Tautological consequence

Summary

In propositional logic, tautological consequence is a strict form of logical consequence in which the tautologousness of a proposition is preserved from one line of a proof to the next. Not all logical consequences are tautological consequences. A proposition $Q$ is said to be a tautological consequence of one or more other propositions ($P_1$, $P_2$, ..., $P_n$) in a proof with respect to some logical system if one is validly able to introduce the proposition onto a line of the proof within the rules of the system; and in all cases when each of ($P_1$, $P_2$, ..., $P_n$) are true, the proposition $Q$ also is true.