What does the ; operator do? I didn’t find it in the book, and I see it described in the Inquisitive Semantics Proposal as follows:
; (P ∨ Q) asks for enough information to conclude, for one of P and Q, that it is true, while presupposing that at least one is. In other words, it asks to know “which one is true?” without the implication that it couldn’t be both, and presupposing that at least one is. The minimally informative answers are “P” and “Q”, and the maximally informative answers are “only P”, “only Q”, and “both P and Q”.
I think I understand from this description what the operator does intuitively, but how is it formally defined? In particular, how does inquisitive semantics treat presuppositions and distinguish them from assertions?
More specifically: if P and Q are informative propositions, then (under my understanding) P ∨ Q asserts that P or Q is true, and asks for an information state specific enough to entail at least one of P or Q. Then how is ; (P ∨ Q) different? Is info(; (P ∨ Q)) the same as info(P ∨ Q), which is info(P) ∪ info(Q)? Or does the fact that “P or Q” is presupposed (rather than asserted?) mean that the informational content is different?
As far as I can tell (@magnap please correct me if I’m wrong), a bare P v Q is more of a meta object, a set of sets of possible worlds. Formally, you need to map it to a proposition/question.
If i recall correctly, Magnap uses ;P := (P | info(P). So we ask for P, presupposing that at least one alternative is true. (Which is different from ?, which also accepts an answer of “none”.)
I really need to find a better source, which I think I have left for myself on the Inquisitive Semantics Proposal page as a TODO, but until I do, here’s a slide show where Ciardelli defines a 0th-order language that is essentially InqB with presuppositions (what he calls $?_c$ (can’t find a subscript c in Unicode) I call ;). Briefly, the presupposition of a sentence is a set of worlds, and the proposition expressed by the sentence is only over that set of worlds, not the entire set of worlds.
The reason I haven’t treated giving a formal definition of it on the Inquisitive Semantics Proposal wiki page as a high priority is that since the current logic of Toaq already uses presupposition, I figured that by saying “presupposition of an inquisitive semantics proposition presupposes its informative content; from there on deal with it the usual way” I could save myself some effort and avoid accidentally formalizing it in a way that conflicts with current Toaq logic (especially since I am very vague on the degree to which it is formalized at least in terms of any specific model).
As for how how presupposition is distinguished from assertion, I am not aware that the “restriction on the set of possible worlds” way it’s done in inquisitive semantics is any different from what I understand to be the usual way.
Yes. Informational content and presupposition are accounted for separately.
P ∨ Q is a “hybrid” sentence: both informative (asserting “P or Q”) and inquisitive (asking which). ;(P ∨ Q) is not a hybrid sentence: it inquisitive but non-informative, since info(; (P ∨ Q)) is a tautology over the set of worlds where it is defined: those where info(P ∨ Q).
The downward closed set-of-sets is not a meta object in inquisitive semantics, it’s what propositions are. But you are right that just because we know what proposition a sentence denotes doesn’t mean we know what it means to say it (is it a question? an assertion of (for a hybrid sentence) knowledge yet partial ignorance? a wish, or command, for the world to be put into a certain state?).
Pedantically, ; and ? sometimes have the exact same effect, specifically on a non-informative proposition: none.
This was very helpful—I now know that your model differs from the one presented in Ciardelli et al. (2018) and have a good mental model of how exactly it differs.
I agree. After reading how presupposition is denoted, it seems pretty straightforward to me. But I hadn’t thought of that definition (probably just because I didn’t think to try to extend the model to include presupposition), and it was helpful to see it written out explicitly so I know for certain that I have a good idea of it. So thanks again for your explanation!