission-giving reading that it has in (i) and the `may' in (iv) cannot have the epistemic reading that it has in (iii). I take it that by `de se-readings' Geurts refers to uses of modals that directly express the states of mind of the speaker. 220 of an antecedent would make the same semantic contribution as the corresponding material implication. Since the dynamic account allows us to dene an update by a conditional! as an update by2(:_ ) (see (Corollary 2) in Chapter 3), we could say that the antecedent creates a context in which the conditional is, as it were, de-modalized, i.e. stripped o the box. 11 If the actual contribution of a conditional embedded in the antecedent of another conditional were the corresponding material implication, we could at least explain in what sense the antecedent of the conditional imposes a (hypothetical) restriction on the information state. But as I said in my `disclaimer', this is a rather speculative hypothesis about the meaning of conditionals in antecedents. If sound, it might help the proponent of CUS + to leave the awkward stipulation that antecedents must always be conditional- free behind. Assessing its merits in a more detailed way goes beyond the scope of the current project. 11 It is much harder to say what the result of `de-modalizing' epistemic modals would be. What is the meaning of `must' and `might' embedded in an antecedent? Even if they, pace Geurts, lose their de se-meaning, what meaning do they acquire? Also, what is the meaning of a negated conditional embedded in an antecedent? Many questions remain open. 221 6.4 Remarks on the logic of CUS + In section 5.2.2 I have oered a denition of validity for CUS + . In this section I will brie y survey some features ofL O and its logic. Most of my remarks will be promissory. A fully adequate treatment of the logic generated by CUS + models would start with either designing an axiomatic system for the language with epistemic modals and probability operators,L O , or identifying a set of rules for a natural deduction-style system of derivation. I shall take a glance at the fragment of the languageL O that can be used for reasoning from uncertain premises. Interestingly, the languageL O faces some syn- tactic restrictions at the level of formulae embedded under the probability operators. Whereas ^ (! ) is an expression ofL O , indeed of its fragmentL ! , its `uncer- tain' would-be counterpartO(^ (! )) is not an expression ofL O . This feature of the language is puzzling, especially since the restriction seems somewhat arbitrary. Yet, it is important to remind the reader that the expressive limitations that occur at the level ofL O are less severe than the ones that Adams (and with him defenders of SCE) are facing with respect to the language involving the conditional. 222 6.4.1 Transmission Principles Before I turn to the resources of the probabilistic fragment of the language I would like to call the reader's attention to a couple of valid arguments of CUS + that connect the non-probabilistic fragment ofL O with its probabilistic fragment, i.e. arguments that show how the two fragments interact: 12 Lemma 3 (Probable Modus Ponens) Probable Modus Ponens is valid in CUS + , i.e. for all x, the inference fromO x and ! toO x is valid. 13 Proof : See Appendix Lemma 4 (Probable Modus Tollens) Probable Modus Tollens is not valid in CUS + i.e. the inference from:O x and ! to:O x is invalid. Proof : See Appendix 12 Next to Probable Modus Ponens and Probable Modus Tollens, another important argument that shows how the fragments interact is the following (valid) argument: 2O3. It says that `probably' is intermediate in logical strength with respect to the other two epistemic modals. We also have2 =O 1 , connecting the likelihood-modal with `must'. 13 It is trivially valid if x = 0. If x = 0, prexingO x makes any formula valid. For example, we have that:O x ^:. 223 Counterexamples to Modus Tollens have sometimes been mistaken for counterex- amples to Modus Ponens. 14 Suppose we are temporally located in fall 1980 and speculating about the outcome of the presidential election. stands for `A Republi- can wins the 1980 presidential election'. Suppose! stands for `If Reagan doesn't win, Anderson will win'. 15 Now, given available evidence about various polls, it is unlikely (in 1980) that Anderson is going to win, given that Reagan won't win. Yet it is both likely that a Republican is going to win and that Anderson is going to win if a republican is going to win and Reagan is not going to win. The proof showing that Modus Tollens tout court is not valid in CUS and CUS + is very similar to the proof showing that Probable Modus Tollens is not valid in CUS and CUS + (It is a special case of that proof). The failure of Modus Tollens also shows what goes wrong with the proof of God's existence discussed in Chapter 1: The inference from: and:!:(! ) to:: is not valid. 6.4.2 Expressive limitations The update-semantics dened forL O gives rise to a rich and formally interesting logic for conditionals, a logic that can only be hinted at in this work. As we have 14 See Gillies (2004) pp. 8-10 for a nice discussion of the sources of this confusion. 