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I finally read the article Is there a Logic of Imperatives? Conifold showed me and it elicited the question, for me, whether imperative programming is a form of imperative logic at all? The essay took to task the idea that in natural language, there are pure imperative inferences and highlighted how grammatically peculiar many attempted examples are. In fact while reading it, I was stricken by the extremely funny way it made its points, but now anyway I noticed that the grammatical peculiarities that would be present in an ordinary language imperative inference are also present, implicitly, in what I can remember, personally, from times that I used programming languages in some form or other.

It would seem then, that in imperative programming, syntactic warping is just bypassed by fiat, maybe. On the other hand, if computer programming involves a distinct kind of mathematics, and if classical set theory is directly correlated only with propositional logic, then there seems as if there possibly should be a form of set theory involving imperatives. My theory is that erotetic logic (logic of questions) "bridges the gap" here. The epistemic-imperative theory of questions is not completely correct about the reduction of erotetic functions to epistemic imperatives inasmuch as, for example, a prescriptive question such as, "Do this?" would not be a request for knowledge in the same way that it would be if the question directly affixed itself to a proposition about compliance with that imperative, say.

However, there is a fundamental unity of the declarative, imperative, and interrogative syntaxes inasmuch as, at least according to some of the research I've done, those three forms of syntax or sentence type are the only ones universal for known human languages. To reflect this unity in set theory, I have been trying to model the powerset operation in erotetic form as the ability to derive a new question from any set of answers given at some time, such that the answer to this new question cannot be derived from that set of answers.

Are there any existing works along these lines?

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Intuitionism has some of this content, if looked at from an unusual direction. You can see it as a philosophy of mathematics as a whole that reflects proving as a communication process with questions and answers, as opposed to a static configuration of absolute facts. And that includes set-theory.

(One reason one needs such a thing is that the static-information view leads to contradictions like Russell's paradox. And the traditional response seems phony: If all the sets in the world exist together, there is no good reason to imagine that something tentative and basically procedural, like the ZF axioms under classical logic, actually describe what is going on.)

In intuitionistic situations, infinities cannot be seen as completed wholes, but only as querying processes probing a structure as needed. A real number is not seen as a predetermined set of digits but as a freely-flowing sequence that can always provide a new digit -- but only if you ask.

This means that proofs involving infinite sets of real numbers must be pursued in a very different way. It changes the nature of real-analytic proofs from operations on a fixed object with multiple infinite dimensions, taking place in a declarative language into a sort of parallel program with callbacks. Of course, the latter can be reduced to the former in any finite situation (or we would have no formal semantics, which generally reduces processes to first-order logic.) But this does not apply to the infinite case.

And even in the finite case, it is just not the way we choose to look at real, constructive algorithms most of the time outside of mathematics. For most programmers, the world is not a database with preconditions, only the data is. Programs have steps more often than they have rules. Mathematically oriented designers largely think of moves away from this and toward the declarative view as progress, but the recent revival of NoSQL databases and agent-oriented searches like map-reduce worms controverts that view. So from a 'true-believer' point of view, this is just a bias inculcated by exposure to classical logic.

The constructions needed to deal with this kind of restriction effectively can be interesting in-and-of-themselves. You might want to look at the work of Steven Kleene.

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  • I have tried to incorporate Brouwer's notion of a "freely creating subject" into my model. But the way I've approached this is by reinterpreting the Kantian question of transfinite synthesis as a problem about the structure of the entire universe of sets: not that specific infinities are physically unreachable, the problem is that absolute infinity is as such inaccessible, except in free will and deontic information. [Moreover, I think we have a "logical" intuition of ~~A=A, or at least such an intuition is as plausible as Brouwer's primordial duality.] – Kristian Berry Nov 17 '19 at 21:54

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