Csdrhrt

A $vDash $B can be read in words as:

1. A entails B


2. B is a semantic consequence of A


3. A models B

The first two are fine. But the third one seems a bit counter-intuitive to me. Somehow, I can't reconcile the term 'model' (used colloquially) with the notion of entailment or consequence.

Why do mathematicians and logicians use the term 'model' in such contexts? How is the term associated with the idea of entailment? And even more bizarre to me is the use of the $vDash$ symbol in different contexts. For example, let A be an assignment, and F is an atomic formula. So that A $vDash$ F (again, read as A 'models' F) means that A assigns a truth value of 1 to F. Again, this is counter-intuitive, and often had me wondering: "what is the underlying concept or idea behind it?"

First, as another user has said, it is important to note that there is an important distinction between $Gamma models phi$, where $Gamma$ is a set of formulas, and $mathcal{A} models phi$, where $mathcal{A}$ is a structure. As far as I know, the word "models" is used only in the latter case. Since your question was originally about the relation between the "common" use of the word "model" and the logician's use, here I'll focus on this relationship. For more details about the history of the word "model" and its (scientific) uses, I strongly recommend Roland Müller's essay "The Notion of a Model: A Historical Overview", from which I gathered most of my information about this.

Anyway, according to Müller, one of the common uses of the word "model" is in the sense of "prototype" or "original", as when a person serves as a model for a painting, or a small-scale sculpture serves as the model for a larger construction (say, a cathedral). Notice that the salient feature here is less the idea that the "model" is the original, and more the idea that we use it to read off important properties of the object being modeled. This is still in use as when we talk in logic about using a "toy model" for studying a theory; in fact, it seems that nineteen centuries geometers, such as Plücker, literally used such toy models (three-dimensional objects) to aid in the study of geometrical theories. So, at first, the word "model" was linked to this idea of a small-scale prototype which we use to "control" for the desired properties.

Later, however, as Müller emphasizes, geometers began to encounter objects which were difficult to model in this way. In particular, both projective and hyperbolic geometries posed problems for those who wanted to build small-scale objects to serve as models for their study. Indeed, it's not entirely clear how to create a small-scale Klein bottle, or more complicated geometric objects. So, according to Müller, geometers came up with two solutions for this problem: pseudo-models and abstract models.

Pseudo-models are models that introduce systematic distortions in order to aid visualization. That is, you create an object which distorts the original one, but whose distortion is controlled, so that you know exactly which features are being distorted. In other words, we know exactly how to pass from the distorted feature to the undistorted one and vice-versa. Müller example is Poincaré's models of the hyperbolic plane using the circle or the half-plane.

The other device is an abstract model, in which we don't attempt to create a physical model or prototype, but instead attempt to give an abstract description of a mathematical structure which has the desired properties. Although Müller doesn't make the link very explicitly, it seems to me that this latter device is connected with the rise of set-theory and structural descriptions in the work of Riemann and Dedekind. The latter, especially, seemed to think of his "Systems" in this way.

Müller also deals with how the word was eventually introduced by Tarski and Robinson in order to give birth to model theory, but the important point for this question is this: when we say that $mathcal{A} models phi$, we are not saying that $mathcal{A}$ is a model of $phi$ in the sense of a representation of $phi$. Rather, we are saying that $mathcal{A}$ is a model, or prototype, from which we read off the property that $phi$. So $mathcal{A}$ is a model of $phi$ in the sense that some people speculate that Da Vinci himself is the model of the Mona Lisa: it is the original, or prototype, which Da Vinci studied in order to read the relevant properties which allowed him to paint the Mona Lisa. Alternatively, and more liberally, you can also say that $mathcal{A}$ is a model of $phi$ in the sense that a given three-dimensional object (say, a small-scale pyramid) is an object (a model) which allows us to study properties of a geometrical concept (say, the concept of tetrahedron).

So much for the word "model". Now, why use the symbol $models$ for different things? Well, there is a clear relation between them: $Gamma models phi$ iff for every $mathcal{A}$, $mathcal{A} models Gamma$ implies $mathcal{A} models phi$. So the overload is harmless (or so it seems to me).

The corresponding usage of "model" is e.g. "this program models the behaviour of the bees". $A$ is treated as an environment wherein $B$ is true.

The 'reading' #3 is a different thingy from reading #2 and #1.

As far as I know, the double turnstile ($vDash$) has two uses.

In your reading #3, $A$ is some 'structure', which can be called a 'model', an interpretation, or an 'assignment', while $B$ is a propositional formula. The notation is more like $mathcal{A}vDash B$ or $mathcal{M}vDash B$, and it should be read/understood as "The structure $mathcal{A}$ (or $mathcal{M}$) models B" or "B is true in $mathcal{A}$ (or $mathcal{M}$)."

For example, in classical propositional logic, an assignment $mathcal{A}$ is a two-valued function $A:Brightarrow {top,bot}$ where $B$ is a propositional formula.

In readings #1 and #2, $A$ and $B$ are both propositional formulas. That is the 'semantic consequence' that you understand. The notation is now $AvDash B$. Moreover, this means that for all assignments $mathcal{A}$, if $mathcal{A}vDash A$, then $mathcal{A}vDash B$.