It's that, and that's it! Exhaustivity and homogeneity presuppositions in clefts (and definites)

This paper proposes a way to encode exhaustivity in clefts as a presupposition, something which has been claimed to be adequate, but never successfully implemented. We furthermore show that the facts that prompted the need for such an analysis carry over to identity sentences with definite DPs and propose a way to achieve the same presuppositions for definite DPs. 
 
http://dx.doi.org/10.3765/sp.6.6 
 
 BibTeX info

1 Background: Exhaustivity in clefts A cleft of the form It is x that P not only expresses that x has property P , but also that x is the only individual to have P , i. e. that x exhaustively identifies P (in the relevant contextual domain).Call that the Exhaustivity Claim.
1.1 Exhaustivity is not part of the asserted/at-issue content As first argued in Halvorsen 1978: §1.4.2 and discussed in more detail in Horn 1981, the Exhaustivity Claim associated with clefts does not behave like a part of the at issue content, i. e. it does not appear to be asserted.This is particularly evident when we compare clefts to parallel sentences with only: (1) a.It was Fred she invited.b.She only invited Fred.
Both sentences in (1) convey that she didn't invite anyone other than Fred.
But only in (1b) does this seem to be part of the assertion.This becomes clear when we try to negate either of (1) for the sole reason that the Exhaustivity Claim is false (examples modelled on Horn 1981, exx. (11-12)): (2) a. #Bob knew she invited Fred, but he didn't know it was Fred she invited.b. Bob knew she invited Fred, but he didn't know she only invited Fred.
(3) a. #It wasn't Fred she invited.She also invited Gord.b. #It wasn't Fred she invited.She invited Fred and Gord. 1 c.She didn't only invite Fred.She also invited Gord.d.She didn't only invite Fred.She invited Fred and Gord.
(4) shows the perhaps simplest version of the contrast: (4) a. #She invited Fred, but it wasn't Fred she invited.b.She invited Fred, but she didn't invite only Fred.
The contrasts in (2)-(4) makes sense if the Exhaustivity Claim is part of the assertion in only-sentences, but isn't in clefts.Before going on, let us note that the negated clefts are usually taken to imply an existential statement, e. g. that she invited someone in (3a) and (3b), which is commonly analyzed as an existential presupposition of clefts in general.But note, too, that such an existential implication, whatever its nature, does not explain any of the contrasts above.We will assume for the sake of concreteness that clefts do have an existential presupposition in addition to the Exhaustivity Claim, but nothing we say about exhaustivity hinges on that.Some further comments on the existential implication will be made in section 6.

Exhaustivity presuppositions?
Given the data above, it is occasionally claimed that the Exhaustivity Claim is presupposed rather than asserted in clefts. 2 But what is meant by this?Clearly, (5a/b) do not presuppose (5c) (see also Halvorsen 1978: 15, Atlas & Levinson 1981: 24f.): (5) a.It wasn't Fred she invited.b.Was it Fred she invited?c.She invited Fred and no-one else.
A similar objection can be raised against the idea that exhaustivity is conventionally implicated, since conventional implicatures are standardly held to survive under negation, question formation, etc. 3  One could argue that the presupposition is instead that she invited exactly one individual, and more generally, that it was x that P presupposes that P denotes a singleton set.This would successfully explain the ill-formedness of (2a), (3a) and (3b): Since Bob is presupposed to know that she invited exactly one person, he must therefore know that she invited only Fred if he knows that she invited Fred; similarly, a speaker who presupposes that she invited only one person (though not necessarily Fred) can't then assert that she invited Fred and Gord.
This explanation, however, wrongly predicts that (6) should be as bad as (3a) and (3b), since in (6) the second sentence, too, would contradict the singularity presupposition of the first: (6) It wasn't Fred she invited.She invited Bob and Gord.
2 E. g.Wedgewood 2007: 215, echoed approvingly in Hedberg in press: 4. The problems for this view summarized in this section are for the most part already pointed out by Halvorsen (1978: 15ff.) and and Atlas & Levinson (1981: 24f.).While Halvorsen 1978 explicitly leaves problems for further research, Atlas & Levinson 1981 attempt a solution, which, however, fails to generalize, see the appendix.
3 This argument is already put forward by Horn (1981) against Halvorsen.Horn's own account of the Exhaustivity Claim as a conversational implicature falls short of providing a convincing derivation of the implicature and fails to explain its non-cancellability.

Daniel Büring and Manuel Križ
Moreover, the phenomena can be replicated with plural clefts:4 (7) It wasn't Fred and Sue she invited.#She invited Fred, Sue and Gord.
(8) #John knew she invited Fred and Sue, but he didn't know it was Fred and Sue she invited.
Evidently, the first sentence in (7) and the second clause in (8) can't be said to presuppose that (John knows that) she invited exactly two people.Yet the sentences clearly imply that she invited Fred, Sue, and no-one else.So exhaustivity can't be derived from a singularity presupposition.
On the other hand, any parallel 'plural exhaustivity' would amount to nothing more than existence: there will be a maximal group of individuals that she invited whenever she invited someone.Clearly, presuppositions of this kind aren't going to give us exhaustivity in the plural case, and hence seem unpromising to help us in general.
Our problem, then, is that exhaustivity is somehow implied in clefts, but it seems to be neither asserted, nor presupposed or implicated.

