Denotation - Logic I am currently studying the proof of the undecidability of dyadic logic from the book Computability and Logic (see section 21.3), written by George S. Boolos and John P. Burgess. I came across the notion of 'denotation' but I do not know what this means. See the following fragment of the proof:

Could someone please explain this to me?
Thank you in advance!
 A: Here's what the paragraph above means.
Dyadic logic in Computability and Logic has only two-place predicates and does not have $=$.
In the chapter 12, where the canonical domains theorem is introduced (I can't find the theorem itself, 12.18 appears to be an exercise (?)), we have the Löwenheim-Skolem theorem. Additionally, without $=$, we can't restrict the size of our domain to be finite.
These two facts together give us the ability to build a new model $M'$ whose domain is $\mathbb{N}$ given a model $M$ over any domain. $M'$ and $M$ are elementarily equivalent.
A denotation is an object in the domain, or an element of some larger set built using the domain, that gives a symbol its meaning.
If $x$ is a constant or a variable, the denotation of $x$ in $(M, \vec{v})$ is an element of the domain of $M$. $x$'s denotation is provided by $M$ if and only if $x$ is a constant. $x$'s denotation is provided by $\vec{v}$ if and only if $x$ is a variable.
If $x$ is an $n$-ary predicate, then the denotation of $x$ is an element of $2^{M^n}$, where $2^A$ denotes the powerset of $A$ and $M^n$ denotes the  Cartesian product of the domain of $M$ with itself $n$ times.
If $x$ is an $n$-ary function, then the denotation of $x$ is the denotation of the $n{+}1$-ary relation corresponding to the graph of the function.
The second part of the paragraph is describing a procedure for taking a ternary predicate $P$ in the original model $\mathcal{M}$ and paraphrasing it away using binary relations only.
$P(x_1, x_2, x_3)$ is a well-formed formula in $\mathcal{M}$ headed by a ternary predicate.
$\exists w \mathop. Q_1(w, x_1) \land Q_2(w, x_2) \land Q_3(w, x_3) \land P^*(w)$ is a paraphrase in $\mathcal{M}^*$. In this case, $w$ is a number and the set of all values of $w$ that satisfy $P^*(w)$ plays a role similar to the graph of $P$. $Q_1$, $Q_2$, and $Q_3$ give us the ability to ask whether specific numbers in our domain are specific places within the triple $w$. For example $Q_2((3, 4, 5), 4)$ is true, but $Q_2((3, 4, 5), 3)$ is false.
