How Serious Is the Paradox of Serious Possibility?

Transcription

1 How Serious Is the Paradox of Serious Possibility? Simone Duca Ruhr University Bochum Hannes Leitgeb Ludwig Maximilian University The so-called Paradox of Serious Possibility is usually regarded as showing that the standard axioms of belief revision do not apply to belief sets that are introspectively closed. In this article we argue to the contrary: we suggest a way of dissolving the Paradox of Serious Possibility so that introspective statements are taken to express propositions in the standard sense, which may thus be proper members of belief sets, and accordingly the normal axioms of belief revision apply to them. Instead the paradox is avoided by making explicit, for any occurrence of an introspective modality in the object language, the belief state to which this occurrence refers; this will make it impossible for any doxastic modality to refer to two distinct belief sets within one and the same context of doxastic appraisal. By this move the standard derivation of a contradiction from the theory of belief revision in the presence of introspectively closed belief sets does not go through any more, and indeed the premisses of the Paradox of Serious Possibility become jointly consistent once they are reformulated with our amended introspective modalities only. Additionally, we present a probabilistic version of the Paradox of Serious Possibility which can be avoided in a perfectly analogous manner. In the literature on qualitative belief change, doxastic possibility is often referred to as serious possibility, following a usage introduced by Levi (see Levi 1984, 1996). In particular, for any sentence A, A counts as a serious possibility for an agent if and only if its negation is not included in the agent s current belief set. In turn, the belief set of a rational agent consists precisely of those sentences the negations of which are not serious possibilities. In a similar way, or so it seems at least at first glance, one can analyse not just plain factual beliefs about the world, but also (doxastic-)modal beliefs. After all, an agent might just as well exclude certain propositions about its own belief states as serious possibilities. Accordingly, the belief set of a rational introspective agent will include doi: /mind/fzs042

2 2 Simone Duca and Hannes Leitgeb introspective statements the negations of which are not serious possibilities. However, if the standard theory of qualitative belief revision the so-called AGM account developed by Carlos Alchourrón, Peter Gärdenfors, and David Makinson (1985) (see Hansson 1999 for an overview) is applied to belief sets that contain straightforwardly introspective or reflective claims, then general wisdom has it that contradictory conclusions are bound to follow, and accordingly the AGM postulates need to be weakened in the presence of such introspective information. The so-called Paradox of Serious Possibility is maybe the best-known example of a difficulty of this kind. 1 In this article we will suggest a way in which this paradox can be avoided without curtailing the axioms of belief revision. Instead we will suggest that one should remove an ambiguity that affects the meaning of the introspective doxastic operators that are employed in the original formulation of the paradox, which, as we will show, is going to leave the standard axioms of belief revision perfectly consistent even if applied to introspectively closed belief sets. Let us make the context in which the Paradox of Serious Possibility emerges more precise by introducing some formal conventions. In the following, A is intended to run over the sentences that any of the agents that we are going to consider might or might not believe to be true, where such sentences are assumed to be included in a fixed formal language the logical vocabulary of which consists (at least) of the logical connectives of propositional logic and the two modal operators. and -. In our context,. will be used to express it is believed (by the agent in question) that, while - is interpreted as it is a serious possibility (for the agent in question) that. If a formula does not include either of these two modalities, we call that formula (purely) factual. For the paradox below it will not be important if modal operators can be iterated or not; even simple applications of introspective operators to purely factual formulas will suffice for the problem. K is intended to range over all belief sets that our rational agent in question could possibly have, where a belief set is a set of formulas that is meant to capture what the agent believes to be true at a time when its mental state has (in some idealized sense) that belief 1 The rich literature on the paradoxical effects of combining (a variant of ) the Ramsey test for conditionals with standard AGM belief revision for languages that include such Ramsey test conditionals could be counted as another family of such problems if the conditionals in question are interpreted as introspective statements. But of course it is controversial whether conditionals ought to be interpreted in this manner. For more on this, see Gärdenfors 1986, 1988, and Lindström and Rabinowicz 1995.

