INTENTIONAL IDENTITY REVISITED

INT E NT IO NA L ID EN T IT Y RE VI SI T ED A H T I PI ETA R IN E N The problem of intentional identity, as originally oVered by Peter Geach, says that there can be an anaphoric link between an inde nite term and a pronoun across a sentential boundary and across propositional attitude contexts, where the actual existence of an individual for the inde nite term is not presupposed. In this paper, a semantic resolution to this elusive puzzle is suggested, based on a new quanti ed intensional logic and game-theoretic semantics (GTS) of imperfect information. This constellation leads to an expressive intensional language with a property of informational independence, argued to produce a purely semantic explication to intentional identity statements. One consequence is that various extra-logical and pragmatic factors become of secondary concern; it is possible to solve the puzzle by logico-semantic methods, albeit somewhat radically renewed ones.

1.

A P r o blem

Let us envisage the possibility that I can refer to what you think of without believing or thinking of anything particular about the objects of your thought. I just intend to refer to what you are referring to, although I do not have to have complete or even partial information about the contents of your thoughts, or any special access to your cerebration identifying the content of your beliefs. I can try to report this set-up by saying that you are thinking of some objects, and I am thinking of those objects too. Now if our thought-objects are said to refer to the same things, how does this come about in our semantics and in the meaning-analysis of our language? Geach (1967) introduced into the linguistic and philosophical discussion the following sentences. (1)

Hob thinks a witch has blighted Bob’s mare, and Nob wonders whether she (the same witch) has killed Cob’s sow (Geach 1967, p. 628).

For simplicity, we shall consider: (2)

Hob thinks that a witch has blighted Bob’s mare and Nob thinks that she (the same witch) has killed Cob’s sow.

Nordic Journal of Philosophical Logic, Vol. 6, No. 2, pp. 147–188. © 2001 Taylor & Francis.

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a h t i p i e t a r i ne n

It is evident that these two sentences, for our purpose at least, create the same problems. Analysing (2) and nding an accurate symbolization for it has posed various diYculties ever since its introduction. The core of the problem is what Geach coined intentional identity : some sentences of natural language seem to assert an identity between objects of thoughts of distinct agents, objects that are somehow meant to be focused to the same thing, as expressed by an indirect-speech clause and an anaphoric pronoun. Unfortunately, no formalization of such a notion of focus or anaphoric intention has been found satisfactory so far, for the following reasons. Binding: The sentence (2) creates a binding problem across the sentential boundary, because the inde nite term a witch and the anaphoric pronoun she reside within the scopes of distinct propositional attitudes Hob thinks that and Nob thinks that, namely attitudes located within separate sentences. Non-iteration: A proper reading of (2) should not entail that Nob thinks that the witch he has in mind blighted Bob’s mare, neither should it entail that Nob thinks that Hob thinks the witch blighted Bob’s mare. In other words, no excess thoughts about the witch’s harassing should occur, and so no iteration of attitude operators should be permitted. Non-speci city: The attitude verbs in (2) associated with a witch and she govern these terms such that an existentially quanti ed phrase remains within the scopes of the attitudinal verbs. From a logical point of view, the sentence needs to have non-speci c (or de dicto) attitude construction that is viable only when exhibiting a ‘‘narrow scope’’ organization of quanti ed phrases. Non-actuality: Since the object term in the rst conjunct is an inde nite description, the reading that Geach was concerned was one according to which no witches need to exist nor any speci c actual individual need to occupy either Hob or Nob’s mind. Putting these four conditions together, we get Geach’s preferred reading (or preferred understanding) of intentional identity. One can now try to nd suitable formalizations that would ful l these requirements using some rst-order intensional language. The possibilities that come to mind are presumably the following (the key—T H : ‘‘Hob thinks that’’; TN : ‘‘Nob thinks that’’; Wx : ‘‘x is a witch’’; Bx: ‘‘x blighted Bob’s mare’’; Kx: ‘‘x killed Cob’s sow’’): (3) Zx(W x 9 T H Bx 9 T N Kx ), (4) Zx(T H (Wx 9 Bx ) 9 T N Kx ), (5) T H Zx(W x 9 Bx 9 T N Kx ) ,

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T H Zx(W x 9 Bx ) 9 T N Zx(W x 9 Kx ) .

The problem with formulas (3) and (4) is obvious. There is an existential quanti cation Zx into the attitude contexts T H and T N which, in the usual semantics of intensional logics, amounts to a de re reading: the formulas would say that there exists some particular individual occupying Hob and Nob’s minds. It does not translate into inde nite description that is required by the preferred reading. In (5), the same problem recurs with respect to Nob’s thoughts, since although we have placed TH outside the binding scope of the existential quanti er, the quanti er would still quantify over Nob’s thought operator T N . One could try to argue a way out by saying that in (5), the referential pronoun she does not really require a narrow scope reading of the quanti ed phrase in the same way as the inde nite a witch does, and is merely a term that could be recovered from the context as meaning the witch who blighted Bob’s mare. This answer is unsatisfactory, however, because we do not get Geach’s preferred reading anymore. Namely, the non-iteration condition saying that the latter agent does not believe anything about the thoughts of the former agent is violated. Furthermore, the problem of unlicensed nesting would occur, because TN occurs within the scope of T H . It seems that the remaining possibility to overcome the scoping problem is to detach the conjunct into two separate blocks as is done in (6). However, no pretence makes the two quanti ed variables in (6) refer to the same individual (as required per any coherent reading of the sentence), and the variables could as well be renamed. But trying to modify the formula of renamed variables, presumably by adding an identity clause to account for the intentional assimilation of thought-objects, would not solve the problem of intentional identity either, since there would then be a free variable in the second conjunct: (7)

T H Zx (Wx 9 Bx ) 9 T N Zy (x 5 y 9 W y 9 Ky ).

This free variable x can be quanti ed away only by widening the binding scope of the quanti er Zx, amounting to (8)

T H Zx (Wx 9 Bx 9 T N Zy (x 5 y 9 W y 9 Ky )).

In (8), however, the ‘‘quanti cation into’’ the scope of T N is resumed, as well as the nesting of attitudes, and the original problems raise their ugly heads again. A proposal to analyse (2) was made by Geach himself (Geach 1967, pp. 630–631), who suggested that the pronoun could be understood not as a quanti cational object, but as what he calls a pronoun of laziness. Lazy pronouns act as mere substitutes for entire de nite descriptions, in

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this context the de nite description is the witch who blighted Bob’s mare. Sentence (2) would then be understood as: (9)

Hob thinks that a witch has blighted Bob’s mare, and Hob thinks that the witch who blighted Bob’s mare killed Cob’s sow.

Geach rejected the use of lazy pronouns as preserving the meaning of the preferred reading of the original sentence, however, because there is no trace of Nob’s thoughts about Bob’s mare in the original sentence. This is essentially what we attempted to do as well, when trying to rescue formula (5). Moreover, according to Geach, it does not help to say that Hob and Nob meant to refer to the same person, since the preferred reading does not imply Hob or Nob knowing whom or what they meant or had in mind when their state of mind was described by the original Hob–Nob report. Geach (1967, p. 630), apparently frustrated, noted that: .. . [(2)] raises [a] ... diYculty: a pronoun in one indirect-speech clause is on the face of it bound to a quanti er phrase in another such oblique context; the scope of the quanti ed phrase thus seems both to lie wholly within the earlier oblique context and to cover something in the later context. I cannot even sketch a structure of operators that would make good logical sense of this ... Similar sentiments are echoed in King 1993, p. 62: .. . there is no way to give the quanti er the scope required to bind the pronoun and get [Geach’s preferred reading].

Such a structure that Geach and others lacked will be developed, analysed, discussed, and its correctness argued for in this paper. To summarize, we need to accomplish two steps. Find an accurate logical symbolization for (2). If traditional rst-order intensional logic is not adequate, some alternative method needs to be motivated and defended. Explain, within such language, the fundamental puzzle of how the two attitudes can be focused toward the same object, although there does not need to be anything or anyone in particular in that focus. Before undertaking these, let us review some previous attempts put forward to tackle the puzzle. 2.

Some P revious Attempts

2.1. Backward-looking Operators Saarinen (1979) attempts to give an account of Geach’s puzzle using quanti ed modal semantics augmented with backward-looking operators. Saarinen uses backward-looking operators to make concrete the idea that

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although modal operators may syntactically subside within the scopes of other modal operators, they do not have to do so semantically. One can pick an individual in an accessible world and refer to it as a member of some previous world.1 Applied to Hob–Nob sentences, a backward-looking operator in front of TN will draw it from the scope of the previous operator TH , while the pronoun within the scope of T N remains bound by the preceding existential quanti ers. More concretely, (2) will become: (10)

T H Zx(W x 9 Bx 9 D T H T N (Wx 9 Kx )).

In semantic terms, the meaning of the backward-looking operator D T H in (10) is that the evaluation for the subformula T N (Wx 9 Kx) following DT H takes place in the world where the evaluation for the sentence tagged with the intensional operator TH (as indicated by the subscript in D) took place. Semantics for backward-looking operators can be given by games, prescribing when a play of the game will return to one of the earlier coordinates, where the evaluation will then proceed for the rest of the formula. These operators deliver interesting semantics, but as Saarinen notes, contra Geach’s understanding, they do not make Nob’s thought genuinely inde nite. The thought remains speci c (de re), the reason being that the interpretation generates a (at least perspective) cross-world identi cation over Nob’s thought-alternatives. This is because every object satisfying the predicates Wx and Bx in Hob’s thought-alternatives has to be an object satisfying Wx and Kx in any one of Nob’s thought-alternatives. Even though such objects are not cross-identi ed over Hob’s worlds, they are bound to be trans-worldly elements in Nob’s worlds. And as known from the usual conception of quanti ed intensional semantics, an identi cation process across a set of alternatives can be taken to amount to a version of de re reading of expressions. Saarinen goes on to oVer two further ‘‘perspectival de re’’ readings and two ‘‘descriptive de re’’ readings to overcome this problem at least partially. He concludes, however, that ‘‘there is no one reading which is the reading intended by Geach’’ (Saarinen 1979, p. 310). These further readings do not actually give rise to the necessity of backward-looking operators, but

1

Ideas similar to backward-looking behaviour can be found in temporal logics with operators such as ‘‘since’’, ‘‘until’’, ‘‘then’’ and ‘‘now’’. The latter two are closely related to actuality operators, also looking back to the worlds introduced earlier in discourse. They are customarily given semantics in terms of multi-dimensional modal logics.

