Formulae with symbol outside the language of a structure

Question

A basic question, but it's a little bit confusing for me. Given languages $L \subset L'$, an $L'$ formula $\phi$ and an $L$-structure $M$ and its $L'$-reduct $M'$ how do I define satisfaction in a case, where I don't have an interpretation of a given symbol contained in a formula, e.g.

symbol $0$ is in $L'$ but not in $L$ and we have formula $x + 0 = x$ obviously satisfiable in $(N, +, 0)$. Now is it satisfiable in $(N, +)$ or not? Or it just makes no sense to consider such a formula in this reduct?

Sorry if I'm being incoherent, first-timer here...

Answer 1

You are confusing some things. There is no $L'$-reduct of $M$, $L'$ contains more symbols than $L$. So if you are given an $L'$-structure $M'$ then you can get an $L$-reduct $M$ by forgetting the extra symbols in $L'$.

To take your example, let $L' = \{+, 0\}$ and $L = \{+\}$. Then $M' = (\mathbb{N}, +, 0)$ is an $L'$-structure. Forgetting the extra symbols, in this case just the $0$ symbol, we get an $L$-structure $M = (\mathbb{N}, +)$. So $(\mathbb{N}, +)$ is a reduct of $(\mathbb{N}, +, 0)$, and not the other way around.

Now, about evaluating formulas. We cannot evaluate an $L'$-formula in an $L$-structure, exactly because of the issue you encountered: the formula may use symbols from $L'$ that are not in $L$. However, because all the symbols of $L$ are also in $L'$, it makes sense to evaluate an $L$-formula in an $L'$-structure. There are two ways to make this precise:

1. Since all the symbols in $L$ are also in $L'$, we could argue that the set of $L$-formulas is a subset of $L'$-formulas. In other words, we view any $L$-formula simultaneously as an $L'$-formula. Then we can just evaluate such a formula as usual in an $L'$-structure.
2. Given an $L'$-structure $M'$ we can take its reduct $M$, by forgetting the extra symbols (see the above example). Then we can simply evaluate an $L$-formula in $M'$ by taking its evaluation in $M$.

Note that both ways of doing this are equivalent, so it is maybe more a matter of perspective or preference.