15 For younger generations and people not familiar with McGee's example: John B. Anderson was a Republican also-ran who ran as an independent candidate and obtained 6.6% of the vote. 224 seen, the CUS fragment allows us to embed conditionals under negation, disjunction, conjunction and in the consequents of conditionals. I have also tentatively sketched a strategy for extending CUS (and theL ! -fragment of CUS + ) to conditionals em- bedded in antecedents. Based on these observations we can conclude that CUS allows us to show that conditionals behave syntactically almost exactly like other sentences. CUS (and theL ! -fragment of CUS + ) is able to make sense of reasoning with embedded conditionals. Some other embeddings contained in the fragmentL ! that I haven't discussed but appealed to at various points are embeddings of conditionals under epistemic modals like `2' and `3'. Moreover, once we move from theL ! to theL O -fragment of CUS + , we notice that we can embed conditionals under probability-operators like O as well. But this is where the road gets slippery. There is a clear limit to what kind of sentences we may embed under probability-operators likeO x . And given the assumptions made about the interpretation of the formal system in Chapter 4, this unfortunately means that there is a clear limit to how good CUS + is at modeling uncertain inference. In fact, there are uncertain inferences involving conditionals that cannot be mod- eled in CUS even though the corresponding `certain' inferences can be modeled in CUS + and, for that matter, CUS. To give an example, in CUS + we cannot model 225 the state of being at least 80% condent in ^ ( ! ), even though we can model the state of being certain that^( ! ). The reason is that probability-operators do not allow us to embed anything that does not semantically correspond to a set of worlds or an intension. Probability-operators measure probabilities and we cannot assign probabilities to compounds of conditionals. That not dening probabilities for sentences like^ ( ! ) may have advantages was one of the major takeaways of the last section of Chapter 2. The fact that we can embed conditionals but nothing containing conditionals under `O' is simply the result of the stipulation (Wide Scope) made in Chapter 4. According to (Wide Scope) conditionals always take wide scope. This stipulation has some clear benets. It makes (Probable Exportation-Importation) true by stip- ulation. Principle 15 (Probable Exportation - Importation) For all x:O x (! ( ! )) =O x (^ !) These limitations may seem as ad hoc as SCE's claim that conditionals do not embed at all. But even if they sound ad hoc, they are not philosophically unmoti- vated. One lesson that we have to learn, even if we don't fully grasp the reasons for why we have to learn it, is that we cannot engage in the ction that conditionals behave just like ordinary declarative sentences forever. At some point that helpful 226 ction has to come to an end. Somewhere the line has to be drawn. I have decided to draw it at the point at which we start treating conditionals and epistemic modals as singling out bits of information that we can be (more or less) condent in, just like we are condent in other pieces of information. Or, to put it a little more ambiguously, I have drawn the line at the point at which condence becomes iy. Maybe I should have been more audacious and pushed the line out a little further. Maybe, but the triviality results discussed in Chapter 2 show that pushing that line out too far is risky and adding epistemic modals to the language doesn't make the endeavor any less risky. 16 This is one of the points at which I wholeheartedly agree with Adams. Adams and his followers were haunted by many problems, but they avoided the specter of triviality. His getting around the results can be regarded as a reward for opting out of the ction that conditionals behave like ordinary sentences very early on, maybe too early on. By drawing a maybe arbitrary line I hope to be able to cash in some of the same rewards. 16 As Willer (Willer (2011b)) shows, probabilistic theories of conditionals that (i) assign probabil- ities to epistemic modals, (ii) dene the probability of a conditional to be equal to its conditional probability and (iii) allow epistemic modals to be interdenable with the conditional are incom- patible with data about what inferences speakers take to be valid. This shows that one cannot have one's cake and eat it. The current proposal avoids these threats of inconsistency since it does not assign probabilities to either conditionals or epistemic modals. Proponents of Simple Credal Expressivism, on the other hand, are forced to come up with some ad hoc reason for giving up one of (i)-(iii). 227 6.5 Appendix to Chapter 6 Proof 4 (_ (: !) _ _) To see this, pick an arbitrary information state C and focus on the case in which the update by _ (: ! ) does not crash (the case in which it crashes is trivial). First case: C"_ (: !) = C. For this to be true, we have to have that C":(:^ (: !3:)) = C (by pushing negation through to the consequent) and hence that for enough non-empty subsets y of C that cover all of C we havejyj CL " (:^ ( !3:) =;. Hence we know that C itself is such that C CL " (:^( !3:) =;. But that just means that the set of probability functions in C that are rst updated by: and then put to the test of whether they assign 1 to:2 if updated by: is empty. So if that is the case the set of probability functions in C that are rst updated by:, then by: and eventually by: is also going to be empty, so C will support the disjunction. Second case: C"_(: !) = C , where C is a subset of C. Then an analogous argument shows that C has to support the rst disjunct, i.e. and hence the entire disjunction. Proof 5 ( _:_2 _ (!) ) It suces to show that there can be a state that does not support _ (! ). For there to be such a state we know that the result of updating C by (:^:(! ) must not be the empty set. But that just means that the result of updating C by (:^ (! 