Many modern theories of indicative conditionals treat them as restricted epistemic necessity modals. This view, however, faces two problems. First, indicative conditionals do not behave like necessity modals in embedded contexts, e.g., under ‘might’ and ‘probably’: in these contexts, conditionals do not contribute a universal quantification over epistemic possibilities. Second, when we assess the probability of a conditional, we do not assess how likely it is that the consequent is epistemically necessary given the antecedent. I propose a semantics which solves both problems, while still accounting for the data that motivated the necessity modal view. The account is based on the idea that the semantics of conditionals involves only a restriction of the relevant epistemic state, and no quantification over epistemic possibilities. The relevant quantification is contributed by an attitude parameter in the semantics, which is shifted by epistemic modals. If the conditional is asserted, the designated attitude is acceptance, which contributes a universal quantification, producing the effect of a restricted necessity modal.

Many modern theories of indicative conditionals treat them as restricted epistemic necessity modals. This family includes dynamic semantics accounts (

Restriction: restrict the set of epistemic possibilities to the

Quantification: check that

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

This view is motivated by a compelling analysis of how indicative conditionals are assessed, known as the

In possible world semantics, an information state

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

(1) | If Alice left, she went to London; #but it might be that she left and went to Paris. |

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):

(2) | It might be that if Alice left she went to London. |

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

The second problem, pointed out by Edgington (

(3) | If the coin was tossed, it landed heads. |

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

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

It accounts for the data that motivate the Box View. In particular, it predicts that a conditional

It accounts for the peculiar way in which conditionals embed under epistemic modals. This is achieved without

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

The account builds on the idea that epistemic sentences express attitudes towards a truth-conditional content, assessed relative to an information state

If-clauses are devices for restricting the relevant information state: to express an attitude towards

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

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

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.

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

There are several reasons for treating indicative conditionals as restricted epistemic necessity claims. One was discussed in the introduction: the Box View seems to arise in a natural way from the plausible Ramsey test view of how we assess conditionals.

A second reason is that a plain indicative conditional

(4) | a. | If Alice left, she went to London. | |

b. | If Alice left, she must have gone to London. |

The simplest explanation for this is that (4-a) and (4-b) mean exactly the same. If they do, since (4-b) is manifestly a restricted epistemic necessity claim, so is (4-a).

A third reason (cf.

(5) | If Alice left, she went to London; #but it might be that she left and went to Paris. |

This is just what we would expect on the Box View: if

When epistemic modals scope above conditionals, they seem to remove the universal force from the conditional and replace it with their own quantificational force. As an example, consider a conditional in the scope of an epistemic ‘might’:

(6) | It might be that if Alice left she went to London. |

Intuitively, (6) is a conditional possibility claim: it says that, among the epistemic possibilities where Alice left, there are some where she went to London. This is not what the Box View would lead us to expect. According to that view, (6) should be a second-order epistemic statement: it is possible that, relative to the worlds where Alice left, it is necessary that she went to London.

Moreover, (6) seems to convey exactly the same as (7) (cf.

(7) | If Alice left, it might be that she went to London. |

This, too, is puzzling from the perspective of the Box View: if conditionals contribute a universal quantification, then

This phenomenon is not restricted to epistemic ‘might’. For instance, consider conditionals embedded under a probability modal:

(8) | It is probable that if Alice left she went to London. |

What (8) means is not that it is probable that Alice being in London is epistemically necessary on the supposition that she left; rather, what (8) means is just that Alice being in London is probable on the supposition that she left. Again, this does not involve any epistemic necessity; as in the case of ‘might’, the conditional does not seem to be contributing any universal quantifier over epistemic possibilities.

Again, a sentence like (8), where ‘probably’ embeds a conditional, sounds fully equivalent to (9), where ‘probably’ occurs in the consequent, which is unexpected if the conditional contributes a universal quantification.

