On fatal competition and the nature of distributive inferences

Abstract

Denić (2018, 2019, To appear) observes that the availability of distributive inferences—for sentences with disjunction embedded in the scope of a universal quantifier—depends on the size of the domain quantified over as it relates to the number of disjuncts. Based on her observations, she argues that probabilistic considerations play a role in the computation of implicatures. In this paper we explore a different possibility. We argue for a modification of Denić’s generalization, and provide an explanation that is based on intricate logical computations but is blind to probabilities. The explanation is based on the observation that when the domain size is no larger than the number of disjuncts, universal and existential alternatives are equivalent if distributive inferences are obtained. We argue that under such conditions a general ban on ‘fatal competition’ (Magri 2009a,b, Spector 2014) is activated, thereby predicting distributive inferences to be unavailable.

Appendices

Appendix A: Pruning

In Sect. 6, we identified the question a sentence addresses with a superset of its set of formal alternatives. However, as is known since Horn (1972), some formal alternatives of a sentence can sometimes be ignored (‘pruned’), depending on the context. As we see below, the picture we drew in this paper becomes more complicated once we allow for pruning to shrink the set of alternatives against which Question-Sentence Correspondence is evaluated.

One way to avoid these complications could be to simply assume that Sentence-Question Correspondence applies as stated in (24) (i.e., considering all formal alternatives), even when alternatives are pruned. This assumption, however, raises empirical questions; as we pointed out in Sect. 6 (fn. 21), an empirical issue arises whenever an implicature is not associated with a weak alternative because the strong alternative is taken to be irrelevant. There is a simple way to avoid this issue by assuming a version of (24) which is based on the idea of partition by exhaustification from Fox (2018), as in (44).

• (44)

  

(44) does not require the set of alternatives of a sentence to be a partition of some subset of the context set, but rather merely requires that it be a set which would yield such a partition if exhaustification applied. As a result, it does not mandate applying $\mathit {\mathcal {E}xh}_{}$, nor does it put any constraints on pruning if $\mathit {\mathcal {E}xh}_{}$ does in fact apply.

While this looks like a reasonable choice to make, in this appendix we would like to consider the possibility that pruning does affect Sentence-Question Correspondence, and that it requires the set of alternatives which are left after pruning to be a partition of some subset of the context set. In Sect. A.1, we aim to show that our proposal can be maintained even if Sentence-Question Correspondence is affected by pruning, once an apparently unrelated issue with FCIs is taken into account. In Sect. A.2, we further suggest a pruning-based restatement of our proposal which aims to account for the felicity of some sentences which have contextually contradictory alternatives.

1.1 A.1 Pruning and distinctness

If Sentence-Question Correspondence is taken to be affected by pruning, then coming up with a specific theory of pruning becomes crucial. If pruning of any alternative is always possible, we can no longer rule out cases where some and all sentences are equivalent, as we will always be able to prune the equivalent alternative and avoid a violation of Sentence-Question Correspondence; and if pruning is never possible, then implicatures should always be obligatory, contrary to fact. So if Sentence-Question Correspondence applies after pruning, we need to weaken our constraint as in (45), which assumes that pruning is possible but limited to irrelevant alternatives, relying on Lewis’s (1988) notion of relevance in (46) (mainly following Magri 2009a,b, Fox and Katzir 2011):

• (45)

  

• (46)

  

Support for the assumption that (46) governs prunability comes from several sources. Magri argues (based on the data in (23)) that when two alternatives are contextually equivalent it’s impossible to prune one without the other, which is a consequence of (45) when relevance is defined as in (46). In the same spirit, Fox and Katzir argue that an alternative cannot be pruned when it’s in the Boolean closure of the set of alternatives which haven’t been pruned, which is also a consequence of (45) when relevance is defined as in (46).

Let us show first how this proposal explains the basic case of oddness discussed by Magri in (23). The contextual equivalence of some and all makes sure (just like in Magri 2009a,b) that the some alternative cannot be ignored once all is relevant. And as before, Q cannot contain both alternatives and still be a partition. Furthermore, (46) renders contradictions relevant (as noted by Lewis 1988), which is why $\mathit {\mathcal {E}xh}_{}$(some) (=some ∧¬ all) also cannot be ignored in this case (see Bar-Lev 2023 for elaboration on the effects of the relevance of contradictions on pruning).

