This paper presents a language-independent proof system for reachability properties of programs written in non-deterministic (e.g., concurrent) languages, referred to as all-path reachability logic. It derives partial-correctness properties with all-path semantics (a state satisfying a given precondition reaches states satisfying a given postcondition on all terminating execution paths). The proof system takes as axioms any unconditional operational semantics, and is sound (partially correct) and (relatively) complete, independent of the object language. The soundness has also been mechanized in Coq. This approach is implemented in a tool for semantics-based verification as part of the K framework (http://kframework.org)