More evidence: conditional questions
So far, we focused on declarative sentences. But if-clauses occur in other kinds of sentences as well, as witnessed by the following interrogative:Footnote 23
Since the restrictor view aspires to be a fully general account of if-clauses, we should be able to provide an analysis of (21) based on the assumption that the if-clause is a restrictor. But since there is no obvious operator to restrict, the challenge we faced for bare conditionals also arises here.
Suppose one resorts again to a covert epistemic necessity modal. Then (21) asks which individual x is such that the evidence implies that if there is an accomplice, it is x. This is not the right result. To see why, suppose we are sure the evidence is compatible with different people being the accomplice. Then we are certain that there is no individual x such that the evidence implies that if there is an accomplice, it is x. So, we should perceive (21) as being just as defective as (22):
But this is not the case. In the described situation, (21) seems a perfectly meaningful question to entertain.
This speaks against (21) being epistemically modalized: intuitively, (21) is not a question about the evidence, but a question about the facts. When asking (21), what one is asking is who the accomplice actually is. However, one is asking this under the supposition that there is an accomplice. In other words, by asking (21), one first sets up a hypothetical context by restricting to words where Axton has an accomplice. Then one asks, relative to this context, who this accomplice is.
Assuming this analysis is on the right track, the question is how to derive it compositionally, on the assumption that if-clauses are restrictors.
On the view that if-clauses restrict operators, it is not clear how to do that. What operator would the if-clause in (21) be restricting?
By contrast, the view that if-clauses can restrict an information state parameter delivers the above prediction in a natural way. Let us see how. Suppose, following recent work on inquisitive semantics (Ciardelli et al. 2018), that the semantics of interrogatives is given by specifying resolution conditions relative to an information state. That is, interrogatives \(\textsf {Q}\) are interpreted by a map of the form \([\![\textsf {Q}]\!]^{s,\dots }\), which takes the value 1 when Q is resolved by the information available in s. For instance, the question ‘who is the accomplice?’ is resolved in a state s just in case all the live possibilities \(w\in L_s\) agree on who the accomplice is (i.e., the same individual is the accomplice in all the live possibilities).
Now consider (21): syntactically, it involves an if-clause modifying an interrogative main clause. That is, (21) is naturally analyzed as a conditional \(\textsf {A}\Rightarrow \textsf {Q}\) with an interrogative consequent. We can analyze the if-clause as a restrictor of the information state parameter, obtaining:
$$\begin{aligned}{}[\![\textsf {A}\Rightarrow \textsf {Q}]\!]^{s,\dots }=[\![\textsf {Q}]\!]^{s+\textsf {A},\dots } \end{aligned}$$
This says that to resolve a conditional interrogative is to resolve the consequent under the assumption of the antecedent. This seems correct. E.g., this predicts that (21) is resolved in a state s if all the live possibilities in s in which Axton has an accomplice agree on who this accomplice is.
Now suppose that, in typical situations, by asking a question one requests information that resolves the question. Then we predict that by asking a conditional question, one requests information that resolves the consequent under the assumption of the antecedent. I.e., one formulates the request for information specified by the consequent (in our example, identify the accomplice), but only in restriction to the antecedent worlds (those in which there is an accomplice). This vindicates the intuition described above about the effect of asking of a conditional question.
Thus, looking at conditional questions provides an independent source of evidence for the view that if-clauses need not always restrict operators, but may instead restrict an information state parameter of the semantics.
Truth ascriptions
In the proposal we just described, non-factual sentences do not express propositions, but a more general sort of contents, that we called x-contents. As we described above, x-contents are things towards which one can have an attitude. However, they are not evaluable in terms of truth at a world—at least, not when they do not correspond to a proposition.
However, in natural language we do ascribe truth to claims involving expressive conditionals and modals. Consider, for instance:
Here, the anaphor ‘that’ presumably refers to the content that Alice expressed by her modal claim. But this content is not a proposition, so it cannot be true or false. Is Bob just making a bad category mistake?
Before sketching a response to the objection, notice that the worry is not specific to our view. Rather, it arises for anyone who holds that the function of certain declarative sentences is not to describe the world. For instance, it is famously a matter of debate whether the function of evaluative claims and moral claims is to describe facts about the world, or rather to manifest certain attitudes. Yet the following dialogue is unremarkable.
