2.1 Grosz (2012)

Grosz (2012: §6.2) postulates two versions of an exclusive particle, which he calls ONLY₁ and ONLY₂, and uses the latter to account for the minimal sufficiency reading (ONLY₁ is used for the exclusive reading, and doesn’t concern us here). He gives the following denotation to the ONLY₂ reading of an exclusive particle (adapted from Grosz 2012: 216).

Grosz’s main interest is to account for uses of exclusive particles like only in optative constructions across languages using (6), about which we have nothing to say in this article. Yet, he explicitly claims in §6.2.2.2 that the minimal sufficiency reading of just is to be accounted for by (6) as well (while remarking at the same time that such readings of only are restricted in English; see the discussion at the end of this paper). For example, he would analyze (2), repeated here, as follows.

Grosz would assume that just here takes a propositional scope, and operates on focus alternatives that include (7), of which (7a) is the prejacent (Note that by convention the prejacent is included as a focus alternative to itself).

Just yields a scalar presupposition with respect to some salient scale, which is presumably a scale of likelihood or expectedness in this case. Thus, the scalar presupposition amounts to that the prejacent (7a) is relatively unlikely or surprising among the focus alternatives, which captures the scalar component of the minimal sufficiency reading of this example. Note also that assuming that the numerals in these focus alternatives receive lower-bounded (‘at least’) readings (see Section 3.1 for more on this), (7a) entails the other alternatives, so the additive component of the minimal sufficiency reading will be captured as well, although it should be noted that this explanation of the additivity cannot apply to other examples.

However, this analysis, as it stands now, overgenerates. Notice, in particular, that the scalar presupposition that just is purported to trigger in (2) is very similar to what is standardly assumed for even (more on this in Section 3.2). If such a reading is available for just, then a reading of (8a) is predicted that is be similar to the reading of (8b), but this reading does not seem to be available.

Specifically, let us assume as before that just in (8a) takes scope over the rest of the sentence. Also, suppose that the alternatives to Patrick are Andy, Bojan, and James. Then the ONLY₂ reading of (8a) should have a scalar presupposition that it was relatively unlikely that Patrick was in the room, compared to these other people. However, this is intuitively what (8b) can mean, which suggests that the relevant scale of likelihood or unexpectedness could in principle be made contextually accessible, and (8a) cannot mean the same thing. Rather, (8a) only has an exclusivity reading, which entails that no one else is in the room. This problem therefore illustrates that Grosz’s (2012) account at least needs to restrict the distribution of ONLY₂, but it is not obvious to us how that can be done.

Another similar issue arises with a more ad hoc scale like the one used for (5).

For just in this example, the following sentences are reasonable focus alternatives together with the scale where they are ordered in this order.

Note that this scale is not necessarily a scale of likelihood, because if Patrick is (known to be) in his 20s, for example, (9c) and (9d) are not very likely to be true, but (5) is still felicitous even in such a context. Rather, the relevant scale is perhaps that of academic ranks. But then, the same ordering of academic jobs should be able to order other sentences, for example, (10).

Consequently, the prediction is that a sentence like (11) should have a scalar reading based on the ordering in (10), but this does not seem to be the case. Rather, (11) only has an exclusivity reading.

These issues show that Grosz’s (2012) account, as it stands, is too liberal with respect to the distribution of the scalar reading, and it needs to somehow restrict the range of scales that just can access in yielding a scalar presupposition. However, in light of the above examples, it is not clear to us how such a constraint can be formulated. In particular, we do not see how this account can explain the contrast between (2) and (8) with respect to the availability of a minimal sufficiency reading.

2.2 Coppock & Beaver (2014)

Coppock & Beaver (2014) also attempt to account for the minimal sufficiency reading as a scalar reading of just, but unlike Grosz (2012), they attempt to do so with a general lexical entry for all uses of exclusive particles, given in (12), which is adapted from Coppock & Beaver (2014: 394). For them, an exclusivity particle, represented here as ONLY, presupposes minimality and asserts maximality. Just like Grosz, they assume that a context specifies an order, ≥, among the contextually relevant focus alternatives, and CQ here refers to the set of sentences on the scale formed by ≥, which is taken to represent the current question under discussion.

