The definition of interpretation in a Kripke model collides with my intuition of what it should do

Question

In Lindröm and Segerberg (2007) exposition of a Kripke model, with frame $F= \langle W,D,R,E,w_0\rangle$, they define an interpretation $I$ as

a family of functions $I_w$, where $w$ ranges over $W$, such that $I_w$ assigns a subset $I_w$$(P)$ of $D^n$ to each n-ary predicate constant $P$ of $L$ and an element $I_w$$(c)∈D$ to each individual constant $c$ of $L$.

Intuitively, $I$ should give us for every world the set of elements of $D$ that satisfies $P$ in that world. In this sense, I would have expected $I_w$(P) to be a set of ordered subsets of $D$ that satisfy $P$ in $w$. Now, my intuition of what $I$ should do does not correspond to how $I$ is defined. Shouldn't $I$ be defined as something along the lines of:

a family of functions $I_w$, where $w$ ranges over $W$, such that $I_w$ assigns a set $I_w$$(P)$ of ordered subsets $D^n$ to each n-ary predicate constant $P$ of $L$ etc.etc.?

Is my intuition of what $I$ should do wrong?

Answer 1

The $w$ indexes elements in $W$, where the elements of $W$ are traditionally called possible worlds.

In each world $w \in W$ you have an "interpretation function" $I_w$ which interpret the constant $c$ of the language with an "object" $I_w(c) \in D$ and each ($n$-ary) predicate constant $P$ with a $n$-ary relation in $D$.

Please, notice that, being $P$ an $n$-ary predicate, its interpretation must be an $n$-ary relation in $D$, i.e. a subset of $D^n$.

There are no "deviations" from the standard semantics for f-o languages; the only addition is that there is no more a single domain (a "world") but a whole family of worlds.

From : Alan Berger (editor), Saul Kripke (2011), Chapter 5 : Kripke Models by John Burgess, page 119-on. See page 134:

Now a Kripke model for modal predicate logic will consist of five components, $\mathcal M = ( X , a , R , D , I )$. Here, as with modal sentential logic, $X$ will be a set of indices [your $W$], $a$ a designated index [your $w_0$], and $R$ a relation on indices [the "accessibility" relation]. As for $D$ and $I$ , the former will be a function assigning each $x \in X$ and set $D_x$ , the domain at index $x$ , while the latter will be a function assigning to each $x \in X$ and each predicate $F$ a relation $F_x^I$ , the interpretation of $F$ at $x$, of the appropriate number of places [emphasis mine].