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are nonetheless argued to hold, provided that one undoes the existential import of referentially interpreted existential quanti ers.2 In sum, Saarinen’s theory does not work, although the reasons for its failure are somewhat diVerent from the criticism levelled against it in the literature. Also, one can argue that since there does not appear to be traces of backward-looking operators in the syntax of natural language, their legitimacy in the object language becomes questionable. 2.2. Belief Objects Edelberg (1986) presents a ‘‘new puzzle of intentional identity’’, purported to modify intentional identity statements so that all attempts to give a quanti cational interpretation of them could be frustrated. In particular, the new puzzle is calculated to show that both Saarinen’s referentially interpreted quanti ers and substitutional quanti cation in Pendlebury 1982 are either implausible or at best seriously insuYcient. Edelberg’s new puzzle is easily stated, and it does not require the minute narratives given in Edelberg 1986 in order to see what is going on. The goal is to create a background situation in which the original Hob–Nob sentence can be false but where a version with inversed conjuncts becomes true. Let us call the sentence related to Edelberg’s new puzzle the Nob–Hob (not: Hob–Nob!) sentence and a story behind it the Nob–Hob situation (this is a modi cation of Edelberg’s mayor–commissioner example, Edelberg 1986, pp. 13–17). The Nob–Hob sentence is: (11)

Nob thinks that a witch has killed Cob’s sow, and Hob thinks that she has blighted Bob’s mare.

The Nob–Hob situation would be this. Hob, for some reason or another, knows something has happened to Cob’s sow. He also thinks, as the Nob–Hob sentence indeed suggests, that the same witch who did something to Cob’s sow, blighted Bob’s mare (no lazy pronouns here, however). Nob, on the other hand, only thinks that a witch has killed Cob’s sow. Should he know anything about what happened to Bob’s mare, he would make no connection between the two events. Given this relatively unwrinkled background situation, Edelberg argues that the original Hob–Nob sentence is false in it, while the inversed (11) comes out as true. This is because in (11) Hob is thinking that it must be the same witch behind the two events (whoever she is), whereas Nob is taken to be completely ignorant of the latter incident. In (2), however, it 2

Saarinen’s proposed referential method of interpreting quanti ers can also be contrasted with Pendlebury 1982, where it is suggested that quanti ers in intentional identity statements are to be interpreted substitutionally rather than referentially.


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is Hob who does not have any thoughts about possibly existing links between the two events. In particular, he does not have any information about the status of Cob’s sow. Therefore, given this background situation, the Hob–Nob and Nob–Hob sentences have diVerent truth-values, and hence the conjunction behaves non-commutatively. The rst thing to note is that independently of whether the proposed ‘‘new puzzle’’ is taken to be valid or not, it is not likely that this puzzle has enough power to imply that all quanti cational interpretations of intentional identities become fallacious, contrary to what Edelberg purports to have shown. What does follow is that conjunction in intentional identity statements may not always be commutative. However, Edelberg thinks that the unmanageableness of the resulting logic speaks against such a property (Edelberg 1986, p. 14). Yet, the commutation properties of sentential connectives are well-studied and common features in a number of logics, let alone natural language. Furthermore, contrary to what Edelberg and some others have claimed, this puzzle is not the primary reason why Saarinen’s theory of backwardlooking operators turns out to be unviable. The problem with backwardlooking operator theory is its commitment to invoke some method of cross-world identi cation with respect to Nob’s alternatives, which then directly runs counter to the desired understanding of intentional identities. Quite apart from the various implications of the puzzle, one can also raise a doubt about its initial viability. For suppose, now for the sake of the argument, that one can establish, after some stage-setting, that there are reasons to take the commutativity of conjunction in intentional identity statements as problematic, reasons strong enough to make the Hob–Nob sentence false and the Nob–Hob one true. Then it is not diYcult to see that similar problems may undermine extensional sentences involving anaphora as well, making one extensional sentence to have diVerent truthconditions from its commuted part, while preserving the same meaning. For consider simply: ‘‘A witch has blighted Bob’s mare and she has killed Cob’s sow’’ versus ‘‘A witch has killed Cob’s sow and she has blighted Bob’s mare’’. Re ect this against the stage-setting situation ‘‘Horse-Hecate killed Cob’s sow and blighted Bob’s mare, and Dog-Hecate killed Cob’s sow but didn’t blight Bob’s mare’’. Now how does it follow that such a phenomenon in extensional sentences can also count against the behaviour of conjunction in intentional identities, and especially against the status of quanti cation, any quanti cation, in such identities? Nonetheless, Edelberg takes the new puzzle to have demonstrated that pronouns in intentional identity statements cannot behave quanti cationally, and intends to dispense with quanti cation by resorting to the notions of belief objects and counterparts of belief objects. He takes these

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objects to serve as proper referents of inde nite descriptions. Thus a witch would have a referent a standing for Hob’s belief object, and the anaphoric pronoun she would have a referent b standing for Nob’s belief object and the counterpart of a. The speci c counterpart relation is taken to be denoted by a # b. It follows that the Hob–Nob sentence becomes: (12) ZaZb ((T H W a 9 B a) 9 (T N W b 9 K b ) 9 a # b). This approach is not unproblematic, however. First, why is it that in (12), the second existential quanti er has wide scope over the rst attitude operator and not only over the second one, as one would expect from the symmetric nature of sentences? Second, Edelberg does not really explain how the counterpart relation works, or what the nature and behaviour of belief objects is. According to him, the concept used in assessing counterpart relation pertains to the ‘‘rough similarity of explanatory role’’ (Edelberg 1992, p. 576), which is taken to mean that belief objects are counterparts if they are introduced to explain the counterpart data in roughly the same way. Indeed, the range of such an obscure condition is necessarily limited. The ensuing perspectival semantics (as diVering from a more realist concept of intensional semantics) is further developed in Edelberg 1995. In perspectivalist semantics, sentences are evaluated relative to certain contextual parameters, whereas a realist account tries to dispense with such notions. Perspectivalism brings in pragmatic factors, including speaker’s reference to objects and speaker’s meaning in general, in distinction to a more objective quanti cational and semantic uses of singular inde nites. Unfortunately, attempts that try to bring pragmatic components in tend to confuse the puzzle rather than solve or even clarify it, at least if one is, as we are, interested in the precise semantic conditions when the original reading of intentional identities come out as true and when they come out as false. 2.3. Formal Pragmatics Pragmatic aspects of anaphoric references in attitude contexts are key elements in van Rooy 2000, van Rooy and Zimmermann 1996 and Dekker and van Rooy 1998. These papers address some of the questions raised by Edelberg’s theory. According to van Rooy and Zimmermann (1996) in particular, Hob and Nob’s beliefs are about a single thing external to both believers. Therefore the reading of (2) evokes a de re or wide scope belief ascription after all, but the content of the anaphoric pronoun is diVerent from what could be expected from traditional modal semantics. In these approaches, beliefs have their origin in the same object, but this object may be an external event. Belief objects of individual agents are then


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related to this unique external event. What explains the sameness of the belief relation in diVerent points of evaluation is that there is an additional property ‘‘being a cause of’’, taken to be a function from ‘‘thinking of an event e’’ to the ‘‘object which caused the event e’’ (see van Rooy and Zimmermann 1996, p. 134). Formally, what is being proposed is that the Hob–Nob sentence would really be: (13) ZePe 9 T H (Zy ( Cause( y, e) 9 W y) 9 By ) 9 T N Ky . In this account, some extra-linguistic contextual ‘‘discourse structures’’ such as properties of being an event (ZePe) are crucial. We are thus dealing with formal pragmatics, making extensive use of the resources of discourse referents introduced in the discourse that embeds intentional identities. The counterpart relation similar to that of Edelberg’s is now explained by using the idea of discourse referents: if two belief objects emerge from the same source, namely if there is a factual link from a ‘‘de re event’’ or from a ‘‘thought of a de re event’’ to an actual object (the witch), then the two belief objects are counterparts of each other. These ideas, although drawing in unnecessarily complicated extralogical material, might be worth pursuing further if the key question is about how Hob and Nob came to believe what they do. But that is not the question Geach asked. He asked about the logical structure of operators that would make good logical sense of the sentence. Useful pragmatic analyses may enter when at least some semantic questions are detained (of course semantics and pragmatics can still be interrelated). Further, the proposed formalization does not seem to hold water. In (13), the last two occurrences of the variable y are free. If they are tried to be quanti ed away, the scoping problems, namely the speci city and iteration, raise their heads again. This might be an oversight, but an explanation is needed utcumque ceciderit. Even though this technical defect could be met, there is a further problem. For how would one now plausibly paraphrase (13)? It appears to say the following. (14)

As regards some event, Hob thinks that a witch who blighted Bob’s mare caused the event, and Nob thinks that she (the same witch) killed Cob’s sow who caused the event.