In addition, Coppock & Beaver assume that for the minimal sufficiency reading, just takes non-sentential scope. For example, for (2), repeated below, they would assume that just takes the DP, one cat, as its prejacent.

In order to give a full compositional account of an example like this, Coppock & Beaver postulate some type-shifting operations, but we think it’s not necessary to go into these technical details here (but note that we will make use of similar mechanisms in our account presented in the next section), as the more or less intuitive characterization of their analysis below is enough to understand the heart of the problem.

If the prejacent of just is one_F cat in (2), as Coppock & Beaver assume, then, the focus alternatives will look like (13).

These alternatives can be order by something like remarkableness such that (13a) is the least remarkable member of the scale. With this scale, just triggers a presupposition that the prejacent one cat is low on this scale. Then the overall meaning of the sentence in (2) would be something along the lines of ‘Being one cat is relatively unremarkable, and something that is one cat and no more than one cat will make Patrick happy’.

Note that it is crucial here that just is interpreted within the subject DP, because if the focus alternatives included the VP as well and looked like (7), then the scalar presuppose would be that it is unremarkable for one cat to make Patrick happy, which is intuitively the opposite of what we want to derive. Furthermore, since Coppock & Beaver encode maximality in the semantics of just, so as to uniformly account for different uses of exclusive particles, the maximality in this case would amount to the negation of the focus alternatives formed with higher numerals. That is, it would negate that two or more cats will make Patrick happy, which is the opposite of the additivity observed with the minimal sufficiency reading of (2).

This analysis, however, also suffers from an overgeneration problem. The main issue comes from the crucial assumption that just takes the subject DP as its argument and introduces a scalar inference at this level. That is, because this assumption, it is predicted that the material in the VP should not matter for the availability of the minimal sufficiency reading, which is not borne out. More concretely, as already pointed out Section 1, (1a), repeated below, does not have a minimal sufficiency reading, and its only available reading is an exclusivity reading.

The problem for Coppock & Beaver is that the subject DP is exactly the same as in (2), so just should be able to make use of the same scale as above, in which case (1a) should be able to mean ‘Something that is as unremarkable as one cat is in the room’, without an exclusivity inference (so the sentence should be compatible with there being more than one cat), contrary to fact.

The same point can be made with the following pair of sentences: (3a) has a minimal sufficiency reading, while (14) does not.

Thus, Coppock & Beaver’s account is simply unable to capture the fact that the VP portion of the sentence matters for the availability of minimal sufficiency readings, due to the crucial assumption that just takes scope at the DP-level. It seems to us that their analysis of the minimal sufficiency reading is hard to maintain.

3.1 The semantics of just

Let us start with the semantics of just. The two previous analyses we discussed in the previous section assume that it can have a scalar reading, at least in sentences like (5). While this seems to be undeniable, the problems we pointed out for them, as well as the observations made earlier in this section, suggest that the minimal sufficiency reading might not be a special case of it. And at least in sentences like (4a) and (15a), which contain an overt even, the scalar component of the meaning seems to be coming from even, rather than from just. For this reason, we will analyze just in the relevant examples to be expressing an exclusivity reading, which can be characterized as in (17).³ Alt(φ) here is the contextually relevant set of focus alternatives to ‘φ’.

Note that for the minimal sufficiency reading of (4a), the prejacent of just should not be ‘one_F cat will make Patrick happy’, because the exclusive reading in that case would entail two cats will not make Patrick happy. We therefore assume that just should take scope right above or within the subject DP. This assumption is similar to Coppock & Beaver’s (2014) but we are crucially assuming that just has an exclusivity reading here, rather than a scalar reading.

In fact, such a DP-level reading of the exclusivity reading of just is independently observed. Consider (18), for example, which is ambiguous with respect to the scope of the exclusivity inference of just.

If the exclusivity inference of just is understood at the sentential scope, the example will entail that nothing other than the relevant teapot costs £3. But this example has another reading, according to which buying that teapot alone without anything else will cost £3. This latter reading, unlike the former reading, is compatible with another item—for example, a milk jar—costing £3 as well. It should also be noted that this reading is not a minimal sufficiency reading, as it doesn’t have a scalar component (although it could also have a minimal sufficiency reading; we will come back to this point in Section 5). Rather it is a kind of exclusivity reading in the sense that it is excluding other items from the total price.⁴

In order to work out the composition of the DP-level exclusivity inference of just, we will type-generalize the meaning given in (17) above. This can easily be done following the standard recipe (as is done in Rooth 1985, for example). Specifically, we define conjoinable types as follows.

