Dependence and Independence

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.