Tautological consequence

In propositional logic, tautological consequence is a strict form of logical consequence[1] 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 ${\displaystyle Q}$ is said to be a tautological consequence of one or more other propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) 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 those one or more other propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) are true, the proposition ${\displaystyle Q}$ also is true.

Another way to express this preservation of tautologousness is by using truth tables. A proposition ${\displaystyle Q}$ is said to be a tautological consequence of one or more other propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) if and only if in every row of a joint truth table that assigns "T" to all propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) the truth table also assigns "T" to ${\displaystyle Q}$.