Restriction without Quantification: Embedding and Probability for Indicative Conditionals

1. Introduction

Many modern theories of indicative conditionals treat them as restricted epistemic necessity modals. This family includes dynamic semantics accounts (Gillies 2004; 2009; Starr 2014a; Willer 2014; 2018) and semi-dynamic information-based accounts (Bledin 2014; Gillies 2009; 2010; Moss 2015; Punčochář & Gauker 2020; Yalcin 2007).¹ The central idea, which I will refer to as the Box View, is the following:²

Box View: the semantics of an indicative conditional $p\Rightarrow q$ involves two components:

1. Restriction: restrict the set of epistemic possibilities to the $p$ -worlds;

2. Quantification: check that $q$ is true at every world in the restricted set.

The conditional is true/accepted iff the check in the second step is successful.³

This view is motivated by a compelling analysis of how indicative conditionals are assessed, known as the Ramsey test view (after a remark in Ramsey 1929). The idea is that, in order to assess $p\Rightarrow q$ in an information state $s$ , we proceed in two steps: first, we add the antecedent $p$ to $s$ , resulting in a hypothetical state $s[p]$ ; second, we check whether we accept the consequent $q$ in this hypothetical state; if (and only if) so, the conditional is accepted in $s$ .

In possible world semantics, an information state $s$ determines a set of worlds, namely, those worlds which are possible according to the available information. Adding $p$ to $s$ corresponds to restricting to those worlds in which $p$ is true; thus, the resulting hypothetical state is the set $s[p]$ of $p$ -worlds in $s$ . Checking whether $p$ is accepted in this hypothetical state amounts to checking whether $q$ is true in all the worlds in $s[p]$ . Thus, the two steps of the Ramsey test procedure correspond exactly to the two components of the semantics of conditionals as postulated by the Box View.

The Box View of conditionals also makes a number of welcome predictions. For instance, it explains why a discourse like (1) sounds contradictory.

However, the view also faces two serious problems. The first has to do with embeddings of conditionals under epistemic modals. When embedded, conditionals do not behave like necessity modals. For instance, consider (2):

Given the Box View, we would expect (2) to be a higher-order epistemic claim: it is possible that the consequent is necessary given the antecedent. This, however, is not what (2) means. What (2) means is simply that the consequent is possible given the antecedent—i.e., that if Alice left, she might have gone to London.

But how is this possible? If the conditional is a restricted necessity claim, how can the outer modal turn it into a restricted possibility claim? If the embedded conditional involves a universal quantification over antecedent worlds, how can the external modal prevent this quantification, and replace it with an existential quantification?⁴

The second problem, pointed out by Edgington (2014), concerns probability judgments.⁵ Consider (3):

If all we know is that the coin is fair, it is natural to judge (3) to be 50% probable. This is not the probability that it is epistemically necessary that the coin landed heads if tossed. That probability has got to be zero, since we know for sure that it is not epistemically necessary that the coin lands heads if tossed. This is unexpected: if a conditional is a restricted epistemic necessity claim, its probability should just be the probability that this epistemic necessity claim obtains.

In this paper, I propose an account of conditionals, modals, probabilities, and probabilistic vocabulary that achieves the following:

1. It accounts for the data that motivate the Box View. In particular, it predicts that a conditional $p\Rightarrow q$ is acceptable iff $q$ follows on the supposition of $p$ , vindicating the Ramsey test idea, and that $p\Rightarrow q$ is logically inconsistent with $\diamond (p\wedge \neg q)$ .

2. It accounts for the peculiar way in which conditionals embed under epistemic modals. This is achieved without ad-hoc syntactic stipulations, in particular, without denying that conditionals can take scope with respect to modals.

3. It predicts that the probability of a factual conditional is the conditional probability of the consequent given the antecedent. Crucially, this result is achieved without ad-hoc stipulations about probabilities of conditionals. Rather, it is derived on the basis of the semantics of conditionals and of a general definition of probability applying to all sentences alike.

The account builds on the idea that epistemic sentences express attitudes towards a truth-conditional content, assessed relative to an information state $s$ . The contribution of epistemic modals is that of indicating the relevant attitude. Thus, our treatment of epistemic modals fits within the expressivist line (see also Hawke & Steinert-Threlkeld 2021; Moss 2015; Willer 2013; Yalcin 2007; 2011), but it differs from its predecessors in that, as we shall see, the connection between epistemic modals and attitudes is reflected in the semantics in a more direct way.