(9) | If Alice left, it is probable that she went to London. |

The point that we just illustrated for

(10) | a. | It is obvious that if Alice left she went to London. |

b. | If Alice left, it is obvious that she went to London. |

Summing up, then, when embedded under epistemic modals, conditionals do not seem to contribute a universal quantification over epistemic alternatives. Moreover, conditionals seem to commute with epistemic modals. These facts are surprising from the perspective of the Box View analysis. This is not to say that one could not develop a theory that accounts for these observations while holding on to the Box View. But this does pose a challenge: existing Box View accounts do not provide a general explanation of these data, and it is not obvious how they could be modified to provide one.

Imagine that a fair die has just been rolled, but the result has not been revealed yet. Our friend Alice makes the following claim:

(11) | If the outcome is even, it is above three. |

Clearly, the claim is a guess—she cannot be sure of what she is saying. But it

This observation is puzzling from the perspective of the Box View. According to that view, what Alice has claimed is that in all epistemic possibilities in which the outcome is even, the outcome is above three. This is simply not the case: it is definitely possible that the die landed on two, and thus, that the outcome is even but not above three. Therefore, from the perspective of the Box View, it seems that Alice’s claim should have no likelihood whatsoever.

To put it another way: according to the Box View, (11) means the same as (12).

(12) | If the outcome is even, it must be above three. |

But, unlike (11), (12) is

It seems that, when we assess the probability of a conditional

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

where |

The full language

where |

An information state is a map

A world

If

The probability of a proposition

Notice that the set of live possibilities after the supposition is

The possible values for the attitude parameter

We now have all ingredients to recursively specify the semantics.

For factual sentences

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

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

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.

What if the antecedent

The corresponding notion of equivalence tracks identity in acceptance conditions:

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,

Let us now turn to the predictions that

Consider the acceptance conditions that

Thus,

We also predict the inconsistency of

Thus, it is impossible for a state to accept simultaneously

Finally, consider again:

(13) | a. | If Alice left, she went to London. |

b. | If Alice left, she must have gone to London. |

To see that the equivalence holds, notice that the two conditionals are admitted in the same states (those in which

We can, thus, account for the fact that (13-a) and (13-b) can be inferred from each other, and that whenever one fully accepts one of these sentences, one also fully accepts the other. Moreover, given the connection between assertion and acceptance, we account for the intuition that (13-b) and (13-b) convey the same when asserted.

At the same time,

Thus, consider a state

We saw that both

But, as the derivation shows, the source of the universal quantifier is the acceptance attitude, not the conditional operator. When interpreting a conditional embedded under an epistemic modal, the relevant attitude may be shifted away from acceptance; as a result, no universal quantifier shows up in the semantics. For instance, consider the acceptance conditions for a conditional embedded under ‘might’. Assuming

Thus, we predict that

Which shows that the two formulas are semantically equivalent.

The story with conditionals embedded under ‘probably’ is completely analogous:

Thus,

Moreover, the account just given for

How to characterize the probability

Conditionals—we are assuming—don’t express propositions, so they cannot be assigned probabilities in this way. Still, as we discussed, it seems that we can meaningfully assign probabilities to conditionals. Of course, we could follow Adams (

Moreover, this definition would not go far enough. Consider (14):

(14) | If the die was rolled, then if the outcome was even, it was a six. |

Given a fair die, it seems natural to say that (14) has probability

Attitude semantics allows us to define such a general notion. We will take the probability of a sentence

The probability of a sentence

Let us look at the predictions that this makes. First, we can prove that, as we expect, the probability of a factual sentence

The proof is simple:

Next, consider a factual conditional

the maximum value of

which completes the proof.

