Studia Logica

, Volume 101, Issue 2, pp 399–410

# Dependence and Independence

• Erich Grädel
• Jouko Väänänen
Article

## Abstract

We introduce an atomic formula $${\vec{y} \bot_{\vec{x}}\vec{z}}$$ intuitively saying that the variables $${\vec{y}}$$ are independent from the variables$${\vec{z}}$$ if the variables $${\vec{x}}$$ are kept constant. We contrast this with dependence logic $${\mathcal{D}}$$ based on the atomic formula =$${(\vec{x}, \vec{y})}$$ , actually equivalent to $${\vec{y} \bot_{\vec{x}}\vec{y}}$$ , saying that the variables $${\vec{y}}$$ are totally determined by the variables $${\vec{x}}$$ . We show that $${\vec{y} \bot_{\vec{x}}\vec{z}}$$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that $${\vec{y} \bot_{\vec{x}}\vec{z}}$$ 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 =$${(\vec{x}, \vec{y})}$$ have.

## Keywords

Logics of dependence and independence Team semantics Logics with imperfectinformation Axiomatization of independence

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