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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.
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).
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.
Adding reference
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.
The definition in the entry says essentially:
“Given a modal operator, then a Kripke frame is a binary relation $R$. 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 $R$.”
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 $R$, where $R(w,v)$ is interpreted as asserting that $\lozenge \phi$ holds in world $w \in W$ if $\phi$ holds in world $v \in W$.”
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.
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.
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.
Have re-written and expanded this entry.
Also added the original references:
Saul A. Kripke, A Completeness Theorem in Modal Logic, The Journal of Symbolic Logic 24 1 (1959) 1-14 [doi:10.2307/2964568, jstor:2964568, pdf]
Saul A. Kripke, Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi, Mathematical Logic Quaterly 9 5-6 (1963) 67-96 [doi:10.1002/malq.19630090502]
added also this reference:
and this one:
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