The situation becomes more involved once we consider the case we are interested in, namely the disappearance of DIs when n ≥ |D|. First, since the universal and existential alternatives are not equivalent in this case before exhaustification applies, one might suspect that the existential alternative can be ignored so that it would not be able to block the derivation of DIs if n ≥ |D|. However, note that the existential alternative ∃x(Px∨Qx) is equivalent to the disjunction of the alternatives ∃xPx and ∃xQx. As mentioned above, relevance is closed under Boolean operations (conjunction, negation, and disjunction; see Fox and Katzir 2011); hence, it is impossible for ∃xPx and ∃xQx to be relevant without ∃x(Px∨Qx) being relevant as well (because ∃x(Px∨Qx)⇔∃xPx∨∃xQx). Since the relevance of ∃xPx and ∃xQx is a necessary ingredient for deriving DIs, it is impossible to derive DIs while pruning the existential alternative ∃x(Px∨Qx).

Second, in our discussion of the equivalence between (19) and (20), repeated in (47) and (48), we assumed that the strong exclusive inference in (47-a) is derived.

• (47)

  

• (48)

  

While there is evidence for the existence of the strong exclusive inference (see Chemla and Spector 2011), it is not very robust and is often not derived. Furthermore, the absence of DIs when n ≥ |D| does not seem to depend in any way on deriving the strong exclusive inference. However, the equivalence between (47) and (48) crucially relies on this inference. If (47-a) and (48-a) are replaced with the weak exclusive inference in (49), the equivalence no longer holds.

• (49)

  

We’d like to suggest that this issue has to do with a known problem located elsewhere, namely that FCIs are stronger than the conjunction of the existential disjunctive alternatives. This issue has been brought up by Menéndez-Benito (2010). Consider the following example (Menéndez-Benito demonstrated the problem with her Canasta scenario using free choice items; here we replicate it with Free Choice disjunction instead):³¹

• (50)

  

The oddness of (50) is surprising since the sentence is true both on its ordinary meaning and its strengthened meaning. Specifically, the FCIs in (51) are true in the context:³²

• (51)

  

What seems to us to be the minimal assumption needed in order to capture the surprising oddness of (50) is the following:³³

• (52)

  

(50) violates Distinctness: The set of worlds compatible with the rules where you take the blue card is the same as the set of worlds compatible with the rules where you take the red card.³⁴ Intuitively, in the case of (47) and (48), Distinctness requires that there would be one French friend (who is possibly also Spanish) and another Spanish friend (who is possibly also French). Under the distinctness assumption, (47) and (48) are equivalent even in the absence of the strong exclusive inference, since for (48-b) to be true there would have to be one French friend and another Spanish friend. But if that’s the case then (47-b) is true. More generally, the following holds:

• (53)

  

Given Distinctness, then, our account of the CDG extends to cases which involve pruning (whatever may be the proper explanation for Distinctness; see fn. 36 for one possibility).

1.2 A.2 Felicity with contextually contradictory alternatives

As established, we predict that having contextual contradictions as alternatives should lead to infelicity. Note furthermore that this does not change once we allow for pruning to affect Sentence-Question Correspondence as in (45), since contradictions are always relevant and, as a result, cannot be pruned (Lewis 1988, Bar-Lev 2023). As noted in fn. 22, there are, however, felicitous sentences which have contextual contradictions as alternatives. Consider the following example brought up by Roni Katzir (p.c.):

• (54)

  

We acknowledge that this might be a serious problem for the perspective we proposed in this paper and suggest two tentative paths towards a solution. The first path latches on to a salient difference between the contextual contradictions leading to infelicity (or to blocking DIs) we discussed in this paper and (54), namely that in (54) the contextual contradiction is not the result of an application of $\mathit {\mathcal {E}xh}_{}$. We could propose to exempt alternatives which are contextually trivial from the get go from even being considered. This, in turn, could be achieved by replacing the requirement to have Sentence-Question Correspondence (in any of its versions) with a constraint on exhaustified meanings, as follows:

• (55)

  

This constraint requires that the pointwise application of $\mathit {\mathcal {E}xh}_{}$ over the chosen set of alternatives be a partition, and that its domain must contain all relevant alternatives which are not trivial. (55) still explains the main facts we aim to capture in this paper if taken together with Magri’s assumption that applying $\mathit {\mathcal {E}xh}_{}$ is obligatory. At the same time, it is compatible with the felicity of (54) because it allows pruning of alternatives that are contextually contradictory before exhaustification.³⁵

Our second alternative path towards a solution focuses on a less obvious property of (54), namely that the contradictory status of an (exhaustified) alternative does not follow in any way from the demands it imposes on the prejacent. This raises the following alternative.

