# Empty Context in Deduction Theorem

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 $$\Gamma$$?

The answer to both of your questions is that "$$\langle \rangle \to A$$" doesn't exist. $$\phi \to \psi$$ is a formula iff $$\phi$$ is a formula and $$\psi$$ is a formula, but $$\langle \rangle$$ is not a formula, so $$\langle \rangle \to A$$ isn't, either. $$A\vdash B\ \Leftrightarrow\ \langle\ \rangle\vdash A\rightarrow B$$ is the smallest sequent to which the deduction theorem is applicable.
We do have a generalization in the other direction, though: It holds for arbitrary sequents with $$n$$ pemises that
$$A_1, \ldots, A_n \vdash B \ \Longleftrightarrow \ \vdash (A_1 \land \ldots \land A_n )\to B.$$
The right-hand side $$(A_1 \land \ldots \land A_n )\to B$$ can logically equivalently be expressed as $$A_1 \to (\ldots \to (A_n \to B))$$, i.e. for each premise add one implication. In the limit case with zero premises, there is no implication at all, what you mean by "$$\langle \rangle \to A$$" is just $$A$$, and the two sides of the deduction theorem, $$\vdash A$$ and $$\vdash A$$, coincide.