1) What does individual variable mean ?
It is a term, i.e. a symbol that acts as a name for an object.
Thus, it cannot be a propositional variables, i.e. a symbols that stands for a sentence.
Consider the simple example from first-order language of arithmetic : $(x=0)$.
In this formula $x$ must be replaced by a number in order to give an arithmetical meaning to the formula.
2) In first-order language, can be definition symbol ?
A definition must either introduce a term, i.e. a symbol acting as a name for an object, or a predicate letter, i.e. a symbol naming a property.
Again, examples from first order arithmetic : we start from the basic symbols of the language : $0$ (an individual constant denoting the number $\text {zero}$), the unary function $s(x)$ (the $\text {successor}$ function) and the binary function $+(x,y)$ (the $\text {sum}$ operation, abbreviated with : $(x+y)$).
With them we define the new constant $1$ as $s(0)$.
And we define the new binary predicate $<(n,m)$ (the relation $\text {less than}$, abbreviated with $(n < m)$) as follows :
$(n < m) \text { iff } \exists z \ (m=n+s(z))$.
Your understanding is correct. Symbols have no meaning until they are interpreted.
Yes, $A$ is the domain of the $L$-structure $\alpha$. You specify the language $L$ and then interpret the symbols of the language in $\alpha$. That is why it is called an $L$-structure. You may encounter the notation
$$\alpha:=\left(A,\{R^{\alpha}:R\in L\},\{f^{\alpha}:f\in L\},\{c^{\alpha}:c\in L\}\right)$$
Which is just notation to tell us what the $L$-structure $\alpha$ consists of, as the definition you provided.
Yes, if $f\in L$ is an $n$-ary function symbol, then its interpretation $f^{\alpha}$ is a function with domain $A^{n}$ and range always a subset of the domain of interpretation $A$. Meaning is always with respect to one domain. If the domain changes, the meaning may change. But you cannot have two different meanings within the same domain.
The superscript is notation to tell us we are talking about an element of the domain $A$. If $c\in L$ is a constant symbol, then $c^{\alpha}\in A$ is the element of a corresponding to the interpretation of $c$ in $A$. If $f$ is a function symbol, $f^{\alpha}$ is the function corresponding to the interpretation of $f$, and so on.
No, an algebraic structure consists of a nonempty set $X$ (also called the domain or underlying set), a collection of operations on $X$ (e.g addition and multiplication if applicable), and a finite set of axioms that the operations must satisfy. For example, groups, rings and vector spaces are examples of algebraic structures.
However, an algebraic structure is indeed a particular type of logical structure. For example, consider the language of rings: $L=\left<\dot{0},\dot{+},\dot{-},\dot{\times}\right>$ (where I use dots to emphasize these are symbols of the language). We have not given these symbols meaning yet, and $\dot{0}$ is a constant symbol, $\dot{-}$ is a $1$-ary function symbol, and $\dot{+}$ and $\dot{\times}$ are $2$-ary function symbols. Then, a group with underlying set $G$ is an algebraic structure, and it is also an $L$-structure $\alpha:=(G,0,+,-\times)$ subject to the group axioms (here I am denoting the interpretation of the symbols of the language $L$ simply by removing the dot over them, e.g $\dot{+}^{\alpha}:=+$).
Example: Take as domain $H$ the set of all humans. The language $L$ consists of the following symbols (with their intuitive meaning in parenthesis): 1-ary function symbols $m$ (mother of) and $f$ (father of), $1$-ary relation symbols $M$ (man) and $W$ (woman), binary relation symbols $P$ (parent of), $C$ (child of), $\ell$ (loves), and constant symbols $\{\text{Adam}, \text{Mary}, \ldots\}$ (names). The $L$-structure
$$\alpha=(H,m^{\alpha},f^{\alpha},M^{\alpha}, W^{\alpha}, C^{\alpha}, \ell^{\alpha}, \{\text{Mary}^{\alpha}, \text{Adam}^{\alpha},\ldots\})$$
is not algebraic. Symbols are interpreted according to their intuitive meaning. For example, $P^{\alpha}$ is the $2$-ary relation "parent of" in the domain $H$ of all humans, and $\text{Mary}^{\alpha}$ is a human named Mary.
Example: Consider the language of sets $L=\{\in\}$ (i.e. $\in$ is a binary relation symbol). The set of real numbers $\mathbb{R}$ with $\in$ interpreted in the usual way is an $L$-structure. But it does not have a defined algebraic structure. If we add to $L$ the symbols $+$, $-$, $\cdot$, $0$, $1$ defined and interpreted in the usual way (i.e. $+$, and $\cdot$ are interpreted as the usual binary operations) and we add the field axioms, then in the augmented language the $L$-structure now has an algebraic structure defined- that of a field.
A final point: For any $L$-structure $\alpha$, an assignment is a function $a:\{\text{set of variables of $L$}\}\to A$. Relative to the assignment $a$ of $\alpha$ you will see notation like $c^{\alpha}[a]$, or $f^{\alpha}[a]$ denoting the interpretation of the symbols in $A$ relative to the assignment $a$ of the variables.
Best Answer
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.
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 $a$.
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.