Logical formalisms equivalent to weighted automata have been the topic of numerous research papers in the recent years. It started with the seminal result by Droste and Gastin on weighted logics over semir-ings for words. It has been extended in two dimensions by many authors. First, the weight domain has been extended to valuation monoids, valuation structures, etc., to capture more quantitative properties. Along another dimension, different structures such as ranked or unranked trees, nested words, Mazurkiewiz traces, etc., have been considered. The long and involved proofs of equivalences in all these papers are implicitely based on the same core arguments. This article provides a meta-theorem which unifies these different approaches. Towards this, we first introduce a core weighted logic with a minimal number of features and a simplified syntax. Then, we define a new semantics for weighted automata and weighted logics in two phases—an abstract semantics based on multisets of weight structures (independent of particular weight domains) followed by a concrete semantics. We show at the level of the abstract semantics that weighted automata and core weighted logic have the same expressive power. We show how previous results can be recovered from our result by logical reasoning. In this paper, we prove the meta-theorem for words, ranked and unranked trees, showing the robustness of our approach.