If-clauses are devices for restricting the relevant information state: to express an attitude towards $p\Rightarrow q$ in state $s$ is to express the same attitude towards $q$ in the hypothetical state $s[p]$ . Thus, conditionals are essentially devices to express conditional attitudes—attitudes subordinated to a supposition. This so-called suppositional view of conditionals has been defended in detail, notably by Edgington (1986; 1995; 2014), based on ideas from Adams (1975) (see also Bennett 2003). The view is widely appreciated for its psychological plausibility (as it links conditionals to the all-important process of supposing), its accurate empirical predictions (see, e.g., Evans & Over 2004), and its generality (as it extends in a natural way beyond statements, to an analysis of conditional questions and commands). However, acceptance of the suppositional view has been hindered by the lack of a precise compositional semantics. The present theory could be seen as filling this gap, supplying a specific formal implementation.

Crucially, on the proposed theory, the conditional operator does not contribute any quantification. Rather, what contributes the relevant quantification is the attitude expressed towards the conditional. I will propose that assertion is associated with the attitude of acceptance, which contributes a universal quantification over epistemic alternatives. In this way, it will turn out that the information conveyed by asserting a non-modal conditional $p\Rightarrow q$ is the one predicted by the Box View, i.e., that all $p$ -possibilities are $q$ -possibilities. However, this result is obtained by dividing labor: the restriction to $p$ -worlds is contributed by the conditional operator, while the universal quantification is contributed by the acceptance attitude associated with assertion.

This difference is crucial when the conditional is not asserted, but occurs embedded under another operator, like ‘might’: in this case, the higher operator may shift the attitude being expressed away from acceptance. As a consequence, no universal quantification shows up in the semantics of the resulting sentence. As we will see, this provides a solution to the puzzle of embedded conditionals.

The situation is similar when the relevant conditional is not asserted, but assessed for probability. Since no acceptance attitude is involved in the process, no universal quantification over epistemic alternatives shows up. Instead, all the conditional contributes is the restriction to the antecedent worlds: assessing the probability of $p\Rightarrow q$ then amounts to assessing the probability of $q$ relative to the $p$ -worlds, i.e., to assessing the conditional probability of $q$ given $p$ . As we will see, this provides a solution to the puzzle of probabilities of conditionals.

Thus, the main conclusion of the paper is summarized in its title: although the Box View is right as a view about the acceptance conditions of indicative conditionals, it is wrong about how these conditions come about; what the conditional operator itself contributes is only a restriction of the relevant set of possibilities, and not also a quantification over the restricted set.

The paper is structured as follows. In Section 2 I discuss the considerations that motivate the Box View and the two problems it faces. In Section 3 I propose a new account of the semantics of indicative conditionals and epistemic modals. In Section 4 I show that the account solves the two problems under consideration, while retaining the attractions of the Box View. In Section 5 I discuss how to extend the system to capture objective readings of epistemic modals and conditionals, as well as the interaction of these constructions with negation. In Section 6 I discuss similarities and differences with Kratzer’s restrictor theory. Section 7 concludes with some considerations on the proposal and prospects for future work.

2. The Box View: Attractions and Problems

In this section, I will spell out in more detail the attractive features of the Box View, which we will aim to preserve, and the problems it faces, which we will aim to overcome. I will use $\Rightarrow$ to denote the indicative conditional construction, and $\diamond$ , $\square$ , and $\mathsf{P}$ to denote the epistemic modals ‘might’, ‘must’, and ‘probably’.⁶

3. Attitude Semantics

In this section I lay out a new account of the semantics of conditionals and epistemic modals. I will refer to this account as Attitude Semantics, abbreviated as $\mathsf{AS}$ .

Language. To present the theory explicitly, I will work with a formal language. The base layer of the language is a set ${\mathcal{L}}_0$ of factual sentences, the semantics of which can be given in terms of truth-conditions relative to possible worlds. For our purposes, it does not matter what ${\mathcal{L}}_0$ is, but for the sake of concreteness, I will take ${\mathcal{L}}_0$ to be the language of propositional logic based on a set $\mathcal{P}=\{p,q,\dots \}$ of atomic sentences.