Thus, now we can explain

Thus,

Notice that the same definition allows us to associate probabilities with iterated conditionals

Since the probability of a sentence in a state is defined in purely semantic terms, the probability of

Finally, consider the modalized conditional

Thus, for

Thus,

(15) | a. | If the outcome is even, it is above three. |

b. | If the outcome is even, it must be above three. |

Summing up, then,

There is a large literature on triviality results, which shows that Adams’ Thesis along with some other assumptions about probabilities of conditionals leads to absurd conclusions (see

provided

Does this position amounts to a rejection of standard probability theory? In my view, it does not: instead, it calls for a more careful understanding of the role of probability theory. Probability theory is not concerned with language, but with the obtaining of certain “events”, which on one interpretation can be identified with propositions. Thus, what probability theory gives us are constraints on the admissible ways to assign probabilities to propositions; what

To conclude, let me discuss a common objection to views like the present one. According to this objection, the axioms of probability theory are constitutive of the concept of probability; therefore, if a certain quantity that attaches to sentences does not obey these axioms, that quantity cannot properly be called

I have two things to say in response. First: what we call the magnitude

Second, I think it is in fact justified to call

We treated epistemic modals as devices to express attitudes towards truth-conditional contents. Sometimes, however, epistemic modals, like other modals, are used to describe facts about the world. For instance, (16) seems to be a descriptive statement about the information available to Alice.

(16) | According to Alice, the butler might be the murderer. |

Some theories of epistemic vocabulary (e.g.,

(17) | According to Alice, the butler is the murderer. |

We will enrich our formal language with a set

Given a sentence

Let us illustrate the clause by looking at the examples above, repeated below:

(18) | a. | According to Alice, the butler is the murderer. | |

b. | According to Alice, the butler might be the murderer. |

Our semantics delivers the following truth-conditions for these sentences:

So, (18-a) is true if Alice’s information implies that the butler is the murder, while (18-b) is true if Alice’s information is compatible with the butler being the murder. These are good predictions.

Notice that, in the interpretation of (18-b), our clause for epistemic modals as attitude shifters is exploited to produce a truth-conditional result. Thus, the idea that epistemic modals are shifters of the attitude parameter does not stand in contrast to the fact that epistemic modals have objective, truth-conditional readings; instead, it can be exploited to derive these readings.

Also, crucially, (18-b) is correctly predicted to involve not two epistemic quantifiers—one associated with the anchoring and one with the modal—but a single quantifier, whose kind (existential) is determined by the modal, and whose range (Alice’s state) is determined by the anchor. This elegant interplay is made possible by the fact that no universal quantifier is directly built into the meaning of the anchoring operator. Rather, the anchoring operator by default sets the attitude parameter to acceptance; if no modal operator interferes, as in (18-a), the acceptance parameter ultimately provides a universal quantifier over epistemic alternatives; however, if a modal operator is present, as in (18-b), it may shift the attitude parameter away from acceptance; as a consequence, no universal quantification will show up in the resulting reading. Thus, the same fundamental idea which allowed us to solve the problem of embedded conditionals also does some work in other places, such as the interpretation of modals in anchored contexts.

Notice that all the phenomena concerning conditionals that we discussed above also arise when conditionals occur in an anchored context. For instance, (19-a) and (19-b) seem to mean exactly the same.

(19) | a. | According to Alice, it might be that if Bob left he went to London. | |

b. | According to Alice, if Bob left it might be that he went to London. |

The explanations that we saw for the un-anchored case all carry over. For instance, (19-a) and (19-b) are indeed predicted to be semantically equivalent. Here is the proof that they have the same truth-conditions, where the second equivalence uses Fact 4.3:

This shows that the solution to the embedding problem that

Occurrences of modals can be anchored not just when the sentence contains an explicit ‘according to’ locution, but also when an anchor is implicit. This allows us to make sense of the possibility of nestings of epistemic modals. E.g., suppose we observe a detective at work, not knowing exactly what evidence she has gathered. It seems that, in this context, we can use (20) to convey that the detective might have gathered enough evidence to conclude that the butler did it.

