Peter Pagin

Philippe De Brabanter

Herb Clark

Pure Quotation and General Compositionality *

Starting from the familiar observation that no straightforward treatment of pure quotation can be compositional in the standard (homomorphism) sense, we introduce general compositionality, which can be described as compositionality that takes linguistic context into account. A formal notion of linguistic context type is developed, allowing the context type of a complex expression to be distinct from those of its constituents.

We formulate natural conditions under which an ordinary meaning assignment can be non-trivially extended to one that is sensitive to context types and satisfies general compositionality. As our main example we work out a Fregean treatment of pure quotation, but we also indicate that the method applies to other kinds of context, e.g. intensional contexts.

The Problem

A straightforward treatment of quotation cannot be compositional. Consider the simplest case: pure quotation in written language by means of quote marks (we shall consistently use single quotes):

(1) a. 'Cicero' has six letters b. 'farfalla' is Italian for butterfly c. 'str jd e' is not a sentence d. 'ℵ' is a Hebrew letter Although Cicero is Tully, and thus 'Cicero' and 'Tully' have the same meaning (taking the reference of names as their meaning), (1a) is not synonymous (on any account of meaning) with

(2) 'Tully' has six letters This argument is familiar to everyone, but the observation about compositionality doesn't depend on taking the meaning of a name to be its bearer. Much less is required. To see this, let us be precise about what we mean by 'straightforward' here: A straightforward account of (the use of quote marks in) pure quotation is one which (a) takes the quoted phrase to be a syntactic constituent of the the quoting phrase (the quoted phrase surrounded by quote marks), and (b) allows at least one case of two syntactically distinct and quotable expressions having the same semantic interpretation (meaning). 1Then it should be uncontroversial that no such account can be compositional: By (a), application of quote marks is a syntactic rule, on a par with other syntactic rules used for building or analyzing sentences, so compositionality requires that there be a corresponding semantic operation yielding the meaning of the quoting expression from the meaning of the quoted expression and the quote marks rule, but (b) provides a counter-example. 2 Indeed, non-compositionality is a main motive behind the wide variety of theories attempting to reanalyze quotation so that the quoted phrase is not a constituent of the quoting phrase. For example, the proper name theory (Tarski [START_REF] Tarski | The concept of truth in formalized languages[END_REF], Quine [START_REF] Quine | Mathematical Logic[END_REF]) treats the quoting phrase as atomic or unanalyzable, no more containing the quoted phrase than the word 'Quine' contains 'in'. Other theories have analyzed quoting phrases as descriptions of how to form the quoted phrase from atomic expressions (letters or words) by means of concatenation (Tarski [38], Quine [START_REF] Quine | Word and Object[END_REF], Geach [START_REF] Geach | Mental Acts[END_REF], Werning [START_REF] Werning | Right and wrong reasons for compositionality[END_REF]). 3 Especially popular among philosophers is Davidson's demonstrative theory in [5], which takes (the occurrence of) the quote marks to be a demonstration of a token, the type of which is what the sentence is about. 4 Although reference to the quoted expression occurs on this analysis, it is performed by a demonstrative device -the quote marks themselves -not by an expression containing the quoted expression as a part. There is a vast literature on the demonstrative theory; among recent 2 At least on the plausible assumption that if you enclose two expressions with distinct surface forms within quote marks, the resulting two quoting expressions never have the same meaning. For example, 'brother' and 'male sibling' (quote marks included) are trivially nonsynonymous on any reasonable account of meaning, regardless of whether the quoted expressions are held to have the same meaning or not. That's just because no reasonable semantics should take two (non-indexical) referring expressions that refer to distinct objects to have the same meaning. 3 Cappelen & Lepore [START_REF] Cappelen | Quotation[END_REF] surveys theories of quotation. As these authors note, the proper name theory and the description theory fail to do justice to essential features of quotation in natural languages (this was however not the ambition of Tarski or Quine), and are now out of fashion. Werning [START_REF] Werning | Right and wrong reasons for compositionality[END_REF] is a recent attempt to revive the description theory. Like all description theories, Werning's account is not straightforward in our sense (requirement (a)

is not satisfied). 4 For example, (1b) is analyzed as (i) farfalla. That / The expression of which this is a token is Italian for butterfly works, see, for example, Cappelen & Lepore [START_REF] Cappelen | Language Turned on Itself. The Semantics and Pragmatics of Metalinguistic Discourse[END_REF] and Predelli [START_REF] Predelli | The demonstrative theory of quotation[END_REF].

Another popular approach, called the use theory or -misleadingly -the identity theory, takes quotation to be a different use of words than the normal one, and quote marks as indicators of this use. The idea goes back to the medieval distinction between suppositio materialis and suppositio formalis. In modern times it occurs with Frege (see below); more recent variants can be found with Searle [START_REF] Searle | Speech Acts: An Essay in the Philosophy of Language[END_REF], Washington [START_REF] Washington | The identity theory of quotation[END_REF], Saka [START_REF] Saka | Quotation and the use-mention distinction[END_REF], Recanati [START_REF] Recanati | Open quotation[END_REF], [START_REF] Recanati | Open quotation revisited[END_REF], and many others. As we will see, this idea is not necessarily incompatible with approaches taking quoting expressions to name quoted expressions, but many of its proponents take it to be radically different. 5 5 E.g. Searle [START_REF] Searle | Speech Acts: An Essay in the Philosophy of Language[END_REF]. Recanati [START_REF] Recanati | Open quotation[END_REF], [START_REF] Recanati | Open quotation revisited[END_REF] combines a pragmatic and a referential view: Pure quotations (closed, in his terminology) are referential singular terms, and he explicitly raises the issue of compositionality of the corresponding semantics, as we do in the present paper.

Open quotations (e.g. indirect or mixed cases), on the other hand, do not refer; the words have their ordinary meaning but the speech act is different.

Some thoroughly pragmatic accounts (Clark and Gerrig [START_REF] Clark | Quotations as demonstrations[END_REF]) deny that pure quotation is ever done by exhibiting the quoted object within some quotational device. The idea is instead that quotation depicts selected aspects of the object, even in quotation of written text. For instance, one may both include and exclude the case of the letters (upper, lower) as part of what is depicted. Moreover, C&G stress that the so-called verbatim assumption, that the reporter commits himself to repeating the actual words spoken (p. 795, actually a quote from Leech), is not in general respected in actual quotations, partly because it conflicts with the permission to select features.

We are here dealing only with written quotation of written text, and we attempt to provide a semantics for a quotation device that operates under the verbatim assumption (a difficult enough task). It is clear that other features than verbatim wording can be selected for quoting, e.g. letter case, font size, or typeface. This can be accommodated in our formal model by means of encoding the relevant properties in the grammatical terms (see Section One may discern some recent dissatisfaction with complicated analyses of what seems to be -in the pure case -a very simple phenomenon: quote marks are a productive and transparent device for forming names of linguistic entities, names that refer to these entities in just the same way as other names refer to non-linguistic entities. For example, this eliminates any problem whatsoever with iterated quotation, something which at least is an issue with other approaches. 6 Intuitively, this seems correct: there is no difference between quoting expressions surrounded by quote marks and quoting any other expressions.

In other words, pure quotation should be straightforward. One elegant recent account along these lines is given in Potts [START_REF] Potts | The dimensions of quotation[END_REF]; indeed Potts deals not only with pure quotation but also direct and indirect discourse as well as mixed cases.

But, as we just saw, with a straightforward analysis the problem of compositionality returns. On this point, the relevant literature is unclear. Semanticists accounting for quotation, including Potts, generally claim that their accounts are compositional. Sometimes, the reason such a claim doesn't conflict with our initial observation is that the account is not straightforward in our sense. 7 But

proper. And therefore we need not deviate from the straightforward view that quotation is a syntactic operation. 6 It is not obvious how Davidson's original account indicated in footnote 4 can deal with demonstrations within demonstrations, as in (i) "farfalla" refers to 'farfalla', and although Cappelen & Lepore [START_REF] Cappelen | Language Turned on Itself. The Semantics and Pragmatics of Metalinguistic Discourse[END_REF], Predelli [START_REF] Predelli | The demonstrative theory of quotation[END_REF], and others suggest mechanisms for doing this, these are certainly more complex than the straightforward approach.

the treatment in Potts [START_REF] Potts | The dimensions of quotation[END_REF] is straightforward. It could of course be the case that different notions of compositionality are at stake. In fact, Potts replied to us (p.c.) that he required no more than that 'the syntax and the semantics work together in tandem', a slogan taken from the Introduction to Barker and Jacobson [START_REF]Direct Compositionality[END_REF]. Here, on the other hand, although we have not (yet) given a precise definition, we have taken for granted that a necessary condition for compositionality is that substitution of synonyms preserves meaning. That this condition is violated in straightforward accounts of quotation seems uncontroversial, as noted. But the slogan just quoted says something much weaker, and has no implications at all for substitution of synonyms. 8 itself). 8 Potts's grammar generates triples P = Π; Σ; α : σ , where Π is a phonological representation, Σ a syntactic category, and α a semantic representation of type σ. Letting SEM( Π; Σ; α : σ ) = α, the semantic interpretation function Potts (in fact) uses is

µ(P) = [[SEM(P)]]
where Given that the triples are the linguistic objects that get quoted, requirement (a) of a straightforward semantics is satisfied. But requirement (b) is also satisfied: Let P be the triple corresponding to Cicero and Q the one corresponding to Tully. Then µ(P) = µ(Q) ∈ De according to Potts's truth definition. But, by the above, µ(q(P)) = µ(q(Q)).

