Intuition behind a structure of a language in mathematical logic [long read but simple]

Question

I do not get the intuition behind the structure (I think it's called "interpretation" in other texts) of a first order language. Here I use the following definition, given in Shoenfield's Introduction to Mathematical Logic:

I do get the definitions, but I am at loss to why such definitions were invented in the first place, and some things are unclear to me. Specifically:

1. I do not that the paragraph "We want to define a formula $A$ [called "well-closed formulas", or "wfs," in some texts"] to be valid... this leads us to the followong definitions." Could you provide a specific example that explains the intuition behind the definitions?

2. It's written "for each individual $a$ of $\alpha$, 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?

3. If $A$ is $p$ with $p$ a $0$-ary predicate symbol, then $\alpha(A)= T$ iff $\alpha(\emptyset)$ belongs to the predicate. However, we haven't defined what $\alpha(\emptyset)$ is,unless we are making another assumption that there exist an individual of $|\alpha|$ that has $\emptyset$ as its name. Am I mistaken?

Answer 1

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.

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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 $a$.

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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.