In quantified
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
, the Barcan formula and the converse Barcan formula (more accurately, schemata rather than formulas) (i) syntactically state principles of 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
Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quant ...
, in the first extensions of modal propositional logic to include quantification.
[Journal of Symbolic Logic (1946),11 and (1947), 12 under Ruth C. Barcan]
Related formulas include the
Buridan formula.
The Barcan formula
The Barcan formula is:
:
.
In
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
, the schema reads: If every x is necessarily F, then it is necessary that every x 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
In analytic philosophy, actualism is the view that everything there ''is'' (i.e., everything that has ''being'', in the broadest sense) is actual. Another phrasing of the thesis is that the domain of unrestricted quantification ranges over al ...
—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.
Converse Barcan formula
The converse Barcan formula is:
:
.
It is equivalent to
:
.
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 exist. The converse Barcan formula is taken to be more plausible than the Barcan formula.
See also
*
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
References
External links
''Barcan both ways''by Melvin Fitting
''Contingent Objects and the Barcan Formula''by Hayaki Reina
{{logic-stub
Modal logic