Note here the distinction between recursive and compositional semantics. Potts's semantics is perfectly recursive: his rules allow you to compute the meanings of any well-formed triple in a finite number of steps. But this is not enough for compositionality, which requires that meanings of complex expressions be determined by the meanings of their immediate constituents (and the rule applied), whereas a recursive definition allows you to use not only those meanings but the constituents themselves as arguments of the composition operation. (In addition, it requires, in contrast with compositionality, that the composition operation is recursive; see Pagin & Westerståhl [START_REF] Pagin | Compositionality I: Definitions and variants[END_REF], Section 3.2.) This feature is essential for a (straightforward) formulation of the semantic rule for quotation, but, as we have been at pains to emphasize, the result is that substitution of synonyms does not preserve meaning, i.e. compositionality fails.

This brings us to an important point. To make a meaningful claim about (non-)compositionality -one that has a chance of being true or false -two things need to be in place. First, you should have a particular syntax or grammar in mind, and a particular semantic interpretation. When a semanticist presents a fragment with a model-theoretic interpretation, this is usually not a problem, as long as it is clear that the claim concerns that syntactic analysis and that notion of meaning (as opposed to, say, English). The same surface strings may generate a counter-example to compositionality under one analysis (or one notion of meaning) but not under another. Second, you should have a clear notion of compositionality in mind. The slogan about syntax and semantics working in tandem is too vague to serve as a definition. 9

Should we then despair of a compositional and straightforward treatment of pure quotation? With the standard notion of compositionality, the answer is Yes. In this paper we show, however, that there is an extended but still precise sense of compositionality in which the straightforward account is compositional.

We call this general compositionality.

The quickest way to grasp this notion is to relate compositionality to context dependence. Consider first the familiar case of the extra-linguistic context dependence of indexicals and demonstratives The standard notion of composi-One might still think that since neither Potts's semantics nor ours is standardly compositional, there is no principled reason to prefer ours. However, beside the wide applicability of general compositional semantics, we can briefly point to another reason that because of space limitations we cannot here go deeply into. Semantic computational complexity, in the sense of [START_REF] Pagin | Communication and the complexity of semantics[END_REF], [START_REF] Pagin | Compositionality, computability, and complexity[END_REF], does not increase with general compositionality compared with standard compositionality. By contrast, Potts's semantics for quotation has the effect that iterated quotation does increase computational complexity dramatically, because it leads to semantic values with iterated triples: triples that have other triples as their third element (and so on). 9 We thus disagree with the view of compositionality expressed in Dowty [START_REF] Dowty | Compositionality as an empirical problem[END_REF], i.e. that the idea is fundamentally vague. We have no quarrel with Dowty's interesting discussion of how to treat various linguistic phenomena compositionally (the bulk of their papers), but we don't think compositionality is a vague notion. On the contrary, treating the grammar and the semantic interpretation as parameters, there are just a couple of notions of compositionality around, each of them precise (see our [START_REF] Pagin | Compositionality I: Definitions and variants[END_REF] for an overview).

tionality applies directly to Kaplan's notion of character. Character is assigned to expressions, so we can ask if the character of a complex expression is determined by the characters of its immediate constituents (and the mode of composition). But this doesn't immediately apply to content, since content is assigned not to an expression but to an expression and a context. Nevertheless, no one takes this fact alone to show that content is not compositional. Indeed, Kaplan himself states explicitly in [START_REF] Kaplan | Demonstratives: An essay on the semantics, logic, metaphysics, and epistemology of demonstratives and other indexicals[END_REF] that his account of content is compositional, and Lewis in [START_REF] Lewis | Index, context, and content[END_REF] makes compositionality a necessary requirement on any assignment of semantic values. But, although only implicitly acknowledged by these authors, this requires an extended notion of contextual compositionality, one that takes the presence of the extra context argument into account. 10 Our claim in this paper is that with a corresponding notion of linguistic context, the straightforward account of quotation can be made compositional, in the general, contextual sense. We shall not here discuss the intrinsic merits of compositionality. One could make an exception for quotation, and content oneself with a recursive but non-compositional semantics for a straightforward treatment of this phenomenon, as Potts de facto does (see footnote 8). That would be consistent with the view that quotation is a linguistic phenomenon without much interest. Such a view is not uncommon in the literature, but it is not Potts's view; on the contrary, he thinks quotation is a 'hugely important matter for linguistic theory' ( [START_REF] Potts | The dimensions of quotation[END_REF], p. 425). Likewise, Cappelen and Lepore find it 'one of the most difficult and interesting topics in the philosophy of language' ([3], opening paragraph).

We agree that the way speakers 'talk about the words themselves' (see the quote from Frege below) is of significant interest, and we shall here take it for granted that a compositional account is worthwhile. And just as the presence of indexicals and demonstratives motivates an extension of the notion of composi-10 Several writers are aware that the notion of compositionality needs to be extended to take context into account. Partee [START_REF] Partee | Compositionality[END_REF] has an extensive discussion without however arriving at explicit proposals. As we will see in Section 3 below, the formulation of context-dependent compositionality is slightly more subtle than one imagines.

tionality to take extra-linguistic context into account, the presence of quotation motivates a generalization of compositionality that takes linguistic context into account. This can be seen already from Frege's familiar remark:

If words are used in the ordinary way, what one intends to speak of is their reference. It can also happen, however, that one wishes to talk about the words themselves or their sense. This happens, for instance, when the words of another are quoted. One's own words then first designate words of the other speaker, and only the latter have their usual reference. We then have signs of signs. In writing, the words are in this case enclosed in quotation marks. Accordingly, a word standing between quotation marks must not be taken to have its ordinary reference. (Frege [8], pp. 58-9)

As Cappelen & LePore [START_REF] Cappelen | Quotation[END_REF] note, this passage, which seems to be just about all Frege ever said about quotation, is probably too cryptic to be taken as a clear proposal. 11 But we shall interpret it as a combination of the use theory and the straightforward theory. It is straightforward in that the quoting phrase refers to the quoted phrase, but it is a use theory in that the quoted phrase itself, in the quotation context, has another meaning than it usually does: it refers to itself. So the quote marks are a syntactic operator, but they also signal a context change.

However, Frege also indicates that this sort of context dependence is not limited to quotation. He mentions talking about the senses of words, and we will see in Section 6 that one can construe his notion of indirect sense and reference as an attempt to give a general compositional semantics for intensional contexts. In fact, although in this paper we focus on quotation, there are several other applications. Moreover, we will see (Section 5.3) that the notion of general compositionality can be formulated in an equivalent way which doesn't mention contexts at all: the idea is to let a semantics consist of a set of meaning 11 See Parsons [START_REF] Parsons | What do quotation marks name? Frege's theory of quotations and that clauses[END_REF] for a discussion of possible interpretations.

assignments rather than just one. This gives some indication of the robustness of the notion, and also paves the way for further applications.

We begin, in the next section, by presenting standard compositionality at a sufficiently abstract and precise level and, in the section after that, the extension of this concept to deal with non-linguistic context. Then we discuss linguistic context, and propose a formal version of this notion, which is used for the first formulation of general compositionality. We state the equivalence of this formulation to the second one (proofs of this and some other facts are relegated to Appendix 2) as well as some other facts about general compositionality. We also suggest a way to make precise the conditions under which an ordinary semantics can be said to generalize to a semantics which is compositional in the new sense. Finally, we show that the approach to quotation we take Frege to recommend is indeed a compositional generalization of a straightforward account of pure quotation, and in Appendix 1 we explain why an alternative approach that has been suggested would not be superior.12

Ordinary Compositionality

Compositionality is not a demand for a certain format of the grammar or the syntax-semantics interface. It is a condition that may or may not be satisfied by any semantically interpreted grammar in (almost) any format. This is clear already from the informal formulation of the compositionality principle, in either its functional or its substitutional version:

(PoC-F) The meaning of a complex expression is determined by the meanings of its immediate constituents and the mode of composition.

(PoC-S) Appropriate substitution of (not necessarily immediate) constituents with the same meaning preserves meaning.

These formulations are of course relative to given specifications of meaning, constituent, etc., but they are independent of how those specifications are made.

For any assignment of semantic values to any set of structured expressions (expressions equipped with a notion of constituent or sub-expression), the issue of compositionality can be raised. Indeed, for (PoC-S) we don't even need to specify what the values are, only when two expressions have the same value. It follows that general facts about compositionality, such as the ones about quotation established in this paper, are (to a large extent) independent of particular formats chosen for syntax and semantics.

A formal framework

To make (PoC-F) and (PoC-S) precise we shall use the following framework. 13 Think of (surface) expressions as strings, and complex expressions as generated from atomic ones via grammar rules, which we simply take to be partial functions from tuples of expressions to expressions. There is no requirement on what these functions are; in particular, they are not restricted to concatenation of strings. Partiality is used to respect syntactic categories: rather than taking these as primitive, we let grammar rules be undefined for arguments of the wrong kind. To illustrate, rules like

NP -→ Det N S -→ NP VP
correspond to binary partial functions, say α, β, such that, if few, cat, and bark are atoms, one derives the complex expression (sentence) Few cats bark, by first applying α to few and cat, and then applying β to the result of that and bark.

These functions are necessarily partial; for example, β is undefined whenever its second argument is cat.

Thus, let a grammar 13 See Hodges [START_REF] Hodges | Formal features of compositionality[END_REF] or Westerståhl [START_REF] Westerståhl | On the compositional extension problem[END_REF] for more detailed accounts, and Pagin and Westerståhl [START_REF] Pagin | Compositionality I: Definitions and variants[END_REF] and [START_REF] Pagin | Compositionality II: Arguments and problems[END_REF] for further discussion and motivation.

E = (E, A, Σ)

be a partial algebra, where E is a set of expressions (strings), A its subset of atoms, and Σ is a set that, for each required n ≥ 1, has a subset of partial functions from E n to E, and is such that E is generated from A via Σ.