Then we define generalized entailment, generalized conjunction, and generalized negation as follows.

Using these notions, the type-generalized version of (17) can be stated as (23).⁵ For a set S whose members are of the same conjoinable type, we write ∏S for the grand generalized conjunction of the members of S.

Using this semantics, the sentential-level exclusivity inference of (18) is derived as follows. In fact, there are two ways of deriving it. One is by assuming that just simply takes sentential scope (potentially via covert movement or some other scope taking mechanism). The other way is to assume that the subject DP this teapot denotes a generalized quantifier via the usual Lift operation that turns an entity into a generalized quantifier, i.e. a ⟼ λP. P(a)=1, and just applies to this generalized quantifier, turning it to the generalized quantifier that is true of any predicate that is true of the relevant teapot and nothing else, i.e. λP. P(t)=1 and for other relevant entities x, P(x)=1, where t is the relevant teapot. This latter derivation is important for later discussion in this paper, and should be emphasized at this point: even if just is syntactically at the DP-level, the exclusivity inference can arise at the sentential level.

In order to derive the DP-level exclusivity reading of just in (18), we have to make assumptions about plurality, because intuitively, what is being excluded here is the possibility that the subject denotes a plural entity. To this end, we will adopt Winter’s (2001) approach to plurality (see also Champollion 2016), which allows us to straightforwardly use the type-general semantics of just given in (23).⁶

Winter (2001) represents plural entities as sets of entities and treats predicates such as costs £3 as true of such pluralities qua sets of entities (in this case including singleton sets), and we will follow him here, although this assumption is not crucial for us. What is important is that he proposes a way to account for the non-distributive reading of conjoined proper names like Anna and Becky with generalized conjunction ⊓ (rather than a separate plurality forming conjunction). The derivation for this example goes as follows. First we lift the two individuals to quantifiers, and conjoining them via ⊓, which results in the distributive generalized quantifier, λP. P(a) = 1 and P(b) = 1, where a is the denotation of Anna and b is the denotation of Becky. To get the non-distributive reading out of this, the minimality-operator (min-operator) is applied to this generalized quantifier, which yields a predicate of predicates that is only true of the minimal ones with respect to ⇒ among the predicates that this generalized quantifier is true of, i.e. min(Q):= λ P. Q(P) = 1 and there is no P′ such that Q(P’) = 1 and P′ ⇒ P. For the example at hand, it will be a predicate that is only true of the predicate that is only true of a and b, i.e. λP. P = [λx. x = a or x = b]. Then existential type-shift applies to this set of predicates turning it into a plural existential quantifier, i.e. $\mapsto \lambda Q.\hspace{0.17em}\exists X(\chi (X))=1$ and Q(X)=1, where χ(X) is the characteristic function of the set X. This can now combine with a non-distributive predicate which takes a set of entities as an argument. We will henceforth call the combination of the two operations, the min-operator + the existential type-shift, the plurality-forming type-shift.

Assuming this theory of plurality, we can account for the DP-level exclusivity reading of just in (18) as follows. First, just applies to the lifted version of the individual denoted by this teapot, which yields the generalized quantifier λP. P(t) = 1 and there’s no other relevant entity x such that P(x) = 1, where t is the entity that this teapot refers to and x ranges over the referents of its focus alternatives. Then via the plurality-forming type-shift, we obtain the plural quantifier: λQ. Q({t}) = 1. This captures the reading of (18) that we are after.

With this analysis of just at hand, let us now come back to (4a), repeated here.

As explained above, just here should trigger an exclusivity inference at the DP-level. To derive it, we have to make assumptions about the semantics of one cat and its focus alternatives. We assume that DPs with bare numerals like one cat have existential meanings. In many examples such DPs receive a maximal, bilateral (‘exactly’) reading, which could be captured by analyzing the denotation of one cat as (24a) and that of two cats as (24b).