3 How Serious Is the Paradox of Serious Possibility? 3 set as one of its components. Belief sets will always be assumed to be deductively closed as understood by propositional logic; accordingly, consistency and inconsistency will always be understood as being given by propositional logic. In view of the modal operators employed, one should actually strengthen the notion of deductive closure by turning to some standard system of modal (doxastic) logic, but as it turns out this is not really necessary for the paradoxical result that we will be interested in to go through. 2 Given these preliminary remarks, and in line with the formal treatment in Fuhrmann 1989 whose exposition of the paradox we are going to follow, let the set of introspective judgements with respect to a belief set be defined as follows: Definition 1 (The set of introspective judgements with respect to K ): For any given belief set K, let Poss(K) be the (uniquely determined) set of sentences satisfying: (a) (b) ;K, ;A : A2 K T.A 2 Poss(K) ( Positive Introspection ) ;K, ;A : A =2 K T - A 2 Poss(K) ( Negative Introspection ) (c) Poss(K) is the smallest set satisfying conditions (a) (b) 3 Definition 1 has a straightforward metalinguistic reading. When one is talking about a particular belief set, it can always be determined whether a given sentence is or is not an introspective judgement with respect to that belief set. Accordingly, by means of the definition above, one can always generate a meta-belief-set Poss(K), for any given belief set K, which encodes what the agent can or cannot believe rationally as determined by K. However, it also seems to be perfectly acceptable to include such introspective judgements within the agent s belief sets themselves. After all, nothing seems to prevent us from assuming some members of belief sets to be well-formed sentences 2 In fact there is not a single step in the proof of the paradoxical result below in which the presence of a formula in a belief set is derived from the presence of other formulas in the same belief set on the basis of some logical rule of classical propositional logic. What we do need for our version of the paradox is that all logical truths are members of any belief set whatever the logic in question is like and that the only inconsistent belief set is identical to the whole object language. On the other hand, taking all belief sets to be deductively closed in the sense of classical propositional logic is just what the standard theory of belief revision does, which is why it will be convenient for us to take the same for granted in what follows. 3 Notice that Poss(K) is not required to include K as a subset. As a matter of fact, the two sets could even be disjoint, that is, Poss(K) \ K = [, and if K did not include any modal formulas at all, then they would be disjoint.

4 4 Simone Duca and Hannes Leitgeb which contain. or -as primitive operators. All we need to do is to integrate Poss(K) inside K. This can be achieved formally in the following way: Definition 2 (Closure under Poss): A belief set K is closed under Poss iff Poss(K) " K In fact, we would expect all perfectly rational agents who come equipped with the necessary introspective capacities to have belief sets which are closed introspectively in this sense. The corresponding operation of Closure under Poss, that is, taking the least superset of K that is closed under Poss, seems equally innocuous. Nevertheless, as we shall soon see, rationally revising belief sets which are closed under Poss is a non-trivial task. 4 Indeed, the Paradox of Serious Possibility is usually regarded as showing that the standard AGM axioms of belief revision do not apply if an agent s belief sets are assumed to be closed in this introspective manner. In this article we will argue to the contrary: we will present a way of dissolving the Paradox of Serious Possibility in which introspective statements are taken to express propositions in the standard sense and so may be proper members of belief sets (contrary to the way in which, for example, Levi 1996 would have it); as a consequence, then, the normal axioms of belief revision will be seen to apply to them. Instead the paradox is avoided by making explicit, for any occurrence of it is believed that (.) or it is a serious possibility that (-) in a formula of the object language, the belief state to which this occurrence refers; this will make it impossible for any doxastic modality to refer to two distinct belief sets within one and the same context of doxastic appraisal. We show that by this move the standard derivation of a contradiction from the theory of belief revision in the presence of introspectively closed belief sets does not go through any more, and indeed the premisses of the Paradox of Serious Possibility become jointly consistent once they are reformulated with amended introspective modalities of the form it is believed-in-k that (. K ) or it is a serious-possibility-in-k that (- K ) only. Additionally, we present a probabilistic version of the Paradox of Serious Possibility which can be avoided in a perfectly analogous manner. The remedy we suggest is even more natural in a probabilistic context, since probability theorists 4 Note that unlike closure under logical consequence, the operation of closure under Poss is not a closure operation in the Tarskian sense. In particular, monotonicity fails in view of the Negative Introspection clause (b) of definition 1.

5 How Serious Is the Paradox of Serious Possibility? 5 usually avoid from the start object-linguistic probabilistic operators whose reference is determined by the semantic context, whereas modal logicians are used to evaluating modal operators in semantic contexts which are only specified metalinguistically (hence the original formulation of the paradox in terms of such modalities). This is the structure of the paper: in section 1 we will present the postulates on which the Paradox of Serious Possibility rests, we will derive the usual contradictory conclusion from them, and we will mention briefly some suggestions in the literature on how this conclusion could be avoided; our own suggestion is going to differ from all of them. In section 2 we will introduce a new probabilistic version of the paradox, which will show that the theory of probabilistic belief change is not better off if introspective claims are allowed to enter the object language and if subjective probability measures are assumed to satisfy certain introspection postulates. At the same time we believe the derivation of the contradiction in the probabilistic context makes it (if only slightly) easier to understand what is going wrong in the original qualitative or modal Paradox of Serious Possibility. After having explained our diagnosis of the problem in section 3, we will show in section 4 that neither the qualitative nor the probabilistic version of the paradoxical derivation remains valid when the assumptions of the paradox are reformulated in terms of introspective modalities which state explicitly to which belief sets they refer. Section 5 adds to this result by presenting two concrete models of belief revision for introspective belief sets two amongst many other possible models in which the premisses of the paradox taken together turn out to be perfectly consistent once they have been reformulated by means of the amended modalities. Section 6 sums up what has been achieved and enumerates some open questions which arise from our discussion and which point towards future work on introspective belief revision. 1. The qualitative Paradox of Serious Possibility We turn first to the postulates on belief sets and belief change from which Fuhrmann derives his version of the paradox (see Fuhrmann 1989, pp ). The specification qualitative in the section title is meant to distinguish the postulates below from the probabilistic postulates that we will introduce later when we deduce a probabilistic