But if so, one could object that this reading implies that there exists some one event both Hob and Nob have in mind that the suspected witch generated. Consequently, this witch-generating event being the same, together with the alleged existence of a factual link between the event and the object, produce the reading which says that there now exists some one person or a thing Hob and Nob had in mind. But this is precisely what

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they do not have according to Geach’s original preferred understanding, since Hob and Nob may just think that there was a witch faYng about, and that their suspicions, as Geach puts it, ‘‘had not yet settled on a particular person’’ (Geach 1967, p. 629). The later works of Dekker and van Rooy (1998) and van Rooy (2000) aim at developing further on some of the pragmatic aspects of intentional identities, relating properties of belief objects that are introduced by asserting Hob–Nob sentences to discourse objects that have been studied in relation to other cross-speaker anaphora in general. These papers do not address the de ciencies identi ed above, however. In particular, in all of these papers, Edelberg’s new puzzle is taken to be ‘‘intuitively acceptable’’ (van Rooy 2000, p. 160), although the reasons for having such intuitions are not spelt out. But then these theories fail to use the puzzle for Edelberg’s purposes, namely to criticise Saarinen’s theory of backwardlooking operators. In addition, van Rooy (2000) argues that the pragmatic theories use the notion of common ground, which means that the belief objects of diVerent agents can be counterparts with each other if ‘‘two agents have communicated with each other about a certain object, and for that reason parts of what they believe have a common ground’’ (van Rooy 2000, p. 167). The common ground is thus shared because of the presence of communicative channels, creating a shared belief object for Hob and for Nob. As far as Hob–Nob sentences are concerned, however, there is no such communication. If some means of communication are created in the background narrative or even explicitly in the sentence itself, we are not dealing with the original puzzle, but with some other that lets pragmatics in without genuinely attempting to analyse and understand Geach’s reading. 2.4. Generalized Schemes A diVerent but related theory is given in King 1993, which develops a theory of context-dependent quanti ers and suggests that these quanti ers can deal with anaphoric references within attitude contexts. King’s idea is that anaphoric pronouns behave like context-dependent quanti ers, which are quanti ers whose main features are extracted from antecedent quanti er phrases (and so this idea puts these quanti ers roughly into the same category with discourse referential quanti ers). There are three main purposes for the theory of context-dependent quanti ers to accomplish: (1) to determine whether the pronoun has universal or existential force, (2) to determine the relative scope of the pronoun and (3) to determine the range of the contextually dependent pronoun. The interplay of intentional operators and antecedent quanti ers is taken into account by a method of checking whether the antecedent takes wide or


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narrow scope with respect to the intensional operator. There is some evident ambiguity in determining scopes of these contextually dependent quanti ers, but King sidesteps such ambiguity by assuming that sometimes the non-linguistic context of utterance can determine quanti er scope. As applied to the Hob–Nob sentence, the theory comes to this. First, one forks the sentence into two discourse constituents. (a) (b)

Hob thinks that a witch has blighted Bob’s mare. Nob thinks that she killed Cob’s sow.

The pronoun she in (b) functions as a context-dependent quanti er. The indeterminate a witch, not denoting anyone in particular, says that the quanti er phrase associated with it takes narrow scope with respect to the intensional operator Hob thinks that. Hence, according to the principle of scope determination, the context-dependent quanti er associated with she must also behave similarly and take narrow scope with respect to the associated operator Nob thinks that. The symbolization of the latter sentence, under this prediction, becomes: (15)

T N Zx (Wx 9 Bx 9 Kx ).

However, an immediate rejoinder to this is that against the preferred reading sentence (15) plainly is false, since Nob is not supposed to have any thoughts about Bob’s mare, and hence the sentence does not capture Geach’s preferred reading rejecting lazy pronouns. King’s attempted answer is that it should not come as a surprise that there are readings of natural language sentences that can be both true and false, once the situation behind them is spelled out in detail. This much may be true in natural language, but it does not answer why King permits the use of lazy pronouns. As to the new puzzle of intentional identity, King takes it that against a certain background story the Hob–Nob sentence can be judged ‘‘to be true in situations in which they are, strictly speaking, false’’ (King 1993, p. 77). Now there is some truth in this remark, especially since in belief or thought-contexts, people often are committed to opposing or inconsistent propositions. The entire theory of modal neighbourhood semantics can be seen as a development to deal with this, and there is a whole range of various modal semantics to deal with inconsistent belief sentences. But if one reconstructs intentional identity as saying something about knowledge or some other attitudes that would express certainties, there is less credence to the claim that some readings of epistemic intentional identity sentences can simply be both true and false. Without going into the details of such examples here, suYce it to mention that by means of game-theoretic semantics (GTS) to be presented

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below one can virtually see how non-coherent sentences arise, when the class of games comprises of non-strictly competitive activities (see Pietarinen 2000). Then it is evident that in such non-strictly competitive games, the background situation can operate in favour of both players, that is, both players can rely on various strategic winning resources in their attempt to uncover the truth-value of the sentence (while a form of ex falso, expressing assertions about games, does indeed obtain). One of the virtues of King’s approach is that it introduces new generalized intentional identity sentences, namely sentences involving also universally quanti ed phrases and complex combinations of quanti ed phrases. According to King, a generalized intentional identity sentence is of the form: (16)

T1 S 1 9 T2 S 2 , where S1 contains quanti ed phrases and S2 contains at least one pronoun anaphoric to some quanti er in S1 . One of King’s examples of generalized schemes in English is: (17)

Winston believes that every successful movie1 stars a famous female actor2 and Emmett believes that she2 is responsible for its1 success.

(The co-indexing is used to illustrate the desired anaphoric links.) These generalized sentences need to recount similar problems as the original Hob– Nob sentence, but naturally they introduce some further complications. An example of such a complication is that since in generalized sentences one is not dealing exclusively with inde nites, the treatment of intentional identity as undoing the existential import of the existential quanti cation might not work in sentences involving also universal phrases (like (17)). Such existential assumptions can enter semantics in the form of a special existence predicate Ex, expressing quanti cation over actual individuals and not just possible ones.3 Existential quanti ers with existential presuppositions would then be read as Zx(Ex 9 ...), and the de dicto versus de re distinction would dissolve into the matter of existence. More precisely, both attitude attributions can receive a wide scope interpretation, but the de re cases have to be augmented with existential presuppositions, whereas for de dicto or non-speci c sentences such presuppositions are not needed. With existence predicates, one can try to represent Hob–Nob sentences by a wide scope quanti cation without falling prey to an outright de re interpretation. That is, the lack of an existential predicate indicates that existential quanti cation does not commit one to the actual existence of 3 In modal logic, existence predicates are used in simulating the varying domains semantics and actual individuals when the semantics at hand in fact has common domains.


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an individual or an individual concept, and so a version of de dicto interpretation can remain to be denoted as a wide scope existential sequence. Hob–Nob sentences would then be of the form Zx(T H (Wx 9 Bx ) 9 T N Kx ).

So just adding Ex would now make this sentence to have speci cally denoting existential quanti ers. But this works only for simple sentences, not generalized ones. Oksanen (2000) has developed a theory of intentional identity using the resources of GTS. In his proposal, one deals with anaphoric reference in attitude contexts by resorting to the notions of subgames and choice sets relativized to intensional contexts (see also Sandu 1997). There is a multitude of other work relevant to intentional identities. An early attempt by Dennett (1968) doubts the coherence of Geach’s readings in the rst place, and so does Lycan (1994, pp. 124–129). McKinsey (1984) is also sceptical about the intended understanding of these sentences, arguing for certain types of lazy pronouns in cases where sentences might introduce descriptive mental references (this work seems to have anticipated discourse representation theories). For other studies, see Asher 1987, Castan˜eda 1974, Crimmins 1998, Hazen 1995, Ranta 1994, pp. 157–160, Roberts 1996, Slater 1988, 2000, Zimmermann 1999. An interesting historical exposition of Saint Thomas Aquinas’ (1225–1274) use of intentional identity can be found in Klima 2000. 3.

Generalizing Generalized Sentences

Before turning to our analysis of intentional identities, it is useful to investigate some further variations and generalizations to the basic Geachtype intentional identity. The purpose here is not to try to solve the problems related to the semantic account of these new generalized sentences but just to describe them; the sentences are interesting and important to be analysed in some other occasion. Such analyses often need new tools such as GTS that is extended to cover generalized quanti ers (see Pietarinen 2001), and they occasionally need to address diYcult questions concerning plural anaphora in attitude contexts, as well as various interplays between anaphora and negation. King considered strings with nested determiners and pronouns, where all the determiners are either existential or universal quanti ers. A natural generalization of this idea would now expand to two dimensions: we would want to have other determiners such as generalized quanti ers, and we would want to have more than two agents, that is, more than two distinct attitudes being directed toward one or many objects.

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The rst dimension generates sentences such as (18)–(20): (18) (19) (20)

Hob thinks that more than three witches blighted Bob’s mare, and Nob thinks that they killed Cob’s sow. Hob thinks that at most three witches blighted Bob’s mare, and Nob thinks that they killed Cob’s sow. Hob thinks that few witches blighted Bob’s mare, and Nob thinks that they killed Cob’s sow.

These sentences pose the same problems Geach was concerned with, but introduce some additional ones. In (18), there is a maximal reference anaphora they which refers to the monotonically increasing determiner more than three; it thus refers to all witches even though there were more than three of them. Likewise, the pronoun they in sentence (19) refers to the monotonically decreasing determiner at least three, but this time it behaves as a minimal reference anaphora. In (20), few is a monotonically decreasing determiner, and anaphoric reference is legitimate because the plural pronoun can indeed obtain a minimal reference. It is conceivable to have generalized quanti ers also in the plural pronoun: (21)

Hob thinks that many witches were blighting Bob’s mare, and Nob thinks that most of them were killing Cob’s sow.

But consider: (22)

*Hob thinks that no witch blighted Bob’s mare, and Nob thinks that she killed Cob’s sow.

Here the anaphoric pronoun lacks a reference (is not coreferential with its ‘‘head’’), although the sentence contains a perfectly legitimate monotonically decreasing determiner no. For a diVerent example, consider the following: (23)

?Hob thinks that only a witch blighted Bob’s mare, and Nob thinks that she killed Cob’s sow.

Here the descriptive content of two belief clauses is diVerent. In the antecedent clause, Hob’s belief is focused toward the operator only rather than the (inde nite) object, whereas in the consequent, Nob’s belief is about the object of the pronoun. It is diYcult to say, given only this much of information, whether Nob has a belief that ‘‘only she killed Cob’s sow’’, because only combines with the particular believer or an attitude of an agent rather than with the object. Indeed, the item only is customarily not


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to be regarded as a determiner but rather as a noun phrase modi er. Other modi ers can also be envisaged, such as even.4 An analysis is also needed for adverbials inside attitude phrases:5 (24)

Hob thinks that a witch usually blights Bob’s mare, and Nob thinks that she sometimes kills Bob’s mare. (25) Hob thinks that a witch usually blights Bob’s mare, and Nob thinks that she always kills Bob’s mare. The second dimension adds to the stock of intentional identities complex sentences: (26)

Hob thinks that a witch has blighted Bob’s mare, Nob thinks that she has killed Cob’s sow, and Rob thinks that she has stolen Mob’s heifer.

