User:Magnap/Inquisitive Semantics Proposal

Inquisitive semantics provides a framework that we can use to give well-defined and reasonable semantics to questions (both direct and indirect) in Toaq.

Background

Prerequisites

This page assumes some background in first-order logic and possible-world semantics. If you know enough to get the gist of the basics of Toaq semantics, you should be good to go. If not, here's very briefly what you need to know: we can have a model of propositional logic where we represent each proposition by the set of "possible worlds" where it is true, where a "possible world" for this model is some object that completely determines the truth value of every proposition. Then, to judge truth, we distinguish a possible world that we call the real world, and say that a proposition is true if the set that represents it contains the real world. Then, we have ways to interpret $\lor$, $\neg$, and the other symbols of propositional logic as functions on sets, such that we can translate any statement in propositional logic to a set of possible worlds, in such a way that it "is true" (contains the real world) exactly when the statement is true. Oh, and this generalizes to first-order logic as well. Finally, if we have functions, we don't need to explicitly have sets as a "type of object": we can represent a set as a function that tells you whether some object is in the set or not, and you can "get the set back" by applying that function to everything and only keeping the things it returns true on. That way, you can choose to think of a proposition as either a set of possible worlds, or as a function from possible worlds to a truth value.

Inquisitive semantics

In inquisitive semantics, rather than being subsets of all possible worlds (of type $\left\langle {\text{s}},{\text{t}}\right\rangle$), propositions are taken to be nonempty sets of subsets of all possible worlds (of type $\left\langle \left\langle {\text{s}},{\text{t}}\right\rangle ,{\text{t}}\right\rangle$), downward closed under containment (meaning that, for some proposition ${\text{P}}$, if ${\text{Q}}\in {\text{P}}$, then for all ${\text{R}}\subseteq {\text{Q}}$, ${\text{R}}\in {\text{P}}$). In other words, they are specially-structured sets of what we are used to thinking of as propositions, which will be called truth-conditional propositions in the rest of this text. In other words, "proposition" will refer to propositions in the inquisitive semantics sense unless otherwise specified. On the basis of this idea of a proposition, a semantics for first-order logic can be built, that, if we take truth for a proposition ${\text{P}}$ to be $\left\{{\text{w}}\right\}\in {\text{P}}$, is extensionally equivalent to classical predicate logic.

For an explanation of this semantics, see the book chapter "A first-order inquisitive semantics". Briefly:

• A proposition in the truth-conditional sense (the set of possible worlds where it holds) is lifted into the semantics by taking it to its powerset
• Negation takes a proposition to its complement
• Conjunction is intersection
• Disjunction is union
• Universal quantification is conjunction over all individuals
• Existential quantification is disjunction over all individuals
• (for implication, look at the paper, I don't have a brief explanation for it)

With propositions being downward closed sets, we can always represent them in terms of their maximal elements, which are those elements which are not subsets of any other element. If you think of inquisitive propositions as sets of truth-conditional propositions, the maximal elements are those elements that don't imply any other element. The maximal elements of a proposition are called its alternatives, and are meaningful to inquisitive semantics. A proposition asserts that the real world is in at least one of the (likely overlapping) alternatives, and simultaneously raises an issue which can be settled by any truth-conditional proposition that provides enough information to conclude, for at least one alternative, that the current world is in it. If asked as a question, such a proposition would ask for the issue to be settled, but it could also, for example, be used as an imperative, requesting the issue to be settled one way or another. However, in an abuse of notation, this proposal will occasionally describe propositions as "asking questions" as a convenient shorthand for "raising an issue which would be settled by any truth-conditional proposition that would give enough information to establish at least one alternative as containing the current world".

