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nLab > Latest Changes: Kripke frame

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  1. Clarify by replacing nonsensical “n-unary” and “n-binary”.

    Mark John Hopkins

    diff, v22, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2023

    I have touched wording, hyperlinking and formatting throughout the entry.

    But this entry is still lacking content. E.g. it defines a Kripke frame to be exactly a binary relation and then leaves it at that.

    In particular the comments in the References-section seem out of place until there is some content here that could possibly be erroneous.

    diff, v23, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 22nd 2023

    Right, something of a concept with an attitude. Perhaps the entry could make clearer that Kripke frames are a component part of Kripke models, as described at geometric model for modal logics (contrasted with algebraic model for modal logics).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2023

    The entry geometric model for modal logics is only marginally better: Who spots the connection between the modal operator and those relations (hidden in the fifth item of “Satisfaction”, using undeclared notation) without which there is no meaning to the definitions.

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMay 22nd 2023

    Adding reference

    • Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber (2013). Kripke’s Worlds: An Introduction to Modal Logics via Tableaux. Springer. ISBN 978-3764385033. (doi:10.1007/978-3-7643-8504-0)

    diff, v24, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 23rd 2023

    Have added something to the introduction, along the lines of #3.

    diff, v25, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2023

    But the entry still just declares the bare concept and not its actual attitude.

    Why not add a line connecting the modal operator to the relation. Without that discussed, it seems all pointless.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 23rd 2023

    Better now?

    diff, v27, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2023

    The definition in the entry says essentially:

    “Given a modal operator, then a Kripke frame is a binary relation RR. Period.”

    This is not contentful. For it to be contentful, there needs to be a statement that refers back to the modal operator in the assumption.

    Otherwise the content of the entry is logically equivalent to: “Given a pink elephant, a Kripke frame is a binary relation RR.”

    So at the very least it needs to say something like this:

    “Given a modal operator ◊\lozenge, then a Kripke frame is a set equipped with a binary relation RR, where R(w,v)R(w,v) is interpreted as asserting that ◊ϕ\lozenge \phi holds in world w∈Ww \in W if ϕ\phi holds in world v∈Wv \in W.”

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 23rd 2023

    So that needs changing. A Kripke frame isn’t defined relative to a given modal operator. It just is a non-empty set with a binary relation. A Kripke model then adds to a frame a valuation, a map from propositional variables to subsets of worlds. Then we can speak of what it means to say of a model that at some given world some modal proposition is true.

    It really is a concept with an attitude, the attitude being how it will be taken up by the concept of a model, and the satisfaction relation relative to a model.

    I’ll see what I can do to make this clear.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 23rd 2023

    It really doesn’t deserve any length this entry, so I’ve pruned it.

    diff, v28, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2023

    Hm, I didn’t think that removing information was the way to go in the face of too little information. :-)

    If it helps, imagine a student asking you for what a Kripke frame is, how would they ever understand it from this entry?

    But never mind, evidently I should go and edit myself instead of trying to make somebody else do it. I’ll try to find the time later, at some point.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 23rd 2023

    Well I think I’ve made it a lot clearer that it’s a subsidiary concept. It only makes proper sense when taken up in the larger notion of a Kripke model. It’s like we’re dwelling on the contents of a page for the carrier set of a group, and you’re saying there must be more to it than the carrier set being a non-empty set.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2023

    Have re-written and expanded this entry.

    Also added the original references:

    diff, v30, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2023
    • (edited Jul 16th 2023)

    added also this reference:

    • Saul A. Kripke, Semantical Considerations on Modal Logic, Acta Philosophical Fennica 16 (1963) 83-94 [pdf]

    diff, v32, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2023

    and this one:

    • Saul A. Kripke, Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi, in The Theory of Models (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [doi:10.1016/B978-0-7204-2233-7.50026-5]

    diff, v32, current

  17. Added explicit definition in terms of coalgebras for the powerset functor (and thus, the definition of morphisms of Kripke frames) and a brief of discussion of duality results.

    Evan Washington

    diff, v39, current

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