However, it has been debated in the literature whether the maximality of bare numerals should be accounted for semantically like this, or it should be derived as a scalar implicature from non-maximal, lower-bounded (‘at least’) readings of bare numerals as in (25) (cf. Horn 1972; Geurts 2006; Breheny 2008; Spector 2013).⁷

Notice that if DPs with bare numerals receive bilateral readings like (24), the contribution of just would be vacuous, because, for example, (24a) and (24b) are already incompatible with each other (in the sense of generalized contradiction, ⊥, as defined in footnote 5). As a vacuous use of just seems to be generally infelicitous, let us assume the lower-bounded semantics of numerals given in (25) here. In fact, it seems that all accounts of bare numerals assume that the lower-bounded semantics is at least available, if not by default. Then, just one cat will denote the generalized quantifier in (24a). This can further undergo the plurality-forming type-shift, yielding the plural quantifier, λQ. ∃X[X is a singleton set containing one cat and Q(X) = 1].⁸

The DP-level exclusivity reading of just in (15a) can be analyzed analogously, but to save space, we will not spell it out here.

Having analyzed the semantic contribution of just in (4a) and (15a), we will now turn to the semantics of even.

3.2 The semantics of even

Even is standardly analyzed as a focus sensitive particle that introduces two presuppositions based on focus alternatives, a scalar presupposition and an additive presupposition (Karttunen & Peters 1979; Rooth 1985; Kay 1990; Wilkinson 1996; Crnič 2011). To illustrate, consider (26).

This sentence has a scalar presupposition that Marten was unlikely to dance, and also an additive presupposition that someone else also danced. Since these inferences are absent from the version of the sentence without even, they should arise from even. We can formalize the semantic contribution of even as follows.

Two caveats are in order here. Firstly, proposals in the literature differ as to how exactly the scalar presupposition in (27a) is characterized, e.g. in terms of likelihood, noteworthiness, or something more general (Rooth 1985; Kay 1990; Herburger 2000; Greenberg 2018). We are certainly not in a position to solve this debate, and we will try to make sure that our analysis of the minimal sufficiency reading does not crucially hinge on one particular view on this point. Secondly, it is known that the additive presupposition (27b) is not always observed, which has led some to propose that it is not inherent to the semantics of even, but as far as we can see, there is no theory at the moment that can fully explain when exactly the additive presupposition arises (see Rullmann 1997; 2007 and Crnič 2011 for attempts). For this reason, we will stick to the above ‘classical analysis’, where it is encoded in the semantics of even.

Now, to see how the above semantics works, let us apply it to (26) above. The focus alternatives to the prejacent ‘Marten_F danced’ will be, for example, ‘Paul danced’, ‘Sophie danced’, ‘Sascha danced’, etc. The scalar presupposition amounts to that it was relatively unlikely among these alternatives that Marten would dance and the additive presupposition to that at least one more person danced, which capture the observed reading of (26).

Let us now come back to (4a).

As discussed above, the exclusivity inference of just takes DP-level scope in this example. For even, we assume that it takes sentential scope for the following reason. Recall at this point the problem for Coppock & Beaver (2014). Under their account of the minimal sufficiency reading, the scalar meaning comes from just, which takes a DP-level scope, and for that reason it is oblivious to the material outside the local DP. But that was shown to result in an overgeneration problem, because the VP portion of the sentence matters for the availability of a minimal sufficiency reading. The sentence in (4a) contains an overt scalar particle even, but if it introduces a scalar inference at the level of the subject DP, then it will run into essentially the same problem. That is, the contrast between (4a) and (4b) will be difficult to explain.

For this reason, it is crucial that even somehow sees the VP material as well. The most natural way is to assume that it takes sentential scope here. Note that we do not make particular commitments about the syntax of these sentences. It is possible that even is structurally located within the DP on the surface or it in fact sits above the rest of the sentence. In the former case, even would have to take wider scope than its syntactic position, which could be done with a standard mechanism of scope shift (e.g. Quantifier Raising, or type-shift).