6 6 Simone Duca and Hannes Leitgeb analogue of theorem 1. We will first lay down the postulates and then explain them briefly. The postulates derive from the standard AGM account of belief revision (see also Gärdenfors 1988) except that belief sets are not purely factual as in the original approaches to belief revision, but are instead required to be introspectively closed in the sense explained above. We are going to refer to any structure consisting of a class of possible belief sets together with a revision operator defined on any pair hk, Ai, where K is a belief set in that class and A is a sentence in the given object language (with. and -), as a qualitative beliefchange model; ifk is such a belief set and A is such a sentence, then K *A is another belief set in the same class. Variables such as K and A are really meant to range over the corresponding belief sets and formulas as determined by a given belief-change model, however, we will normally suppress any explicit reference to belief-change models in any of the postulates on belief-change, as is common in the literature on qualitative belief revision. Only in section 5, where we introduce models for our set of postulates with the amended introspective modalities, will we say more about qualitative belief-change models as objects in their own right. It seems natural to assume that belief-change models that belong to rational introspective agents ought to satisfy the following desiderata: (IC) 'K, 'A :A=2K & A =2K (P) ;K, ;A : A =2K T K " K *A (Poss) (a) ;K, ;A :A2K T.A2K (b) ;K, ;A :A=2K T - A2K (S) ;K, ;A :A2 K *A (C) (DC) ;K, ;A :K ÞL, 6 A T K *A ÞL ;K, ;A :.A 2 K, - A 2 K T K = L The incompleteness postulate (IC) demands the existence of at least one belief set that is incomplete with respect to the sentences of the language over which our metavariable A ranges. It is a very weak non-triviality assumption; if (IC) failed, the agent would be opinionated as to every sentence of the language in every belief set it could possibly have, which clearly would be very odd.

7 How Serious Is the Paradox of Serious Possibility? 7 The preservation principle (P) deals with an agent s belief revision operator *: it states that if A is consistent with K, nothing has to be removed from a belief set K in order to obtain the very belief set K *A that is meant to differ from K only in as much as is needed to consistently include A. Another way of putting this is to say that nothing should be removed from K if A is a serious possibility, that is, it is not believed by the agent that A. It is worth pointing out that in order for the paradox below to go through, the variable A may well be restricted to purely factual formulas (without modalities) in (P) as well as in all other postulates; however, if K " K *A in (P) is unpacked in terms of ;B :B2 K T B 2 K *A, then B will have to be taken to range over all formulas whatsoever in order for the theorem below to apply, including formulas with modal operators. The upshot of this is that the Paradox of Serious Possibility is not a paradox of revision by introspective information, but it is a paradox of what happens to introspective information if one revises by straightforwardly factual propositions. (Poss) specifies the conditions by which one can obtain the set of introspective judgements with respect to any given belief set K and any sentence A. K is assumed to be closed under Poss as explained in the last section. In our statement of (Poss), and in contrast with Fuhrmann s own version of (Poss) (Fuhrmann 1989, p. 119), we have merged the Closure-under-Poss condition Poss(K) " K from our definition 2 with (a) (c) of definition 1, while at the same time eliminating all references to Poss itself; this is merely for the sake of simplicity, since no explicit reference to the Poss operation is needed for the formulation of the paradox below. 5 Both (P) and (Poss) have been questioned in various ways (see, for instance, Gärdenfors 1986 and Fuhrmann 1989, to name just two); we shall return to this. One requirement that seems somewhat less controversial, at least as far as the topic of introspective belief revision is concerned, is (S). The rationale behind it is of course that one wants the revision of a belief set K by A to be successful in the sense that A should be a member of the revised belief set K *A; but even this postulate might be taken to amount to a problematic presumption if it is interpreted as claiming that one s data in question never could be flawed in any 5 Neither will there be any need, given our purposes, to make any further reference to a probabilistic version of Poss.

8 8 Simone Duca and Hannes Leitgeb way. We are going to ignore this sort of worry in what follows, since it is orthogonal to the problem of the Paradox of Serious Possibility. Finally, (C) and (DC) deal with the consistency or inconsistency of certain belief sets under certain assumptions. In particular, (C) states that if K is consistent and the negation of A is not logically true, then the revision of K by A is not L, that is, the trivial belief set consisting of all sentences whatsoever. Since belief sets are assumed to be deductively closed under propositional logic, L is of course the uniquely determined inconsistent belief set, that is, the whole object language in question. The modal (in-)consistency postulate, (DC) with D standing for doxastic is a requirement that in Fuhrmann s original derivation followed from defining one modal operator in terms of the other (and negation). It states a sufficient condition for the inconsistency of belief sets, such that an agent cannot believe consistently that the negation of A is both impossible (.A) and possible (- A). As observed by several authors, these postulates (or variations thereof ) lead to a contradictory conclusion (see Fuhrmann 1989, Levi 1996, and Rott 1989): this is the so-called Paradox of Serious Possibility. Fuhrmann provides two proofs for it. The first one is carried out in terms of the contraction and expansion operations of the theory of belief revision, while the second one is given by using the revision operator directly. We will only consider the latter version of the proof. Our preference for the revision operator version is due to the fact that revision better suits our aim of probabilifying the paradox in the next section, in view of the fact there is a clear and straightforward probabilistic counterpart of revision, that is, conditionalization. In the proof, we refer to the previous list of qualitative postulates when we justify the different proof steps. Theorem 1 (The Paradox of Serious Possibility): No qualitative belief-change model can satisfy all of the postulates (IC), (P), (Poss), (S), (C), and (DC). Proof: In accordance with (IC), we consider a (deductively closed) belief set K which is not opinionated with respect to some sentence A. (1) A =2 K, from (IC) (2) A =2 K, from (IC) (3) - A 2 K, from (1), by (Poss-b)

