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

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 (which will unfortunately be necessary to understand this proposal unless or until I repeat some of the exposition here), see the book chapter "A first-order inquisitive semantics". 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 $\lambda {\text{P}}.!{\text{P}}$ is just $\lambda {\text{P}}.\neg \neg {\text{P}}$. Note that this dramatically increases the number of distinct logical connectives (please don't be foreshadowing...).

TODO proposal broad strokes

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

TODO proposal specifics

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?