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In software languages, we have a small set of terms to indicate a rule: if, then, else, upon, while, otherwise, case, etc.

In specification writing, we use terms like: "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL"

In contract law, you'll see: "have to", "is mandatory", "is required", etc.

Question: Has there been any formal analysis on the 'language of rules'?

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Sentence logic, aka propositional logic, studies sentence connectives And, Or, Implies, Not, and for modal logic, Necessarily, Possibly. Is that what you mean by "rule"?

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    Yes. Unfortunately, I'm finding silo solutions (modals only, first-order-logic only, ...) and not a general treatise on the subject. Trying to see if anyone has holistically analyzed the silo solutions. May 10 '19 at 16:00
  • I don't know what "silo solution" means. There are loads of textbooks on logic. It's a big subject. The generalization to quantifiers is called "predicate logic", and first order logic refers to a predicate logic with quantification over individuals but not over predicates. I don't know how to interpret what you want in a "holistic" treatment.
    – Greg Lee
    May 10 '19 at 21:28
  • Yes, and many of the books are sitting on my desk (each representing a small portion of the problem); hence, I'm asking if anyone has attempted to create a comprehensive framework for identifying logic in natural language. Such an analysis would be able to reverse engineer anything from a legal contract, to the rules of tennis, to the rules for posting on Twitter, ... to the rules of X. May 10 '19 at 22:42
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    One of the reasons why the literature is restricted to first-order predicate calculus is because second-order (which would quantify over predicates as well as arguments, and is clearly involved in human cognition and generalization) has been shown to be logically inconsistent. Since formal logic is just a stick-figure model of human cognition whose value lies in its automaticity and consistency, this is not useful.
    – jlawler
    May 11 '19 at 14:11
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    @jlawler, first order predicate logic has been shown to be complete and consistent (by Gödel), but higher order logics cannot be shown to be both complete and consistent (also shown by Gödel). But this is not to say that higher order logics are inconsistent.
    – Greg Lee
    May 11 '19 at 14:27

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