A few theorems I read somewhere online say that in a tautology, you can replace any statement by another statement to get another tautology. For instance, we have the following tautology:
$[A \land (A \rightarrow B)] \rightarrow B$
The statement $A$ can be replaced by $B \rightarrow C$ and we have that the following is still a tautology:
$[(B \rightarrow C) \land ((B \rightarrow C) \rightarrow B)] \rightarrow B$
However, in the first tautology, what if you were to replace the statement $A \rightarrow B$ by statement $C$?
We would end up having $[A \land C] \rightarrow B$.
If you were to assume that $A$ and $C$ are both true, then that doesn't give us the value of $B$ (true or false), unless we were to write out a truth table with $A$, $B$, and $C$ having their own columns. In the case that $B$ is false and $A \land C$ are false, then $[A \land C] \rightarrow B$ wouldn't be a tautology.
Can someone please clarify the meaning of this for me? Thanks in advance.
It means you can replace any of the atomic statements (those represented by a single letter) by another statement.