9 How Serious Is the Paradox of Serious Possibility? 9 (4) ;B :B2 K T B 2 K *A, from (2), by (P) and def. of " (5) - A 2 K *A, from (3) and (4) (6) A2 K *A, by (S) (7).A 2 K *A, by (6) and (Poss-a) (8) K *A=L, from (5) and (7) with (DC) (9) 6 A, from (2) (10) K ÞL, from (1) (11) K *A ÞL, from (9), (10), and (C) (12) Contradiction! from (8) and (11). Since the postulates above are at least prima facie plausible, and since the conclusion that follows logically from them is contradictory, this is a paradox. There is a variety of responses to the paradox which can be found in the literature (see Fuhrmann 1989 for a tentative list of possible ways out). To the best of our knowledge, they all differ from the diagnosis and from the remedy that we are going to put forward. The classic reaction to the paradox has been provided by Levi (1984): for him, sentences such as -A and.a are not truth-value-bearing, that is, they do not express propositions (at least from the viewpoint of the introspective agent in question). Therefore, they should not be included in the belief set K, in contrast with purely factual sentences A that speak about the world which may be members of belief sets. Instead, modal sentences express higher-order judgements which are accepted by an agent in some merely pragmatic manner and which belong to a metacorpus M(K) that is not to be incorporated into K. As a consequence, if one collects together all sentences introspective or not which are accepted by an agent at a time, then the standard axioms of AGM belief revision cannot be assumed to apply to all of them, since the underlying justification of the AGM axioms in terms of manipulations of the set of doxastically possible worlds of an agent at a time requires belief sets to be given semantically by propositions, that is, such sets of possible worlds. 6 If -A and.a were expressing 6 This becomes very clear from the representation theorem in Grove (1988) which shows that every belief revision operator * (for purely factual formulas) which satisfies the AGM postulates can be represented in terms of a spheres system or linear preorder of possible

10 10 Simone Duca and Hannes Leitgeb propositions, that is, if they were true or false at worlds in the same sense as a purely factual sentence A is, then they should be proper members of belief sets in the same sense as A, and hence the axioms of belief revision ought to apply to them just as much as they apply to purely factual sentences. But according to Levi, -A and.a differ from A in this respect; while all the standard AGM axioms of belief revision apply to the members of K, this is not so for the members of M(K). Indeed most of the proposals on how to avoid the paradox which one can find in the literature are interpreted best as suggestions for how to revise introspective acceptance sets which at the same time include sentences that express propositions (such as A) and sentences which do not (such as -A and.a). The postulates of revision that one is then permitted to apply to such acceptance sets are strictly weaker than the standard AGM axioms. Fuhrmann s own way out (Fuhrmann 1989, p. 132) also develops along the idea that Poss-closed belief sets do not obey the same laws as purely non-modal belief sets. In particular, he questions the validity of the Preservation principle (P) as governing revision of the former kind of sets. As a consequence, he proposes a new definition of K *A in terms of serious possibility, based on the equation K *A ¼ K - A where is the belief contraction operator. This provides a way of satisfying the decomposition principle according to which every legitimate belief revision is decomposable into a sequence of expansions and contractions. Another way out consists in rejecting (Poss), as stated above, as a valid criterion for including modal judgements in belief sets. Some other, more refined, criterion which may not lead to paradoxical results may well be available. Our own strategy does fall into this third category. However, we do not so much attack the original statement of (Poss) on the grounds of rejecting introspective belief sets themselves in any way, but rather we want to replace the original formulation of (Poss) by a syntactically more explicit version of it. Roughly, our general aim will be to show that one can refrain from banning modal judgements from belief sets (as Levi does) by simply indexing the corresponding modal operators in order to specify which belief sets they refer to (see Sect. 3.4 for the details). This way of defusing worlds, such that the least sphere or rank in any such semantic structure corresponds to the set of worlds in which all and only the members of the agent s current belief set are true.

