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I mean this as a linguistic question, not a cultural one. Operators in modal logic, like must and should exist in English, and many other languages -- but not in all of them. I mean: Modality is not the same across all languages.

So what does this mean, if anything, for the adequacy of modal logic and other formalisms?

  • This sounds like a kind of claim made by social constructionists, but it is hard to tell what they mean specifically in this case without looking at the reference where you encountered the claim. Could you include it into the post? – Conifold Aug 11 '17 at 19:00
  • Neither is the notion of inanimate plurals. Chinese has them only on personal nouns and pronouns, and not on inanimate nouns or on verbs. Does that make the notion of multiplicity of elements in a set somehow unhelpful in formal systems? I can't catch your drift either. – user9166 Aug 11 '17 at 22:39
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    Modality is not the same across all modal logics, either. Nor are there clean and neat correspondences between all natural language modals and particular modal logics. Even the logical rendition of English language modals is varied and difficult. – Dennis Aug 12 '17 at 2:16
  • I find it hard to believe that there are some human languages that forego modality, do you have any examples of this? – Mozibur Ullah Aug 12 '17 at 4:45
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    @MoziburUllah : indeed, all languages have modality of some kind (so far as I know), but not all modal operators in a given language have the same semantic scope across languages. Some languages, for example have more than one kind of "must", others conflate "should" with "must" and so on... – Teusz Aug 12 '17 at 9:45
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You ask what the implications are for formal modal logic of there being differences among languages in what modal terms they have — or differences with respect to terms we might treat as modal, since it depends on how you define modality.

There are no such implications, because the question mixes together two things which logicians keep separate. Terms like "must" and "should" are not operators in modal logic. The strong and weak modal operators of typical modal logic are defined formally, in relation to other symbols. "Must" and "should" are part of interpretations sometimes assigned to these formal operators for particular purposes. It is only in interpretations that the modal operators are connected with words in natural languages. And no interpretation is part of the formal definition of the modal operators.

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    Interpretation may not be a part of formal definitions but it is certainly decisive in what formal definitions are made, and what axioms they enter. Formal logic is after all meant to model or improve on natural reasoning. Kripke's justification of modal semantics, for example, explicitly relies on "modal intuitions" which he takes to be general and reality based. If they turn out to be Eurocentric or in some other way non-objective there would be serious implications for formal modal logic, it seems. Quine, who thought that "modal intuitions" are misguided, described them rather dramatically. – Conifold Aug 11 '17 at 20:33
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    @Conifold The claim about the purpose of formal logic is debatable and debated. Everything following that concerns interpretations. Modal semantics is a function of interpretation. – ChristopherE Aug 12 '17 at 1:00
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    @Conifold subsequent work on modal semantics has moved beyond and generalized Kripke semantics in a way that makes his original intuition largely irrelevant. Subsequent work on general frames, metaframes, fibre bundles, etc., have been motivated largely by desire to achieve certain formal properties like completeness with respect to a characterizable class of frames and study of duality. – Dennis Aug 12 '17 at 2:19
  • Isn't this exactly the problem? As Beziau pointed out, if modal logic is a formal study of unary operators then "modal logic includes paraconsistent logic, since negations are unary operators. Nobody seems to be aware of this fact and this shows very well that most of the people working in modal logic do not really think about the interpretation of the modal operators." The practical result is the de facto fallback on Kripke's metaphysics. And in turn, dissatisfaction with it motivated recent upsurge of alternative modal logics. – Conifold Aug 12 '17 at 23:08

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