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 [7] 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.Of the numerous uses of the word "dependence" we focus on the concept of an attribute 1 depending on a number of other similar attributes when we observe the world. We call these attributes variables. We follow the approach of [7] and focus on the strongest form of dependence, namely functional dependence.
In this paper, we study logics of dependence on the propositional level. We prove that several interesting propositional logics of dependence, including propositional dependence logic, propositional intuitionistic dependence logic as well as propositional inquisitive logic, are expressively complete and have disjunctive or conjunctive normal forms. We provide deduction systems and prove the completeness theorems for these logics.
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