Dependence and Independence

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