Dependence and Independence

摘要

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.