In modal logic, why are models ordered sets?

Question

I just started undergrad math, so I only have a fuzzy idea of what a model is. I'm learning about modal logic in one of my classes. Our text describes modal logic as operating in a model defined as an ordered set < W, R, v> (correct me if I'm wrong), such that W is the set of all possible worlds (every possible world is a set of positive literals), R is a subset of the Cartesian product of the worlds that W comprises, and v is a binary function that assigns takes a world and a given literal and assigns 'true' to the literal if the literal is an element of that world.

Why are these three things in an ordered set? Moreover, why are they even in a set at all?

Thank you. -Hal

Answer 1

See in SEP Modal Logic, Ch.6 : Possible Worlds Semantics :

In propositional logic, a valuation of the atomic sentences (or row of a truth table) assigns a truth value (T or F) to each propositional variable $p$. Then the truth values of the complex sentences are calculated with truth tables. In modal semantics, a set $W$ of possible worlds is introduced.

A valuation then gives a truth value to each propositional variable for each of the possible worlds $w \in W$.

This means that value assigned to $p$ for world $w$ may differ from the value assigned to $p$ for another world $w′$.

The truth value of the atomic sentence $p$ at world $w$ given by the valuation $v$ may be written $v(p, w)$. Given this notation, the truth values (T for true, F for false) of complex sentences of modal logic for a given valuation $v$ (and member $w$ of the set of worlds $W$) may be defined by the following truth clauses.

$(\lnot) \ \ v(\lnot A, w)=T$ iff $v(A, w)=F$.

$(→) \ \ v(A→B, w)=T$ iff $v(A, w)=F$ or $v(B, w)=T$.

$(□) \ \ v(□A, w)=T$ iff for every world $w′ \in W$, $v(A, w′)=T$.

Clauses $(\lnot)$ and $(→)$ simply describe the standard truth table behavior for negation and material implication respectively.

According to $(□), □A$ is true (at a world $w$) exactly when $A$ is true in all possible worlds.

In order to correctly describe the properties of various "modalities" (temporal, deontic, etc.) it is necessary to "give a structure" to the set $W$ of possible worlds; to do this, it is used a binary relation $R$ defined on the set $W$.

According to the properties of $R$ (symmetry, reflexivity, transitivity) it is possible to characterize different modal logics.

A frame $\langle W, R \rangle$ is a pair consisting of a non-empty set $W$ (of worlds) and a binary relation $R$ on $W$.

A model $\langle \mathcal F, v \rangle$ consists of a frame $\mathcal F$, and a valuation $v$ that assigns truth values to each atomic sentence at each world in $W$.

Summing up, we have that a model is a (ordered) triple :

$\langle \langle W, R \rangle, v \rangle$ i.e. $\langle W, R, v \rangle$.

There is no "magic" here : we have only to recall that in set-theory the pair $\{ x,y \}$ and the ordered pair $\langle x,y \rangle$ are different, because $\{ x,y \} = \{ y,x \}$ while, in general : $\langle x,y \rangle \ne \langle y,x \rangle$.

This is a simple way to track the different roles played in the semantics by the set of worlds $W$, the relation between them $R$, and the function $v : L \times W$, where $L$ is the set of propositional variables of the language.