Possibility configurations

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A proposition p is slightly possible in w with respect to modal base f and an ordering source g iff (i) p is compatible with ∩f (i.e. p is a contextual possibility);

-And-

(ii) ¬p is humanly necessary in w with respect to maxg(∩f) (i.e. the max set)

Since a max set is a subset of a set of accessible worlds, it of course presupposes the existence of a set of accessible worlds. Since both sets must exist for human modality to exist, if we adopt human modality semantics we are free to assume definitions which are relative to only the larger set (contextual impossibility/possibility/necessity), only the smaller set (human necessity/possibility) or both (slight possibility). Note that slight possibility is the only definition which makes direct reference to both sets²⁷. Although it has been largely overlooked in the literature and even by its creator, we will see that slight possibility plays a large role in the communication of belief states regarding given propositions in discourse.

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Mustc ¬p tells us that ¬p is contextually necessary, and so by definition that there is no p-world in the set of accessible worlds. In order to felicitously utter Mustc ¬p, Speaker's belief state must correspond to Row 1; Speaker must take p to be contextually impossible and ¬p to be both contextually possible and contextually necessary. The reverse also holds, so that mustc p tells us that Speaker's belief state corresponds to Row 2. Put briefly, Mustc gives rise to a single possibility configuration regarding the proposition it operates over (i.e. the prejacent).

Unlike contextual necessity, contextual possibility is consistent with multiple possibility configurations. Mayc

p tells us that p is contextually possible. Since contextual possibility requires there to be at least one p-world in the set of accessible worlds, by definition ¬p cannot be necessary, eliminating Row 1. However, Speaker's belief state may correspond to either Row 2 or 3. Row 3 corresponds to the option where both p and ¬p are contextually possible.

Thus, we end up with three possible belief states or possibility configurations regarding a given proposition.

Table 1 Possibility configurations for singly relative modality Contextually

Impossible

Contextually Possible Contextually Necessary

mustc not mayc mustc

1 p ¬p ¬p

2 ¬p p p

3 ¬p , p

The following definition is intended to be general enough to use for both singly and doubly-relative modality.

The portions related to doubly-relative modality will be explained below.

Def. 14 Possibility Configuration

For any proposition p, the possibility configuration of p shall refer to the status of p and ¬p regarding

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whether each is compatible with or entailed by maxg(∩f) (i.e. the max set) and/or ∩f (i.e. the set of accessible worlds).

Double relativity increases the number of possible possibility configurations for any given proposition. When we add a subset of the set of accessible worlds to the semantics of modality and thus make it doubly-relative, we essentially leave Row 1 and 2 the same but split Row 3 into three. This is reflected in Table 2 below, which shows the full range of possibility configurations allowed by double-relativity. Each row, 1-5, can be taken to represent a maximal belief state regarding a proposition to which a doubly-relative modal can make reference by virtue of its semantics²⁸. The English word may is taken to be ambiguous between contextual possibility mayc and human possibility mayh.

Table 2 The possibility configurations of doubly-relative modality Contextually

Impossible

Contextually Possible

Slightly Possible

Humanly Possible Humanly Necessary

Contextually Necessary

mustc not mayc mayh musth mustc

1 p ¬p ¬p ¬p ¬p

2 ¬p p p p p

3 ¬p , p ¬p p p

4 ¬p , p p ¬p ¬p

5 ¬p , p ¬p , p

28 A given individual will often not have a fully-fleshed out belief state. This corresponds to Row 3 in Table 1, or to undecided between 3,4 and 5 in Table 2. This will be important for the informativity of modal assertions. Of course, it is also possible for a given individual's belief state to be more fine-grained than either singly- or doubly-relative modality conveys. We are not interested in a precise characterization of internal belief states, only the portion of them that an individual can systematically express using modal forms. In the final chapter, deciding on a possibility configuration for the common ground will be proposed as how to "settle an issue/QUD" brought up by a modal assertion, similar to how a polar question is "settled" by deciding whether a proposition should be true, false, or undecided in the common ground (cf. Farkas & Bruce, 2009).

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As above, must and may serve to partially indicate which belief state Speaker has. However, the increase in configurations also results in less determinative power for the linguistic expressions must and may. With respect to Table 1, may was ambiguous but must was determinative. With respect to Table 2, even must is ambiguous.

 Given a single instance of must p, we can determine that Speaker's beliefs correspond to either Configuration 2 or 3: 2 for contextual must and either 2 or 3 for human must.

 To determine which, we need more information. One way is to see if may is useable.

 In the case of Configuration 2, p is (both a human and) a contextual necessity, so ¬p is a contextual impossibility.

 If this is the configuration corresponding to Speaker's beliefs, Speaker should not be able to use must p and may ¬p without contradicting himself or changing his mind.

 In the case of Configuration 3, p is a human necessity but not a contextual necessity.

 ¬p is a contextual possibility but not a human possibility, that is, it is a slight possibility²⁹.

 If this is the configuration corresponding to Speaker's beliefs, Speaker should be able to assert must p intended as human necessity and a may ¬p intended as contextual possibility without contradicting himself.

 Therefore, if Speaker can assert must p and may ¬p without contradicting himself, his belief state corresponds to Configuration 3. If he cannot, it corresponds to Configuration 2.

This setup allows us a very simple way to verify that double relativity is necessary. According to singly-relative modality, a must p assertion is determinative and should never be compatible with a may ¬p assertion. In following, if we can find an instance where a single utterer can assert must p and may ¬p (or the reverse: must ¬p and may p) without contradicting himself, we need the ambiguity offered by double relativity.

Thus, we want to show that Prediction 1 can be wrong while Prediction 2 is not.

Prediction 1 (singly-relative): must p contradicts ¬p

29 There is no linguistic expression dedicated to slight possibility, so it must be deduced through a combination of other operators.

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Prediction 2 (doubly-relative): must p is compatible with may ¬p³⁰

Both singly and doubly-relative modality allow Prediction 1, but only doubly-relative modality allows Prediction 2. If in discourse, Prediction 1 covers all of the entailment relations that we need, then we have to go elsewhere for evidence in favor of double-relativity. If, on the other hand, Prediction 2 is true, we will have found convincing evidence for the need for double-relativity which is based in linguistic fact rather than on a subjective sense of strength or weakness.

von Fintel & Gillies (2010) claim the following sentences support a single relativity definition for epistemic necessity.

(12) It must be raining, but #perhaps³¹ it isn’t raining.

(13) Perhaps it isn’t raining, #but it must be.

(14) There is a vanishingly small chance that it isn’t raining, #but it must be.

In reference to these examples, they point out that even though perhaps ¬p does not remove all ¬p worlds from the modal base, it is still contradictory to must p. Under the assumption that perhaps ¬p is weak (contextual), they posit this as evidence that must cannot be relative to a smaller set than the set of accessible worlds. However, if perhaps ¬p is taken to denote mayh ¬p, it says there is at least one ¬p world in the max set. If must p is taken to denote musth, then it says there is no ¬p-world in the max set, so they contradict each other. The observations can thus be explained by Prediction 1 above, which as we saw is compatible with either single or double relativity. Thus, (12) and (13) do not offer evidence either way for our issue. As long as must and perhaps denote relativity to the same set of worlds, they contradict each other.

30 Note that we must show not just that musth p and mayc ¬p are compatible, but that musth p and mayh ¬p are incompatible.

Otherwise the evidence would point not to double relativity but to a break in the duality of may and must such that there are conceptual differences other than force (possibility vs. necessity).

31 In the examples they use perhaps instead of may, but they are proposing a general definition for possibility and necessity operators, so their predictions should be the same for may. One motivation for the choice of perhaps may be to control for the increased salience of may and must as a pair.

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von Fintel and Gillies acknowledge this issue in a footnote, and claim to have ousted the possibility of double relativity definitively with (14). Since a vanishingly small chance that ¬p would certainly reflect weak (mayc) possibility even if we assumed that perhaps denotes strong possibilty (mayh), and it is still contradictory with must p, they conclude that must is strong and, worded in terms of our current investigation, singly-relative. However, what this may reflect is some sort of clausal restriction on the type of modality such that when there are multiple epistemic modals in a given clausal unit one cannot denote singly-relative while the other denotes doubly-relative modality. I offer the following as evidence that that is exactly the case.

(15) John: Where's Bob?

Taro: He may have gone to the store.

Jane: No, look. His brief case is gone. He must have gone to the office.

John: So you're sure he didn't go to the store?

Jane: Well, no. He may (marked intonation) have, but come on! He must have gone to the office.

If we only allow strong necessity, we must assume that Jane, rejecting Taro's suggestion, believes that "Bob went to the store" is contextually impossible. However, if this were the case she would be contradicting herself in her next utterance, where she would have to be agreeing with Taro. However, we can see that she still upholds her original opinion, so she seems to believe her belief state is consistent.

One way for it to be consistent would be to say that "Bob went to the store" and "Bob went to the office" are not mutually exclusive, but if this were the basis for Jane's utterance then we would predict Jane does not disagree with Taro's may, which goes against her utterance of "no". Furthermore, the answer being sought is why Bob is not present, not what his trajectory is, so any proposition offered as an answer should be a competitor entailing the complement of the other competitors.

We can give an adequate explanation if we assume doubly-relative modality. First, we can say that Jane takes Taro's may to denote mayh. Then, in her first utterance, "Bob went to the office" is presented as humanly necessary relative to a

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stereotypical ordering source and "Bob did not go to the office" as only contextually possible. This disagrees with Taro's may given that Taro presents Bob went to the store as a human possibility. Finally, in her second utterance, Bob went to the office is presented as a contextual, not a human, possibility employing mayc. Mayc p does not contradict musth

¬p, so Jane's second utterance need not contradict her first. Also note that Jane's may seems to require marked intonation, suggesting that she intends it to be interpreted differently from the standard interpretation, or perhaps from the interpretation it is given in Taro's utterance³². We might assume the difference is that may in Jane's final utterance is presented as mayc. Since for musth ¬p and mayc p to be consistent, p must be a slight possibility, we can infer that Jane's belief state corresponds with Configuration 4 in Table 2³³. Thus, we have seen that Prediction 1 from above can be false, and that Prediction 2 correct, showing the need for double relativity in the semantics of English modal assertions.

Now consider the following utterance.

(16) John may have won, but I doubt it.

Note that this may also prefers marked intonation or the sentence sounds contradictory. As in (15), marked intonation on may may indicate that it denotes mayc. Given this, the need for marked intonation suggests that doubt cannot pair with mayh. Since a proposition that is slightly possible cannot be humanly possible, this suggests that doubt is a marker of slight possibility. Also note that in interpreting this utterance, our attention goes to what Speaker’s reason for doubting John’s victory might be while the possibility of John winning seems like a grudging concession that cannot be ruled out rather than a possibility that Speaker endorses. Taking John won to be p, Speaker's belief state would correspond to Configuration 4. Again we have evidence for may being used to differentiate between slight and human possibility in discourse, and thus corroboration for double relativity.

32 The need for marked intonation seems to be idiosyncratic to English. The same discourse event did not require marked intonation on may (kamo shirenai) in Japanese.

33 In the discussion of pragmatics in a later chapter, I shall propose and demonstrate that discussion involving modals often involves clarifying and bringing change to the belief state of each discourse participant relative to the five configurations.

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