I think it is fair to doubt whether ‘true’ as used in such responses means the same as ‘true’ in a more substantive philosophical sense. In the latter use, truth is a kind of correspondence between language and the world: a sentence is true if it describes things as being in a way they actually are.
By contrast, it seems plausible that what Bob is doing in (23) and (24) is merely to express his agreement with Alice’s claim—to manifest a certain alignment of attitudes—which is of course possible regardless of whether Alice’s claim is capable of being true in the substantive sense.
This view goes well with a deflationist account of truth ascriptions. According to that view, to claim of something that is true amounts to claiming that very thing. Thus, in (23), Bob’s claim is equivalent to the claim that the butler might have done it, and in (24), it is equivalent to the claim that the tiramisù was delicious. Here is one possible way to implement the idea compositionally: in Sect. 6 we interpreted ‘probable’ as a function from x-contents to x-contents; we can treat ‘true’ as a function of the same kind: the identity function.
$$\begin{aligned} \textsf {true}\quad \leadsto \quad \lambda C.C \end{aligned}$$
This vindicates the deflationist idea that ‘A is true’ is equivalent to A. In the case of the dialogue in (23), Alice claims that \(\Diamond b\), and thereby expresses the x-content \([\![\Diamond b]\!]_{\textsf {AS}}=\{\langle s,a\rangle \mid [\![\Diamond b]\!]^{s,a}=1\}\). In Bob’s utterance, the anaphor ‘that’ refers to that x-content. Applying the truth predicate returns the same x-content, \([\![\Diamond b]\!]_{\textsf {AS}}\). So we predict that by his utterance, Bob is merely endorsing Alice’s modal claim, and not describing it as true in the more substantive sense.
Of course, this story is not the only option available to us. My aim in presenting it is merely to illustrate the point that truth ascriptions like the one in (23) are not necessarily a problem for the present account.Footnote 24
A uniform treatment of modals
In this paper, we have strived to retain a unified account of conditionals, interpreting if-clauses uniformly as restrictors. To be sure, such clauses do not all restrict the same thing. But it seems uncontroversial that that would be too much uniformity to ask, since it is clear from the canonical examples in (1) that if-clauses can restrict different sorts of objects (e.g., a set of occasions in (1-a) and a modal base in (1-b)). What is crucial is that a general scheme underlies all these cases: a clause of the form ‘if \(\alpha \)’ always targets some kind of parameter and restricts it to the objects that satisfy \(\alpha \).
One could, however, complain with some justice that in the process, we have given up a uniform treatment of modals: in the proposed account, expressive modals shift a semantic parameter, while factual modals quantify over accessible worlds. These are two quite different treatments. But it seems implausible to maintain that must and may are simply ambiguous, especially in light of the stability of this ambiguity cross-linguistically.
Fortunately, however, we can in fact isolate a common core to the semantics of must and may across their occurrences. This is given by their semantics as shifters of the attitude parameter. Indeed, a factual occurrence of a modal can be decomposed in two parts: one part is just the bare operator \(\Box \) or \(\Diamond \), interpreted as a shifter of the attitude parameter; the other part is an operator \(@_i\), whose role is to relativize the modal claim to a modal base. More precisely, \(@_i\) operates by shifting the point of evaluation from a world w to the information state \(f_i(w)\).Footnote 25
In this clause, the symbol ‘\(\_\)’ stands for a null value. In order for \([\![\varphi ]\!]^{f_i(w),\_,f}\) to be well-defined, a value for the attitude parameter must be supplied by some modal inside \(\varphi \). If that does not happen, we can stipulate that the result is undefined. We can then derive the semantics that we assigned above to the factual modals \(\Box _i,\Diamond _i\) from the interaction of the operator \(@_i\) and the semantics of \(\Box ,\Diamond \) as attitude shifters:
$$\begin{aligned}&[\![@_i\Box \alpha ]\!]^{w,f}=[\![\Box \alpha ]\!]^{f_i(w),\_,f}=[\![\alpha ]\!]^{f_i(w),\forall ,f}=1\iff \forall w'\in f_i(w):[\![\alpha ]\!]^{w',f}=1\\&[\![@_i\Diamond \alpha ]\!]^{w,f}=[\![\Diamond \alpha ]\!]^{f_i(w),\_,f}=[\![\alpha ]\!]^{f_i(w),\exists ,f}=1\iff \exists w'\in f_i(w):[\![\alpha ]\!]^{w',f}=1 \end{aligned}$$
Thus, we may assume that modals like ‘must’ and ‘may’ make a uniform semantic contribution, but that some of their occurrences are “anchored” by a modal base operator, giving rise to factual readings, while others are “free”, giving rise to expressive readings.Footnote 26
Extension to attitude ascriptions
A puzzle analogous to our Problem 4 above arises for attitude ascriptions. In this section, I show that our solution extends naturally to that case. To see the problem, consider first the following statement.
