I have a small question about Deduction Theorem. According to Deduction Theorem, we have the following: $$A\vdash B\ \Leftrightarrow\ \langle\ \rangle\vdash A\rightarrow B.$$ Here I use $\langle\ \rangle$ to denote an empty context. My questions are related to the use of empty context $\langle\ \rangle$.
[Question 1] By weakening rule, it seems to me that the following should hold, right? $$\langle\ \rangle\vdash A\ \Leftrightarrow\ \langle\ \rangle\vdash\langle\ \rangle\rightarrow A.$$ [Question 2] Then by monotonicity, if $\langle\ \rangle\vdash\langle\ \rangle\rightarrow A$ holds, we also have $$\Gamma\vdash\langle\ \rangle\rightarrow A.$$ But what does $\Gamma\vdash\langle\ \rangle\rightarrow A$ exactly mean? More concretely, I am wondering whether $\Gamma$ really plays a role in $\Gamma\vdash\langle\ \rangle\rightarrow A$; putting it in another way, whether the truth of $A$ in $\Gamma\vdash\langle\ \rangle\rightarrow A$ depends on the assumptions collected in $\Gamma$?