Doubts on the truth table of $\models$ I'm reading Shawn Hedman's A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity:

Definition 1.18 Formula $G$ is a consequence of formula $F$ if for every assignment $A$, if $A\models F$ then $A\models G$. We denote
  this by $F\models G$.
Proposition 1.19 For any formulas $F$ and $G$, $G$ is a consequence of $F$ if and only if $F \rightarrow G$ is a tautology.
Example 1.20 Let $F$ and $G$ be formulas. Each of the following can easily be verified by computing a truth table.
$$(F \wedge G)\models F \tag{1}$$
$$F\models (F \vee G) \tag{2}$$
$$(F\wedge ¬ F )\models G \tag{3}$$

I'm failing to understand why the truth table of $\models$ is so, (I'm not really sure that it exists, but according to the text, I guess it exists). For example, calculating  from $(1)$, I'll obtain the following truth table:

$$\begin{array}[b]{cccc} F & G & (F\wedge G) & (F\wedge G)\models F\\
 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 1\\ 1 & 0 & 0 & 1\\ 1 & 1 & 1 & 1
 \end{array}$$

I've also obtained tautologies by calculating the truth tables for the other examples and considering that both $(1)$, $(2)$ and $(3)$ are convenient examples chosen by the author to demonstrate examples of tautologies, I believe that the truth table of $\models$ is:

$$\begin{array}[b]{ccc} F' & G' & F'\models G'\\ 0 & 1 & 1\\ 1 & 1 & 1
 \end{array}  $$

What's not obvious to me at all is why is that so? why this is the truth table of $\models$?
 A: It is incorrect to think of $\models$ as a truth function (or a relation between truth values).  Instead it is a relation that holds between (a set of) formulas, and a single formula.
The basic idea is as follows:  to see if $\Phi \models \Psi$, find all of the truth assignments which give $\Phi$ the value $\top$ (or $1$).  If in each of these truth assignments $\Psi$ is also given the value $\top$, then the relation holds; otherwise it doesn't.
Let's go through the first example:
$$\begin{array}{cc|c|c}
F & G & F \wedge G & F \\
\hline
\top & \top & \top & \top \\
\top & \bot & \color{red}{\bot} & \color{red}{\top} \\
\bot & \top & \color{red}{\bot} & \color{red}{\bot} \\
\bot & \bot & \color{red}{\bot} & \color{red}{\bot}
\end{array}$$
The only truth assignment that gives $F \wedge G$ the value true also gives $F$ the value $\top$, and so $F \wedge G \models F$.
For the second example:
$$\begin{array}{cc|c|c}
F & G & F & F \vee G \\
\hline
\top & \top & \top & \top \\
\top & \bot & \top & \top \\
\bot & \top & \color{red}{\bot} & \color{red}{\top} \\
\bot & \bot & \color{red}{\bot} & \color{red}{\bot}
\end{array}$$
There are two truth assignments that give $F$ the value $\top$, and in each of these $F \vee G$ is also given the value $\top$, and so $F \models F \vee G$.
And the third example
$$\begin{array}{cc|c|c}
F & G & F \wedge \neg F & G \\
\hline
\top & \top & \color{red}{\bot} & \color{red}{\top} \\
\top & \bot & \color{red}{\bot} & \color{red}{\bot} \\
\bot & \top & \color{red}{\bot} & \color{red}{\top} \\
\bot & \bot & \color{red}{\bot} & \color{red}{\bot}
\end{array}$$
Here no truth assignment gives $F \wedge \neg F$ the value $\top$, and so every time $F \wedge \neg F$ is given the value $\top$ so is $G$, and so $F \wedge \neg F \models G$.
A: You have formulae (eg, $F \lor G$), you have assignments (eg, $F=\top, G= \bot$) and rules for 'propagating' assignments through formulae (eg, the above assigns the value $\top$ to $F \lor G$).
You can create a truth table for a formula which expresses the formula as a boolean function.
A formula is a tautology iff the value is true for any assignment, or equivalently, its truth table has all true results. Note that there is no truth table for being a tautology; it expresses something about the truth table itself. But it makes no sense to try and write down the truth table for a tautology.
Think of $F \models G$ in the same way. It expresses a relationship between the truth tables of $F$ and $G$, but it doesn't make any more sense to talk about the truth table of $\models$ than it does to talk about the truth table of a tautology.  It means that if a line in the truth table for $F$ results in true, then the corresponding line in the truth table for $G$ must also result in true.
Proposition 1.19 shows that the statement $F \models G$ is the same as saying $F \rightarrow G$ is a tautology.