Definition 3.1 (Factual language, ${\mathcal{L}}_0$ )

The full language $\mathcal{L}$ that I will work with is obtained by enriching ${\mathcal{L}}_0$ with operators designed to capture epistemic vocabulary: a binary operator $\Rightarrow$ for the conditional construction; unary operators $\square$ and $\diamond$ for ‘it must be that’ and ‘it might be that’; and a unary operator $\mathsf{P}$ for ‘it is probable that’. To avoid further complications which are not essential to our concerns, we restrict to factual antecedents, and we do not consider Boolean compounds of epistemic sentences.¹²

Definition 3.2 (Epistemic language, $\mathcal{L}$ )

Models. I will assume as background a model $M=\langle W,V\rangle$ that provides a universe $W$ of possible worlds together with a valuation function $V:\mathcal{P}\times W\to \{0,1\}$ which specifies the truth-values of atomic sentences at each possible world. In order to simplify the exposition I will assume that the set $W$ is finite, although this restriction is not essential. The relation of truth between worlds $w\in W$ and factual formulas $\alpha$ , denoted $w\models \alpha$ , is defined as usual; I write $w(\alpha )$ for the truth-value of $\alpha$ in $w$ , and denote by $|\alpha |$ the set of worlds where $\alpha$ is true:

$|\alpha |:=\{w\in W\mid w\models \alpha \}$

Information states. In order to spell out our semantics, we will need a formal notion of information states. Since we are concerned not just with qualitative notions, but also with probabilistic ones, I will take an information state to be a probability distribution on $W$ . Since $W$ is finite, we can represent such a distribution simply as a map which assigns a probability to each possible world.

Definition 3.3 (Information states)

An information state is a map $s:W\to [0,1]$ such that ${\displaystyle {\sum }_{w\in W}s(w)=1}$ .

A world $w$ is ruled out by an information state $s$ if it is assigned probability $0$ . Worlds which are not ruled out are referred to as the live possibilities in $s$ .

Definition 3.4 (Live possibilities)

If $s$ is an information states set of live possibilities is $L(s):=\{w\in W\mid s(w)\ne 0\}$ .

The probability of a proposition $X\subseteq W$ is just the probability that $X$ is true.

Definition 3.5 (Probability of a proposition) If $X\subseteq W,\hspace{0.17em}s(X):={\displaystyle {\sum }_{w\in X}s(w)}$ .

Suppositions. Next, we need a modeling of the process of making an indicative supposition. When we suppose $\alpha$ in a state $s$ , we enter a new state $s[\alpha ]$ in which $\alpha$ is treated as certain. Thus, all worlds in which $\alpha$ is false should be assigned probability $0$ . The relative probabilities of the $\alpha$ -worlds are unaffected by the supposition, and should just be rescaled by a factor $1/s(|\alpha |)$ so that they sum up to 1 again. In other words, supposing can be modeled by the operation of conditionalization. If $\alpha$ is ruled out in $s$ , i.e., if $s$ rules out all $\alpha$ -worlds, then the supposition cannot take place: we then say that $\alpha$ is not supposable in $s$ .¹³¹⁴

Definition 3.6 (Supposing) If $s$ is an information state and $\alpha$ a factual sentence with $s(|\alpha |)\ne 0$ , then $s[\alpha ]$ is the information state defined as follows:

$s[\alpha ](w)=\{\begin{array}{cc}\frac{s(w)}{s(|\alpha |)}& \text{if }w\in |\alpha |\\ 0& \text{if }w\cancel{\in }|\alpha |\end{array}$

Notice that the set of live possibilities after the supposition is $L(s[\alpha ])=L(s)\cap |\alpha |$ .