Towards a solution
The idea we want to explore is that exhaustivity in clefts is indeed presuppositional, but that it is formulated in slightly different terms than one would expect.The presupposition of a cleft of the form it is x that P is that x is not a proper part of the maximal member of P .For distributive predicates like invite, this boils down to the requirement that if x is in P , it is the maximal P , i.e. the sum of all elements in P .
In the concrete case of (9), then, there is a presupposition that Fred is not a proper part of the sum of all invitees.As a result, he is either the sole invitee (=the 'sum' of all invitees), or he was not invited at all.

(9)
It was Fred she invited.
a. Ass: She invited Fred.b.Pres: Fred is not a proper part of the sum of all people invited by her.
We will consider the question how such a presupposition should come about in a moment.Let us first see how it addresses the problems noted in the previous section.First, assertion and presupposition of ( 9) taken together correctly imply exhaustivity: She invited Fred and no-one else.Second, negation: (10) It wasn't Fred she invited.
Ass: She did not invited Fred.Pres: Fred is not a proper part of the sum of all invitees.
From the assertion and presupposition in (10), it does not follow that she invited Fred.Indeed, it doesn't even follow that she invited any single person (so we get (6) right); there may well have been a multitude of invitees, only they couldn't have included Fred.It does, however, follow that she didn't invite Fred (that is just the assertion).If it weren't for the presupposition, then a continuation of the form She invited Fred and . . .would simply lead to a contradiction, but in fact we predict that such a scenario would result in a presupposition failure for the cleft, since if she invited Fred and someone else, then Fred is a proper part of the sum of all invitees.5Third, consider belief contexts: (11) #Bob knew she invited Fred, but he didn't know it was Fred she invited.
By standard assumptions about presupposition projection under verba sentiendi (e. g.Heim 1983), Bob didn't know it was Fred she invited, assuming (16), presupposes that Bob knew that Fred isn't a proper part of the sum of all invitees, i. e. that either only Fred was invited, or Fred was not invited at all, and entails that Bob didn't know that she invited Fred.The latter, evidently, is incompatible with the first sentence in (11), which asserts that Bob knew she invited Fred. 6So the presupposition as made explicit in (9) looks promising for deriving the correct patterns.We now should turn to the question of how such a strange-sounding presupposition comes about, and what its proper formalization is.First, however, we would like to take the reader on a quick excursus.
6 Indeed, the assertion of the first sentence together with the presupposition of the second entails that Bob knew she invited Fred and only Fred.This correctly predicts the oddness of (i), too, because new information can't be entirely presupposed: (i) #Bob knew she invited Fred.Indeed, he even knew that it was Fred she invited.

Daniel Büring and Manuel Križ
3 The connection with definites It has been proposed that a cleft like it was Fred she invited is the same -at some level of linguistic representation -as the parallel sentence with a definite description, the person(s) she invited was Fred. 7(Note that this rendering already takes into consideration that clefts are number-neutral.)Would adopting such a proposal give us the correct behavior for the Exhaustivity Claim (given that definite descriptions on many analyses are presuppositional)?It won't by itself.Without going into too much detail, consider two common interpretations for definite descriptions.On a term (or Fregean) interpretation, 'the individual(s) she invited' denotes a (plural) individual if such an individual exists, and fails to denote otherwise.It correctly follows that 'the individual(s) she invited = Fred' implies (indeed asserts) exhaustivity.However, its negation, too, is defined as long as there is a (possibly plural) individual she invited, and true if that individual is not identical to Fred.This is clearly the case if, for example, she invited Fred and Gord (since Fred ≠ Fred Gord -where is the mereological sum (or fusion) operator of Link 1983). 8It is thus unclear what should be wrong with (3a), (3b), or (4); similar for the belief case.
On a presuppositional generalized quantifier (or Russellian) interpretation of definites, 'the individual(s) she invited' maps a property Q to true if there is an individual X she invited which has Q and presupposes that every individual she invited is a part of X. 'The individual(s) she invited ≠ Fred' (i.e. Q is the property of not being Fred) is thus again true if the maximal individual she invited is not Fred, including the case in which it is the sum of Fred and Gord.So in either case, even if we grant a presuppositional analysis, we don't get exhaustivity to be part of that presupposition.
What was a problem for the analysis of clefts thus turns out to be a problem for the analysis of identity statements with definite descriptions as well, for those pattern just like clefts with respect to the exhaustivity facts (we 7 Akmajian 1970, Han & Hedberg 2008, Harries-Delisle 1978, Hedberg 2000, Percus 1997.The proposal by Atlas & Levinson (1981: 53ff.) is billed as an implementation of this intuition, too; it is dicussed in the appendix.
Exhaustivity and homogeneity presuppositions in clefts use plural examples to avoid the awkward person(s). . .is/are circumscription): (12) #The people she invited weren't Fred and Sue.She invited Fred, Sue and Gord.
(13) #John knew she invited Fred and Sue, but he didn't know the people she invited were Fred and Sue That is to say, the parallelism between clefts and identity statement with definites extends to the status of exhaustivity as non-asserted.The problem lies not with the idea that clefts underlyingly involve definites, but with an inadequate semantics for definites.Rather than presupposing that there is a maximal group of individuals she invited, which is what standard analyses of the people she invited predict (and which comes down to a mere existential presupposition, as discussed above), it should hold here, too, that: (14) The people she invited were Fred and Sue.
Ass: She invited Fred and Sue.
Pres: Fred and Sue are not proper parts of the sum of all invitees.
Let us summarize at this point: We have run into two problems related to exhaustivity, one in clefts, one in identity statements with definite descriptions.In both cases, exhaustivity apparently needs to be coded in a somewhat non-obvious manner with reference to mereological parthood.We believe that this is underlyingly the same phenomenon and thus concur with e.g.Percus (1997), according to whom clefts are statements of the form 'the one(s) who. . .are. . .' at Logical Form.In the following, we will discuss further steps towards an analysis in terms of identity statements.
4 How to encode the exhaustiveness presupposition

