Although we have now seen how the embedding predicates distinguish between true and possible answers, we have not yet taken a look at what kind of relations these predicates actually denote. Observe that in the lexical entry for e. g.knowit is required that “Kxw ∈ Q(w)”; that is, x’s knowledge in worldw has to be an element of the answer set. This is different from merely demandingx knows a certain propositionq. In the latter case,q is one of the (usually) many propositions that x knows, while in (140) x’sentireknowledge is specified. To understand why this is a reasonable requirement, recall that the answer set is downward-closed modulo NFA. Hence,Kxw ∈ Q(w) can be arbitrarily specific. What the lexical entry in (140) does is simply to impose conditions onx’s knowledge; but it is not very demanding:xis free to knowanythingas long as she knows a true answer to the embed- ded question. Ifknowembeds an interrogative,ANS[+/–cmp]ensures that the answer setQ
is downward-closed modulo NFA—which means it obeys the no-false-answers constraint. Accordingly,know-whcomes out as upward-monotonic modulo NFA. In contrast, ifknow embeds a declarative, the answer setQ yielded byANS[+/–cmp] will be vanilla downward- closed—which means the no-false-answers constraint does not apply. Consequently,know- thatcomes out as vanilla upward-monotonic.
A word on the significance of downward-closedness for embedded clauses in general. In
Inqλ
B, sentence denotations, both those of declaratives and of interrogatives, are downward- closed. For one of the reasons behind this design decision, see Section 2.3.1.2. Now, our semantics is compositional: interrogatives and declaratives have the same denotation regard- less whether they appear in a root or an embedded context. It hence goes without saying that the denotations ofembeddedsentences will have to be downward-closed as well. These are stillsentences, however.Answer setsin contrast might be a different matter. In the proposed setup, ANS[+/–cmp] overrides the vanilla downward-closedness of the question denotation
and replaces it by a more restricted version of downward-closedness. As forrestrictingthe downward-closedness, we have just seen how this can be useful. But in fact—why do we
have to make the answer set downward-closed in the first place? Quite simply, for the same reasons that have already motivated the downward-closedness of sentence denotations. That is, it helps us achieve an explanatorily more adequate treatment of clause conjunction—this time in embedded contexts.
Consider e. g. the conjunction of two embedded clauses in (142), a modified version of (57). (142) Mary knows whether John speaks French and that he speaks Russian.
Again, we can capture this conjunction using the uniform lexical entry (143) for and; it applies to both embedded declaratives and embeddded interrogatives (or rather, to their respective “answer sets”).
(143) Tr(and):=λQ〈s,T〉.λQ〈0s,T〉.λws.λp〈s,t〉.Q(w)(p)∧ Q0(w)(p)
The syntactic structure of example (142) is sketched in Figure14.11Note that the denotation of the conjunction is computed simply by intersecting the two conjuncts. Modulo the type- lift, this is the classical treatment of conjunction. As discussed in Section2.3.1.2, being able to use ordinary intersection in an alternative semantics hinges crucially on the fact that the objects that get intersected are downward-closed. If they are not,pointwiseintersection is required instead. In a framework based on set-theory, however, this choice would be less generally motivated since pointwise intersection, other than classical intersection, is not a meet operation.
The answer sets in Figure14are indeed vanilla downward-closed: one stems from a declar- ative clause (it has been laid out above that applying the answer operator to a declarative denotation does not remove any states from this denotation) and the other one stems from an exhaustive question denotation (all states from an exhaustive question denotation satisfy the NFA and thus none of them are excluded from the answer set). However, does the clas- sical treatment of sentence conjunction still work reliably if the answer sets are not vanilla downward-closed, but only downward-closed modulo NFA? Yes. In general, recall that the reason why those pieces of information that are excluded from an answer set by virtue of the NFA need to be excluded is that they are not allowed from entering into the specified relation (e. g. knowledge) with the individual. This basic fact does not change only because we are dealing with sentence conjunction instead of with single sentences: pieces of informa- tion not complying with the NFA have to be excluded. Since such excluded pieces will never be relevant for the specified relation anyway, they need not “participate” in the clause con- junction either. What kind of problem then could arise from the lack of vanilla downward- closedness? It might be the case that, for a given worldw, the intersection of two answer setsQ(w)and Q0(w)is empty. However, it appears that the only worlds in which this can happen are those where the conjunction would give rise to a contradiction; and those cases already come out as undefined by virtue of the factivity presupposition. Consider the following example. Assume that what gets coordinated byand areQ = ANS[+cmp]( )
and Q0 =ANS[+cmp]( ). Then, becauseQ is only downward-closed modulo NFA, we find that is not an element ofQ(w2). IntersectingQ(w2)with would therefore in- deed yield the empty set. However, notice that is not true atw2. Hence,Q0(w
2)simply 11 Although in this example both clauses contain the[+cmp]version of the answer operator, nothing hinges on
Mary knows 〈s,T〉 ( w1 7→ w2 7→ ) 〈s,T〉 ¦ w1,w2 7→ © ANS[+cmp] T that John speaks French and ::〈〈s,T〉,〈〈s,T〉,〈s,T〉〉〉 λQ.λQ0.λw.λp. Q(w)(p)∧ Q0(w)(p) 〈s,T〉 ( w1,w3 7→ w2,w4 7→ ) ANS[+cmp] T whether he speaks Russian
Figure 14: Conjunction of embedded clauses
is not defined (likewise forQ0(w4)). The conjunction ofQandQ0is thus only defined for the worldsw1andw3.
In this section it has been shown how restricted downward-closedness can be used to implement upward-monotonic predicates which obey the no-false-answers constraint. This is exactly what we wanted for verbs likeknowandbe certain: since they only allow[–lit] readings, they are always upward-monotonic. However, our account does not yet take care of emotive predicates and verbs of communication, which—depending on whether they are read deductively or literally—differ in their monotonicity behaviour and their sensitivity to background knowledge. Before turning to the[+/–lit]distinction, though, we will first explore a contrast that shows up already with the above lexical entries—namely the contrast between inquisitiveness holes and inquisitiveness plugs.