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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \text{s}, \text{t} \right\rangle} ), propositions are taken to be nonempty sets of subsets of all possible worlds (of type Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \left\langle \text{s}, \text{t} \right\rangle, \text{t} \right\rangle} ), downward closed under containment (meaning that, for some proposition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}} , if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Q} \in \text{P}} , then for all ${\text{R}}\subseteq {\text{Q}}$ , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}} to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg\neg\text{P} = \text{P}} is not guaranteed. In fact, the non-inquisitive projection operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \text{P}. \neg\neg\text{P}} . Another important operator is the non-informative projection operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ?\text{P}} does not assert anything, but asks (for enough information to conclude) whether Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}} is the case.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P} \lor \text{Q}} asserts that at least one of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}} , only Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Q}} , or both?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle !\left(\text{P} \lor \text{Q}\right)} only asserts that at least one of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Q}} is the case, without asking anything.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}\left(x\right)} , and asks for enough information to identify at least one such Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ?\exists x. \text{P}\left(x\right)} does not assert anything, and asks either to be able to conclude that no Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} exists which satisfies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{P}\left(x\right)} , or to be able to identify at least one Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} which does.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle !\exists x. \text{P}\left(x\right)} is only the assertion of existence.

TODO proposal broad strokes

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

TODO proposal specifics

Show how Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ?} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle !} can be used to build a good semantics for questions, such as with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \text{P}. ?\exists x. \text{P}\left(x\right)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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?