Now, notice that (4a) contains two focus sensitive particles, even and just, but only one focused material, one. We have to say something about how this works, given that Rooth (1992) and many later studies assume that when a focus operator operates on a set of focus alternatives, it ‘eats up’ these focus alternatives, and yields a trivial set of focus alternatives as a consequence. More concretely, for a phrase like ‘just [one_F cat]’, the relevant focus alternatives that just operations on will be a contextually determined subset of {‘ξ cat’|‘ξ’ is a numeral}, but once the meaning of just is computed, a higher phrase cannot refer to these focus alternatives anymore. Thus, it is often assumed that in a configuration like ‘even [just one_F cat will make Patrick happy]’, the set of focus alternatives for even is just a singleton set containing ‘just one_F cat will make Patrick happy’. Of course, if that is the case, then the semantic contribution of even will be vacuous, which needs to be avoided this situation. We will propose a simple solution to this issue in the next subsection and show that the intended reading of (4a) can be derived together with the semantics of just and even introduced above.

3.3 Two focus particles and one focused item

Rooth (1992) introduces the assumption that focus particles eat up focus alternatives, but he does so without presenting empirical evidence for it.⁹ We would like to question this assumption here. In fact, empirical facts independent from minimal sufficiency readings suggest that a single focussed phrase can actually associate with multiple focus sensitive operators, at least on the surface. Such examples are discussed and analyzed by Krifka (1991). Consider the following example taken from Krifka (1991: 131), for example.

Krifka’s analysis of this example is couched in the Structured Meaning approach to focus. We could state our analysis in this alternative approach, but we stick to the Roothian approach we have been assuming here. Recasting Krifka’s idea for such complex foci in a way that is compatible with our framework of choice amounts to dropping Rooth’s aforementioned assumption, and assume instead that focus alternatives can (at least optionally) remain active after the structurally lower focus operator has operated on them.¹⁰ With this revised assumption, the set of focus alternatives for only in (28) will be a subset of {‘drank ξ’|‘ξ’ = ‘water’, ‘beer’, ‘wine’, ‘vodka’, …}, and the set of focus alternatives for even will be a subset of {‘only drank ξ’|‘ξ’ = ‘water’, ‘beer’, ‘wine’, ‘vodka’, …}. Consequently, the scalar inference that even yields is that (John’s) drinking water but nothing else is less likely/more remarkable than (John’s) drinking beer and nothing else, (John’s) drinking wine and nothing else, etc., which captures the intended reading of (28).

We therefore think that there are enough empirical reason to assume that focus alternatives do not necessarily become inactive after a focus operator has operated on them. Now we have all the ingredients to analyze (4a). The focus alternatives for just are, as before, a subset of {‘ξ cat’|‘ξ’ is a numeral}. On the other hand, the focus alternatives for even, which we assume takes sentential scope here, will be sentences like the following:

There can well be more focus alternatives, but without loss of generality we will just use these three here. Note that the predicate will make Patrick happy is a non-distributive predicate, so we assume that the subject DPs in these sentences undergo the plurality-forming type-shift. Crucially, we assume that the plurality-forming type-shift applies after the exclusivity inference is introduced by just, because otherwise the exclusivity inference would take scope at the sentential level (see the discussion in Section 3.2). Consequently, (29a) means that if there is an entity made up of exactly one cat, that entity will make Patrick happy, (29b) means that if there is a plural entity made up of exactly two cats, that plural entity will make Patrick happy, and likewise (29c) means that if there is a plural entity made up of exactly three cats, that plural entity will make Patrick happy.¹¹

With these focus alternatives, the scalar presupposition of even will be that (29a) is relatively unlikely/more remarkable among them, and its additive presupposition will be that at least one more focus alternative is true. These seem to capture the intuitive reading of (4a).

As remarked in passing at the end of Section 3.1, the truth-conditional contribution of just in (4a) is locally meaningful (with the lower-bounded semantics for bare numerals), but truth- conditionally vacuous in the end, due to the plurality-forming type-shift (although it could have other interpretive effects; see Section 5). This leads to a prediction that just is not a necessary component for this reading, which is in fact borne out: As we saw at the beginning of the present section, (16a) has essentially the same reading as (4a).

Our analysis for this sentence proceeds as follows. Here, there is only one focus sensitive operator, even, and the relevant alternatives include the sentences in (30).

As discussed in Section 3.1, bare numerals can have lower-bounded readings or bilateral readings, but as before, the existential quantifier embedded in the plurality-forming type-shift will neutralize their distinction at the global level, and all the sentences in (30) will effectively have lower-bounded (‘at least’) readings.¹² Then the presuppositions that even yields will be exactly the same as in the previous example. That is, the scalar presupposition is that (30a) is less likely/more remarkable than the other alternatives, and the additive presupposition is that at least one more alternative is true.