Following the simple notation suggested by King, the generalized notion of intentional identity, therefore, can be in the form of the following scheme: (27)

T1 S1 9 · ·· 9 Tn Sn . Here Si , i < n contain determiners, and S2 ... Sn contain a pronoun anaphoric to S1 (of course, there may be other anaphoric pronouns in S2 . .. Sn referring to other determiners in S1 ... Sn 1 ). Õ Another generalization to this dimension would iterate attitudes: (28)

Hob thinks that Rob thinks that a witch has blighted Bob’s mare, Nob thinks that Rob thinks that she has killed Cob’s sow, and Rob thinks that she has stolen Mob’s heifer.

In general: (29)

T1 ... T1 k S 1 9 · · · 9 T n 1 . . . T n k S n k . Let us note that also certain simpli cations of original Geach sentences are plausible: (30)

Hob thinks that a witch has blighted Bob’s mare, and he thinks that she has killed Cob’s sow.

However, such single-agent sentences introduce diVerent problems. When we have only one agent, there is no multi-modal cross-world identity to be accounted for. What we need to explain is the behaviour of modal subordination of the pronoun in the secondary subordinate intensional 4

See Pietarinen 2001 for an analysis of even and not even in GTS. Again, Pietarinen (2001) suggests some new game rules for aspectual and other adverbs and various temporal constructions. 5

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context and the interpretation of predicates in both subordinate and antecedent contexts, which are of course contexts of the same agent. It is not truthful to refer to this problem also as the puzzle of intentional identity. In this and similar cases, there is no principled hindrance in considering the pronoun she as a lazy pronoun going proxy for the expression the witch who has blighted Bob’s mare, as is seen to happen in the following sentence: (31)

Hob thinks that a witch has blighted Bob’s mare, and he thinks that the witch who has blighted Bob’s mare has killed Cob’s sow.

For certainly if Hob thinks something about a witch he can, later on, think something else about this witch, his earlier thoughts notwithstanding. As a nal suggestion, the attitude verb can be changed to volitional, doxastic or deontic ones, for instance. In addition, many objectual attitudes (hate, love, hope, expectation, attempt etc.) are conceivable. (32)

Hob is obliged to bring a witch to the court and Nob is permitted to cross-examine her.

Although the generalizations and various modi cations of attitude reports are worth of closer scrutiny, for the sake of keeping matters manageable we shall in the following pages deal almost exclusively with the original two-agent Hob–Nob sentence and Geach’s preferred reading. 4.

Intentional Identity as an Independence P henomenon

The foremost question concerns the possibility of analysing intentional identities in a purely logical and semantic manner, without any need to resort to extra-logical belief objects, discourse elements and discourse referents, or context-dependent quanti ers of similar origin. The answer will be positive. Intentional identities can indeed be so analysed, but it is not possible to accomplish this in traditional modal predicate logic. There are some serious doubts that this can be accomplished even in theories of discourse or other contextually permeated semantics. For one needs to go further beyond the ‘‘Frege–Russell boundary’’ in one fell swoop. This is because the inherent feature in intentional identities (and in other modal anaphora) is the phenomenon of ‘‘quantifying out’’, which roughly means that quanti ers inside modal operators bind variables outside those operators. To adequately cope with this ‘‘quantifying out’’ phenomenon, we need to introduce a new expressive modal language which facilitates the property of informational independence.


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4.1. Language and Semantics Let the syntax of the language L of rst-order intensional logic with vv identity consist of a signature t, a logical vocabulary, and rules for building up formulas. Well-formed formulas (wVs) of L are de ned for each t vv as: Q:: 5 P |T i Q | Yx Q |Zx Q | Q 8 y | "Q | x .

y.

The operator T i denotes a modality which will be read as ‘‘an agent i thinks (that) Q’’, and . is an identity sign. Let Qy, Q × { Yxj , Zyj , T i } be an L -formula in the scope of A 5 vv {T1 ... Tn , Yxk , Zyk } for nite integers i, j, k, n, j Þ k . Then L * consists of vv wVs of L together with vv if B k A, then (Q /B ) y is an L * -formula, Q 1 B. vv * A m odel M for L is a tuple 7 U, F 8 , where U is a t-structure vv 7 W , D j 8 of a signature t of a non-empty set of possible worlds W and v a non-empty world-relative domain D w j . F is an interpretation 7 p, R 8 , where p attaches to each possible world a t-structure together with a valuation from terms to the domain. Hence P p(w i ) means that the predicate P is interpreted in the world w i . The terms ‘‘world’’, ‘‘state’’ and ‘‘alternative’’ are used interchangeably. R 5 { r1 . . . rn } is a set of epistemic accessibility (alternativeness) relations for each agent i 5 1 . . . n, ri k W Ö W . Since the language is multi-modal, a distinct accessibility relation is needed for each distinct modality. If there is only one agent, we can drop the subscript. Let us write w 1 × [w 0 ] i to r express that w 1 is i -accessible from w 0 . No extra conditions are imposed on ri . The interpretation function does not itself provide a way of interpreting individual variables. Therefore we need a valuation g 5 X D w i , which assigns individual variables in X to elements in the domains of wi . Let us extend this valuation to g p(w i ) : T S w i which maps terms to the domains of individuals such that every n -ary function symbol f n has a denotation of an n-ary ( possible partial) operation in a domain D w i , every variable x × X has a denotation g p(w i ) (x ) 5 x p(w i ) , every constant symbol c × C has a denotation g p(w i ) (c) 5 c p(w i ) , and every m-ary predicate symbol in P has a denotation of an m-ary predicate (or relation) in a domain D w i . Every term t 5 f(t 1 .. . tm ) can now be recursively de ned as ) ) ) g p(w i (t ) 5 f p(w i ( g p(w i (t1 ) . . . g p(w i ) (t m )). In addition to a relational t-structure, p now attaches to each possible world a valuation g p(w i ) 5T D w i . The models of L * can be depicted as labelled graphs with a set of vv i nodes W , a pointed arrow over W for each i, and a function g assigning to each atomic P ×W a subset of W .

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We go on to enrich the semantics with a nite number of world lines in a modal model M, extending the valuation g to a ( partial) mapping from worlds to individuals, i.e. to g 5 X D W w i , such that if w ×W and g is a world line, then g (w) × D w . World lines amount to the case where individuals have only local aspects or manifestations of themselves in any particular world. Two world lines may also meet at some world, and then part company. The range of possible methods of cross-identi cation giving rise to world lines includes physically oriented modes (public identi cation) as well as perspectivally oriented modes ( personal identi cation), and can explicitly be denoted to a contextual parameter subscripted to truth-conditions. Such cross-identi cation methods have eVects upon how world lines are being drawn in a model. For instance, perspectivally identi ed individuals are relative to an agent and (causally) to the actual world, while physical ones can coincide with what are known as rigid designators, individuals that are identical in all worlds. For simplicity, such contextuality will be, albeit important, ignored in our formalism.6 The interpretation of the equality sign . (identifying functional) is (M, w 0 , g ) Y x . y iV for some w i , w j × W , Z f Zh such that f (w i ) 5 h(w j ) .

That is, two individuals are identical iV there are world lines f and h that pick the same individuals in w i and in w j . World lines can meet at some world but then pick diVerent individuals in other worlds; the two-place identifying functional operation spells out when they meet. Individuals within a domain of a possible world are local and need to be cross-identi ed in order to be global and speci c. In case there are independent submodels, the atomic formulas will be evaluated against pairs of worlds, or in general against n-tuples of worlds corresponding to each starting point. In the latter case we evaluate formulas in (M, (w 10 , w 20 . . . w n0 ), g ). We will be concerned solely with two agents and so these generalizations will be ignored. 6

The fact that the identi cation methods can in extreme cases be quite curious is shown by one of Oliver Sacks’ patients who, as reported in The Man Who Mistook His Wife for a Hat (Macmillan, London, 1986), when given a rose, could not identify it, describing it as ‘‘.. . about six inches in length . . . A convoluted red form with a linear green attachment’’. The patient’s identi cation was completed after his smelling the object. In the subsequent situation, the same patient described a glove as ‘‘a continuous surface . .. infolded on itself. It appears to have . . . ve outpouchings, if this is the word’’. The glove was nally identi ed when the patient put it on and tried to use it: ‘‘My god, it’s a glove!’’.


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4.2. Intensional Game-Theoretic Semantics An eYcient general approach to game-theoretic actions in logic is to use extensive form game representation. In general, extensive form games capture the sequential structure of players’ strategic decision problems. These games can be represented as ( nite) tree structures, with decision nodes and actions labelling the edges departing from the decision nodes, representing the possible moves made by the players. The game starts at the root of the tree and ends at the terminal nodes. At each non-terminal node or a decision point, a respective player has to make a decision what to choose. The outcome of this decision in a particular play of the game is a choice, and in the game as a whole a move, assigned to the outgoing edges of non-terminal nodes. The following subsections elaborate on the treatment of GTS given in Hintikka and Sandu 1997, for example. 7 4.2.1. Extensive Form Games Perfect information. Let us x a family of actions a, where a nite sequence i n i a 7 8 i = 1 , n × v represents the consecutive actions of players N, a × A . There are no chance moves. An extensive game G of perfect information is a ve-tuple G A 5 7 H , Z, P, N , (u i ) i × N 8 such that H is a set of nite sequences of actions h 5 7 a i 8 ni 1 from A, called game = histories. We require that: – the empty sequence 7 8 is in H ; – if h × H , then any initial segment of h is in H too, i.e. if h 5 7 a i 8 ni 1 × H then pr (h) 5 7 a i 8 ni Õ 11 × H for all n, where pr (h) is the = = immediate predecessor of h ( 5 Æ for h 5 Æ ). Z is a set of maximal histories (complete plays) of the game. If a history h 5 7 a i 8 ni 1 × H can continue as h ¾ 5 7 a i 8 ni +11 × H , h is a non-terminal = = history and a n + 1 × A is a non-terminal element, otherwise they are terminal. Any h × Z is terminal. P 5 H c Z N is the player function which assigns to every non-terminal history a player i × N whose turn is to move. Each u i , i × N is the pay-oÚ function, that is, a function which speci es for each maximal history the pay-oV for player i. For any non-terminal history h × H we de ne A(h ) 5

{x × A | h Ç

x × H }.

7 Extensive forms were rst formulated in von Neumann 1928, although the presentation in Kuhn 1953 has become commonplace.

