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Suppose I have a truth tree for a modal logic statement that is closed or is open but for which I cannot continue drawing worlds, does that mean that there are no other accessible worlds in that modal logic for that statement?

I assume there are no other worlds, but perhaps my view on the accessible relation is limited by my use of truth trees.

This question is related to another one I had about a particular statement in modal system K where I was not sure if I could find a counterexample given a completed, but open truth tree: What is the counterexample in modal system K for "⬜A ➡A"?

I am interested in references providing an answer so I could quote them later and get more detailed information on this question.

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  • Not very clear... The completeness theorem for truth trees is based on the lemma that an open branch that is "finished" (i.e. no more rules are applicable) is satisfiable. Amd a closed branch (i.e. one with a contradiction) is obviously unsatisfiable. – Mauro ALLEGRANZA Nov 18 '18 at 16:24
  • "no other accessible worlds in that modal logic for that statement" - accessibility is a relation between worlds (e.g. w17 is accessible from w12), not between statements and worlds. – Eliran Nov 18 '18 at 17:19
  • @MauroALLEGRANZA Would you have a reference for the completeness theorem for truth trees. I will search for it in the mean time. – Frank Hubeny Nov 18 '18 at 17:41
  • @Eliran I agree that the accessibility relation is between worlds, but the truth tree is built on a statement and different statements have different trees. That is why I am associating this with a statement. – Frank Hubeny Nov 18 '18 at 17:43
  • Classical logic : R.Smullyan, First order logic. – Mauro ALLEGRANZA Nov 18 '18 at 17:43
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Suppose I have a truth tree for a modal logic statement that is closed or is open but for which I cannot continue drawing worlds, does that mean that there are no other accessible worlds in that modal logic for that statement?

No, it merely means that you can not infer that there are any when constructing a proof tree.   Don't confuse a proof tree with a Frame.   A Frame is a particular set of worlds and the accessiblity relation between them.   A proof tree is simply a tool to determine if a set of statements could be satisfiable in a given system.

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  • That makes sense, but I need a reference for this question, preferably one online so I can easily access it and get more information. – Frank Hubeny Nov 19 '18 at 11:42

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