Dependence and Independence

Abstract

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.

Approximation and Dependence via Multiteam Semantics

“…First we notice that the known translation of dependence atoms to independence atoms (see Grädel et al [13]) works also in the probabilistic case. The result then follows since A |= (X,m) =(x, y) iff the right-hand side of (3) holds.…”

“…First note that inclusion and dependence atoms can be expressed in FO(⊥ c ) [9,13]. Also it is easy to see that one can construct existential secondorder logic sentences that capture probabilistic inclusion and conditional independence atoms over teams of fixed domain.…”

Multiverse set theory and absolutely undecidable propositions

“…This is reminiscent of the formulax ?ỹ in [12] (see also [7]). The idea is that M 00 picks the truth value of ' from M and the truth value of from M 0 .…”

“…However, the first order logical consequences of dependence logic sentences can be axiomatized and such axioms are given in [14]. There are many other new atomic formulas that suggest themselves in the team semantics context, for example the independence atom [12] x?y with the meaning that x and y are "independent" (the values of x reveal nothing about the values of y), and the inclusion atom [7] x ✓ y with the meaning that values of x occur also as values of y,…”

A Finite Axiomatization of Conditional Independence and Inclusion Dependencies

“…[24,12] Any dependence (or independence) logic sentence φ is logically equivalent to some existential second-order sentence φ * , and vice versa.…”