Exhaustivity as uniformity in definites
Our formulation of the exhaustivity presupposition may have sounded strange, making reference to mereological parthood.But there is another phenomenon surrounding definites, often considered a presupposition (albeit an easily accommodated one), which has to do with such notions.The sentence The boys went swimming is true if all the boys went swimming, but it is false only if none of them went swimming.It thus seems to have a presupposition that either all or none of the boys went swimming.

Daniel Büring and Manuel Križ
The phenomenon is often called homogeneity, but proposals about it are found in Fodor 1970, Löbner 1987: 83 and 2000: 239ff., as well as -extending to other phenomena than definite descriptions -von Fintel 1997, Schwarzschild 1994: 220, and Gajewski 2005, under various names.We call this the Uniformity presupposition of definites, and give it again a slightly unusual formulation: a sentence of the form [the P] Q presupposes that either the entire group of P s is Q or nothing in Q is a proper part of the maximal P . 9 To be able to state this formally most conveniently, we assume with e. g.Schwarzschild (1994) and Champollion (2010) that natural language predicates are always closed under mereological fusion. 10This means that every non-empty predicate has a maximal element.We therefore don't need the star operators that are commonly used to indicate closure under sum formation.Now we can state the presupposition of definites generally and formally: (15) The P are Q. Ass: Applied to a standard example, this translates to the following: 11 (16) The boys went swimming.
Ass: All the boys went swimming.
Pres: All the boys went swimming or no proper part of the boys went swimming (no swimmer is a proper part of the sum of boys).

This analysis predicts nothing new for the positive case. It does predict that
The boys didn't go swimming implies that none of the boys went swimming, which strikes us as desirable (see also the discussion in the sources quoted above).It also predicts the contrast in ( 17): (17) a. A: The boys went swimming.B: #No, the boys didn't go swimming.Mike stayed at the shore.
In case Q is a distributive predicate, that -even more simply -means that every or no P Qs.Thanks to an anonymous reviewer for making us point this out clearly.That is to say, P (x) ∧ P (y) → P (x y).
We are aware that due to the nature of the operator, this formulation implies an existence presupposition.It is straightforward to give an alternative formalization that doesn't have this effect, based on the (possibly empty) set of maximal P s, which, however, would only hinder the exposition at this point.

6:8
Exhaustivity and homogeneity presuppositions in clefts b.A: The boys went swimming.B: Well, not all the boys went swimming.Mike stayed at the shore.
There are probably other ways to explain this contrast, but at least the general effect of the presupposition in (15) doesn't strike us as problematic in the light of such examples.

Clefts with definites
Applying (15) to clefts, the role of P , the restrictor of the definite, is now played by the relative clause, while the role of Q is taken by the predicate of being identical to the pivot (the constituent that is clefted, in this case a): (18) It was a that P . Ass: This conveniently reduces to the following: Ass:

P = a
Pres: a P The presupposition is precisely the one we formulated in section 2: a is not a proper part of the maximal P .
For reasons that will become clear in sections 5 and 6 below, it is preferable to not speak about the maximal P , but about the set of maximal P s instead.Given our assumption that predicates are closed under mereological fusion, this set will always be a singleton for predicates of individuals.We then end up with the following variant of (19) as our final formulation.(An analogous adjustment could and perhaps should be made for definites.)(20) Ass:

P ( a )
Pres: ∀x ∈ max( P ) : a x (where for any Finally, for ease of use, we roll (20a+b) into a single operator cleft (where the formula between the colon and the period in (21) defines the domain of the partial function, which can be thought of as its presupposition, see e.g.Heim & Kratzer 1998, ch.4.4

