Dependence and Independence

• Published in: Studia logica Vol. 101; no. 2; pp. 399 - 410
• Main Authors: Grädel, Erich, Väänänen, Jouko
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer 01-04-2013; Springer Netherlands
• Subjects: Atoms; Computational Linguistics; Dependence logics; Education; Imperfect information; Intuition; Logic; Logical theorems; Mathematical constants; Mathematical Logic and Foundations; Philosophy; Predicates; Semantics; Team semantics; Axiomatization of independence; Logics of dependence and independence; Logics with imperfectinformation
• ISSN: 0039-3215; 1572-8730
• DOI: 10.1007/s11225-013-9479-2

We introduce an atomic formula $\overrightarrow{\mathrm{y}}{\perp }_{\overrightarrow{\mathrm{x}}}\overrightarrow{\mathrm{z}}$ intuitively saying that the variables $\overrightarrow{\mathrm{y}}$ are independent from the variables $\overrightarrow{\mathrm{z}}$ if the variables $\overrightarrow{\mathrm{x}}$ are kept constant. We contrast this with dependence logic 𝓓 based on the atomic formula $=(\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{y}})$ , actually equivalent to $\overrightarrow{\mathrm{y}}{\perp }_{\overrightarrow{\mathrm{x}}}\overrightarrow{\mathrm{y}}$ , saying that the variables $\overrightarrow{\mathrm{y}}$ are totally determined by the variables $\overrightarrow{\mathrm{x}}$ . We show that $\overrightarrow{\mathrm{y}}{\perp }_{\overrightarrow{\mathrm{x}}}\overrightarrow{\mathrm{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that $\overrightarrow{\mathrm{y}}{\perp }_{\overrightarrow{\mathrm{x}}}\overrightarrow{\mathrm{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using $=(\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{y}})$ have.