Full Idea
Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'.
Gist of Idea
Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true
Source
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
A Reaction
We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another.