• (56)

  

• (57)

  

This weakening of the partition-demand can also capture an additional challenge to our proposal coming from the following example due to Bar-Lev (2023):

• (58)

  

Bar-Lev points out that, just like the deviant examples brought up by Denić in (6-a) and (7-a), (58) is non-contradictory if DIs are derived and contradictory if ignorance inferences are derived (because (58) can only be true if there is at least one parent-child pair in the group, so one cannot believe (58) while being ignorant about the DIs). If it’s impossible to derive neither DIs nor ignorance inferences (as we assumed in this paper following Denić), one may conclude that (58) is fine because DIs can be derived.³⁶

However, Question-Answer correspondence predicts that the derivation of DIs should be blocked, because of alternatives like the one in (59); this alternative is a contextual contradiction in the case of (58), since it is impossible that there be a person in the group who has a parent in the group (∃xPx) without there being a person in the group who has a child in the group (∃xQx).

• (59)

  

Note however that the version of the partition demand in (56) does not block the derivation of DIs here, since (59) remains a contradiction even when the prejacent is removed from the set of alternatives; in this case exhaustification of ∃xPx still entails the contextual contradiction ∃xPx∧¬∃xQx. In other words, this alternative does not engage with the prejacent and as a result does not need (by (56)) to be a member of the set that partitions a subset of the context set.

Appendix B: On the asymmetry between weak and strong alternatives

Consider again the two sentences in (23), repeated here in (60). We marked both sentences as infelicitous, following Magri (2009a,b) and Spector (2014). However, some speakers find that (60-a), although not entirely felicitous, is markedly better than (60-b).

• (60)

  

This contrast intensifies in examples such as (61) (apparently due to Emmanuel Chemla, discussed in Singh 2009, Magri 2011, Schlenker 2012, Spector 2014), where the universal alternative is judged as entirely acceptable and the existential alternative is still infelicitous.

• (61)

  

Our account of the CDG was crucially based on the assumption that both strong and weak alternatives are unacceptable in contexts where they are contextually equivalent (prior to exhaustification). If the account is right, the acceptability of (61-a) in a context where the first sentence in (61) is taken to be common ground (CG) is not expected.

We do not have a complete response to this problem and would like to simply express the hope that the ultimate resolution will be compatible with our proposal. Here we sketch a possible way of thinking about the problem that suggests to us that there might be reasons for optimism. Under our proposal, when the some and the all sentences are formal alternatives of each other and when they are contextually equivalent, both should be unacceptable. When things appear to deviate from this expectation, this, if we are right, must only be an appearance, one that occurs either because there is a parse of the relevant sentences where they are not formal alternatives of each other or because there is a way of thinking about the context where the alternatives are actually not contextually equivalent.

We pursue the latter possibility, building on an observation made by Spector about the nature of the distinction between (60) and (61) (see Singh 2009, who pursues the other possibility). Spector points out that the bit of the common ground that makes (60-a) and (60-b) contextually equivalent is deeply “entrenched”. Specifically, the belief that Italians all come from the same country is almost definitional, virtually on a par with the belief that bachelors are unmarried men. By contrast, the statement about the CG in (60) (that I always give the same grade to all of my students), although a possible characterization of the CG, is not as deeply entrenched.

Imagine, as suggested by Spector, that the contrast between the sentence with a universal and an existential quantifier weakens the more entrenched the homogeneity assumption (HA) is (the bit of common ground that makes the pre-exhaustified universal and existential alternatives contextually equivalent). To capture such a distinction in levels of entrenchment, one might be tempted to think of the CG as containing information about degrees of belief (as in, e.g., Lassiter 2012). Instead, we would like to continue to think of the CG in non-gradable terms: the more entrenched a bit of shared belief is, the more participants in the conversation are likely to take it as CG. In other words, what we would like to claim (following Spector) is that it could sometimes be difficult to ensure that some bit of information explicitly introduced is going to enter the CG. The introduction of information might make it clear that various participants in the conversation believe it to be true. It might even ensure that they all believe it to be true. But this does not necessarily ensure that they thereby fix on the idea that the utterance would be interpreted with this shared belief as CG, especially if exactly the same results could be achieved without taking this shared belief to be part of the CG.³⁷