3:)must not be the 228 empty set (since:(!) and (!3:) are equivalent in the strictest sense, i.e. epistemically equivalent). But the last condition on a state C is clearly satisable, for example if C supports: or if it doesn't support but:_:. Proof 6 (If 2 and 2 we have: (! )_ (!) ( _)) Updating an arbitrary information state by the premise is running two consecutive tests on the state. The only interesting case is the case in which the state survives both tests. That means that C" (! )_ (!) = C, and hence the subset of probability functions y such that y CL ":(! )^:(!) =; is C itself. But an update of C by:(! )^:(!) or, equivalently, by (!3: )^(!3:) cannot result in a crash if C supports both: and:. For if C supported both of these formulae the update by (!3: )^ (!3:) would not result in a crash. Hence, every state that supports the disjunction of conditionals supports the disjunction of their consequents. Notice that this reasoning goes only through if neither entails nor entails , for C can support: without supporting - it may support neither nor: . Proof 7 (Probabilistic Counterpart Arguments) We have to show that (i) if an argument is valid in Adams' Logic, all its counterparts are valid in CUS + and (ii) if an argument is invalid in Adams' Logic, not all its counterparts in CUS + are valid. Let's start with (i). If an argument is valid in Adams' logic, the uncertainty 229 of its conclusion does not exceed the sum of the uncertainties of its premises. Now, if the argument from 1 , 2 ,... n to is valid in Adams' logic, no CUS + -state C is going to accept all the premisesO x 1 1 ,O x 2 2 , ...,O xn n and yet fail to accept the conclusionO y . For if it did all probability functions in C would assign all the premises 1 , 2 ,... n values at least as great as x 1 , x 2 , ....x n but yet fail to assign the conclusion a value that is at least as great as 1 P 0<in (1x i ), which means that the conclusion's uncertainty would be greater than the sum of the uncertainties of the premises. (ii) Per Adams' Logic, if the original argument is invalid it will be the case that for all> 0 there is some probability functionPr proper for all 1 , 2 , n ,...and such that for all i, Pr( i )> 1 - , but Pr( )6 . 17 Given Adams' theorem, all we have to do is construct a singleton-information state C that contains the witness function Pr (while trivially satisfying weak regularity). Then we can select the operatorO 1 and translate theL ) -formulae 1 , 2 , 3 ,...and intoL O -formulaeO 1 1 ,O 1 2 , ...,O 1 n andO 1(n) . We choose the operator such that< 1- (n), otherwise our counterexample will not work. For example, if n = 3, we can choose to be 0.1, because 0.1 < 0.7. Now, updating the state C by all the premisesO 1 1 ,O 1 2 , 17 This follows from Adams' denition of validity, Adams (1975), p.57. 230 ...,O 1 n is still going to result in the original state C. However, since Pr( ) is, ex hypothesi, less than updating the state C byO 1(n) is going to result in a crash. Proof 8 (Proof of Lemma 3) If updating an information state C by the two premises results in a crash, updating the empty set by the conclusion is also going to result in the empty set. So suppose that updating by the two premises does not result in a crash. Suppose that (C"O x )"(! )= C but that C"O x 6= C. Then in must be the case that9 Pr2 (C) and we have that for any such Pr( ) < x. But we know the following aboutC: Since the update byO x did not crash, all members of (C) must assign the factual formula a probability that is greater than x. Hence conditionalizing C on did not result in the empty set, but in a non-empty set C 1 and since the conditional ! passed the test, we know that all probability func- tionsPr C 1 inC 1 assign the value 1. We have to show that at least one probability function Pr C in C assigns a value y such that y > x. But since we know that, by Conditionalization, the `new' probability of , i.e. Pr C 1 equals Pr C (^ ) Pr C () , we know that Pr C ( )> Pr C (), hence Pr C ( )> x. Contradiction. Proof 9 (Proof of Lemma 4) We will outline the construction of a counterexam- ple involving a conditional embedded in the consequent of another conditional. So suppose:O x (! ) holds at some information state C, for some x. That means that C":O x (!) = C. Hence the union of the subsets of C at which an update 231 byO x (! ) crashes is C itself. That in turn means that (C" ) does contain at least one probability function that assigns a value that is less than x. Now suppose further that C is such that ! (! ) holds throughout the state C, as the second premise demands. For that to be the case, has to hold throughout the state (C" )" . In other words, no probability function in (C" )" assigns a probability less than 1, but at least one probability function in the state C" assigns a probability less than x. 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Abstract (if available)