One and the same expression may be generated in more than way, i.e. the grammar may allow structural ambiguity. Also, a semantically relevant element need not be represented in the surface expression. For these reasons, it is not the expressions in E but rather their derivation histories ('analysis trees'), that are assigned semantic values. These derivation histories can be represented by the terms in the partial term algebra corresponding to E. The sentence Few cats bark is a string, but its derivation history is represented by the term

(3) t = β(α(few, cat), bark)
in the term algebra. Grammatical terms are those where all the functions involved are defined for the respective arguments. So t is grammatical but β(α(few, cat), cat) is not. Let GT E be the set of grammatical terms of E.

We shall need to distinguish between the functions α, β ,. . . taking expressions to expressions and the names of these functions used as operations in the term algebra. The convention here will be that α is a name of α. Thus,

α(few, cat) = few cats
is an element of E -the string few cats, 14 but the corresponding term is a formal expression in the term algebra. So rather than (3), we shall write, from now on, (4)

t = β(α(few, cat), bark)
In other words, the α for α ∈ Σ are the operations in the term algebra corresponding to E. If we obey the constraints coming from the partiality of the functions in Σ, we get the grammatical terms. Forgetting those constraints we obtain the set T E of arbitrary terms, obtained by successively applying the operations α to any (term) arguments.

Each grammatical term maps to a unique string in E. In other words, there is a string value function V from GT E to E. For an atomic term like few, we have V (few) = few. To handle lexical ambiguity, we need distinct atomic terms with the same value, say, V (bank 1 ) = V (bank 2 ) = bank. For a complex term

α(t 1 , . . . , t n ) we have V (α(t 1 , . . . , t n )) = α(V (t 1 ), . . . , V (t n ))
where α is defined for the arguments V (t 1 ), . . . , V (t n ) precisely when the term

α(t 1 , . . . , t n ) is grammatical. (So V is a homomorphism from the term algebra
to the expression algebra E.) To illustrate:

V (β(α(few, cat), bark)) = β(V (α(few, cat)), V (bark)) = β(α(V (few), V (cat)), bark) = β(α(few, cat), bark) = β(few cats , bark) = few cats bark
Now we can let a semantics for E simply be a partial function µ from GT E to some set X of semantic values ('meanings'). The domain of µ can be a proper subset of GT E if, for example, some grammatical terms have no semantic value because no meaning of the selected kind fits them, or because none has yet been found (say, the language is still partly unknown to the theorist).

As noted, for compositionality the nature of the meanings doesn't matter, only sameness of meaning. µ induces a partial equivalence relation on E: for

u, t ∈ E, define u ≡ µ t iff µ(u), µ(t) are both defined and µ(u) = µ(t)
We sometimes call such relations synonymies, noting that this is a technical notion, relative to the chosen notion of meaning.

Compositionality

For a given grammar E and semantics µ, we render (PoC-F) as follows:

Funct(µ) For every rule α ∈ Σ there is a meaning operation r α such that if α(u 1 , . . . , u n ) has meaning (belongs to the domain of µ), then

µ(α(u 1 , . . . , u n )) = r α (µ(u 1 ), . . . , µ(u n )).
Here is the formal version of (PoC-S): Note that the formulation Funct(µ) presupposes the Domain Principle (DP):

Subst(≡ µ ) If s[u 1 , . . . ,
u i ≡ µ t i for 1 ≤ i ≤ n, then s[u 1 , . . . , u n ] ≡ µ s[t 1 , . . . ,
Subterms of meaningsful terms are meaningful. The following is well-known (e.g.

Hodges [START_REF] Hodges | Formal features of compositionality[END_REF]):

Fact 1 Under DP, Funct(µ) is equivalent to Subst(≡ µ ).
In some cases, DP seems too strong. For example, a semantics for first-order logic assigning meanings to closed but not to open formulas doesn't satisfy DP.

Likewise, a semantics for quotation that allows quotation of meaningless strings, as in (1c), may appear to violate DP. We will see, however (Sections 5 and In addition to our set GT E of terms corresponding to structured expressions, and our set X of semantic values, there is a set C of contexts. 16 We can then think of the assignment of values in two ways: It is trivial to define µ in terms of ν, and vice versa, by the equation

(7) a. ν : GT E -→ [C -→ X] b. µ : GT E × C -→ X Here f : A -→ B means that f is a (partial)
µ(t, c) = ν(t)(c)
In mathematical parlance, ν is the currying of µ, and µ is the uncurrying of ν.

Nevertheless, compositionality for the two functions is not the same. Indeed, the notion of compositionality from the preceding section applies only to ν. Funct(ν) is well-defined, but for µ we must define compositionality for semantics that take an extra context argument.

There is an obvious way to do this: think of the previous condition as parametric in the context. Just replace µ(t) with µ(t, c) everywhere. We obtain: 17 C-Funct(µ)For every rule α ∈ Σ there is a meaning operation r α such that for every context c, if α(u 1 , . . . , u n ) and u 1 , . . . , u n have meaning in c,

then µ(α(u 1 , . . . , u n ), c) = r α (µ(u 1 , c), . . . , µ(u n , c)).
Now, one might think that this is simply equivalent to Funct(ν). But it isn't.

First, it is easy to give examples where Funct(ν) holds but C-Funct(µ) fails..

Second
i , c 1 ) = µ(t i , c 2 ) for 1 ≤ i ≤ n, then µ(α(u 1 , . . . , u n ), c 1 ) = µ(α(t 1 , . . . , t n ), c 2 ).
But this does not extend to subterms that are more deeply embedded: that would require the subterms of s[u 1 , . . . , u n ] that are not replaced to have the same meaning in c 1 and c 2 , and there is no guarantee for that.

Third, there is a way in which C-Funct(µ) can fail merely by changing the context, without replacing any expressions at all, namely, if each of u 1 , . . . , u n has the same meaning in c 1 and c 2 but α(u 1 , . . . , u n ) doesn't. This is called 17 We make the simplifying assumption that the non-linguistic context c is the same for all parts of a complex term. This is sometimes violated; for example, when the sergeant, pointing in turn at three of the recruits, shouts (i) You, you, and you are volunteers!

We shall not pursue this here. When we come to linguistic context, however, it is a crucial feature that the context of the immediate subterms of a term can be different from the context of the term itself in a way that is semantically significant (see Section 4).

context shift failure in Pagin [START_REF] Pagin | Compositionality and context[END_REF], where possible examples are given. This kind of failure of compositionality is not available for ν.

Fourth, there is a weaker variant of C-Funct(µ), which also seems quite natural:

C-Funct(µ) w For every α ∈ Σ there is a meaning operation r α such that for every context c, if α(u 1 , . . . , u n ) and u 1 , . . . , u n have meaning in c, then

µ(α(u 1 , . . . , u n ), c) = r α (µ(u 1 , c), . . . , µ(u n , c), c).
The Moreover, we shall see that it is this version of contextual compositionality that lends itself to adaptation for linguistic contexts.

The following result explains the logical relations between the notions of compositionality discussed so far, with µ and ν as in [START_REF] Dowty | Compositionality as an empirical problem[END_REF] (see Westerståhl [44] for a proof and discussion):

Fact 2 C-Funct(µ) implies C-Funct(µ) w (equivalently, C-Subst(≡ µ ) w ), which in turn implies Funct(ν) (equivalently, Subst(≡ ν )
), but none of these implications can be reversed.

Linguistic Context

Often the interpretation of a linguistic expression on an occasion depends on other expressions that are used (or just have been used) on the same occasion.

This can happen in two main ways: by linguistic context-dependence proper, and also by way of the influence of linguistic expressions (used on the occasion) on the extra-linguistic context. We may call the latter 'linguistic contextual impact'. Consider [START_REF] Frege | Letter to Russell 28[END_REF] That is a big mouse.

Here it may be, or it may fail to be, the case that a standard or comparison class for size is given in the extra-linguistic context. If there is, e.g. the comparison class of animals in some particular laboratory, and it applies in the case of the utterance of ( 9), then big applies intersectively with respect to mouse. But the occurrence of mouse may also set a new standard of size in the context, whether

or not there was a standard in the immediately preceding context. In that case the complex noun big mouse will apply to anything that (within some relevant domain) is big for a mouse. 'big' still applies intersectively, but the standards have changed. This phenomenon is worthy of further study (see e.g. Recanati [START_REF] Recanati | Compositionality, flexibility and context-dependence[END_REF] for a similar view, and Kennedy [START_REF] Kennedy | Vagueness and grammar: the semantics of relative and absolute gradable adjectives[END_REF] for more on gradable adjectives), but is mentioned here only as a contrast to the present topic, that of linguistic context dependence proper.

Sentential contexts

What is a linguistic context? Here we restrict attention to the context of an expression within a larger expression, notably a sentence. The basic idea is simple enough: take a well-formed complex term s[u] and knock out the constituent

u. What remains, s[. . .], is the linguistic context of that occurrence of u in s[u],
the environment of the argument place. This simple idea needs to be refined, and we shall see how.

Standard examples of linguistic contexts that are relevant to semantics include belief contexts. In a belief sentence like [START_REF] Geach | Mental Acts[END_REF] John believes that Jenny is hungry the embedded sentence [START_REF] Glüer | Proper names and relational modality[END_REF] Jenny is hungry occurs in the belief context [START_REF] Glüer | Relational modality[END_REF] John believes that . . . According to Frege [START_REF] Frege | Über Sinn und Bedeutung[END_REF], that [START_REF] Glüer | Proper names and relational modality[END_REF] occurs in a belief context has an effect on its meaning in that context: it has indirect reference (ungerade Bedeutung).