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A (pure) strategy for a player i is any function fi 5 P Õ

1

({x})

A

such that f i (h) × A(h) , where P Õ 1 ({x}) is the set of all histories where player i is to move. A strategy speci es an action also for histories that may never be reached. A strictly competitive game is a particular case of a game de ned as above, in which N 5 {V, F } and in addition: u V (h) 5

Õ u F (h) ; either u V (h) 5 1 or u V (h) 5 Õ 1 (that is, V either wins or loses); either u F (h) 5 1 or u F (h) 5 Õ 1 (that is, F either wins or loses); for all terminal histories h × Z . Imperfect information. To represent imperfect information by means of extensive form games, let us extend G A to a six-tuple G *A 5 7 H , Z, P, N , L, (u i ) i N , ( Ji ) i N 8 with an additional component × × (Ji ) i N . (Ji )i N is an information partition of P Õ 1 ({x}) (the set of histories × × where i moves) such that for all h , h ¾ × S ji , h Ç x × H if and only if h ¾ Ç x × H , x × A, j 5 1 . . . m , i 5 1 . . . k, m < k. S ji is an information set S ji × ( Ji )i N . × Games are exactly as before, except that now players might not encompass all the information concerning the past. This is achieved by imposing information partition on extensive games, which partitions histories into information sets (equivalence classes). Those histories that belong to the same information set are indistinguishable to players, and thus a player may not be informed which is the actual history that has been played. In imperfect information games, the strategy function is required to be uniform on indistinguishable histories: f (h) 5

f (h ¾ ) for any equivalent histories h, h ¾ × H c Z .

The uniformity property thus is: if h, h ¾ × S ji × ( Ji )i

×N

then f i (h) 5 fi (h ¾ ), for i × N .

4.2.2. Extensive Form Semantic Games Perfect information. Let ‘‘Y + ’’ mean a positive logical consequence (a formula being true in a model), and ‘‘YÕ ’’ mean a negative logical consequence (a formula being false in a model). Let Sub(Q) denote a set of subformulas of Q. An extensive form semantic game G ( Q, g, U) associated with an L -formula Q is exactly like G , except that it has one extra vv element: a labelling function L : H Sub(Q) such that L(7 8 )5

Q (the root); for every terminal history h × Z, L (h) is a literal.


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In addition, the components H , L, P, u V and u F jointly satisfy the following requirements: if L (h) 5 y 8 h or L (h) 5 y 9 h, then h Ç Left × H , h Ç L (hÇ Left) 5 y, and L (hÇ Right ) 5 h; if L (h) 5 y 8 h, then P (h) 5 V ; if L (h) 5 y 9 h, then P (h) 5 F ; for every terminal history h × Z : – if L(h) 5 P and (U, g) Y + P , then u V (h) 5 1 and u F (h) 5 Õ 1; – if L (h) 5 P and (U, g) YÕ P , then u V (h) 5 Õ 1 and u F (h) 5 1.

Right × H ,

The notion of strategy is de ned in the same way as before. A winning strategy for i is a set of strategies f i that leads i to u i (h) 5 1 no matter how the player Õ i (the player other than i ) decides to act. The extensions for the quanti ers are straightforward. An extensive form semantic game G ( Q, g , U) associated with an L -formula Q also has: vv if L (h) 5 Zx Q or L (h) 5 Yx Q, then h Ç a × H for every a × |U |; if L(h) 5 ZxQ, then P (h) 5 V ; if L (h) 5 Yx Q, then P (h) 5 F ; for every terminal history h × Z : – if L (h) 5 Pt1 . . . tm and (U, g ) Y + Pt1 . . . t m , then u V (h) 5 1 and u F (h) 5 Õ 1; – if L (h) 5 Pt 1 . . . t m and (U, g ) YÕ Pt 1 . . . t m , then u V (h) 5 Õ 1 and u F (h) 5 1. Imperfect information. These games stand in similar relation to perfect information extensive form semantic games as imperfect information extensive games stand to perfect information extensive games, that is, players may not be able to distinguish between some histories of the game. This is brought out by the information partition (Ji ) i N as before, where × information sets S ji × ( Ji )i N tell how players are informed when making × their moves: if a player cannot distinguish between the histories at the same information set, he or she is not allowed to know something that has happened earlier in the game. When there are only singleton information sets, i.e. no two histories belong to the same information set, one has perfect information games. The imperfect information semantic games are denoted by G *(Q, g , U). What happens in these semantic games is that the partition is determined by the formulas labelling the decision nodes, and this partition may have diVerent properties depending on what language we are dealing with and what syntactic restrictions there might be. We will in general assume that players V and F denote teams of individual actors V i and F j in order to account for the game-theoretic phenomenon of imperfect recall (see Pietarinen and Sandu 1999).

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4.3. Adding Intensionality Given an extensive form semantic game G *A 5

7 H , Z, P, N , L, (u i ) i × N , ( Ji ) i × N 8 ,

let G *(Q, w, g , U) be an extensive form semantic game for an L * -formula Q, comprising of the following rules: vv 1. if L (h) 5 T i Q and the game has reached w , then h Ç w 1 × H , for all w 1 × [w] i ; r 2. if L (h) 5 (T i /x ) Q and the game has reached w , then h Ç w 1 × H , for all w 1 × [w] i ; r 3. if L (h ) 5 (T i /T 1 . . . T n ) Q, then h Ç w 1 × H , for w 1 ×W ; 4. if L (h ) 5 (T i /B ) Q, then P (h) 5 F ; 5. if L (h ) 5 P and (M, ( w 0 , w 0¾ ), g ) Y + P, then u V (h) 5 1 and u F (h) 5 Õ 1; 6. if L (h ) 5 P and (M, (w0 , w0¾ ), g ) YÕ P , then u V (h) 5 Õ 1 and u F (h) 5 1. Rule 3 overrides 2: if there are independent modalities the game does not need to make a choice from an accessible world but can also jump to a starting point for a respective modality and make a choice there. In words, then, (Q x/T ) Px says that Q x is not within the scope of an attitude operator T , and T 1 (T 2 /T 1 ) Q says that the operator T 2 is not within the scope of the rst operator T 1 . The expression (T /x ) says that the modality is not within the scope of the quanti ers binding the occurrences of x . Semantically, these all mean that in the associated semantic games, the players evaluate sentences by making choices corresponding to the logical components in the sentence, such that players are not informed of some previous choices. This can again be brought out in the sequential extensive form game structure, being re ected in the strategies and their uniformity property. Apart from just being able to analyse some speci c problems of intensionality and anaphora, including the problem of intentional identity, these semantically non-linear languages are useful in analysing the totality of scope phenomena in natural language. 5.

Independence at Work: I

Consider now a formalization of the Hob–Nob sentence in L (33)

T H Zx(W x 9 Bx 9 (T N /T H x )Kx ).

*

vv

:

Here the application of the slash-operator has an eVect that the second thought operator T N is not within the scope of the rst operator TH . Likewise, T N is not within the scope of the rst existential quanti er binding x .


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This approach amounts to a forthright non-compositional treatment of Hob–Nob sentences: the scoping problem is circumvented by explicitly denoting which logical components do not lie inside scopes of some previous components. In other words, they are being regarded as informationally independent. This essentially is what accounts for the ‘‘quantifying out’’ phenomenon. One has to be careful in treating the pronoun she in the Hob–Nob sentence, however. Since we do not want to create a de re reading for the last conjunct, it seems as if T N is to be exempted from the scope of the quanti er that binds the variables in the other conjunct. In other words, we do not want Zx to have wider scope over T N , generating eVects similar to quanti cation into attitude operators, for this would produce a reading not conforming to Geach’s preferred understanding. It is a natural property of intensional logic that when applying informational independence to intensional operators, possible independence aVects the evaluation of an operator rather than those components that exist in its scope. Hence, if an operator is drawn from the scope of some previous quanti ers, these quanti ers still bind the variables within the scope of the operator. This is a reasonable thing to happen. The very notion of informational independence asserts that some logical components can be evaluated independently of some previous component. That is, information that exists at some previous level is not allowed to propagate down in a formula to reach the component that is being scoped out from it. Semantically, information about the choice made for the earlier component cannot be used in a game when making choices for the informationally independent components. In game-theoretic terms, imperfect information aVects only the de nitions of the available strategies for players, not the bindings of variables. This proposal can be made rigorous. There are reasons to believe that the received notion of scope, be it formal or linguistic, is ambiguous in at least two ways. In order to have a viable concept of scope in linguistics, one has to distinguish between two diVerent notions of scope, the priority scope (the relative ordering of logically active expressions), and the binding scope (the segment of a formula where the variables of a quanti cation are bound). This distinction can be put to work, for instance, in analysing anaphoric donkey sentences (see Hintikka 1997, pp. 518–521). According to this proposal, the universal reading for the sentence ‘‘If a man owns a donkey he beats it’’ can be explained as soon as the representation of the sentence has the form: (34) Yx( [ ~( D x 9 O x )] Bx ).

8

In (34), the square brackets denote the priority scope and the ordinary parentheses denote the binding scope. Since the priority scope of the

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inde nite a donkey is con ned to the part of the conditional to which the pronoun is coreferential with, whereas the binding scope extends over the tail of the conditional, the controversial implication that every man beats every donkey he happens to own is blocked, yet the universal translation of an apparent existential reading is legitimately attained.8 These two notions of scope and its cousin of informational independence are related but not the same. For informational independence is the product of the game-theoretic failure of perfect information in a semantic game associated with symbolizations of natural language sentences, and thus informationally independent components may not form parts of the priority scope of the component they are independent of, as informationally independent components may not be ordered by inclusion. Incidentally, this raises the question of the nature of informational independence in classical donkey sentences, for example. With respect to (34), there are three possibilities (ignoring negation) as to how the information ow can be regulated. The information regarding the choice of an individual made by F for the universal quanti er is not passed down to the conjunction, it is not passed to the disjunction, or then it reaches neither. A moment’s re ection shows, however, that none of these yields the desired reading of the donkey sentence. For instance, making the disjunction informationally independent of Yx amounts to (35)

Yx( "( D x 9 O x ))

8 YxBx .

In the account that distinguishes between the two notions of scope, however, only the latter disjunct Bx in (34) is informationally independent of the quanti er, in the sense of exempting the disjunct from the priority scope of Yx . It can be noted that nothing in this approach precludes the situation that Bx is chosen by V in a semantic game before Yx , but how is one then able to say that the variable x in the disjunct is not free? That is, how does the binding scope aVect the variables whose associated moves are made in total ignorance from the quanti ers that are supposed to bind them? These questions cannot be discussed here any further, however. To return to our rst take on intentional identity, the resolution along the 8

It is interesting to note that in LePore and Garson 1983, donkey sentences are treated by sorting ‘‘the scope’’ into the anaphoric and semantical scopes, also one of the early proposals related to ‘‘dynamic semantics’’ being developed around that time. According to this proposal, the anaphoric scope denotes the binding of variables, and the semantical scope determines the truth-values of sentences by showing which logical components have wide scope over the other components. The distinctions LePore and Garson make share some similarities with our proposed ambiguity of scope in natural language. For example, both aim at a resolution of donkey sentences by a method that is not fully compositional.


