It is more helpful to view "predicate logic" as a taxonomic term (the same goes for the term "logic" itself). So the question becomes: what properties of a logic cause us to call it a "predicate logic"?
That's a hard question partially because "logic" itself is so broad. We can identify at least a few common properties, but not every predicate logic will have all of them. The basic examples, of course, are the logics that are called "first-order logic" in the literature. But there are also higher-order logics, modal predicate logics, temporal predicate logics, etc.
Here are a few common traits:
Predicate logics may have variables to range over "individual" objects. There many be more than one sort of "individual".
Predicate logics may have variables that range over higher types or predicates, with syntax to match.
Predicate logics often have quantifiers over the individuals and other sorts of objects
Predicate logics often come with semantics in which the predicate symbols in formulas are interpreted as relations on a set of "individuals".
1) "We want to define a formula $A$ to be valid... this leads us to the followong definitions." Could you provide a specific example that explains the intuition behind the definitions?
The intuition behind the definition of a formula $A$ being valid in a structure $\mathfrak A$ is quite "natural".
A structure is a piece of the "mathematical world" made of objects (e.g. natural numbers), properties (e.g. odd and even) and relations (e.g. less than) between them.
Thus, to interpret a language is to link the symbols of the language to objects and relations of the structure.
In this way, expressions (terms and formulas) of the language, when interpreted, have meaning: terms are names for objects, and formulas are statements expressing facts about objects.
To be valid in $\mathfrak A$ means that, according to the way we have chosen to interpret in $\mathfrak A$ the symbols of the language, the interpeted formula will express a fact that is true in the structure.
2) It's written "for each individual $a$ of $\mathfrak A$, we choose a new constant, called the name of $a$." Where does the new constant come from? Are we assuming descriptive set theory and choose an elment not in the $0$-ary function symbol and adding it to the language being considered?
Correct; for every "object" of the "universe of discourse", i.e. for every element $a$ of the domain of the structure $\mathfrak A$, we add to the language a new constant symbols a whose reference is the object $a$: thus, the symbol a is the "name" in the expanded language $L (\mathfrak A)$ of the object $a$.
3) If $A$ is $p$ with $p$ a $0$-ary predicate symbol, then $\mathfrak A(A)= \text T$ iff $\mathfrak A(\emptyset)$ belongs to the predicate. However, we haven't defined what $\mathfrak A(\emptyset)$ is,unless we are making another assumption that there exist an individual of $|\mathfrak A|$ that has $\emptyset$ its name.
If my memery is sound, the case for $0$-ary predicate symbols is not explicitly discussed in Shoenfield's textbook...
Having said, that, a $0$-ary predicate symbol is a propositional symbol, like those of propositional logic. Thus, the "natural" interpretation is trough truth-values: $\text T, \text F$.
We may choose to map $\text T$ on $|\mathfrak A|$ and $\text F$ on $\emptyset$, and this is consistent with the fact that the interpretation of unary predicate symbols of the language are subsets of the domain of the structure.
Best Answer
Rif. Uwe Schöning, Logic for computer scientists, Birkhauser (1989).
The "usual" approach is to define an interpretation for the predicates, functions and constants symbols of the language.
On top of it, we have to add a mechanism, usually called variable assignment function to assign a "temporary" meaning to the free variables of a formula.
Thus, if we consider the formula $(x=0)$ and interpret it in the domain $\mathbb N$ of natural numbers, we have to consider a variable assignment :
such that, e.g. : $\mu(x)=0$.
In this case, we have $\mathbb N, \mu \vDash (x=0)$.
With a different assignment $\mu'(x)=1$, we will have : $\mathbb N, \mu' \nvDash (x=0)$.
As you can see form Example, page 45 with formula $F = ∀xP(x,f(x)) ∧ Q(g(a,z))$, the author says :
As you can see, the difference is only of terminology.