Barcan formula
In quantified modal logic, the Barcan formula and the converse Barcan formula (more accurately, schemata rather than formulas) (i) syntactically state principles or interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds. The formulas were introduced as axioms by Ruth Barcan Marcus, in the first extensions of modal propositional logic to include quantification.
Related formulas include the Buridan formula, and the converse Buridan formula.
The Barcan formula is:
In English, the schema reads: If everything is necessarily F, then it is necessary that everything is F. It is equivalent to
The Barcan formula has generated some controversy because - in terms of possible world semantics - it implies that all objects which exist in any possible world (accessible to the actual world) exist in the actual world, i.e. that domains cannot grow when one moves to accessible worlds. This thesis is sometimes known as actualism--i.e. that there are no merely possible individuals. There is some debate as to the informal interpretation of the Barcan formula and its converse.
An informal argument against the plausibility of the Barcan formula would be the interpretation of the predicate Fx as "x is a machine that can tap all the energy locked in the waves of the Atlantic Ocean in a practical and efficient way". In its equivalent form above, the antecedent seems plausible since it is at least theoretically possible that such a machine could exist. However, it is not obvious that this implies that there exists a machine that possibly could tap the energy of the Atlantic.
The converse Barcan formula is:
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