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So I'm reading Hodel's Introduction to Mathematical Logic.

Here's a passage:

Let $L$ be a first order language. Then an interpretation $I$ of $L$ consists of:

-Non-empty set $D$ called the domain of $I$;

-For each constant symbol $c$ of $L$, an element $c$ of $D$;

-For each $n$-ary function symbol of $L$, an $n$-ary operation $F$ on $D$;

-For each $n$-ary relation symbol of $L$, an $n$-ary relation $R$ on $D$.

So suppose I have a domain $\{1,2\}$ and relation "is less than" and no function symbol. What about the constant symbols? At first I thought that I would have constant symbols $a,b$, where '$a$' would be assigned $1$, and '$b$' assigned $2$. But then I thought, would it instead just be that I have no constant symbols, and instead just the variables $x,y,z$ that are part of all first order languages (as opposed to specific interpretations with specific constant symbols)?

I guess I'm confused about what a constant symbol is supposed to be used for. In some of the examples of interpretations, '$0$' is a constant symbol interpreted as the natural number $0$ (also $1$ is used as a constant for some examples), but the rest of the numbers aren't considered constants in the interpretation?

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  • $\begingroup$ Every constant symbol of $L$ names some $d \in D$, but its not necessarily the case that every $d \in D$ is named by some constant symbol of $L$. $\endgroup$ – goblin Jun 21 '14 at 0:48
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Constant symbols are a part of the language and may or may not be present. Eg linear order has just a predicate and no constants or functions. Fields and rings usually have $0$ and $1$ as constant symbols, so in an interpretation you need to specify which elements are to interpret these constant symbols. Just as when you define a ring you should say which element os $0$ and which is $1$.

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In your example, you have stated the general definition of interpretation for a first-order language.

An interpretation is relative to a specific language, because it is "made of" two components :

  • a domain of objects, and

  • a mapping from the set of symbols of the language to elements and subsets of the domain.

We have to specify if there are relation symbols, e.g. $<$, and if there are individual constants, e.g. $\overline 1$ and $\overline 2$.

If this is our case, what about an interpretation with domain $D = \{ 1,2 \}$ ?

First, we have to "map" the relation "symbol" on the relation "less than" id $D$, i.e. on the relation $Less \subseteq D \times D = \{ \langle 1,2 \rangle \}$.

Then we have to define a "denotation" for the two constants (assume the "obvious" ones).

Having done this, we may check for the truth-value of a formula in our language like :

$\exists x (\overline 1 < x)$.

This formula is true in our interpretation : it is enough to assign to $x$ the value $2$.

If we consider a "restricted" language without the two individual constants, we cannot use the above formula any more.

But we may have instead :

$\exists x (y < x)$.

It is true when $y$ is interpreted as $1$, while it is false when $y$ is interpreted as $2$.

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A language, say $L$, does not necessarily have constant symbols. However the definition that I know of for defining what holds true inside the structure $\mathfrak{A}$ involves expanding the model in to an $L(A)$ structure. Here $L(A)=L\cup\{c_{a}:a\in{A}\}$ where the $c_{a}$ are distinct constant symbols and $c_{a}^\mathfrak{A}=a$.

So for your example, you would have $L(A)=L\cup\{c_{0},c_{1}\}$ and in your interpretation you would have $c_{0}^{\mathfrak{A}}=0$ and $c_{1}^{\mathfrak{A}}=1$.

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