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lines previously indicated turns on the methods of exempting the latter attitude operator from the priority scope of the former attitude operator and existential quanti er, but not from the binding scope of the existential quanti er. These scopes can be made visible by explicitly applying the bracket notation, whereby the priority scopes (the logical orderings of components) of (33) become T H [Zx[(Wx 9 Bx )]] and T N [Kx] , while the binding scope (the segment of the reachability of quanti ers) remains as Zx((W x 9 Bx ) 9 Kx ) . As soon as these two notions of scope are distinguished, the problem of intentional identity admits of a solution, albeit a noncompositional one. Now, an important thing to note regarding (33) is that it has independent modal operators. This means that neither of them precedes the other. This can also be seen from the fact that the modal depth of both T H and T N is the same, namely zero. Thus either of them can be chosen as the rst element in the game, and the second is then evaluated without any information of the former. Admittedly, we have forced the operators in a sequential, super cially linear format, and indeed the extensive games evaluate them sequentially, but these operator pre xes can as well be viewed as parallel ones, associated with concurrent or simultaneous move games. Therefore a formulation of (33) could also be (36)

TH TN

Zx

(Wx 9 Bx 9 Kx ).

This formula makes it clear that T H and T N are independent, concurrent operators (since the interpretation of atomic formulas is relativized to worlds, the world-relative valuations determine the truth- and falsityconditions).9 A play of G *(Q, w, g, U), Q 5 (33) goes on as follows. 1. F 1 chooses a possible world, say w 1 × [w 0 ] H , and the game continues r with respect to that choice: G *(Zx(W x 9 Bx 9 (T N /T H x )Kx, w 1 , g, U). 2. V chooses an individual from the domain W w 1 (and gives it a name if it already does not have one, say a ), to be the interpretation of the variable x , and the game continues as G *(Wa 9 Ba 9 (T N /T H x )Ka, w 1 , g n {(x, a)}, U). (with the occurrences of x instantiated with the individual named ‘‘a ’’. 9 See Henkin 1961 for the rst suggestion of having partially-ordered quanti ers in logic, cf. Hintikka 1996.

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3. F 1 chooses a conjunct, say (T N /T H x )Ka , and the game continues as G *((T N /T H x )Ka, w 1 , g n {(x, a)}, U). 4. F 2 chooses a possible world, say w ¾1 × [w ¾0 ] N , without being informed r of V ’s previous choices that is, he cannot distinguish between possible worlds F 1 chose for T H (which furthermore may belong to the disjoint submodel if w 0 Þ w 0¾ —hence no nesting of the attitudes), nor between the individuals V chose for x (hence the common focus). The game terminates at G *(Ka, w ¾1 , g n {(x, a)}, U).

Given an interpretation F, if (M, w ¾1 , g n {(x, a)}) Y + Ka, V wins, and if (M, w ¾1 , g n {(x, a)}) YÕ Ka, F wins (so the individual named ‘‘a ’’ has to be de ned and exist in w 1 ). We de ne strategies on all earlier possible moves — if there is imperfect information as indicated by slash-operators, respective arguments are missing from strategies. This describes only one possible play of the game. It can happen, for example, that F , in the third step, chooses Wa 9 Ba , wherefore he continues to play either Wa or Ba . If Wa or Ba is true in w 1¾ then F wins in either case, otherwise V wins. Since the truth of the formula is de ned as the existence of a winning strategy for V and the falsity as the existence of a winning strategy for F , (33) is true in the interpretation iV there exists a winning strategy for V , and false iV there exists a winning strategy for F . More precisely, then, we have that (M, ( w 0 , w ¾0 ), g ) Y + T H Zx(W x 9 Bx 9 (T N /T H x )Kx ) if and only if

Zf which is winning for V such that for all w i × [w 0 ] H , r ) ) (w ) w ¾j × [w ¾0 ] N : W p(w i f (w i ) 9 B p(w i f (w i ) 9 K p ¾ j f (w j¾ ).

r To see this, assume the left-hand side. Then there must be a strategy function f such that for every possible world w 1 and w ¾1 chosen by the opponent, this strategy is de ned only over respective predicates. This is because the function f picks individuals from the domains of the possible worlds w i , depending only on the worlds w i with respect to the worldrelatively interpreted predicates W x and Bx . Similarly, this strategy function picks individuals from the domains of w ¾j , depending only on these worlds, with respect to the world-relatively interpreted predicate Kx . The denotations of these predicates are given by a world-relative valuation, and hence K is being de ned over Nob’s alternatives, and W and B over Hob’s alternatives.


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Since in GTS of imperfect information truth-conditions do not determine formula’s falsity-conditions, we also have that: (M, ( w 0 , w ¾0 ), g ) YÕ T H Zx(W x 9 Bx 9 (T N /T H x )Kx ) if and only if

Zh which is winning for F such that for some h × [w 0 ] H , r h × [w ¾0 ] N Yx 5 "W p(h ) x "B p(h) x "K p(h) x.

8 8 r That is, the sentence is false if no individual satis es W x and Bx in some of Hob’s thought-alternative, and Kx in some of Nob’s thought-alternative. Looking at the information partition of extensive form games, the only non-trivial information set in G *(w, w 0 , g, U) is S 1F 2 , referring to the case that F 2 cannot distinguish between the worlds {w 1 . . . w n } (where for all i 5 1 . .. n, w i × [w 0 ] H ), nor between the individuals in D w i . The extensive r form game is drawn in gure 1, for the mini-domains D w 1 5 {a, b}, D w 2 5 {c, d, e}, possible worlds W 5 {w 0 , w 1 , w 2 , w ¾0 , w ¾1 , w ¾2 }, and the alternativeness relations w 1 , w 2 × [w 0 ] H and w ¾1 , w ¾2 × [w ¾0 ] N . Naturally, r r because this is a concurrent game, it could have started with F 1 ’s choice for T N , followed by F 2 ’s choice for T H , V for Zx and so on. To see how the method of concealing information accounts for the anaphoric reference, it suYces to observe that the same strategy function is used in choosing an individual for x in the last conjunct as for its other

Figure 1.

A segment of the game G *(w, w, g, U), w 5 (33) with an information set S 1F 2 annotated for F 2 .

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occurrences. The variable x is subsumed under the binding scope of T N and Zx , although T N is not within the priority scope of Zx or T H . Hence the strategy function picks the same individual for Kx as for the predicates that fall under Hob’s attitude construction. This function is the winning strategy for V , which spells out the truth-conditions of the sentence. 10 The same tactics works for at least some generalized schemes as well, but further investigation of these sentence has to be left for another occasion. For example, the symbolization of (17) is (37) (we use B i to denote the belief operator of agent i ): (37)

B W Yx ( M x Zy (Fy 9 Syx 9 (B E /B W xy )Ryx )).

Expressed in a strategic form, (37) is true in a given interpretation iV Yw ¾ × [w ¾0 ] E Yx : rW j r (38) M p(w i ) f (w i ) (F p(w i ) g (w i , x ) 9 S p(w i ) g (w i , x ) f (w i ) 9 R p(w j¾ ) g (w j¾ x ) f (w j¾ )),

Z f Zg Yw i × [w 0 ]

where f is a winning strategy for F and g is a winning strategy for V . From (38) it is seen that the strategy functions f and g are de ned only over agent W’s (Winston) belief alternatives with respect to M x, Fy and Syx , and only over the agent E’s (Emmett) alternatives with respect to Ryx , accounting for the anaphoric reference as before. An additional feature here is that the existential quanti er becomes dependent on the universal one, re ected in V ’s strategy. The anaphoric reliance of the pronoun on universal quanti ers is something the usual dynamic approaches to anaphora cannot easily achieve, for universal quanti ers are not taken to pass on bindings and act as proper (grammatical) antecedents for anaphoric pronouns in these theories.11 6.

Independence at Work: II

At this point, our suggested tentative resolution is not without problems, however. Independence has been marked only for the components associated with the falsifying player, meaning that the sentence reduces to the ordinary slash-free rst-order sentence in the sense of being weakly equivalent to it. This is because slashes would aVect only F ’s strategies, and so he has to choose some Nob’s thought-alternative being only partially informed about previous choices. These strategies are not explicitly 10 If we were to use only bracketing notation to denote the logical priorities in addition to binding scope parentheses, the rst move could have been made by F choosing the latter conjunct T N Kx , for example, in which case x in that conjunct would remain without a value. To have independence in the pre x guarantees instantiation by V ’s move for Zx . 11 For a comparison between dynamic theories of meaning and GTS see Sandu 1997.

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indicated in the formula spelling out the truth-conditions. They only show when the sentence is true, that is, what V ’s strategies are. The individual chosen by V (‘‘a ’’ in the example) has to be de ned in Nob’s alternatives w ¾j × [w ¾i ] N for all j, i as well. However, V has one crucial decision point r in the game (which is her only move), whereas F ’s strategic role comes out in his attempts to falsify the sentence based on available information. In the presence of imperfect information, the choice for the individual or the world is hidden from him, and so he cannot have a winning falsifying strategy in this game, although V can have a winning verifying strategy. In the perfect information counterpart, on the other hand, F can know, at each stage of the game, what the previous choices have been. Hence he is able to falsify the sentence by choosing suitable Nob’s alternatives (where the individual is, say, not de ned, or then the predicate Kx with this individual is false). Therefore the Hob–Nob symbolization is not false in the same models as its perfect information counterpart, for the logical equivalence preserves only the positive part of the sentence, that is, is weakly ( positively) equivalent to the unslashed version. Hence we have that: (M, (w 0 , w ¾0 ), g ) Y + T H Zx(W x 9 Bx 9 (T N /T H x )Kx ) if and only if (M, (w 0 , w ¾0 ), g ) Y + T H Zx(W x 9 Bx 9 T N Kx ) (where in the latter, w 0 5 w ¾0 ). Therefore, if we were concerned only on the truth of the sentence, then our resolution may not be satisfactory. Alternatively, then, we can use a symbolization with two variables and an identity statement, together with imperfect information as before. This approach amounts to the following representation of (2): (39)

T H Zx(W x 9 Bx 9 (T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 x .

y )).

