Canonical models for systems

Question

I've been struggling with understanding canonical models for systems. I'll state my question here, but at the end I define some terms in case they're not uniformly defined in Logic.

My text claims that in a canonical model $\vDash^{\mathcal{M}}_\alpha A$ iff $A\in\alpha$. However, how is this not true of all models? I think the right-to-left direction is by definition of $\vDash$. But for the left-to-right direction, how could you have a model satisfying a sentence but not have that sentence in the world that satisfies it? Is it because worlds are not required to be closed under implication, so that for instance $\alpha$ might contain $\mathbb{P}_0$ and $\mathbb{P}_1$ but not $\mathbb{P}_0\land \mathbb{P}_1$?

Model: A model is a tuple $\mathcal{M}=\langle W, ..., P\rangle$ where $W$ is a "world set" which is a subset of the powerset of sentences. The rest is a short-hand notation for a sequence of interpretations $P_n$ which are sets of sentences.

There seems to be the implication, although not formally stated as far as I can tell, that for instance $\mathbb{P}_n\in P_n$. As far as I can tell (in Chellas' modal logic text) there is no explicit restriction on which sets of worlds $P_n$ can be, although in an informal section of the book he says that these are to be regarded as the set of worlds at which sentence $\mathbb{P}_n$ is true, and the worlds are to be regarded as sets of sentences.

$\Sigma$-maximal: A set $\Gamma$ is $\Sigma$-maximal if it is consistent with the system $\Sigma$ but none of its proper extensions is.

Denoted Max$_\Sigma\Gamma$.

Proof set: The proof set of a sentence $A$ is the set $\{\text{Max}_\Sigma\Gamma:A\in\Gamma\}$.

Denoted $|A|_\Sigma$.

Canonical model: A canonical model for a system $\Sigma$ is a model $M=\langle W,...,P\rangle$ where $W=\{\text{Max}_\Sigma\Gamma\}$ and each $P_n=|\mathbb{P}_n|_\Sigma$.

Answer 1

Definition 2.5. [page 35] states the conditions for a formula $A$ to be true at the possible world $\alpha$ in the model $\mathcal M$ starting with :

(1) $\vDash_{\alpha}^{\mathcal M} \mathbb{P}_n \ \text {iff} \ \alpha \in \mathbb{P}_n$, for $n = 0,1,2, \ldots$

where the $\mathbb{P}_i$s are the atomic sentences, and so on.

The conditon for a canonical model [page 60] is :

The chief feature of a canonical model $\mathcal M$ for a system of modal logic $\Sigma$is this: in $\mathcal M$ just those sentences are true at a world ($\Sigma$-maximal set of sentences) as are contained by it; i.e. for every $\alpha$ in $\mathcal M$,

$$\vDash_{\alpha}^{\mathcal M} A \ \text {iff} \ A \in \alpha.$$

Intuitively, a canonical model $\mathcal M$ is a model which falsies all the non-theorems of $\Sigma$.

See page 61 :

Because the worlds in a canonical model for a system of modal logic will always verify just those sentences they contain, it follows that the sentences true in such a model are precisely the theorems of the system. That is to say, if $\mathcal M$ is a canonical model for a system $\Sigma$ then

$$\vDash^{\mathcal M} A \ \text {iff} \ \vdash_{\Sigma} A.$$

This result is the favored way of establishing that a modal logic $\Sigma$ is complete with respect to a class $\mathsf C$ of models.