Semantics. Traditionally, semantics is about specifying truth conditions for sentences in contexts. With much recent work (e.g. Gillies 2004; Hawke & Steinert-Threlkeld 2021; Moss 2015; Punčochář & Gauker 2020; Veltman 1996; Willer 2013; Yalcin 2007), we will assume that epistemic sentences, including sentences headed by epistemic modals as well as indicative conditionals, differ essentially from factual sentences, in that they do not have truth conditions relative to states of affairs. Rather, we will take such sentences to be devices for negotiating the epistemic attitude to be taken towards certain truth-conditional contents.¹⁵ Accordingly, our semantics for the epistemic language $\mathcal{L}$ specifies what information it takes to bear a certain attitude to a sentence $\phi \in \mathcal{L}$ . More specifically, the semantics will take the form of a ternary relation between an information state $s$ , an attitude $A$ , and a sentence $\phi$ :

$s{\models }_A\phi$

The possible values for the attitude parameter $A$ include full acceptance (denoted $\forall$ , and called simply ‘acceptance’ below), compatibility (denoted $\exists$ ), and partial acceptance to degree $x\in [0,1]$ (denoted ${\pi }_x$ ). Other values for the attitude parameter could be considered as well, but the ones above are sufficient for our present purposes.

Definition 3.7. The set of attitude parameters is $\mathsf{At}=\{\forall ,\exists \}\cup \{{\pi }_x\mid x\in [0,1]\}$

We now have all ingredients to recursively specify the semantics.

For factual sentences $\alpha \in {\mathcal{L}}_0$ , full acceptance in $s$ amounts to $s$ assigning credence $1$ to the proposition expressed by $\alpha$ ; compatibility amounts to non-zero credence in that proposition; and partial acceptance to degree $x$ amounts to credence $x$ or higher.

• $s{\models }_{\forall }\alpha \hspace{0.5em}\iff \hspace{0.5em}s(|\alpha |)=1$

• $s{\models }_{\exists }\alpha \hspace{0.5em}\iff \hspace{0.5em}s(|\alpha |)\ne 0$

• $s{\models }_{{\pi }_x}\alpha \hspace{0.5em}\iff \hspace{0.5em}s(|\alpha |)\ge x$

Notice that the first two clauses can be rewritten qualitatively as follows:

• $s{\models }_{\forall }\alpha \hspace{0.5em}\iff \hspace{0.5em}\forall w\in L(s):w(\alpha )=1$

• $s{\models }_{\exists }\alpha \hspace{0.5em}\iff \hspace{0.5em}\exists w\in L(s):w(\alpha )=1$

So, acceptance amounts to truth at all live possibilities, while compatibility amounts to truth at some live possibility.

The semantic role of an epistemic modal is that of indicating the attitude expressed towards the prejacent: acceptance for ‘must’, compatibility for ‘might’, and partial acceptance to a high degree for ‘probably’. Formally, these operators work by shifting the attitude parameter: for every $A\in \mathsf{At}$ ,

• $s{\models }_A\square \phi \hspace{0.5em}\iff \hspace{0.5em}s{\models }_{\forall }\phi$

• $s{\models }_A\diamond \phi \hspace{0.5em}\iff \hspace{0.5em}s{\models }_{\exists }\phi$

• $s{\models }_A\mathsf{P}\phi \hspace{0.5em}\iff \hspace{0.5em}s{\models }_{{\pi }_t}\phi$                          where  $t\in [0,1]$ is a fixed threshold value

Finally, conditionals are interpreted by a generalized Ramsey test clause: to bear an attitude to the conditional is to bear the attitude to the consequent, on the supposition of the antecedent.

• $s{\models }_A\alpha \Rightarrow \phi \hspace{0.5em}\iff \hspace{0.5em}s[\alpha ]{\models }_A\phi$

What if the antecedent $\alpha$ is not supposable in the state $s$ ? Then the conditional cannot be assessed at $s$ . I take this to be a case of presupposition failure.¹⁶ In this case we say that $\alpha \Rightarrow \phi$ is not admitted by $s$ .¹⁷

Entailment and equivalence. Attitude semantics allows us to define different notions of entailment. A salient option is to define entailment as preservation of (full) acceptance (cf. Bledin 2014; Yalcin 2007): an entailment is valid if the conclusion is acceptable in every information state in which the premises are acceptable.¹⁸¹⁹

Definition 3.8 (Logical entailment)

$\mathrm{\Phi }\models \psi \hspace{0.5em}\iff \hspace{0.5em}\forall s:s{\models }_{\forall }\phi \text{ for all }\phi \in \mathrm{\Phi }\text{ implies }s{\models }_{\forall }\psi$

The corresponding notion of equivalence tracks identity in acceptance conditions:

Definition 3.9 (Logical equivalence)

$\phi \equiv \psi \hspace{0.5em}\iff \hspace{0.5em}\forall s:(s{\models }_{\forall }\phi \hspace{0.5em}\iff \hspace{0.5em}s{\models }_{\forall }\psi )$

Notice that two formulas can be logically equivalent although they do not have the same semantics: this can happen if the acceptance conditions for the two formulas are the same, but their conditions relative to some other attitudes are different. To keep logical equivalence and semantic equivalence clearly distinct, it will be useful to introduce a notation, $\phi \doteq \psi$ , for the latter relation:

Definition 3.10 (Semantic equivalence)

$\phi \doteq \psi \hspace{0.5em}\iff \hspace{0.5em}\forall s\forall A:(s{\models }_A\phi \hspace{0.5em}\iff \hspace{0.5em}s{\models }_A\psi )$

Assertion. In the truth-conditional setting, an assertion of $\phi$ is normally construed, along the lines of Stalnaker (1978), as a proposal to the conversational participants to accept the proposition expressed by $\phi$ . In $\mathsf{AS}$ , not all sentences express propositions. However, the semantics gives us a notion of what it is to accept these sentences. Therefore, we do not need the detour through the proposition expressed. We can simply say that an assertion of $\phi$ is a proposal to the conversational participants to coordinate on a state which accepts $\phi$ .²⁰

4. Predictions

Let us now turn to the predictions that $\mathsf{AS}$ makes. First, we will show that it can vindicate the conceptual considerations and the empirical observations that motivated the Box View of conditionals. Then, we will proceed to show how it solves the two problems discussed above.

5. Extensions

6. Comparison with the Restrictor Theory

According to Kratzer’s restrictor theory of conditionals (Kratzer 1986) the embedding problems that we faced in Section 2.2 stem from a fundamental mistake about the syntax of conditionals: there is no operator $\Rightarrow$ corresponding to the ‘if … then’ construction in natural language. Rather, if-clauses spell out the restrictor argument of a modal operator. Thus, for instance, the following sentences do not have the logical forms we have assumed above, but rather the ones given on the right:

In these formulae, the modified operator $O_p$ works like the original operator $O$ , except that the relevant domain is restricted to worlds where $p$ is true.

Moreover, the sentences in (24), though differering in the way the operators appear at surface form, actually correspond to exactly the same logical forms.

Thus, on this approach the problem of why the sentences in (23) sound equivalent to the sentences in (24) vanishes—or, rather, it is transformed from a semantic problem to a problem of syntax-semantics interface.

What about plain conditionals like (25), so-called bare conditionals?

Kratzer postulates that such a sentence actually contains a silent epistemic ‘must’, so its logical form is the same as that of (23-a) and (24-a), namely, ${\square }_{p}q$ . Thus, the equivalence between ‘if A, C’ and ‘if A, must C’ is obtained by stipulation.

Finally, Kratzer’s theory can explain why, although (25) involves a universal quantification over epistemic possibilities, (24-b) does not. According to this theory, appearances are deceiving: (24-b) is not obtained by embedding (25) under ‘might’, but by replacing the silent ‘must’ in the logical form of (25) by a ‘might’. As a result, the logical form of (25) does not occur as a sub-constituent in (24-b). In this way, the restrictor theory dissolves the embedding problem discussed in Section 2.2.

However, the theory still faces the probability problem. To see why, consider again the scenario described in Section 2.3. A fair die was rolled and the outcome has not yet been revealed. Our friend Alice makes the following guess:

Intuitively, though she cannot be sure, what she said is quite likely, since two out of three even outcomes are above three. But now, if the logical form of (26) is ${\square }_{p}q$ , what Alice has asserted is that it is epistemically necessary that the outcome is above three, given that it is even. Since this is clearly not the case, we should judge what she said to have probability zero or near-zero. But that seems wrong.

