TITLE: What is the logical consequence of a tautology
QUESTION [1 upvotes]: I know that the logical consequence of a contradiction is the whole Language $\mathcal{L}$ : $Cn(\bot) = \mathcal{L}$, but what is the logical consequence of a tautology is it $Cn(\top)$ ?

REPLY [2 votes]: The logical consequence of a tautology must be a tautology.
If $A$ is a tautology, it means that $A$ is true in every interpretation, i.e. every interpretation is a model of $A$.
But $A \vDash B$, we have that $B$ must be true in every model of $A$, and thus also $B$ is true in every interpretation.
{"set_name": "stack_exchange", "score": 1, "question_id": 3973989}