An important thing to bear in mind about inquisitive semantics is that it does not give us a Boolean algebra, but only a Heyting algebra, meaning that $\neg \neg {\text{P}}={\text{P}}$ is not guaranteed. In fact, the non-inquisitive projection operator $\lambda {\text{P}}.!{\text{P}}$, which collapses all the alternatives of a proposition into just one (called ${\text{info}}\left({\text{P}}\right)$) which contains them all, thus keeping the assertion the same but ensuring that no question is asked, is just $\lambda {\text{P}}.\neg \neg {\text{P}}$. Another important operator is the non-informative projection operator $\lambda {\text{P}}.?{\text{P}}=\lambda {\text{P}}.{\text{P}}\lor \neg {\text{P}}$, which ensures that a proposition does not assert anything by adding an alternative which covers all worlds that would otherwise have been ruled out. Finally (TODO source! well tbf I made it up initially but it is attested, at least in one presentation by one of the inqsem guys) there's the presupposition operator $\lambda {\text{P}}.;{\text{P}}=\lambda {\text{P}}.\left({\text{P}}|{\text{info}}\left({\text{P}}\right)\right)$.

TODO explain how adding presupposition to InqB would work in more detail? Since this proposal is, after all, suggesting that we base the formal language of our semantics on InqB+presupposition.

Examples

$?{\text{P}}$ does not assert anything, but raises the issue of whether ${\text{P}}$ is the case.

$;\left({\text{P}}\lor {\text{Q}}\right)$ asks for enough information to conclude, for one of ${\text{P}}$ and ${\text{Q}}$, 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 "${\text{P}}$" and "${\text{Q}}$", and the maximally informative answers are "only ${\text{P}}$", "only ${\text{Q}}$", and "both ${\text{P}}$ and ${\text{Q}}$".

$?\left({\text{P}}\lor {\text{Q}}\right)$ asks for enough information to conclude either that ${\text{P}}$ and ${\text{Q}}$ are both false, or that for one of ${\text{P}}$ and ${\text{Q}}$, it is true. The minimally informative answers are "${\text{P}}$", "${\text{Q}}$", and "neither".

$?!\left({\text{P}}\lor {\text{Q}}\right)$ asks only for enough information to conclude either that ${\text{P}}$ and ${\text{Q}}$ are both false, or that at least one of ${\text{P}}$ and ${\text{Q}}$ is true. In other words, it asks "Is it true that ${\text{P}}$-or-${\text{Q}}$?". The minimally informative answers are "yes" and "no".

${\text{P}}\lor {\text{Q}}$ as a question asserts that at least one of ${\text{P}}$ and ${\text{Q}}$ is the case, and asks for enough information to conclude, for one of them, that it is true.

$\left({\text{P}}\lor {\text{Q}}\right)\land {}?{\text{P}}\land {}?{\text{Q}}$ as a question asserts the same, but asks for more information. It raises an issue which can only be settled by fully establishing which of the 3 cases is true: only ${\text{P}}$, only ${\text{Q}}$, or both? In other words, those are the three minimally informative answers.

$!\left({\text{P}}\lor {\text{Q}}\right)$ only asserts that at least one of ${\text{P}}$ and ${\text{Q}}$ is the case, without raising any issue.

$\forall x.?{\text{P}}\left(x\right)$ does not assert anything, and asks, for each $x$, whether ${\text{P}}\left(x\right)$ is the case or not.

$\exists x.{\text{P}}\left(x\right)$ asserts that that an $x$ exists such that ${\text{P}}\left(x\right)$, and raises an issue which can be settled by the proposition ${\text{P}}\left(x\right)$ for any particular $x$. As a question, it thus asks for enough information to identify at least one such $x$.

$?\exists x.{\text{P}}\left(x\right)$ does not assert anything, and asks either to be able to conclude that no $x$ exists which satisfies ${\text{P}}\left(x\right)$, or to be able to identify at least one $x$ which does.

$!\exists x.{\text{P}}\left(x\right)$ is only the assertion of existence.

Proposal aims (as used during the development)

Note: this is an ordered list, in order of decreasing priority.

1. Preserve "declarative Toaq" fully.

No non-question Toaq sentence should have its meaning altered.