Abstract In Iffy Confidence, I investigate indicative conditional sentences and assimilate two non-descriptivist theories of the meaning of these sentences, a simple expressivist theory and a theory using the resources of dynamic semantics. This assimilation is made possible through the construction of a non-standard dynamic semantic framework. The dynamic semantic framework relies on a more fine-grained representation of information, a representation in terms of sets of probability functions, and enables us to vindicate central commitments of both theories while extending the range of sentences involving conditionals that each of the theories makes predictions about. I argue that defenders of non-descriptivism about conditionals should abandon the simple expressivist view and adopt an extended dynamic theory that relies on my construction.

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Asset Metadata

Creator Schmitt, Johannes V. (author)

Core Title Iffy confidence

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Philosophy

Publication Date 08/01/2012

Defense Date 04/25/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag conditional probability,conditionals,credence,dynamic semantics,epistemic modals,expressivism about conditionals,if..then,OAI-PMH Harvest,probability

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Schroeder, Mark (committee chair), Easwaran, Kenny (committee member), Schein, Barry (committee member)

Creator Email johannes.schmitt@uni-konstanz.de,observationselection@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-82653

Unique identifier UC11290087

Identifier usctheses-c3-82653 (legacy record id)

Legacy Identifier etd-SchmittJoh-1100.pdf

Dmrecord 82653

Document Type Dissertation

Rights Schmitt, Johannes V.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA