Assertions, questions, and conjectures

We used to define assertions as non-inquisitive formulas, and question as non- informative ones. That is, speaking in informal but evocative terms, assertions were formulas whose meaning lay entirely on the axis described by empty in- quisitive component, and questions formulas whose meaning lay on the axis of

empty informative component (see figure 2.1.3). This will be similar now, but instead of a plane we will have a cube and one more type of formulas. As- sertions will be formulas that are purely informative, i.e. non-inquisitive and non-suggestive; questions will be purely inquisitive, i.e. non-informative and non-suggestive; finally,conjectureswill be purely suggestive, i.e. non-informative and non-inquisitive.

Let us start considering the notion of assertion. We want assertion to be formulas whose only effect is (at most) to provide information.

Definition 6.2.23 (Assertions). Anassertion is a formula that is neither in- quisitive nor suggestive.

The following proposition insures, among other things, that the definition given here coincides with the one given above, of assertions as formulas with a one- piece proposition. Incidentally, notice that corollaries 6.2.6, 6.2.7, and 6.2.8 supply many examples of assertions.

Proposition 6.2.24(Alternative characterizations of assertions). For any for- mulaϕ, the following are equivalent:

1. ϕis an assertion;

2. ϕhas only one possibility; 3. [[ϕ]] ={|ϕ|};

4. ϕ∼!ϕ.

Proof.

(1)⇒(2) If ϕ is an assertion, then it is not inquisitive, so it has only one maximal possibility, and it is also not suggestive, so all its possibilities are maximal; thereforeϕhas only one possibility.

(2)⇒(3) Follows from the equalityS

[[ϕ]] =|ϕ|(corollary 6.2.4). (3)⇒(1) Immediate by definition of inquisitiveness and suggestiveness. (3)⇔(4) Follows from corollary 6.2.8.

Analogously, questions will be formulas whose sole effect is (at most) to raise an issue.

Definition 6.2.25 (Questions). A question is a formula that is neither infor- mative nor suggestive.

Note that now a formula of the shape ?ϕ will not in general be a question: it will be if and only ifϕis non-suggestive. However, ifχis a meaningful assertion (that is, neither tautological nor contradictory) then ?χ does express the polar question whetherχ.

Finally, we turn to the notion ofconjecture. A conjecture is a formula whose only purpose is (at most) to highlight one or more possible updates, but without either providing or requiring any information.

Definition 6.2.26 (Conjectures). A conjecture is a formula that is neither informative nor inquisitive.

To gather some intuition, let us look at figure 6.1(b). The formulap∨ >, which we abbreviate as3p, has the effect of suggesting, or highlighting, the possibility that p, but without providing any information in regards. As such, it can be taken as a formal counterpart of the natural language sentences “it might be thatp” or “perhapsp”.

In general, for any assertion χ, the formula3χ =χ∨ > does not provide information and does not raise issues, but simply “highlights” the possibility that χ so that it can be compliantly picked by the other participants if they wish; thus, we think of3χ as “it might be thatχ”.

In the previous section we said that the proposition of p∨(p∧q) can be taken to represent the natural language utterances “p, and it might be thatq”, or “p, and perhaps q”. Now this comes out of our interpretation of 3: for, it is easy to see that p∨(p∧q) ∼ p∧3q. In general, if both χ and ξ are assertions andξ(classically) entails χ, thenχ∨ξ∼χ∧3ξ, so at least in this case we have a natural interpretation of the role of non-maximal possibilities as

might-possibilities.

More in general, for an arbitrary formulaϕ,3ϕhas the effect of highlighting the resolutions ofϕwithout providing the information thatϕis true and without necessarilyrequiring a resolution ofϕ.

In order to understand better the last remark, let us consider the difference between the polar question ?p and the formula 3p∨3¬p whose meaning is depicted in figure 6.1(c). The polar question ?pintroduces in the conversation an issue that is only resolved by asserting eitherpor¬p. The conjecture3p∨3¬p, on the other hand, does not introduce an issue: it gives the other participants the possibility to assertpand¬p, but does not require this information, since it is also resolved by not uttering anything (or nodding, or saying “Ok”, or whatever non-informative, non-inquisitive and non-suggestive conversational move > is taken to represent).

Proposition 6.2.27 (Alternative characterizations of conjectures). The fol- lowing are equivalent:

ϕis a conjecture;

I ∈[[ϕ]];

ϕ∼3ϕ;

ϕis supported everywhere.

Proof.

(1)⇒(2) If ϕ is a conjecture, then it is not inquisitive, so it has a greatest possibility, and it is also not informative, so this greatest possibility must coincide withI.

(2)⇒(3) Follows from the fact that [[3ϕ]] = [[ϕ∨ >]] = [[ϕ]]∪[[I]].

(3)⇒(4) Ifϕ∼3ϕthenϕ≡3ϕ=ϕ∨ > ≡ >, soϕis supported everywhere. (4)⇒(1) Immediate.

Observe that, differently from assertions and questions, conjectures admit a characterization in terms of support.

Also, notice the following funny fact: item (4) states that a formula is a conjecture if and only if it used to be a tautology in the maximalization se- mantics. In chapter 2 we refined the classical notion of meaning and we got a new class of meanings, what we called then questions, out of formulas that were classical tautologies; now we refined the notion of meaning even further and we obtained a new class of meanings, namely conjectures, out of what used to be the tautologies.

Finally, we remark three closure properties of the class of conjectures. Remark 6.2.28. For any formulasϕandψ,

1. 3ϕis a conjecture;

2. ifϕand ψare conjectures, then so isϕ∧ψ;

3. if at least one ofϕandψis a conjecture, so isϕ∨ψ; 4. ifψis a conjecture, then so isϕ→ψ.

Proof. I ∈[[>]]⊆[[3ϕ]]. IfI ∈[[ϕ]] and> ∈[[ψ]], thenI =I ∩ I is in [[ϕ∧ψ]] by definition of propositions. IfI ∈[[ϕ]] or I ∈[[ψ]], then I ∈[[ϕ]]∪[[ψ]] = [[ϕ∨ψ]]. Finally if I ∈[[ψ]] then letting f> : [[ϕ]] →[[ψ]] be the function mapping each

s∈[[ϕ]] toI we haveI = Πf>∈[[ϕ→ψ]].

This is in tune with the intuition that utterances like “it might be thatpand it might be thatq”, “p, or it might be thatq” and “ifp, it might be thatq” are all conjectures.

Note on the empty set. In possibility semantics, it is often the case that the empty, inconsistent state shows up among the possibilities for a formula. Obviously, in a conversation, it is never a meaningful option to choose to update the common ground to the inconsistent state. Therefore, in my opinion, the empty state should simply be disregarded when judging the proposal made by a formula.

Having the empty state in the semantics is handy for several purposes, in the first place to make the definition of propositions come out right, in particular as regards negations. But for practical purposes, as soon as a formula has at least another possibility, we should probably not regard the inconsistent state as a suggestion of the formula. We should thus also consider two formulas strongly

equivalent if one only differs from the other only for the presence of the empty state.

There are plenty of examples where this appears in fact to be the reasonable thing to do. For instance, we would expect the formula3p∧3¬pto highlight the possibility thatpand the possibility that¬p, thus being strongly equivalent to3p∨3¬p: this is indeed what we obtain if we disregard the presence of the empty set in [[3p∧3¬p]]. We will see more convincing examples of this need in the first-order case.