11 How Serious Is the Paradox of Serious Possibility? 11 the paradox will allow us to stick to the standard understanding of, and postulates for, belief revision, without having to amend the belief revision operator in any way (as Fuhrmann does): in our theory, introspective statements will also be members of the belief set K, and the standard axioms of belief revision will apply to them just as they apply to purely factual statements. We do not even see any formal or structural similarity between our approach and these other approaches, since the formal properties of belief revision operators that apply to acceptance sets in the sense of these other authors differ from the formal properties that belief revision operators have in our theory; while in our case these formal properties are simply amongst the standard constraints on belief revision as one finds them in the traditional AGM theory, in these other theories the formal properties of belief revision differ from standard AGM. There are some precursors to the idea of indexing: in a probabilistic context, van Fraassen (1975) has suggested that the proposition that is expressed by a conditional depends on the probability measure that is used to evaluate that conditional, and Lindström (1996) makes a similar move in his qualitative theory of conditionals (we will return to Lindström s article at the end of our paper). While our theory will highlight the context sensitivity of introspective modalities just as van Fraassen (1975) and Lindström (1996) emphasized the context sensitivity of (epistemic) conditionals we will not leave their disambiguation to the metalinguistic specification of a context as van Fraassen (1975) and Lindström (1996) did for conditionals. Instead, we will disambiguate the doxastic modalities in question at the level of the object language, by attaching to each occurrence of any such modality an index that will make explicit to which belief set that modal operator occurrence refers The paradox rephrased probabilistically In this section we will prove our Theorem 2, a probabilistic counterpart of Theorem 1. Although this new result is analogous to the qualitative Theorem 1 indeed we will take care to mirror as closely as possible the assumptions and the proof of Theorem 1 when we 7 There is a reference to operator indices in Fuhrmann s paper as well. However, his usage of indexes is justified on the ground that one needs to relativize modal operators whenever more than one doxastic agent is considered at the same time. Instead we intend to specify an explicit relativization of modal operators to the belief states of one and only one agent.

12 12 Simone Duca and Hannes Leitgeb formulate the assumptions and the proof of the new theorem Theorem 2 is interesting in itself: for it shows that quantitative belief update in the standard Bayesian manner is threatened by considerations similar to the ones above if only the object languages on which subjective probability measures are defined are enabled to express statements about degrees of belief, and certain introspective statements about one s assignment of degrees of belief are themselves assumed to be assigned particular probabilities. This is important to note since sometimes it seems to be thought that subjective probability theory is capable of avoiding worries about introspective modalities that do seem to affect the theory of qualitative belief revision; see footnote 1 of Chalmers and Hájek 2007 for an illustrative example in the context of Ramsey-test conditionals with introspective antecedents or consequents. But Theorem 2 will show that probability theory does not necessarily fare better in this respect. Even more importantly for our own purposes in this paper, spelling out the proof of the paradox in probabilistic terms will highlight quite straightforwardly a problem that we claim was latent in the original qualitative paradox. The reason for this slight discrepancy between the qualitative and the quantitative version of the paradox in the eye of the beholder is that while in standard modal logic we are used to letting some metalinguistically specified context determine the meaning of modal operators so that. and -may quantify over different sets of possible worlds relative to different evaluation contexts (which are usually worlds again) this is typically not the case in probability theory. In a statement of the form Cr(A) = x, which says that A has a credence of x according to the credence function Cr, it had better be specified uniquely which credence function one refers to; and probability theorists normally do not burden some context, such as a metalinguistically described possible world, with the job of doing so. But we will argue that the Paradox of Serious Possibility is due precisely to taking occurrences of one and the same indexical introspective modality to stand for different belief states at the same time. This will thus become most obvious in the probabilistic context, even though an analogous observation will be seen later to apply also on the qualitative side. We will be ready to explain our diagnosis of the Paradox of Serious Possibility in full detail in section 3. But now for the probabilistic formulation of the paradox. First of all, we introduce a suitable way of translating certain key expressions

13 How Serious Is the Paradox of Serious Possibility? 13 from the qualitative domain into the probabilistic one. We state the manual and then explain it. 8 Translation manual: A 2 K ^ Pr(A) = 1 6 B 2 K *A 7 ^ 6Pr A (B) = 1 7, (where if Pr(A) > 0, Pr A (B) = Pr(B A) is the obvious choice for the probabilistically updated belief function Pr A ) 6.A 7 ^ 6 Cr(A) = A 7 ^ 6 Cr(A) > 0 7 The underlying idea of the manual is very simple: first of all, believing A is turned into a subjective probability of 1 being assigned to A. Secondly, belief revision corresponds to probabilistic conditionalization; the corresponding translation equivalence is also to be found in Gärdenfors 1986 when he considers the central Bayesian claim concerning belief revisions (p. 88). Note that we will only need to refer to conditionalization on propositions which themselves have some absolute probability greater than 0; nothing like conditionalization on zero sets as permitted by so-called primitive conditional probability measures or Popper functions will be required. Finally, introspective judgements are analysed probabilistically in terms of statements that say that some other statement is assigned a subjective probability of some kind. This is all quite straightforward. However, our usage of the metalinguistic probability or credence function sign Pr( ) vs. the object-linguistic probability or credence function sign Cr( ) needs further explanation: when we state 6 Pr(A) = 1 7, we make a metalinguistic claim, since we are talking about the agent s degree of belief regarding a certain proposition A; this corresponds to the metalinguistic claim 6 A 2 K 7 on the qualitative side. Now we also want to include statements such as 6 Pr(A) = 1 7 in the object language, that is, in the language to the members of which a probability measure such as Pr will assign quantitative degrees of belief. In order to avoid any confusion, we denote the probability function in question by a different symbol, that is, Cr( ), so that we can represent the original metalinguistic claim in the following 8 We use Quine corners when we refer to complex metalinguistic expressions, and we employ quotation marks when we speak about simple metalinguistic expressions. We do not use either of them when we refer to expressions in the object language.

