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Give a brief explanation of inquisitive semantics

Magnap

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 In inquisitive semantics, rather than being subsets of all possible worlds (of type <math>\left\langle \text{s}, \text{t} \right\rangle</math>), propositions are taken to be nonempty ''sets'' of subsets of all possible worlds (of type <math>\left\langle \left\langle \text{s}, \text{t} \right\rangle, \text{t} \right\rangle</math>), downward closed under containment (meaning that, for some proposition <math>\text{P}</math>, if <math>\text{Q} \in \text{P}</math>, then for all <math>\text{R} \subseteq \text{Q}</math>, <math>\text{R} \in \text{P}</math>). 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 <math>\text{P}</math> to be <math>\left\{\text{w}\right\} \in \text{P}</math>, 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 [https://doi.org/10.1093/oso/9780198814788.003.0004 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 <math>\neg\neg\text{P} = \text{P}</math> is not guaranteed. In fact, the non-inquisitive projection operator <math>\lambda \text{P}. !\text{P}</math> ''is'' just <math>\lambda \text{P}. \neg\neg\text{P}</math>. Note that this dramatically increases the number of distinct logical connectives (''please don't be foreshadowing...'').
+For an explanation of this semantics, see [https://doi.org/10.1093/oso/9780198814788.003.0004 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 <math>\neg\neg\text{P} = \text{P}</math> is not guaranteed. In fact, the non-inquisitive projection operator <math>\lambda \text{P}. !\text{P}</math>, 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 <math>\lambda \text{P}. \neg\neg\text{P}</math>. Another important operator is the non-informative projection operator <math>\lambda \text{P}. ?\text{P} = \lambda \text{P}. \text{P} \lor \neg\text{P}</math>, which ensures that a proposition does assert anything by adding an alternative which covers all worlds that would otherwise have been ruled out.
+===Examples===
+<math>?\text{P}</math> does not assert anything, but asks (for enough information to conclude) whether <math>\text{P}</math> is the case.
+<math>\text{P} \lor \text{Q}</math> asserts that at least one of <math>\text{P}</math> and <math>\text{Q}</math> 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.
+<math>\left(\text{P} \lor \text{Q}\right) \land{} ?\text{P} \land{} ?\text{Q}</math> 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 <math>\text{P}</math>, only <math>\text{Q}</math>, or both?
+<math>!\left(\text{P} \lor \text{Q}\right)</math> ''only'' asserts that at least one of <math>\text{P}</math> and <math>\text{Q}</math> is the case, without asking anything.
+<math>\forall x. ?\text{P}\left(x\right)</math> does not assert anything, and asks, for each <math>x</math>, whether <math>\text{P}\left(x\right)</math> is the case or not.
+<math>\exists x. \text{P}\left(x\right)</math> asserts that that an <math>x</math> exists such that <math>\text{P}\left(x\right)</math>, and asks for enough information to identify at least one such <math>x</math>.
+<math>?\exists x. \text{P}\left(x\right)</math> does not assert anything, and asks either to be able to conclude that no <math>x</math> exists which satisfies <math>\text{P}\left(x\right)</math>, or to be able to identify at least one <math>x</math> which does.
+<math>!\exists x. \text{P}\left(x\right)</math> is only the assertion of existence.
 ==TODO proposal broad strokes==