Tautology And Substitution

Question

Let $ \alpha $ be some proposition, and let $ A = \alpha $, and let $ B = \neg \alpha $.

Is the statement $ A \lor B $ a tautology?

Answer 1

Obviously, $α ∨ ¬α$ is a tautology.

But - in general - $A∨B$ is not a tautology.

The "rule" is:

every instance of a tautological schema will be a tautology.

In other words, given a tautological schema, like $\varphi \lor \lnot \varphi$, every formula obtained via substitution from it, like $p_1 \lor \lnot p_1$ or $\forall x Q(x) \lor \lnot \forall x Q(x)$, will be a tautology.