It is sometimes suggested in discussions that the advocate of the restrictor theory could respond by giving the following error theory: when we say that the probability of (26) is 2/3, what we are in fact doing is not judging the probability of the bare conditional (26); instead, we are making an assertion of the form “It is 2/3 probable that if the outcome is even, it is above three”, in which the if-clause restricts the explicit probability operator. But this does not solve the problem, since it does not explain why we would assert this when asked to judge the likelihood of what Alice said. According to the theory, this behavior is perplexing: what Alice said is ${\square }_{p}q$ , but when we are asked how likely that is, we instead make an assertion about the conditional probability of $q$ given $p$ . Why would we do that? The given error theory does not seem to offer a coherent explanation of the way we behave when asked about the probabilities of conditionals (for related discussion of this problem, see also Mandelkern 2018).³³

Let us now examine some similarities and differences between Kratzer’s restrictor theory and attitude semantics. An important similarity is that, in both theories, conditional constructions serve only a restricting function, and do not contribute a quantifier. The source of quantification lies elsewhere. This fundamental point of convergence is what allows both theories to predict, e.g., that the sentence “it might be that if A then C” involves just an existential quantifier over possible worlds, and no universal quantifier.

At the same time, there is a key difference between the two theories. The source of quantification is not the same in both: in Kratzer’s theory, the source of quantification is a modal operator; therefore, we need to assume that a modal operator is always present in a sentence involving conditionals, whence the need to postulate silent necessity modals in the logical form of bare conditionals. Besides lacking independent motivation, this stipulation is empirically problematic, as we saw, in light of probability judgments about conditionals. In $\mathsf{AS}$ , by contrast, the source of quantification is the attitude parameter; while modal operators can shift this parameter, we need not assume that every sentence contains a modal operator. Thus, $\mathsf{AS}$ obviates the need for covert modals. If a sentence does not contain an epistemic modal that fixes the attitude parameter, the relevant quantification is not determined once and for all from within the sentence, but is determined from the outside, by the attitude under consideration. If the sentence is asserted, the relevant attitude is full acceptance, which is responsible for introducing a universal quantifier, producing the same effect as if the sentence had contained a ‘must’. But in case the sentence is assessed for probabilistic acceptance, the process will involve weighing of probabilities rather than universal quantification, thus differing from the assessment of the corresponding ‘must’ sentence. This is why attitude semantics, unlike Kratzer’s version of the restrictor theory, can solve the probability problem.

To conclude, let me point out that, while differing from Kratzer’s specific theory, attitude semantics can itself be seen as embodying a version of the restrictor view of conditionals: if-clauses are merely devices to restrict the relevant set of epistemic possibilities. This may seem too restrictive: as Kratzer (1981) emphasized, in general if-clauses can serve to restrict different things, not just a set of epistemic possibilities. But while the simple theory presented in this paper is not equipped with the resources to deal with if-clauses which perform non-epistemic restrictions, it can be extended with such resources in a natural way, drawing on ideas from the restrictor tradition. Such an extension is described in Ciardelli (in press).

7. Conclusion

According to the Box View which lies at the core of many modern accounts, indicative conditionals are essentially restricted epistemic necessity modals. Two kinds of considerations challenge this view: first, conditionals do not embed under other operators like epistemic necessity modals; and second, the probability of a conditional is not the probability of an epistemic necessity claim. Both observations point to the conclusion that the core semantics of conditionals involves only a restriction of the set of relevant alternatives, and not a universal quantification over the resulting set.

This simple idea, however, is not easy to turn into a concrete proposal. After all, suppose we have used the antecedent to restrict the relevant set, generating a hypothetical information state; what should we then do with the consequent? The standard answer is: check whether it is acceptable—whence the universal quantification. In this paper we saw that another answer is possible: do with the consequent whatever you were originally doing with the conditional. If you were checking if the conditional is acceptable, check if the consequent is acceptable in the hypothetical state; if you were judging the probability of the conditional, judge the probability of the consequent in the hypothetical state; and so on.

We have seen how to implement this idea in a formal system which we called attitude semantics. This system provides a solution to the embedding problem and to the probability problem, while still accounting for the data that motivated the Box View. In more detail, the proposal (i) predicts that the acceptance conditions for a factual conditional $p\Rightarrow q$ accord with the Box View: $p\Rightarrow q$ is fully accepted iff all the epistemically possible $p$ -worlds are $q$ -worlds; (ii) makes $p\Rightarrow q$ and $p\Rightarrow \square q$ logically equivalent, and $p\Rightarrow q$ and $\diamond (p\wedge \neg q)$ logically inconsistent; (iii) explains why $p\Rightarrow q$ fails to contribute a universal quantifier when embedded under epistemic modals; (iv) given an epistemic modal $\mathsf{M}$ , it predicts the commutation $\mathsf{M}(p\Rightarrow q)\equiv p\Rightarrow \mathsf{M}q$ ; (v) in combination with a natural characterization of probability, it vindicates Adams’ thesis, predicting that the probability of $p\Rightarrow q$ is the conditional probability of $q$ given $p$ .