This view gives us a first reason for revising the simple idea: the occurrence of the subject John does not participate in defining the belief context. What matters for being in a belief context is that the expression occurs in the second argument place of the verb believes. What occurs in the first argument place, John, matters to the truth conditions of the sentence as a whole, but has no effect on the semantics for the embedded sentence itself.

For definiteness, let the construction of belief VPs be given by two one-place rules: α bel , where the argument is the sentence term for the that-clause (Complementizer Phrase) argument, and α cp , which takes the embedded sentence as argument and gives the that-clause as value (various other formats are possible).

Figure 1 gives a simplified tree representation.

The belief context is the context of the position of the dominated S node in the tree, where the embedded sentence goes. In the term notation, this is

α bel (α cp (. . .)).
The example illustrates that we should not rule out that what matters to determining the linguistic context can extend beyond the immediate syntactic rule. In this case, two rules are required.

There is a further important complication. On a standard conception, if a complex expression occurs in a belief context, then the constituents of that expression also occur in the belief context. For instance, the name Jenny occurs in the belief context just like the entire sentence [START_REF] Glüer | Proper names and relational modality[END_REF]. So on this conception, being in a particular linguistic context is like being in the scope of an operator.

But this is not quite right, or at least highly misleading, as can be seen by considering iterated belief contexts. In [START_REF] Glüer | General terms and relational modality[END_REF] Emma believes that John believes that Jenny is hungry the name John occurs in a simple belief context, but Jenny occurs in an iterated, and more precisely second degree, belief context. This can matter to semantics. On a common interpretation of Frege, expressions in second degree belief contexts have a doubly indirect reference and a doubly indirect sense. 18 This shows that in general we need to take into account several constituent levels in the syntactic term, and for each level the relevant argument place. Therefore it is appropriate to let linguistic context be defined in terms of operators and 18 There is some textual evidence that this, at least at some point, was Frege's own view.

The following is a quote from a letter to Russell 28 December 1902:

Man kann sagen, dass wir im doppelt Unterstrichenen die ungerade Bedeutung zweiten Grades haben, in dem einmal Unterstrichenen dagegen die ungerade Bedeutung ersten Grades ([9], p. 236).

Here Frege explicitly refers to "an indirect reference of second degree".

argument positions down to the target argument place. We now give a precise version of this idea.

A formal notion of linguistic context

The notion of linguistic context applies to occurrences of (sub)terms. We let o (with subscripts) range over occurrences. 19 We say that ξ is a possible context for t if there is a term s which has a subterm occurrence o of t such that ctx(o, s) = ξ. Thus, CXT E is the set of possible contexts of grammatical terms in E. If ξ is a possible context for some term α(u 1 , . . . , u n ), then ξ (α, i) is a possible context for u i . In this case, we 19 We use an informal notion of term occurrence here. It could be redefined later in terms of the function cxt introduced below. 20 For example, ξ = (α, 1), (β, 2) is a context in each of the terms α(β(t, u), γ(t)), α(β(t, u), t ), and α(β(t, δ(u)), γ(t)). The subterm occurring at context ξ is u in the first two cases, and δ(u) in the third case.

say that ξ allows extension by α.

CXT E may be large -usually infinite -but when studying the effect of linguistic context on meaning we may be interested only in a small number of types of contexts. For example, Frege in the quote earlier in effect isolates three types of context: an 'ordinary' (null) context type (when talking about the references of words), a quotation context type (talking about the words themselves), and an 'intensional' context type (talking about their senses). This motivates the following definition:

(L ctype )
Acontext typing is a partition C of CXT E satisfying the following requirement, where [ξ] is the equivalence class of ξ:

(i) If [ξ] = [ξ ]
, and ξ and ξ both allow extension by an n-ary

grammar rule α, then, [ξ (α, i) ] = [ξ (α, i) ] for 1 ≤ i ≤ n.
The elements of C are called context types.

Requirement (i) says that if two contexts are of the same type, and can be extended in the same way, then the corresponding extensions also have the same type. This means that the context type of a subterm occurrence α(o 1 , . . . , o k ) in s determines the context types of o 1 , . . . , o k in a way which is independent of s, according to the partial function Φ C , which takes a n-ary grammar rule α, a natural number i (an argument place of α), and a context type c to a context type Φ C (α, i, c), and is defined as follows:

(14) Φ C (α, i, [ξ]) =    [ξ (α, i) ] 1≤ i ≤ n,

General Compositionality

The first main idea for an extension of a semantic framework to handle linguistic context dependence is that when a semantic function µ is applied to a term s, the value depends on the relevant context type. The second main idea is that the context type of an immediate subterm of a term s may be different from the context type of s itself. As we shall see, there are two equivalent styles of implementing this: either with a semantic function that takes a context type as a second argument, or instead with a set of ordinary semantic functions, one for each context type.

A formulation in terms of formal contexts

The first style builds on the formal notion of context just introduced. Assume a context typing C of CXT E according to (L ctype ) is given. Just as for extralinguistic context (Section 3), a semantics µ now takes an additional argument:

it is a partial function from GT E × C.

For simplicity, we shall also require that a Generalized Domain Principle, holds for µ:

(GDP µ ) If µ(α(u 1 , . . . , u n ), c) is defined, then µ(u i , Φ C (α, i, c)) is defined for 1 ≤ i ≤ n.
In Section 2.2 we noted that DP is sometimes too demanding. When meaning is allowed to depend on linguistic context, however, the case for a (generalized) domain principle is stronger. Rather than not assigning any meaning at all to some subterms of a complex term, we can assign them a different meaning when they are in that context (type). In Section 7 we will see an example how this works out, in a way consistent with GDP µ . Now, the compositionality requirement is similar to C-Funct(µ) w , the dif-ference being that the context types of the arguments of complex terms may change; indeed the function Φ C assigns those types. The linguistic contexts of the arguments are different from the context of the term itself, and that difference may be semantically relevant. For example, the context type of the complex term may be a quotation type.

LC-Funct(µ, C)

For every α ∈ Σ there is an operation r α such that for every

c ∈ C, if µ(α(u 1 , . . . , u n ), c) is defined, then (i) µ(α(u 1 , . . . , u n ), c) = r α (µ(u 1 , c 1 ), . . . , µ(u n , c n ), c),
where

c i = Φ C (α, i, c), 1 ≤ i ≤ n.

A formulation with sets of semantic functions

In the second style, we keep the standard notion of a semantics as an assignment of meanings to grammatical terms, but we use several such semantic functions.

Intuitively, there is one function for each context type, but this formulation doesn't require a formal notion of context. We simply suppose that a set S of (ordinary) semantic functions is given, together with a selection function Ψ; a partial function from triples of a grammar rule, a natural number, and an element of S, to S. The Generalized Domain Principle now takes the form:

(GDP S ) If µ ∈ S and µ(α(u 1 , . . . , u n )) is defined, then Ψ(α, i, µ)(u i ) is de- fined, 1 ≤ i ≤ n.
We also assume that there is a designated (null) function µ d ∈ S, the default semantics corresponding to the null context type. µ d , however, plays no explicit role in the compositionality condition, which, for S satisfying GDP S , becomes:

LC-Funct(S, Ψ) For every α ∈ Σ and every µ ∈ S there is an operation r α,µ

such that if µ(α(u 1 , . . . , u n )) is defined, then (ii) µ(α(u 1 , . . . , u n )) = r α,µ (µ 1 (u 1 ), . . . , µ n (u n )),
where

µ i = Ψ(α, i, µ), 1 ≤ i ≤ n.

Some facts

The two generalized versions of contextual compositionality have standard compositionality as a special case, in the sense of the next, easily verified, fact.

Fact 3

If µ is a partial function from GT E such that DP holds, then Funct(µ) is equivalent to each the following:

(a) LC-Funct({µ}, Ψ 0 ), where Ψ 0 (α, i, µ) = µ. (b) LC-Funct(µ * , {CXT E }),
) = µ(t) (if µ(t) is defined).
Next, we can show that LC-Funct(µ, C) and LC-Funct(S, Ψ) are equivalent in the following sense: from CXT E to S inductively by:

Proposition 4 (a) If LC-Funct(µ, C) holds, define, for c ∈ C, µ c (t) = µ(t, c) (if defined), and let Ψ(α, i, µ c ) = µ ΦC (α,i,c) (if Φ C (α, i, c) is defined), 1 ≤ i ≤ n. Then
         F ( ) = µ d (the null function in S) F (ξ (α, i) ) = Ψ(α, i, F (ξ)), if for some term α(u 1 , . . . , u n ), F (ξ)(α(u 1 , . . . , u n )) is defined (otherwise undefined)
The functions in S themselves are taken to be the context types, or more exactly, their inverse images under F : let The proof of this result is given in Appendix 2. In view of Proposition 4, we shall refer to LC-Funct(µ, C) and LC-Funct(S, Ψ) indiscriminately as general compositionality, or g-compositionality.

[ξ] = {ξ : F (ξ ) is defined iff F (ξ)

Synonymies

With the general versions of functional compositionality, we obtain, for each choice of (S, Ψ), a variety of synonymy relations. For every µ ∈ S there is a corresponding synonymy relation ≡ µ . This gives synonymy relative to members of S. But we can also distinguish three absolute notions. First, define

u i ≡ S des u j iff u i ≡ µ d u j (µ d is the designated function in S)
Call this designated synonymy. Another natural candidate is total synonymy:

u i ≡ Stot u j iff for every µ ∈ S, u i ≡ µ u j
As is intuitively clear, and as will follow from the semantics of the following section, if there is a quotation context type in E, then there will be no nontrivial total synonymy pairs: a term s is totally synonymous only with itself.