2. Minimal changes to existing question grammar

Ideally none, depending on my comfort level (and that of the community when I ask!) with new mandatory grammar words vs adding implicit behavior. Ideally I can find default behavior that is simultaneously least informative and least inquisitive, allowing Toaq users to add it on demand, while never accidentally implying something they didn't mean to nor asking for information they didn't necessarily require.

3. Opening up new possibilities for communication

Propositions like ${\text{P}}\lor {\text{Q}}$ and ${\text{P}}\to {}?{\text{Q}}$ ("would ${\text{Q}}$ necessarily be true if ${\text{P}}$ were?") are useful-seeming, delightful, and intriguing to me, and I want to preserve as many of them in the semantics of Toaq clauses as possible, albeit only to such an extent that Toaq remains a language learnable, understandable, and usable by humans.

4. Simplicity

A Heyting algebra as opposed to a Boolean one means a lot of new no longer equivalent logic constructions, among these a whole host of possible connectives. Let's aim for the smaller side when picking a subset of those that must nevertheless not only be sufficient for natural communication, but allow expressing every possible statement without too much hassle. NAND is functionally complete on its own for Boolean logic but you don't want to speak a language whose only connective is NAND.

TODO proposal details

Lift Toaq's current first-order truth-conditional semantics into first-order inquisitive semantics. This is pretty easy: take truth-conditional propositions to their powersets, and to presuppose a proposition ${\text{P}}$, presuppose ${\text{info}}\left({\text{P}}\right)$.

The complementizer ma will gain semantics, specifically $\lambda {\text{P}}.?!P$.

We'll probably want prefixes that apply to quantifiers and apply one of the following to the result of applying the quantifier (in other words, applying them outside the quantifier): $!$, $;$, $?$, $?!$. Maybe we can get by with just putting those words into $\Sigma$ (in addition to being prefixes), or maybe we'll want a $\Sigma _{?}$ phrase with scope outside of quantifiers. TODO TBD. We could introduce a new (unprefixed, single-syllable, so as to be short and easy to use) determiner $;\exists$ which would be like hú or ké without having to decide if you want it to be endophoric or exophoric. Maybe it would be enough to just have determiners $\exists$, $;\exists$, $?\exists$, and $!\exists$, and we wouldn't need any prefixes? Finally, the hí problem: if we want to remain backwards compatible with existing Toaq Delta wh-questions, we must assign it a meaning. Unfortunately, whether or not wh-questions generate presuppositions (in our notation, whether hí means $;\exists$ or $?\exists$) is a highly contested issue that we would be forced to pick a side on...

We might also want new or different connectives, or at least to give specific denotations for them. For example, we probably want ró to create two alternatives, which means we'd want it to be $\left(P\land \neg {\text{Q}}\right)\lor \left(\neg {\text{P}}\land {}{\text{Q}}\right)$. Other potentially quite useful connectives are (sometimes writing them out in a more verbose way than necessary to make their alternatives clearer):

• $?{\text{P}}\land {}?{\text{Q}}$ (one or the other or both or neither)
• $\left(P\land {\text{Q}}\right)\lor \left(P\land \neg {\text{Q}}\right)\lor \left(\neg {\text{P}}\land {\text{Q}}\right)$ (one or the other or both)
• $\left(P\land \neg {\text{Q}}\right)\lor \left(\neg {\text{P}}\land {\text{Q}}\right)\lor \left(\neg {\text{P}}\land \neg {\text{Q}}\right)$ (one or the other or neither)

And so many more, this is rich bikeshedding territory and luckily more can easily be added later.

Another concern is rí which is weird in the refgram, allowing you to answer with a connective!? I think we should fix that one up somehow. We almost certainly need a denotation that generates a non-informative proposition and doesn't presuppose anything, but there are many of those. Two reasonably intuitive options would be $?\left({\text{P}}\lor {\text{Q}}\right)$ and $?{\text{P}}\land {}?{\text{Q}}$.