Dependence and Independence

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.