Barcan formula
Encyclopedia
In quantified modal logic
, the Barcan formula and the converse Barcan formula (more accurately, schemata rather than formulae) (i) syntactically state principles or interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds. The formulae 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.
.
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 it implies that all objects which exist in every 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.
.
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 formula is valid in the frame. It states that domains cannot shrink as one moves to accessible worlds, i.e. that individuals cannot cease to be possible. The converse Barcan formula is taken to be more plausible than the Barcan formula.
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, the Barcan formula and the converse Barcan formula (more accurately, schemata rather than formulae) (i) syntactically state principles or interchange between quantifiers and modalities; (ii) semantically state a relation between domains of possible worlds. The formulae were introduced as axioms by Ruth Barcan Marcus
Ruth Barcan Marcus
Ruth Barcan Marcus is the American philosopher and logician after whom the Barcan formula is named. She is a pioneering figure in the quantification of modal logic and the theory of direct reference...
, 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
The Barcan formula is:.In English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
, 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 it implies that all objects which exist in every 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
Actualism
In contemporary analytic philosophy, actualism is a position on the ontological status of possible worlds that holds that everything that exists is actual. Another phrasing of the thesis is that the domain of unrestricted quantification ranges over all and only actual existents...
--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.
Converse Barcan formula
The converse Barcan formula is:.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 formula is valid in the frame. It states that domains cannot shrink as one moves to accessible worlds, i.e. that individuals cannot cease to be possible. The converse Barcan formula is taken to be more plausible than the Barcan formula.
External links
- Barcan both ways by Melvin Fitting
- Contingent Objects and the Barcan Formula by Hayaki Reina