The alternative branching representation for this is: (40)

TH

Zx

TN

Zy

(Wx 9 Bx 9 W y 9 Ky 9 x .

y ).

Since the interpretation of predicates is world-relative, W x and Bx (40) fall under the scope of T N , and W y and Ky fall under the scope of T H . This is obvious because the interpretation of predicates with the mapping p(w i ) attaches a relational t-structure to each alternative w i ×W , including the (world-relative) valuation for predicates, functions, constants and variables, and so the interpretations for predicates can vary from world to world, the fact that explicitly is denoted in the truth- and falsity-conditions.

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It is seen from (39) that both variables in the identity clause are bound, and that the identity sign resides within the scopes of the intensional operators in the sense that the values for x depend only on Hob’s thoughtworlds, that is, it is de ned with respect to all alternative worlds w i × W , w i × [w 0 ] H . Likewise, the values for y are de ned for those w ¾j ×W for r which w j¾ × [w 0¾ ] N , that is, they are de ned only with respect to Nob’s r thought-world alternatives. The evaluation of (39) in GTS goes on as follows. Let w be (39). One possible play of G *(w, w, g, U) on the structure U, with respect to w, and with a valuation g for free variables and alternativeness relations rH and rN on the set of possible worlds is: 1. F 1 chooses a possible world, say w 1 × [w 0 ] H , and the game continues r as G *(Zx (Wx 9 Bx 9 (T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 x . y )), w 1 , g, U). 2. V 1 chooses an individual from the domain D w 1 , say a , to be the interpretation of x , and the game continues as G *(Wa 9 Ba 9 (T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 a . y ), w 1 , g n {(x, a)}, U), with the occurrences of x instantiated by an individual named ‘‘a ’’. 3. F 1 chooses a conjunct, say (T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 a . y ), and the game continues as G *((T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 a . y ), w 1 , g n {(x, a)}, U). 4. F 2 chooses a possible world w ¾1 × [w ¾0 ] N (without being informed of V ’s r choice for x , and without being informed of F 1 ’s choice of a possible world), and the game continues as G *((Zy/T H x ) (Wy 9 Ky 9 a . y ), w ¾1 , g n {(x, a)}, U). 5. V 2 chooses an individual from the domain D w ¾1 , say b ¾ , to be the interpretation of y (b ¾ has to be de ned and exist in w ¾1 ), and the game continues as G *(Wb ¾ 9 Kb¾ 9 a. b¾ ), w¾1 , g n {(x, a), ( y, b)}, U). 6. F (having full information) chooses one of the conjuncts, say Kb ¾ , and the game ends as G *(Kb ¾ , w ¾1 , g n {(x, a), ( y, b)}, U).

Given an interpretation F, if (M, w ¾1 , g n {(x, a), ( y, b)}) Y + Kb ¾ , V wins the play, and if (M, w ¾1 , g n {(x, a), ( y, b)}) YÕ Kb ¾ , F wins it.

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i nt e nt i o na l i d e nt i t y r e v i si t e d The formula (39) has the following truth-condition: (M, (w 0 , w ¾0 ), g ) Y +

T H Zx(W x 9 Bx 9 (T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 x .

y ))

if and only if Wp

(w i )

Z fZh such that for all w i × [w 0 ] H , w ¾j × [w ¾0 ] N : r r ) ) ) f (w i ) 9 B p(w i f (w i ) 9 W p(w ¾ j h(w ¾j ) 9 K jp(w ¾ j h(w ¾ ) 9 f (w i ) 5 h(w ¾j ),

where { f, h} is a winning strategy for V , and w i × [w 0 ] H and r w j¾ × [w ¾0 ] N for all positive integers i, j . r Likewise, (39) is false iV there exists a winning strategy { f, h} for F : (M, (w 0 , w ¾0 ), g ) YÕ

T H Zx (Wx 9 Bx 9 (T N /T H x ) (Zy/T H x ) (Wy 9 Ky 9 x .

y ))

if and only if

Z f Zh Yx Yy such that for some f × [w 0 ] H , h × [w ¾0 ] N : r r "W p(f ) x 8 "B p(f ) x 8 "W p(h ) y 8 "K p(h ) y 8 "(x 5 y),

where { f, g} is a winning strategy for F . In words, (39) is false iV no single individual both is a witch and has blighted Bob’s mare in some of Hob’s thought-alternative and is a witch and has killed Cob’s sow in some of Nob’s alternative. From these strategic forms it can be seen that x and y are de ned over Hob and Nob’s thought-worlds, respectively, and that the extensions of all predicates and relations fall under the right intensions. It is also seen that for truth- and falsity-conditions, it does not matter where the identity clause is being located, insofar as it is being bound by the existential quanti ers. Expressed in the alternative Henkin pre x form, we thus have that: (M, ( w 0 , w ¾0 ), g ) Y +

TH

Zx

TN

Zy

(Wx 9 Bx 9 W y 9 Ky 9 x .

y)

if and only if Z f Zh such that for all w i × [w 0 ] H , w ¾j × [w ¾0 ] N : r ) r ) ) ) W p(w i f (w i ) 9 B p(w i f (w i ) 9 W p(w j¾ h(w j¾ ) 9 K p(w j¾ h (w j¾ ) 9 f (w i ) 5 h(w ¾j ),

where { f, h} is a winning strategy for V . Also: (M, ( w 0 , w ¾0 ), g ) YÕ

TH

Zx

TN

Zy

(Wx 9 Bx 9 W y 9 Ky 9 x .

y)

if and only if Z f Zh such that for some f × [w 0 ] H , h × [w ¾0 ] N Yx Yy : r r "W p(f ) x 8 "B p(f ) x 8 "W p(h ) y 8 "K p(h ) y 8 "(x 5 y),

where { f, h} is a winning strategy for F .

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The corresponding extensive form game is drawn in gure 2, for the mini-domains of D w 1 5 {a, b}, D w 2 5 {d, e, f } and D w 1¾ 5 {a ¾ , b ¾ }, possible worlds W 5 {w 0 , w 1 , w 2 , w ¾0 , w ¾1 , w ¾2 }, and the alternativeness relations w 1 , w 2 × [w 0 ] H and w ¾1 , w ¾2 × [w ¾0 ] N . r r In this game, the crucial point is when V making the latter move has to get by with less information than in the classical perfect information counterpart, as she should be able to produce an individual satisfying the identity clause without being informed about the course of events chosen by f (neither is she supposed to have information about her own previous choice

Figure 2.

A segment of the game G *(w, w, g, U), w 5 (39) with (nonstandard) information sets S 1F 2 and S 1V 2 .


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of the individual for the rst existential quanti er). This game has two non-standard information sets, one for F 2 and one for V 2 .12 A model M where (39) is true is given in gure 3. It is not necessary to have detached disjoint submodels in order to arrive at correct truth- and falsity-conditions for intentional identities, however. If we are dealing with, say, complex nested sentences, disjointedness may disappear. For consider: (41)

Hob thinks Rob thinks a witch has blighted Bob’s mare, and that Nob thinks she (the same witch) has killed Cob’s sow.

Figure 3.

A model M for (39).

12 By non-standard information sets we mean that information sets do not have to be marked at the same levels where the corresponding players make their choices. The traditional way of de ning imperfect information games imposes the condition that the players can observe the set of immediately available choices, and on the basis of this information, could deduce at which decision node they are inside an information set. In our games, the set of available actions is not constant, since there can be a varying number of accessible worlds from the hidden worlds. In dynamic game-theoretic evaluation process, one may have to mark information sets to nodes whose departing actions refer to the history of the game and not to the current action, so the number of actions would be irrelevant for the decision in any case.

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We would formalize this as (42)

TH

TR

Zx

TN

Zy

(Wx 9 Bx 9 W y 9 Ky 9 x .

y),

being true in (M, w 0 , g ). 7.

Discussion

7.1. General Consequences The approach given in this paper solves the problem of intentional identity for good. In (39), all the original requirements for Geach’s preferred reading are satis ed, for only non-speci c individuals are concerned in regard to Hob and Nob’s thoughts (the non-speci city condition), without commitment to the existence of witches (the non-actuality condition), no free variables occur, while the anaphoric reference is attained (the binding condition), and no excess thoughts are created (the non-iteration condition). The semantics advocated here is one where players do not choose world lines but individuals. These world lines are nonetheless being drawn according to players’ strategic actions, although their actual graphics rests on agents’ knowledge and the various identifying modes they exhibit. Indeed, it is questionable to let players to choose world lines in a game, since players do not have (and do not need to have) identifying capabilities themselves; those capabilities belong to the resources of agents whose attitudes our language tries to capture. An outright persistence of individuals through a sequence of courses of events is not a matter of those who try to verify or falsify utterances speaking of such individuals, because an evaluation process is a stepwise activity operating within a single possibility at a time. It is a conceptual confusion to install world lines into players’ semantic resource, a confusion related to mixing knowledge and information of players with those of agents’. Based on arguments given in this paper, it is to be suspected that any otherwise successful proposal to capture the semantic behaviour of intentional identities, a proposal that would appear to dispense with the use of information hiding or branching formulas but which nonetheless would resort to Skolem functions and their world line re ections, is after all likely to accommodate imperfect information. Such imperfectness would merely be in the guise of these (uniform) Skolem functions. For if one does not use world hiding of any kind, Skolem functions picking witches in one of Hob’s worlds would necessarily depend on worlds of the other agent, that


