# If I cannot draw a world in a truth tree in modal logic does that mean that no other worlds exist for that statement?

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.

• 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. Commented Nov 18, 2018 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.
– E...
Commented Nov 18, 2018 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. Commented Nov 18, 2018 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. Commented Nov 18, 2018 at 17:43
• Classical logic : R.Smullyan, First order logic. Commented Nov 18, 2018 at 17:43