Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in literature a same simple formalization of interrogative affirmations, that is questions, as a universal algebra. There is a pretty large domain of research, erotetic logic, but I did not encounter a simple "algebra of questions", same simple as Boolean algebra. This sounds strange to me due to the argument below.

In one of his publications in computer science, Donald Knuth treated questions as ideals of a Boolean algebra - a treatment which sounds very natural. Namely, he treated a question $Q$ as the set of all answers to it, and such answers make up an ideal. Really, for any two propositions $X$ and $Y$ the following two conditions are satisfied: (a) if $X$ implies $Y$, and $Y$ replies to question $Q$, then it is natural to consider $X$ also an anwer to $Q$. (b) if both $X$ and $Y$ reply to $Q$, then also $ X\lor Y$ reply to $Q$. This leads to the idea that questions make up a special universal algebra correlated with a boolean algebra and Stone theorem must play a role in defining this algebra and this correlation. Due its strong correlation with Boolean algebra, this univeral algebra must play same key role in algebra as the Boolean algebra, and not only serve the role of formalizing the questions.

  • Is there any research of an "algebra of questions" treated as ideals of a Boolean algebra?

  • What kind of universal algebra make up the ideals of a Boolean algebra?

  • If such universal algebras are defined, is there a theorem "inverse" to Stone representation theorem, which allow for such an algebra $A$ to build a Boolean algebra $B$ "of replies to questions from $A$"?

I am aware that some of my questions sound naive.

  • $\begingroup$ well, the ideals of a Boolean algebra form a base for the corresponding Stone space. Next, the ideals form an algebraic directed-complete poset, or an algebraic lattice if we admit the trivial ideal. It's not realy clear what you are looking for, but these observations are fairly trivial. $\endgroup$ – Andrej Bauer Jun 26 '14 at 7:10
  • $\begingroup$ This looks very similar to what is often known as "Medvedev's logic of finite problems" in the context of intermediate logics. $\endgroup$ – François G. Dorais Jul 26 '14 at 16:00
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    $\begingroup$ Ioachim, could you say more about your erotetic logic? $\endgroup$ – Włodzimierz Holsztyński Aug 25 '14 at 1:42
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    $\begingroup$ @Andrej Actually ideals form not just a base, the lattice of ideals is isomorphic to the lattice of all open sets of the Stone space. $\endgroup$ – მამუკა ჯიბლაძე Oct 24 '14 at 5:55
  • $\begingroup$ As მამუკა ჯიბლაძე says, the ideals of a Boolean algebra correspond to the open subsets of the Stone space. Thus, like the open subsets of any topological space, they form a complete Heyting algebra. $\endgroup$ – Andreas Blass Oct 24 '14 at 13:53

I'm not certain exactly what you would consider a "question," but here's a few things that spring to mind:

  • Notions in intuitionistic and modal logic, coming from Kolmogorov, Goedel, Kleene, and presumably others: Kolmogorov's logic of problems, Goedel's modal logic of provability, and Kleene's notion of realizability.

  • Around computability and reverse mathematics, mass problems and (the more specialized, and probably less relevant, but still interesting) degrees of provability.

Presumably you're interested in the algebraic side of things, about which I know less, but I believe that these have been studied algebraically to varying degrees. (I'll add references when I have a bit more time, in a few hours.)

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