(20) | It might be that the butler must be the murderer. |

This can be predicted if we analyze (20) as involving an implicit anchor to the detective’s information for the second modal: indeed, the formula

Notice that, intuitively, a reading where at least one modal is anchored seems to be the only option for (20). Here is a possible explanation: in

Finally, let me briefly note that the strategy described in this section can also be used to give an account of attitude verbs. The idea would be that an attitude verb operates similarly to the anchoring operator above: it selects its own information state, its own quantificational force, and checks whether the prejacent is supported at that state with that force. It seems natural to think, for instance, that attitude verbs like

Let us now turn to negation. Though not a modal operator, various observations suggest that negation also commutes with the conditional, i.e., that

(21) | a. | I doubt that if Alice took the test she passed. | |

b. | I believe that if Alice took the test she failed. |

(22) | a. | Nobody passed if they took the test. | |

b. | Everybody failed if they took the test. |

The two sentences of each pair seem to convey the same. Assuming that ‘doubt’ is equivalent to ‘believe not’, ‘nobody’ to ‘everyone not’, and ‘fail’ to ‘not pass’, these equivalences can be accounted for, provided that

An asset of attitude semantics is that it can be extended in a natural way with an account of negation that predicts this commutation. Let us see how.

First, we extend the syntax given in Definition 3.2 by allowing all formulas of our epistemic language

Let us start from factual formulas

Next, take modals: as before, modals act as shifters of the attitude parameter; moreover, they behave in a dual way on the positive and negative side.

As before, conditionals simply restrict the evaluation state to the antecedent worlds, leaving everything else the same.

Finally, negation flips positive and negative support:

Now let us look briefly at how negation behaves in this system.

First, since we also have negation as a truth-conditional operator, in principle we might have a conflict between the result we get when we interpret negation at the truth-conditional level, as reversing the truth-value of a formula at each world, and the result we get when we interpret it at the support level, as reversing the polarity of support. However, one can check that, in fact, the two interpretations lead to the same results. For instance, consider the case of positive acceptance. If we interpret

If we interpret

Second, negation interacts elegantly with the modalities: indeed,

As an illustration, here is the proof that the formulas

Finally, negation commutes with the conditional.

Here is the proof. Again, I spell out only the case of positive support, but the case of negative support is completely analogous.

According to Kratzer’s restrictor theory of conditionals (

(23) | a. | If Alice left, she must have gone to London. | |

b. | If Alice left, she might have gone to London. | ||

c. | If Alice left, she probably went to London. |

In these formulae, the modified operator

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

(24) | a. | It must be that if Alice left she went to London. | |

b. | It might be that if Alice left she went to London. | ||

c. | It is probable that if Alice left she went to London. |

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

(25) | If Alice left, she went to London. |

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,

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:

(26) | If the outcome is even, it is above three. |

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

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

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

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:

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

In order to achieve these results, we adopted a novel semantic architecture: sentences are interpreted relative to two semantic parameters: an information state

These merits are shared by some previous expressivist accounts of epistemic modals, in particular the one of Yalcin (

Turning from modals to conditionals,

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

Second, as we know from Lewis (

(27) | Usually, if a man buys a horse, he pays cash. |

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.

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:

(28) | a. | It is probable that if Alice had left she would have gone to London. |

b. | If Alice had left, it is probable that she would have gone to London. |

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

Then we have:

Thus, the commutation of

Moreover, consider the probability issue. It seems that the probability of a subjunctive conditional

The restrictor account of Kratzer (

In this paper I focus on indicative conditionals, although the puzzles that I will raise concern subjunctive conditionals as well. The solution that I propose extends straightforwardly to the subjunctive case. However, making subjunctive assumptions involves a different hypothetical process than making indicative assumptions; since spelling out the details of this process is orthogonal to our main concerns, I will leave subjunctive conditionals mostly out of consideration here, coming back to them only in the conclusion section.

In some of the theories mentioned above, the view is implemented as giving truth conditions for conditionals relative to a world with an associated set of epistemic possibilities; in other theories, conditionals lack truth-conditions, and the semantics delivers acceptance conditions relative to an information state.