Because of this, we may be interested in a concept of total synonymy except for quotation, i.e. synonymy in all pure use-contexts (as opposed to quotation contexts, where expressions are both used and mentioned). Call this strong notion use synonymy (which for languages without quotation coincides with total synonymy); like the others, it is a partial equivalence relation on the set of grammatical terms.

Finally, let us consider the substitution version of general compositionality.

We use the version LC-Funct(µ, C). First, modify the notation s[u 1 , . . . , u n ] as follows: let

s = s[(u 1 , ξ 1 ), . . . , (u n , ξ n )]
indicate that, for 1 ≤ i ≤ n, at context ξ i in s there is an occurrence of u i , where ξ i and ξ j are pairwise disjoint for i = j, i.e. neither is an extension of the other. Then

t = s[(t 1 , ξ 1 ), . . . , (t n , ξ n )]
is the unique term obtained from s by replacing the occurrence of u i at ξ i by t i .

The disjointness assumption entails that the result is independent of whether the replacements are done simultaneously or sequentially.

We also need recourse to the function Θ C from ( 15) in Section 4.2. Then we can state:

LC-Subst(µ, C) If s = s[(u 1 , ξ 1 ), . . . , (u n , ξ n )] and t = s[(t 1 , ξ 1 ), . . . , (t n , ξ n )]
are both µ-meaningful in the context type c ∈ C, and if, for

1 ≤ i ≤ n, µ(u i , c i ) = µ(t i , c i ), where c i = Θ C (c, ξ i ), then µ(s, c) = µ(t, c).
This says that, for any c, the semantic values of two terms are the same in context type c if one results from the other by means of substitution of subterms that are pairwise µ-equivalent in the type of context where they occur.

Fact 5

If (GDP µ ) holds, LC-Funct(µ, C) is equivalent to LC-Subst(µ, C).

The left-to-right direction of Fact 5 is proved by means of induction over term complexity. The right-to-left direction follows immediately from the special case when the substitution terms are the immediate subterms. Typically, however, we are initially given a grammar E and an initial single unary semantic function µ for E. In some cases, for instance with a straightforward account of quotation, µ is non-compositional, as we have seen. We are then interested in whether there is a proper corresponding µ-related pair (µ , C)

Compositional Generalization

or (S, Ψ) that satisfies LC-Funct(µ, C) or LC-Funct(S, Ψ), respectively. We use the following terminology:

(16) (µ , C) generalizes µ iff for all t, µ(t) = µ (t, [ ]). Similarly, (S, Ψ)
generalizes µ iff µ is the designated function in S.

Now we are asking when µ is in a relevant sense compositionally generalizable.

At first blush, the relevant sense seems easy to spell out. In the LC-Funct(S, Ψ) format, the first suggestion would be: [START_REF] Kennedy | Vagueness and grammar: the semantics of relative and absolute gradable adjectives[END_REF] µ is compositionally generalizable iff LC-Funct(S, Ψ) holds for some pair (S, Ψ) generalizing µ.

But this requirement is empty: in the sense of [START_REF] Kennedy | Vagueness and grammar: the semantics of relative and absolute gradable adjectives[END_REF], any semantic function µ is generalizable. To see this, define the function ν by ν(t) = µ(t), t . Let S = {µ, ν}, and Ψ(α, i, µ) = Ψ(α, i, ν) = ν, for any α and i. Then LC-Funct(S, Ψ) holds. 21 In the above definition of ν, general compositionality is achieved by way of 21 Proof. We must find, for each α, meaning operations rα,µ and rα,ν that satisfy the compositional recursion equations. Let

rα,µ(m 1 , . . . , mn) = µ(α(π 2 (m 1 ), . . . , π 2 (mn))) rα,ν (m 1 , . . . , mn) = µ(α(π 2 (m 1 ), . . . , π 2 (mn))), α(π 2 (m 1 ), . . . , π 2 (mn))
where π 2 is a right projection function: π 2 ( x, y ) = y. Then, with ν, S, and Ψ as above:

(i) For any unary function µ, LC-Funct(S, Ψ) holds.

(MAX) If ξ is a context such that for all terms s containing ξ, s[(u, ξ)] ≡ µ s[(t, ξ)], then u ≡ µ [ξ] t.
Informally, if u and t make the same contribution to the meaning of complex expressions in which they occur, in a certain linguistic context, then their meaning in the type of that context is the same. This is similar to Hodges' notion of a Fregean extension (or full abstraction; see Hodges [START_REF] Hodges | Formal features of compositionality[END_REF]), which can be seen as expressing a version of Frege's Context Principle, but this time for meanings that are sensitive to linguistic context.

Suppose the trivial generalization of µ into S = {µ, ν} above satisfies MAX.

Since for all distinct terms t, u, ν(t) = ν(u), this means that in the ν context type, to which every embedded (non-null) context belongs according to the associated Ψ, distinct terms are semantically non-equivalent. By MAX, it follows that for any distinct terms t, u, and for any embedded context ξ where t and u can occur, there is a term s such that

s[(u, ξ)] ≡ µ s[(t, ξ)]
As a consequence, if for any two terms t and u, µ(t) = µ(u), then there is substitution failure in every embedded linguistic context (although not for every term that contains it). µ is then non-compositional, not just with respect to some operator α, but with respect to every syntactic operator. Although such semantic functions are possible, we seem not to find them in natural language.

Generating a language L by adding quotation to a language L does make L non-compositional if there are non-trivial synonymies in L, but only because of the quotation contexts. It does not make all the operators in the L fragment non-compositional as well. So, for an ordinary semantics µ, generalizing it into S = {µ, ν} as above does not satisfy MAX. MAX seems to be a reasonable requirement on generalization.

However, even under MAX, any unary semantic function µ turns out to be generalizable to some (S, Ψ) satisfying provided there is no limitation on the number of context types:

Fact 6

For any unary semantics µ, there is (S, Ψ) generalizing µ such both MAX and LC-Funct(S, Ψ) hold.

The key to the proof, which is given in Appendix 2, is to use many context types; in fact, we let each context be its own type. Even though that is hardly a situation one encounters in practice, there is thus reason to introduce a second requirement on compositional generalizability.

The type of a context ξ may in principle depend on all the operators in ξ (recall that the formal notion of a context is a sequence starting with the null context, followed by operator-index pairs), in the order in which they occur, or on any particular subset of the set of operators in ξ, again in their order in ξ.

In the simplest case, however, only one operator matters. We can spell this out as follows.

Call an operator-index pair (α, i) is a switcher (relative to a context typing C) iff there are at least two context types, and for any two contexts ξ, ξ in which a term α(t 1 , . . . , t n ) can occur,

[ξ (α, i) ] = [ξ (α, i) ]
This means that there is a context type c such that whatever the context type of (a grammatical occurrence of) α(t 1 , . . . , t n ), the context type of the corresponding occurrence of t i is c. Let us also say that (α, i) is a keeper iff there are at least two context types, and for any admissible context ξ,

[ξ (α, i) ] = [ξ]
This means that (α, i) never changes context type. We can now define a firstgrade g-compositional semantics (S, Ψ) to be a g-compositional semantics where every operator-index pair is either a switcher or a keeper. In such a semantics, the context type of a context ξ is determined by the last switcher pair in the sequence that constitutes ξ, or else is the default type if there is no switcher in the sequence. If all pairs are switcher pairs, then the type of a context ξ (α, i) is always determined by (α, i), irrespective of ξ .

We can analogously define a second-grade g-compositional semantics by defining switcher pairs of operator-index pairs. In this case, the type of a con- Clearly, along this parameter, the simplest g-compositional semantics are the first-grade ones. The number of context types for such semantics is at most equal to the number of operator-index pairs. A particular operator-index pair is responsible for the context type. The number of rules can be kept low, and no more than one operator-index pair needs to be kept in memory at any single recursion step of processing the semantics. This indicates that the following terminology is appropriate: 22(GEN) µ is compositionally generalizable iff there is a first-grade pair (S, Ψ)

generalizing µ such that LC-Funct(S, Ψ) and MAX hold.

GEN is not an empty requirement. A proof of the following is outlined in Appendix 2:

Fact 7

For each n there is a grammar E and a semantics µ for E which is not generalizable by any nth-grade pair (S, Ψ) satisfying LC-Funct and MAX.

The g-compositional semantics for pure quotation given in the next section satisfies GEN. The one-place quotation operator always triggers a quotation context type. All other operator-index pairs are keepers, preserving either the quotation context type or the default type. In particular, there is no switcher that takes the semantics out of a pure quotation context, which seems in accordance with ordinary intuitions (but see note 27). 23 requirement. 23 General compositional semantics appear in earlier work by Kathrin Glüer and Peter Pagin.

[11] presents a semantics accounting for the behavior of proper names in modal contexts. The idea is to keep proper names as non-rigid designators, with non-constant intensions, and achieve the rigidity-like effect by means of the interaction of the modal operator with the term. The modal operator then acts as a switcher, creating an actualist context where the name reference is the value of its intension in the actual world.

The resulting semantics is equivalent to a standard semantics with rigid designators w.r.t.

truth, and almost equivalent w.r.t. logical consequence, as shown in [START_REF] Glüer | Relational modality[END_REF]. In [START_REF] Glüer | Proper names and relational modality[END_REF], an appendix considers the addition of a naive belief operator that switches from an actualist context type back to the default possibilist context type.

The idea has also been extended to handle general terms in modal contexts in [START_REF] Glüer | General terms and relational modality[END_REF]. In unpublished work ([21]), Pagin develops a more adequate g-compositional account of belief sentences. In all these cases, GEN is in fact satisfied, although this condition is not mentioned.

