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Full Idea
A propositional logic sentence is a 'logical consequence' of a set of sentences (written Γ |= φ) if for every admissible truth-assignment all the sentences in the set Γ are true, then φ is true.
Gist of Idea
Γ |= φ for sentences if φ is true when all of Γ is true
Source
José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4)
Book Ref
Zalabardo,José L.: 'Introduction to the Theory of Logic' [Westview 2000], p.51
A Reaction
The definition is similar for predicate logic.