181

is, it would receive the worlds of the other agent as input. But this would imply at once that such a function picks an individual in any of these latter worlds as well, in a way that would make the individual to coincide with another Skolem function. But then that individual becomes cross-identi ed over Nob’s worlds, and this is precisely what we need to avoid in order to respect Geach’s understanding of the identities. If the latter worlds are not given as input, one enters into the joyful sovereignty and independence of some alternative worlds and their denizens. Our informational independence and imperfect information theory is seen to share some similarities with the backward-looking operator approach, but it is more exible than that, for it can implement the needed backward references as informational hiding, while ultimately dispensing with syntactic operators. When compared to some other proposals, it is seen that no postulation of extra-logical belief objects (Edelberg 1986, 1992) or additional discourse referents or speaker’s perspective (Dekker and van Rooy 1998) are needed. The new puzzle of intentional identity in Edelberg 1986, which assumes non-commutativity of conjunction, does not arise in our game-theoretic setting. This is because the new puzzle presupposes a wide-scope interpretation of inde nites, whereas the proposal here is thoroughly ‘‘quantifying-in’’ resistant: no wide scope quanti cation with respect to thought operators takes place in our examples. In particular, attempts based on some version of dynamic predicate logic, for example, may falter on some further variants of intentional identity, such as those where the alleged ‘‘grammatical head’’ for the pronoun is a universally quanti ed phrase. This is because the anaphoric pronoun cannot refer to such antecedents since universal quanti ers are not supposed to pass on bindings in dynamic logics. Since our solution does not turn on these presuppositions of dynamic semantics, it remains immune to this problem. The analysis advocated here also suggests that the complaints about the lack of sensitivity of possible-worlds theories to such logical phenomena as partiality or dynamic modes of evaluation need to be reconsidered. Semantic games with imperfect information can indeed generate partiality in a natural way, and may even turn out to be superior to partial modal logics (see the papers in Doherty 1996), because partiality is not any more based on assumptions about partial models but can arise at the level of complex sentences. On the other side, dynamics remains an intrinsic feature of games, already involving constant updating of information (see Sandu and Pietarinen 2001). Although our resolution is non-compositional, one can devise

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compositional versions by extending compositional semantics for rst-order imperfect information logic to possible-worlds semantics.13 Let us also nally note one other manifestation of the independence property in modal logic, provided by the McKinsey axiom Q Q.14 In the light of independent modalities, this axiom Q to Q is licensed preexpresses the fact that the inference from cisely when is independent of . That independence indeed is patent can be veri ed from the relational translation of this axiom to the background ( nite-variable) rst-order language, which gives a parallel Henkin quanti er. 7.2. Negated Sentences: a Test Case? It might be argued against our approach that it does not give a right predication for negated versions of Hob–Nob sentences. For consider the following: (43)

Hob does not think that a witch has blighted Bob’s mare, and Nob does not think that she has killed Cob’s sow.

What has been done here is to modify the original sentence to have verbal negation for intensional verbs in both conjuncts. Does the anaphoric pronoun lack a reference, making the sentence ungrammatical? One can contrast this with extensional contexts: (44)

*A man does not walk in the park. He whistles.

As has often been observed, this sentence is ungrammatical because the pronoun he lacks an antecedent, usually taken to be a result of a negation creating a barrier to the existence of such a link, making the sentence impossible to have coreferential anaphora. The problem with these sentences is then said to be to nd of an explanation as to why negation renders the existence of anaphora impossible, and this has caused puzzlement.15 13

For compositional semantics for rst-order imperfect information logic, see Hodges 1997a,b, Caicedo and Krynicki 1999, and for propositional imperfect information logic, see Sandu and Pietarinen 2001. 14 Roughly, means ‘‘modal necessity’’ and means ‘‘modal possibility’’. About this axiom and its properties (see Goldblatt 1991, Wang 1992). 15 In GTS, the explanation uses the nobby idea of having a storage of individuals, a choice set (cf. a storage of discourse individuals in a discourse representation theory, and the impossibility of anaphora in the case of an inde nite term being located under the scope of negation, which results from the failure of introducing a discourse referent), where stages of storages represent discourse situations at which choices of elements and additions to the storage are taking place.


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However, the sentence (43) cannot be ruled out as outright ungrammatical, since the intensional verbs of thinking or believing do not commit speakers to the non-existence of inde nites in verbally negated expressions. What is more, if need be, the pronoun in this sentence merely refers to an imaginary individual, of which Hob might think she did not blight Bob’s mare. Thus the sentence cannot be put on a par with (44), or with one where the second sentence is ‘‘He does not whistle’’. As to the negated objects, however, the situation is diVerent and the sentence can indeed become ungrammatical, as previously was observed to happen for (22). As already noted, if we were to assume an existential presupposition at the level of intensional contexts, we would gain a surprisingly simple treatment of Hob–Nob sentences. One would just undo the existential import of the existential quanti cation, by using an explicit existential predicate Ex to express the fact that the denotation of x exists, namely when the quanti er quanti es over actual individuals and not only possible ones. Existential quanti ers with existential presuppositions would then become Zx (Ex 9 . . .), representing Hob–Nob sentences with a wide scope quanti cation without falling prey to an outright de re interpretation, i.e. the lack of presence of an existential predicate indicates that existential quanti cation does not commit one to the actual existence of an individual. However, such existential presuppositions are not necessary in the theory of intentional identity. They are not very useful, either, since they do not provide general solutions. For instance, they do not help us in analysing generalized sentences containing, say, universal determiners. In addition, such presuppositions cannot be taken to be obvious or unproblematic because it is not clear that Hob and Nob can always be forced to have at least some beliefs toward witches (the more sober they are the less they would think the reason behind the unhappy incidents was a witch, so to speak). The proper treatment of negation in the presence of informational independence is not a trivial matter, however. This is because negation bifurcates into two diVerent items with diVering characteristics. These diVerences can nonetheless be turned into virtues, because one can apply a test to Hob–Nob sentences to nd out whether informational independence genuinely takes place in the resolution advocated here. The test follows the suggestion in Barwise 1979, p. 57. Essentially, it says that if a sentence of English is an unambiguous example of informational independence not reducible to ordinary linear notation, then its ‘‘negation normal form’’ should be rather peculiar. Admittedly, the notion of negation normal form in natural language is rather vague when compared to it in logic, but the key idea is to try to push the sentential negation deeper in the sentence to the positions normally occupied by verbal (quanti-

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er) negations such that no subject noun phrase has narrow scope with respect to negation. For example, the sentences (45), (46) and (47) would yield to the negation normal forms (48), (49) and (50), respectively. (45) (46) (47) (48) (49) (50)

It is not the case that everyone cheated in the exam. It is not true that no one loves no one. It is not the case that Mary believes that someone cheated in the exam. Someone did not cheat in the exam. Someone loves no one. Mary does not believe that someone cheated in the exam.

Now, consider the sententially negated denial of (2): (51)

?It is not the case that Hob thinks that a witch has blighted Bob’s mare and Nob thinks that she has killed Cob’s sow.

As seen from this sentence, it indeed becomes diYcult to paraphrase it as (43). For consider the corresponding negation normal form with respect to our resolution. If we try to put (51) into a negation normal form, the only possibility is to consider (43), but there we have a verbal negation which is not equivalent in meaning to (51). 8.

Conclusion

The problem of intentional identity can be analysed and resolved in intensional logic accommodating informational independence. Because such language has not been considered in the literature before, it is not surprising that existing treatments do not prove satisfactory (at least without trying to jumble pragmatic features up in the analysis). Edelberg’s theory operates inside standard formal languages and hesitates to cross the ‘‘Frege–Russell boundary’’. Hence his methodological purism causes the system to remain in Hob’s pound of limited expressive capacity so to speak. When presented the original puzzle, Geach did not have resources at hand that would have necessitated him to take the required radical departure from extant formalisms. The departure is seen to take a syntactic step from linear to non-linear concurrent formulas, and a semantic leap from perfect information to imperfect information and independent models. The mechanism of world lines becomes increasingly important, for they are drawn so as to relate aspects of individuals in distinct worlds of two agents who may have epistemically detached models, but never between two worlds compatible with respective attitudes of the same agent. How these lines come about and how they can be varied is best studied within other elds such as epistemology and cognitive science, and so this points


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up a subject that deserves a separate study. After all, there is a whole range of cognitive methods that people use when identifying things in addition to non-cognitive methods. In general, our approach is seen to manifest yet another dynamic aspect of epistemic logic. The methods and concepts used in our analysis are warranted in other respects too. Modal semantics is well studied, and the phenomenon of informational independence can be seen to form a natural ingredient of quanti ed modal logics, over and above the old perturbations about mixing quanti cation with modal operators. Besides, in game theory the notion of imperfect information plays a major role. The convergence of the two is likely to have repercussions to natural language semantics as well as to the philosophy of propositional attitudes in general, and intentional identities are but one example of a range of concepts in the need of such analyses. There are thus prospects for demystifying what the locutions such as ‘‘mentalist processes’’, ‘‘having a common focus’’ or ‘‘what goes on in agents’ minds’’ might mean in our language, things often discussed in relation to various attitude puzzles, and often claimed to need some extralogical thought-objects or a full-toned investigation of the epistemic resonance of discourse referents. For agents’ independent thought-spheres can after all be related and focused toward shared individuals, and this can be done without dispatching any disorderly linguistic or metaphysical cargo into the vessel of meaning and ontology. R efer enc es Asher, N. 1987. A typology for attitude verbs and their anaphoric properties. Linguistics and Philosophy, vol. 10, pp. 125–197. Barwise, J. 1979. On branching quanti ers in English. Journal of Philosophical Logic, vol. 8, pp. 47–80. Caicedo, X. and Krynicki, M. 1999. Quanti ers for reasoning with imperfect information and S11 -logic. In W. A. Carnielli, and I. M. L. D’Ottaviano (eds), Advances in Contemporary Logic and Computer Science. American Mathematical Society–Contemporary Mathematics, vol. 235, pp. 17–31. Castan˜eda, H.-N. 1974. Thinking and the structure of the world. Philosophia, vol. 4, pp. 3–40. Crimmins, M. 1998. Hesperus and phosphorus: sense, pretense, and reference. The Philosophical Review vol. 107, pp. 1–47. Dekker, P. and van Rooy, R. 1998. Intentional identity and information exchange. In R. Cooper and T. Gamkrelidze (eds), Proceedings of the Second Tbilisi Symposium on Language, Logic and Computation. Tbilisi State University, Tbilisi, in press.

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