This statement ascribes a conditional belief: it describes Charlie as believing, conditionally on the supposition that the butler did it, that he did it with a knife. It is a strength of the restrictor theory that this can be predicted, by treating the if-clause as a restrictor of the attitude verb.
However, the problem with anaphora discussed above also strikes here. Consider:
Intuitively, what Bob is saying here could be rephrased as in (25). But in the standard version of the restrictor view, it is not clear how this reading might come about. If ‘so’ picks out the content of the bare conditional, which in that view is a certain modal proposition, then (26-b) is predicted to ascribe to Charlie a categorical belief in a modal proposition, rather than a conditional belief in a non-modal proposition.
Our solution extends naturally to this variant of the puzzle. Suppose we treat the belief operator \(\textsf {Bel}\) as similar to \(@_i\) above: the role of \(\textsf {Bel}\) is to shift the point of interpretation from a world w to an information state (the doxastic state \(s_x^w\) of the relevant agent at w) and an attitude (say, probabilistic acceptance, \(\pi \), but this won’t matter for our purposes):Footnote 27
$$\begin{aligned}{}[\![\textsf {Bel}(x,\varphi )]\!]^{w,f}=[\![\varphi ]\!]^{s_x^w,\pi ,f} \end{aligned}$$
In words, x believes \(\varphi \) in case x’s doxastic state probabilistically accepts \(\varphi \). One way to obtain this result compositionally is to analyze the belief operator \(\textsf {Bel}\) as a function from x-contents C and agents x to propositions:
$$\begin{aligned} \textsf {Bel}\;\leadsto \;\lambda C.\lambda x.\{w\mid \langle s_x^w,\pi \rangle \in C\} \end{aligned}$$
Now we are in a position to deal with the little dialogue above. Alice asserts \(b\Rightarrow k\), and thereby expresses the x-content \([\![b\Rightarrow k]\!]_{\textsf {AS}}\). The anaphor ‘so’ in Bob’s utterance picks out this content. Bob’s claim is then associated with the following proposition:
$$\begin{aligned} \textsf {Bel}([\![b\Rightarrow k]\!]_{\textsf {AS}})(c)= & {} \{w\mid \langle s_c^w,\pi \rangle \in [\![b\Rightarrow k]\!]_{\textsf {AS}}\}\\= & {} \{w\mid [\![b\Rightarrow k]\!]^{s_c^w,\pi }=1\}\\= & {} \{w\mid [\![k]\!]^{s_c^w+b,\pi }=1\}\\= & {} \{w\mid (s_c^w+b)([\![k]\!])\ge t\} \end{aligned}$$
So, Bob’s sentence is true in case Charlie has a certain conditional belief: namely, in case his doxastic state updated with the supposition that the butler did it results in a hypothetical state that assigns high probability to the proposition that he did it with a knife. This is the result we wanted.
Again, notice that two things were crucial to achieve this result: first, the content of the bare conditional \(b\Rightarrow k\) is not a proposition, so that to ascribe belief in this content is not to ascribe belief in a proposition (instead, it is to ascribe belief in a proposition given another); and second, this content does not involve any modal, so that the resulting belief ascription does not involve any layer of modality beyond the one provided by the attitude verb itself.
It is worth noting that this strategy extends beyond belief ascriptions. For instance, one could give a similar account of ascriptions involving want, provided one extends the repertoire of attitudes in the semantics to include non-cognitive attitudes like desire.