Parthood and collectivity
Looking at (20)/( 21), note that we stated the presupposition as 'a is not a proper part of a maximal P (which, in the case of predicates of individuals, reduces to '. . . of the sum of all P s'), rather than the seemingly simpler 'a is not in P or it is a maximal P '.12These two options are indeed equivalent as long as we consider distributive predicates only, since it will hold for any distributive predicate P that a is a proper part of P iff a ∈ P .But once we take non-distributive predicates into account, our proposal actually makes different predictions from the one stated in terms of a ∈ P .Consider a predicate like carry the piano, which can hold of either atoms or sums, and is not distributive.Assume Bill and Fred carried the piano together, and neither of them did alone, nor did anyone else.We believe that ( 22) should suffer a presupposition failure in this case.(22) It was Bill who carried the piano.
On our account in (20), this is predicted.Given that Bill isn't a (the) maximal piano-carrier -he didn't carry a piano alone and exclusively -, the presupposition of the cleft requires that he not be a mereological part of the maximal piano-carrier, i. e. that he wasn't involved in any piano-carrying at all.But he did in our scenario, because he carried a piano together with Fred, so the presupposition fails.If, on the other hand, we used the presupposition that if Bill carried the piano, he is the maximal piano carrier, we would predict (22) to simply be false, since the singular individual Bill is not in the extension of carried the piano (which contains just the plural individual of Bill and John), so that the conditional presupposition would be true because its antecedent is false. 13n sum, we believe that distributive and collective predicates behave This is because Bill Fred is not in the extension of carried the piano, and is also not part of anything in that set, i. e. is not a part of Bill.The result strikes us as the correct one.

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Exhaustivity and homogeneity presuppositions in clefts identically in clefts: If a is a proper part of those who Q, it was a that Qed is undefined, rather than false.Stating the presupposition of clefts in terms of part-hood to a maximal Q, as we did in (20), rather than in terms of having Q, correctly derives this parallelism.
The analogy we draw between clefts and definites predicts that the latter also show corresponding effects with collective predicates.In particular, the two sentences in ( 23 So far, we have described exhaustivity, via uniformity, as a presupposition, which is also the stance taken in much of the literature on the latter phenomenon.In fact, however, they are not particularly prototypical presuppositions.If we tell you that the boys went swimming, you need not really have assumed previously that all of the boys did the same thing.This is also evident in clefts.In order to be informed by us that it was Fred who was invited, it is does not seem necessary for you to have any prior beliefs about Fred; in particular, you need not have had any commitment to the position that while Fred may or may not have been invited, if he was, then he was the only one.So while clearly not at-issue content, the Exhaustivity Claim still seems to be something that is communicated, as opposed to presupposed.So should exhaustivity in clefts and definites rather be modelled as a conventional implicature?Like presuppositions, conventional implicature are not part of the assertion, yet they can, and routinely are, used to convey new information; this seems to be the right mix of properties.However, it turns out that the Exhaustivity Claim differs from conventional implicatures in at least one crucial respect: it is strictly local.

Daniel Büring and Manuel Križ
For example, (25) ascribes to Peter the belief that no-one other than Fred was invited.As a consequence, (26) sounds contradictory: (25) Peter believes that it was Fred she invited.
(26) I think only one person was invited.Peter believes that it was Fred who was invited, #and that Sue and Gord were, too.
In particular, the second sentence in (26) can't be interpreted to mean that Peter believes that Fred and Sue and Gord were invited, whereas the Exhaustivity Claim -that if Fred was invited, no one else was -would be ascribed to the speaker.In contrast, as observed by Potts (2007: 477), a conventional implicature, e. g. triggered by an appositive, can be attributed to the speaker, even if embedded under a verb of saying or thinking: Sheila says that Chuck, a confirmed psychopath, is fit to watch the kids.
That is to say, the claim that Chuck is a confirmed psychopath (the conventional implicature) need not be part of what Sheila said (and hence she doesn't have to hold the strange attitude that a confirmed psychopath is fit to watch the kids) (these are the 'editorial comments' in Bach 1999: 339).
Presumably it is a hallmark of presuppositions, as opposed to conventional implicatures, that they need to be met in the local context in which the sentence triggering them is evaluated.Since this appears to be the case for the Exhaustivity Claim, witness (26), we choose to characterize it as a presupposition.The reason why it often seems to be communicated content, we suggest, is that this presupposition is just extremely easy to accomodate.This is similar to the way in which definite descriptions are often used to inform the hearer of the existence of the described object.In principle, however, any dimension of meaning that delivers locality in this sense would be appropriate.
This concludes the presentation of our main proposal and the motivation for it.In the following sections, we will extent our treatment to more complex examples and discuss its predictions (section 5), and add some more speculative remarks (section 6).

