 Independence logic is strictly more expressive than dependence logic: indeed, it lacks the downwards closure property, and the dependence atom \(\eqord(\vec x, y)\) is equivalent to the independence atom \(y \mathop{\bot_{\vec x}} y\). Furthermore, it can also be shown (Galliani &amp; Väänänen 2014) that conditioned independence atoms \(\vec x \mathop{\bot_{\vec y}} \vec z\) can be defined in terms of unconditional independence atoms \(\vec x \mathop{\bot} \vec y\).