For recent discussion of this puzzle, see Gillies (

See also Mandelkern (

That the conditional construction corresponds to a binary operator at logical form is a non-trivial assumption, which is challenged by Kratzer (

In many implementations of the Box View, the existential quantification associated with

Could one respond by claiming that the relevant reading of (6) results from

(i) I’ll go ahead and say something which I can’t be sure of, but which I think is a possibility: if Smith stole the jewels, she left the country.

Here, the speaker appears to be explicitly claiming the epistemic possibility of a conditional. It seems really implausible that the possibility operator takes syntactic scope below the if-clause. Yet, the result is still unambiguously a claim of conditional possibility.

I am only aware of two attempts to account for the observations above while preserving some version of the Box View. The first is due to Gillies (

For an overview of the classical experimental literature, see Evans and Over (

To articulate this point a bit further: in theories based on the Box View, such as Yalcin (

The reason to restrict to factual antecedents is that it is just not clear how the process of supposing epistemic sentences works (though see

As an indicative assumption: one can of course suppose

There might be reasons to allow for the possibility of supposing

And for describing the properties of different bodies of information, as in the sentence “According to the detective, the butler might be the murderer.” We will come back to this descriptive function of epistemic modals in Section 5.1.

This is in line with a tradition that treats indicative conditionals as presupposing the epistemic possibility of their antecedent. See, among others, von Fintel (

The way to do this precisely is to define a notion of admittance of a sentence

Thus, for instance,

Taking into account admittance, this notion can be articulated in two versions. The strong version: whenever all the premises are admitted and accepted, the conclusion is also admitted and accepted. The weak version: whenever all the premises are admitted and accepted, if the conclusion is admitted, then it is also accepted. The latter is a notion of Strawson entailment in the sense of von Fintel (

This is not the only interesting notion of consequence that we can define in

To this basic effect of assertion we may add another: by asserting

Given our semantics, if the set

On the idea that Adams’ thesis can be explained in terms of the restricting role of if-clauses, see also Egré and Cozic (

Kaufmann (

This illustrates a general property of sentences involving epistemic modals: they are either fully accepted by the state, or incompatible with it (something similar holds for modal sentences in dynamic semantics: they are always either accepted or rejected by a state). I take this to be a good prediction: while it is possible to be in a state in which one is unsure about, say, whether it might be the case that

A reviewer asks, quite naturally, whether the map

The objection goes back to Lewis, who made this point against Adams (

But if it be grated that the ‘probabilities’ of conditionals do not obey the standard laws, I do not see what is to be gained by insisting on calling them ‘probabilities’. (

Thanks to Matt Mandelkern for this terminological suggestion.

Thanks to Shane Steinert-Threlkeld for drawing my attention to this independent motivation.

Notice however that this kind of view, unlike the one advocated here, still faces the probability problem.

A reviewer also suggests the intriguing hypothesis that an attitude verb like

Examples include data semantics (

As before, we take

These considerations might raise the question of what, according to our own account, is the object whose probability we are judging—given that it is not a proposition. The quick answer is that it is the content expressed by Alice’s utterance. This content can be identified with a set of state-attitude pairs, namely,

This is not to say that sentences involving adverbial quantifiers don’t express propositions: they do. The idea is the following: a basic sentence

See, among others, Williams (

Many thanks to two anonymous reviewers and an anonymous editor for their extremely valuable comments and suggestions. Also, thanks to the following people for comments on preliminary versions of the paper or discussions of the ideas presented here: Maria Aloni, Fabrizio Cariani, Paul Egré, Peter Hawke, Justin Khoo, Luca Incurvati, Hannes Leitgeb, Matthew Mandelkern, Adrian Ommundsen, Floris Roelofsen, Robert van Rooij, Paolo Santorio, Katrin Schulz, Shane Steinert-Threlkeld, Frank Veltman, and Seth Yalcin.