6:12
Exhaustivity and homogeneity presuppositions in clefts 5 Extensions Our semantics for clefts from section 4.2 requires that the pivot is an individual and that the meaning of the relative clause is a predicate of individuals.There also has to be a notion of parthood that is defined for individuals.But there are many clefts the pivots of which are not individuals.For instance, it is perfectly unremarkable to have a quantifier in that position. (28) It was a hat that I bought.
For (28) and many cases like it, it would be semantically adequate to simply quantify into the cleft after raising the quantifier, as illustrated in ( 29).This gives us the right result: there is a hat x that was among my acquisitions, and that hat is not a proper part of the 'maximal acquisition', so it must be the maximal acquisition, i. e. the one thing that I bought.( 29) ist predicted to be undefined if I also bought something else, and false if I only bought one or more non-hats.
In other cases, however, quantifying in will not yield the correct result.To focus on a particularly blatant failure, it was pointed out to us by an anonymous reviewer that DPs with just are perfectly acceptable and informative in clefts, and so are, we believe, those with only, which we will assume to be synonymous for our purposes: The problem is this: On the one hand, sentences with only are generally held to presuppose their so-called prejacent, i. e. the content of the parallel clause without only (or something implying that, see below): not only Fred bought a copy of my book implies that Fred bought a copy of my book (Horn 1969, Rooth 1985, a. m. o.).On the other hand, according to the analysis developed here, a cleft of the form It was X who P presupposes that if X is (part of) a P -er, X is the maximal P -er: It was(n't) Fred who bought a copy of my book implies that if 6:13

Daniel Büring and Manuel Križ
Fred bought a copy, no one else did.
Taken together, these two presuppositions seem to entail that Fred bought a copy (only) and that no one else did (cleft, together with first presupposition); and this is in fact just what we'll get formally if we scope out only Fred.But this seems implausible for (30), because it would leave nothing for it to assert.And it seems downright disastrous for ( 31 What we need to do is therefore to extend the proposal to pivots whose meanings are quantificational, which poses the challenge of defining a notion of parthood among quantifiers, as the cleft-operator relies on that notion.

Extensional quantifiers
It is easy to obtain the structure required for our cleft operator by type-lifting the cleft predicate P to λP.P(P ), i. e. the set of quantifiers that are true of the predicate.Assuming that we have a notion of parthood for quantifiers, we can then apply our meaning schema ( 20): (36) It was Q (et)t that P (et) .
Ass: [λP.P(P )](Q) (≡ Q(P )) 'Q is true of P ' Pres: ∀Q ∈ max(λP.P(P )) : Q Q 'Q is not a part of any maximal quantifier that is true of P ' We now see the reason why in section 4.2, we chose to speak about the pivot being among the maximal P s, and not the maximal P , even though the latter would have been more natural for individual pivots: one and the same situation can be described with several quantificational clefts, e.g.(37b-d): ( Based on such a notion of exhaustivity, we can now define parthood among quantifiers as follows: 6:16 Exhaustivity and homogeneity presuppositions in clefts (39) A quantifier P is a proper part of a quantifier Q (written P Q) iff 1. ∃P : P(P ) ∧ exh(Q, P ), and 2. ∀P : Q(P ) → ¬exh(P, P ).
Condition 1 says that the smaller quantifier is consistent with the exhaustivized version of the larger. 19 Condition 2 requires that the larger quantifier entail that the smaller quantifier doesn't hold exhaustively.It straightforwardly implements the idea mentioned above that the larger quantifier somehow says that additional individuals are involved which are not required by the smaller quantifier.
Note that it follows that being exhaustively true of a predicate P and being among the maximal quantifiers that are true of P are equivalent, which is intuitively what we want. 20  Let us now illustrate how this definition does what it is supposed to.We want (40) to have a presupposition failure if I bought anything in addition to the hat, so what we need is that a hat is a part of quantifiers like two hats , more than one hat , and a hat and a coat . (40) It's a hat that I bought.
We can easily see that this is the case.Looking at the parthood relation between a hat and two hats , the witnesses for Condition 1 are just sets that contain two hats.Since a hat is exhaustively true only of singletons of hats, and the quantifier two hats is not true of singletons of hats, we immediately see that Condition 2 is fulfilled as well.What we do not want 19 Again, we could not express this with the seemingly more straightforward ∀P : Q(P ) → P(P ), because of collective P s.A boy should be part of a boy and a girl, but from A boy and a girl carried the piano upstairs it does not necessarily follow that A boy carried the piano upstairs, in case boy and girl only jointly carried the piano; Condition 1 is therefore weakened to say that this is the case for some predicates (namely all distributive ones).
20 Proof: Left to right: Assume that P exhausts P , but that there is a quantifier Q P that is also true of P .The contradiction with Condition 2 of parthood is immediately visible, P being the counterexample to the universal quantification.
Right to left: Assume that P is a maximal quantifier that is true of P , but that it does not exhaust P .For every predicate P , there is the quantifier Q = λP .P = P , which holds of P and exhausts P .We can now show that P is a part of Q and thus is not a maximal quantifier that is true of P , contrary to the assumption.Condition 1 of quantifier parthood is obviously fulfilled, with P being a witness.Since Q is true of only one predicate, namely P , of which P is not exhaustively true, Condition 2 holds as well.