14 14 Simone Duca and Hannes Leitgeb object-linguistic fashion: Cr(A) = 1. This corresponds then to the qualitative object-linguistic claim.a. In either case we refer to the same function, that is, if 6 Pr(A) = 1 7 is true, to the subjective probability function that assigns degree of belief 1 to proposition A; the difference merely lies in the linguistic level on which we refer to this function. Analogously, Cr(A) > 0 takes the place of the qualitative object-linguistic statement -A. 9 It is well known that bridge principles between qualitative and quantitative belief such as the ones expressed by our translation manual above are far from being unproblematic. For instance, it seems that one can believe that one will be home from a conference this evening, even when one is not certain of that, and hence even when one would not be willing to accept each and every possible bet on that proposition. But for the purposes of this paper the bridge principles above will prove to be sufficient in order to mimick the closure properties of qualitative introspective belief within the boundaries of elementary probability theory (and there are authors like Isaac Levi who in fact hold the bridge principles above to be true). Note that by the laws of probability, for every probability measure Pr, Pr(A) = 1 if and only if Pr( A) = 0, and Pr(A) > 0 if and only if Pr( A) < 1, which yields further equivalent ways of expressing belief and serious possibility in probabilistic terms ways which we will use in the following (as in the proof below) without further comment. One final remark on this: it is also well known that if probability measures are to be updated by introspective information, it is no longer clear whether the equation update = conditionalization can be sustained. Fortunately, this is not an issue in our context: the proof of theorem 1 does not rely on any revision by a statement involving. or -, and accordingly, the probabilistic version of the paradox does not rely on updating by statements involving Cr either. On the basis of our translation manual, we can now formulate probabilistic counterparts of our qualitative postulates from above. A ranges over sentences in a fixed object language with the same 9 Strictly speaking, things are actually slightly more complicated: syntactically, Pr is a metalinguistic variable for functions, that is, certain sets. And Cr will in fact not be regarded as a complete symbol in the object language at all. Instead, Cr( )=1 and Cr( )> 0 will be our proper logical symbols in the object language; if A is a formula of the object language, then both Cr(A) = 1 and Cr(A) > 0 will be formulas of the object language as well, just as.a and -A are formulas if A is. Any function referred to by Pr will assign numerical values to formulas, including formulas of the form Cr(A) = 1 and Cr(A) > 0. However, we will keep speaking loosely of Pr and Cr as function signs in the rest of this paper, for the sake of convenience.

15 How Serious Is the Paradox of Serious Possibility? 15 vocabulary and syntactic closure conditions as before, except that. gets replaced by Cr( )=1 and - by Cr( )> 0, as explained. Pr runs over a class of belief functions, which we assume to be probability measures except that we allow for one trivial belief function L which is defined by L(A) = 1 for all A, and which takes the place of the unique trivial inconsistent belief set L from before. By a quantitative belief-change model we mean any structure consisting of such a class of subjective probability measures and L, together with an update operation that maps any pair hpr, Ai of a belief function Pr in that class and a sentence A to the updated belief function Pr A in the very same class. In the current probabilistic context, it is actually rather clear what belief revision or update will have to be like, since one may stick to the standard Bayesian assumption that revision will be given by standard probabilistic conditionalization of a belief function Pr in light of some formula A (where we will only need to look at cases in which Pr(A) > 0), with the sole exception that the update of L on a consistent A has to be defined to be some probability measure in the given class (and hence differs from L). Indeed, if belief update is assumed to be defined by conditionalization, then some of the postulates below will simply follow from it. Once again, we will not make the reference to quantitative belief-change models explicit in any of our probabilistic postulates below. While we restrict ourselves to belief functions which are probability measures (apart from L), nothing even remotely like the full strength of the axioms of probability theory will be needed in order to derive the paradoxical result below. This corresponds to a remark that we made earlier when we explained our qualitative postulates: closure under logical consequence is not actually needed in order to obtain the paradox in its qualitative version. But just as it was handy to assume belief sets to be deductively closed, it is now convenient to suppose that our quantitative belief functions (except for one) satisfy the standard axioms of probability theory. These are thus the probabilistic translations of our qualitative postulates: (IC y ) 'Pr, 'A :0 < Pr(A) < 1 (P y ) ;Pr, ;A :Pr( A) < 1 T ;B[Pr(B) = 1 T Pr A (B) = 1] (Poss y ) (a) ;Pr, ;A :Pr(A) = 1 T Pr[Cr(A) = 1]=1 (b) ;Pr, ;A :Pr(A) < 1 T Pr[Cr(A) < 1]=1

