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

This page assumes some background in classical 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 classical 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 classical 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.

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. 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 (possibly overlapping) alternatives, and simultaneously asks for enough information to conclude, for at least one alternative, that the real world is in it.

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 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 assert anything by adding an alternative which covers all worlds that would otherwise have been ruled out.

Examples

$?{\text{P}}$ does not assert anything, but asks (for enough information to conclude) whether ${\text{P}}$ is the case.

${\text{P}}\lor {\text{Q}}$ 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. In other words, it asks to know "which one is true?" without the implication that it couldn't be both.

$\left({\text{P}}\lor {\text{Q}}\right)\land {}?{\text{P}}\land {}?{\text{Q}}$ asserts the same, but asks for more information. It asks (for enough information) to be able to conclude exactly which of the 3 cases is true: only ${\text{P}}$, only ${\text{Q}}$, or both?

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

$\forall x.?{\text{P}}\left(x\right)$ 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 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.

TODO proposal details

Lift Toaq's current first-order truth-conditional semantics into first-order inquisitive semantics.

TODO Show how $?$ and $!$ can be used to build a good semantics for questions, such as with $\lambda {\text{P}}.?\exists x.{\text{P}}\left(x\right)$ and $\lambda {\text{P}}.\forall x.?{\text{P}}\left(x\right)$ hí variants for exhaustivity. Do we end up needing the completeness operators? For embedded clauses only or also main clauses?