Comparing answers

Instead of defining a notion of partial answerhood as such, in terms of exclud- ing certain possibilities, we concentrate on a notion which compares indicatives as to how completely and precisely they answer a certain question. First we define an auxiliary comparative notion of informativeness, which leaves precision out of consideration.

Definition 4.21(Informativeness).

(i) φgives a partial true answer to ?ψ inwin M iff[φ]M∩[?ψ]M,w6=∅.

(ii) φis a more informative answer to?ψ inM thanφ0 iff ∀w∈M: ifφ gives

a partial true answer to?ψ inM inw, thenφ0 does, too.

The auxiliary notion of giving a partial true answer is a very weak one:φgives a partial true answer inwinM iffφoverlaps with the block in the partition in which

wis situated. In particular, it is not required that φitself be true inw, only that it be compatible with the actual true answer. Neither is it required thatφexclude any possible answers. Thus, even the tautology counts as a partial true answer.

Notice that the comparative notion of being a more informative answer does favor indicatives which exclude more possible answers. The contradiction then turns out to be the most informative answer to any question. Disregarding that, the most informative answers to ?ψ are complete answers, i.e., thoseφ such thatφ|=?ψ. If

φandφ0 imply the same possible answer to ?ψ, they count as equally informative relative to the question. In terms of the absolute notion of informativeness, i.e., entailment, the one may be more informative than the other, or the one may imply the negation of the other, or they may be incomparable. Ifφandφ0 imply different possible answers to ?ψ, they are unrelated with respect to their informativeness relative to ?ψ.

Partial true answerhood and relative informativeness are put to use in the fol- lowing definition, which also takes the precision of answers into consideration (the obvious relativization to a modelM and a worldware omitted).

Definition 4.22(Comparing answers).

Letφandφ0 give a true partial answer to?ψ. Thenφis a better answer to?ψthan

φ0 iff

(i) φis a more informative answer to?ψ thanφ0; or

(ii) φ andφ0 are equally informative answers to ?ψ and φ is properly entailed

byφ0.

According to the first clause, among the true partial answers the more informative ones, which exclude more possible answers, are preferred. The second clause favors weaker answers among equally informative ones. The sum effect is that the answer that equals that block in the partition which contains the actual world is the most preferred. Next are those answers which are inside that block, but do not fill it

completely. This means that such answers contain additional information that the interrogative does not ask for. This lack of precision is not as harmless as it may seem. An answer whichgivesa complete (or partial) true answer need not express a true proposition itself. The precise true and complete answer which fills the whole block is guaranteed to do so. Stronger, and hence over-informative answers, can express false propositions themselves. I.e., although with respect to the question posed by the interrogative they provide correct information, at the same time they may provide incorrect information with respect to some other question. This is precisely why the comparison of answers favors weaker propositions among the ones that give the same answer.

Of course, the available information may simply not support such a complete answer, in which case a partial answer is all that can be offered. The effect of the comparison is that among the answers supported by the information, those are selected that are incompatible with all possible answers incompatible with the infor- mation. There again, even though the available information may support stronger propositions, the comparison prefers the unique partial answer that completely fills the union of the blocks corresponding to the possible answers compatible with the information. Again, there is a good reason for this. The information as a whole may be incorrect at some points. But still, it may support a true partial (or complete) answer to some questions. Providing more information than an interrogative asks for, means answering other questions at the same time. But the answer given to those might very well be false. The comparison cautiously tries to prevent this.

Besides partiality of information there is another reason why proper complete (or partial) answers may not be available: it might be that they are not expressible in the language. (Or not expressible relative to the information. In discussing the notion of answerhood, we have seen that in the predicate logical case this can easily occur.) In these cases, too, we have to make do with partial or over-informative an- swers which are expressible, among which a choice needs to be made. In particular, to be able to provide proper answers, expressions are needed that (with respect to the information available to the questioner) rigidly identify (sets of) objects. If such expressions are lacking, a comparison among the available answers is called for. Hence, such a comparison is a necessary ingredient of any theory of questions and answers.44

4.5.5. Remark on natural language

It should be noted that, unlike in natural language, in this predicate logical system querying over variables is unrestricted. In natural language, however, this almost never the case. A wh-phrase is usually of the form which cn, and even for those

phrases which lack on overtcn, such as who,what, it can be argued that they are

in fact restricted.

Taking our lead from quantification in standard predicate logic, we might think that restricted querying can be expressed by means of conjunction: ‘WhichP are

Q?’ would be turned into ?x(P x∧Qx). In most cases this representation is adequate, but there is a snag here. Consider the following pair of sentences:45 _{‘Which men} are bachelors?’, ‘Which bachelors are bachelors?’. Clearly, the first sentence poses a non-trivial question, which is adequately represented by ?x(M x∧Bx). The second sentence, is trivial: it asks the tautological question. However, its representation, ?x(Bx∧Bx), is equivalent to ?xBx, which is not trivial at all: it asks who the bachelors are. Clearly, a real extension of the syntax of the representation language is called for, if we are to deal with these cases.

*

Up to this point we have been concerned mainly with developing an argument that purports to show that a semantic analysis of interrogatives is a viable subject. The outlines of one particular approach, based on the three Hamblin postulates, have been sketched, using two simple formal languages as pedagogical devices. But other approaches have been developed in the literature. In the remainder of this chapter we will discuss them under two headings: logical and computational theories, and linguistic theories.