A question on the Substitution for predicate letters theorem as in Kleene Introduction to Metamathematics (1952)

Question

I'm self studying Kleene's IM (1952). On page 149 Kleene defines a name form thus:

"Let $x_1, ... x_n$ be distinct variables, and $A(x_1, ... x_n)$ a formula. When we are interpreting $A(x_1, ... x_n)$ by a predicate, or performing formal operations with it which are in keeping with an interpretation by a predicate (even though the interpretation is not involved in the formal operations), we call $A(x_1, ... x_n)$ a name form in $x_1, ... x_n$ as the name form variables ..."

My first question here is which are the formal operations which are in keeping with an interpretation by a predicate and which are not in keeping with an interpretation by a predicate? If, as he says, the interpretation is not involved in the formal operations, why should there even be formal operations in or out of keeping with with the interpretation?

On page 157/8 is goes on with the following definitions in preparation for the SUBSTITUTION FOR PREDICATE LETTERS Theorem (Theorem 15 on page 159):

"By an occurrence of a predicate letter $P(a_1, ... a_n)$ with attached variables in a predicate letter formula $E$ we shall mean a (consecutive) part of E of the form $P(t_1, ... t_n)$ where $t_1, ... t_n$ are terms. A predicate letter formula $E$ is said to be a predicate letter formula in the distinct predicate letters

(1) $P_1(a_1, ... a_{n_1})$, ..., $P_m(a_1, ... a_{n_m})$, where $n_1, ... n_m \geq 0; m \geq 1$

if no predicate letter other than (1) occur in $E$.

The substitution of formulas

(2) $A_1(a_1, ... a_{n_1})$, ..., $A_m(a_1, ... a_{n_m})$

considered as name forms in the respective variables shown, for the predicate letters (1) in $E$ (with result $E^*$) shall consist in replacing, simultaneously for each $j(j=1, ..., m)$, each occurrence $P_j(t_1, ... a_{t_j})$ of $P_j(a_1, ... a_{n_j})$ in $E$ by $A_j(t_1, ... t_{n_j})$."

My question here is, why the expressions in (2) have to be name forms and not just formulas?

Thanks for your help.

Answer 1

See Stephen Cole Kleene, Mathematical logic (1967, Dover reprint), page 77:

The issue is about the proper way to use symbols to represent a predicate.

The predicate (also called propositional function) named by "$x < y$" can also be named simply "$<$".

The issue is similar to that concerning e.g. the $\sin$ function, that usually we write as $\sin(x)$.

A notation for predicate using variables we may call a name form for the predicate.

See page 78: the "official" syntax is made of ions $P(-,-)$ for e.g. binary predicates (with two argument-places) and (individual) variables: $x_1,x_2,x_3$.

Thus, an atom (or prime formula) $P(x_1,x_2)$ is a ion "applied" to variable.

IMO, with a name form $P(x,y)$ Kleene means a "generic" name for the ion $P(-,-)$, understanding that it is different from the atom.

See page 78:

We call $P(x,y,z), P(y,z,x)$, etc. different name forms for the same 3-place prime predicate expression or ion $P(-,-,-)$.

And see page 98 for the Substitution theorem:

Say $E$ is a formula which contains only the distinct ions having the respective name forms

$$\text {(1) } P_1(w_1,\ldots,w_{P_1}), \ldots P_N(w_1,\ldots,w_{P_1})$$

[...] The substitution of formulas

$$\text {(2) } A_1(w_1,\ldots,w_{P_1}), \ldots A_N(w_1,\ldots,w_{P_1})$$

for (1) in $E$ with result $E^*$ [...]