A G-compositional and Straightforward

Account of Pure Quotation

Our final task is to show that a straightforward semantics for pure quotation (Section 1) is compositionally generalizable in the sense of GEN. We formulate this as a compositional extension result: given any semantics µ which is compositional in the ordinary sense, there is a natural way to extend it to a semantics which handles pure quotation straightforwardly and is compositionally generalizable. 24 We use the formulation LC-Funct(S, Ψ), but Proposition 4 provides an immediate translation to the version with explicit context types.

Let, then, E = (E, A, Σ) be a given grammar and µ a semantics for E such that Funct(µ) holds. There is a string value function V from GT E to E (Section where µ is µ extended to the new terms, and V the corresponding string value function. Roughly, the claim is that the set S = {µ , V } (with a suitable selection function) is g-compositional and satisfies GEN with respect to µ .

However, to deal with sentences like those in (1), Section 1, we need to quote not only well-formed expressions but also, say, single letters of some alphabet, expressions in other languages or dialects, misspellings, even arbitrary concatenations of letters. A general theory of quotation must take care to handle such cases correctly. Since we are restricting attention here to pure quotation, we shall simply add a mechanism for quoting arbitrary finite strings of letters from some suitable alphabet L, which is assumed to contain the letters occurring 24 A compositional extension result has the form: if µ is a compositional semantics for E, there is a (possibly unique) extension µ of µ satisfying a certain property. Hodges [START_REF] Hodges | Formal features of compositionality[END_REF] and Westerståhl [START_REF] Westerståhl | On the compositional extension problem[END_REF] prove theorems concerning the extension from a partial to a total semantics.

Westerståhl [START_REF] Westerståhl | On the compositionality of idioms; an abstract approach[END_REF] gives extension results for adding idiomatic expressions to a language.

in atoms of E as well as a space symbol and the quote marks. We then have

A ⊆ E ⊆ L *
, where L * is the set of all finite strings of letters. Some strings in L * are well-formed, most strings aren't, but in principle we should be able to quote all of them.

One way to achieve this is to add all strings of the form 'u', when u ∈ L * , as new atoms. This is the proper name theory of quotation, and we indicated in Section 1 why it is unsatisfactory: (a) we get infinitely many primitive expressions, and (b) quoting expressions have no structure. To remove (a), we generate all strings from letters and the concatenation operation. (b) is removed by using κ whenever we want to quote something (grammatical or not), thus keeping the account straightforward. 25In more detail, we extend the grammar E as follows. Let the new set of atoms be

(19) A = A ∪ L (assuming A ∩ L = ∅).
     V (a) = a, for a ∈ A V (δ(t 1 , . . . , t n )) = δ(V (t 1 ), . . . , V (t n )), for δ ∈ Σ (23)
It is now fairly clear how the given semantics µ can be extended to a semantics µ suitable for E . We adjoin L * to the given domain of interpretation M (assuming M ∩ L * = ∅), and adapt the given composition functions r α for α ∈ Σ to functions r α that work for the new terms. The meaning of terms of the form κ(t) was given in (18b), but µ is undefined for string terms.

As we have emphasized, the extended semantics µ is not compositional : one cannot substitute synonymous terms in the range of κ and expect meaning to be preserved. However, (S, Ψ) is g-compositional, where S = {µ , V }, and the selection function Ψ is defined, for δ ∈ Σ and ν ∈ S, by

Ψ(δ, i, ν) =    V if δ = κ ν otherwise (24)
That is, µ only switches to V when something is quoted, and then every more deeply embedded subterm is also evaluated with V . 27Next, LC-Funct(S, Ψ) requires us to define operations r δ,ν for each δ ∈ Σ and ν ∈ S. Specifically, we must satisfy

(25) V (δ(t 1 , . . . , t n )) = r δ,V (V (t 1 ), . . . , V (t n ))
so it follows from (23) that we should set (26) r δ,V = δ for all δ ∈ Σ . In particular, when δ = κ we then have

(27) r κ,V (V (t)) = κ(V (t)) = 'V (t)'
this will apply only in iterated quotation contexts (see the example below). As to r δ,µ , we set

     r α ,µ (m 1 , . . . , m n ) = r α (m 1 , . . . , m n ), for α ∈ Σ r κ,µ (m) = m (28)
(We don't need to define r cc,µ , since µ (cc(t, u)) is always undefined.)

In particular, we get

(29) µ (κ(t)) = r κ,µ (V (t)) = V (t)
in accordance with [START_REF] Lewis | Index, context, and content[END_REF].

To see how S and Ψ work, consider sentence [START_REF] Pagin | General compositionality and belief sentences[END_REF], and suppose, for simplicity of illustration, that this string is the value of the term t = isa (κ(α (John, β (like, κ(Mary)))), sentence)

(so sentences of the form 'NP is a N' are analyzed as isa(NP,N)). Its designated meaning is calculated, using ( 23) -( 29), as shown in Figure 2. Here we have We can summarize our extension result for quotation as follows:

µ (t) = r isa ,µ (µ (κ(α (John, β (like, κ(Mary))))), µ (sentence)) = r isa (µ (κ(α (John, β (like, κ(Mary))))), SENT) = r isa (V (α (John, β (like, κ(Mary)))), SENT) = r isa (α (V (John), V (β (like, κ(Mary)))), SENT) = r isa (α (John, β (V (like), V (κ(Mary)))), SENT) = r isa (α (John, β (like, κ(V (Mary)))), SENT) = r isa (α (

Fact 8

Suppose Funct(µ) holds, and µ is extended to a semantics µ for E , which is generalized to (S, Ψ) as above.

(a) µ handles pure quotation correctly, i.e. it gives the intended meanings to sentences containing such quotation.

(b) µ extends µ in the sense that when t ∈ GT E , µ (t) = µ(t).

(c) If µ satisfies DP, then GDP S and LC-Funct(S, Ψ) hold, even though Funct(µ ) in general fails.

(d) (S, Ψ) satisfies GEN: it is first-grade and MAX holds (the semantics is hyperdistinct in quotation contexts while µ itself applies in the null context type.)

In the absence of a detailed specification of the given grammar E, we have only been able to indicate certain parts of the extension, but it should be fairly clear how to apply these indications to specific grammars.

The extended semantics allows quotation of an arbitrary string z in L * , using κ(s), where s is a string term such that V (s) = z. So it can handle all the examples (1). If z is also the string value of a µ -meaningful grammatical term t, we can quote t instead, with the same result: µ (κ(t)) = µ (κ(s)). Using t is then more natural, and will be important, for example, when extending the account to deal with mixed quotation. 28 28 What about quoting ill-formed expressions, such as 'eckullectic' uttered by Bush, as discussed in Shan [START_REF] Shan | The character of quotation[END_REF]? There seem to be three alternatives: (a) the string is in fact part of Bush's vocabulary, in which case it is well-formed, but part of an unusual idiolect; (b) the utterance was a temporary performance slip, and the reporter uses the string literally to represent Bush's speech performance, but then it is not a case of quotation proper, rather a case of depicting speech by other means (see note 5); (c) the report about Bush really means that he made an utterance that could be reproduced by a standard-rule pronunciation of the string 'eckullectic', this is the case we handled in this section. Shan deals with case (a) by adding a parallel language (Bush English) to the fragment, specifying formally how it can be accessed from the main language. We have not attempted anything similar here.

every expression can be quoted. One can think of this carried out in such a way that each term t of the old grammar now comes in two versions, t d and t q . However, the distribution of these subscripts is not arbitrary. For example, one should not allow a d(efault) subscript inside a complex term with a q(uote) subscript. To be sure, it is easy to state the rule governing the correct distribution; the rule would exactly mirror our selection function Ψ in Section 7. In other words, the result would essentially be a notational variant of our proposal. Where we let µ evaluate t in context type c (or use more than one meaning assignment), the alternative account, using a meaning assignment ν, say, would evaluate t c : ν(t c ) = µ(t, c).

But here is the crucial point. It might seem that, in spite of the problems already mentioned, the ambiguity account has the advantage of being able to rely on the standard notion of compositionality, since there is just one meaning assignment, and it applies just to (indexed) terms. If that were true, it would indeed be an advantage. But it isn't: the alternative account would still only be general compositional. To see this, recall that LC-Funct(µ, C) requires that to α corresponds an operation r α such that for every context type c, µ(α(u 1 , . . . , u n ), c) = r α (µ(u 1 , c 1 ), . . . , µ(u n , c n ), c),

where c i = Φ C (α, i, c). The alternative account would need a similar requirement for the meaning of α(u 1 , . . . , u n ) c . This has to take the form [START_REF] Quine | Word and Object[END_REF] ν(α(u 1 , . . . , u n ) c ) = r α (ν(u 1c 1 ), . . . , ν(u nc n ), c) Thus, we still need a rule to give us the indices c 1 , . . . , c n and, more importantly, we still need the semantic operation to take c as an argument. There is no operation s α such that (30) can be written ν(α(u 1 , . . . , u n ) c ) = s α (ν(u 1c 1 ), . . . , ν(u nc n ))

In other words, the meaning of a complex expression is not determined by the meaning of its immediate parts and the mode of composition (α), you also need and F (ξ (α, i) ) are both undefined. This proves (i).

Next, it clearly follows from our definitions that the function ν is well-defined, i.e. that if [ξ] = [ξ ], then, for any term t, ν(t, ξ) is defined iff ν(t, ξ ) is defined, and when both are defined they have the same value.

We must also verify that GDP ν holds. Suppose ν(α(u 1 , . . . , u n ), [ξ]) is defined. Then F (ξ) is defined and ν(α(u 1 , . . . , u n ), [ξ]) = F (ξ)(α(u 1 , . . . , u n )). By GDP S and the definition of F , Ψ(α, i, F (ξ))(u i ) = F (ξ (α, i) )(u i ) is defined.

