In predicate logic, you have quantifiers, a structure and a model, and something called (in Dutch) "een bedeling", which I will call "mapping" (since I have no idea what it is called in English). This mapping is a function that maps a variable to an object in the domain, and works like this:
Let b be the mapping with b(x) = 1 and b(y) = 2. Then b[x $\mapsto$ 7](x) = 7 and b[x $\mapsto$ 7](y) = 2.
In the book I'm reading, it says the following:
"If we want to determine the interpretation of a formula $\phi$ with help of the truth definition (which is an inductive definition that in a finite number of steps awards a semantic value to all formulas), then the only thing we need to know (apart from the interpretation function I), is what the mapping b does with the free variables in $\phi$."
"For sentences, i.e. 'closed formulas', formulas without free variables, the mapping does not matter. 'A sentence is true' is equivalent with 'It is true under a mapping', and with 'It is true under all mappings'."
Then the book gives an example (I give a snippet):
Let $M$ be a model with D = $\langle \mathbb{Q}, < \rangle$ and I(*R*) = '<'. Let b be a mapping with $b(x_1) = 4$. We've then got:
$M, b \models \forall{y} (Rx_1y \rightarrow \exists{z}(Rx_1y \wedge Rzy))$ $\\ \equiv \verb#for all q in # \mathbb{Q}: M, b[y \mapsto q] \models (Rx_1y \rightarrow \exists{z}(Rx_1y \wedge Rzy))$
I'm having a really hard time understanding the differences and similarities are between the mapping and quantifiers. What does
"For sentences, i.e. 'closed formulas', formulas without free variables, the mapping does not matter. 'A sentence is true' is equivalent with 'It is true under a mapping', and with 'It is true under all mappings'."
actually say? And why, in the example snippet given above, is there a mapping that works with a quantifier? I thought they were two different things? Is a mapping a quantifier, can it work alone or not? I'm completely miffed by it!