The other example containing both even and just, namely (15a), and its variant without just, (16b), can be analyzed analogously, although more contextual information is needed to determine the relevant set of focus alternatives in these cases.

For the sake of illustration, let us consider the following focus alternatives for even. Just will be included for (15a) but not for (16b):

We again assume that just introduces an exclusivity inference at the DP-level and since the predicate is collective, the whole DP undergoes the plurality-forming type-shift, due to which the exclusivity inference is neutralized, as in the previous examples. Consequently, each sentence in (31) has the same truth-conditions with or without just. Now, the scalar presupposition of even amounts to that (31a) is relatively unlikely/remarkable, and the additive presupposition to that at least one of (31b) and (31c) is also true, which capture the observed readings of (15a) and (16b).

3.4 Minimal sufficiency with covert even

Having explained how the examples that contain both an overt even and just can be analyzed, we are now ready to account for our initial examples, (2) and (3a), which do not contain an overt even. We simply propose that these sentences contain a covert version of even, which we assume has the same semantics as its overt counterpart (cf. Chierchia 2013). Then they can be analyzed in a completely parallel fashion to what we have just demonstrated above.

We furthermore would like to convince the reader that it is not too far-fetched to postulate a covert version of even. A growing body of research, at least since Groenendijk & Stokhof (1984), suggests that there is a mechanism that gives rise to an exclusivity inference without an overt exclusive particle, although there is a heated debate about the nature of that mechanism in the current theoretical literature, especially with respect to whether or not the mechanism is purely pragmatic or something that is present at the syntax-semantics interface (see, for example, Chierchia et al. 2012, Geurts 2010, and references therein).

Whatever the true nature of the mechanism, exclusivity interpretations are clearly available for sentences like the following, despite the fact that they do not contain an exclusive particle.

We need not go into the theoretical debate about exclusivity inferences here (see the references above), but what is relevant for our purposes is Chierchia’s (2013) idea that a similar mechanism exists that gives rise to scalar inferences.¹³ To illustrate this, Chierchia (2013: 109) mentions the following example.

Here, the second sentence can be paraphrased by ‘Imagine that even my ex_F came’, and has a scalar inference that is similar in nature to the scalar inference of even.

Parallel to the debate on exclusivity inferences mentioned above, the question then arises as to whether such scalar inferences without an over scalar particle should be understood in terms of some pragmatic reasoning or in terms of a covert operator that is present at the syntax-semantics interface.

In this connection, it should be noted that previous studies on exclusivity inferences have raised a number of examples that seem to involve an exhaustivity inference below some linguistic operator, which have been used to argue for the operator-based view, and simultaneously against the pragmatic view of exclusivity inferences. However, whether or not these examples are actually problematic for a pragmatic analysis has been highly controversial (see, for example, Sauerland 2004; Russell 2006; Geurts 2010). Importantly for us, similar examples of embedded scalar inferences can be constructed, as an anonymous reviewer pointed out to us. For instance, the second sentence of (33) seems to involve the scalar inference under imagine, and also, more importantly for our interests here, the following sentence seems to have a minimal sufficiency reading under the scope of doubt.

If one assumes that covert even is a pragmatic mechanism of some sort, one will certainly need an analysis of how it can take scope below operators like doubt.

While acknowledging the importance of this observation, however, we cannot evaluate the feasibility of the pragmatic view with respect to such sentences, in the absence of an explicitly formulated pragmatic theory of scalar inferences. In fact, it is not necessary to assume that a pragmatic mechanism cannot affect the semantics locally within the scope of an operator like doubt, even for exclusivity inferences (although such an assumption is certainly reasonable for purely Gricean conversational implicatures; cf. Geurts’s 2010 discussion of pragmatic enrichment of lexical meanings). For this reason, we are reluctant to reject the pragmatic view altogether at this point. But as far as the present paper is concerned, the only crucial assumption is that such a mechanism exists (and can interact with linguistic operators), and we will not decide whether that is done via a covert operator in the syntax/semantics, or via some pragmatic mechanism. That being said, for the ease of exposition, we will speak for the rest of this paper as if the relevant mechanism is a silent operator that has the same semantics as even, and call it ‘covert even‘.