Thus, each ν(u i , [ξ (α, i) ]) is defined. Again, this reasoning is independent of the choice of context in [ξ]. Thus, GDP ν holds. Then (cf. Hodges [START_REF] Hodges | Formal features of compositionality[END_REF], Lemma 1), we have

≡ µ [ξ] = ≡ ξ
Also,

µ [ ] = µ so (S, Ψ) generalizes µ, with S = {µ [ξ] : ξ ∈ CXT E }, and

Ψ(α, i, µ [ξ] ) = µ [ξ (α,i) ]
Now it is immediate from (31) that MAX holds. To show compositionality, we prove: But this means that (34) entails LC-Subst(µ, C), or if you wish the correspond-ing principle for (S, Ψ). Note here that LC-Subst(µ, C) is about simultaneous substitution of terms, but in view of (31), if t ≡ ξ ξ u, substitution of u for t can never lead from a meaningful to a meaningless term. Therefore, the substitutions in LC-Subst(µ, C) can be performed one by one, 29 and (34) says that meaning is preserved at each step. Thus, LC-Funct(S, Ψ) holds as well (cf. We claim that µ is not compositionally generalizable. Assume, for reductio, that there is a pair (S, Ψ) generalizing µ that satisfies GEN. Observe first that since in the context ξ = (α, 1) we have It then follows by MAX that 29 Hodges [START_REF] Hodges | Formal features of compositionality[END_REF] notes that a sufficient condition for this is the Husserl property, i.e. that synonymous terms are meaningful in the same linguistic contexts. The semantics given in [START_REF] Recanati | Open quotation[END_REF] clearly has the Husserl property.

  [[•]] gives model-theoretic denotations via a recursive truth definition. Quotation is the following syntactic rule, for P = Π; Σ; α : σ , q(P) = Π; Σ; 'P' : u with the corresponding semantic interpretation rule µ(q(P)) = [[SEM(q(P))]] = [['P']] = P ∈ Du where Du is the set of grammatical triples (treated as a basic domain besides De and Dt).

  function from the set A to the set B, [A -→ B] is the set of such functions, and A × B is the set of pairs (a, b) such that a ∈ A and b ∈ B. So ν is a meaning assignment of the kind treated above: it assigns values directly to terms, although these values are now themselves functions. It corresponds to what Kaplan called character. µ, on the other hand, assigns values to 'expressions-in-context'. Is this more than a notational difference, more than two ways to think of character?

Figure 1 :

 1 Figure1: A phrase structure tree for[START_REF] Geach | Mental Acts[END_REF].

GDP

  S and LC-Funct(S, Ψ) hold for S = {µ c : c ∈ C}. (b) Conversely, suppose LC-Funct(S, Ψ) holds. Define a partial mapping F

  is defined, and then F (ξ ) = F (ξ)} and let C = {[ξ] : ξ ∈ CXT E }. Define a semantics ν, taking a term t and a context type [ξ] as arguments, by ν(t, [ξ]) = F (ξ)(t), if both F (ξ) and F (ξ)(t) are defined (otherwise undefined). Then GDP ν is satisfied, C is a context typing of CXT E , and LC-Funct(ν, C) holds.

  LC-Funct(µ, C) holds for a pair (µ, C) of a binary function µ and a context typing C. Similarly, LC-Funct(S, Ψ) holds for a pair (S, Ψ) of a set S of unary functions and a selection function Ψ (for LC-Funct(µ, C) the function Φ C is determined by the typing itself according to[START_REF] Hinzen | The Oxford Handbook of Compositionality[END_REF]).

2 29hyperdistinctness:

 2 if ν(t) = ν(u), then t = u. This also ensures ordinary compositionality, in a degenerate way (cf. (5) and (6) in Section 2.2). It is reasonable to block this move by requiring that a semantic function for a particular context type respect the semantic equivalences in that context for the original function µ. Staying in the LC-Funct(S, Ψ) format, recall from Proposition 4 that each function in S has the form µ c , for some context type c in the corresponding typing. The requirement can then be stated as follows (with the notation from Section 5.4):

  text ξ is determined by the last switcher pair of operator-index pairs in the sequence that constitutes ξ. Similarly, switcher triples determine context type in a third-grade semantics. And so on. A Fregean semantics for belief contexts, where each iteration of the belief operator switches to a new type in an infinite hierarchy of indirect context types, is an ω-grade semantics.

2. 1 )

 1 . Now add a new syntactic operation κ that puts quote marks around any expression e, and interprets the result as referring to e:[START_REF] Lewis | Index, context, and content[END_REF] a. κ(e) = 'e' (i.e. the string leftquote e rightquote) b. µ (κ(t)) = V (t)

Figure 2 :

 2 Figure 2: Evaluation of (21)

GDP

  S such that LC-Funct(S, Ψ) holds. First, note that F as defined in (b) is a well-defined partial function, by induction over the length of ξ: if it is clear for which contexts of length at most k F is defined and what its values are, and η is a context of length k + 1, then η has the form ξ (α, i) . If F (ξ) is undefined, or if F (ξ) is an element of S which is not defined for any term of theform α(u 1 , . . . , u n ), then F (η) is undefined. Otherwise, F (η) = Ψ(α, i, F (ξ));this is defined by GDP S .Next, note that the definition of the sets [ξ] is formulated in such a waythat if F (ξ) is defined, [ξ] = {ξ : F (ξ ) = F (ξ)}, whereas if F (ξ) is undefined, [ξ] = {ξ : F (ξ ) is undefined}. Thus, C is a partition of CXT E . We must check that C is a context typing, i.e. that condition (i) in (L ctype ) holds. So suppose [ξ] = [ξ ]. If F (ξ) is defined, then so is F (ξ ) and F (ξ ) = F (ξ). If there is a term α(u 1 , . . . , u n ) such that F (ξ)(α(u 1 , . . . , u n )) is defined, then F (ξ (α, i) ) is defined, and Ψ(α, i, F (ξ)) = Ψ(α, i, F (ξ )), so F (ξ (α, i) ) = F (ξ (α, i) ), i.e. [ξ (α, i) ] = [ξ (α, i) ], as desired.If there is no such term, then F (ξ (α, i) ) and F (ξ (α, i) ) are both undefined, so again [ξ (α, i) ] = [ξ (α, i) ]. Finally, if F (ξ) is undefined, so is F (ξ ), and again F (ξ (α, i) )

Finally, take a

  rule α and a context type c ∈ C such that ν(α(u 1 , . . . , u n ), c) is defined. Let c = [ξ]. Using LC-Funct(S, Ψ) and the definitions of F , i fξ = and µ(t) is defined [[t]] ξ if ξ = and t ∈ dom(≡ ξ ) undefined otherwise

( 34 )

 34 If s contains ξ, and if t ≡ ξ ξ u, then s[(t, ξ)] ≡ ξ s[(u, ξ)]. For let s be any term containing the context ξ . Then, for all terms v, s = s [(s[(v, ξ)], ξ )] contains the context ξ ξ, and s [(s[(v, ξ)], ξ )] = s [(v, ξ ξ)] By assumption and (31), s [(t, ξ ξ)] ≡ µ s [(u, ξ ξ)], and so s [(s[(t, ξ)], ξ )] ≡ µ s [(s[(u, ξ)], ξ )]. Since s was arbitrary, it follows by (31) that s[(t, ξ)] ≡ ξ s[(u, ξ)]. This proves (34). Now, in the chosen context typing C, the function Θ C from (15) in Section 4.2 is simply given by Θ C ([ξ ], ξ) = [ξ ξ]

Proposition 4 and Fact 5.) 2 Proof of Fact 7 :

 27 We prove the case n = 1 (i.e. GEN) and indicate how the idea extends to other n. Consider a grammar with three atomic terms a, b, c and one one-place operator α, and let µ be a total semantics such that a ≡ µ b ≡ µ c, α(a) ≡ µ α(b) ≡ µ α(c), and α(α(a)) ≡ µ α(α(b)).

  α(b) = α(b)[(b, ξ)] ≡ µ α(b)[(c, ξ)] = α(c), it follows by LC-Funct(S, Ψ) (or the corresponding substitution condition) that b ≡ µ [ξ] c, where µ [ξ] = Ψ(α, 1, µ). Since b ≡ µ c, the pair (α, 1) is not a keeper, whence it must be a switcher, which means that: (35) For any function ν in S, Ψ(α, 1, ν) = µ ξ . Next, by assumption, for any term s containing ξ we have s[(a, ξ)] ≡ µ s[(b, ξ)]

  The simplest case is that of basic indexicals. A sentence ϕ containing such indexicals does not express a complete proposition, something that is true or false, but utterances or occurrences of ϕ do. The context of the utterance or occurrence is needed to fix the reference of the indexicals. Thus, we can distinguish the linguistic meaning of the sentence (type), and the content or proposition that various uses of it express. But what does it mean to say that this kind of linguistic meaning is or is not compositional? Without commitment to any view of what contexts are, the abstract situation is as follows.

	Funct(µ) and Subst(≡ µ ), like their informal counterparts (PoC-F) and (PoC-positional treatment.
	S), express the core meaning of compositionality. For compositionality to have
	practical import, one also needs to require that the meaning operations r α
	are computable in some suitable sense. There are several ways in which basic
	compositionality can be strengthened. But note that (PoC-F) -by far the
	most common informal version of the principle in the literature -only says
	that the meaning of a complex phrase is determined by the meanings of its
	constituents (and the mode of composition); it says nothing about how it is
	Context-dependence is ubiquitous in natural languages, and there is a current
	7), that when linguistic context is taken into account, DP is a less demanding debate on how much of these phenomena are amenable to a systematic or com-
	requirement, and indeed our account handles quotation of arbitrary strings while
	satisfying a generalized version of DP.

determined. And the equivalent (PoC-S) doesn't even mention functionality or determination. As a consequence, we have (under DP):

(5)

If µ is one-one, i.e. if no two distinct grammatical terms have the same meaning, then µ is compositional. This is because under the assumption, ≡ µ is the identity relation, so Subst(≡ µ ) holds trivially.