16 16 Simone Duca and Hannes Leitgeb (S y ) ;Pr, ;A :Pr(A) > 0 T Pr A (A) = 1 (C y ) ;Pr, ;A :Pr ÞL, 6 A T Pr A ÞL (PC y ) ;Pr, ;A :Pr(Cr(A) = 1)=1, Pr(Cr(A) < 1)=1 T Pr = L (IC y ) should be unproblematic by now, however, a few of the other postulates may be in need of further clarification. Gärdenfors (1986) states a probabilistic version of (P) which is equivalent to our (P y ): (P y ) actually follows from the axioms of probability if Pr A is given by conditionalization (and (P y ) holds vacuously for L). Note that even if we axiomatized probability measures in terms of the standard theory of primitive conditional probability measures (Popper functions), we would still get (P y ) for free, if Pr A is defined through conditionalization. Finally, it is worth pointing out again that, in order for the paradox to go through, the variable A may be taken to range over all purely factual formulas (without occurrences of Cr expressions) in (P y ) as in all other postulates; but in ;B[Pr(B) = 1 T Pr A (B) = 1] the metavariable B needs to range over all formulas whatsoever, notably those that contain Cr somewhere. While (Poss y ) should be clear from our previous explanation of the translation manual and the role of Cr therein, our postulate (S y )is nothing but a slight weakening of the actual probabilistic translation of the qualitative postulate (S) : strictly, 6 ;Pr;A :Pr(A A) = 1 7 would encapsulate the proper translation, but we will not need the full strength of the actual translation, and the weaker (S y ) allows for conditional probabilities only to be defined in cases in which one conditionalizes on statements with non-zero probability, which is convenient if we like to define Pr A by standard conditionalization by means of the standard ratio formula. In any case, our (S y ) condition then follows again from the standard axioms of probability (and even the proper probabilistic translation of (S) would still follow from the theory of Popper functions). Finally, (C y ) and (PC y ) P for probabilistic (in-)consistency are the probabilistic counterparts of (C) and (DC): updating a subjective probability measure by a statement that is not logically false ought to lead to another probability measure and hence not to the trivial belief function L, 10 and no belief function except for L should 10 If conditional probabilities are given by the standard ratio formula, then this is not completely unproblematic, since there might be sentences which are not logically false but which still have probability 0, in which case standard conditionalization would not be defined

17 How Serious Is the Paradox of Serious Possibility? 17 assign a degree of belief of 1 to sentences Cr(A) = 1 and Cr(A) < 1 simultaneously. We are ready to state a probabilistic version of Theorem 1. Fuhrmann himself suggests that any operator that satisfies his definition of reflective modality would instantiate the Paradox of Serious Possibility. In fact, he claims that that would be the case also with some subjective probability operator (Fuhrmann 1989, p. 131). This is exactly what we have in mind when we formulate the following result. But we also want to show that even though Theorem 2 is a proper theorem some of the postulates on which it is based are flawed for almost trivial reasons which might become slightly clearer from the probabilistic version of the paradox than from its original qualitative variant. The problem that we want to address will be highlighted in the proof below. As we will see, once this flaw is patched up, the paradox cannot be reinstantiated. Theorem 2: No quantitative belief-change model can satisfy all of the postulates (IC y ), (P y ), (Poss y ), (S y ), (C y ), and (PC y ). Proof: In accordance with (IC y ), we consider a probability measure Pr which is not opinionated with respect to some sentence A. Then it follows: (1) Pr(A) < 1, from (IC y ) (2) Pr(A) > 0, from (IC y ) (3) Pr(Cr(A) < 1)=1, from (1) and (Poss-b y ) (4) ;B[Pr(B) = 1 T Pr A (B) = 1], from (2) and (P y ) (5) Pr A (Cr(A) < 1)=1, from (3) and (4) (6) Pr A (A) = 1, from (2) and (S y ) (7) Pr A ð CrðAÞ ¼1Þ ¼1, by (6) and (Poss-a y ) 11 (8) Pr A = L, from (5) and (7) with (PC y ) (while Popper-function conditionalization would be). (C y ) makes sure that even in such a case the result of the update operation will be some probability measure again. 11 This is the step where our worries regarding the proof, or rather regarding the assumptions on which it rests, show up most clearly (as highlighted by the box). In the next section, we will explain this in detail.