6:17
Daniel Büring and Manuel Križ is for a hat to be a part of less than three hats , for that would mean that a hat can never be a cleft pivot: after all, its exhaustivized form, which is asserted in the cleft, implies less than three hats .We would therefore always have to use less than three hats in a cleft describing such a situation.Fortunately, Condition 2 is violated in this case, because there are predicates of which both quantifiers are true exhaustively: singletons of hats.
Then there are negative existentials like no boy.These can be proven to never participate in any parthood relations.They cannot ever be part of any other quantifier because for them, exhaustification is vacuous, but the conjunction of the two conditions requires that there is a predicate for which the smaller quantifier is true, but not exhaustively so (the witness for the existential quantifier in Condition 1).Hence the negative existential cannot be such a smaller quantifier.Neither can anything be a part of, say, no boy .In order for P to be a part of no boy , it would have to be the case that exh(P, P ) implies that there is a boy in P (by contraposition of Condition 2).At the same time Condition 1 requires that P be true of at least one predicate that does not contain any boys.Call that one predicate Q.A proper definition of exhaustivity (the ones that we are aware of do so) will entail that if a quantifier is true of a predicate P , then there is a P ⊆ P such that the quantifier is exhaustively true of P .This means that there is a subset of Q, which, of course, also contains no boys, of which P is exhaustively true.This contradicts the previous inference that any predicate that P is true of exhaustively must contain a boy.Hence, P cannot be a part of no boy .
The prediction that this makes for clefts is that clefts with negative existentials as pivots do not, in fact, have a presupposition; at least no presupposition beyond maybe an existence presupposition, which is not our concern.This strikes us as correct, as we cannot find anything related to exhaustivity in (41).( 41) It was no boy that came.

Only explained
We are now in a position to explain the behavior of only in clefts, which ostensibly makes the exhaustivity presupposition disappear.To that end, assume that only DP is a quantifier with the meaning λP : DP (P ).exh( DP , P ); that is to say, it presupposes that the quantifier DP is true of the predicate, and asserts that it is exhaustively true.Since iteration of the exhaustivity operator

6:18
Exhaustivity and homogeneity presuppositions in clefts is vacuous, it follows that only DP is exhaustively true of any predicate that it is true of.As we saw above in the case of negative existentials, this means that only DP cannot be a part of any other quantifier, since the definition of parthood requires, per Condition 1, that a quantifier that is part of another quantifier be true, but not exhaustively true, of at least one predicate (of which the other is exhaustively true).It must be noted, though, that it is possible for other quantifiers to be a part of only DP, namely exactly those that that are part of DP anyway. 21Per our semantics for clefts, (42) has the assertive meaning component (42a) and the presupposition (42b).Since only John is not part of any other quantifier, the presupposition (42b) is a tautology, which makes it effectively vanish.But of course ( 42), as a whole, does have a presupposition: that John came.This is because only John is a partial function such that (42a) is only defined if John came.As a result, clefts with only DP as their pivot end up equivalent to ordinary predication (with only).
We are furthermore in a position to explain the difference between (43a) and ( 43b only DP to DP: Assume that Q is part of only DP .Condition 1 for Q DP is fulfilled because only DP (P ) entails exh( DP , P ).Furthermore, DP holds only of predicates that are supersets of those that only DP holds of (by the definition of exhaustivity).Since Q doesn't hold exhaustively of any set that only DP holds of, it also doesn't hold of a superset of such a set.Hence, Q is not true of any predicate that DP is true of.

