Questions as games

By denoting questions as mathematical games, we can give a nice semantics to questions with hí under a quantifier, like Chuabo tú hí? "What's everyone called?" in Toaq. (We're assuming we don't want that to mean, "which name does everyone share?")

Background

There is an interpretation of logic formulas with ∀ and ∃ as mathematical games, played against a hypothetical opponent whose every move you must have a plan for. A proof of the formula is a strategy for winning at the game.

Consider the theorem: ${\color {red}\forall a\in \mathbb {N} :\color {blue}\exists b\in \mathbb {N} :\color {red}\forall c\in \mathbb {N} :}~~(a-c)c\leq b.$

We can think of this as: "if I hand you an $a$, can you always reply with a $b$ so that no matter what $c$ I then hand you, $(a-c)c\leq b$?" The ∀s are my turns and the ∃s are your turns.

The answer is yes: when handed $a$, we can reply with $b=a^{2}$. This is our "strategy" for winning the game. If you work out the math, you get a proof of the theorem.

The equivalence between logical statements and games is something like:

Logical expression	Rules of the corresponding game
∀x ∈ S: …	Your opponent gets to choose an x from S, and then …
∃x ∈ S: …	You get to choose an x from S, and then …
¬ …	You win if game "…" is unwinnable for you. Otherwise, your opponent wins.
P(x,y,z)	If P(x,y,z) is true, you win. Otherwise, your opponent wins.

Questions

We can imagine a denotation for questions in Toaq that turns them into these mathematical games.

Then answers correspond to strategies that win these games.

For example, ⟦Mıo hí raı⟧ is a game where you win if you can name something that's blue. The strategy of "picking séoq" wins this game, at least during the day.

Asking a question — saying X móq? — means you'd like your interlocutor to describe a strategy that wins at game X. An indirect question like dua jí ꝡä X means "I have a winning strategy for game X".

More interestingly, ⟦Geı tú poq hí raı⟧ is a game where you win if, no matter what person P I throw at you, you can tell me what they're wearing.

Then ⟦Geı tú poq hí raı móq?⟧ is asking the interlocutor to give enough info, like Geı mí Sara sá lena, rú geı mí Jaq sá tıshatı, from which we can derive a strategy to win at that game.

Toaq phrase	The corresponding game
tú hao nä…	The "opponent" gets to choose a hao.
sá hao nä…	The strategy gets to choose a hao, but it need not be named. (?)
hí hao nä…	The strategy must name any (or all) hao.
sentence without quantifiers	If the sentence is now true, you win. Otherwise, your opponent wins.