How do I intuitively make sense of "logically implies" in first order logic

Question

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 $\Delta$.

Answer 1

You write in particular:

I would like to know why we are demanding "for each model of $\Delta$" and not discussing truth in a particular model of $\Delta$.

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.

Answer 2

This is just the basic concept of logical implication: Something (call it $X$) logically implies something else (call it $Y$) if and only if $Y$ is true whenever $X$ is true.

Consider this. In our world, grass is green, and snow is white. Now, would you sat that grass being green logically implies that snow it white just because there is a world in which both statements are true? Of course not. And of course grass being green does not logically imply snow being white: you can imagine a logically possible world where grass is green, but snow is not white.

In other words, $X$ does not imply $Y$ if there is some world where $X$ is true but $Y$ is false. But that means that $X$ does imply $Y$ when there is no such world ... which is to say that in every world where $X$ is true, $Y$ will also be true.

Answer 3

It might help if we think about when $\Delta \models \Gamma$ is false. Let's consider the contrapositive of the forall definition and call it (I).

$$ \textbf{(I)}\;\;\; \Delta \not\models \Gamma \;\;\text{iff}\;\; \text{there exists a model $M$ such that $M \models \Delta$ and $M \not\models \Gamma$ } $$

If we only looked at a single not-fixed model for $\models$, then the contrapositive would be as follows

$$ \textbf{(II)}\; \textbf{BAD} \;\;\; \Delta \not\models \Gamma \;\;\text{iff} \;\; \text{for all models $M$ where $M \models \Delta$, $M \not\models \Gamma$ holds} $$

The statement (II) is not compatible with the fact that implication only requires a single counterexample to be false.

If we look at a single fixed model $M$, then we're not talking about a relationship between $\Gamma$ and $\Delta$ that always holds, but rather a relationship between them in $M$ alone.