Editing Method of analytic tableaux (section)

===Formula-deleting tableau===

A method for avoiding formulae referring to different worlds interacting in the wrong way is to make sure that all formulae of a branch refer to the same world. This condition is initially true as all formulae in the set to be checked for consistency are assumed referring to the same world. When expanding a branch, two situations are possible: either the new formulae refer to the same world as the other one in the branch or not. In the first case, the rule is applied normally. In the second case, all formulae of the branch that do not also hold in the new world are deleted from the branch, and possibly added to all other branches that are still relative to the old world.

As an example, in [[S5 (modal logic)|S5]] every formula <math>\Box A</math> that is true in a world is also true in all accessible worlds (that is, in all accessible worlds both <math>A</math> and <math>\Box A</math> are true). Therefore, when applying <math>\frac{\neg \Box B}{\neg B}</math>, whose consequence holds in a different world, one deletes all formulae from the branch, but can keep all formulae <math>\Box A</math>, as these hold in the new world as well. In order to retain completeness, the deleted formulae are then added to all other branches that still refer to the old world.