2.1 B.1 All sentences sometimes acceptable despite the introduction of HA

We would like to suggest that the utterance in (61-a) is acceptable simply because we can understand it as one that is intended to be interpreted against a CG that does not already entail HA. Having a speaker assert HA (without a hearer protesting) might not be sufficient to ensure that HA enters the CG. A speaker might, still, intend for the (a) sentence to be interpreted against a weaker CG that does not entail HA, especially as this speaker can achieve exactly the same results without HA as CG; specifically, the speaker can expect to achieve a CG where HA is established as a consequence of the assertion. (Remember that the all sentence entails HA: saying that all students received an A entails that they all got the same grade.)³⁸

The remaining problem is to explain why the (b) sentence is unacceptable under the same circumstances, and why DIs are categorically blocked when |D| is not greater than the number of disjuncts.

2.2 B.2 Some sentences never acceptable with the introduction of HA

To understand why this strategy might not be available for the (b) sentence, it is useful to begin with the reasoning outlined in Magri’s work. Specifically, when (b) is uttered, we can assume with Magri that there is an obligatory $\mathit {\mathcal {E}xh}_{}$ that is inserted at the clausal level (below K if Meyer’s approach to ignorance inferences is adopted). If the all alternative is not pruned, we will have a straightforward explanation for unacceptability: the resulting meaning will contradict a shared belief, namely HA, the belief that all of the students got the same grade. This will lead to an anomaly despite our assumption that the shared belief need not be part of the CG. So, a parse in which the all alternative is not pruned is unacceptable for trivial reasons.

The more challenging case is the parse in which the all alternative is pruned. In a context in which some and all are contextually equivalent, such a parse will be unavailable for the reasons articulated by Magri. (Specifically, pruning an alternative is only possible if the alternative is taken to be irrelevant given the topic of conversation. But this is impossible given that the some alternative is taken to be relevant and that relevance is taken to be closed under contextual equivalence.) But this reasoning is not obviously helpful, given our assumption that the CG need not entail HA (despite the fact that HA is introduced as a shared belief). We would like to suggest that there is nevertheless a potential difficulty with the utterance of the (b) sentence even when the all alternative is pruned.

We would like to suggest that in general speakers and hearers need to be able to coordinate on the goal of the assertion—the CG that the speaker intends to achieve with her final assertion (CG$_{\text{goal}}$). The difficulty we see with the utterance of the (b) sentence is related to this requirement. Does the speaker intend for CG$_{\text{goal}}$ to be a CG that entails HA? If so, this would not be achieved by simple (Stalnakerian) update given the necessary assumption that HA is not part of the input context-set. If not, what kind of CG is the speaker interested in establishing? It is hard to see how the speaker and the hearer could coordinate on the goal of the assertion.

The situation with the all sentence is very different, as already mentioned. With the all sentence a decision to interpret the sentence against a CG that does not entail HA leads to exactly the same CG that would be reached if the sentence is interpreted against a CG that does entail HA, hence there is no difficult coordination problem to solve. This line of reasoning can be simplified with the following principle:

• (62)

  

This line of reasoning accounts for the disappearance of the contrast in (61) and crucially for the observation that it is weakened the more entrenched HA is. Specifically, if we can’t conceive of a world in which HA is false, the relevant principles of grammar will reveal themselves. So the more entrenched HA is, the closer our observations will be to the idealized prediction.³⁹

2.3 B.3 Back to the CDG

What remains is to explain why the CDG feels like a categorical constraint. Here we would simply like to refer to the level of entrenchment involved. It is simply impossible to imagine possibilities in which the most basic laws of arithmetic do not hold true. And, as we’ve seen, imagining such possibilities would be necessary to make the all and the some alternatives not equivalent (when inferences about each other are ignored) when |D| is not greater than the number of the disjuncts. More specifically, it is simply impossible for the existential FC meaning to be true while the universal alternative is false: if one of my two children lives in Europe and one lives in North America, it is impossible (if two means two) that the universal alternative is false, that is, that I have a child who lives neither in Europe nor in North America.