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Barcan formula

In quantified modal logic, the Barcan formula and the converse Barcan formula state possible relationships between quantifiers and modalities. The Barcan formula is a natural axiom to consider in quantified forms of Clarence Irving Lewis's modal logic S5. It was first discussed by Ruth Barcan Marcus, for whom it is named.

Related formulas include the Buridan formula, and the converse Buridan formula.

The Barcan formula

The Barcan formula is:

$forall x Box Fx rightarrow Box forall x Fx$.

In English, the statement read: If everything is necessarily F, then it is necessary that everything is F. The Barcan formula has generated some controversy because it implies that all objects which exist in every possible world (accessible to the actual world) exist in the actual world. In other words, the domain of any accessible possible world is a subset of the domain of the actual world. This condition on domains is known as anti-monotonicity. (Anti-monotonicity and the Barcan formula are not equivalent in all modal systems.)

Converse Barcan formula

The converse Barcan formula is:

$Box forall x Fx rightarrow forall x Box Fx$.

The formula implies that everything that exists in the actual world exists in all every world (accessible to the actual world). The corresponding condition on domains is called 'monotonicity': the domain of each world is a subset of the domain of world that is accessible to it.

If a frame is based on a symmetric accessibility relation, then the Barcan formula will be valid in the frame if, and only if, the converse Barcan formulas is valid in the frame.

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