Difference between $\mathscr B$ and $\mathscr B[b_1, ... , b_k]$ in first order logic.

Question

This is from "Introduction to Mathmatical logic-forth edition" by Elliot Mendelson.

In page 60 of the book, it says:

A wf is true for the interpretation M (written $\vDash _M\mathscr B $) if and only if every sequence in $\Sigma$ satisfies $\mathscr B$.

But in page 62 it says:

...we shall say that every $k$-tuple $\langle b_1, ... , b_k \rangle$ in this relation satisfies $\mathscr B(x_{i_1}, ... , x_{i_k})$ in the interpretation $M$; this will be written as $\vDash_M \mathscr B[b_1, ... , b_k]$

My question is, what is the difference between $\mathscr B$ and $\mathscr B[b_1, ... , b_k]$?

Lets say $\mathscr B[b_1, ... , b_k] = \mathscr C$ .In the book, $\vDash _M\mathscr B $ means that every possible denumerable sequence of the elements of $D$ (here $D$ is the "Domain of disclosure" in $M$) satisfies $\mathscr B$. So, because the $\vDash$ notation is used in that way, I assume that $\mathscr C$ is a first order logic wf and $\vDash_M \mathscr C$ means that "every possible denumerable sequence of the elements of $D$ (here $D$ is the "Domain of disclosure" in $M$) satisfies $\mathscr C$ . But the problem is , I have no idea what this wf $\mathscr C$ actually is.

Answer 1

The basic definition is that of the satisfaction relation (page 60), that means:

"intuitively, a sequence $s = (s_1, s_2, \ldots)$ satisfies a wf $\mathscr B$ if and only if, when, for each $i$, we replace all free occurrences of $x_i$ (if any) in $\mathscr B$ by a symbol representing [the objcet of the domain of the interpretation] $s_i$, the resulting proposition is true under the given interpretation."

The relation is expressed by the following symbol: $s \vDash_M \mathscr B$.

Consider a simple example in the language of arithmetic. Formula $(x_1=0)$ will be satisfied in $\mathbb N$ by the sequence $s$ such that $s(x_1)=0$.

Now, the example of page 62 consider the general case of a formula $\mathscr B(x_{i_1},\ldots,x_{i_k})$ with $k$ free variables (listed in increasing order; see the specifications of the language: page 57).

Again, consider as formula $\mathscr B (x_1,x_2)$ the simple example: $(x_1+x_2 \le 5)$.

This formula "specifies" a binary relation in $\mathbb N \times \mathbb N$, i.e. the set of all pairs of natural numbers $n,m$ such that their sum is less-or-equal to five. Thus, $(1,1)$ and $(3,2)$ will belong to that set of pairs, because in each case the two numbers of the pairs satisfy the formula.

According to the previous definition, we have that for sequence $s_1$ such that $s_1(x_1)=s_1(x_2)=1$ and $s_2$ such that $s_2(x_1)=3$ and $s_2(x_2)=2$ we have:

$s_1 \vDash_{\mathbb N} (x_1+x_2 \le 5) \text { and } s_2 \vDash_{\mathbb N} (x_1+x_2 \le 5)$.

If we call $R$ the binary relation on $\mathbb N$ such that: $R = \{ (n,m) \mid n+m \le 5 \}$, we have that

every $2$-tuple (pair) $\langle b_1, b_2 \rangle$ in the relation $R$ satisfies formula $\mathscr B(x_1,x_2)$ in the interpretation $\mathbb N$; this will be written as

$\vDash_{\mathbb N} \mathscr B[b_1,b_2]$.

Now,

what is the difference between $\mathscr B$ and $\mathscr B[b_1,\ldots,b_k]$?

The first one is a symbol in the meta-language denoting a formula with its free variables: $\mathscr B(x_{i_1},\ldots,x_{i_k})$.

The expression $\vDash_M \mathscr B[b_1,\ldots,b_k]$ instead, is an expression of the meta-language referrin to a property of the "interpreted formula": it is an "extension" of the original notion of satisfaction and means that the sequence $s$ that maps the variables $(x_{i_1},\ldots,x_{i_k})$ to a $k$-uple $(b_1,\ldots,b_k)$, i.e. where $s(x_{i_j})=b_j$, satisfies formula $\mathscr B$ in the interpretation $M$.