Here is a definition of a derivation
How can we, for the first time, obtain a derivation with a hypothesis to use, for example, $(2\rightarrow)$?
More specifically, to prove, for example, $\vdash\phi\rightarrow(\psi\rightarrow\phi)$, I need to have a derivation with $\phi$ as a premise and $\psi\rightarrow\phi$ as a conclusion. But the only way to get $\psi\rightarrow\phi$ as a conclusion is to use a derivation with a premise (or a hypothesis, that's a little confusing, I'm not sure whether the author means the same thing for the words hypothesis and premise). But how can we introduce a derivation with a hypothesis for the first time according to the definition? I think the definition is not formal enough. There are no formal definitions of a premise, conclusion, or hypothesis above. Could someone clarify that, please?
P.S. Sorry for my English, I'm not a native speaker. Correct if there are some mistakes. I don't want to spend time polishing irrelevant things. Thank you.
By the first rule: The formula $\phi$ in the definition is for hypotheses.
Often the terms "hypothesis" and "premise" are used interchangeably; "assumption" is another term. To add to the confusion, there are two meanings of "premise" and "conclusion" in deduction systems, one which applies to derivations (i.e. the trees introduced in the definition you showed) and one which applies to rules (individual derivation steps (2) in the definition). In the context of natural deduction, it is possible to distinguish between:
A. For derivations:
hypotheses, which are leave nodes in the tree (introduced by rule one), which may or may not be discharged later,
premises, which appear on the left hand side of the turnstile in the derivability claim, and which may occur as undischarged hypotheses in the derivation.
Informally, hypotheses are temporary assumptions introduced in the proof procedure, whereas premises are the assumptions an argument finally depends on. (Though note that in classical logic, we may add arbitrary premises to the argument even when they are not actually needed in the proof, which is why the definition later will say that the undischarged hypotheses need only be a subset of the premises.)
The conclusion is then simply the root node of a derivation (the bottom most formula in the tree), corresponding to the right hand side of the turnstile.
B. For rules:
Premises are the formulas above the horizontal bar of a given rule, and the conclusion is the formula below. Hypotheses are separate from this: they are not rule applications but constitute an second kind of deduction step.
It contains only the hypothesis in the sense of which parts of the given definition of derivation (namely (1)), and only one premise and conclusion in the sense outlined first, i.e. leaf and root nodes of the tree. It does not contain any premises and conclusion in the second sense, as there are no rules (2)).
Right; it is implicitly assumed that $\mathcal{D}$ may be empty, so $\dfrac{\phi \quad \phi'}{\phi \land \phi'}$ (without anything more above $\phi'$ and $\phi'$) constitutes a derivation, and we can build up from there. In this case, $\phi$ and $\phi'$ would be hypotheses as defined in (1), and at the same time, they are premises of the rule (2$\land$(i)). The $\mathcal{D}$ here essentially just means that $\phi$ and $\phi'$ have been legally obtained; they may be the result of some rule applications (2), but they may also be licensed by (1). We start the derivation with hypotheses (1) and then incrementally plug together the derivation tree from rules (2).