18 18 Simone Duca and Hannes Leitgeb (9) 6 A, from (2) (10) Pr ÞL, from (1) (11) Pr A ÞL, from (9), (10) and (C y ) (12) Contradiction! from (8) and (11). 3. Our diagnosis In the previous section, we stated the probabilistic counterpart of the original qualitative version of the Paradox of Serious Possibility. Though we proved a theorem, the conclusion of the theorem is contradictory, and indeed some of the postulates on which that conclusion was based, in particular (Poss), are not as plausible as they might have seemed at first glance. Indeed, when the proof above proceeds to step (7), things go awfully wrong, as far as the intended interpretation of the assumptions is concerned on which this step rests. We recapitulate the underlying reasoning: in line (3) the object linguistic function sign Cr is introduced. Obviously, Cr is used to represent on the object language level the belief function that is denoted by Pr in the metalanguage, for it is the fact that 6 Pr(A) < 1 7 which gets introspected and which thus yields 6 Pr(Cr(A) < 1)=1 7 by (Poss-b y ). In (5) Cr can still be taken to denote the same belief function Pr on the object language level, even though in the metalinguistic statement Pr has been replaced by a sign for the updated belief function Pr A via step (4). Nothing has gone wrong so far: Cr denotes Pr, it is just that the updated belief function Pr A is also capable of representing a (true) statement about the original function Pr, that is, it assigns 1 to Cr(A) < 1. (6) is perfectly fine, too. But now in (7), (Poss-a y ) is used to introspect the statement 6 Pr A (A) = 1 7 : when doing so and this is the problem that we want to address Cr is now taken to represent Pr A, since it is a statement about Pr A upon which (7) reflects. Hence, one and the same sign in the object language, that is, Cr, is meant to stand for two different belief functions, that is, Pr and Pr A, which are of course denoted by two distinct expressions in the metalanguage. No wonder this leads to a paradox: the classical fallacy of equivocation rears its ugly head again! This is our diagnosis of the paradox. Why is it that this problem becomes more apparent in the probabilistic version than in the qualitative version? In subjective

19 How Serious Is the Paradox of Serious Possibility? 19 probability theory, one deals with probability functions which assign numbers to statements; while it is possible to represent facts about probability functions within the very statements to which one intends to assign degrees of belief, it must be clear at any time which function sign denotes which measure. It is not part of the tradition of probability theory to leave it to some semantic context to determine which probability measure gets denoted by some indexical expression for belief functions; in fact, indexical names for probability measures are normally not used at all. However, since the great success of modal logic we have become used to expressing all sorts of modal claims in a contextual fashion: when we say.a (and accordingly for -A) whether this is meant to formalize it is necessary that A, it is believed that A, or something else we leave it to the semantic context to decide what the operator. refers to; so,.a is really short for: in this world, it is necessary that A, in this belief state, it is believed that A, and the like. In the standard semantics of modal logic, that is, possible worlds semantics, this does not lead to any trouble, for it is not the case that evaluations of.a relative to two different contexts ever get mixed up; every evaluation is relative to precisely one possible world, which represents a unique semantic context. But this mixing up of contexts is exactly what is going on in the paradoxical derivation above: within one and the same context, one sign in the object language is meant to stand for two different doxastic states. We maintain that every probability theorist will find this unacceptable in a context such as the one of Theorem 2. But then this should be equally unacceptable in the context of Theorem 1. Note that it is not the indexicality of the modal and probabilistic operators in our two derivations above just by itself that leads to the problem: it is the indexicality in conjunction with an epistemic context (that is, doxastic evaluations according to K *A and Pr A ) in which properties of two distinct belief states are referred at one and the same time. If introspective operators were used which referred to belief states by means of some non-indexical expressions such as proper names, then the problematic ambiguity would not arise. But the fatal equivocation could just as well be avoided if introspective operators referred to belief states on the basis of some sort of indexical expressions, as long as the agent managed not to use the same indexical expression for two different belief states. In the next section, we will reconstruct the paradox once again in qualitative and in probabilistic terms. But this time we will take care to specify the relevant belief state that any occurrence of one of our

20 20 Simone Duca and Hannes Leitgeb modal operators. and - or any occurrence of our probabilistic function sign Cr refers to. Once their reference has been disambiguated, the paradox dissolves. 4. The paradox exposed 4.1 Qualitatively Our aim is now to make the reference of modal operators explicit in the object language by stating in the object language to which belief set an occurrence of a modal operator in a formula refers. For instance, in the qualitative postulate (Poss-a), that is, 6 ;K, ;A :A2 K T.A 2 K 7, the operator. belongs to, or refers to, the belief set K, since A is assumed to be a member of K, and this gets reflected at the level of the object language. Therefore, one cannot jump from the belief state K to, say, the belief state K *A, without redefining the context of. accordingly. Formally, this can be achieved by relativizing the operator. to the belief set that it belongs to: for example, if K differs from K *A, we will distinguish the operator. K from the operator. K *A; just as K and 6 K *A 7 differ qua metalinguistic expressions, the two operators will differ qua object-linguistic symbols. The indices do not refer to the agent that entertains certain beliefs nor to the moment in time at which they are entertained. They will only specify to which belief set a particular modal operator occurrence refers. Later we will make this conception of indexed operators syntactically perfectly precise. There are different options here: for instance, we may want to allow for infinitely many indices representing one and the same belief set; this possibility should not come as a great surprise: for instance, if. K corresponds to being believed in K, then so should. K *(C C),. (K *(C C))*(C C), and so forth. Or we reserve one and only one syntactically primitive index for each belief set, and those will be all the indices that are employed. (In the first model in Sect. 5 we will follow the latter strategy, in the second model of that section we opt for the former strategy.) But for now we will presuppose a purely intuitive understanding of such indexed operators when we introduce the corresponding amended qualitative postulates that we will use in the following proof (attempt). They mirror the postulates given in section 1, but with the obvious difference that they specify the reference of any occurrence of a modal operator. The idea behind this is of course that the introduction of these indices in the postulates will