In order to achieve these results, we adopted a novel semantic architecture: sentences are interpreted relative to two semantic parameters: an information state $s$ , and an attitude $A$ . Epistemic modals are treated as attitude shifters. This reflects the idea that epistemic modals like ‘might’, ‘must’, and ‘probably’, at least in some of their uses, do not contribute to the truth-conditions of the sentence but are, instead, ways of expressing an attitude towards the prejacent. This idea is related to a tradition that views epistemic modals as force modifiers (see, among others Bybee, Perkins, & Pagliuca 1994; Huddleston & Pullum 2002; Price 1983; Schnieder 2010), whose role is to modulate the “force” with which a proposition is put forward. To see how this idea is vindicated in our setting, consider for example an assertion of “probably $p$ ” $(\mathsf{P}p)$ . With such an assertion, one is proposing to adopt a state that fully accepts this sentence. However, to fully accept $\mathsf{P}p$ is not to fully accept a proposition; rather, it is to bear a certain attitude to the proposition $\left|p\right|$ , namely, partial acceptance to a high degree. Thus, when we compare an assertion of $p$ with an assertion of “probably $p$ ”, we see that the speaker is putting on the table each time the same proposition $\left|p\right|$ , but recommending a different attitude towards it: full acceptance in one case, partial acceptance in the other. At the same time, we see that in our proposal, epistemic modals do make a compositional semantic contribution: this is crucial to accounting for their role in embedded contexts, as we illustrated in Section 5.1. Thus, $\mathsf{AS}$ vindicates some intuitions that motivated the force modifier view, while avoiding the main criticism usually moved to the view—namely, that it leaves us unequipped to account for epistemic modals embedded in truth-conditional constructions (von Fintel & Gillies 2007; MacFarlane 2011).

These merits are shared by some previous expressivist accounts of epistemic modals, in particular the one of Yalcin (2007; 2011). However, in Yalcin’s account, modals are treated as contributing a quantification over epistemic alternatives; by contrast, in $\mathsf{AS}$ the role of epistemic operators is to shift a semantic parameter of the evaluation. This parameter, in turn, contributes the relevant quantification, but that happens only once a factual formula is eventually interpreted. The difference is immaterial when we look at plain epistemic statements like $\diamond p$ , since then the shift of the relevant parameter and the transformation of this parameter into a quantifier occur directly in a sequence, producing the same overall effect as if $\diamond$ had directly introduced the quantifier. But the difference becomes crucial when we look at epistemic operators scoping above conditionals, as in $\diamond (p\Rightarrow q)$ : in this case, first $\diamond$ shifts the attitude parameter to $\exists$ ; then the conditional updates the information state parameter from $s$ to $s[p]$ ; and only at this stage, when it comes to assessing $q$ , the attitude parameter contributes an existential quantification over the worlds in $s[p]$ . Thus, the relevant quantification only shows up when we finally assess the consequent, after the restriction has taken place.

Turning from modals to conditionals, $\mathsf{AS}$ can be seen as providing a reconciliation of two different traditions, the Box View tradition and the probabilistic tradition (Adams 1965; 1975; Bennett 2003; Edgington 1986; 1995), which focused on somewhat different aspects of the semantics to conditionals. In $\mathsf{AS}$ , both can be recovered: the semantics of $p\Rightarrow q$ as given by $\mathsf{AS}$ predicts at the same time that (i) the conditional is logically incompatible with $\diamond (p\wedge \neg q)$ , and that (ii) its probability is the conditional probability of $q$ given $p$ . This shows that the insights from the two traditions need not lead to conflicting views about the semantics of conditionals, but can be seen as two different facets of a single semantics.

To conclude, let me mention three directions for further work. First, it would be interesting to study in more detail the formal and logical features of $\mathsf{AS}$ , especially since such an investigation would give us more insight into the repercussions of having an extra attitude parameter in the semantics.

