What does rule schematic mean?

Question

While I'm studying the mathematical logic, the book says "Each rule of such a calculus either says that certain strings belong to $Z$, or else permits the passage from certain strings $\zeta_1,\cdots,\zeta_n$ to a new string $\zeta$ in the sense that, if $\zeta_1,\cdots,\zeta_n$ all belong to $Z$, then $\zeta$ also belongs to $Z$. The way such rules work is made clear when we write them schematically, as follows: $$\frac{\zeta_1,\cdots,\zeta_n}{\zeta}.$$"

$Z$ is a set of terms or folmulae. I know the rule and calculus. And I do understand the former, but can't figure it out what latter means. How can one says that if $\zeta_1,\cdots,\zeta_n$ all belong to $Z$, then $\zeta$ also belongs to $Z$? How does it work and why do we define this?

Answer 1

Z isn't just the set of axioms. Z is the set of all terms or formulae which can get deduced in the calculus. Or in other words one might say that Z consists of the set of consequences of the axiom(s) and rule(s) of the logic. Formulae and terms can belong to Z which are not axioms. They can belong to Z by application of the rule(s) of inference. In other words, such formulae and terms can belong to Z because we can construct formal proofs of those formulae and terms using the rule(s) of inference and/or axiom(s) of the system.

For example suppose our logic has formulae defined as:

1. All lower case letters of the Latin alphabet are formulae.
2. If $\alpha$ and $\beta$ are formulae, then so is C$\alpha$$\beta$. (not all of these formulae are in Z... but every formula in Z satisfies this definition)

two axioms:

1. CpCqp
2. CCpCqrCCpqCpr

the following rule of detachment, and the rule of uniform substitution for any formula in Z.

  α  Cαβ
  -------
    β

CpCqp, and CCpCqrCCpqCpr belong to Z. Now CCpCqpCCpqCpp belongs to Z by axiom 2 and uniform substitution (r got substituted with p in axiom 2). Now since CCpCqpCCpqCpp (Cαβ in this case) an CpCqp (α in this case) belong to Z, by detachment we can say that CCpqCpp (β in this case) belongs to Z. Now since CCpqCpp belongs to Z, and we have uniform substitution as a rule, substituting q with Cqp in CCpqCpp, we infer that CCpCqpCpp belongs to Z. Since CpCqp belongs to Z also, by detachment we can say that Cpp belongs to Z.

Answer 2

These rules [see Heinz-Dieter Ebbinghaus & Jörg Flum & Wolfgang Thomas, Mathematical Logic (2nd ed), page 17] are not deduction rules, but formation (or production) rules.

They license the passage from certain adimissible strings of symbols to a new admissible string.

In the calculus of terms [see page 18], they permit to pass from the empty word to a new word consisting in a single variable $x$ or contstant $c$ and permit to pass from an $n$-uple of already "produced" terms : $t_0, \ldots, t_{n+1}$ to a new term $ft_0, \ldots, t_{n+1}$ by way of an $n$-ary function symbols.

In he same way it is possible to set up a calculus of formulas, rewriting the clauses of Def 3.2 [page 15].