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User:Magnap/Inquisitive Semantics Proposal

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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 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 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 \lor} , 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} , 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.

Inquisitive semantics

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 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} \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 not 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 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} , 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}\left(x\right)} is the case or not.

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)} asserts that that an 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 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.

Proposal aims (not final)

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 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}} 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{P} \to{} ?\text{Q}} ("would 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}} necessarily be true 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{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 (I predict) will be doable with a minimum of hassle.

TODO 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?

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