This paper develops a very simple probabilistic dynamic formalism for a semantics of epistemic modal expressions that is essentially expressivist in nature, with a notion of information states based on probability spaces. It differs from existing approaches to this task (in particular [22] and [12]) in certain features that make it more appropriate for modelling actual conversational behavior, especially as regards ignorance and unawareness. It will be argued that the latter is important for an adequate treatment of the alleged non-monotonicity of conditionals. Directions for further research that follow from the approach will be pointed out.
Definition 1. A probability space over possible worlds is a triple hS, S , P i, where S is a set of worlds, S is some Boolean algebra over S, and P is a probability function on S .
Let our language L be a simple propositional language with modal operators ✸, ✷ and △ (the latter for probably) and a conditional connective ≻. We call a formula non-modal if it contains none of these.
Definition 2. A model for L is a pair hW, J·Ki, where W is a set of possible worlds and J·K an interpretation function that assigns to every non-modal formula of the language a subset of W , subject to the usual conditions for ∧ and ∨.
Definition 3. We define an information state (or context) c to be a pair hS , Pi, where S is a boolean algebra over W and P is a set of functions P such that hW, S , P i is a probability space.
A preliminary definition of update, which serves to illustrate what properties we want it to have, is given in the following: Definition 4. (to be revised) If φ is a non-modal formula, then the update of a context c is defined as follows:1
c[φ] := {P (·|φ)|P ∈ c ∧ P (φ) 6= 0} c[✸φ] := {P ∈ c|P (φ) > 0} c[✷φ] := {P ∈ c|P (φ) = 1} c[△φ] := {P ∈ c|P (φ) > .5} c[φ ≻ ψ] := {P ∈ c|{P }[φ][ψ] = {P }[φ]} Definition 5. A formula φ is accepted in an information state c (c φ) if c[φ] = c 6= ∅. It is consistent with c if c[φ] 6= ∅ and inconsistent if it is not consistent. It is compatible with c if there is a sequence Δ (which may include φ) of formulae such that successive update with the members of Δ yields a c′ in which φ is accepted. φ is incompatible with c if it is not compatible.
The recursive definition of negation in such a system is a tricky matter due to the
qualitative difference between conditionalization and eliminative update. It is not even
clear that the definition of negation really needs to be given recursively instead of as a
set of rules for negative update, as in [
We use a slightly simplified version of Schmitt’s notion of a Bayesian closure of a set of probability functions. It is the set that contains all the functions in the original set, and all those that are the result of conditionalizing a function in the original set on some non-modal sentence in the language.
Definition 6. The Bayesian closure ccl of a context c is defined as {P |∃P ′ ∈ c : ∃φ ∈ L : P (·) = P ′(·|φ)}.
Schmitt proves that the function (·)∗ is the reverse of the Bayesian closure function if c fulfills the condition of weak regularity: if a function in c assigns 1 to any formula φ, then all functions in c do. Our contexts are, in general, not weakly regular, but as will be clear in a moment, that is not a problem for our final definition of update.
1We write P (φ) for P (JφK). Furthermore, since S is not affected by updates at this point, we will for ease of exposition pretend that a context is just a set of probability functions. Definition 7. The ∗-function returns for every set C of probability functions that is closed under conditionalization in the above sense the unique weakly regular set of probability functions that C is the Bayesian closure of.
C∗ = {P ∈ C|¬∃P ′ ∈ C : ∃φ ∈ L : P (·) = P ′(·|φ)}
On the level of Bayesian closures, a recursive definition for negation can be given. These are Schmitt’s update rules: Definition 8. The update of a Bayesian closure C with a formula is defined as follows, where φ is a non-modal formula and ψ is an arbitrary formula.
C ↑ φ := {P ∈ C|∃P ′ ∈ C : P (·) = P ′(·|φ)} C ↑ ¬ψ := {P ∈ C|∃y ⊆ C : P ∈ y ∧ ycl ↑ φ = ∅} C ↑ ✸φ := {P ∈ C|C ↑ ¬φ 6= C} C ↑ ✷φ := {P ∈ C|C ↑ φ = C}
C ↑ △φ := {P ∈ C|∀P ∈ C∗ : P (φ) > .5}
C ↑ (φ ≻ ψ) := {P ∈ c|(C ↑ φ) ↑ ψ = C ↑ φ}
But Bayesian closures are not what correctly represents our information, and furthermore, the update with modals here is only a test. What we will do is to make the update of a context distributive: we take the singleton of every function in P , form the Bayesian closure of it, apply Schmitt’s rules to it, feed it to the ∗-function, and collect all the results.
Definition 9. The update of a context c with an arbitrary formula φ (written c[φ]) is given by the following:
c[φ] = [{P|∃P ∈ c : P = ({P }cl ↑ φ)∗}
The effects of this update are generally as in Definition 4. Now the negation of a formula φ is accepted in c iff φ is inconsistent with c.2 Norm of Assertion. A speaker S may assert φ iff φ is accepted in their information state cS.
Norm of Contradiction. A hearer H may contradict a speaker’s assertion of φ iff φ is incompatible with their information state cH . 2 2.1
These definitions allow us to capture elegantly a number of linguistic phenomena. The
first (perhaps more of a cognitive phenomenon) is what is called question sensitivity
2We cannot, however, derive the fact, pointed out in [
Using sets of probability functions is the standard way of capturing that probability
assignments are typically vague (cf. e. g. [
It could be suggested that possibility operators embedded under know have no semantic effect (at least at the level of at-issue content), so that an agent who says she doesn’t know if something is possible is just expressing that she is ignorant as to whether it does in fact obtain. However, evidence for an analysis along the lines presented above comes from the fact that while the agent may, in a given situation, either say that something is possible or that she doesn’t know if it’s possible, she can’t say both at the same time; she has to decide how to present herself:
3A reviewer suggested that this might just be a case of a familiarity presupposition of a proper name. But that is not an alternative explanation: the fact that his probability function is undefined for sentences involving the name is just an effect of the agent’s being unfamiliar with it! In addition, it is a well-known fact from the literature of belief that it does often not seem appropriate to ascribe beliefs about a matter that the agent is simply not thinking of, even if she would have no trouble forming them were the proposition to come to his mind. See section 3 for some further discussion of this.
4Naturally, there is also higher-order vagueness, which will be ignored. (1) ??John might have cancer, so they ran a test to rule that out. I haven’t seen the results yet, so I don’t know if he might have cancer.
Our analysis is similar in spirit to the treatment that the phenomenon receives in [
The reader may have been puzzled by the norm of contradiction given above; the usual way to state it is, of course, to say that a hearer may (or even must) contradict an assertion of φ if φ is inconsistent with their information state. There are, however, some cases where a hearer can neither accept an assertion nor contradict it. Assume that A assign’s to John’s having been there probabilities in the interval [.6, .9]. (2)
B: He must be.
A1: Why? A2:??I see.
A3:??No.
We can make sense of this if we assume that A is in an information state where the range of probabilities assigned to John’s having been there (which we abbreviate as χ) is, say, [.6, .8]. If he updated with ✷χ, he would end up with the absurd context, so he cannot just accept B’s assertion. On the other hand, he cannot contradict it either, because his information state is, presumably, not incompatible with ✷χ: from 5Note that in order for this to work technically, the person’s probability estimate about the contents of the lab report also have to be totally vague. But that seems about right: she is not disposed to speculate about the lab report. Rather, if questions about its content come up, she would try to settle them by having a look at it. an information state that accepts the negation of a modal formula, one can, in the usual case, reach one that accepts its unnegated version through update with certain non-modal formulae, i. e. conditionalization on the right kind of evidence. That is why, in the most typical case, the only route open to A is to ask for such evidence.
As predicted, unmodalized assertion behaves differently:
B: Yes, he was there.
A1: I see.
To be sure, there are cases that can be updated with ✷φ directly, viz. if the hearer’s information state contains a P such that P (φ) = 1. In that case, we would expect her to just accept the assertion. Indeed, if a hearer is very weakly opinionated about φ, or thinks it possible that φ follows from what she knows, then such a reaction seems appropriate intuitively. Only when the hearer does have an opinion about the probability of φ, the question for evidence is necessary.
Of course, the classic kind of examples of “subjective” use of epistemic modals are covered by our theory as well, as well as the usual asymmetries. (4)
B: No, they can’t be, I still had them when we came into the house.
B’: #Okay, but I know that they’re not there.
Here, speaker A asserts a possibility on the basis of her own information state, and
speaker B, following the norm of rejection, denies it. Her information state assigns
probability 0 to the keys’ being in the car, so she cannot update in any way to make
A’s assertion accepted in his information state, thus fulfilling the norm of rejection.
What she cannot do is accept A’s assertion and point out that she, on the other hand,
knows better — such a reaction is just not sanctioned by the norms of conversation.
(after [
#A: John isn’t in the garden. B: He might be.
Our system also includes the explanation from [
In (5b), this is not an option. Of course, (5b) is not an impossible sequence in actual discourse, but the point is that B is prompting A to revise her information state — by removing the information that John isn’t in the garden — and not merely to update.
6Contra Yalcin ([
The technical setup we have seen might look very similar to that of [
Yalcin’s reason for sticking with a quantificational semantics for modals, is given
in [
There is a well-known kind of conjunctions of conditionals which are not reversible, sometimes called Sobel sequences. The classic example is this: (6) (7)
If the USA were to throw its nukes into the sea tomorrow, there would be war. Of course, if the USA and all the other superpowers were to throw their nukes into the sea tomorrow there would be peace. #If the USA and all the other superpowers were to throw their nukes into the sea tomorrow, there would be peace. Of course, if the USA were to throw its nukes into the sea tomorrow, there would be war.
Sobel sequences are generally discussed in the context of counterfactual conditionals,
but they are just as possible with epistemic conditionals as with counterfactuals.7 In
7The existence of indicative Sobel sequences is in fact acknowledged in [
If the USA threw its nukes into the sea, there was war. Of course, if the USA and all the other superpowers threw their nukes into the sea, there was peace. #If the USA and all the other superpowers threw their nukes into the sea, there was peace. Of course, if the USA threw its nukes into the sea, there was war. Intuitively, it is clear what happens here: at first, the speaker didn’t even consider the possibility that all superpowers could throw their weapons into the sea. Then it comes to her attention and she corrects herself.
The traditional analysis (e. g. [
Theories based on domain widening have a problem: they cannot distinguish between disregarding the possibility of φ (i. e. failure to consider it) and accepting that φ is impossible. Assume that in a context9 c1, a speaker has accepted if φ, ψ. At the same time, she is disposed to accept if φ and χ, then ¬ψ (which is the situation in Sobel sequences), update with which would put her in the information state c2. On the domain restriction theory, it follows that in c1, no χ-worlds are in the relevant domain. But then in c1, ¬✸χ is accepted, and so is if φ, ¬χ. But this is not what we want: a speaker who has failed to consider a possibility is not necessarily disposed to accept its negation.
Conversely, domain restriction theories predict that propositions deemed to be
impossible can still feature as the antecedent of a subsequent conditional. But sequences
like the following have a strong contradictory flavor:
(10) #The USA can’t have thrown their weapons into the sea, because human nature
wouldn’t have allowed them to; of course, if they did, there was peace.
It suggests itself to apply the ideas in [
In our approach, there is a notion of insensitivity that captures the fact that
disregarding a possibility is to disregard its negation and various combinations of it with
8Schulz’s ([
Our update rules do not change S , which means that we cannot model becoming sensitive to a question in the course of a conversation. This is obviously inadequate. In fact, when a question is mentioned in a conversation that we weren’t sensitive to, we adjust our information state often (though not always) silently. In our formalization, this would mean that if an agent is insensitive to φ and φ (or some modalized version of it) is mentioned, they extend S to the smallest superset of itself that contains JφK and still fulfills all the closure conditions that come with being part of a probability space. Of course, the probability functions in P would have to be altered so as to assign some value to φ. This could be done by just extending them (which will result in some constraints on the probability of the new φ), but this is quite obviously inadequate: upon becoming aware of a possibility, we sometimes lower the probability we assign to other possibilities. This is also what is needed in Sobel sequences. For assume that in c, φ ≻ ψ is accepted and φ and ψ are non-modal formulae, so that P(ψ|φ) = 1. As long we are not allowed to alter existing probability assignments, that will not change no matter what probability we assign to the new possibility χ, since P(φ|φ ∧ χ) = 1, and so P(ψ|φ ∧ χ). But the conditional that prompted us to become sensitive to χ was (φ ∧ χ) ≻ ψ, with which our context is then inconsistent.
What is missing is a representation of the background knowledge on the basis of
which we alter probability assignments when we become aware of an additional
possibility; presumably the kind of information that is encoded in generic conditionals.11
The modeling of this is a challenging task: one may suspect with [
There are other issues we cannot treat with our simplistic approach; in particular,
the evidential effect of modals. While ✷φ and φ have different update effects and
deniability conditions, they still have identical acceptance conditions, and so should
be assertable by an agent in the same circumstances (and be interchangeable under
attitude operators and in the consequent of conditionals). However, as noted in [
Context: Rain is falling outside and the agent is looking out of the window. a. It’s raining outside.
b. #It must be raining outside.
10This is the reason why we want to distinguish insensitivity to φ from P(φ) = [
11Of course, one could just use an ordered set of contexts with successively larger event spaces S , but that would be entirely uninteresting.
Peter believes that aliens must have been here. (12b) clearly implies that Peter’s belief about the presence of aliens is inferred, and that he neither actually saw, nor, indeed, hallucinated them. (12a) is neutral about either possibility. That ✷φ and φ are therefore not in complementary distribution either is what makes the incorporation of the evidential impact of modals very nontrivial. One can of course extend information states to include a special body of directly evidenced propositions, but the intricate interplay of implicatures and presuppositions with respect to these will then have to be explored. This we leave to future research as well.
In summary, we have demonstrated that the probabilistic dynamic approach has a number of useful features. Modeling an information state as a vague probability space over possible worlds, we have provided a conspicuous picture of the workings of modals in conversation regarding acceptance, rejection and to some extent the information conferred by them, and makes use of the formal possibility of a totally vague probability distribution to model an agent who presents herself having no commitment at all as to the probability of a proposition. Furthermore, a superior account of question sensitivity falls out directly from the formal tools that are employed.
We have reached the limits of our simple model at two points, giving rise to questions for further research. First, the evidential effect of modals is not captured, which points to the necessity of extending information states to designate some propositions of special status so that evidential presuppositions may be formulated. Second, while we have seen that our notion of insensitivity is in fact superior to that employed in standard accounts of Sobel sequences of conditionals, we have not ourselves provided an analysis of these. It turned out that in order to treat the dynamics of sensitivity that are involved, a representation of generic probabilistic knowledge is needed, and it stands to reason that this ultimately leads us deep into cognitive science.