Second, as we know from Lewis (1975), if-clauses can be used to restrict not only the domain of quantification of modals, but also that of other operators, in particular, adverbs such as sometimes, usually, and always, as in (27):

It would be interesting to explore to what extent the proposal made here can be extended to deal with the interaction between conditionals and such adverbs, which seems to present many of the same puzzles that we discussed above. It seems that the same basic semantic architecture can be used, but the fundamental ingredients of the semantics must be construed somewhat differently: the analogue of the attitude parameter in these sentences would be a “frequency parameter”, which the adverb shifts, and the analogue of an information state would be a domain of occasions/situations which the conditional restricts.³⁴ The fact that conditionals are treated as mere restricting devices is crucial in this setting as well. Indeed, Gillies (2010) gives a Box View account of conditionals which he argues to shares the benefits of the restrictor theory in accounting for the way in which if-clauses restrict epistemic modals. However, Khoo (2011) shows that this account does not extend to adverbs of quantification: the problem, as Khoo clearly shows, is precisely that in order to get the right predictions we need to get rid of the universal quantification introduced by the conditional, and it is not clear how to do that. In the present approach, this problem does not arise, since conditionals do not introduce a universal quantification to begin with.

Finally, the proposal should be extended to cover subjunctive conditionals. These conditionals raise problems analogous to those discussed here for indicatives.

For instance, (28-a) and (28-b) seem to mean the same:

The solution presented in this paper extendeds straightforwardly to this case. The crucial difference with the indicative case is that, for subjunctive conditionals, the hypothetical state resulting from an assumption is not obtained just by conditionalization, but must be computed by a different procedure, perhaps involving causal reasoning or a similarity ordering. The specifics of this procedure, however, do not matter to our solution of the embedding problem. Suppose $\Rrightarrow$ stands for the subjunctive conditional and $s\langle \alpha \rangle$ for the result of supposing $\alpha$ as a subjunctive assumption in state $s$ . Suppose the semantics of $\Rrightarrow$ is given by a Ramsey-test clause analogous to the one for $\Rightarrow$ :

$\begin{array}{ccc}s{\models }_A\alpha \Rrightarrow \phi & \hspace{0.5em}\iff \hspace{0.5em}& s\langle \alpha \rangle {\models }_A\phi \end{array}$

Then we have:

$\begin{array}{lll}s{\models }_A\mathsf{P}(p\Rrightarrow q)\hfill & \hspace{0.5em}\iff \hspace{0.5em}\hfill & s{\models }_{{\pi }_t}p\Rrightarrow q\hfill \\ \hfill & \hspace{0.5em}\iff \hspace{0.5em}\hfill & s\langle p\rangle {\models }_{{\pi }_t}q\hfill \\ \hfill & \hspace{0.5em}\iff \hspace{0.5em}\hfill & s\langle p\rangle {\models }_A\mathsf{P}q\hfill \\ \hfill & \hspace{0.5em}\iff \hspace{0.5em}\hfill & s{\models }_{A}p\Rrightarrow \mathsf{P}q\hfill \end{array}$

Thus, the commutation of $\mathsf{P}$ and $\Rrightarrow$ is predicted.

Moreover, consider the probability issue. It seems that the probability of a subjunctive conditional $p\Rrightarrow q$ is just the probability that $q$ is true in the state $s\langle p\rangle$ resulting from the counterfactual assumption of $p$ .³⁵ This can be predicted in the same way as we did for epistemic conditionals in Section 4.3. Indeed, we have $s{\models }_{{\pi }_x}p\Rrightarrow q\hspace{0.17em}\hspace{0.5em}\iff \hspace{0.5em}\hspace{0.17em}s\langle p\rangle {\models }_{{\pi }_x}q\hspace{0.17em}\hspace{0.5em}\iff \hspace{0.5em}s\langle p\rangle (|q|)\ge x$ . Since ${\mathbb{P}}_s(p\Rrightarrow q)$ is defined as the largest $x$ such that $s{\models }_{{\pi }_x}p\Rrightarrow q$ , we have that ${\mathbb{P}}_s(p\Rrightarrow q)=s\langle p\rangle (|q|)$ , as expected.