Although Funct(µ) and Subst(≡ µ ) without extra conditions are weak, they are not trivial: there are grammars and semantics for which they fail (e.g. Potts's semantics for quotation mentioned in footnote 8.) It is another matter that any semantics µ for E is recoverable from some compositional semantics µ ; for a simple example, let (6) µ (t) = µ(t), t (when defined)

µ is one-one, so Subst(≡ µ ) holds, but this in no way makes Subst(≡ µ ) trivial.

15

3 Extra-linguistic Context Dependence

  For any term s, and any occurrence o of a subterm of s, we define inductively the (linguistic) context of o in s, cxt(o, s), as a finite sequence ξ = (α 1 , i 1 ), . . . , (α n , i

n ) of pairs consisting of a grammar rule and a natural number. ( is now concatenation of sequences.)

(L cxt ) (i) cxt(s, s) = (the null context). (ii) If cxt(α(o 1 , . . . , o k ),s)=ξ then cxt(o i ,s)=ξ (α,i) , 1≤ i ≤ k. (iii) CXT E = range(cxt) (the set of contexts in E).

So the context of o in s encodes the path in the tree corresponding to the term s that starts with the node s and ends with the node o. Clearly, every subterm occurrence in s is in a unique context in s, and distinct subterm occurrences are in distinct contexts in s. The same context ξ can be a context in many different terms,

20

but for any grammatical term s, ξ determines at most one subterm occurrence. If cxt(o, s) = ξ, and we replace o by another term, we obtain a term s (which may or may not be grammatical), differing from s in having the new term in the same context as o, i.e. ξ.

A quotable expression is one that can occur within quote marks as well as without them.So (b) is the very weak requirement that one's notion of meaning allows some non-trivial synonymies. Note that if a semantics disallows non-trivial sameness of meaning, it is trivially compositional (see(5) in Section

below), regardless of how it treats quotation, so the issue of whether a particular account of quotation is compositional or not becomes void.

This holds for Davidsonian variants, for the account in Werning[START_REF] Werning | Right and wrong reasons for compositionality[END_REF], and also for the treatment in Shan[START_REF] Shan | The character of quotation[END_REF]. Shan mentions that feature (a) fails in his fragment, i.e. that no rule allows to you to enclose an arbitrary expression in quote marks; he also notes that this is somewhat counter-intuitive (p. XXX). In fact, his account of mixed quotation is nonstandard also in the sense of not satisfying the requirement, mentioned in footnote 2, that two syntactically distinct quoting expressions cannot have the same meaning. This is because the meaning he assigns a quoting expression isn't sensitive to the quoted expression itself, but only to the quoting context, which is what allows his account to be compositional. As to Shan's account of pure quotation (his section 4.1), it is compositional for the trivial reason that there are no distinct synonymous quoted expressions (since each such expression means

A sketch of the ideas in this paper appeared in Pagin & Westerståhl[START_REF] Pagin | Compositionality I: Definitions and variants[END_REF] and[START_REF] Pagin | Compositionality II: Arguments and problems[END_REF].

More correctly, we should write the string value of α(few, cat) as few cats, where ' ' denotes word space and ' ' concatenation, but the simplified notation used here is easier to read.

See Westerståhl[START_REF] Westerståhl | On mathematical proofs of the vacuity of compositionality[END_REF], and for further discussion of the alleged triviality of compositionality, Pagin and Westerståhl[START_REF] Pagin | Compositionality II: Arguments and problems[END_REF].

Note that X may in turn consist of functions. For example, in standard possible-world semantics, X contains intensions, which are functions from possible worlds to ordinary extensions. In relativist variants, these functions have further arguments, such as times, locations, assessors, standards, etc. We then get analogous issues for the compositionality of intension, and how such compositionality is related to that discussed below. These questions are treated inWesterståhl [44], but they will not be important here.

In Pagin and Westerståhl[START_REF] Pagin | Compositionality I: Definitions and variants[END_REF], we distinguish 1st level compositionality, where the meaning of a complex term depends on the meanings of its immediate subterms, from 2nd level compositionality, where also the meanings of the immediate subterms of the immediate subterms may be required, and so on. This is somewhat analogous to the grade distinction here, and standard compositionality is 1st level, which is yet another reason for choosing the formulation in GEN. Also, many semantics where linguistic context is allowed to play a role in fact satisfy GEN; see the next section,

and note 23. The ω-grade Fregean semantics, on the other hand, requires infinitely many context types, which threatens to trivialize the compositionality

Thus, the task of quoting arbitrary strings leads us to something like the description theory of quotation, and in this respect our account is similar to the one in Werning[START_REF] Werning | Right and wrong reasons for compositionality[END_REF]. But the crucial difference is that we combine this with a standard syntactic quotation operator.

Strictly speaking, E , GT E , and V should be defined by simultaneous induction.

So the semantics is first-grade in the sense of Section 6. In a slightly different, but still first-grade semantics we can add an operator that switches back from V to µ , e.g. if one wants a mechanism for quantifying into quotation contexts.

The research leading to these results has received funding from the European Communitys Seventh Framework Programme FP7/2007-2013 under grant agreement no. FP7-238128. The first author's work on this paper has also received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 229 441 -CCC, and from his research fellowship at IEA-Paris, spring 2010. The second author gratefully acknowledges support from the Swedish Research Council.

Further directions

The work begun here could be extended in at least two directions. First, one should try to cover other uses of quotation, such as the interplay between direct and indirect discourse, or mixed quotation (as in Quine says that quotation 'has the overwhelming practical convenience of visible reference' ). Indeed, no account of quotation would be complete without this. Our aim in this paper has not been to give a complete account of quotation, however, but to present the notion of general compositionality, and to show that even for the simple case of pure quotation, this kind of compositionality is all one can hope for.

Second, as we have indicated, linguistic context dependence as outlined here can be exploited for other constructions, notably various kinds of intensional contexts.

Appendix 1: Another approach?

Do we really need the apparatus of Sections 4 -7 to take care of pure quotation compositionally? One referee suggested that we could avoid it by treating words as ambiguous between a default and a quote reading, so that the lexicon provides, for example, instead of one atomic term Cicero, two terms, Cicero d and Cicero q with the same surface form:

We have three problems with this idea. First, it is not a straightforward account (since Cicero d is not a constituent of Cicero q or of any term referring to Cicero). Second, it is counterintuitive. While there are good reasons to let the lexicon distinguish two or more meanings of bank, for example, there are no good reasons why the lexicon should care about quotation. Although the same word can sometimes be used to refer to a man and sometimes to itself, this has nothing to do with its lexical meaning but is a systematic feature of the quotation context. Third, it is not enough to introduce such massive ambiguity for lexical items: the same must be done for all complex terms generated by the grammar, since the context type (index). This is not compositionality, but general compositionality.

Summing up, we can see no way in which the alternative would be superior, but at least two ways in which it would be inferior. But the main point is that even if we choose the alternative, we still need the notion of general compositionality, and thus the machinery developed in Sections 4 -7.

Appendix 2: Some proofs

Proof of Proposition 4: (a): Assume that µ is a partial function from GT E × C satisfying GDP µ , where C is a context typing of CXT E such that LC-Funct(µ, C) holds, and that µ c for c ∈ C and Ψ are as described in (a), with

This shows that GDP S holds. Also, by LC-Funct(µ, C) we have

where s α,µ c is defined by

Note that s α,µ c is well-defined, since it follows from the above that if µ c = µ c , then r α (m 1 , . . . , m n , c) = r α (m 1 , . . . , m n , c ). This proves LC-Funct(S, Ψ).

(b): Now suppose S is a set of functions and Ψ a selection function satisfying Φ C , we calculate:

where the operation s α is defined by

These calculations are independent of the choice of ξ, as long as F (ξ) is defined, which is an assumption. Therefore, s α is well-defined. This shows that LC-Funct(ν, C) holds.

2

Proof of Fact 6: Let µ be given, and let the context typing C be such that each context is its own type, i.e. [ξ] = {ξ}. Define, for each ξ, the relation ≡ ξ by [START_REF] Recanati | Open quotation[END_REF] u ≡ ξ t iff for all terms s containing ξ, s [(u, ξ) Then let, for each ξ, µ [ξ] be a semantics whose corresponding synonymy relation we must, as above, have a ≡ µ [ξ ] b, where µ [ξ ] = Ψ(α, 1, µ [ξ] ). But from [START_REF] Searle | Speech Acts: An Essay in the Philosophy of Language[END_REF] it follows that µ [ξ ] = µ [ξ] , which contradicts [START_REF] Shan | The character of quotation[END_REF]. Thus, µ is not first-grade compositionally generalizable.

We can easily extend this substitution failure format to yield a function µ that is not generalizable into a second-grade g-compositional semantics: just start with four (µ-)synonymous atoms, of which three are synonymous under α, of which two are synonymous under α(α), which in turn are not synonymous under α(α(α)). And so on.

2

The construction in this proof gives a natural generalization of the substitution failure format, where standard substitution failure is the first element of this sequence, with two terms that are atomically synonymous but not under α.

This also speaks in favor of the naturalness of the generalization requirements in GEN. We may note that Frege's infinite hierarchy of indirect reference can provide substitution failure for generalization of any finite grade.