Intensional clefts
It is also possible to cleft the quantificational complements of opaque verbs: (44) It's a hat that I want.
Assuming a standard Hintikka-style analysis, where want takes a quantifier intension as its complement, we have the right types here to apply our cleft schema.All we need is a notion of parthood among quantifier intensions.This is easily supplied: (45) A quantifier intension P is a proper part of a quantifier intension Q (written P Q) iff 1. ∃w, P : P w (P w ) ∧ exh(Q w , P w ), and 2. ∀w, P : Q w (P w ) → ¬exh(P w , P w ).
For all practical purposes, this results in the same parthood relations as we saw in the last section for quantifier extensions. 22Consequently, (44) suffers a presupposition failure in a situation where I want both a hat and a coat.Crucially, a hat is not a part of a green hat : any (concept of a) singleton of a green hat is a counterexample to parthood Condition 2. So a cleft like (44) does not presuppose that I want nothing more specific than just any hat.It may well be uttered in a situation where I actually desire a green hat in particular.This would be difficult to account for without recourse to the notion of parthood between quantifiers. 23 6 Speculative remarks: exhaustivity and existence So far we haven't talked about the existence presupposition, except that we assumed it to be there in addition to the exhaustivity claim, which was formulated so as to not entail it.In this section we want to point out a few phenomena and contrasts that follow from exhaustivity, assuming there is no existence presupposition.For example, consider the following dialog (where the first sentence is to be intoned as an alternative question): 24 In principle, we could subsume the extensional case under this if we abstracted over quantifier intensions in the relative clause.If a definite article is used, the sentence as a whole seems to presuppose something about the altercation in question (e. g. it has been described and discussed before); pretheoretically, the indefinite makes sense because the sentence introduces the altercation as a hypothetical: if there were an (not: the) altercation, I would regret that.
In certain cases, notably with event nominals, though, a definite is used: (50) Be careful where you dig!I would regret the destruction of the old bench.
Crucially, this sentence does not seem to presuppose anything about the destruction of the old bench; like (49), it introduces a hypothetical: if the bench were destroyed, I would regret that.Based on these intuitions (which closely follow the analysis in Schueler 2008), it seems that the definite in (50) asserts, rather than presupposes, existence the same way the indefinite in (49) does.
What does set ( 49) and ( 50) apart is of course that the destruction of the old bench is by necessity a unique event, while the altercation isn't.In other words, the destruction of the bench licenses a presupposition to the effect that if there is a destruction of the bench, that would be the exhaustive/unique destruction of the bench.This seems remarkably similar to the presupposition we proposed for definites and clefts above.
What can we conclude from the data in this section?Arguably, good is done by an exhaustivity presupposition along the lines explored here.However, the effects discussed in this section (unlike those connected to identity statements and simple clefts discussed initially) are only explained by the exhaustivity presupposition if we assume at the same time that there is no existence presupposition (which would seem to be violated in all the cases discussed).
On the other hand, there is strong evidence that clefts and definites do have existential presuppositions (for clefts see e. g. the arguments in Dryer 1996, Rooth 1999), so we can't very well dismiss them generally.Perhaps a more cautious approach is to say that existence presupposition and exhaustivity presupposition in definites (and clefts) are independent of each other.Under certain, as of yet ill-understood, circumstances, the existence presupposition can somehow be ignored, while the exhaustivity presupposition is still active.These would be the cases discussed in this In this note, we have proposed a new exhaustivity presupposition for clefts.That presupposition amounts to the statement that any individual in the domain of the function denoted by it was. . .that P is not a proper part of the maximal individual in the extension of P , which entails that if the individual is in P , then it is the maximal P .This was shown to explain the peculiar status of exhaustivity in clefts: it is not asserted, yet seemingly disappears under negation and similar presupposition filters.Our formal account also makes correct non-trivial predictions for collective predicates.
Furthermore, the account seems promising with respect to extension to clefting of elements that are of higher type and do not denote individuals.With a proper notion of parthood for such elements, it turns out that it straightforwardly captures the clefting of the quantifier objects of intensional verbs as in It was a hat that I wanted.Analyzing only DP as a quantifier, the use of higher types also allows for an explanation of the very peculiar behavior of clefts with only, in which the usual presupposition of clefts seems to be wholly absent.
We also showed that identity statements with definite descriptions show the same presuppositional behavior as clefts and can be treated in parallel, which is in line with the claims in e.g.Percus 1997.However, our account does not in principle depend on this connection.
Finally, we added some data in which the exhaustivity presupposition contemplated here, which does not, in its strict formulation, entail any kind of existence presupposition, would seem to explain a number of surprising contrasts.Their analysis, however, ultimately hinges on a proper treatment of the existential presupposition in definites and clefts, which we did not discuss at any length.
26 Cancellation of the existence presupposition in a cleft has also been argued for a. o. by Delahunty (1981), Gazdar (1979), Halvorsen (1978), Hedberg (in press), Keenan (1971), Levinson (1983) using examples like (i) (this one from Hedberg's paper): (i) You believe that Mary kissed someone in this room.But it wasn't Joe that she kissed, and it wasn't Rita, and clearly it wasn't Bill, and there hasn't been anyone else here.Therefore, Mary didn't kiss anybody in this room.
Whether this is the same effect as in the examples in the main text, however, and when it occurs, is not clear to us.(We discuss this proposal here since it is the only detailed one we know of that neither makes exhaustivity an entailment, nor a singularity presupposition/implicature, so comparison may be instructive.)According to Atlas & Levinson (1981), ( 51) has the logical form in (51a) (adapted to our typographic conventions): So if in fact John and Gord kissed Mary, (54a)/(55) is false (since not every Mary-kisser is distinct from John), as is its non-negated counterpart (51) (since not every Mary kisser is John).This agrees with the present account on truth conditions, but predicts the negated subset cases, which we have be undefined and all other analysis have as true, to be false.But even granting the effect of the Principle of Informativeness (which, it should be noted, does look a bit like presuppositionalizing all but the B part in the meaning of γ), it is less clear that it correctly treats (56), which, recall,

