The notation $A \Rightarrow B$ can mean that $A \rightarrow B$ is a tautology. For example $\neg A \vee C \Rightarrow A \rightarrow (B \vee C)$. The question is whether it is possible to have $A \Rightarrow B$ or not. The reason why I am asking is that I saw an expression, which in my mind applies two different meanings to the symbol '$\Rightarrow$', namely that of implication ($\rightarrow$) and that of logical truth ($\Rightarrow$). I do not remember the exact expression, but it looked something like this:
$[(A \Rightarrow B) \vee \neg B] \Rightarrow \neg A$
I think what is meant is this:
$[(A \rightarrow B) \vee \neg B] \Rightarrow \neg A$
However I could be wrong. However the question is that you are supposed to show that the expression is tautological with the help of a truth table. So I think the expression $[(A \Rightarrow B) \vee \neg B] \Rightarrow \neg A$ ascribes two different meanings to the symbol '$\Rightarrow$'. The question: Is this an ambiguous use of the symbol '$\Rightarrow$' or is it correct to write $[(A \Rightarrow B) \vee \neg B] \Rightarrow \neg A$ with the context I provided?