Barcan formula

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. ^[1]

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 schema reads: If everything is necessarily F, then it is necessary that everything is F. It is equivalent to

$\Diamond\exists xFx\to\exists x\Diamond Fx$.

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.

Converse Barcan formula

The converse Barcan formula is:

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

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.

References

1. ^ Journal of Symbolic Logic (1946),11 and (1947), 12 under Ruth C. Barcan

External links

• Barcan both ways by Melvin Fitting
• Contingent Objects and the Barcan Formula by Hayaki Reina

This logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Logic stubs Modal logic Wikimedia Foundation. 2010. Игры ⚽ Нужно решить контрольную? Leyte Chess prodigy Look at other dictionaries: Barcan formula — A fundamental thesis in quantified modal logic, first isolated by the 20th century American philosopher Ruth Barcan Marcus. It was originally the schema that ⋄(∃x )A x strictly implies (∃x )⋄Ax (informally: if possibly something is A, then… …   Philosophy dictionary Ruth Barcan Marcus — (born 1921 in Bronx, New York) 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. She has written seminal papers on… …   Wikipedia MARCUS, RUTH BARCAN — (1921– ), U.S. logician and philosopher who played a key role in many of the philosophical debates of the second half of the 20th century. Born and educated in New York City, Ruth Barcan received her B.A. in mathematics and philosophy from New… …   Encyclopedia of Judaism List of topics in logic — This is a list of topics in logic.See also: List of mathematical logic topicsAlphabetical listAAbacus logic Abduction (logic) Abductive validation Affine logic Affirming the antecedent Affirming the consequent Antecedent Antinomy Argument form… …   Wikipedia Max Cresswell — Max John Cresswell (* 19. November 1939 in Wellington) ist ein neuseeländischer Philosoph und Logiker, der sich insbesondere mit der Philosophie der Logik, Modallogik und formaler Semantik beschäftigt. Daneben hat Cressell auch zur Philosophie… …   Deutsch Wikipedia De dicto et de re — sont deux locutions distinguant deux modalités importantes des énoncés, ainsi que les raisonnements qui s ensuivent. Ces distinctions relèvent de la logique, de la rhétorique, de la linguistique et de la métaphysique. La traduction littérale de… …   Wikipédia en Français List of philosophy topics (A-C) — 110th century philosophy 11th century philosophy 12th century philosophy 13th century philosophy 14th century philosophy 15th century philosophy 16th century philosophy 17th century philosophy 18th century philosophy 19th century philosophy220th… …   Wikipedia De dicto and de re — are two phrases used to mark important distinctions in intensional statements, associated with the intensional operators in many such statements. The distinctions are most recognized in philosophy of language and metaphysics. The literal… …   Wikipedia 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… …   Wikipedia Tautología — Para otros usos de este término, véase Tautología (retórica). En lógica, una tautología (del griego ταυτολογία, decir lo mismo ) es una fórmula bien formada de un sistema de lógica proposicional que resulta verdadera para cualquier… …   Wikipedia Español 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”