6:24
Exhaustivity and homogeneity presuppositions in clefts should not be counted as true in Scenario 1: Gus didn't know that it was John that Mary kissed. [ and Fred who carried the piano.
This, we believe, is the case.In a scenario where the piano was in fact carried by Bill, Fred and Mary together, both sentences are undefined, rather than false.Conversely, if Bill and Fred carried the piano, then (24b) is just like (24a): false.(24) a.It was Bill, Fred and Mary that carried the piano.b.The people who carried the piano were Bill, Fred and Mary.4.4 Presupposition, conventional implicature, or what?
), which clearly should not presuppose that Fred and no one but Fred bought a copy: Horn 1996, Ippolito 2008, McCawley 1981)t, and it seems plausible that indeed both (31) and (30) presuppose that.So even if we availed ourselves of one of the weaker lexical presuppositions occasionally proposed for only (e. g.Horn 1996, Ippolito 2008, McCawley 1981), as long as we concede that in our specific examples the sentence as a whole implies that Fred bought a copy of my book, the problem remains.14 It rather seems to be the Exhaustivity Claim connected with clefts that needs to disappear in these examples: if Fred and Marge wanted it, (32a) is clearly true, where (32b) is truth-value-less (as discussed at length above): (32) a.It wasn't only Fred who wanted it.b.It wasn't Fred who wanted it.So (32a) (and arguably clefts with only in general) should not presuppose anything to the effect that if Fred wanted it, no one else did.It seems like the quantifier denoted by only Fred, not an individual variable bound by it, should be the argument to cleft here.
(Groenendijk & Stokhof 1984, Sevi 2005neralized quantifier P is a part of one Q if the latter implies that more individuals are involved.Fortunately, to get that, we can make use of a different notion of exhaustivity as discussed in the context of answers to questions(Groenendijk & Stokhof 1984, Sevi 2005, van Rooij & Schulz 2006 and others).Exhaustivity in this sense is a relation between a quantifier and a predicate, and what it says is, intuitively, that the predicate is true of a set of individuals that is sufficient to make the quantifier true of it, and of no other individuals.We will leave open the precise definition of exhaustivity and will just write exh(Q, P ), but we assume that a theory of exhaustivity in this sense delivers, among others, the following as theorems (that of Sevi 2005 does so; van Rooij & Schulz 2006 yields all except (38f) 17 ): 16 We 'decompose' want in the fashion of Hintikka 1969 here in order to make it clearer what we need to quantify over.
It was John or Mary that came.b.It was John or Mary or both that came.(43a)suffers a presupposition failure if both John and Mary came, while (43b) is true in such a context.The reason is that John or Mary is not exhaustively true of a predicate which holds of both, while John or Mary or both is.
21 Proof: DP to only DP: Assume that Q is part of DP .Since exh( DP , P ) entails only DP (P ), Condition 1 for Q only DP is fulfilled.Since only DP (P ) entails DP (P ), Condition 2 is also fulfilled.
E. g. with the conditional exhaustivity presupposition that was suggested in Büring 2011.Boldface indicates stress.John who called Mary.b.John is the one who called Mary.John called Mary, and Sue called Bill.b. #It was John who called Mary, and it was Sue who called Bill.Unlike (47a), (47b) is only fine if it was presupposed that Mary and Bill received calls (or at any rate were supposed to be called).While this would follow from an existence presupposition for the . . .who called John/Bill clause in the clefts, this can't be the story in light of (46) above., on the other hand, wouldn't be met in a run-of-the-mill scenario of the sort: for each of a given group of people, tell me who they called (i. e. what appears to be the neutral meaning conveyed by a question like (47)).If on the other hand it was known that Mary and Bill were supposed to be called, the presupposition would again be met.A second class of examples relates to talk about hypothetical or potential individuals, this time involving definites.As pointed out by David Schueler (p.c. and Schueler 2008), existential nominals in subjunctive hypotheticals usually have to be indefinite, as in (49): 25 Such vice versa clefts are discussed in Ball & Prince 1978, Hedberg in press: §3.1, Carlson 1983: 234; see particularly Hedberg's paper for more examples and discussion.
Scenario 1:] Gus knew that John kissed Mary, but didn't know that no one else kissed Mary.Clearly Gus does not know (53) in Scenario 1, so prima facie, (56) is predicted to be true.But as discussed, (56) is intuitively true only if Gus didn't know that John kissed Mary; and that statement is wrapped into the universal statement in Atlas & Levinson's logical form, so it is at best unclear how this should come about on their account.In addition, and more clearly, the account in Atlas & Levinson 1981 fails in plural cases like (57): (57) It was John, Bob and Gord that Mary kissed.Atlas & Levinson (1981) don't give a precise characterization of what sequences (which they also call 'vectors' and 'lists') are, but note that (58) could only be true if none of [John,Bob], [John,Gord], [Bob,Gord], [John], [Bob], or [Gord] make λx.kiss(Mary, x) true.But if kiss corresponds to natural language kiss, clearly the entailments from kiss(Mary, [John,Bob,Gord]) to kiss(Mary, [John,Bob]) etc. should hold, regardless of what exactly a sequence is.The only way the entailment would not hold is if kiss were itself already 'exhaustified', i. e. held of the maximum sequence of kissers only.But this would not only beg the question, it would make (55) true even if John and Gord kissed Mary (since the only value for x then would be [John,Gord] (not John, or [John] etc.), and obviously [John,Gord] ≠ John), setting us back to square one.