The implication (if $P$ then $Q$) is true IFF the set of assignments (propositional, first order, whatever depending on the setting) which satisfy $P$ is a subset of the set of assignments which satisfy $Q$.
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The implication (if $P$ then $Q$) is true IFF the set of assignments (propositional, first order, whatever depending on the setting) which satisfy $P$ is a subset of the set of assignments which satisfy $Q$.