I am reading Friendly Introduction to Mathematical Logic by Christopher Leary and Lars Kristiansen. In Definition 1.9.1., the authors define logically implies for two sets of $\mathcal{L}$-sentences $\Delta$ and $\Gamma$ as follows:
We will say $\Delta$ logically implies $\Gamma$ and write $\Delta \models\Gamma$ if for each $\mathcal{L}$-structure $\mathfrak{U}$, if $\mathfrak{U}\models\Delta$ then $\mathfrak{U}\models \Gamma$.
The definition seems okay but it would really help me to understand as to why we are making such a definition and what the motivation really is behind this definition. I would like to know why we are demanding "for each model of $\Delta$" and not discussing truth in a particular model of $\Delta$.
Best Answer
You write in particular:
Often logic is introduced in the context of understanding some fixed structure such as the semiring of natural numbers $\mathbb{N}$ or (stretching the term "structure" a bit) the universe of set theory $V$. However, that's not something we have to do. The apparatus of first-order logic provides us with a broad collection of sentences which make sense (whether true or false) in arbitrary structures, and we care right at the outset about the general behavior of this apparatus. The definition of logical entailment reflects this level of generality. We may later consider modifications of the entailment relation (although spoiler alert, none of them seem to have the same ultimate staying power), but to start with we should at least understand the "no-unnecessary-restrictions" situation.
It may help at this point to look at a particular mathematical claim in this context. Take for example the statement "If every element has order $2$, then every pair of elements commutes" in the context of group theory. The class of groups is axiomatizable: there is a set $\Delta_{grp}$ of first-order sentences such that the models of $\Delta_{grp}$ are exactly the groups. The fact that the statement above is true in every group amounts to the following logical entailment: $$\Delta_{grp}\models[\forall x(x*x=e)\rightarrow\forall x,y(x*y=y*x)].$$ Note that it's important that we range over all $\mathfrak{U}$s here; we're really trying to make a statement about arbitrary models of $\Delta_{grp}$ here.
Ultimately, I consider it a mistake to think about model theory as being oriented towards individual structures at all; model theory, in my opinion, is better thought of as the study of certain classes of, operations on, dividing lines amongst, and other things around structures in general.