Confused about interpretation and satisfiability in first order logic.

Question

In my textbook("Introduction to mathmatical logic" by Elliot Mendelson , there is a definition of satisfiability:-

If $\mathscr B$ is an atomic wf $A^k_n(t_1 , ... , t_n)$ and $(A^n_k)^M$ is the corresponding n-place relation of the interpretation , then a sequence $s = (s_1 , s_2 , ...)$ satisfies $\mathscr B$ if and only if $(A^n_k)^M(s^*(t_1) , ... , s^*(t_n))$.

Shouldn't it say "iff $(A^n_k)^M(s^*(t_1) , ... , s^*(t_n))$ is true" or "iff $(A^n_k)^M(s^*(t_1) , ... , s^*(t_n))$ holds" ? maybe it is saying implicitly $(A^n_k)^M(s^*(t_1) , ... , s^*(t_n))$ is a theorem.But where are the axioms and rules from which it can be even deduced? I am confused.
For me it is kind of like saying "the painting is good iff $3$"

Answer 1

The logical form of the definition of satisfiability is: (the sentence) “Plato is a Philosopher “ is true iff Plato is a philosopher.

Thus, the locution “is satisfiable”, that like “is true” applies to a linguistic expression, is on the left side of the definition: formula ... is satisfiable iff..., while on the right side we have a fact, to which “it holds” applies, but it is redundant.

See A.Tarski.

Answer 2

An $n-$place relation is just a set of $n-$tuples. The $n-$tuples in the set are the ones the relation is true for. As an example, a nice one is $\lt$ over the integers. In the language of set theory, $\lt$ is a set of ordered pairs. We have $(1,2) \in \lt, (2,1000)\in \lt, (4,2) \not \in \lt$ and so on. From this point of view, $(A^n_k)^M(s^*(t_1) , ... , s^*(t_n))\iff (s^*(t_1) , ... , s^*(t_n))\in (A^n_k)^M$

When you are discussing satisfiability you don't talk about deduction rules. You just invent a structure and show that it satisfies the axioms. For example, if I claim $\Bbb \{{Z/34353Z},+\}$ is a group, I just have to show that it satisfies the group axioms. I exhibit an identity, inverses for each element, and associativity and I am done.